# bibtab

bibtype: misc
filename: fparith.bib
label: Demmel:2014:THS
author: James Demmel and Hong Diep Nguyen
title: Toward hardware support for Reproducible BLAS
pages: 105:1--105:??
day: 24
month: sep
monthnumber: 9
year: 2014
bibdate: Mon Jan 04 15:49:09 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.01.04 15:49:09 ???
URL: http://bebop.cs.berkeley.edu/reproblas/img/pdf.png
acknowledgement: ack-nhfb
howpublished: SCAN 2014 talk slides.
rawauthor: James Demmel and Hong Diep Nguyen
rawtitle: Toward hardware support for Reproducible {BLAS}
entry:
@Misc{Demmel:2014:THS,
  author =       "James Demmel and Hong Diep Nguyen",
  title =        "Toward hardware support for Reproducible {BLAS}",
  howpublished = "SCAN 2014 talk slides.",
  day =          "24",
  month =        sep,
  year =         "2014",
  bibdate =      "Mon Jan 04 15:49:09 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://bebop.cs.berkeley.edu/reproblas/img/pdf.png",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Doerr:2014:RRP
author: Benjamin Doerr and Magnus Wahlstrom
title: Randomized Rounding in the Presence of a Cardinality Constraint
journal: j-ACM-J-EXP-ALGORITHMICS
volume: 19
number: 1
pages: 1.2:1--1.2:??
month: may
monthnumber: 5
year: 2014
DOI: https://doi.org/10.1145/2594409
ISSN: 1084-6654 OR 10846654
ISSNL: 1084-6654 OR 10846654
bibdate: Wed May 21 14:36:05 MDT 2014
bibtimestamp: 2014.05.21 14:36:05 MDT
fjournal: Journal of Experimental Algorithmics (JEA)
acknowledgement: ack-nhfb
articleno: 1.2
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J430
rawauthor: Benjamin Doerr and Magnus Wahlstr{\"o}m
rawtitle: Randomized Rounding in the Presence of a Cardinality Constraint
bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jea.bib

abstract:
We consider the problem of generating randomized roundings that satisfy a single cardinality constraint and admit Chernoff-type large deviation bounds for weighted sums of the variables. That this can be done efficiently was proven by Srinivasan [2001], a different approach was later given by the first author [Doerr 2006]. In this work, we (a) present an improved version of the bitwise derandomization given by Doerr, (b) give the first derandomization of Srinivasan's tree-based randomized approach and prove its correctness, and (c) experimentally compare the resulting algorithms. Our experiments show that adding a single cardinality constraint typically reduces the rounding errors and only moderately increases the running times. In general, our derandomization of the tree-based approach is superior to the derandomized bitwise one, while the two randomized versions produce very similar rounding errors. When implementing the derandomized tree-based approach, however, the choice of the tree is important.

rawabstract:
We consider the problem of generating randomized roundings that satisfy a single cardinality constraint and admit Chernoff-type large deviation bounds for weighted sums of the variables. That this can be done efficiently was proven by Srinivasan [2001], a different approach was later given by the first author [Doerr 2006]. In this work, we (a) present an improved version of the bitwise derandomization given by Doerr, (b) give the first derandomization of Srinivasan's tree-based randomized approach and prove its correctness, and (c) experimentally compare the resulting algorithms. Our experiments show that adding a single cardinality constraint typically reduces the rounding errors and only moderately increases the running times. In general, our derandomization of the tree-based approach is superior to the derandomized bitwise one, while the two randomized versions produce very similar rounding errors. When implementing the derandomized tree-based approach, however, the choice of the tree is important.

entry:
@Article{Doerr:2014:RRP,
  author =       "Benjamin Doerr and Magnus Wahlstr{\"o}m",
  title =        "Randomized Rounding in the Presence of a Cardinality
                 Constraint",
  journal =      j-ACM-J-EXP-ALGORITHMICS,
  volume =       "19",
  number =       "1",
  pages =        "1.2:1--1.2:??",
  month =        may,
  year =         "2014",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2594409",
  ISSN =         "1084-6654",
  ISSN-L =       "1084-6654",
  bibdate =      "Wed May 21 14:36:05 MDT 2014",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jea.bib",
  abstract =     "We consider the problem of generating randomized
                 roundings that satisfy a single cardinality constraint
                 and admit Chernoff-type large deviation bounds for
                 weighted sums of the variables. That this can be done
                 efficiently was proven by Srinivasan [2001], a
                 different approach was later given by the first author
                 [Doerr 2006]. In this work, we (a) present an improved
                 version of the bitwise derandomization given by Doerr,
                 (b) give the first derandomization of Srinivasan's
                 tree-based randomized approach and prove its
                 correctness, and (c) experimentally compare the
                 resulting algorithms. Our experiments show that adding
                 a single cardinality constraint typically reduces the
                 rounding errors and only moderately increases the
                 running times. In general, our derandomization of the
                 tree-based approach is superior to the derandomized
                 bitwise one, while the two randomized versions produce
                 very similar rounding errors. When implementing the
                 derandomized tree-based approach, however, the choice
                 of the tree is important.",
  acknowledgement = ack-nhfb,
  articleno =    "1.2",
  fjournal =     "Journal of Experimental Algorithmics (JEA)",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J430",
}


bibtype: article
filename: fparith.bib
label: Drane:2014:SCF
author: T. A. Drane and T. M. Rose and G. A. Constantinides
title: On the Systematic Creation of Faithfully Rounded Truncated Multipliers and Arrays
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 10
pages: 2513--2525
month: oct
monthnumber: 10
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.126
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Nov 06 07:29:34 2014
bibtimestamp: 2014.11.06 07:29:34 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: T. A. Drane and T. M. Rose and G. A. Constantinides
rawtitle: On the Systematic Creation of Faithfully Rounded Truncated Multipliers and Arrays
keywords:
arbitrary array; bit vectors; correct rounding; correct-by-construction; Data-path design; elemental semiconductors; faithfully rounded truncated arrays; faithfully rounded truncated multipliers; fixed point arithmetic; fixed-point multiplication; floating-point arithmetic; hardware description languages; HDL code; heuristically inspired schemes; high-speed arithmetic; Monte Carlo methods; Monte Carlo simulation; multiplying circuits; parallel circuits; partial product array; Si; silicon; silicon area; systematic creation; truncated multiplier schemes; word length 32 bit; worst-case analysis; worst-case error

entry:
@Article{Drane:2014:SCF,
  author =       "T. A. Drane and T. M. Rose and G. A. Constantinides",
  title =        "On the Systematic Creation of Faithfully Rounded
                 Truncated Multipliers and Arrays",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "10",
  pages =        "2513--2525",
  month =        oct,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.126",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Nov 06 07:29:34 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "arbitrary array; bit vectors; correct rounding;
                 correct-by-construction; Data-path design; elemental
                 semiconductors; faithfully rounded truncated arrays;
                 faithfully rounded truncated multipliers; fixed point
                 arithmetic; fixed-point multiplication; floating-point
                 arithmetic; hardware description languages; HDL code;
                 heuristically inspired schemes; high-speed arithmetic;
                 Monte Carlo methods; Monte Carlo simulation;
                 multiplying circuits; parallel circuits; partial
                 product array; Si; silicon; silicon area; systematic
                 creation; truncated multiplier schemes; word length 32
                 bit; worst-case analysis; worst-case error",
}


bibtype: article
filename: fparith.bib
label: Du:2014:AEP
author: Peibing Du and Hao Jiang and Lizhi Cheng
title: Accurate Evaluation of Polynomials in Legendre Basis
journal: j-J-APPL-MATH
volume: 2014
pages: 742538:1--742538:13
year: 2014
DOI: https://doi.org/10.1155/2014/742538
ISSN: 1110-757X (print), 1687-0042 (electronic) OR 1110757X (print), 16870042 (electronic)
ISSNL: 1110-757X OR 1110757X
bibdate: Mon Mar 19 14:19:47 2018
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2018.03.19 14:19:47 ???
URL: https://www.hindawi.com/journals/jam/2014/742538/
fjournal: Journal of Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.hindawi.com/journals/jam/
rawauthor: Peibing Du and Hao Jiang and Lizhi Cheng
rawtitle: Accurate Evaluation of Polynomials in {Legendre} Basis
entry:
@Article{Du:2014:AEP,
  author =       "Peibing Du and Hao Jiang and Lizhi Cheng",
  title =        "Accurate Evaluation of Polynomials in {Legendre}
                 Basis",
  journal =      j-J-APPL-MATH,
  volume =       "2014",
  pages =        "742538:1--742538:13",
  year =         "2014",
  DOI =          "https://doi.org/10.1155/2014/742538",
  ISSN =         "1110-757X (print), 1687-0042 (electronic)",
  ISSN-L =       "1110-757X",
  bibdate =      "Mon Mar 19 14:19:47 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.hindawi.com/journals/jam/2014/742538/",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Applied Mathematics",
  journal-URL =  "http://www.hindawi.com/journals/jam/",
}


bibtype: article
filename: fparith.bib
label: Dumas:2014:NRI
author: Jean-Guillaume Dumas
title: On Newton--Raphson Iteration for Multiplicative Inverses Modulo Prime Powers
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 8
pages: 2106--2109
month: aug
monthnumber: 8
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.94
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Aug 25 08:24:32 2014
bibtimestamp: 2014.08.25 08:24:32 ???
note: See corrections cite Walther:2019:VNR.
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Jean-Guillaume Dumas
rawnote: See corrections \cite{Walther:2019:VNR}.
rawtitle: On {Newton--Raphson} Iteration for Multiplicative Inverses Modulo Prime Powers
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

entry:
@Article{Dumas:2014:NRI,
  author =       "Jean-Guillaume Dumas",
  title =        "On {Newton--Raphson} Iteration for Multiplicative
                 Inverses Modulo Prime Powers",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "8",
  pages =        "2106--2109",
  month =        aug,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.94",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Aug 25 08:24:32 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  note =         "See corrections \cite{Walther:2019:VNR}.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Duracz:2014:PFI
author: Jan Duracz and Michal Konecny
title: Polynomial function intervals for floating-point software verification
journal: j-ANN-MATH-ARTIF-INTELL
volume: 70
number: 4
pages: 351--398
month: apr
monthnumber: 4
year: 2014
CODEN: AMAIEC
DOI: https://doi.org/10.1007/s10472-014-9409-7
ISSN: 1012-2443 (print), 1573-7470 (electronic) OR 10122443 (print), 15737470 (electronic)
ISSNL: 1012-2443 OR 10122443
bibdate: Sat Jun 8 08:22:14 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2024.06.08 08:22:14 ???
fjournal: Annals of Mathematics and Artificial Intelligence
keywords: PolyPaver open-source software tool
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10472
rawauthor: Jan Duracz and Michal Kone{\v{c}}n{\'y}
rawtitle: Polynomial function intervals for floating-point software verification
entry:
@Article{Duracz:2014:PFI,
  author =       "Jan Duracz and Michal Kone{\v{c}}n{\'y}",
  title =        "Polynomial function intervals for floating-point
                 software verification",
  journal =      j-ANN-MATH-ARTIF-INTELL,
  volume =       "70",
  number =       "4",
  pages =        "351--398",
  month =        apr,
  year =         "2014",
  CODEN =        "AMAIEC",
  DOI =          "https://doi.org/10.1007/s10472-014-9409-7",
  ISSN =         "1012-2443 (print), 1573-7470 (electronic)",
  ISSN-L =       "1012-2443",
  bibdate =      "Sat Jun 8 08:22:14 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Annals of Mathematics and Artificial Intelligence",
  journal-URL =  "http://link.springer.com/journal/10472",
  keywords =     "PolyPaver open-source software tool",
}


bibtype: article
filename: fparith.bib
label: Gilani:2014:EEP
author: Syed Zohaib Gilani and Nam Sung Kim and Michael Schulte
title: Energy-Efficient Pixel-Arithmetic
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 8
pages: 1--1
month: aug
monthnumber: 8
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2325827
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Aug 25 08:24:32 2014
bibtimestamp: 2014.08.25 08:24:32 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Syed Zohaib Gilani and Nam Sung Kim and Michael Schulte
rawtitle: Energy-Efficient Pixel-Arithmetic
entry:
@Article{Gilani:2014:EEP,
  author =       "Syed Zohaib Gilani and Nam Sung Kim and Michael
                 Schulte",
  title =        "Energy-Efficient Pixel-Arithmetic",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "8",
  pages =        "1--1",
  month =        aug,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2325827",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Aug 25 08:24:32 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Gladshtein:2014:DBP
author: Michael Gladshtein
title: Delay-based processing-in-wire for design of QCA serial decimal arithmetic units
journal: j-JETC
volume: 10
number: 2
pages: 13:1--13:??
month: feb
monthnumber: 2
year: 2014
DOI: https://doi.org/10.1145/2564927
ISSN: 1550-4832 (print), 1550-4840 (electronic) OR 15504832 (print), 15504840 (electronic)
ISSNL: 1550-4832 OR 15504832
bibdate: Fri Feb 28 17:06:25 MST 2014
bibtimestamp: 2014.02.28 17:06:25 MST
fjournal: ACM Journal on Emerging Technologies in Computing Systems (JETC)
acknowledgement: ack-nhfb
articleno: 13
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J967
rawauthor: Michael Gladshtein
rawtitle: Delay-based processing-in-wire for design of {QCA} serial decimal arithmetic units
bibsource:
http://www.acm.org/pubs/contents/journals/jetc/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jetc.bib

abstract:
Quantum-dot cellular automata (QCA) technology is now considered to be one of the prospective technologies for a nanocomputer creation. The physical properties of QCA and its expanding range of computer applications make it expedient to use the novel paradigm of nanocomputer architecture: serial decimal storage-transfer-processing. The delay-based encoding of decimal digits allows the use a delay element as a main element of QCA serial arithmetic units. The simple implementation of the delay element by a short length of QCA wire results in reduction of complexity and of the area required for a QCA circuit. The theoretical basics of delay-based processing-in-wire and design examples of QCA serial decimal arithmetic units are presented.

rawabstract:
Quantum-dot cellular automata (QCA) technology is now considered to be one of the prospective technologies for a nanocomputer creation. The physical properties of QCA and its expanding range of computer applications make it expedient to use the novel paradigm of nanocomputer architecture: serial decimal storage-transfer-processing. The delay-based encoding of decimal digits allows the use a delay element as a main element of QCA serial arithmetic units. The simple implementation of the delay element by a short length of QCA wire results in reduction of complexity and of the area required for a QCA circuit. The theoretical basics of delay-based processing-in-wire and design examples of QCA serial decimal arithmetic units are presented.

entry:
@Article{Gladshtein:2014:DBP,
  author =       "Michael Gladshtein",
  title =        "Delay-based processing-in-wire for design of {QCA}
                 serial decimal arithmetic units",
  journal =      j-JETC,
  volume =       "10",
  number =       "2",
  pages =        "13:1--13:??",
  month =        feb,
  year =         "2014",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2564927",
  ISSN =         "1550-4832 (print), 1550-4840 (electronic)",
  ISSN-L =       "1550-4832",
  bibdate =      "Fri Feb 28 17:06:25 MST 2014",
  bibsource =    "http://www.acm.org/pubs/contents/journals/jetc/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jetc.bib",
  abstract =     "Quantum-dot cellular automata (QCA) technology is now
                 considered to be one of the prospective technologies
                 for a nanocomputer creation. The physical properties of
                 QCA and its expanding range of computer applications
                 make it expedient to use the novel paradigm of
                 nanocomputer architecture: serial decimal
                 storage-transfer-processing. The delay-based encoding
                 of decimal digits allows the use a delay element as a
                 main element of QCA serial arithmetic units. The simple
                 implementation of the delay element by a short length
                 of QCA wire results in reduction of complexity and of
                 the area required for a QCA circuit. The theoretical
                 basics of delay-based processing-in-wire and design
                 examples of QCA serial decimal arithmetic units are
                 presented.",
  acknowledgement = ack-nhfb,
  articleno =    "13",
  fjournal =     "ACM Journal on Emerging Technologies in Computing
                 Systems (JETC)",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J967",
}


bibtype: article
filename: fparith.bib
label: Goualard:2014:HDY
author: Frederic Goualard
title: How do you compute the midpoint of an interval?
journal: j-TOMS
volume: 40
number: 2
pages: 11:1--11:25
month: feb
monthnumber: 2
year: 2014
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2493882
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Fri Mar 14 06:30:41 MDT 2014
bibtimestamp: 2014.03.14 06:30:41 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 11
journalURL: https://dl.acm.org/loi/toms
rawauthor: Fr{\'e}d{\'e}ric Goualard
rawtitle: How do you compute the midpoint of an interval?
abstract:
The algorithm that computes the midpoint of an interval with floating-point bounds requires some careful devising to handle all possible inputs correctly. We review several implementations from prominent C/C++ interval arithmetic packages and analyze their potential failure to deliver the expected results. We then show how to amend them to avoid common pitfalls. The results presented are also relevant to noninterval arithmetic computation such as the implementation of bisection methods. Enough background on IEEE 754 floating-point arithmetic is provided for this article to serve as a practical introduction to the analysis of floating-point computation.

rawabstract:
The algorithm that computes the midpoint of an interval with floating-point bounds requires some careful devising to handle all possible inputs correctly. We review several implementations from prominent C/C++ interval arithmetic packages and analyze their potential failure to deliver the expected results. We then show how to amend them to avoid common pitfalls. The results presented are also relevant to noninterval arithmetic computation such as the implementation of bisection methods. Enough background on IEEE 754 floating-point arithmetic is provided for this article to serve as a practical introduction to the analysis of floating-point computation.

entry:
@Article{Goualard:2014:HDY,
  author =       "Fr{\'e}d{\'e}ric Goualard",
  title =        "How do you compute the midpoint of an interval?",
  journal =      j-TOMS,
  volume =       "40",
  number =       "2",
  pages =        "11:1--11:25",
  month =        feb,
  year =         "2014",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2493882",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Fri Mar 14 06:30:41 MDT 2014",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "The algorithm that computes the midpoint of an
                 interval with floating-point bounds requires some
                 careful devising to handle all possible inputs
                 correctly. We review several implementations from
                 prominent C/C++ interval arithmetic packages and
                 analyze their potential failure to deliver the expected
                 results. We then show how to amend them to avoid common
                 pitfalls. The results presented are also relevant to
                 noninterval arithmetic computation such as the
                 implementation of bisection methods. Enough background
                 on IEEE 754 floating-point arithmetic is provided for
                 this article to serve as a practical introduction to
                 the analysis of floating-point computation.",
  acknowledgement = ack-nhfb,
  articleno =    "11",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: fparith.bib
label: Graillat:2014:MRE
author: Stef Graillat and Vincent Lefevre and Jean-Michel Muller
title: On the maximum relative error when computing $ x^n $ in floating-point arithmetic
journal: arXiv.org
pages: ??--??
day: 11
month: feb
monthnumber: 2
year: 2014
bibdate: Wed Sep 30 12:43:49 MDT 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.09.30 12:43:49 MDT
URL: http://arxiv.org/abs/1402.2991
subject: Numerical Analysis (cs.NA); Numerical Analysis (math.NA)
acknowledgement: ack-nhfb
rawauthor: Stef Graillat and Vincent Lef{\`e}vre and Jean-Michel Muller
rawtitle: On the maximum relative error when computing $ x^n $ in floating-point arithmetic
abstract:
In this paper, we improve the usual relative error bound for the computation of $ x^n $ through iterated multiplications by x in binary floating-point arithmetic. The obtained error bound is only slightly better than the usual one, but it is simpler. We also discuss the more general problem of computing the product of n terms.

rawabstract:
In this paper, we improve the usual relative error bound for the computation of $ x^n $ through iterated multiplications by x in binary floating-point arithmetic. The obtained error bound is only slightly better than the usual one, but it is simpler. We also discuss the more general problem of computing the product of n terms.

entry:
@Article{Graillat:2014:MRE,
  author =       "Stef Graillat and Vincent Lef{\`e}vre and Jean-Michel
                 Muller",
  title =        "On the maximum relative error when computing $ x^n $
                 in floating-point arithmetic",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "??--??",
  day =          "11",
  month =        feb,
  year =         "2014",
  CODEN =        "????",
  ISSN =         "????",
  ISSN-L =       "????",
  bibdate =      "Wed Sep 30 12:43:49 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://arxiv.org/abs/1402.2991",
  abstract =     "In this paper, we improve the usual relative error
                 bound for the computation of $ x^n $ through iterated
                 multiplications by x in binary floating-point
                 arithmetic. The obtained error bound is only slightly
                 better than the usual one, but it is simpler. We also
                 discuss the more general problem of computing the
                 product of n terms.",
  acknowledgement = ack-nhfb,
  subject =      "Numerical Analysis (cs.NA); Numerical Analysis
                 (math.NA)",
}


bibtype: inproceedings
filename: fparith.bib
label: Hormigo:2014:ODC
author: J. Hormigo and J. Villalba
editor: Michael B. Matthews
title: Optimizing DSP circuits by a new family of arithmetic operators
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 871--875
year: 2014
DOI: https://doi.org/10.1109/ACSSC.2014.7094576
ISBN: 1-4799-8297-0, 1-4799-8298-9, 1-4799-8295-4 OR 1479982970, 1479982989, 1479982954
ISBN13: 978-1-4799-8297-4 OR 9781479982974
ISSN: 1058-6393 OR 10586393
LCCN: TK7801
bibdate: Sun Jun 19 14:59:21 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.06.19 14:59:21 ???
acknowledgement: ack-nhfb
rawauthor: J. Hormigo and J. Villalba
raweditor: Michael B. Matthews
rawtitle: Optimizing {DSP} circuits by a new family of arithmetic operators
booktitle:
Proceedings of the 48th Asilomar Conference on Signals, Systems, and Computers, November 2--5, 2014, Pacific Grove, California

rawbooktitle:
{Proceedings of the 48th Asilomar Conference on Signals, Systems, and Computers, November 2--5, 2014, Pacific Grove, California}

entry:
@InProceedings{Hormigo:2014:ODC,
  author =       "J. Hormigo and J. Villalba",
  editor =       "Michael B. Matthews",
  booktitle =    "{Proceedings of the 48th Asilomar Conference on
                 Signals, Systems, and Computers, November 2--5, 2014,
                 Pacific Grove, California}",
  title =        "Optimizing {DSP} circuits by a new family of
                 arithmetic operators",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "871--875",
  year =         "2014",
  DOI =          "https://doi.org/10.1109/ACSSC.2014.7094576",
  ISBN =         "1-4799-8297-0, 1-4799-8298-9, 1-4799-8295-4",
  ISBN-13 =      "978-1-4799-8297-4",
  ISSN =         "1058-6393",
  LCCN =         "TK7801",
  bibdate =      "Sun Jun 19 14:59:21 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: techreport
filename: fparith.bib
label: Jeannerod:2014:REF
author: Claude-Pierre Jeannerod and Siegfried M. Rump
title: On relative errors of floating-point operations: optimal bounds and applications
type: Preprint
pages: 871--875
year: 2014
bibdate: Mon May 30 07:31:09 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.05.30 07:31:09 ???
acknowledgement: ack-nhfb
rawauthor: Claude-Pierre Jeannerod and Siegfried M. Rump
rawtitle: On relative errors of floating-point operations: optimal bounds and applications
entry:
@TechReport{Jeannerod:2014:REF,
  author =       "Claude-Pierre Jeannerod and Siegfried M. Rump",
  title =        "On relative errors of floating-point operations:
                 optimal bounds and applications",
  type =         "Preprint",
  number =       "??",
  institution =  "????",
  address =      "????",
  month =        "????",
  year =         "2014",
  bibdate =      "Mon May 30 07:31:09 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: misc
filename: fparith.bib
label: Johansson:2014:PMP
author: Fredrik Johansson and The mpmath Development Team
title: pkg mpmath: a Python library for arbitrary-precision floating-point arithmetic
pages: 871--875
year: 2014
bibdate: Wed Apr 24 13:50:47 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/python.bib
bibtimestamp: 2024.04.24 13:50:47 ???
URL: http://mpmath.org/
acknowledgement: ack-nhfb
howpublished: Web site
rawauthor: Fredrik Johansson and {The mpmath Development Team}
rawtitle: \pkg{mpmath}: a {Python} library for arbitrary-precision floating-point arithmetic
entry:
@Misc{Johansson:2014:PMP,
  author =       "Fredrik Johansson and {The mpmath Development Team}",
  title =        "\pkg{mpmath}: a {Python} library for
                 arbitrary-precision floating-point arithmetic",
  howpublished = "Web site",
  year =         "2014",
  bibdate =      "Wed Apr 24 13:50:47 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/python.bib",
  URL =          "http://mpmath.org/",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Joldes:2014:CRF
author: Mioara Joldes and Jean-Michel Muller and Valentina Popescu
editor: IEEE
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 63--67
year: 2014
DOI: https://doi.org/10.1109/ASAP.2014.6868632
ISBN: 1-4799-3608-1 OR 1479936081
ISBN13: 978-1-4799-3608-3 OR 9781479936083
bibdate: Fri Sep 29 10:47:32 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.09.29 10:47:32 ???
acknowledgement: ack-nhfb
rawauthor: Mioara Joldes and Jean-Michel Muller and Valentina Popescu
raweditor: {IEEE}
booktitle:
2014 IEEE 25th International Conference on Application-Specific Systems, Architectures and Processors. 18-20 June 2014. Zurich, Switzerland

title:
On the computation of the reciprocal of floating point expansions using an adapted Newton--Raphson iteration

rawbooktitle:
{2014 IEEE 25th International Conference on Application-Specific Systems, Architectures and Processors. 18-20 June 2014. Z{\"u}rich, Switzerland}

rawtitle:
On the computation of the reciprocal of floating point expansions using an adapted {Newton--Raphson} iteration

entry:
@InProceedings{Joldes:2014:CRF,
  author =       "Mioara Joldes and Jean-Michel Muller and Valentina
                 Popescu",
  editor =       "{IEEE}",
  booktitle =    "{2014 IEEE 25th International Conference on
                 Application-Specific Systems, Architectures and
                 Processors. 18-20 June 2014. Z{\"u}rich, Switzerland}",
  title =        "On the computation of the reciprocal of floating point
                 expansions using an adapted {Newton--Raphson}
                 iteration",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "63--67",
  year =         "2014",
  DOI =          "https://doi.org/10.1109/ASAP.2014.6868632",
  ISBN =         "1-4799-3608-1",
  ISBN-13 =      "978-1-4799-3608-3",
  bibdate =      "Fri Sep 29 10:47:32 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Joldes:2014:SSH
author: Mioara Joldes and Valentina Popescu and Warwick Tucker
title: Searching for Sinks for the Henon Map using a Multiple-precision GPU Arithmetic Library
journal: j-COMP-ARCH-NEWS
volume: 42
number: 4
pages: 63--68
year: 2014
CODEN: CANED2
DOI: https://doi.org/10.1145/2693714.2693726
ISSN: 0163-5964 (print), 1943-5851 (electronic) OR 01635964 (print), 19435851 (electronic)
ISSNL: 0163-5964 OR 01635964
bibdate: Wed Dec 3 16:18:50 MST 2014
bibtimestamp: 2014.12.03 16:18:50 MST
fjournal: ACM SIGARCH Computer Architecture News
remark: HEART '14 conference proceedings.
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J89
rawauthor: Mioara Joldes and Valentina Popescu and Warwick Tucker
rawtitle: Searching for Sinks for the {H{\'e}non} Map using a Multiple-precision {GPU} Arithmetic Library
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigarch.bib

abstract:
Today, GPUs represent an important hardware development platform for many problems in dynamical systems, where massive parallel computations are needed. Beside that, many numerical studies of chaotic dynamical systems require a computing precision higher than common floating point (FP) formats. One such application is locating invariant sets for chaotic dynamical systems. In particular, we focus on rigorously proving the existence of stable periodic orbits for the Henon map for parameter values close to the classical ones. For that, we present a multiple-precision floating-point arithmetic library in CUDA programming language for the NVIDIA GPU platform. Our library extends the precision using so-called FP expansions, where a number is represented as the unevaluated sum of standard machine precision FP numbers. This format offers the advantage of using directly available and highly optimized hardware FP operations. We generalize algorithms used by multiple-precisions libraries such as Bailey's QD, or the analogue GPU version, GQD.

rawabstract:
Today, GPUs represent an important hardware development platform for many problems in dynamical systems, where massive parallel computations are needed. Beside that, many numerical studies of chaotic dynamical systems require a computing precision higher than common floating point (FP) formats. One such application is locating invariant sets for chaotic dynamical systems. In particular, we focus on rigorously proving the existence of stable periodic orbits for the H{\'e}non map for parameter values close to the classical ones. For that, we present a multiple-precision floating-point arithmetic library in CUDA programming language for the NVIDIA GPU platform. Our library extends the precision using so-called FP expansions, where a number is represented as the unevaluated sum of standard machine precision FP numbers. This format offers the advantage of using directly available and highly optimized hardware FP operations. We generalize algorithms used by multiple-precisions libraries such as Bailey's QD, or the analogue GPU version, GQD.

entry:
@Article{Joldes:2014:SSH,
  author =       "Mioara Joldes and Valentina Popescu and Warwick
                 Tucker",
  title =        "Searching for Sinks for the {H{\'e}non} Map using a
                 Multiple-precision {GPU} Arithmetic Library",
  journal =      j-COMP-ARCH-NEWS,
  volume =       "42",
  number =       "4",
  pages =        "63--68",
  year =         "2014",
  CODEN =        "CANED2",
  DOI =          "https://doi.org/10.1145/2693714.2693726",
  ISSN =         "0163-5964 (print), 1943-5851 (electronic)",
  ISSN-L =       "0163-5964",
  bibdate =      "Wed Dec 3 16:18:50 MST 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigarch.bib",
  abstract =     "Today, GPUs represent an important hardware
                 development platform for many problems in dynamical
                 systems, where massive parallel computations are
                 needed. Beside that, many numerical studies of chaotic
                 dynamical systems require a computing precision higher
                 than common floating point (FP) formats. One such
                 application is locating invariant sets for chaotic
                 dynamical systems. In particular, we focus on
                 rigorously proving the existence of stable periodic
                 orbits for the H{\'e}non map for parameter values close
                 to the classical ones. For that, we present a
                 multiple-precision floating-point arithmetic library in
                 CUDA programming language for the NVIDIA GPU platform.
                 Our library extends the precision using so-called FP
                 expansions, where a number is represented as the
                 unevaluated sum of standard machine precision FP
                 numbers. This format offers the advantage of using
                 directly available and highly optimized hardware FP
                 operations. We generalize algorithms used by
                 multiple-precisions libraries such as Bailey's QD, or
                 the analogue GPU version, GQD.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGARCH Computer Architecture News",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J89",
  remark =       "HEART '14 conference proceedings.",
}


bibtype: article
filename: fparith.bib
label: Korzen:2014:PPP
author: Marcin Korzen and Szymon Jaroszewicz
title: PaCAL: A Python Package for Arithmetic Computations with Random Variables
journal: j-J-STAT-SOFT
volume: 57
number: 10
pages: ??--??
month: may
monthnumber: 5
year: 2014
CODEN: JSSOBK
ISSN: 1548-7660 OR 15487660
ISSNL: 1548-7660 OR 15487660
bibdate: Mon Jun 16 11:01:52 MDT 2014
bibtimestamp: 2014.06.16 11:01:52 MDT
URL: http://www.jstatsoft.org/v57/i10
fjournal: Journal of Statistical Software
acknowledgement: ack-nhfb
journalURL: http://www.jstatsoft.org/
rawauthor: Marcin Korze{\'n} and Szymon Jaroszewicz
rawtitle: {PaCAL}: A {Python} Package for Arithmetic Computations with Random Variables
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib

entry:
@Article{Korzen:2014:PPP,
  author =       "Marcin Korze{\'n} and Szymon Jaroszewicz",
  title =        "{PaCAL}: A {Python} Package for Arithmetic
                 Computations with Random Variables",
  journal =      j-J-STAT-SOFT,
  volume =       "57",
  number =       "10",
  pages =        "??--??",
  month =        may,
  year =         "2014",
  CODEN =        "JSSOBK",
  ISSN =         "1548-7660",
  ISSN-L =       "1548-7660",
  bibdate =      "Mon Jun 16 11:01:52 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jstatsoft.bib;
                 https://www.math.utah.edu/pub/tex/bib/prng.bib",
  URL =          "http://www.jstatsoft.org/v57/i10",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Statistical Software",
  journal-URL =  "http://www.jstatsoft.org/",
  pubdates =     "Submitted 2012-02-14; Accepted 2013-07-21",
}


bibtype: inproceedings
filename: fparith.bib
label: Leeser:2014:MIR
author: Miriam Leeser and Sayan Mukherjee and Jaideep Ramachandran and Thomas Wahl
editor: IEEE
title: Make it real: Effective floating-point reasoning via exact arithmetic
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 1--4
year: 2014
DOI: https://doi.org/10.7873/DATE.2014.130
ISBN: 1-4799-3297-3, 3-9815370-2-5 OR 1479932973, 3981537025
ISBN13: 978-1-4799-3297-9, 978-3-9815370-2-4 OR 9781479932979, 9783981537024
LCCN: TK7870 .D467 2014
bibdate: Sat Jun 4 17:09:03 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.06.04 17:09:03 ???
URL: http://ieeexplore.ieee.org/servlet/opac?punumber=6784162
acknowledgement: ack-nhfb
rawauthor: Miriam Leeser and Sayan Mukherjee and Jaideep Ramachandran and Thomas Wahl
raweditor: {IEEE}
rawtitle: Make it real: Effective floating-point reasoning via exact arithmetic
booktitle:
Design, Automation and Test in Europe Conference and Exhibition (DATE), Dresden, Germany March 24--28, 2014

rawbooktitle:
{Design, Automation and Test in Europe Conference and Exhibition (DATE), Dresden, Germany March 24--28, 2014}

entry:
@InProceedings{Leeser:2014:MIR,
  author =       "Miriam Leeser and Sayan Mukherjee and Jaideep
                 Ramachandran and Thomas Wahl",
  editor =       "{IEEE}",
  booktitle =    "{Design, Automation and Test in Europe Conference and
                 Exhibition (DATE), Dresden, Germany March 24--28,
                 2014}",
  title =        "Make it real: Effective floating-point reasoning via
                 exact arithmetic",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "1--4",
  year =         "2014",
  DOI =          "https://doi.org/10.7873/DATE.2014.130",
  ISBN =         "1-4799-3297-3, 3-9815370-2-5",
  ISBN-13 =      "978-1-4799-3297-9, 978-3-9815370-2-4",
  LCCN =         "TK7870 .D467 2014",
  bibdate =      "Sat Jun 4 17:09:03 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/servlet/opac?punumber=6784162",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Lei:2014:FIS
author: Yuanwu Lei and Lei Guo and Yong Dou and Sheng Ma and Jinbo Xu
title: FPGA Implementation of a Special-Purpose VLIW Structure for Double-Precision Elementary Function
journal: j-TRETS
volume: 7
number: 2
pages: 8:1--8:??
month: jun
monthnumber: 6
year: 2014
DOI: https://doi.org/10.1145/2617594
ISSN: 1936-7406 (print), 1936-7414 (electronic) OR 19367406 (print), 19367414 (electronic)
ISSNL: 1936-7406 OR 19367406
bibdate: Mon Jun 30 18:26:23 MDT 2014
bibtimestamp: 2014.06.30 18:26:23 MDT
fjournal: ACM Transactions on Reconfigurable Technology and Systems (TRETS)
acknowledgement: ack-nhfb
articleno: 8
journalURL: http://portal.acm.org/toc.cfm?id=J1151
rawauthor: Yuanwu Lei and Lei Guo and Yong Dou and Sheng Ma and Jinbo Xu
abstract:
In the current article, the capability and flexibility of field programmable gate-arrays (FPGAs) to implement IEEE-754 double-precision floating-point elementary functions are explored. To perform various elementary functions on the unified hardware efficiently, we propose a special-purpose very long instruction word (VLIW) processor, called DP_VELP. This processor is equipped with multiple basic units, and its performance is improved through an explicitly parallel technique. Pipelined evaluation of polynomial approximation with Estrin's scheme is proposed, by scheduling basic components in an optimal order to avoid data hazard stalls and achieve minimal latency. The custom VLIW processor can achieve high scalability. Under the control of specific VLIW instructions, the basic units are combined into special-purpose hardware for elementary functions. Common elementary functions are presented as examples to illustrate the design of elementary function in DP_VELP in detail. Minimax approximation scheme is used to reduce degree of polynomial. Compromise between the size of lookup table and the latency is discussed, and the internal precision is carefully planned to guarantee accuracy of the result. Finally, we create a prototype of the DP_VELP unit and an FPGA accelerator based on the DP_VELP unit on a Xilinx XC6VLX760 FPGA chip to implement the SGP4/SDP4 application. Compared with previous researches, the proposed design can achieve low latency with a reasonable amount of resources and evaluate a variety of elementary functions with the unified hardware to satisfy the demands in scientific applications. Experimental results show that the proposed design guarantees more than 99% of correct rounding. Moreover, the SGP4/SDP4 accelerator, which is equipped with 39 DP_VELP units and runs at 200 MHz, outperforms the parallel software approach with hyper-thread technology on an Intel Xeon Quad E5620 CPU at 2.40 GHz by a factor of 7X.

rawabstract:
In the current article, the capability and flexibility of field programmable gate-arrays (FPGAs) to implement IEEE-754 double-precision floating-point elementary functions are explored. To perform various elementary functions on the unified hardware efficiently, we propose a special-purpose very long instruction word (VLIW) processor, called DP_VELP. This processor is equipped with multiple basic units, and its performance is improved through an explicitly parallel technique. Pipelined evaluation of polynomial approximation with Estrin's scheme is proposed, by scheduling basic components in an optimal order to avoid data hazard stalls and achieve minimal latency. The custom VLIW processor can achieve high scalability. Under the control of specific VLIW instructions, the basic units are combined into special-purpose hardware for elementary functions. Common elementary functions are presented as examples to illustrate the design of elementary function in DP_VELP in detail. Minimax approximation scheme is used to reduce degree of polynomial. Compromise between the size of lookup table and the latency is discussed, and the internal precision is carefully planned to guarantee accuracy of the result. Finally, we create a prototype of the DP_VELP unit and an FPGA accelerator based on the DP_VELP unit on a Xilinx XC6VLX760 FPGA chip to implement the SGP4/SDP4 application. Compared with previous researches, the proposed design can achieve low latency with a reasonable amount of resources and evaluate a variety of elementary functions with the unified hardware to satisfy the demands in scientific applications. Experimental results show that the proposed design guarantees more than 99\% of correct rounding. Moreover, the SGP4/SDP4 accelerator, which is equipped with 39 DP_VELP units and runs at 200 MHz, outperforms the parallel software approach with hyper-thread technology on an Intel Xeon Quad E5620 CPU at 2.40 GHz by a factor of 7X.

rawtitle:
{FPGA} Implementation of a Special-Purpose {VLIW} Structure for Double-Precision Elementary Function

entry:
@Article{Lei:2014:FIS,
  author =       "Yuanwu Lei and Lei Guo and Yong Dou and Sheng Ma and
                 Jinbo Xu",
  title =        "{FPGA} Implementation of a Special-Purpose {VLIW}
                 Structure for Double-Precision Elementary Function",
  journal =      j-TRETS,
  volume =       "7",
  number =       "2",
  pages =        "8:1--8:??",
  month =        jun,
  year =         "2014",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2617594",
  ISSN =         "1936-7406 (print), 1936-7414 (electronic)",
  ISSN-L =       "1936-7406",
  bibdate =      "Mon Jun 30 18:26:23 MDT 2014",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/trets.bib",
  abstract =     "In the current article, the capability and flexibility
                 of field programmable gate-arrays (FPGAs) to implement
                 IEEE-754 double-precision floating-point elementary
                 functions are explored. To perform various elementary
                 functions on the unified hardware efficiently, we
                 propose a special-purpose very long instruction word
                 (VLIW) processor, called DP_VELP. This processor is
                 equipped with multiple basic units, and its performance
                 is improved through an explicitly parallel technique.
                 Pipelined evaluation of polynomial approximation with
                 Estrin's scheme is proposed, by scheduling basic
                 components in an optimal order to avoid data hazard
                 stalls and achieve minimal latency. The custom VLIW
                 processor can achieve high scalability. Under the
                 control of specific VLIW instructions, the basic units
                 are combined into special-purpose hardware for
                 elementary functions. Common elementary functions are
                 presented as examples to illustrate the design of
                 elementary function in DP_VELP in detail. Minimax
                 approximation scheme is used to reduce degree of
                 polynomial. Compromise between the size of lookup table
                 and the latency is discussed, and the internal
                 precision is carefully planned to guarantee accuracy of
                 the result. Finally, we create a prototype of the
                 DP_VELP unit and an FPGA accelerator based on the
                 DP_VELP unit on a Xilinx XC6VLX760 FPGA chip to
                 implement the SGP4/SDP4 application. Compared with
                 previous researches, the proposed design can achieve
                 low latency with a reasonable amount of resources and
                 evaluate a variety of elementary functions with the
                 unified hardware to satisfy the demands in scientific
                 applications. Experimental results show that the
                 proposed design guarantees more than 99\% of correct
                 rounding. Moreover, the SGP4/SDP4 accelerator, which is
                 equipped with 39 DP_VELP units and runs at 200 MHz,
                 outperforms the parallel software approach with
                 hyper-thread technology on an Intel Xeon Quad E5620 CPU
                 at 2.40 GHz by a factor of 7X.",
  acknowledgement = ack-nhfb,
  articleno =    "8",
  fjournal =     "ACM Transactions on Reconfigurable Technology and
                 Systems (TRETS)",
  journal-URL =  "http://portal.acm.org/toc.cfm?id=J1151",
}


bibtype: article
filename: fparith.bib
label: Lindstrom:2014:FRC
author: Peter Lindstrom
title: Fixed-Rate Compressed Floating-Point Arrays
journal: j-IEEE-TRANS-VIS-COMPUT-GRAPH
volume: 20
number: 12
pages: 2674--2683
month: dec
monthnumber: 12
year: 2014
CODEN: ITVGEA
DOI: https://doi.org/10.1109/TVCG.2014.2346458
ISSNL: 1077-2626 OR 10772626
bibdate: Thu Feb 12 16:40:54 MST 2015
bibtimestamp: 2015.02.12 16:40:54 MST
URL: http://csdl.computer.org/csdl/trans/tg/2014/12/06876024-abs.html
fjournal: IEEE Transactions on Visualization and Computer Graphics
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2945
rawauthor: Peter Lindstrom
rawtitle: Fixed-Rate Compressed Floating-Point Arrays
ISSN:
1077-2626 (print), 1941-0506 (electronic), 2160-9306 OR 10772626 (print), 19410506 (electronic), 21609306

bibsource:
https://www.math.utah.edu/pub/tex/bib/datacompression.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetransviscomputgraph.bib

entry:
@Article{Lindstrom:2014:FRC,
  author =       "Peter Lindstrom",
  title =        "Fixed-Rate Compressed Floating-Point Arrays",
  journal =      j-IEEE-TRANS-VIS-COMPUT-GRAPH,
  volume =       "20",
  number =       "12",
  pages =        "2674--2683",
  month =        dec,
  year =         "2014",
  CODEN =        "ITVGEA",
  DOI =          "https://doi.org/10.1109/TVCG.2014.2346458",
  ISSN =         "1077-2626 (print), 1941-0506 (electronic), 2160-9306",
  ISSN-L =       "1077-2626",
  bibdate =      "Thu Feb 12 16:40:54 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/datacompression.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetransviscomputgraph.bib",
  URL =          "http://csdl.computer.org/csdl/trans/tg/2014/12/06876024-abs.html",
  abstract-URL = "http://csdl.computer.org/csdl/trans/tg/2014/12/06876024-abs.html",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Visualization and Computer
                 Graphics",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2945",
}


bibtype: article
filename: fparith.bib
label: Long:2014:SIF
author: Fan Long and Stelios Sidiroglou-Douskos and Deokhwan Kim and Martin Rinard
title: Sound input filter generation for integer overflow errors
journal: j-SIGPLAN
volume: 49
number: 1
pages: 439--452
month: jan
monthnumber: 1
year: 2014
CODEN: SINODQ
DOI: https://doi.org/10.1145/2578855.2535888
ISSN: 0362-1340 (print), 1523-2867 (print), 1558-1160 OR 03621340 (print), 15232867 (print), 15581160
ISSNL: 0362-1340 OR 03621340
bibdate: Tue Mar 4 17:04:57 MST 2014
bibtimestamp: 2014.03.04 17:04:57 MST
note: POPL '14 conference proceedings.
fjournal: ACM SIGPLAN Notices
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J706
rawauthor: Fan Long and Stelios Sidiroglou-Douskos and Deokhwan Kim and Martin Rinard
rawnote: POPL '14 conference proceedings.
rawtitle: Sound input filter generation for integer overflow errors
abstract:
We present a system, SIFT, for generating input filters that nullify integer overflow errors associated with critical program sites such as memory allocation or block copy sites. SIFT uses a static program analysis to generate filters that discard inputs that may trigger integer overflow errors in the computations of the sizes of allocated memory blocks or the number of copied bytes in block copy operations. Unlike all previous techniques of which we are aware, SIFT is sound --- if an input passes the filter, it will not trigger an integer overflow error at any analyzed site. Our results show that SIFT successfully analyzes (and therefore generates sound input filters for) 56 out of 58 memory allocation and block memory copy sites in analyzed input processing modules from five applications (VLC, Dillo, Swfdec, Swftools, and GIMP). These nullified errors include six known integer overflow vulnerabilities. Our results also show that applying these filters to 62895 real-world inputs produces no false positives. The analysis and filter generation times are all less than a second.

rawabstract:
We present a system, SIFT, for generating input filters that nullify integer overflow errors associated with critical program sites such as memory allocation or block copy sites. SIFT uses a static program analysis to generate filters that discard inputs that may trigger integer overflow errors in the computations of the sizes of allocated memory blocks or the number of copied bytes in block copy operations. Unlike all previous techniques of which we are aware, SIFT is sound --- if an input passes the filter, it will not trigger an integer overflow error at any analyzed site. Our results show that SIFT successfully analyzes (and therefore generates sound input filters for) 56 out of 58 memory allocation and block memory copy sites in analyzed input processing modules from five applications (VLC, Dillo, Swfdec, Swftools, and GIMP). These nullified errors include six known integer overflow vulnerabilities. Our results also show that applying these filters to 62895 real-world inputs produces no false positives. The analysis and filter generation times are all less than a second.

entry:
@Article{Long:2014:SIF,
  author =       "Fan Long and Stelios Sidiroglou-Douskos and Deokhwan
                 Kim and Martin Rinard",
  title =        "Sound input filter generation for integer overflow
                 errors",
  journal =      j-SIGPLAN,
  volume =       "49",
  number =       "1",
  pages =        "439--452",
  month =        jan,
  year =         "2014",
  CODEN =        "SINODQ",
  DOI =          "https://doi.org/10.1145/2578855.2535888",
  ISSN =         "0362-1340 (print), 1523-2867 (print), 1558-1160
                 (electronic)",
  ISSN-L =       "0362-1340",
  bibdate =      "Tue Mar 4 17:04:57 MST 2014",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigplan2010.bib",
  note =         "POPL '14 conference proceedings.",
  abstract =     "We present a system, SIFT, for generating input
                 filters that nullify integer overflow errors associated
                 with critical program sites such as memory allocation
                 or block copy sites. SIFT uses a static program
                 analysis to generate filters that discard inputs that
                 may trigger integer overflow errors in the computations
                 of the sizes of allocated memory blocks or the number
                 of copied bytes in block copy operations. Unlike all
                 previous techniques of which we are aware, SIFT is
                 sound --- if an input passes the filter, it will not
                 trigger an integer overflow error at any analyzed site.
                 Our results show that SIFT successfully analyzes (and
                 therefore generates sound input filters for) 56 out of
                 58 memory allocation and block memory copy sites in
                 analyzed input processing modules from five
                 applications (VLC, Dillo, Swfdec, Swftools, and GIMP).
                 These nullified errors include six known integer
                 overflow vulnerabilities. Our results also show that
                 applying these filters to 62895 real-world inputs
                 produces no false positives. The analysis and filter
                 generation times are all less than a second.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGPLAN Notices",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J706",
}


bibtype: article
filename: fparith.bib
label: Lupon:2014:SHS
title: Speculative hardware / software co-designed floating-point multiply-add fusion
journal: j-COMP-ARCH-NEWS
volume: 42
number: 1
pages: 623--638
month: mar
monthnumber: 3
year: 2014
CODEN: CANED2
DOI: https://doi.org/10.1145/2654822.2541978
ISSN: 0163-5964 (print), 1943-5851 (electronic) OR 01635964 (print), 19435851 (electronic)
ISSNL: 0163-5964 OR 01635964
bibdate: Thu Sep 4 07:12:13 MDT 2014
bibtimestamp: 2014.09.04 07:12:13 MDT
fjournal: ACM SIGARCH Computer Architecture News
remark: ASPLOS '14 conference proceedings.
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J89
rawtitle: Speculative hardware\slash software co-designed floating-point multiply-add fusion
author:
Marc Lupon and Enric Gibert and Grigorios Magklis and Sridhar Samudrala and Raul Martinez and Kyriakos Stavrou and David R. Ditzel

abstract:
A Fused Multiply-Add (FMA) instruction is currently available in many general-purpose processors. It increases performance by reducing latency of dependent operations and increases precision by computing the result as an indivisible operation with no intermediate rounding. However, since the arithmetic behavior of a single-rounding FMA operation is different than independent FP multiply followed by FP add instructions, some algorithms require significant revalidation and rewriting efforts to work as expected when they are compiled to operate with FMA --- a cost that developers may not be willing to pay. Because of that, abundant legacy applications are not able to utilize FMA instructions. In this paper we propose a novel HW/SW collaborative technique that is able to efficiently execute workloads with increased utilization of FMA, by adding the option to get the same numerical result as separate FP multiply and FP add pairs. In particular, we extended the host ISA of a HW/SW co-designed processor with a new Combined Multiply-Add (CMA) instruction that performs an FMA operation with an intermediate rounding. This new instruction is used by a transparent dynamic translation software layer that uses a speculative instruction-fusion optimization to transform FP multiply and FP add sequences into CMA instructions. The FMA unit has been slightly modified to support both single-rounding and double-rounding fused instructions without increasing their latency and to provide a conservative fall-back path in case of misspeculation. Evaluation on a cycle-accurate timing simulator showed that CMA improved SPECfp performance by 6.3% and reduced executed instructions by 4.7%.

rawabstract:
A Fused Multiply-Add (FMA) instruction is currently available in many general-purpose processors. It increases performance by reducing latency of dependent operations and increases precision by computing the result as an indivisible operation with no intermediate rounding. However, since the arithmetic behavior of a single-rounding FMA operation is different than independent FP multiply followed by FP add instructions, some algorithms require significant revalidation and rewriting efforts to work as expected when they are compiled to operate with FMA --- a cost that developers may not be willing to pay. Because of that, abundant legacy applications are not able to utilize FMA instructions. In this paper we propose a novel HW/SW collaborative technique that is able to efficiently execute workloads with increased utilization of FMA, by adding the option to get the same numerical result as separate FP multiply and FP add pairs. In particular, we extended the host ISA of a HW/SW co-designed processor with a new Combined Multiply-Add (CMA) instruction that performs an FMA operation with an intermediate rounding. This new instruction is used by a transparent dynamic translation software layer that uses a speculative instruction-fusion optimization to transform FP multiply and FP add sequences into CMA instructions. The FMA unit has been slightly modified to support both single-rounding and double-rounding fused instructions without increasing their latency and to provide a conservative fall-back path in case of misspeculation. Evaluation on a cycle-accurate timing simulator showed that CMA improved SPECfp performance by 6.3\% and reduced executed instructions by 4.7\%.

rawauthor:
Marc Lupon and Enric Gibert and Grigorios Magklis and Sridhar Samudrala and Ra{\'u}l Mart{\'\i}nez and Kyriakos Stavrou and David R. Ditzel

entry:
@Article{Lupon:2014:SHS,
  author =       "Marc Lupon and Enric Gibert and Grigorios Magklis and
                 Sridhar Samudrala and Ra{\'u}l Mart{\'\i}nez and
                 Kyriakos Stavrou and David R. Ditzel",
  title =        "Speculative hardware\slash software co-designed
                 floating-point multiply-add fusion",
  journal =      j-COMP-ARCH-NEWS,
  volume =       "42",
  number =       "1",
  pages =        "623--638",
  month =        mar,
  year =         "2014",
  CODEN =        "CANED2",
  DOI =          "https://doi.org/10.1145/2654822.2541978",
  ISSN =         "0163-5964 (print), 1943-5851 (electronic)",
  ISSN-L =       "0163-5964",
  bibdate =      "Thu Sep 4 07:12:13 MDT 2014",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigarch.bib",
  abstract =     "A Fused Multiply-Add (FMA) instruction is currently
                 available in many general-purpose processors. It
                 increases performance by reducing latency of dependent
                 operations and increases precision by computing the
                 result as an indivisible operation with no intermediate
                 rounding. However, since the arithmetic behavior of a
                 single-rounding FMA operation is different than
                 independent FP multiply followed by FP add
                 instructions, some algorithms require significant
                 revalidation and rewriting efforts to work as expected
                 when they are compiled to operate with FMA --- a cost
                 that developers may not be willing to pay. Because of
                 that, abundant legacy applications are not able to
                 utilize FMA instructions. In this paper we propose a
                 novel HW/SW collaborative technique that is able to
                 efficiently execute workloads with increased
                 utilization of FMA, by adding the option to get the
                 same numerical result as separate FP multiply and FP
                 add pairs. In particular, we extended the host ISA of a
                 HW/SW co-designed processor with a new Combined
                 Multiply-Add (CMA) instruction that performs an FMA
                 operation with an intermediate rounding. This new
                 instruction is used by a transparent dynamic
                 translation software layer that uses a speculative
                 instruction-fusion optimization to transform FP
                 multiply and FP add sequences into CMA instructions.
                 The FMA unit has been slightly modified to support both
                 single-rounding and double-rounding fused instructions
                 without increasing their latency and to provide a
                 conservative fall-back path in case of misspeculation.
                 Evaluation on a cycle-accurate timing simulator showed
                 that CMA improved SPECfp performance by 6.3\% and
                 reduced executed instructions by 4.7\%.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGARCH Computer Architecture News",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J89",
  remark =       "ASPLOS '14 conference proceedings.",
}


bibtype: article
filename: fparith.bib
label: Marche:2014:VFB
author: Claude Marche
title: Verification of the functional behavior of a floating-point program: an industrial case study
journal: j-SCI-COMPUT-PROGRAM
volume: 96 (part 3)
pages: 279--296
day: 15
month: dec
monthnumber: 12
year: 2014
CODEN: SCPGD4
ISSN: 0167-6423 (print), 1872-7964 (electronic) OR 01676423 (print), 18727964 (electronic)
ISSNL: 0167-6423 OR 01676423
bibdate: Fri Oct 24 05:57:57 MDT 2014
bibtimestamp: 2014.10.24 05:57:57 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0167642314001671
fjournal: Science of Computer Programming
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01676423/
rawauthor: Claude March{\'e}
rawtitle: Verification of the functional behavior of a floating-point program: an industrial case study
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib

entry:
@Article{Marche:2014:VFB,
  author =       "Claude March{\'e}",
  title =        "Verification of the functional behavior of a
                 floating-point program: an industrial case study",
  journal =      j-SCI-COMPUT-PROGRAM,
  volume =       "96 (part 3)",
  number =       "??",
  pages =        "279--296",
  day =          "15",
  month =        dec,
  year =         "2014",
  CODEN =        "SCPGD4",
  ISSN =         "0167-6423 (print), 1872-7964 (electronic)",
  ISSN-L =       "0167-6423",
  bibdate =      "Fri Oct 24 05:57:57 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0167642314001671",
  acknowledgement = ack-nhfb,
  fjournal =     "Science of Computer Programming",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01676423/",
}


bibtype: article
filename: fparith.bib
label: Milicevic:2014:PAO
author: Aleksandar Milicevic and Daniel Jackson
title: Preventing arithmetic overflows in Alloy
journal: j-SCI-COMPUT-PROGRAM
volume: 94 (part 2)
pages: 203--216
day: 15
month: nov
monthnumber: 11
year: 2014
CODEN: SCPGD4
DOI: https://doi.org/10.1016/j.scico.2014.05.009
ISSN: 0167-6423 (print), 1872-7964 (electronic) OR 01676423 (print), 18727964 (electronic)
ISSNL: 0167-6423 OR 01676423
bibdate: Mon Sep 22 08:04:04 MDT 2014
bibtimestamp: 2014.09.22 08:04:04 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0167642314002470
fjournal: Science of Computer Programming
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01676423/
rawauthor: Aleksandar Milicevic and Daniel Jackson
rawtitle: Preventing arithmetic overflows in {Alloy}
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib

entry:
@Article{Milicevic:2014:PAO,
  author =       "Aleksandar Milicevic and Daniel Jackson",
  title =        "Preventing arithmetic overflows in {Alloy}",
  journal =      j-SCI-COMPUT-PROGRAM,
  volume =       "94 (part 2)",
  number =       "??",
  pages =        "203--216",
  day =          "15",
  month =        nov,
  year =         "2014",
  CODEN =        "SCPGD4",
  DOI =          "https://doi.org/10.1016/j.scico.2014.05.009",
  ISSN =         "0167-6423 (print), 1872-7964 (electronic)",
  ISSN-L =       "0167-6423",
  bibdate =      "Mon Sep 22 08:04:04 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0167642314002470",
  acknowledgement = ack-nhfb,
  fjournal =     "Science of Computer Programming",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01676423/",
}


bibtype: misc
filename: fparith.bib
label: Moler:2014:CCFa
author: Cleve Moler
title: Cleve's Corner: Floating Point Numbers
pages: 203--216
day: 7
month: jul
monthnumber: 7
year: 2014
bibdate: Sat Jan 19 18:59:13 2019
bibtimestamp: 2019.01.19 18:59:13 ???
URL: https://blogs.mathworks.com/cleve/2014/07/07/floating-point-numbers/
acknowledgement: ack-nhfb
howpublished: MathWorks Web site.
rawauthor: Cleve Moler
rawtitle: {Cleve's Corner}: Floating Point Numbers
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/m/moler-cleve-b.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Misc{Moler:2014:CCFa,
  author =       "Cleve Moler",
  title =        "{Cleve's Corner}: Floating Point Numbers",
  howpublished = "MathWorks Web site.",
  day =          "7",
  month =        jul,
  year =         "2014",
  bibdate =      "Sat Jan 19 18:59:13 2019",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/m/moler-cleve-b.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://blogs.mathworks.com/cleve/2014/07/07/floating-point-numbers/",
  acknowledgement = ack-nhfb,
}


bibtype: misc
filename: fparith.bib
label: Moler:2014:CCFb
author: Cleve Moler
title: Cleve's Corner: Floating Point Denormals, Insignificant But Controversial
pages: 203--216
day: 21
month: jul
monthnumber: 7
year: 2014
bibdate: Sat Jan 19 18:38:12 2019
bibtimestamp: 2019.01.19 18:38:12 ???
acknowledgement: ack-nhfb
howpublished: MathWorks Web site.
rawauthor: Cleve Moler
rawtitle: {Cleve's Corner}: Floating Point Denormals, Insignificant But Controversial
URL:
https://blogs.mathworks.com/cleve/2014/07/21/floating-point-denormals-insignificant-but-controversial-2/

entry:
@Misc{Moler:2014:CCFb,
  author =       "Cleve Moler",
  title =        "{Cleve's Corner}: Floating Point Denormals,
                 Insignificant But Controversial",
  howpublished = "MathWorks Web site.",
  day =          "21",
  month =        jul,
  year =         "2014",
  bibdate =      "Sat Jan 19 18:38:12 2019",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/m/moler-cleve-b.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://blogs.mathworks.com/cleve/2014/07/21/floating-point-denormals-insignificant-but-controversial-2/",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Mukhopadhyay:2014:EMP
author: Debapriyay Mukhopadhyay and Subhas C. Nandy
title: Efficient multiple-precision integer division algorithm
journal: j-INFO-PROC-LETT
volume: 114
number: 3
pages: 152--157
month: mar
monthnumber: 3
year: 2014
CODEN: IFPLAT
DOI: https://doi.org/10.1016/j.ipl.2013.10.005
ISSN: 0020-0190 (print), 1872-6119 (electronic) OR 00200190 (print), 18726119 (electronic)
ISSNL: 0020-0190 OR 00200190
bibdate: Mon Dec 9 09:33:47 MST 2013
bibtimestamp: 2013.12.09 09:33:47 MST
URL: http://www.sciencedirect.com/science/article/pii/S0020019013002627
fjournal: Information Processing Letters
keywords: computational arithmetic; cryptography; division algorithm; normalization
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00200190
rawauthor: Debapriyay Mukhopadhyay and Subhas C. Nandy
rawtitle: Efficient multiple-precision integer division algorithm
bibsource:
http://www.sciencedirect.com/science/journal/00200190; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/infoproc2010.bib

note:
This paper provides a correction to the algorithm presented in cite Huang:2005:EMP, and also supplies a complicated correctness proof.

rawnote:
This paper provides a correction to the algorithm presented in \cite{Huang:2005:EMP}, and also supplies a complicated correctness proof.

entry:
@Article{Mukhopadhyay:2014:EMP,
  author =       "Debapriyay Mukhopadhyay and Subhas C. Nandy",
  title =        "Efficient multiple-precision integer division
                 algorithm",
  journal =      j-INFO-PROC-LETT,
  volume =       "114",
  number =       "3",
  pages =        "152--157",
  month =        mar,
  year =         "2014",
  CODEN =        "IFPLAT",
  DOI =          "https://doi.org/10.1016/j.ipl.2013.10.005",
  ISSN =         "0020-0190 (print), 1872-6119 (electronic)",
  ISSN-L =       "0020-0190",
  bibdate =      "Mon Dec 9 09:33:47 MST 2013",
  bibsource =    "http://www.sciencedirect.com/science/journal/00200190;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/infoproc2010.bib",
  note =         "This paper provides a correction to the algorithm
                 presented in \cite{Huang:2005:EMP}, and also supplies a
                 complicated correctness proof.",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0020019013002627",
  acknowledgement = ack-nhfb,
  fjournal =     "Information Processing Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00200190",
  keywords =     "computational arithmetic; cryptography; division
                 algorithm; normalization",
}


bibtype: inproceedings
filename: fparith.bib
label: Muller:2014:MRE
author: Jean-Michel Muller
booktitle: INVA Conference Proceedings, Tokyo, Japan, 2014
title: On the maximum relative error when computing iterated integer powers in floating-point arithmetic
pages: ??--??
year: 2014
bibdate: Mon May 30 10:37:15 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.05.30 10:37:15 ???
remark: See later paper cite Graillat:2015:MRE.
acknowledgement: ack-nhfb
rawauthor: Jean-Michel Muller
rawbooktitle: {INVA} Conference Proceedings, Tokyo, Japan, 2014
rawtitle: On the maximum relative error when computing iterated integer powers in floating-point arithmetic
entry:
@InProceedings{Muller:2014:MRE,
  author =       "Jean-Michel Muller",
  editor =       "????",
  booktitle =    "{INVA} Conference Proceedings, Tokyo, Japan, 2014",
  title =        "On the maximum relative error when computing iterated
                 integer powers in floating-point arithmetic",
  publisher =    "????",
  address =      "????",
  pages =        "??--??",
  year =         "2014",
  bibdate =      "Mon May 30 10:37:15 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  remark =       "See later paper \cite{Graillat:2015:MRE}.",
}


bibtype: article
filename: fparith.bib
label: Murakami:2014:CRN
author: Hiroshi Murakami
journal: j-ACM-COMM-COMP-ALGEBRA
volume: 48
number: 3/4
pages: 134--136
month: sep
monthnumber: 9
year: 2014
DOI: https://doi.org/10.1145/2733693.2733711
ISSN: 1932-2232 (print), 1932-2240 (electronic) OR 19322232 (print), 19322240 (electronic)
ISSNL: 1932-2232 OR 19322232
bibdate: Thu Feb 5 17:11:08 MST 2015
bibtimestamp: 2015.02.05 17:11:08 MST
fjournal: ACM Communications in Computer Algebra
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
rawauthor: Hiroshi Murakami
title:
Calculation of Rational Numbers in an Interval Whose Denominator is the Smallest by using FP Interval Arithmetic

bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib

abstract:
The continued fraction expansion method is a fast solver to find a rational number in a given real interval whose denominator is the smallest. A simple implementation of the CF expansion method which uses floating point numbers as real numbers has a possibility to give a wrong answer by the effect of numerical round-off errors. In this paper, we show a modification of the algorithm of the CF expansion method so that it uses floating point (FP) intervals as replacements of real numbers. By this modified algorithm, the answer is obtained only when its correctness is guaranteed and the possibility to give a wrong answer is eliminated.

rawabstract:
The continued fraction expansion method is a fast solver to find a rational number in a given real interval whose denominator is the smallest. A simple implementation of the CF expansion method which uses floating point numbers as real numbers has a possibility to give a wrong answer by the effect of numerical round-off errors. In this paper, we show a modification of the algorithm of the CF expansion method so that it uses floating point (FP) intervals as replacements of real numbers. By this modified algorithm, the answer is obtained only when its correctness is guaranteed and the possibility to give a wrong answer is eliminated.

rawtitle:
Calculation of Rational Numbers in an Interval Whose Denominator is the Smallest by using {FP} Interval Arithmetic

entry:
@Article{Murakami:2014:CRN,
  author =       "Hiroshi Murakami",
  title =        "Calculation of Rational Numbers in an Interval Whose
                 Denominator is the Smallest by using {FP} Interval
                 Arithmetic",
  journal =      j-ACM-COMM-COMP-ALGEBRA,
  volume =       "48",
  number =       "3/4",
  pages =        "134--136",
  month =        sep,
  year =         "2014",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2733693.2733711",
  ISSN =         "1932-2232 (print), 1932-2240 (electronic)",
  ISSN-L =       "1932-2232",
  bibdate =      "Thu Feb 5 17:11:08 MST 2015",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigsam.bib",
  abstract =     "The continued fraction expansion method is a fast
                 solver to find a rational number in a given real
                 interval whose denominator is the smallest. A simple
                 implementation of the CF expansion method which uses
                 floating point numbers as real numbers has a
                 possibility to give a wrong answer by the effect of
                 numerical round-off errors. In this paper, we show a
                 modification of the algorithm of the CF expansion
                 method so that it uses floating point (FP) intervals as
                 replacements of real numbers. By this modified
                 algorithm, the answer is obtained only when its
                 correctness is guaranteed and the possibility to give a
                 wrong answer is eliminated.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Communications in Computer Algebra",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
}


bibtype: article
filename: fparith.bib
label: Nannarelli:2014:GEI
author: Alberto Nannarelli and Peter-Michael Seidel and Ping Tak Peter Tang
title: Guest Editors' Introduction: Special Section on Computer Arithmetic
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 8
pages: 1852--1853
month: aug
monthnumber: 8
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2331711
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Aug 25 08:24:32 2014
bibtimestamp: 2014.08.25 08:24:32 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Alberto Nannarelli and Peter-Michael Seidel and Ping Tak Peter Tang
rawtitle: {Guest Editors}' Introduction: Special Section on Computer Arithmetic
entry:
@Article{Nannarelli:2014:GEI,
  author =       "Alberto Nannarelli and Peter-Michael Seidel and Ping
                 Tak Peter Tang",
  title =        "{Guest Editors}' Introduction: Special Section on
                 Computer Arithmetic",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "8",
  pages =        "1852--1853",
  month =        aug,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2331711",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Aug 25 08:24:32 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Neto:2014:PUP
author: Joao Carlos Neto and Alexandre Ferreira Tenca and Wilson Vicente Ruggiero
title: A Parallel and Uniform $k$-Partition Method for Montgomery Multiplication
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 9
pages: 2122--2133
month: sep
monthnumber: 9
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.89
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Sep 9 06:54:10 2014
bibtimestamp: 2014.09.09 06:54:10 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Joao Carlos Neto and Alexandre Ferreira Tenca and Wilson Vicente Ruggiero
rawtitle: A Parallel and Uniform $k$-Partition Method for {Montgomery} Multiplication
entry:
@Article{Neto:2014:PUP,
  author =       "Joao Carlos Neto and Alexandre Ferreira Tenca and
                 Wilson Vicente Ruggiero",
  title =        "A Parallel and Uniform $k$-Partition Method for
                 {Montgomery} Multiplication",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "9",
  pages =        "2122--2133",
  month =        sep,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.89",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Sep 9 06:54:10 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Nguyen:2014:RED
author: Trung Duc Nguyen and Rodney Van Meter
title: A Resource-Efficient Design for a Reversible Floating Point Adder in Quantum Computing
journal: j-JETC
volume: 11
number: 2
pages: 13:1--13:??
month: nov
monthnumber: 11
year: 2014
DOI: https://doi.org/10.1145/2629525
ISSN: 1550-4832 (print), 1550-4840 (electronic) OR 15504832 (print), 15504840 (electronic)
ISSNL: 1550-4832 OR 15504832
bibdate: Wed Nov 5 18:01:28 MST 2014
bibtimestamp: 2014.11.05 18:01:28 MST
fjournal: ACM Journal on Emerging Technologies in Computing Systems (JETC)
acknowledgement: ack-nhfb
articleno: 13
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J967
rawauthor: Trung Duc Nguyen and Rodney {Van Meter}
rawtitle: A Resource-Efficient Design for a Reversible Floating Point Adder in Quantum Computing
bibsource:
http://www.acm.org/pubs/contents/journals/jetc/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jetc.bib

abstract:
Reversible logic has applications in low-power computing and quantum computing. However, there are few existing designs for reversible floating-point adders and none suitable for quantum computation. In this article, we propose a resource-efficient reversible floating-point adder, suitable for binary quantum computation, improving the design of Nachtigal et al. [2011]. Our work focuses on improving the reversible designs of the alignment unit and the normalization unit, which are the most expensive parts. By changing a few elements of the existing algorithm, including the circuit designs of the RLZC (reversible leading zero counter) and converter, we have reduced the cost by about 68%. We also propose quantum designs adapted to use gates from fault-tolerant libraries. The KQ for our fault-tolerant design is almost 60 times as expensive as for a 32-bit fixed-point addition. We note that the floating-point representation makes in-place, truly reversible arithmetic impossible, requiring us to retain both inputs, which limits the sustainability of its use for quantum computation.

rawabstract:
Reversible logic has applications in low-power computing and quantum computing. However, there are few existing designs for reversible floating-point adders and none suitable for quantum computation. In this article, we propose a resource-efficient reversible floating-point adder, suitable for binary quantum computation, improving the design of Nachtigal et al. [2011]. Our work focuses on improving the reversible designs of the alignment unit and the normalization unit, which are the most expensive parts. By changing a few elements of the existing algorithm, including the circuit designs of the RLZC (reversible leading zero counter) and converter, we have reduced the cost by about 68\%. We also propose quantum designs adapted to use gates from fault-tolerant libraries. The KQ for our fault-tolerant design is almost 60 times as expensive as for a 32-bit fixed-point addition. We note that the floating-point representation makes in-place, truly reversible arithmetic impossible, requiring us to retain both inputs, which limits the sustainability of its use for quantum computation.

entry:
@Article{Nguyen:2014:RED,
  author =       "Trung Duc Nguyen and Rodney {Van Meter}",
  title =        "A Resource-Efficient Design for a Reversible Floating
                 Point Adder in Quantum Computing",
  journal =      j-JETC,
  volume =       "11",
  number =       "2",
  pages =        "13:1--13:??",
  month =        nov,
  year =         "2014",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2629525",
  ISSN =         "1550-4832 (print), 1550-4840 (electronic)",
  ISSN-L =       "1550-4832",
  bibdate =      "Wed Nov 5 18:01:28 MST 2014",
  bibsource =    "http://www.acm.org/pubs/contents/journals/jetc/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jetc.bib",
  abstract =     "Reversible logic has applications in low-power
                 computing and quantum computing. However, there are few
                 existing designs for reversible floating-point adders
                 and none suitable for quantum computation. In this
                 article, we propose a resource-efficient reversible
                 floating-point adder, suitable for binary quantum
                 computation, improving the design of Nachtigal et al.
                 [2011]. Our work focuses on improving the reversible
                 designs of the alignment unit and the normalization
                 unit, which are the most expensive parts. By changing a
                 few elements of the existing algorithm, including the
                 circuit designs of the RLZC (reversible leading zero
                 counter) and converter, we have reduced the cost by
                 about 68\%. We also propose quantum designs adapted to
                 use gates from fault-tolerant libraries. The KQ for our
                 fault-tolerant design is almost 60 times as expensive
                 as for a 32-bit fixed-point addition. We note that the
                 floating-point representation makes in-place, truly
                 reversible arithmetic impossible, requiring us to
                 retain both inputs, which limits the sustainability of
                 its use for quantum computation.",
  acknowledgement = ack-nhfb,
  articleno =    "13",
  fjournal =     "ACM Journal on Emerging Technologies in Computing
                 Systems (JETC)",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J967",
}


bibtype: article
filename: fparith.bib
label: Pedram:2014:AAF
author: Ardavan Pedram and Andreas Gerstlauer and Robert A. van de Geijn
title: Algorithm, Architecture, and Floating-Point Unit Codesign of a Matrix Factorization Accelerator
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 8
pages: 1854--1867
month: aug
monthnumber: 8
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2315627
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Aug 25 08:24:32 2014
bibtimestamp: 2014.08.25 08:24:32 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Ardavan Pedram and Andreas Gerstlauer and Robert A. van de Geijn
rawtitle: Algorithm, Architecture, and Floating-Point Unit Codesign of a Matrix Factorization Accelerator
entry:
@Article{Pedram:2014:AAF,
  author =       "Ardavan Pedram and Andreas Gerstlauer and Robert A.
                 van de Geijn",
  title =        "Algorithm, Architecture, and Floating-Point Unit
                 Codesign of a Matrix Factorization Accelerator",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "8",
  pages =        "1854--1867",
  month =        aug,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2315627",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Aug 25 08:24:32 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Piso:2014:OAE
author: D. Piso and J. D. Bruguera
title: Obtaining Accurate Error Expressions and Bounds for Floating-Point Multiplicative Algorithms
journal: j-COMP-J
volume: 57
number: 2
pages: 319--331
month: feb
monthnumber: 2
year: 2014
CODEN: CMPJA6
DOI: https://doi.org/10.1093/comjnl/bxs170
ISSN: 0010-4620 (print), 1460-2067 (electronic) OR 00104620 (print), 14602067 (electronic)
ISSNL: 0010-4620 OR 00104620
bibdate: Mon Feb 3 17:03:04 MST 2014
bibtimestamp: 2014.02.03 17:03:04 MST
URL: http://comjnl.oxfordjournals.org/content/57/2/319.full.pdf+html
fjournal: The Computer Journal
acknowledgement: ack-nhfb
journalURL: http://comjnl.oxfordjournals.org/
onlinedate: January 17, 2013
rawauthor: D. Piso and J. D. Bruguera
rawtitle: Obtaining Accurate Error Expressions and Bounds for Floating-Point Multiplicative Algorithms
bibsource:
http://comjnl.oxfordjournals.org/content/57/2.toc; https://www.math.utah.edu/pub/tex/bib/compj2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Piso:2014:OAE,
  author =       "D. Piso and J. D. Bruguera",
  title =        "Obtaining Accurate Error Expressions and Bounds for
                 Floating-Point Multiplicative Algorithms",
  journal =      j-COMP-J,
  volume =       "57",
  number =       "2",
  pages =        "319--331",
  month =        feb,
  year =         "2014",
  CODEN =        "CMPJA6",
  DOI =          "https://doi.org/10.1093/comjnl/bxs170",
  ISSN =         "0010-4620 (print), 1460-2067 (electronic)",
  ISSN-L =       "0010-4620",
  bibdate =      "Mon Feb 3 17:03:04 MST 2014",
  bibsource =    "http://comjnl.oxfordjournals.org/content/57/2.toc;
                 https://www.math.utah.edu/pub/tex/bib/compj2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://comjnl.oxfordjournals.org/content/57/2/319.full.pdf+html",
  acknowledgement = ack-nhfb,
  fjournal =     "The Computer Journal",
  journal-URL =  "http://comjnl.oxfordjournals.org/",
  onlinedate =   "January 17, 2013",
}


bibtype: misc
filename: fparith.bib
label: Regan:2014:GAD
author: Rick Regan
title: GCC Avoids Double Rounding Errors With Round-To-Odd
pages: 319--331
day: 15
month: jan
monthnumber: 1
year: 2014
bibdate: Fri Dec 08 14:41:07 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/gnu.bib
bibtimestamp: 2023.12.08 14:41:07 ???
URL: https://www.exploringbinary.com/gcc-avoids-double-rounding-errors-with-round-to-odd/
keywords: correct rounding; floating-point arithmetic; round to odd
acknowledgement: ack-nhfb
howpublished: Web site
rawauthor: Rick Regan
rawtitle: {GCC} Avoids Double Rounding Errors With Round-To-Odd
entry:
@Misc{Regan:2014:GAD,
  author =       "Rick Regan",
  title =        "{GCC} Avoids Double Rounding Errors With
                 Round-To-Odd",
  howpublished = "Web site",
  day =          "15",
  month =        jan,
  year =         "2014",
  bibdate =      "Fri Dec 08 14:41:07 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/gnu.bib",
  URL =          "https://www.exploringbinary.com/gcc-avoids-double-rounding-errors-with-round-to-odd/",
  acknowledgement = ack-nhfb,
  keywords =     "correct rounding; floating-point arithmetic; round to
                 odd",
}


bibtype: article
filename: fparith.bib
label: Revol:2014:NRP
author: Nathalie Revol and Philippe Theveny
title: Numerical Reproducibility and Parallel Computations: Issues for Interval Algorithms
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 8
pages: 1915--1924
month: aug
monthnumber: 8
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2322593
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Aug 25 08:24:32 2014
bibtimestamp: 2014.08.25 08:24:32 ???
fjournal: IEEE Transactions on Computers
keywords: floating-point arithmetic; reproducible arithmetic
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Nathalie Revol and Philippe Theveny
rawtitle: Numerical Reproducibility and Parallel Computations: Issues for Interval Algorithms
entry:
@Article{Revol:2014:NRP,
  author =       "Nathalie Revol and Philippe Theveny",
  title =        "Numerical Reproducibility and Parallel Computations:
                 Issues for Interval Algorithms",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "8",
  pages =        "1915--1924",
  month =        aug,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2322593",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Aug 25 08:24:32 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "floating-point arithmetic; reproducible arithmetic",
}


bibtype: article
filename: fparith.bib
label: Riemens:2014:TSA
author: Danny P. Riemens and Georgi N. Gaydadjiev and Chris I. de Zeeuw and Christos Strydis
title: Towards scalable arithmetic units with graceful degradation
journal: j-TECS
volume: 13
number: 4
pages: 87:1--87:??
month: feb
monthnumber: 2
year: 2014
DOI: https://doi.org/10.1145/2499367
ISSN: 1539-9087 (print), 1558-3465 (electronic) OR 15399087 (print), 15583465 (electronic)
ISSNL: 1539-9087 OR 15399087
bibdate: Tue Mar 11 18:33:06 MDT 2014
bibtimestamp: 2014.03.11 18:33:06 MDT
fjournal: ACM Transactions on Embedded Computing Systems
acknowledgement: ack-nhfb
articleno: 87
journalURL: http://portal.acm.org/browse_dl.cfm?&idx=J840
rawauthor: Danny P. Riemens and Georgi N. Gaydadjiev and Chris I. de Zeeuw and Christos Strydis
rawtitle: Towards scalable arithmetic units with graceful degradation
bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/tecs.bib

abstract:
This article presents a new family of scalable arithmetic units (ScAUs) targeting resource-constrained, embedded devices. We, first, study the performance, power, area and scalability properties of general adders. Next, suitable error-detection schemes for low-power embedded systems are discussed. As a result, our ScAUs are enhanced with a suitable error-detection scheme, resulting in a Parity-Checked ScAU (PCScAU) design. The PCScAU strikes a flexible trade-off between space and time redundancy, offering dependability similar to high-end techniques for the area and power cost of low-end approaches. An alternative design, the Precision-Scalable Arithmetic Unit (PScAU) maintains throughput with degraded precision in case of hardware failures. The PScAU is targeting dependable applications where latency rather than numerical accuracy is more important. The PScAU's downscaled mode is also interesting for runtime thermal management due to its advantageous power consumption. We implemented and synthesized the PCScAU, PScAU and a few important reference designs (double-, triple- and quadruple-modular-redundancy adders with/without input gating) in 90- nm UMC technology. Overall, the PC-ScAU ranks first in 9 out of 10 power-delay-area (PDA)-product variants. It exhibits 16% area savings and 12% performance speedup for 7% increase in total power consumption, compared to the cheapest form of conventional hardware replication with the same fault coverage. The PDA product of the PCScAU is, thus, reduced by 21%. It is interesting that, while total power slightly increases, the PCScAU static power in fact decreases by 14%. Therefore, for newer technology nodes where the static power component is significant, the PCScAU can also achieve-next to performance and area --- significant power improvements.

rawabstract:
This article presents a new family of scalable arithmetic units (ScAUs) targeting resource-constrained, embedded devices. We, first, study the performance, power, area and scalability properties of general adders. Next, suitable error-detection schemes for low-power embedded systems are discussed. As a result, our ScAUs are enhanced with a suitable error-detection scheme, resulting in a Parity-Checked ScAU (PCScAU) design. The PCScAU strikes a flexible trade-off between space and time redundancy, offering dependability similar to high-end techniques for the area and power cost of low-end approaches. An alternative design, the Precision-Scalable Arithmetic Unit (PScAU) maintains throughput with degraded precision in case of hardware failures. The PScAU is targeting dependable applications where latency rather than numerical accuracy is more important. The PScAU's downscaled mode is also interesting for runtime thermal management due to its advantageous power consumption. We implemented and synthesized the PCScAU, PScAU and a few important reference designs (double-, triple- and quadruple-modular-redundancy adders with/without input gating) in 90- nm UMC technology. Overall, the PC-ScAU ranks first in 9 out of 10 power-delay-area (PDA)-product variants. It exhibits 16\% area savings and 12\% performance speedup for 7\% increase in total power consumption, compared to the cheapest form of conventional hardware replication with the same fault coverage. The PDA product of the PCScAU is, thus, reduced by 21\%. It is interesting that, while total power slightly increases, the PCScAU static power in fact decreases by 14\%. Therefore, for newer technology nodes where the static power component is significant, the PCScAU can also achieve-next to performance and area --- significant power improvements.

entry:
@Article{Riemens:2014:TSA,
  author =       "Danny P. Riemens and Georgi N. Gaydadjiev and Chris I.
                 de Zeeuw and Christos Strydis",
  title =        "Towards scalable arithmetic units with graceful
                 degradation",
  journal =      j-TECS,
  volume =       "13",
  number =       "4",
  pages =        "87:1--87:??",
  month =        feb,
  year =         "2014",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2499367",
  ISSN =         "1539-9087 (print), 1558-3465 (electronic)",
  ISSN-L =       "1539-9087",
  bibdate =      "Tue Mar 11 18:33:06 MDT 2014",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/tecs.bib",
  abstract =     "This article presents a new family of scalable
                 arithmetic units (ScAUs) targeting
                 resource-constrained, embedded devices. We, first,
                 study the performance, power, area and scalability
                 properties of general adders. Next, suitable
                 error-detection schemes for low-power embedded systems
                 are discussed. As a result, our ScAUs are enhanced with
                 a suitable error-detection scheme, resulting in a
                 Parity-Checked ScAU (PCScAU) design. The PCScAU strikes
                 a flexible trade-off between space and time redundancy,
                 offering dependability similar to high-end techniques
                 for the area and power cost of low-end approaches. An
                 alternative design, the Precision-Scalable Arithmetic
                 Unit (PScAU) maintains throughput with degraded
                 precision in case of hardware failures. The PScAU is
                 targeting dependable applications where latency rather
                 than numerical accuracy is more important. The PScAU's
                 downscaled mode is also interesting for runtime thermal
                 management due to its advantageous power consumption.
                 We implemented and synthesized the PCScAU, PScAU and a
                 few important reference designs (double-, triple- and
                 quadruple-modular-redundancy adders with/without input
                 gating) in 90- nm UMC technology. Overall, the PC-ScAU
                 ranks first in 9 out of 10 power-delay-area
                 (PDA)-product variants. It exhibits 16\% area savings
                 and 12\% performance speedup for 7\% increase in total
                 power consumption, compared to the cheapest form of
                 conventional hardware replication with the same fault
                 coverage. The PDA product of the PCScAU is, thus,
                 reduced by 21\%. It is interesting that, while total
                 power slightly increases, the PCScAU static power in
                 fact decreases by 14\%. Therefore, for newer technology
                 nodes where the static power component is significant,
                 the PCScAU can also achieve-next to performance and
                 area --- significant power improvements.",
  acknowledgement = ack-nhfb,
  articleno =    "87",
  fjournal =     "ACM Transactions on Embedded Computing Systems",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?&idx=J840",
}


bibtype: article
filename: fparith.bib
label: Rohn:2014:VLD
author: Jiri Rohn
title: Verification of Linear (In)Dependence in Finite Precision Arithmetic
journal: j-MATH-COMPUT-SCI
volume: 8
number: 3--4
pages: 323--328
month: sep
monthnumber: 9
year: 2014
DOI: https://doi.org/10.1007/s11786-014-0196-7
ISSN: 1661-8270 (print), 1661-8289 (electronic) OR 16618270 (print), 16618289 (electronic)
ISSNL: 1661-8270 OR 16618270
bibdate: Fri Feb 13 07:10:04 MST 2015
bibtimestamp: 2015.02.13 07:10:04 MST
URL: http://link.springer.com/article/10.1007/s11786-014-0196-7
fjournal: Mathematics in Computer Science
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11786
rawauthor: Jiri Rohn
rawtitle: Verification of Linear (In){Dependence} in Finite Precision Arithmetic
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1661-8270&volume=8&issue=3; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-sci.bib

entry:
@Article{Rohn:2014:VLD,
  author =       "Jiri Rohn",
  title =        "Verification of Linear (In){Dependence} in Finite
                 Precision Arithmetic",
  journal =      j-MATH-COMPUT-SCI,
  volume =       "8",
  number =       "3--4",
  pages =        "323--328",
  month =        sep,
  year =         "2014",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s11786-014-0196-7",
  ISSN =         "1661-8270 (print), 1661-8289 (electronic)",
  ISSN-L =       "1661-8270",
  bibdate =      "Fri Feb 13 07:10:04 MST 2015",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1661-8270&volume=8&issue=3;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/math-comput-sci.bib",
  URL =          "http://link.springer.com/article/10.1007/s11786-014-0196-7",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics in Computer Science",
  journal-URL =  "http://link.springer.com/journal/11786",
}


bibtype: article
filename: fparith.bib
label: Roux:2014:IDR
author: Pierre Roux
title: Innocuous Double Rounding of Basic Arithmetic Operations
journal: Journal of Formalized Reasoning
volume: 7
number: 1
pages: 323--328
year: 2014
DOI: https://doi.org/10.6092/issn.1972-5787/4359
ISSN: 1972-5787 OR 19725787
ISSNL: 1972-5787 OR 19725787
bibdate: Mon Jan 11 08:08:43 2021
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2021.01.11 08:08:43 ???
URL: https://hal.archives-ouvertes.fr/hal-01091186
keywords: double rounding
acknowledgement: ack-nhfb
rawauthor: Pierre Roux
rawtitle: Innocuous Double Rounding of Basic Arithmetic Operations
abstract:
Double rounding occurs when a floating-point value is first rounded to an intermediate precision before being rounded to a final precision. The result of two such consecutive roundings can differ from the result obtained when directly rounding to the final precision. Double rounding practically happens, for instance, when implementing the IEEE 754 binary32 format with an arithmetic unit performing operations only in the larger binary64 format, such as done in the PowerPC or x87 floating-point units. It belongs to the folklore in the floating-point arithmetic community that double rounding is innocuous for the basic arithmetic operations (addition, division, multiplication, and square root) as soon as the final precision is about twice larger than the intermediate one. This paper addresses the formal proof of this fact considering underflow cases and its extension to radices other than two.

rawabstract:
Double rounding occurs when a floating-point value is first rounded to an intermediate precision before being rounded to a final precision. The result of two such consecutive roundings can differ from the result obtained when directly rounding to the final precision. Double rounding practically happens, for instance, when implementing the IEEE 754 binary32 format with an arithmetic unit performing operations only in the larger binary64 format, such as done in the PowerPC or x87 floating-point units. It belongs to the folklore in the floating-point arithmetic community that double rounding is innocuous for the basic arithmetic operations (addition, division, multiplication, and square root) as soon as the final precision is about twice larger than the intermediate one. This paper addresses the formal proof of this fact considering underflow cases and its extension to radices other than two.

entry:
@Article{Roux:2014:IDR,
  author =       "Pierre Roux",
  title =        "Innocuous Double Rounding of Basic Arithmetic
                 Operations",
  journal =      "Journal of Formalized Reasoning",
  volume =       "7",
  number =       "1",
  year =         "2014",
  DOI =          "https://doi.org/10.6092/issn.1972-5787/4359",
  ISSN =         "1972-5787",
  ISSN-L =       "1972-5787",
  bibdate =      "Mon Jan 11 08:08:43 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://hal.archives-ouvertes.fr/hal-01091186",
  abstract =     "Double rounding occurs when a floating-point value is
                 first rounded to an intermediate precision before being
                 rounded to a final precision. The result of two such
                 consecutive roundings can differ from the result
                 obtained when directly rounding to the final precision.
                 Double rounding practically happens, for instance, when
                 implementing the IEEE 754 binary32 format with an
                 arithmetic unit performing operations only in the
                 larger binary64 format, such as done in the PowerPC or
                 x87 floating-point units. It belongs to the folklore in
                 the floating-point arithmetic community that double
                 rounding is innocuous for the basic arithmetic
                 operations (addition, division, multiplication, and
                 square root) as soon as the final precision is about
                 twice larger than the intermediate one. This paper
                 addresses the formal proof of this fact considering
                 underflow cases and its extension to radices other than
                 two.",
  acknowledgement = ack-nhfb,
  keywords =     "double rounding",
}


bibtype: article
filename: fparith.bib
label: Schkufza:2014:SOF
author: Eric Schkufza and Rahul Sharma and Alex Aiken
title: Stochastic optimization of floating-point programs with tunable precision
journal: j-SIGPLAN
volume: 49
number: 6
pages: 53--64
month: jun
monthnumber: 6
year: 2014
CODEN: SINODQ
DOI: https://doi.org/10.1145/2666356.2594302
ISSN: 0362-1340 (print), 1523-2867 (print), 1558-1160 OR 03621340 (print), 15232867 (print), 15581160
ISSNL: 0362-1340 OR 03621340
bibdate: Fri Sep 26 07:38:28 MDT 2014
bibtimestamp: 2014.09.26 07:38:28 MDT
fjournal: ACM SIGPLAN Notices
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J706
rawauthor: Eric Schkufza and Rahul Sharma and Alex Aiken
rawtitle: Stochastic optimization of floating-point programs with tunable precision
abstract:
The aggressive optimization of floating-point computations is an important problem in high-performance computing. Unfortunately, floating-point instruction sets have complicated semantics that often force compilers to preserve programs as written. We present a method that treats floating-point optimization as a stochastic search problem. We demonstrate the ability to generate reduced precision implementations of Intel's handwritten C numeric library which are up to 6 times faster than the original code, and achieve end-to-end speedups of over 30% on a direct numeric simulation and a ray tracer by optimizing kernels that can tolerate a loss of precision while still remaining correct. Because these optimizations are mostly not amenable to formal verification using the current state of the art, we present a stochastic search technique for characterizing maximum error. The technique comes with an asymptotic guarantee and provides strong evidence of correctness.

rawabstract:
The aggressive optimization of floating-point computations is an important problem in high-performance computing. Unfortunately, floating-point instruction sets have complicated semantics that often force compilers to preserve programs as written. We present a method that treats floating-point optimization as a stochastic search problem. We demonstrate the ability to generate reduced precision implementations of Intel's handwritten C numeric library which are up to 6 times faster than the original code, and achieve end-to-end speedups of over 30\% on a direct numeric simulation and a ray tracer by optimizing kernels that can tolerate a loss of precision while still remaining correct. Because these optimizations are mostly not amenable to formal verification using the current state of the art, we present a stochastic search technique for characterizing maximum error. The technique comes with an asymptotic guarantee and provides strong evidence of correctness.

entry:
@Article{Schkufza:2014:SOF,
  author =       "Eric Schkufza and Rahul Sharma and Alex Aiken",
  title =        "Stochastic optimization of floating-point programs
                 with tunable precision",
  journal =      j-SIGPLAN,
  volume =       "49",
  number =       "6",
  pages =        "53--64",
  month =        jun,
  year =         "2014",
  CODEN =        "SINODQ",
  DOI =          "https://doi.org/10.1145/2666356.2594302",
  ISSN =         "0362-1340 (print), 1523-2867 (print), 1558-1160
                 (electronic)",
  ISSN-L =       "0362-1340",
  bibdate =      "Fri Sep 26 07:38:28 MDT 2014",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigplan2010.bib",
  abstract =     "The aggressive optimization of floating-point
                 computations is an important problem in
                 high-performance computing. Unfortunately,
                 floating-point instruction sets have complicated
                 semantics that often force compilers to preserve
                 programs as written. We present a method that treats
                 floating-point optimization as a stochastic search
                 problem. We demonstrate the ability to generate reduced
                 precision implementations of Intel's handwritten C
                 numeric library which are up to 6 times faster than the
                 original code, and achieve end-to-end speedups of over
                 30\% on a direct numeric simulation and a ray tracer by
                 optimizing kernels that can tolerate a loss of
                 precision while still remaining correct. Because these
                 optimizations are mostly not amenable to formal
                 verification using the current state of the art, we
                 present a stochastic search technique for
                 characterizing maximum error. The technique comes with
                 an asymptotic guarantee and provides strong evidence of
                 correctness.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGPLAN Notices",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J706",
  received =     "PLDI '14 conference proceedings.",
}


bibtype: article
filename: fparith.bib
label: Shukla:2014:LLH
author: R. Shukla and K. C. Ray
title: Low Latency Hybrid CORDIC Algorithm
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 12
pages: 3066--3078
month: dec
monthnumber: 12
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.173
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Dec 4 10:36:57 2014
bibtimestamp: 2014.12.04 10:36:57 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: R. Shukla and K. C. Ray
rawtitle: Low Latency Hybrid {CORDIC} Algorithm
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib

keywords:
adders; Approximation algorithms; communication systems; Computer architecture; coordinate rotational digital computer; CORDIC algorithm; Delays; digital arithmetic; Digital computers; digital computers; double step branching; fast adders; first order hardware architecture; hardware complexity; hybrid CORDIC algorithm; image processing; low latency; low latency hybrid CORDIC algorithm; Mathematical model; radix-4; redundant arithmetic; scale factor calculation; signal processing; Signal processing algorithms

entry:
@Article{Shukla:2014:LLH,
  author =       "R. Shukla and K. C. Ray",
  title =        "Low Latency Hybrid {CORDIC} Algorithm",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "12",
  pages =        "3066--3078",
  month =        dec,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.173",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Dec 4 10:36:57 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "adders; Approximation algorithms; communication
                 systems; Computer architecture; coordinate rotational
                 digital computer; CORDIC algorithm; Delays; digital
                 arithmetic; Digital computers; digital computers;
                 double step branching; fast adders; first order
                 hardware architecture; hardware complexity; hybrid
                 CORDIC algorithm; image processing; low latency; low
                 latency hybrid CORDIC algorithm; Mathematical model;
                 radix-4; redundant arithmetic; scale factor
                 calculation; signal processing; Signal processing
                 algorithms",
}


bibtype: article
filename: fparith.bib
label: Toronto:2014:PAF
author: Neil Toronto and Jay McCarthy
title: Practically Accurate Floating-Point Math
journal: j-COMPUT-SCI-ENG
volume: 16
number: 4
pages: 80--95
month: jul
monthnumber: 7
year: 2014
CODEN: CSENFA
DOI: https://doi.org/10.1109/MCSE.2014.90
ISSN: 1521-9615 (print), 1558-366x (electronic) OR 15219615 (print), 1558366x (electronic)
ISSNL: 1521-9615 OR 15219615
bibdate: Thu Sep 4 08:43:09 2014
bibtimestamp: 2014.09.04 08:43:09 ???
fjournal: Computing in Science and Engineering
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5992
rawauthor: Neil Toronto and Jay McCarthy
rawtitle: Practically Accurate Floating-Point Math
bibsource:
https://www.math.utah.edu/pub/tex/bib/computscieng.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Toronto:2014:PAF,
  author =       "Neil Toronto and Jay McCarthy",
  title =        "Practically Accurate Floating-Point Math",
  journal =      j-COMPUT-SCI-ENG,
  volume =       "16",
  number =       "4",
  pages =        "80--95",
  month =        jul # "\slash " # aug,
  year =         "2014",
  CODEN =        "CSENFA",
  DOI =          "https://doi.org/10.1109/MCSE.2014.90",
  ISSN =         "1521-9615 (print), 1558-366x (electronic)",
  ISSN-L =       "1521-9615",
  bibdate =      "Thu Sep 4 08:43:09 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/computscieng.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Computing in Science and Engineering",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5992",
}


bibtype: article
filename: fparith.bib
label: Vazquez:2014:FRM
author: Alvaro Vazquez and Elisardo Antelo and Javier D. Bruguera
title: Fast Radix-$ 10 $ Multiplication Using Redundant BCD Codes
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 8
pages: 1902--1914
month: aug
monthnumber: 8
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2315626
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Aug 25 08:24:32 2014
bibtimestamp: 2014.08.25 08:24:32 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Alvaro Vazquez and Elisardo Antelo and Javier D. Bruguera
rawtitle: Fast Radix-$ 10 $ Multiplication Using Redundant {BCD} Codes
entry:
@Article{Vazquez:2014:FRM,
  author =       "Alvaro Vazquez and Elisardo Antelo and Javier D.
                 Bruguera",
  title =        "Fast Radix-$ 10 $ Multiplication Using Redundant {BCD}
                 Codes",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "8",
  pages =        "1902--1914",
  month =        aug,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2315626",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Aug 25 08:24:32 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Wang:2014:CFA
author: Dong Wang and Milos D. Ercegovac and Yang Xiao
title: Complex Function Approximation Using Two-Dimensional Interpolation
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 12
pages: 2948--2960
month: dec
monthnumber: 12
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.181
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Dec 4 10:36:57 2014
bibtimestamp: 2014.12.04 10:36:57 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Dong Wang and Milo{\v{s}} D. Ercegovac and Yang Xiao
rawtitle: Complex Function Approximation Using Two-Dimensional Interpolation
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib

keywords:
2D convolution algorithm; 2D interpolation; Approximation error; ASIC; bipartite schemes; bivariate functions; coefficient table; complex exponential; complex function approximation; complex function evaluation; complex reciprocal; Complex reciprocal; Computational complexity; cubic interpolation; exponential functions; field programmable gate arrays; FPGA; Function approximation; generic hardware architecture; interpolation; interpolation degree; interpolation kernels; Lagrange interpolation; Lagrangian functions; linear interpolation; lookup tables; memory requirements; multipartite schemes; quadratic interpolation; Quadratic programming; table lookup; tabulated function; two-dimensional interpolation

entry:
@Article{Wang:2014:CFA,
  author =       "Dong Wang and Milo{\v{s}} D. Ercegovac and Yang Xiao",
  title =        "Complex Function Approximation Using Two-Dimensional
                 Interpolation",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "12",
  pages =        "2948--2960",
  month =        dec,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.181",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Dec 4 10:36:57 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "2D convolution algorithm; 2D interpolation;
                 Approximation error; ASIC; bipartite schemes; bivariate
                 functions; coefficient table; complex exponential;
                 complex function approximation; complex function
                 evaluation; complex reciprocal; Complex reciprocal;
                 Computational complexity; cubic interpolation;
                 exponential functions; field programmable gate arrays;
                 FPGA; Function approximation; generic hardware
                 architecture; interpolation; interpolation degree;
                 interpolation kernels; Lagrange interpolation;
                 Lagrangian functions; linear interpolation; lookup
                 tables; memory requirements; multipartite schemes;
                 quadratic interpolation; Quadratic programming; table
                 lookup; tabulated function; two-dimensional
                 interpolation",
}


bibtype: article
filename: fparith.bib
label: Wang:2014:RBR
author: PengFei Wang and JianPing Li
journal: arXiv.org
pages: 1--8
day: 18
month: oct
monthnumber: 10
year: 2014
bibdate: Wed Sep 30 12:43:49 MDT 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.09.30 12:43:49 MDT
URL: http://arxiv.org/abs/1410.4919
subject: Chaotic Dynamics (nlin.CD); Numerical Analysis (cs.NA)
acknowledgement: ack-nhfb
rawauthor: PengFei Wang and JianPing Li
title:
On the relation between reliable computation time, float-point precision and the Lyapunov exponent in chaotic systems

abstract:
The relation among reliable computation time, Tc, float-point precision, K, and the Lyapunov exponent, lambda , is obtained as Tc= (lnB/lambda )K+C, where B is the base of the float-point system and C is a constant dependent only on the chaotic equation. The equation shows good agreement with numerical experimental results, especially the scale factors.

rawabstract:
The relation among reliable computation time, Tc, float-point precision, K, and the Lyapunov exponent, {\lambda}, is obtained as Tc= (lnB/{\lambda})K+C, where B is the base of the float-point system and C is a constant dependent only on the chaotic equation. The equation shows good agreement with numerical experimental results, especially the scale factors.

rawtitle:
On the relation between reliable computation time, float-point precision and the {Lyapunov} exponent in chaotic systems

entry:
@Article{Wang:2014:RBR,
  author =       "PengFei Wang and JianPing Li",
  title =        "On the relation between reliable computation time,
                 float-point precision and the {Lyapunov} exponent in
                 chaotic systems",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--8",
  day =          "18",
  month =        oct,
  year =         "2014",
  CODEN =        "????",
  ISSN =         "????",
  ISSN-L =       "????",
  bibdate =      "Wed Sep 30 12:43:49 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://arxiv.org/abs/1410.4919",
  abstract =     "The relation among reliable computation time, Tc,
                 float-point precision, K, and the Lyapunov exponent,
                 {\lambda}, is obtained as Tc= (lnB/{\lambda})K+C, where
                 B is the base of the float-point system and C is a
                 constant dependent only on the chaotic equation. The
                 equation shows good agreement with numerical
                 experimental results, especially the scale factors.",
  acknowledgement = ack-nhfb,
  subject =      "Chaotic Dynamics (nlin.CD); Numerical Analysis
                 (cs.NA)",
}


bibtype: article
filename: fparith.bib
label: Yao:2014:NRP
author: Gavin Xiaoxu Yao and Junfeng Fan and Ray C. C. Cheung and Ingrid Verbauwhede
title: Novel RNS Parameter Selection for Fast Modular Multiplication
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 8
pages: 2099--2105
month: aug
monthnumber: 8
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.92
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Aug 25 08:24:32 2014
bibtimestamp: 2014.08.25 08:24:32 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Gavin Xiaoxu Yao and Junfeng Fan and Ray C. C. Cheung and Ingrid Verbauwhede
rawtitle: Novel {RNS} Parameter Selection for Fast Modular Multiplication
entry:
@Article{Yao:2014:NRP,
  author =       "Gavin Xiaoxu Yao and Junfeng Fan and Ray C. C. Cheung
                 and Ingrid Verbauwhede",
  title =        "Novel {RNS} Parameter Selection for Fast Modular
                 Multiplication",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "8",
  pages =        "2099--2105",
  month =        aug,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.92",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Aug 25 08:24:32 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: inproceedings
filename: fparith.bib
label: Zafar:2014:HAD
author: Saad Zafar and Raviteja Adapa
booktitle: 2014 International Conference on Advances in Electrical Engineering (ICAEE)
title: Hardware architecture design and mapping of ``Fast Inverse Square Root'' algorithm
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 1--4
month: jan
monthnumber: 1
year: 2014
DOI: https://doi.org/10.1109/icaee.2014.6838433
bibdate: Wed Dec 20 07:29:37 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.12.20 07:29:37 ???
acknowledgement: ack-nhfb
rawauthor: Saad Zafar and Raviteja Adapa
rawbooktitle: 2014 International Conference on Advances in Electrical Engineering {(ICAEE)}
rawtitle: Hardware architecture design and mapping of ``{Fast Inverse Square Root}'' algorithm
abstract:
The Fast Inverse Square Root algorithm has been used in 3D games of past for lighting and reflection calculations, because it offers up to four times performance gains. This paper presents a hardware implementation of the algorithm on an FPGA board by designing the complete architecture and successfully mapping it on Xilinx Spartan 3E after thorough functional verification. The results show that this implementation provides a very efficient single-precision floating point inverse square root calculator with practically accurate results being made available after just 12 short clock cycles. This performance measure is far superior to the software counterpart of the algorithm, and is not processor dependent like rsqrtss of x86 SSE instruction set. Results of this work can aid FPGA based vector processors or graphic processing units with 3D rendering. The hardware design can also form part of a larger floating point arithmetic unit for dedicated reciprocal square root calculations.

rawabstract:
The Fast Inverse Square Root algorithm has been used in 3D games of past for lighting and reflection calculations, because it offers up to four times performance gains. This paper presents a hardware implementation of the algorithm on an FPGA board by designing the complete architecture and successfully mapping it on Xilinx Spartan 3E after thorough functional verification. The results show that this implementation provides a very efficient single-precision floating point inverse square root calculator with practically accurate results being made available after just 12 short clock cycles. This performance measure is far superior to the software counterpart of the algorithm, and is not processor dependent like rsqrtss of x86 SSE instruction set. Results of this work can aid FPGA based vector processors or graphic processing units with 3D rendering. The hardware design can also form part of a larger floating point arithmetic unit for dedicated reciprocal square root calculations.

entry:
@InProceedings{Zafar:2014:HAD,
  author =       "Saad Zafar and Raviteja Adapa",
  booktitle =    "2014 International Conference on Advances in
                 Electrical Engineering {(ICAEE)}",
  title =        "Hardware architecture design and mapping of ``{Fast
                 Inverse Square Root}'' algorithm",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "1--4",
  month =        jan,
  year =         "2014",
  DOI =          "https://doi.org/10.1109/icaee.2014.6838433",
  bibdate =      "Wed Dec 20 07:29:37 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The Fast Inverse Square Root algorithm has been used
                 in 3D games of past for lighting and reflection
                 calculations, because it offers up to four times
                 performance gains. This paper presents a hardware
                 implementation of the algorithm on an FPGA board by
                 designing the complete architecture and successfully
                 mapping it on Xilinx Spartan 3E after thorough
                 functional verification. The results show that this
                 implementation provides a very efficient
                 single-precision floating point inverse square root
                 calculator with practically accurate results being made
                 available after just 12 short clock cycles. This
                 performance measure is far superior to the software
                 counterpart of the algorithm, and is not processor
                 dependent like rsqrtss of x86 SSE instruction set.
                 Results of this work can aid FPGA based vector
                 processors or graphic processing units with 3D
                 rendering. The hardware design can also form part of a
                 larger floating point arithmetic unit for dedicated
                 reciprocal square root calculations.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Ahmadifar:2015:NRN
author: H. Ahmadifar and G. Jaberipur
title: A New Residue Number System with $5$-Moduli Set: $ 2^2 q$, $ 2^q pm 3$, $ 2^q pm 1$
journal: j-COMP-J
volume: 58
number: 7
pages: 1548--1565
month: jul
monthnumber: 7
year: 2015
CODEN: CMPJA6
DOI: https://doi.org/10.1093/comjnl/bxu084
ISSN: 0010-4620 (print), 1460-2067 (electronic) OR 00104620 (print), 14602067 (electronic)
ISSNL: 0010-4620 OR 00104620
bibdate: Tue Jul 28 10:40:55 MDT 2015
bibtimestamp: 2015.07.28 10:40:55 MDT
URL: http://comjnl.oxfordjournals.org/content/58/7/1548
fjournal: Computer Journal
acknowledgement: ack-nhfb
journalURL: http://comjnl.oxfordjournals.org/
onlinedate: September 2, 2014
rawauthor: H. Ahmadifar and G. Jaberipur
rawtitle: A New Residue Number System with $5$-Moduli Set: $ 2^{2 q}$, $ 2^q \pm 3$, $ 2^q \pm 1$
bibsource:
http://comjnl.oxfordjournals.org/content/58/7.toc; https://www.math.utah.edu/pub/tex/bib/compj2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Ahmadifar:2015:NRN,
  author =       "H. Ahmadifar and G. Jaberipur",
  title =        "A New Residue Number System with $5$-Moduli Set: $
                 2^{2 q}$, $ 2^q \pm 3$, $ 2^q \pm 1$",
  journal =      j-COMP-J,
  volume =       "58",
  number =       "7",
  pages =        "1548--1565",
  month =        jul,
  year =         "2015",
  CODEN =        "CMPJA6",
  DOI =          "https://doi.org/10.1093/comjnl/bxu084",
  ISSN =         "0010-4620 (print), 1460-2067 (electronic)",
  ISSN-L =       "0010-4620",
  bibdate =      "Tue Jul 28 10:40:55 MDT 2015",
  bibsource =    "http://comjnl.oxfordjournals.org/content/58/7.toc;
                 https://www.math.utah.edu/pub/tex/bib/compj2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://comjnl.oxfordjournals.org/content/58/7/1548",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Journal",
  journal-URL =  "http://comjnl.oxfordjournals.org/",
  onlinedate =   "September 2, 2014",
}


bibtype: techreport
filename: fparith.bib
label: Ahrens:2015:ERF
author: P. Ahrens and H. D. Nguyen and J. Demmel
title: Efficient Reproducible Floating Point Summation and BLAS
type: Report
number: UCB/EECS-2015-229
institution: EECS Department, University of California, Berkeley
address: Berkeley, CA, USA
pages: 1548--1565
day: 8
month: dec
monthnumber: 12
year: 2015
bibdate: Mon Jan 04 15:39:44 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.01.04 15:39:44 ???
URL: http://www.eecs.berkeley.edu/Pubs/TechRpts/2015/EECS-2015-229.html
keywords: accurate dot product; accurate summation
acknowledgement: ack-nhfb
rawauthor: P. Ahrens and H. D. Nguyen and J. Demmel
rawtitle: Efficient Reproducible Floating Point Summation and {BLAS}
abstract:
We define reproducibility to mean getting bitwise identical results from multiple runs of the same program, perhaps with different hardware resources or other changes that should ideally not change the answer. Many users depend on reproducibility for debugging or correctness. However, dynamic scheduling of parallel computing resources, combined with nonassociativity of floating point addition, makes attaining reproducibility a challenge even for simple operations like summing a vector of numbers, or more complicated operations like the Basic Linear Algebra Subprograms (BLAS). We describe an algorithm that computes a reproducible sum of floating point numbers, independent of the order of summation. The algorithm depends only on a subset of the IEEE Floating Point Standard 754-2008. It is communication-optimal, in the sense that it does just one pass over the data in the sequential case, or one reduction operation in the parallel case, requiring an ``accumulator'' represented by just 6 floating point words (more can be used if higher precision is desired). The arithmetic cost with a 6-word accumulator is $ 7 n $ floating point additions to sum $n$ words, and (in IEEE double precision) the final error bound can be up to $ 10^-8$ times smaller than the error bound for conventional summation. We describe the basic summation algorithm, the software infrastructure used to build reproducible BLAS (ReproBLAS), and performance results. For example, when computing the dot product of 4096 double precision floating point numbers, we get an $ 4 times $ slowdown compared to Intel Math Kernel Library (MKL) running on an Intel Core i7-2600 CPU operating at 3.4 GHz and 256 KB L2 Cache.

rawabstract:
We define reproducibility to mean getting bitwise identical results from multiple runs of the same program, perhaps with different hardware resources or other changes that should ideally not change the answer. Many users depend on reproducibility for debugging or correctness. However, dynamic scheduling of parallel computing resources, combined with nonassociativity of floating point addition, makes attaining reproducibility a challenge even for simple operations like summing a vector of numbers, or more complicated operations like the Basic Linear Algebra Subprograms (BLAS). We describe an algorithm that computes a reproducible sum of floating point numbers, independent of the order of summation. The algorithm depends only on a subset of the IEEE Floating Point Standard 754-2008. It is communication-optimal, in the sense that it does just one pass over the data in the sequential case, or one reduction operation in the parallel case, requiring an ``accumulator'' represented by just 6 floating point words (more can be used if higher precision is desired). The arithmetic cost with a 6-word accumulator is $ 7 n $ floating point additions to sum $n$ words, and (in IEEE double precision) the final error bound can be up to $ 10^{-8}$ times smaller than the error bound for conventional summation. We describe the basic summation algorithm, the software infrastructure used to build reproducible BLAS (ReproBLAS), and performance results. For example, when computing the dot product of 4096 double precision floating point numbers, we get an $ 4 \times $ slowdown compared to Intel Math Kernel Library (MKL) running on an Intel Core i7-2600 CPU operating at 3.4 GHz and 256 KB L2 Cache.

entry:
@TechReport{Ahrens:2015:ERF,
  author =       "P. Ahrens and H. D. Nguyen and J. Demmel",
  title =        "Efficient Reproducible Floating Point Summation and
                 {BLAS}",
  type =         "Report",
  number =       "UCB/EECS-2015-229",
  institution =  "EECS Department, University of California, Berkeley",
  address =      "Berkeley, CA, USA",
  day =          "8",
  month =        dec,
  year =         "2015",
  bibdate =      "Mon Jan 04 15:39:44 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.eecs.berkeley.edu/Pubs/TechRpts/2015/EECS-2015-229.html",
  abstract =     "We define reproducibility to mean getting bitwise
                 identical results from multiple runs of the same
                 program, perhaps with different hardware resources or
                 other changes that should ideally not change the
                 answer. Many users depend on reproducibility for
                 debugging or correctness. However, dynamic scheduling
                 of parallel computing resources, combined with
                 nonassociativity of floating point addition, makes
                 attaining reproducibility a challenge even for simple
                 operations like summing a vector of numbers, or more
                 complicated operations like the Basic Linear Algebra
                 Subprograms (BLAS). We describe an algorithm that
                 computes a reproducible sum of floating point numbers,
                 independent of the order of summation. The algorithm
                 depends only on a subset of the IEEE Floating Point
                 Standard 754-2008. It is communication-optimal, in the
                 sense that it does just one pass over the data in the
                 sequential case, or one reduction operation in the
                 parallel case, requiring an ``accumulator'' represented
                 by just 6 floating point words (more can be used if
                 higher precision is desired). The arithmetic cost with
                 a 6-word accumulator is $ 7 n $ floating point
                 additions to sum $n$ words, and (in IEEE double
                 precision) the final error bound can be up to $
                 10^{-8}$ times smaller than the error bound for
                 conventional summation. We describe the basic summation
                 algorithm, the software infrastructure used to build
                 reproducible BLAS (ReproBLAS), and performance results.
                 For example, when computing the dot product of 4096
                 double precision floating point numbers, we get an $ 4
                 \times $ slowdown compared to Intel Math Kernel Library
                 (MKL) running on an Intel Core i7-2600 CPU operating at
                 3.4 GHz and 256 KB L2 Cache.",
  acknowledgement = ack-nhfb,
  keywords =     "accurate dot product; accurate summation",
}


bibtype: techreport
filename: fparith.bib
label: Ahrens:2015:RPM
author: Peter Ahrens
title: Reproducible Parallel Matrix-Vector Multiply
type: CS 267 final report
institution: Department of Computer Science, University of California, Berkeley
address: Berkeley, CA, USA
pages: 1548--1565
day: 11
month: may
monthnumber: 5
year: 2015
bibdate: Mon Jan 04 15:55:37 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.01.04 15:55:37 ???
URL: http://bebop.cs.berkeley.edu/reproblas/docs/reports/PeterAhrensCS267FinalReport.pdf
acknowledgement: ack-nhfb
rawauthor: Peter Ahrens
rawtitle: Reproducible Parallel Matrix-Vector Multiply
entry:
@TechReport{Ahrens:2015:RPM,
  author =       "Peter Ahrens",
  title =        "Reproducible Parallel Matrix-Vector Multiply",
  type =         "{CS 267} final report",
  institution =  "Department of Computer Science, University of
                 California, Berkeley",
  address =      "Berkeley, CA, USA",
  day =          "11",
  month =        may,
  year =         "2015",
  bibdate =      "Mon Jan 04 15:55:37 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://bebop.cs.berkeley.edu/reproblas/docs/reports/PeterAhrensCS267FinalReport.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Aktan:2015:MEA
author: Mustafa Aktan and Dursun Baran and Vojin G. Oklobdzija
title: Minimizing Energy by Achieving Optimal Sparseness in Parallel Adders
crossref: Muller:2015:ISC
pages: 10--17
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.13
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Mustafa Aktan and Dursun Baran and Vojin G. Oklobdzija
rawtitle: Minimizing Energy by Achieving Optimal Sparseness in Parallel Adders
entry:
@InProceedings{Aktan:2015:MEA,
  author =       "Mustafa Aktan and Dursun Baran and Vojin G.
                 Oklobdzija",
  title =        "Minimizing Energy by Achieving Optimal Sparseness in
                 Parallel Adders",
  crossref =     "Muller:2015:ISC",
  pages =        "10--17",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.13",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: inproceedings
filename: fparith.bib
label: Andrysco:2015:SFP
title: On subnormal floating point and abnormal timing
crossref: IEEE:2015:ISS
pages: 623--639
year: 2015
DOI: https://doi.org/10.1109/SP.2015.44
bibdate: Mon Feb 10 08:54:26 2020
bibtimestamp: 2020.02.10 08:54:26 ???
URL: https://github.com/kmowery/libfixedtimefixedpoint
acknowledgement: ack-nhfb
rawtitle: On subnormal floating point and abnormal timing
author:
Marc Andrysco and David Kohlbrenner and Keaton Mowery and Ranjit Jhala and Sorin Lerner and Hovav Shacham

bibsource:
https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

abstract:
We identify a timing channel in the floating point instructions of modern x86 processors: the running time of floating point addition and multiplication instructions can vary by two orders of magnitude depending on their operands. We develop a benchmark measuring the timing variability of floating point operations and report on its results. We use floating point data timing variability to demonstrate practical attacks on the security of the Firefox browser (versions 23 through 27) and the Fuzz differentially private database. Finally, we initiate the study of mitigations to floating point data timing channels with libfixedtimefixedpoint, a new fixed-point, constant-time math library. Modern floating point standards and implementations are sophisticated, complex, and subtle, a fact that has not been sufficiently recognized by the security community. More work is needed to assess the implications of the use of floating point instructions in security-relevant software.

rawabstract:
We identify a timing channel in the floating point instructions of modern x86 processors: the running time of floating point addition and multiplication instructions can vary by two orders of magnitude depending on their operands. We develop a benchmark measuring the timing variability of floating point operations and report on its results. We use floating point data timing variability to demonstrate practical attacks on the security of the Firefox browser (versions 23 through 27) and the Fuzz differentially private database. Finally, we initiate the study of mitigations to floating point data timing channels with libfixedtimefixedpoint, a new fixed-point, constant-time math library. Modern floating point standards and implementations are sophisticated, complex, and subtle, a fact that has not been sufficiently recognized by the security community. More work is needed to assess the implications of the use of floating point instructions in security-relevant software.

rawauthor:
Marc Andrysco and David Kohlbrenner and Keaton Mowery and Ranjit Jhala and Sorin Lerner and Hovav Shacham

entry:
@InProceedings{Andrysco:2015:SFP,
  author =       "Marc Andrysco and David Kohlbrenner and Keaton Mowery
                 and Ranjit Jhala and Sorin Lerner and Hovav Shacham",
  title =        "On subnormal floating point and abnormal timing",
  crossref =     "IEEE:2015:ISS",
  pages =        "623--639",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/SP.2015.44",
  bibdate =      "Mon Feb 10 08:54:26 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We identify a timing channel in the floating point
                 instructions of modern x86 processors: the running time
                 of floating point addition and multiplication
                 instructions can vary by two orders of magnitude
                 depending on their operands. We develop a benchmark
                 measuring the timing variability of floating point
                 operations and report on its results. We use floating
                 point data timing variability to demonstrate practical
                 attacks on the security of the Firefox browser
                 (versions 23 through 27) and the Fuzz differentially
                 private database. Finally, we initiate the study of
                 mitigations to floating point data timing channels with
                 libfixedtimefixedpoint, a new fixed-point,
                 constant-time math library. Modern floating point
                 standards and implementations are sophisticated,
                 complex, and subtle, a fact that has not been
                 sufficiently recognized by the security community. More
                 work is needed to assess the implications of the use of
                 floating point instructions in security-relevant
                 software.",
  acknowledgement = ack-nhfb,
  URL =          "https://github.com/kmowery/libfixedtimefixedpoint",
}


bibtype: inproceedings
filename: fparith.bib
label: Aneesh:2015:HHM
author: R. Aneesh and Patil Vinayak and M. P. Sobham and A. David Selvakumar
title: HMFPCC: --- Hybrid-mode floating point conversion co-processor
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 623--639
month: jan
monthnumber: 1
year: 2015
DOI: https://doi.org/10.1109/vlsi-sata.2015.7050482
bibdate: Sat Jun 4 17:15:13 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.06.04 17:15:13 ???
acknowledgement: ack-nhfb
rawauthor: R. Aneesh and Patil Vinayak and M. P. Sobham and A. David Selvakumar
rawtitle: {HMFPCC}: --- Hybrid-mode floating point conversion co-processor
booktitle:
2015 International Conference on VLSI Systems, Architecture, Technology and Applications (VLSI-SATA)

rawbooktitle:
{2015 International Conference on VLSI Systems, Architecture, Technology and Applications (VLSI-SATA)}

entry:
@InProceedings{Aneesh:2015:HHM,
  author =       "R. Aneesh and Patil Vinayak and M. P. Sobham and A.
                 David Selvakumar",
  booktitle =    "{2015 International Conference on VLSI Systems,
                 Architecture, Technology and Applications
                 (VLSI-SATA)}",
  title =        "{HMFPCC}: --- Hybrid-mode floating point conversion
                 co-processor",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  month =        jan,
  year =         "2015",
  DOI =          "https://doi.org/10.1109/vlsi-sata.2015.7050482",
  bibdate =      "Sat Jun 4 17:15:13 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: misc
filename: fparith.bib
label: Anonymous:2015:EFP
author: Anonymous
title: The Evils of Floating Point, and the Joys of Unum
pages: 623--639
day: 24
month: mar
monthnumber: 3
year: 2015
bibdate: Sat Apr 02 16:43:03 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.04.02 16:43:03 ???
URL: http://vrworld.com/2015/03/24/the-evils-of-floating-point-and-the-joys-of-unum/
acknowledgement: ack-nhfb
howpublished: Web document
rawauthor: Anonymous
rawtitle: The Evils of Floating Point, and the Joys of Unum
entry:
@Misc{Anonymous:2015:EFP,
  author =       "Anonymous",
  title =        "The Evils of Floating Point, and the Joys of Unum",
  howpublished = "Web document",
  day =          "24",
  month =        mar,
  year =         "2015",
  bibdate =      "Sat Apr 02 16:43:03 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://vrworld.com/2015/03/24/the-evils-of-floating-point-and-the-joys-of-unum/",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Bailey:2015:HPA
author: David H. Bailey and Jonathan M. Borwein
title: High-precision arithmetic in mathematical physics
journal: Mathematics
volume: 3
number: 2
publisher: Multidisciplinary Digital Publishing Institute
pages: 337--367
year: 2015
bibdate: Sat Jun 4 17:21:23 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.06.04 17:21:23 ???
acknowledgement: ack-nhfb
rawauthor: David H. Bailey and Jonathan M. Borwein
rawtitle: High-precision arithmetic in mathematical physics
entry:
@Article{Bailey:2015:HPA,
  author =       "David H. Bailey and Jonathan M. Borwein",
  title =        "High-precision arithmetic in mathematical physics",
  journal =      "Mathematics",
  volume =       "3",
  number =       "2",
  publisher =    "Multidisciplinary Digital Publishing Institute",
  pages =        "337--367",
  year =         "2015",
  bibdate =      "Sat Jun 4 17:21:23 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: misc
filename: fparith.bib
label: Bailey:2015:NRH
author: David H. Bailey
title: Numerical reproducibility in high-performance computing
pages: 337--367
day: 19
month: nov
monthnumber: 11
year: 2015
bibdate: Tue Nov 05 15:38:26 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2019.11.05 15:38:26 ???
URL: https://www.nist.gov/sites/default/files/documents/itl/ssd/is/NRE-2015-02-dhb-num-repro.pdf
acknowledgement: ack-nhfb
howpublished: 24 lecture slides
rawauthor: David H. Bailey
rawtitle: Numerical reproducibility in high-performance computing
entry:
@Misc{Bailey:2015:NRH,
  author =       "David H. Bailey",
  title =        "Numerical reproducibility in high-performance
                 computing",
  howpublished = "24 lecture slides",
  day =          "19",
  month =        nov,
  year =         "2015",
  bibdate =      "Tue Nov 05 15:38:26 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.nist.gov/sites/default/files/documents/itl/ssd/is/NRE-2015-02-dhb-num-repro.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Bajard:2015:RAA
author: Jean-Claude Bajard and Julien Eynard and Nabil Merkiche and Thomas Plantard
crossref: Muller:2015:ISC
pages: 113--120
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.30
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Jean-Claude Bajard and Julien Eynard and Nabil Merkiche and Thomas Plantard
title:
RNS Arithmetic Approach in Lattice-Based Cryptography: Accelerating the ``Rounding-off'' Core Procedure

rawtitle:
{RNS} Arithmetic Approach in Lattice-Based Cryptography: Accelerating the {``Rounding-off''} Core Procedure

entry:
@InProceedings{Bajard:2015:RAA,
  author =       "Jean-Claude Bajard and Julien Eynard and Nabil
                 Merkiche and Thomas Plantard",
  title =        "{RNS} Arithmetic Approach in Lattice-Based
                 Cryptography: Accelerating the {``Rounding-off''} Core
                 Procedure",
  crossref =     "Muller:2015:ISC",
  pages =        "113--120",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.30",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: fparith.bib
label: Bankas:2015:NMA
author: Edem Kwedzo Bankas and Kazeem Alagbe Gbolagade
journal: j-COMP-J
volume: 58
number: 7
pages: 1566--1572
month: jul
monthnumber: 7
year: 2015
CODEN: CMPJA6
DOI: https://doi.org/10.1093/comjnl/bxu089
ISSN: 0010-4620 (print), 1460-2067 (electronic) OR 00104620 (print), 14602067 (electronic)
ISSNL: 0010-4620 OR 00104620
bibdate: Tue Jul 28 10:40:55 MDT 2015
bibtimestamp: 2015.07.28 10:40:55 MDT
URL: http://comjnl.oxfordjournals.org/content/58/7/1566
fjournal: Computer Journal
acknowledgement: ack-nhfb
journalURL: http://comjnl.oxfordjournals.org/
onlinedate: October 9, 2014
rawauthor: Edem Kwedzo Bankas and Kazeem Alagbe Gbolagade
title:
New MRC Adder-Based Reverse Converter for the Moduli Set $ 2^n $, $ 2^2 n + 1 - 1 $, $ 2^2 n + 2 - 1 $

bibsource:
http://comjnl.oxfordjournals.org/content/58/7.toc; https://www.math.utah.edu/pub/tex/bib/compj2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

rawtitle:
New {MRC} Adder-Based Reverse Converter for the Moduli Set $ 2^n $, $ 2^{2 n + 1} - 1 $, $ 2^{2 n + 2} - 1 $

entry:
@Article{Bankas:2015:NMA,
  author =       "Edem Kwedzo Bankas and Kazeem Alagbe Gbolagade",
  title =        "New {MRC} Adder-Based Reverse Converter for the Moduli
                 Set $ 2^n $, $ 2^{2 n + 1} - 1 $, $ 2^{2 n + 2} - 1 $",
  journal =      j-COMP-J,
  volume =       "58",
  number =       "7",
  pages =        "1566--1572",
  month =        jul,
  year =         "2015",
  CODEN =        "CMPJA6",
  DOI =          "https://doi.org/10.1093/comjnl/bxu089",
  ISSN =         "0010-4620 (print), 1460-2067 (electronic)",
  ISSN-L =       "0010-4620",
  bibdate =      "Tue Jul 28 10:40:55 MDT 2015",
  bibsource =    "http://comjnl.oxfordjournals.org/content/58/7.toc;
                 https://www.math.utah.edu/pub/tex/bib/compj2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://comjnl.oxfordjournals.org/content/58/7/1566",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Journal",
  journal-URL =  "http://comjnl.oxfordjournals.org/",
  onlinedate =   "October 9, 2014",
}


bibtype: techreport
filename: fparith.bib
label: Biancolin:2015:HAE
author: David Biancolin and Jack Koenig
title: Hardware Accelerator for Exact Dot Product
type: Report
institution: ASPIRE Laboratory, University of California, Berkeley
address: Berkeley, CA, USA
pages: 1566--1572
day: 19
month: jun
monthnumber: 6
year: 2015
bibdate: Fri Jun 19 14:30:46 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.06.19 14:30:46 ???
keywords: accurate floating-point dot product; accurate floating-point summation
acknowledgement: ack-nhfb
rawauthor: David Biancolin and Jack Koenig
rawtitle: Hardware Accelerator for Exact Dot Product
abstract:
In this paper we present a coprocessor capable of computing a dot product exactly by use of a ``complete register'' (CR) that encodes a fixed point representation of the complete IEEE754 double precision space. We explore the design space of the coprocessor by running simulations on large numbers of distinct configurations. Since only the accumulation register is represented exactly, we demonstrate that EDP is realizable in silicon, requiring additional 11% over Rocket's area. In addition, the accelerator showed speedups of $3$--$ 6 times $ over a conventional dot product and matrix multiplication while providing both exactness and reproducibility.

rawabstract:
In this paper we present a coprocessor capable of computing a dot product exactly by use of a ``complete register'' (CR) that encodes a fixed point representation of the complete IEEE754 double precision space. We explore the design space of the coprocessor by running simulations on large numbers of distinct configurations. Since only the accumulation register is represented exactly, we demonstrate that EDP is realizable in silicon, requiring additional 11\% over Rocket's area. In addition, the accelerator showed speedups of $3$--$ 6 \times $ over a conventional dot product and matrix multiplication while providing both exactness and reproducibility.

entry:
@TechReport{Biancolin:2015:HAE,
  author =       "David Biancolin and Jack Koenig",
  title =        "Hardware Accelerator for Exact Dot Product",
  type =         "Report",
  number =       "??",
  institution =  "ASPIRE Laboratory, University of California,
                 Berkeley",
  address =      "Berkeley, CA, USA",
  day =          "19",
  month =        jun,
  year =         "2015",
  bibdate =      "Fri Jun 19 14:30:46 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "In this paper we present a coprocessor capable of
                 computing a dot product exactly by use of a ``complete
                 register'' (CR) that encodes a fixed point
                 representation of the complete IEEE754 double precision
                 space. We explore the design space of the coprocessor
                 by running simulations on large numbers of distinct
                 configurations. Since only the accumulation register is
                 represented exactly, we demonstrate that EDP is
                 realizable in silicon, requiring additional 11\% over
                 Rocket's area. In addition, the accelerator showed
                 speedups of $3$--$ 6 \times $ over a conventional dot
                 product and matrix multiplication while providing both
                 exactness and reproducibility.",
  acknowledgement = ack-nhfb,
  keywords =     "accurate floating-point dot product; accurate
                 floating-point summation",
}


bibtype: inproceedings
filename: fparith.bib
label: Boldo:2015:FVP
author: Sylvie Boldo and Guillaume Melquiond
title: Formal Verification of Programs Computing the Floating-Point Average
crossref: Butler:2015:FMS
number: 9407
pages: 17--32
year: 2015
DOI: https://doi.org/10.1007/978-3-319-25423-4_2
bibdate: Fri Jan 31 15:41:04 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.01.31 15:41:04 ???
URL: https://hal.inria.fr/hal-01174892; https://link.springer.com/chapter/10.1007/978-3-319-25423-4_2
acknowledgement: ack-nhfb
rawauthor: Sylvie Boldo and Guillaume Melquiond
rawtitle: Formal Verification of Programs Computing the Floating-Point Average
entry:
@InProceedings{Boldo:2015:FVP,
  author =       "Sylvie Boldo and Guillaume Melquiond",
  title =        "Formal Verification of Programs Computing the
                 Floating-Point Average",
  crossref =     "Butler:2015:FMS",
  number =       "9407",
  pages =        "17--32",
  year =         "2015",
  DOI =          "https://doi.org/10.1007/978-3-319-25423-4_2",
  bibdate =      "Fri Jan 31 15:41:04 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://hal.inria.fr/hal-01174892;
                 https://link.springer.com/chapter/10.1007/978-3-319-25423-4_2",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Boldo:2015:SSD
author: Sylvie Boldo
editor: Sergiy Bogomolov and Matthieu Martel
title: Stupid is as Stupid Does: Taking the Square Root of the Square of a Floating-Point Number
journal: j-ELECT-NOTES-THEOR-COMP-SCI
volume: 317
pages: 50--55
day: 18
month: nov
monthnumber: 11
year: 2015
DOI: https://doi.org/10.1016/j.entcs.2015.10.004
ISSN: 1571-0661 OR 15710661
ISSNL: 1571-0661 OR 15710661
bibdate: Sat Jun 04 16:01:38 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.06.04 16:01:38 ???
fjournal: Electronic Notes in Theoretical Computer Science
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/15710661
rawauthor: Sylvie Boldo
raweditor: Sergiy Bogomolov and Matthieu Martel
rawtitle: Stupid is as Stupid Does: Taking the Square Root of the Square of a Floating-Point Number
booktitle:
Proceedings of the Seventh and Eighth International Workshop on Numerical Software Verification, volume 317 of Electronic Notes in Theoretical Computer Science, Seattle, 2015

remark:
The author proves in Coq that the square root of the square of a floating-point number $x$ is always $ |x|$, and thus, that $ x / sqrt x^2 + y^2$ lies in $ [ - 1, 1]$, PROVIDED that the base is $ beta = 2$, the precision is $ p > 1$, and rounding is either the IEEE 754 default of round-to-nearest-ties-to-even, or round-to-nearest-ties-away-from-zero. She gives simple examples for bases 10 and 1000 where the result lies outside $ [ - 1, 1]$.

rawbooktitle:
Proceedings of the Seventh and Eighth International Workshop on Numerical Software Verification, volume 317 of Electronic Notes in Theoretical Computer Science, Seattle, 2015

entry:
@Article{Boldo:2015:SSD,
  author =       "Sylvie Boldo",
  editor =       "Sergiy Bogomolov and Matthieu Martel",
  booktitle =    "Proceedings of the Seventh and Eighth International
                 Workshop on Numerical Software Verification, volume 317
                 of Electronic Notes in Theoretical Computer Science,
                 Seattle, 2015",
  title =        "Stupid is as Stupid Does: Taking the Square Root of
                 the Square of a Floating-Point Number",
  journal =      j-ELECT-NOTES-THEOR-COMP-SCI,
  volume =       "317",
  pages =        "50--55",
  day =          "18",
  month =        nov,
  year =         "2015",
  DOI =          "https://doi.org/10.1016/j.entcs.2015.10.004",
  ISSN =         "1571-0661",
  ISSN-L =       "1571-0661",
  bibdate =      "Sat Jun 04 16:01:38 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Electronic Notes in Theoretical Computer Science",
  journal-URL =  "http://www.sciencedirect.com/science/journal/15710661",
  remark =       "The author proves in Coq that the square root of the
                 square of a floating-point number $x$ is always $ |x|$,
                 and thus, that $ x / \sqrt {x^2 + y^2}$ lies in $ [ -
                 1, 1]$, PROVIDED that the base is $ \beta = 2$, the
                 precision is $ p > 1$, and rounding is either the IEEE
                 754 default of round-to-nearest-ties-to-even, or
                 round-to-nearest-ties-away-from-zero. She gives simple
                 examples for bases 10 and 1000 where the result lies
                 outside $ [ - 1, 1]$.",
}


bibtype: article
filename: fparith.bib
label: Boldo:2015:VCF
author: Sylvie Boldo and Jacques-Henri Jourdan and Xavier Leroy and Guillaume Melquiond
title: Verified Compilation of Floating-Point Computations
journal: j-J-AUTOM-REASON
volume: 54
number: 2
pages: 135--163
month: feb
monthnumber: 2
year: 2015
CODEN: JAREEW
DOI: https://doi.org/10.1007/s10817-014-9317-x
ISSN: 0168-7433 (print), 1573-0670 (electronic) OR 01687433 (print), 15730670 (electronic)
ISSNL: 0168-7433 OR 01687433
bibdate: Sat Apr 2 10:51:13 MDT 2016
bibtimestamp: 2016.04.02 10:51:13 MDT
URL: http://link.springer.com/article/10.1007/s10817-014-9317-x
fjournal: Journal of Automated Reasoning
acknowledgement: ack-nhfb
ajournal: J. Autom. Reason.
journalURL: http://link.springer.com/journal/10817
rawauthor: Sylvie Boldo and Jacques-Henri Jourdan and Xavier Leroy and Guillaume Melquiond
rawtitle: Verified Compilation of Floating-Point Computations
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jautomreason.bib

entry:
@Article{Boldo:2015:VCF,
  author =       "Sylvie Boldo and Jacques-Henri Jourdan and Xavier
                 Leroy and Guillaume Melquiond",
  title =        "Verified Compilation of Floating-Point Computations",
  journal =      j-J-AUTOM-REASON,
  volume =       "54",
  number =       "2",
  pages =        "135--163",
  month =        feb,
  year =         "2015",
  CODEN =        "JAREEW",
  DOI =          "https://doi.org/10.1007/s10817-014-9317-x",
  ISSN =         "0168-7433 (print), 1573-0670 (electronic)",
  ISSN-L =       "0168-7433",
  bibdate =      "Sat Apr 2 10:51:13 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jautomreason.bib",
  URL =          "http://link.springer.com/article/10.1007/s10817-014-9317-x",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Autom. Reason.",
  fjournal =     "Journal of Automated Reasoning",
  journal-URL =  "http://link.springer.com/journal/10817",
}


bibtype: inproceedings
filename: fparith.bib
label: Brain:2015:AFS
author: Martin Brain and Cesare Tinelli and Philipp Ruemmer and Thomas Wahl
title: An Automatable Formal Semantics for IEEE-754 Floating-Point Arithmetic
crossref: Muller:2015:ISC
pages: 160--167
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.26
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Martin Brain and Cesare Tinelli and Philipp Ruemmer and Thomas Wahl
rawtitle: An Automatable Formal Semantics for {IEEE-754} Floating-Point Arithmetic
entry:
@InProceedings{Brain:2015:AFS,
  author =       "Martin Brain and Cesare Tinelli and Philipp Ruemmer
                 and Thomas Wahl",
  title =        "An Automatable Formal Semantics for {IEEE-754}
                 Floating-Point Arithmetic",
  crossref =     "Muller:2015:ISC",
  pages =        "160--167",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.26",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: inproceedings
filename: fparith.bib
label: Brunie:2015:CGM
author: Nicolas Brunie and Florent de Dinechin and Olga Kupriianova and Christoph Lauter
title: Code Generators for Mathematical Functions
crossref: Muller:2015:ISC
pages: 66--73
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.22
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Nicolas Brunie and Florent de Dinechin and Olga Kupriianova and Christoph Lauter
rawtitle: Code Generators for Mathematical Functions
entry:
@InProceedings{Brunie:2015:CGM,
  author =       "Nicolas Brunie and Florent de Dinechin and Olga
                 Kupriianova and Christoph Lauter",
  title =        "Code Generators for Mathematical Functions",
  crossref =     "Muller:2015:ISC",
  pages =        "66--73",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.22",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: fparith.bib
label: Chiang:2015:UFP
author: Wei-Fan Chiang and Ganesh Gopalakrishnan and Zvonimir Rakamaric
title: Unsafe Floating-point to Unsigned Integer Casting Check for GPU Programs
journal: j-ELECT-NOTES-THEOR-COMP-SCI
volume: 317
pages: 1--12
day: 18
month: nov
monthnumber: 11
year: 2015
DOI: https://doi.org/10.1016/j.entcs.2015.10.005
ISSN: 1571-0661 OR 15710661
ISSNL: 1571-0661 OR 15710661
bibdate: Fri Apr 17 18:06:24 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.04.17 18:06:24 ???
fjournal: Electronic Notes in Theoretical Computer Science
remark: Not yet listed at journal Web site (up to volume 311 5-Jan-2015).
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/15710661
rawauthor: Wei-Fan Chiang and Ganesh Gopalakrishnan and Zvonimir Rakamari{\'c}
rawtitle: Unsafe Floating-point to Unsigned Integer Casting Check for {GPU} Programs
note:
Also presented at NSV 2015: 8th International Workshop on Numerical Software Verification 2015, Seattle, WA, USA.

URL:
http://formalverification.cs.utah.edu/papers/nsv15-unsafe-fp2ui.pdf; http://nsv2015.informatik.uni-freiburg.de/

rawnote:
Also presented at NSV 2015: 8th International Workshop on Numerical Software Verification 2015, Seattle, WA, USA.

entry:
@Article{Chiang:2015:UFP,
  author =       "Wei-Fan Chiang and Ganesh Gopalakrishnan and Zvonimir
                 Rakamari{\'c}",
  title =        "Unsafe Floating-point to Unsigned Integer Casting
                 Check for {GPU} Programs",
  journal =      j-ELECT-NOTES-THEOR-COMP-SCI,
  volume =       "317",
  number =       "??",
  pages =        "1--12",
  day =          "18",
  month =        nov,
  year =         "2015",
  DOI =          "https://doi.org/10.1016/j.entcs.2015.10.005",
  ISSN =         "1571-0661",
  ISSN-L =       "1571-0661",
  bibdate =      "Fri Apr 17 18:06:24 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  note =         "Also presented at NSV 2015: 8th International Workshop
                 on Numerical Software Verification 2015, Seattle, WA,
                 USA.",
  URL =          "http://formalverification.cs.utah.edu/papers/nsv15-unsafe-fp2ui.pdf;
                 http://nsv2015.informatik.uni-freiburg.de/",
  acknowledgement = ack-nhfb,
  fjournal =     "Electronic Notes in Theoretical Computer Science",
  journal-URL =  "http://www.sciencedirect.com/science/journal/15710661",
  remark =       "Not yet listed at journal Web site (up to volume 311
                 5-Jan-2015).",
  xxjournal-url = "http://www.elsevier.nl/locate/entcs",
}


bibtype: article
filename: fparith.bib
label: Collange:2015:NRP
author: Sylvain Collange and David Defour and Stef Graillat and Roman Iakymchuk
title: Numerical reproducibility for the parallel reduction on multi- and many-core architectures
journal: j-PARALLEL-COMPUTING
volume: 49
pages: 83--97
month: nov
monthnumber: 11
year: 2015
CODEN: PACOEJ
DOI: https://doi.org/10.1016/j.parco.2015.09.001
ISSN: 0167-8191 (print), 1872-7336 (electronic) OR 01678191 (print), 18727336 (electronic)
ISSNL: 0167-8191 OR 01678191
bibdate: Thu Oct 29 17:53:44 MDT 2015
bibtimestamp: 2015.10.29 17:53:44 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0167819115001155
fjournal: Parallel Computing
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01678191/
rawauthor: Sylvain Collange and David Defour and Stef Graillat and Roman Iakymchuk
rawtitle: Numerical reproducibility for the parallel reduction on multi- and many-core architectures
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/parallelcomputing.bib

keywords:
accuracy; correct rounding; correctly-rounded floating-point sums; CPU; error-free transformations; floating-point arithmetic; GPU; long accumulator; multi- and many-core architectures; parallel floating-point summation; reproducibility; Xeon Phi

entry:
@Article{Collange:2015:NRP,
  author =       "Sylvain Collange and David Defour and Stef Graillat
                 and Roman Iakymchuk",
  title =        "Numerical reproducibility for the parallel reduction
                 on multi- and many-core architectures",
  journal =      j-PARALLEL-COMPUTING,
  volume =       "49",
  number =       "??",
  pages =        "83--97",
  month =        nov,
  year =         "2015",
  CODEN =        "PACOEJ",
  DOI =          "https://doi.org/10.1016/j.parco.2015.09.001",
  ISSN =         "0167-8191 (print), 1872-7336 (electronic)",
  ISSN-L =       "0167-8191",
  bibdate =      "Thu Oct 29 17:53:44 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/parallelcomputing.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0167819115001155",
  acknowledgement = ack-nhfb,
  fjournal =     "Parallel Computing",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01678191/",
  keywords =     "accuracy; correct rounding; correctly-rounded
                 floating-point sums; CPU; error-free transformations;
                 floating-point arithmetic; GPU; long accumulator;
                 multi- and many-core architectures; parallel
                 floating-point summation; reproducibility; Xeon Phi",
}


bibtype: misc
filename: fparith.bib
label: Cowlishaw:2015:GDA
author: Mike Cowlishaw
title: General Decimal Arithmetic
pages: 83--97
year: 2015
bibdate: Mon Apr 25 17:12:00 2022
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2022.04.25 17:12:00 ???
URL: http://speleotrove.com/decimal/
remark: Previous versions 1981, 1997, 2008.
acknowledgement: ack-nhfb
howpublished: Web site.
rawauthor: Mike Cowlishaw
rawtitle: General Decimal Arithmetic
entry:
@Misc{Cowlishaw:2015:GDA,
  author =       "Mike Cowlishaw",
  title =        "General Decimal Arithmetic",
  howpublished = "Web site.",
  year =         "2015",
  bibdate =      "Mon Apr 25 17:12:00 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://speleotrove.com/decimal/",
  acknowledgement = ack-nhfb,
  remark =       "Previous versions 1981, 1997, 2008.",
}


bibtype: article
filename: fparith.bib
label: Coxon:2015:MMP
author: Nicholas Coxon
title: Montgomery's method of polynomial selection for the number field sieve
journal: j-LINEAR-ALGEBRA-APPL
volume: 485
pages: 72--102
day: 15
month: nov
monthnumber: 11
year: 2015
CODEN: LAAPAW
ISSN: 0024-3795 (print), 1873-1856 (electronic) OR 00243795 (print), 18731856 (electronic)
ISSNL: 0024-3795 OR 00243795
bibdate: Sat Oct 10 17:54:20 MDT 2015
bibtimestamp: 2015.10.10 17:54:20 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0024379515004395
fjournal: Linear Algebra and its Applications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00243795/
rawauthor: Nicholas Coxon
rawtitle: {Montgomery}'s method of polynomial selection for the number field sieve
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/linala2015.bib

entry:
@Article{Coxon:2015:MMP,
  author =       "Nicholas Coxon",
  title =        "{Montgomery}'s method of polynomial selection for the
                 number field sieve",
  journal =      j-LINEAR-ALGEBRA-APPL,
  volume =       "485",
  number =       "??",
  pages =        "72--102",
  day =          "15",
  month =        nov,
  year =         "2015",
  CODEN =        "LAAPAW",
  ISSN =         "0024-3795 (print), 1873-1856 (electronic)",
  ISSN-L =       "0024-3795",
  bibdate =      "Sat Oct 10 17:54:20 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/linala2015.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0024379515004395",
  acknowledgement = ack-nhfb,
  fjournal =     "Linear Algebra and its Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00243795/",
}


bibtype: article
filename: fparith.bib
label: Damouche:2015:TPC
author: N. Damouche and M. Martel and A. Chapoutot
title: Transformation of a PID Controller for Numerical Accuracy
journal: j-ELECT-NOTES-THEOR-COMP-SCI
volume: 317
pages: 47--54
month: nov
monthnumber: 11
year: 2015
DOI: https://doi.org/10.1016/j.entcs.2015.10.006
ISSN: 1571-0661 OR 15710661
bibdate: Wed Oct 16 18:53:52 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2019.10.16 18:53:52 ???
fjournal: Electronic Notes in Theoretical Computer Science
remark: Creative Commons license.
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/15710661
rawauthor: N. Damouche and M. Martel and A. Chapoutot
rawtitle: Transformation of a {PID} Controller for Numerical Accuracy
abstract:
Numerical programs performing floating-point computations are very sensitive to the way formulas are written. Several techniques have been proposed concerning the transformation of expressions in order to improve their accuracy and now we aim at going a step further by automatically transforming larger pieces of code containing several assignments and control structures. This article presents a case study in this direction. We consider a PID controller and we transform its code in order to improve its accuracy. The experimental data obtained when we compare the different versions of the code (which are mathematically equivalent) show that those transformations have a significant impact on the accuracy of the computations.

rawabstract:
Numerical programs performing floating-point computations are very sensitive to the way formulas are written. Several techniques have been proposed concerning the transformation of expressions in order to improve their accuracy and now we aim at going a step further by automatically transforming larger pieces of code containing several assignments and control structures. This article presents a case study in this direction. We consider a PID controller and we transform its code in order to improve its accuracy. The experimental data obtained when we compare the different versions of the code (which are mathematically equivalent) show that those transformations have a significant impact on the accuracy of the computations.

entry:
@Article{Damouche:2015:TPC,
  author =       "N. Damouche and M. Martel and A. Chapoutot",
  title =        "Transformation of a {PID} Controller for Numerical
                 Accuracy",
  journal =      j-ELECT-NOTES-THEOR-COMP-SCI,
  volume =       "317",
  pages =        "47--54",
  month =        nov,
  year =         "2015",
  DOI =          "https://doi.org/10.1016/j.entcs.2015.10.006",
  ISSN =         "1571-0661",
  bibdate =      "Wed Oct 16 18:53:52 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Numerical programs performing floating-point
                 computations are very sensitive to the way formulas are
                 written. Several techniques have been proposed
                 concerning the transformation of expressions in order
                 to improve their accuracy and now we aim at going a
                 step further by automatically transforming larger
                 pieces of code containing several assignments and
                 control structures. This article presents a case study
                 in this direction. We consider a PID controller and we
                 transform its code in order to improve its accuracy.
                 The experimental data obtained when we compare the
                 different versions of the code (which are
                 mathematically equivalent) show that those
                 transformations have a significant impact on the
                 accuracy of the computations.",
  acknowledgement = ack-nhfb,
  fjournal =     "Electronic Notes in Theoretical Computer Science",
  journal-URL =  "http://www.sciencedirect.com/science/journal/15710661",
  remark =       "Creative Commons license.",
}


bibtype: techreport
filename: fparith.bib
label: deDinechin:2015:FPH
author: Florent de Dinechin
title: On fixed-point hardware polynomials
type: Technical Report
institution: INSA, CITI Lab, Universite de Lyon
address: Lyon, France
pages: 47--54
month: oct
monthnumber: 10
year: 2015
bibdate: Mon Feb 10 07:54:27 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.02.10 07:54:27 ???
URL: https://hal.inria.fr/hal-01214739
acknowledgement: ack-nhfb
rawauthor: Florent de Dinechin
rawtitle: On fixed-point hardware polynomials
abstract:
Polynomial approximation is a general technique for the evaluation of numerical functions of one variable. This article addresses the automatic construction of fixed-point hardware polynomial evaluators. By systematically trying to balance the accuracy of all the steps that lead to an architecture, it simplifies and improves the previous body of work covering polynomial approximation, polynomial evaluation, and range reduction. This work is supported by an open-source implementation.

rawabstract:
Polynomial approximation is a general technique for the evaluation of numerical functions of one variable. This article addresses the automatic construction of fixed-point hardware polynomial evaluators. By systematically trying to balance the accuracy of all the steps that lead to an architecture, it simplifies and improves the previous body of work covering polynomial approximation, polynomial evaluation, and range reduction. This work is supported by an open-source implementation.

entry:
@TechReport{deDinechin:2015:FPH,
  author =       "Florent de Dinechin",
  title =        "On fixed-point hardware polynomials",
  type =         "Technical Report",
  institution =  "INSA, CITI Lab, Universit{\'e} de Lyon",
  address =      "Lyon, France",
  month =        oct,
  year =         "2015",
  bibdate =      "Mon Feb 10 07:54:27 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://hal.inria.fr/hal-01214739",
  abstract =     "Polynomial approximation is a general technique for
                 the evaluation of numerical functions of one variable.
                 This article addresses the automatic construction of
                 fixed-point hardware polynomial evaluators. By
                 systematically trying to balance the accuracy of all
                 the steps that lead to an architecture, it simplifies
                 and improves the previous body of work covering
                 polynomial approximation, polynomial evaluation, and
                 range reduction. This work is supported by an
                 open-source implementation.",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: deDinechin:2015:HIF
author: Florent de Dinechin and Matei Istoan
title: Hardware Implementations of Fixed-Point Atan2
crossref: Muller:2015:ISC
pages: 34--41
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.23
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Florent de Dinechin and Matei Istoan
rawtitle: Hardware Implementations of Fixed-Point {Atan2}
entry:
@InProceedings{deDinechin:2015:HIF,
  author =       "Florent de Dinechin and Matei Istoan",
  title =        "Hardware Implementations of Fixed-Point {Atan2}",
  crossref =     "Muller:2015:ISC",
  pages =        "34--41",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.23",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: misc
filename: fparith.bib
label: Demmel:2015:CFP
author: James Demmel and Hong Diep Nguyen and Peter Ahrens
title: Cost of Floating-Point Reproducibility
pages: 34--41
day: 20
month: nov
monthnumber: 11
year: 2015
bibdate: Tue Nov 05 15:43:15 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2019.11.05 15:43:15 ???
URL: https://www.nist.gov/sites/default/files/documents/itl/ssd/is/NRE-2015-07-Nguyen_slides.pdf
acknowledgement: ack-nhfb
howpublished: 33 lecture slides
rawauthor: James Demmel and Hong Diep Nguyen and Peter Ahrens
rawtitle: Cost of Floating-Point Reproducibility
entry:
@Misc{Demmel:2015:CFP,
  author =       "James Demmel and Hong Diep Nguyen and Peter Ahrens",
  title =        "Cost of Floating-Point Reproducibility",
  howpublished = "33 lecture slides",
  day =          "20",
  month =        nov,
  year =         "2015",
  bibdate =      "Tue Nov 05 15:43:15 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.nist.gov/sites/default/files/documents/itl/ssd/is/NRE-2015-07-Nguyen_slides.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Demmel:2015:PRS
author: J. Demmel and Hong Diep Nguyen
title: Parallel Reproducible Summation
journal: j-IEEE-TRANS-COMPUT
volume: 64
number: 7
pages: 2060--2070
month: jul
monthnumber: 7
year: 2015
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2345391
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sat Aug 1 08:53:41 MDT 2015
bibtimestamp: 2015.08.01 08:53:41 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J. Demmel and Hong Diep Nguyen
rawtitle: Parallel Reproducible Summation
entry:
@Article{Demmel:2015:PRS,
  author =       "J. Demmel and Hong Diep Nguyen",
  title =        "Parallel Reproducible Summation",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "64",
  number =       "7",
  pages =        "2060--2070",
  month =        jul,
  year =         "2015",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2345391",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sat Aug 1 08:53:41 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Denis:2015:VCF
author: Christophe Denis and Pablo De Oliveira Castro and Eric Petit
title: Verificarlo: checking floating point accuracy through Monte Carlo Arithmetic
journal: arXiv.org
pages: ??--??
day: 4
month: sep
monthnumber: 9
year: 2015
bibdate: Wed Sep 30 12:43:49 MDT 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.09.30 12:43:49 MDT
URL: http://arxiv.org/abs/1509.01347
subject: Mathematical Software (cs.MS); Numerical Analysis (cs.NA)
acknowledgement: ack-nhfb
rawauthor: Christophe Denis and Pablo {De Oliveira Castro} and Eric Petit
rawtitle: {Verificarlo}: checking floating point accuracy through {Monte Carlo} Arithmetic
abstract:
Numerical accuracy of floating point computation is a well studied topic, but which has not made its way to the end-user in scientific computing. With the recent requirements for code modernization to exploit new highly parallel hardware and perform higher resolution computation, this has become one of the critical issues to address. To democratize numerical accuracy analysis, it is important to propose tools and methodologies to study large use cases in a reliable and automatic way. In this paper, we propose verificarlo, an extension to the LLVM compiler to automatically use Monte Carlo Arithmetic in a transparent way for the end-user. It supports all the major languages including C, C++ and Fortran. We also illustrate the fact that unlike source-to-source approaches, our implementation captures the influence of compiler optimizations on the numerical accuracy. Finally, we illustrate on various use cases how Monte Carlo Arithmetic using the verificarlo tool outperforms the existing approaches and is a step toward automatic numerical analysis of large scientific applications.

rawabstract:
Numerical accuracy of floating point computation is a well studied topic, but which has not made its way to the end-user in scientific computing. With the recent requirements for code modernization to exploit new highly parallel hardware and perform higher resolution computation, this has become one of the critical issues to address. To democratize numerical accuracy analysis, it is important to propose tools and methodologies to study large use cases in a reliable and automatic way. In this paper, we propose verificarlo, an extension to the LLVM compiler to automatically use Monte Carlo Arithmetic in a transparent way for the end-user. It supports all the major languages including C, C++ and Fortran. We also illustrate the fact that unlike source-to-source approaches, our implementation captures the influence of compiler optimizations on the numerical accuracy. Finally, we illustrate on various use cases how Monte Carlo Arithmetic using the verificarlo tool outperforms the existing approaches and is a step toward automatic numerical analysis of large scientific applications.

entry:
@Article{Denis:2015:VCF,
  author =       "Christophe Denis and Pablo {De Oliveira Castro} and
                 Eric Petit",
  title =        "{Verificarlo}: checking floating point accuracy
                 through {Monte Carlo} Arithmetic",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "??--??",
  day =          "4",
  month =        sep,
  year =         "2015",
  CODEN =        "????",
  ISSN =         "????",
  ISSN-L =       "????",
  bibdate =      "Wed Sep 30 12:43:49 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://arxiv.org/abs/1509.01347",
  abstract =     "Numerical accuracy of floating point computation is a
                 well studied topic, but which has not made its way to
                 the end-user in scientific computing. With the recent
                 requirements for code modernization to exploit new
                 highly parallel hardware and perform higher resolution
                 computation, this has become one of the critical issues
                 to address. To democratize numerical accuracy analysis,
                 it is important to propose tools and methodologies to
                 study large use cases in a reliable and automatic way.
                 In this paper, we propose verificarlo, an extension to
                 the LLVM compiler to automatically use Monte Carlo
                 Arithmetic in a transparent way for the end-user. It
                 supports all the major languages including C, C++ and
                 Fortran. We also illustrate the fact that unlike
                 source-to-source approaches, our implementation
                 captures the influence of compiler optimizations on the
                 numerical accuracy. Finally, we illustrate on various
                 use cases how Monte Carlo Arithmetic using the
                 verificarlo tool outperforms the existing approaches
                 and is a step toward automatic numerical analysis of
                 large scientific applications.",
  acknowledgement = ack-nhfb,
  subject =      "Mathematical Software (cs.MS); Numerical Analysis
                 (cs.NA)",
}


bibtype: article
filename: fparith.bib
label: Dietz:2015:UIO
author: Will Dietz and Peng Li and John Regehr and Vikram Adve
title: Understanding Integer Overflow in C / C++
journal: j-TOSEM
volume: 25
number: 1
pages: 2:1--2:29
month: dec
monthnumber: 12
year: 2015
CODEN: ATSMER
DOI: https://doi.org/10.1145/2743019
ISSN: 1049-331X (print), 1557-7392 (electronic) OR 1049331X (print), 15577392 (electronic)
ISSNL: 1049-331X OR 1049331X
bibdate: Thu Dec 3 09:54:19 MST 2015
bibtimestamp: 2015.12.03 09:54:19 MST
fjournal: ACM Transactions on Software Engineering and Methodology
acknowledgement: ack-nhfb
articleno: 2
journalURL: https://dl.acm.org/loi/cacm0
rawauthor: Will Dietz and Peng Li and John Regehr and Vikram Adve
rawtitle: Understanding Integer Overflow in {C\slash C++}
bibsource:
http://www.acm.org/pubs/contents/journals/tosem/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/tosem.bib

abstract:
Integer overflow bugs in C and C++ programs are difficult to track down and may lead to fatal errors or exploitable vulnerabilities. Although a number of tools for finding these bugs exist, the situation is complicated because not all overflows are bugs. Better tools need to be constructed, but a thorough understanding of the issues behind these errors does not yet exist. We developed IOC, a dynamic checking tool for integer overflows, and used it to conduct the first detailed empirical study of the prevalence and patterns of occurrence of integer overflows in C and C++ code. Our results show that intentional uses of wraparound behaviors are more common than is widely believed; for example, there are over 200 distinct locations in the SPEC CINT2000 benchmarks where overflow occurs. Although many overflows are intentional, a large number of accidental overflows also occur. Orthogonal to programmers' intent, overflows are found in both well-defined and undefined flavors. Applications executing undefined operations can be, and have been, broken by improvements in compiler optimizations. Looking beyond SPEC, we found and reported undefined integer overflows in SQLite, PostgreSQL, SafeInt, GNU MPC and GMP, Firefox, LLVM, Python, BIND, and OpenSSL; many of these have since been fixed.

rawabstract:
Integer overflow bugs in C and C++ programs are difficult to track down and may lead to fatal errors or exploitable vulnerabilities. Although a number of tools for finding these bugs exist, the situation is complicated because not all overflows are bugs. Better tools need to be constructed, but a thorough understanding of the issues behind these errors does not yet exist. We developed IOC, a dynamic checking tool for integer overflows, and used it to conduct the first detailed empirical study of the prevalence and patterns of occurrence of integer overflows in C and C++ code. Our results show that intentional uses of wraparound behaviors are more common than is widely believed; for example, there are over 200 distinct locations in the SPEC CINT2000 benchmarks where overflow occurs. Although many overflows are intentional, a large number of accidental overflows also occur. Orthogonal to programmers' intent, overflows are found in both well-defined and undefined flavors. Applications executing undefined operations can be, and have been, broken by improvements in compiler optimizations. Looking beyond SPEC, we found and reported undefined integer overflows in SQLite, PostgreSQL, SafeInt, GNU MPC and GMP, Firefox, LLVM, Python, BIND, and OpenSSL; many of these have since been fixed.

entry:
@Article{Dietz:2015:UIO,
  author =       "Will Dietz and Peng Li and John Regehr and Vikram
                 Adve",
  title =        "Understanding Integer Overflow in {C\slash C++}",
  journal =      j-TOSEM,
  volume =       "25",
  number =       "1",
  pages =        "2:1--2:29",
  month =        dec,
  year =         "2015",
  CODEN =        "ATSMER",
  DOI =          "https://doi.org/10.1145/2743019",
  ISSN =         "1049-331X (print), 1557-7392 (electronic)",
  ISSN-L =       "1049-331X",
  bibdate =      "Thu Dec 3 09:54:19 MST 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/tosem/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/tosem.bib",
  abstract =     "Integer overflow bugs in C and C++ programs are
                 difficult to track down and may lead to fatal errors or
                 exploitable vulnerabilities. Although a number of tools
                 for finding these bugs exist, the situation is
                 complicated because not all overflows are bugs. Better
                 tools need to be constructed, but a thorough
                 understanding of the issues behind these errors does
                 not yet exist. We developed IOC, a dynamic checking
                 tool for integer overflows, and used it to conduct the
                 first detailed empirical study of the prevalence and
                 patterns of occurrence of integer overflows in C and
                 C++ code. Our results show that intentional uses of
                 wraparound behaviors are more common than is widely
                 believed; for example, there are over 200 distinct
                 locations in the SPEC CINT2000 benchmarks where
                 overflow occurs. Although many overflows are
                 intentional, a large number of accidental overflows
                 also occur. Orthogonal to programmers' intent,
                 overflows are found in both well-defined and undefined
                 flavors. Applications executing undefined operations
                 can be, and have been, broken by improvements in
                 compiler optimizations. Looking beyond SPEC, we found
                 and reported undefined integer overflows in SQLite,
                 PostgreSQL, SafeInt, GNU MPC and GMP, Firefox, LLVM,
                 Python, BIND, and OpenSSL; many of these have since
                 been fixed.",
  acknowledgement = ack-nhfb,
  articleno =    "2",
  fjournal =     "ACM Transactions on Software Engineering and
                 Methodology",
  journal-URL =  "https://dl.acm.org/loi/cacm0",
}


bibtype: article
filename: fparith.bib
label: Ebergen:2015:RDA
author: J. Ebergen and N. Jamadagni
title: Radix-$2$ Division Algorithms with an Over-Redundant Digit Set
journal: j-IEEE-TRANS-COMPUT
volume: 64
number: 9
pages: 2652--2663
year: 2015
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2366738
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Sep 28 12:03:47 MDT 2015
bibtimestamp: 2015.09.28 12:03:47 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J. Ebergen and N. Jamadagni
rawtitle: Radix-$2$ Division Algorithms with an Over-Redundant Digit Set
entry:
@Article{Ebergen:2015:RDA,
  author =       "J. Ebergen and N. Jamadagni",
  title =        "Radix-$2$ Division Algorithms with an Over-Redundant
                 Digit Set",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "64",
  number =       "9",
  pages =        "2652--2663",
  month =        "????",
  year =         "2015",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2366738",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Sep 28 12:03:47 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: inproceedings
filename: fparith.bib
label: El-Razouk:2015:NBL
author: Hayssam El-Razouk and Arash Reyhani-Masoleh
title: New Bit-Level Serial $ rm GF (2^m) $ Multiplication Using Polynomial Basis
crossref: Muller:2015:ISC
pages: 129--136
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.11
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Hayssam El-Razouk and Arash Reyhani-Masoleh
rawtitle: New Bit-Level Serial {$ {\rm GF} (2^m) $} Multiplication Using Polynomial Basis
entry:
@InProceedings{El-Razouk:2015:NBL,
  author =       "Hayssam El-Razouk and Arash Reyhani-Masoleh",
  title =        "New Bit-Level Serial {$ {\rm GF} (2^m) $}
                 Multiplication Using Polynomial Basis",
  crossref =     "Muller:2015:ISC",
  pages =        "129--136",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.11",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: fparith.bib
label: Elsayed:2015:NPE
author: Essam Elsayed and Hatem M. El-Boghdadi
title: A novel power-efficient multi-operand digit-multiplier using reconfiguration and clock gating
journal: j-J-SUPERCOMPUTING
volume: 71
number: 7
pages: 2539--2564
month: jul
monthnumber: 7
year: 2015
CODEN: JOSUED
DOI: https://doi.org/10.1007/s11227-015-1403-2
ISSN: 0920-8542 (print), 1573-0484 (electronic) OR 09208542 (print), 15730484 (electronic)
ISSNL: 0920-8542 OR 09208542
bibdate: Sat Aug 8 12:23:11 MDT 2015
bibtimestamp: 2015.08.08 12:23:11 MDT
URL: http://link.springer.com/article/10.1007/s11227-015-1403-2
fjournal: The Journal of Supercomputing
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11227
rawauthor: Essam Elsayed and Hatem M. El-Boghdadi
rawtitle: A novel power-efficient multi-operand digit-multiplier using reconfiguration and clock gating
bibsource:
http://link.springer.com/journal/11227/71/7; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jsuper.bib

entry:
@Article{Elsayed:2015:NPE,
  author =       "Essam Elsayed and Hatem M. El-Boghdadi",
  title =        "A novel power-efficient multi-operand digit-multiplier
                 using reconfiguration and clock gating",
  journal =      j-J-SUPERCOMPUTING,
  volume =       "71",
  number =       "7",
  pages =        "2539--2564",
  month =        jul,
  year =         "2015",
  CODEN =        "JOSUED",
  DOI =          "https://doi.org/10.1007/s11227-015-1403-2",
  ISSN =         "0920-8542 (print), 1573-0484 (electronic)",
  ISSN-L =       "0920-8542",
  bibdate =      "Sat Aug 8 12:23:11 MDT 2015",
  bibsource =    "http://link.springer.com/journal/11227/71/7;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jsuper.bib",
  URL =          "http://link.springer.com/article/10.1007/s11227-015-1403-2",
  acknowledgement = ack-nhfb,
  fjournal =     "The Journal of Supercomputing",
  journal-URL =  "http://link.springer.com/journal/11227",
}


bibtype: article
filename: fparith.bib
label: Flocke:2015:AAE
author: N. Flocke
journal: j-TOMS
volume: 41
number: 4
pages: 30:1--30:24
month: oct
monthnumber: 10
year: 2015
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2699468
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Oct 26 17:31:15 MDT 2015
bibtimestamp: 2015.10.26 17:31:15 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 30
journalURL: https://dl.acm.org/loi/toms
rawauthor: N. Flocke
title:
Algorithm 954: an Accurate and Efficient Cubic and Quartic Equation Solver for Physical Applications

bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
We report on an accurate and efficient algorithm for obtaining all roots of general real cubic and quartic polynomials. Both the cubic and quartic solvers give highly accurate roots and place no restrictions on the magnitude of the polynomial coefficients. The key to the algorithm is a proper rescaling of both polynomials. This puts upper bounds on the magnitude of the roots and is very useful in stabilizing the root finding process. The cubic solver is based on dividing the cubic polynomial into six classes. By analyzing the root surface for each class, a fast convergent Newton--Raphson starting point for a real root is obtained at a cost no higher than three additions and four multiplications. The quartic solver uses the cubic solver in getting information about stationary points and, when the quartic has real roots, stable Newton--Raphson iterations give one of the extreme real roots. The remaining roots follow by composite deflation to a cubic. If the quartic has only complex roots, the present article shows that a stable Newton--Raphson iteration on a derived symmetric sixth degree polynomial can be formulated for the real parts of the complex roots. The imaginary parts follow by solving suitable quadratics.

rawabstract:
We report on an accurate and efficient algorithm for obtaining all roots of general real cubic and quartic polynomials. Both the cubic and quartic solvers give highly accurate roots and place no restrictions on the magnitude of the polynomial coefficients. The key to the algorithm is a proper rescaling of both polynomials. This puts upper bounds on the magnitude of the roots and is very useful in stabilizing the root finding process. The cubic solver is based on dividing the cubic polynomial into six classes. By analyzing the root surface for each class, a fast convergent Newton--Raphson starting point for a real root is obtained at a cost no higher than three additions and four multiplications. The quartic solver uses the cubic solver in getting information about stationary points and, when the quartic has real roots, stable Newton--Raphson iterations give one of the extreme real roots. The remaining roots follow by composite deflation to a cubic. If the quartic has only complex roots, the present article shows that a stable Newton--Raphson iteration on a derived symmetric sixth degree polynomial can be formulated for the real parts of the complex roots. The imaginary parts follow by solving suitable quadratics.

rawtitle:
{Algorithm 954}: an Accurate and Efficient Cubic and Quartic Equation Solver for Physical Applications

entry:
@Article{Flocke:2015:AAE,
  author =       "N. Flocke",
  title =        "{Algorithm 954}: an Accurate and Efficient Cubic and
                 Quartic Equation Solver for Physical Applications",
  journal =      j-TOMS,
  volume =       "41",
  number =       "4",
  pages =        "30:1--30:24",
  month =        oct,
  year =         "2015",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2699468",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Oct 26 17:31:15 MDT 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "We report on an accurate and efficient algorithm for
                 obtaining all roots of general real cubic and quartic
                 polynomials. Both the cubic and quartic solvers give
                 highly accurate roots and place no restrictions on the
                 magnitude of the polynomial coefficients. The key to
                 the algorithm is a proper rescaling of both
                 polynomials. This puts upper bounds on the magnitude of
                 the roots and is very useful in stabilizing the root
                 finding process. The cubic solver is based on dividing
                 the cubic polynomial into six classes. By analyzing the
                 root surface for each class, a fast convergent
                 Newton--Raphson starting point for a real root is
                 obtained at a cost no higher than three additions and
                 four multiplications. The quartic solver uses the cubic
                 solver in getting information about stationary points
                 and, when the quartic has real roots, stable
                 Newton--Raphson iterations give one of the extreme real
                 roots. The remaining roots follow by composite
                 deflation to a cubic. If the quartic has only complex
                 roots, the present article shows that a stable
                 Newton--Raphson iteration on a derived symmetric sixth
                 degree polynomial can be formulated for the real parts
                 of the complex roots. The imaginary parts follow by
                 solving suitable quadratics.",
  acknowledgement = ack-nhfb,
  articleno =    "30",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: fparith.bib
label: Frechtling:2015:MMS
author: Michael Frechtling and Philip H. W. Leong
title: MCALIB: Measuring Sensitivity to Rounding Error with Monte Carlo Programming
journal: j-TOPLAS
volume: 37
number: 2
pages: 5:1--5:??
month: apr
monthnumber: 4
year: 2015
CODEN: ATPSDT
DOI: https://doi.org/10.1145/2665073
ISSN: 0164-0925 (print), 1558-4593 (electronic) OR 01640925 (print), 15584593 (electronic)
ISSNL: 0164-0925 OR 01640925
bibdate: Thu Apr 16 18:32:12 MDT 2015
bibtimestamp: 2015.04.16 18:32:12 MDT
fjournal: ACM Transactions on Programming Languages and Systems
acknowledgement: ack-nhfb
articleno: 5
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J783
rawauthor: Michael Frechtling and Philip H. W. Leong
rawtitle: {MCALIB}: Measuring Sensitivity to Rounding Error with {Monte Carlo} Programming
bibsource:
http://www.acm.org/pubs/contents/journals/toplas/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toplas.bib

abstract:
Runtime analysis provides an effective method for measuring the sensitivity of programs to rounding errors. To date, implementations have required significant changes to source code, detracting from their widespread application. In this work, we present an open source system that automates the quantitative analysis of floating point rounding errors through the use of C-based source-to-source compilation and a Monte Carlo arithmetic library. We demonstrate its application to the comparison of algorithms, detection of catastrophic cancellation, and determination of whether single precision floating point provides sufficient accuracy for a given application. Methods for obtaining quantifiable measurements of sensitivity to rounding error are also detailed.

rawabstract:
Runtime analysis provides an effective method for measuring the sensitivity of programs to rounding errors. To date, implementations have required significant changes to source code, detracting from their widespread application. In this work, we present an open source system that automates the quantitative analysis of floating point rounding errors through the use of C-based source-to-source compilation and a Monte Carlo arithmetic library. We demonstrate its application to the comparison of algorithms, detection of catastrophic cancellation, and determination of whether single precision floating point provides sufficient accuracy for a given application. Methods for obtaining quantifiable measurements of sensitivity to rounding error are also detailed.

entry:
@Article{Frechtling:2015:MMS,
  author =       "Michael Frechtling and Philip H. W. Leong",
  title =        "{MCALIB}: Measuring Sensitivity to Rounding Error with
                 {Monte Carlo} Programming",
  journal =      j-TOPLAS,
  volume =       "37",
  number =       "2",
  pages =        "5:1--5:??",
  month =        apr,
  year =         "2015",
  CODEN =        "ATPSDT",
  DOI =          "https://doi.org/10.1145/2665073",
  ISSN =         "0164-0925 (print), 1558-4593 (electronic)",
  ISSN-L =       "0164-0925",
  bibdate =      "Thu Apr 16 18:32:12 MDT 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toplas/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toplas.bib",
  abstract =     "Runtime analysis provides an effective method for
                 measuring the sensitivity of programs to rounding
                 errors. To date, implementations have required
                 significant changes to source code, detracting from
                 their widespread application. In this work, we present
                 an open source system that automates the quantitative
                 analysis of floating point rounding errors through the
                 use of C-based source-to-source compilation and a Monte
                 Carlo arithmetic library. We demonstrate its
                 application to the comparison of algorithms, detection
                 of catastrophic cancellation, and determination of
                 whether single precision floating point provides
                 sufficient accuracy for a given application. Methods
                 for obtaining quantifiable measurements of sensitivity
                 to rounding error are also detailed.",
  acknowledgement = ack-nhfb,
  articleno =    "5",
  fjournal =     "ACM Transactions on Programming Languages and
                 Systems",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J783",
}


bibtype: article
filename: fparith.bib
label: Froggatt:2015:EAU
author: Terry Froggatt
title: An Error in the Ada Universal Arithmetic Package
journal: j-SIGADA-LETTERS
volume: 35
number: 2
pages: 14--14
month: aug
monthnumber: 8
year: 2015
CODEN: AALEE5
DOI: https://doi.org/10.1145/2903260.2903263
ISSN: 1094-3641 (print), 1557-9476 (electronic) OR 10943641 (print), 15579476 (electronic)
ISSNL: 1094-3641 OR 10943641
bibdate: Mon Jun 20 18:46:44 MDT 2016
bibtimestamp: 2016.06.20 18:46:44 MDT
fjournal: ACM SIGAda Ada Letters
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/citation.cfm?id=J32
rawauthor: Terry Froggatt
rawtitle: An Error in the {Ada} Universal Arithmetic Package
note:
See cite Fisher:1984:UAP. The 32-year-old error is a test with digit $t$ that has tt if (t > BASE), but the operator should instead be tt >=.

abstract:
An Ada Universal Arithmetic package was published over three decades ago in Ada Letters. It was incorporated into several Ada compilers, and it is one of the few compiler components which will have required no change as the Ada language has evolved. The purpose if this note is to draw attention to an error in that package, which may or may not have been corrected in the compilers, but which has never subsequently been corrected within Ada Letters.

rawabstract:
An Ada Universal Arithmetic package was published over three decades ago in Ada Letters. It was incorporated into several Ada compilers, and it is one of the few compiler components which will have required no change as the Ada language has evolved. The purpose if this note is to draw attention to an error in that package, which may or may not have been corrected in the compilers, but which has never subsequently been corrected within Ada Letters.

rawnote:
See \cite{Fisher:1984:UAP}. The 32-year-old error is a test with digit $t$ that has {\tt if (t > BASE)}, but the operator should instead be {\tt >=}.

entry:
@Article{Froggatt:2015:EAU,
  author =       "Terry Froggatt",
  title =        "An Error in the {Ada} Universal Arithmetic Package",
  journal =      j-SIGADA-LETTERS,
  volume =       "35",
  number =       "2",
  pages =        "14--14",
  month =        aug,
  year =         "2015",
  CODEN =        "AALEE5",
  DOI =          "https://doi.org/10.1145/2903260.2903263",
  ISSN =         "1094-3641 (print), 1557-9476 (electronic)",
  ISSN-L =       "1094-3641",
  bibdate =      "Mon Jun 20 18:46:44 MDT 2016",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigada.bib",
  note =         "See \cite{Fisher:1984:UAP}. The 32-year-old error is a
                 test with digit $t$ that has {\tt if (t > BASE)}, but
                 the operator should instead be {\tt >=}.",
  abstract =     "An Ada Universal Arithmetic package was published over
                 three decades ago in Ada Letters. It was incorporated
                 into several Ada compilers, and it is one of the few
                 compiler components which will have required no change
                 as the Ada language has evolved. The purpose if this
                 note is to draw attention to an error in that package,
                 which may or may not have been corrected in the
                 compilers, but which has never subsequently been
                 corrected within Ada Letters.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGAda Ada Letters",
  journal-URL =  "http://portal.acm.org/citation.cfm?id=J32",
}


bibtype: article
filename: fparith.bib
label: Fu:2015:ABE
author: Zhoulai Fu and Zhaojun Bai and Zhendong Su
title: Automated backward error analysis for numerical code
journal: j-SIGPLAN
volume: 50
number: 10
pages: 639--654
month: oct
monthnumber: 10
year: 2015
CODEN: SINODQ
DOI: https://doi.org/10.1145/2858965.2814317
ISSN: 0362-1340 (print), 1523-2867 (print), 1558-1160 OR 03621340 (print), 15232867 (print), 15581160
ISSNL: 0362-1340 OR 03621340
bibdate: Tue Feb 16 12:01:43 MST 2016
bibtimestamp: 2016.02.16 12:01:43 MST
fjournal: ACM SIGPLAN Notices
remark: OOPSLA '15 conference proceedings.
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J706
rawauthor: Zhoulai Fu and Zhaojun Bai and Zhendong Su
rawtitle: Automated backward error analysis for numerical code
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigplan2010.bib

abstract:
Numerical code uses floating-point arithmetic and necessarily suffers from roundoff and truncation errors. Error analysis is the process to quantify such uncertainty in the solution to a problem. Forward error analysis and backward error analysis are two popular paradigms of error analysis. Forward error analysis is more intuitive and has been explored and automated by the programming languages (PL) community. In contrast, although backward error analysis is more preferred by numerical analysts and the foundation for numerical stability, it is less known and unexplored by the PL community. To fill the gap, this paper presents an automated backward error analysis for numerical code to empower both numerical analysts and application developers. In addition, we use the computed backward error results to also compute the condition number, an important quantity recognized by numerical analysts for measuring how sensitive a function is to changes or errors in the input. Experimental results on Intel X87 FPU functions and widely-used GNU C Library functions demonstrate that our analysis is effective at analyzing the accuracy of floating-point programs.

rawabstract:
Numerical code uses floating-point arithmetic and necessarily suffers from roundoff and truncation errors. Error analysis is the process to quantify such uncertainty in the solution to a problem. Forward error analysis and backward error analysis are two popular paradigms of error analysis. Forward error analysis is more intuitive and has been explored and automated by the programming languages (PL) community. In contrast, although backward error analysis is more preferred by numerical analysts and the foundation for numerical stability, it is less known and unexplored by the PL community. To fill the gap, this paper presents an automated backward error analysis for numerical code to empower both numerical analysts and application developers. In addition, we use the computed backward error results to also compute the condition number, an important quantity recognized by numerical analysts for measuring how sensitive a function is to changes or errors in the input. Experimental results on Intel X87 FPU functions and widely-used GNU C Library functions demonstrate that our analysis is effective at analyzing the accuracy of floating-point programs.

entry:
@Article{Fu:2015:ABE,
  author =       "Zhoulai Fu and Zhaojun Bai and Zhendong Su",
  title =        "Automated backward error analysis for numerical code",
  journal =      j-SIGPLAN,
  volume =       "50",
  number =       "10",
  pages =        "639--654",
  month =        oct,
  year =         "2015",
  CODEN =        "SINODQ",
  DOI =          "https://doi.org/10.1145/2858965.2814317",
  ISSN =         "0362-1340 (print), 1523-2867 (print), 1558-1160
                 (electronic)",
  ISSN-L =       "0362-1340",
  bibdate =      "Tue Feb 16 12:01:43 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigplan2010.bib",
  abstract =     "Numerical code uses floating-point arithmetic and
                 necessarily suffers from roundoff and truncation
                 errors. Error analysis is the process to quantify such
                 uncertainty in the solution to a problem. Forward error
                 analysis and backward error analysis are two popular
                 paradigms of error analysis. Forward error analysis is
                 more intuitive and has been explored and automated by
                 the programming languages (PL) community. In contrast,
                 although backward error analysis is more preferred by
                 numerical analysts and the foundation for numerical
                 stability, it is less known and unexplored by the PL
                 community. To fill the gap, this paper presents an
                 automated backward error analysis for numerical code to
                 empower both numerical analysts and application
                 developers. In addition, we use the computed backward
                 error results to also compute the condition number, an
                 important quantity recognized by numerical analysts for
                 measuring how sensitive a function is to changes or
                 errors in the input. Experimental results on Intel X87
                 FPU functions and widely-used GNU C Library functions
                 demonstrate that our analysis is effective at analyzing
                 the accuracy of floating-point programs.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGPLAN Notices",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J706",
  remark =       "OOPSLA '15 conference proceedings.",
}


bibtype: inproceedings
filename: fparith.bib
label: Fukushima:2015:PFCc
author: Toshio Fukushima
title: Precise and Fast Computation of Elliptic Integrals and Functions
crossref: Muller:2015:ISC
pages: 50--57
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.15
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Toshio Fukushima
rawtitle: Precise and Fast Computation of Elliptic Integrals and Functions
entry:
@InProceedings{Fukushima:2015:PFCc,
  author =       "Toshio Fukushima",
  title =        "Precise and Fast Computation of Elliptic Integrals and
                 Functions",
  crossref =     "Muller:2015:ISC",
  pages =        "50--57",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.15",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: inproceedings
filename: fparith.bib
label: Gerard:2015:CDR
author: Benoit Gerard and Jean-Gabriel Kammerer and Nabil Merkiche
title: Contributions to the Design of Residue Number System Architectures
crossref: Muller:2015:ISC
pages: 105--112
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.25
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Beno{\^\i}t G{\'e}rard and Jean-Gabriel Kammerer and Nabil Merkiche
rawtitle: Contributions to the Design of Residue Number System Architectures
entry:
@InProceedings{Gerard:2015:CDR,
  author =       "Beno{\^\i}t G{\'e}rard and Jean-Gabriel Kammerer and
                 Nabil Merkiche",
  title =        "Contributions to the Design of Residue Number System
                 Architectures",
  crossref =     "Muller:2015:ISC",
  pages =        "105--112",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.25",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: fparith.bib
label: Gorgin:2015:CXH
author: S. Gorgin and G. Jaberipur
title: Comment on ``High-Speed Parallel Decimal Multiplication With Redundant Internal Encodings''
journal: j-IEEE-TRANS-COMPUT
volume: 64
number: 1
pages: 293--294
month: jan
monthnumber: 1
year: 2015
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.160
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Wed Jan 21 08:56:46 2015
bibtimestamp: 2015.01.21 08:56:46 ???
note: See cite Han:2013:HSP.
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: S. Gorgin and G. Jaberipur
rawnote: See \cite{Han:2013:HSP}.
rawtitle: Comment on {``High-Speed Parallel Decimal Multiplication With Redundant Internal Encodings''}
keywords:
Adders; Computer science; Computers; critical delay path; Decimal arithmetic; Delays; digital arithmetic; Educational institutions; Encoding; high speed parallel decimal multiplication; Logic gates; matrix multiplication; parallel decimal multiplication; parallel processing; redundant internal encodings; redundant partial products; redundant representation; signed-digit partial product

entry:
@Article{Gorgin:2015:CXH,
  author =       "S. Gorgin and G. Jaberipur",
  title =        "Comment on {``High-Speed Parallel Decimal
                 Multiplication With Redundant Internal Encodings''}",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "64",
  number =       "1",
  pages =        "293--294",
  month =        jan,
  year =         "2015",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.160",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Wed Jan 21 08:56:46 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  note =         "See \cite{Han:2013:HSP}.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "Adders; Computer science; Computers; critical delay
                 path; Decimal arithmetic; Delays; digital arithmetic;
                 Educational institutions; Encoding; high speed parallel
                 decimal multiplication; Logic gates; matrix
                 multiplication; parallel decimal multiplication;
                 parallel processing; redundant internal encodings;
                 redundant partial products; redundant representation;
                 signed-digit partial product",
}


bibtype: inproceedings
filename: fparith.bib
label: Gouicem:2015:MMD
author: Mourad Gouicem
title: Modular Multiplication and Division Algorithms Based on Continued Fraction Expansion
crossref: Muller:2015:ISC
pages: 137--143
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.21
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Mourad Gouicem
rawtitle: Modular Multiplication and Division Algorithms Based on Continued Fraction Expansion
entry:
@InProceedings{Gouicem:2015:MMD,
  author =       "Mourad Gouicem",
  title =        "Modular Multiplication and Division Algorithms Based
                 on Continued Fraction Expansion",
  crossref =     "Muller:2015:ISC",
  pages =        "137--143",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.21",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: fparith.bib
label: Graillat:2015:ECF
author: Stef Graillat and Christoph Lauter and Ping Tak Peter Tang and Naoya Yamanaka and Shin'ichi Oishi
title: Efficient Calculations of Faithfully Rounded $ l_2$-Norms of $n$-Vectors
journal: j-TOMS
volume: 41
number: 4
pages: 24:1--24:20
month: oct
monthnumber: 10
year: 2015
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2699469
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Oct 26 17:31:15 MDT 2015
bibtimestamp: 2015.10.26 17:31:15 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 24
journalURL: https://dl.acm.org/loi/toms
rawauthor: Stef Graillat and Christoph Lauter and Ping Tak Peter Tang and Naoya Yamanaka and Shin'ichi Oishi
rawtitle: Efficient Calculations of Faithfully Rounded $ l_2$-Norms of $n$-Vectors
bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
In this article, we present an efficient algorithm to compute the faithful rounding of the $ l_2 $-norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.

rawabstract:
In this article, we present an efficient algorithm to compute the faithful rounding of the $ l_2 $-norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.

entry:
@Article{Graillat:2015:ECF,
  author =       "Stef Graillat and Christoph Lauter and Ping Tak Peter
                 Tang and Naoya Yamanaka and Shin'ichi Oishi",
  title =        "Efficient Calculations of Faithfully Rounded $
                 l_2$-Norms of $n$-Vectors",
  journal =      j-TOMS,
  volume =       "41",
  number =       "4",
  pages =        "24:1--24:20",
  month =        oct,
  year =         "2015",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2699469",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Oct 26 17:31:15 MDT 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "In this article, we present an efficient algorithm to
                 compute the faithful rounding of the $ l_2 $-norm of a
                 floating-point vector. This means that the result is
                 accurate to within 1 bit of the underlying
                 floating-point type. This algorithm does not generate
                 overflows or underflows spuriously, but does so when
                 the final result calls for such a numerical exception
                 to be raised. Moreover, the algorithm is well suited
                 for parallel implementation and vectorization. The
                 implementation runs up to 3 times faster than the
                 netlib version on current processors.",
  acknowledgement = ack-nhfb,
  articleno =    "24",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: fparith.bib
label: Graillat:2015:MRE
author: Stef Graillat and Vincent Lefevre and Jean-Michel Muller
journal: j-NUMER-ALGORITHMS
volume: 70
number: 3
pages: 653--667
month: nov
monthnumber: 11
year: 2015
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-015-9967-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Sun Oct 25 07:27:50 MDT 2015
bibtimestamp: 2015.10.25 07:27:50 MDT
URL: http://link.springer.com/article/10.1007/s11075-015-9967-8
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Stef Graillat and Vincent Lef{\`e}vre and Jean-Michel Muller
title:
On the maximum relative error when computing integer powers by iterated multiplications in floating-point arithmetic

bibsource:
http://link.springer.com/journal/11075/70/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

remark:
The authors show via a complex multipage proof that the iterated product for $ x^n $ in p-bit binary arithmetic with default IEEE 754 rounding (to nearest with ties to even) produces a worst-case relative error in the product that is no larger than $ (n - 1) u $, where $ u = 2^-p $ is the rounding unit.

rawtitle:
On the maximum relative error when computing integer powers by iterated multiplications in floating-point arithmetic

entry:
@Article{Graillat:2015:MRE,
  author =       "Stef Graillat and Vincent Lef{\`e}vre and Jean-Michel
                 Muller",
  title =        "On the maximum relative error when computing integer
                 powers by iterated multiplications in floating-point
                 arithmetic",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "70",
  number =       "3",
  pages =        "653--667",
  month =        nov,
  year =         "2015",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-015-9967-8",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Sun Oct 25 07:27:50 MDT 2015",
  bibsource =    "http://link.springer.com/journal/11075/70/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-015-9967-8",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  remark =       "The authors show via a complex multipage proof that
                 the iterated product for $ x^n $ in p-bit binary
                 arithmetic with default IEEE 754 rounding (to nearest
                 with ties to even) produces a worst-case relative error
                 in the product that is no larger than $ (n - 1) u $,
                 where $ u = 2^{-p} $ is the rounding unit.",
}


bibtype: article
filename: fparith.bib
label: Graillat:2015:NVC
author: Stef Graillat and Fabienne Jezequel and Romain Picot
title: Numerical validation of compensated summation algorithms with stochastic arithmetic
journal: j-ELECT-NOTES-THEOR-COMP-SCI
volume: 317
pages: 55--69
day: 18
month: nov
monthnumber: 11
year: 2015
DOI: https://doi.org/10.1016/j.entcs.2015.10.007
ISSN: 1571-0661 OR 15710661
ISSNL: 1571-0661 OR 15710661
bibdate: Sat Jun 4 17:40:10 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.06.04 17:40:10 ???
fjournal: Electronic Notes in Theoretical Computer Science
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/15710661
rawauthor: Stef Graillat and Fabienne J{\'e}z{\'e}quel and Romain Picot
rawtitle: Numerical validation of compensated summation algorithms with stochastic arithmetic
entry:
@Article{Graillat:2015:NVC,
  author =       "Stef Graillat and Fabienne J{\'e}z{\'e}quel and Romain
                 Picot",
  title =        "Numerical validation of compensated summation
                 algorithms with stochastic arithmetic",
  journal =      j-ELECT-NOTES-THEOR-COMP-SCI,
  volume =       "317",
  pages =        "55--69",
  day =          "18",
  month =        nov,
  year =         "2015",
  DOI =          "https://doi.org/10.1016/j.entcs.2015.10.007",
  ISSN =         "1571-0661",
  ISSN-L =       "1571-0661",
  bibdate =      "Sat Jun 4 17:40:10 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Electronic Notes in Theoretical Computer Science",
  journal-URL =  "http://www.sciencedirect.com/science/journal/15710661",
}


bibtype: inproceedings
filename: fparith.bib
label: Gupta:2015:DLL
author: Suyog Gupta and Ankur Agrawal and Kailash Gopalakrishnan and Pritish Narayanan
editor: Francis Bach and David Blei
title: Deep learning with limited numerical precision
publisher: JMLR.org
pages: 1737--1746
year: 2015
bibdate: Fri Sep 22 17:39:05 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.09.22 17:39:05 ???
acknowledgement: ack-nhfb
rawauthor: Suyog Gupta and Ankur Agrawal and Kailash Gopalakrishnan and Pritish Narayanan
raweditor: Francis Bach and David Blei
rawtitle: Deep learning with limited numerical precision
booktitle:
ICML'15: Proceedings of the 32nd International Conference on International Conference on Machine Learning: Lille, France, July 6--11, 2015

rawbooktitle:
{ICML'15: Proceedings of the 32nd International Conference on International Conference on Machine Learning: Lille, France, July 6--11, 2015}

entry:
@InProceedings{Gupta:2015:DLL,
  author =       "Suyog Gupta and Ankur Agrawal and Kailash
                 Gopalakrishnan and Pritish Narayanan",
  editor =       "Francis Bach and David Blei",
  booktitle =    "{ICML'15: Proceedings of the 32nd International
                 Conference on International Conference on Machine
                 Learning: Lille, France, July 6--11, 2015}",
  title =        "Deep learning with limited numerical precision",
  publisher =    "JMLR.org",
  address =      "????",
  pages =        "1737--1746",
  year =         "2015",
  bibdate =      "Fri Sep 22 17:39:05 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: fparith.bib
label: Gustafson:2015:EEU
author: John L. Gustafson
title: The End of Error: Unum Computing
publisher: pub-CHAPMAN-HALL
address: pub-CHAPMAN-HALL:adr
pages: xx + 416
year: 2015
ISBN: 1-4822-3986-8, 1-4822-3987-6 OR 1482239868, 1482239876
ISBN13: 978-1-4822-3986-7, 978-1-4822-3987-4 OR 9781482239867, 9781482239874
LCCN: QA275 .G928 2015
bibdate: Thu Jan 18 06:44:33 2018
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2018.01.18 06:44:33 ???
acknowledgement: ack-nhfb
rawauthor: John L. Gustafson
rawtitle: The End of Error: Unum Computing
abstract:
Written by one of the foremost experts in high-performance computing and the inventor of Gustafson's Law, The End of Error: Unum Computing explains a new approach to computer arithmetic: the universal number (unum). The unum encompasses all IEEE floating-point formats as well as fixed-point and exact integer arithmetic. This new number type obtains more accurate answers than floating-point arithmetic yet uses fewer bits in many cases, saving memory, bandwidth, energy, and power. Richly illustrated in color, this groundbreaking book is accessible to anyone who uses computers for technical calculations.

rawabstract:
Written by one of the foremost experts in high-performance computing and the inventor of Gustafson's Law, The End of Error: Unum Computing explains a new approach to computer arithmetic: the universal number (unum). The unum encompasses all IEEE floating-point formats as well as fixed-point and exact integer arithmetic. This new number type obtains more accurate answers than floating-point arithmetic yet uses fewer bits in many cases, saving memory, bandwidth, energy, and power. Richly illustrated in color, this groundbreaking book is accessible to anyone who uses computers for technical calculations.

entry:
@Book{Gustafson:2015:EEU,
  author =       "John L. Gustafson",
  title =        "The End of Error: Unum Computing",
  publisher =    pub-CHAPMAN-HALL,
  address =      pub-CHAPMAN-HALL:adr,
  pages =        "xx + 416",
  year =         "2015",
  ISBN =         "1-4822-3986-8, 1-4822-3987-6",
  ISBN-13 =      "978-1-4822-3986-7, 978-1-4822-3987-4",
  LCCN =         "QA275 .G928 2015",
  bibdate =      "Thu Jan 18 06:44:33 2018",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Written by one of the foremost experts in
                 high-performance computing and the inventor of
                 Gustafson's Law, The End of Error: Unum Computing
                 explains a new approach to computer arithmetic: the
                 universal number (unum). The unum encompasses all IEEE
                 floating-point formats as well as fixed-point and exact
                 integer arithmetic. This new number type obtains more
                 accurate answers than floating-point arithmetic yet
                 uses fewer bits in many cases, saving memory,
                 bandwidth, energy, and power. Richly illustrated in
                 color, this groundbreaking book is accessible to anyone
                 who uses computers for technical calculations.",
  acknowledgement = ack-nhfb,
  xxabstract =   "In this groundbreaking book, the author --- a
                 world-renowned expert in high-performance computing
                 (HPC) and inventor of Gustafson's Law --- presents his
                 new approach to computer arithmetic: the unum. The
                 universal number, or unum format, encompasses all IEEE
                 floating-point formats as well as fixed-point and exact
                 integer arithmetic. This approach obtains more accurate
                 answers than floating-point arithmetic yet uses fewer
                 bits in many cases, saving memory, bandwidth, energy,
                 and power. Accessible to researchers in HPC and related
                 areas, the book provides a Mathematica\TM{} notebook
                 for the code examples.",
  xxtitle =      "A New Number Format: The Unum",
}


bibtype: inproceedings
filename: fparith.bib
label: Gustafson:2015:KTE
author: John Gustafson
title: Keynote Talk: The End of Numerical Error
crossref: Muller:2015:ISC
pages: 74--74
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.22
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
note: Abstract only.
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: John Gustafson
rawnote: Abstract only.
rawtitle: Keynote Talk: The End of Numerical Error
entry:
@InProceedings{Gustafson:2015:KTE,
  author =       "John Gustafson",
  title =        "Keynote Talk: The End of Numerical Error",
  crossref =     "Muller:2015:ISC",
  pages =        "74--74",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.22",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  note =         "Abstract only.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: incollection
filename: fparith.bib
label: Hamming:2015:DN
author: R. W. Hamming
title: On the distribution of numbers
crossref: Swartzlander:2015:CAa
pages: 321--337
year: 2015
DOI: https://doi.org/10.1142/9789814651578
bibdate: Tue Aug 7 07:53:59 2018
bibtimestamp: 2018.08.07 07:53:59 ???
acknowledgement: ack-nhfb
authordates: Richard Wesley Hamming (1915--1998)
rawauthor: R. W. Hamming
rawtitle: On the distribution of numbers
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/h/hamming-richard-w.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

abstract:
This paper examines the distribution of the mantissas of floating-point numbers and shows how the arithmetic operations of a computer transform various distributions toward the limiting distribution $$ r(x) = 1 / (x ln b) qquad (1 / b leq x leq 1) $$ (where $b$ is the base of the number system). The paper also gives a number of applications to hardware, software, and general computing which show that this distribution is not merely an amusing curiosity. A brief examination of the distribution of exponents is included.

keywords:
Digital arithmetic; Floating point numbers; Limiting distributions; Number system, Application programs; Numbering systems, Arithmetic operations

rawabstract:
This paper examines the distribution of the mantissas of floating-point numbers and shows how the arithmetic operations of a computer transform various distributions toward the limiting distribution $$ r(x) = 1 / (x \ln b) \qquad (1 / b \leq x \leq 1) $$ (where $b$ is the base of the number system). The paper also gives a number of applications to hardware, software, and general computing which show that this distribution is not merely an amusing curiosity. A brief examination of the distribution of exponents is included.

entry:
@InCollection{Hamming:2015:DN,
  author =       "R. W. Hamming",
  title =        "On the distribution of numbers",
  crossref =     "Swartzlander:2015:CAa",
  pages =        "321--337",
  year =         "2015",
  DOI =          "https://doi.org/10.1142/9789814651578",
  bibdate =      "Tue Aug 7 07:53:59 2018",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/h/hamming-richard-w.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "This paper examines the distribution of the mantissas
                 of floating-point numbers and shows how the arithmetic
                 operations of a computer transform various
                 distributions toward the limiting distribution $$ r(x)
                 = 1 / (x \ln b) \qquad (1 / b \leq x \leq 1) $$ (where
                 $b$ is the base of the number system). The paper also
                 gives a number of applications to hardware, software,
                 and general computing which show that this distribution
                 is not merely an amusing curiosity. A brief examination
                 of the distribution of exponents is included.",
  acknowledgement = ack-nhfb,
  author-dates = "Richard Wesley Hamming (1915--1998)",
  keywords =     "Digital arithmetic; Floating point numbers; Limiting
                 distributions; Number system, Application programs;
                 Numbering systems, Arithmetic operations",
}


bibtype: inproceedings
filename: fparith.bib
label: Hart:2015:EDC
author: William Bruce Hart
title: Efficient Divide-and-Conquer Multiprecision Integer Division
crossref: Muller:2015:ISC
pages: 90--95
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.19
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22; division; multiple-precision arithmetic
acknowledgement: ack-nhfb
rawauthor: William Bruce Hart
rawtitle: Efficient Divide-and-Conquer Multiprecision Integer Division
entry:
@InProceedings{Hart:2015:EDC,
  author =       "William Bruce Hart",
  title =        "Efficient Divide-and-Conquer Multiprecision Integer
                 Division",
  crossref =     "Muller:2015:ISC",
  pages =        "90--95",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.19",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22; division; multiple-precision arithmetic",
}


bibtype: misc
filename: fparith.bib
label: Higham:2015:MCT
author: Nicholas J. Higham
title: Matrix Computation Toolbox
pages: 90--95
year: 2015
bibdate: Sat Apr 01 08:14:19 2017
bibtimestamp: 2017.04.01 08:14:19 ???
URL: http://www.ma.man.ac.uk/~higham/mctoolbox
keywords: floating-point arithmetic; MATLAB
acknowledgement: ack-nhfb
howpublished: Web site.
rawauthor: Nicholas J. Higham
rawtitle: Matrix Computation Toolbox
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/h/higham-nicholas-john.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib

abstract:
The Matrix Computation Toolbox is a collection of MATLAB M-files containing functions for constructing test matrices, computing matrix factorizations, visualizing matrices, and carrying out direct search optimization. Various other miscellaneous functions are also included. This toolbox supersedes the author's earlier Test Matrix Toolbox (final release 1995).par The toolbox was developed in conjunction with the book Accuracy and Stability of Numerical Algorithms (SIAM, Second edition, August 2002, xxx+680 pp.). That book is the primary documentation for the toolbox: it describes much of the underlying mathematics and many of the algorithms and matrices (it also describes many of the matrices provided by MATLAB's gallery function).

rawabstract:
The Matrix Computation Toolbox is a collection of MATLAB M-files containing functions for constructing test matrices, computing matrix factorizations, visualizing matrices, and carrying out direct search optimization. Various other miscellaneous functions are also included. This toolbox supersedes the author's earlier Test Matrix Toolbox (final release 1995).\par The toolbox was developed in conjunction with the book Accuracy and Stability of Numerical Algorithms (SIAM, Second edition, August 2002, xxx+680 pp.). That book is the primary documentation for the toolbox: it describes much of the underlying mathematics and many of the algorithms and matrices (it also describes many of the matrices provided by MATLAB's gallery function).

entry:
@Misc{Higham:2015:MCT,
  author =       "Nicholas J. Higham",
  title =        "Matrix Computation Toolbox",
  howpublished = "Web site.",
  year =         "2015",
  bibdate =      "Sat Apr 01 08:14:19 2017",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/h/higham-nicholas-john.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib",
  URL =          "http://www.ma.man.ac.uk/~higham/mctoolbox",
  abstract =     "The Matrix Computation Toolbox is a collection of
                 MATLAB M-files containing functions for constructing
                 test matrices, computing matrix factorizations,
                 visualizing matrices, and carrying out direct search
                 optimization. Various other miscellaneous functions are
                 also included. This toolbox supersedes the author's
                 earlier Test Matrix Toolbox (final release
                 1995).\par

                 The toolbox was developed in conjunction with the book
                 Accuracy and Stability of Numerical Algorithms (SIAM,
                 Second edition, August 2002, xxx+680 pp.). That book is
                 the primary documentation for the toolbox: it describes
                 much of the underlying mathematics and many of the
                 algorithms and matrices (it also describes many of the
                 matrices provided by MATLAB's gallery function).",
  acknowledgement = ack-nhfb,
  keywords =     "floating-point arithmetic; MATLAB",
}


bibtype: article
filename: fparith.bib
label: Holzmann:2015:B
author: Gerard J. Holzmann
title: Out of Bounds
journal: j-IEEE-SOFTWARE
volume: 32
number: 6
pages: 24--26
month: nov
monthnumber: 11
year: 2015
CODEN: IESOEG
DOI: https://doi.org/10.1109/MS.2015.147
ISSN: 0740-7459 (print), 1937-4194 (electronic) OR 07407459 (print), 19374194 (electronic)
ISSNL: 0740-7459 OR 07407459
bibdate: Tue Nov 17 07:21:14 MST 2015
bibtimestamp: 2015.11.17 07:21:14 MST
URL: http://csdl.computer.org/csdl/mags/so/2015/06/mso2015060024.html
fjournal: IEEE Software
acknowledgement: ack-nhfb
journalURL: http://www.computer.org/portal/web/csdl/magazines/software
journalabr: IEEE Software
rawauthor: Gerard J. Holzmann
rawtitle: Out of Bounds
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeesoft.bib

remark:
This article discusses three cases of serious impact of undetected integer overflow: (a) the loss of control of the Deep Impact mission space probe to visit an asteroid; (b) the Boeing 787 Dreamliner (composite materials) generator control unit (GCU) that must now be rebooted at least every 248 days to avoid possibly causing a plane crash; (c) the Planetary Society's LightSail project to test a solar sail in space. It also discusses a failure of NASA's Mars lander, the Curiosity rover, due to storage overflow. Reprinted in booktitle Computing Edge, January 2016, pp. 50--52.

entry:
@Article{Holzmann:2015:B,
  author =       "Gerard J. Holzmann",
  title =        "Out of Bounds",
  journal =      j-IEEE-SOFTWARE,
  volume =       "32",
  number =       "6",
  pages =        "24--26",
  month =        nov # "\slash " # dec,
  year =         "2015",
  CODEN =        "IESOEG",
  DOI =          "https://doi.org/10.1109/MS.2015.147",
  ISSN =         "0740-7459 (print), 1937-4194 (electronic)",
  ISSN-L =       "0740-7459",
  bibdate =      "Tue Nov 17 07:21:14 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeesoft.bib",
  URL =          "http://csdl.computer.org/csdl/mags/so/2015/06/mso2015060024.html",
  abstract-URL = "http://csdl.computer.org/csdl/mags/so/2015/06/mso2015060024-abs.html",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Software",
  journal-URL =  "http://www.computer.org/portal/web/csdl/magazines/software",
  journalabr =   "IEEE Software",
  remark =       "This article discusses three cases of serious impact
                 of undetected integer overflow: (a) the loss of control
                 of the Deep Impact mission space probe to visit an
                 asteroid; (b) the Boeing 787 Dreamliner (composite
                 materials) generator control unit (GCU) that must now
                 be rebooted at least every 248 days to avoid possibly
                 causing a plane crash; (c) the Planetary Society's
                 LightSail project to test a solar sail in space. It
                 also discusses a failure of NASA's Mars lander, the
                 Curiosity rover, due to storage overflow. Reprinted in
                 \booktitle{Computing Edge}, January 2016, pp. 50--52.",
}


bibtype: article
filename: fparith.bib
label: Hsiao:2015:TSR
author: Shen-Fu Hsiao and Po-Han Wu and Chia-Sheng Wen and Pramod Kumar Meher
journal: j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS
volume: 62
number: 5
pages: 466--470
month: may
monthnumber: 5
year: 2015
DOI: https://doi.org/10.1109/TCSII.2014.2386232
ISSN: 1549-7747 (print), 1558-3791 (electronic) OR 15497747 (print), 15583791 (electronic)
ISSNL: 1549-7747 OR 15497747
bibdate: Sat Feb 08 10:23:05 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.02.08 10:23:05 ???
URL: https://ieeexplore.ieee.org/abstract/document/6998028
fjournal: IEEE Transactions on Circuits and Systems II: Express Briefs
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/issues?punumber=8920
rawauthor: Shen-Fu Hsiao and Po-Han Wu and Chia-Sheng Wen and Pramod Kumar Meher
title:
Table Size Reduction Methods for Faithfully Rounded Lookup-Table-Based Multiplierless Function Evaluation

keywords:
Accuracy; Adders; adders; Approximation methods; Bipartite; Computer Arithmetic; computer arithmetic; Function Evaluation; function evaluation; graph theory; Indexes; lookup-table-based multiplierless function evaluation; multioperand adder; Multipartite; multipartite; multipartite method; performance evaluation; precision width; Silicon compounds; table decomposition; table lookup; Table lookup; table size; table size reduction method; Table-Based Design; table-based design; table-lookup-and-addition methods

rawtitle:
Table Size Reduction Methods for Faithfully Rounded Lookup-Table-Based Multiplierless Function Evaluation

entry:
@Article{Hsiao:2015:TSR,
  author =       "Shen-Fu Hsiao and Po-Han Wu and Chia-Sheng Wen and
                 Pramod Kumar Meher",
  title =        "Table Size Reduction Methods for Faithfully Rounded
                 Lookup-Table-Based Multiplierless Function Evaluation",
  journal =      j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS,
  volume =       "62",
  number =       "5",
  pages =        "466--470",
  month =        may,
  year =         "2015",
  DOI =          "https://doi.org/10.1109/TCSII.2014.2386232",
  ISSN =         "1549-7747 (print), 1558-3791 (electronic)",
  ISSN-L =       "1549-7747",
  bibdate =      "Sat Feb 08 10:23:05 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://ieeexplore.ieee.org/abstract/document/6998028",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Circuits and Systems II: Express
                 Briefs",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/issues?punumber=8920",
  keywords =     "Accuracy; Adders; adders; Approximation methods;
                 Bipartite; Computer Arithmetic; computer arithmetic;
                 Function Evaluation; function evaluation; graph theory;
                 Indexes; lookup-table-based multiplierless function
                 evaluation; multioperand adder; Multipartite;
                 multipartite; multipartite method; performance
                 evaluation; precision width; Silicon compounds; table
                 decomposition; table lookup; Table lookup; table size;
                 table size reduction method; Table-Based Design;
                 table-based design; table-lookup-and-addition methods",
}


bibtype: article
filename: fparith.bib
label: Hutter:2015:MMA
author: Michael Hutter and Peter Schwabe
title: Multiprecision multiplication on AVR revisited
journal: j-J-CRYPTO-ENG
volume: 5
number: 3
pages: 201--214
month: sep
monthnumber: 9
year: 2015
DOI: https://doi.org/10.1007/s13389-015-0093-2
ISSN: 2190-8508 (print), 2190-8516 (electronic) OR 21908508 (print), 21908516 (electronic)
ISSNL: 2190-8508 OR 21908508
bibdate: Sat Aug 8 10:42:46 MDT 2015
bibtimestamp: 2015.08.08 10:42:46 MDT
URL: http://link.springer.com/article/10.1007/s13389-015-0093-2
fjournal: Journal of Cryptographic Engineering
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/13389
rawauthor: Michael Hutter and Peter Schwabe
rawtitle: Multiprecision multiplication on {AVR} revisited
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jcryptoeng.bib

keywords:
AVR ATmega 8-bit microcontroller; Karatsuba multiplication; microcontroller; multiprecision multiplication

entry:
@Article{Hutter:2015:MMA,
  author =       "Michael Hutter and Peter Schwabe",
  title =        "Multiprecision multiplication on {AVR} revisited",
  journal =      j-J-CRYPTO-ENG,
  volume =       "5",
  number =       "3",
  pages =        "201--214",
  month =        sep,
  year =         "2015",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s13389-015-0093-2",
  ISSN =         "2190-8508 (print), 2190-8516 (electronic)",
  ISSN-L =       "2190-8508",
  bibdate =      "Sat Aug 8 10:42:46 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcryptoeng.bib",
  URL =          "http://link.springer.com/article/10.1007/s13389-015-0093-2",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Cryptographic Engineering",
  journal-URL =  "http://link.springer.com/journal/13389",
  keywords =     "AVR ATmega 8-bit microcontroller; Karatsuba
                 multiplication; microcontroller; multiprecision
                 multiplication",
}


bibtype: misc
filename: fparith.bib
label: Iakymchuk:2015:EEB
author: Roman Iakymchuk and Sylvain Collange and David Defour and Stef Graillat
title: ExBLAS --- Exact BLAS
pages: 201--214
year: 2015
bibdate: Sat Oct 31 07:25:14 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.10.31 07:25:14 ???
acknowledgement: ack-nhfb
howpublished: Web site.
rawauthor: Roman Iakymchuk and Sylvain Collange and David Defour and Stef Graillat
rawtitle: {ExBLAS} --- Exact {BLAS}
URL:
https://exblas.lip6.fr/; https://www.nist.gov/sites/default/files/documents/itl/ssd/is/NRE-2015-04-iakymchuk.pdf

abstract:
ExBLAS stands for Exact (fast, accurate, and reproducible) Basic Linear Algebra Subprograms.par The increasing power of current computers enables one to solve more and more complex problems. This, therefore, requires to perform a high number of floating-point operations, each one leading to a round-off error. Because of round-off error propagation, some problems must be solved with a longer floating-point format.par As Exascale computing is likely to be reached within a decade, getting accurate results in floating-point arithmetic on such computers will be a challenge. However, another challenge will be the reproducibility of the results --- meaning getting a bitwise identical floating-point result from multiple runs of the same code --- due to non-associativity of floating-point operations and dynamic scheduling on parallel computers.par ExBLAS aims at providing new algorithms and implementations for fundamental linear algebra operations --- like those included in the BLAS library --- that deliver reproducible and accurate results with small or without losses to their performance on modern parallel architectures such as Intel Xeon Phi many-core processors and GPU accelerators. We construct our approach in such a way that it is independent from data partitioning, order of computations, thread scheduling, or reduction tree schemes.

rawabstract:
ExBLAS stands for Exact (fast, accurate, and reproducible) Basic Linear Algebra Subprograms.\par The increasing power of current computers enables one to solve more and more complex problems. This, therefore, requires to perform a high number of floating-point operations, each one leading to a round-off error. Because of round-off error propagation, some problems must be solved with a longer floating-point format.\par As Exascale computing is likely to be reached within a decade, getting accurate results in floating-point arithmetic on such computers will be a challenge. However, another challenge will be the reproducibility of the results --- meaning getting a bitwise identical floating-point result from multiple runs of the same code --- due to non-associativity of floating-point operations and dynamic scheduling on parallel computers.\par ExBLAS aims at providing new algorithms and implementations for fundamental linear algebra operations --- like those included in the BLAS library --- that deliver reproducible and accurate results with small or without losses to their performance on modern parallel architectures such as Intel Xeon Phi many-core processors and GPU accelerators. We construct our approach in such a way that it is independent from data partitioning, order of computations, thread scheduling, or reduction tree schemes.

entry:
@Misc{Iakymchuk:2015:EEB,
  author =       "Roman Iakymchuk and Sylvain Collange and David Defour
                 and Stef Graillat",
  title =        "{ExBLAS} --- Exact {BLAS}",
  howpublished = "Web site.",
  year =         "2015",
  bibdate =      "Sat Oct 31 07:25:14 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://exblas.lip6.fr/;
                 https://www.nist.gov/sites/default/files/documents/itl/ssd/is/NRE-2015-04-iakymchuk.pdf",
  abstract =     "ExBLAS stands for Exact (fast, accurate, and
                 reproducible) Basic Linear Algebra Subprograms.\par

                 The increasing power of current computers enables one
                 to solve more and more complex problems. This,
                 therefore, requires to perform a high number of
                 floating-point operations, each one leading to a
                 round-off error. Because of round-off error
                 propagation, some problems must be solved with a longer
                 floating-point format.\par

                 As Exascale computing is likely to be reached within a
                 decade, getting accurate results in floating-point
                 arithmetic on such computers will be a challenge.
                 However, another challenge will be the reproducibility
                 of the results --- meaning getting a bitwise identical
                 floating-point result from multiple runs of the same
                 code --- due to non-associativity of floating-point
                 operations and dynamic scheduling on parallel
                 computers.\par

                 ExBLAS aims at providing new algorithms and
                 implementations for fundamental linear algebra
                 operations --- like those included in the BLAS library
                 --- that deliver reproducible and accurate results with
                 small or without losses to their performance on modern
                 parallel architectures such as Intel Xeon Phi many-core
                 processors and GPU accelerators. We construct our
                 approach in such a way that it is independent from data
                 partitioning, order of computations, thread scheduling,
                 or reduction tree schemes.",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Iakymchuk:2015:ERA
author: Roman Iakymchuk and Sylvain Collange and David Defour and Stef Graillat
booktitle: NRE: Numerical Reproducibility at Exascale, Austin, TX, USA, November 2015
title: ExBLAS: Reproducible and Accurate BLAS Library
pages: ??--??
year: 2015
bibdate: Mon Feb 10 06:22:16 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.02.10 06:22:16 ???
acknowledgement: ack-nhfb
rawauthor: Roman Iakymchuk and Sylvain Collange and David Defour and Stef Graillat
rawbooktitle: {NRE: Numerical Reproducibility at Exascale, Austin, TX, USA, November 2015}
rawtitle: {ExBLAS}: Reproducible and Accurate {BLAS} Library
entry:
@InProceedings{Iakymchuk:2015:ERA,
  author =       "Roman Iakymchuk and Sylvain Collange and David Defour
                 and Stef Graillat",
  editor =       "????",
  booktitle =    "{NRE: Numerical Reproducibility at Exascale, Austin,
                 TX, USA, November 2015}",
  title =        "{ExBLAS}: Reproducible and Accurate {BLAS} Library",
  publisher =    "????",
  address =      "????",
  pages =        "??--??",
  year =         "2015",
  bibdate =      "Mon Feb 10 06:22:16 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: fparith.bib
label: IEEE:2015:ISI
author: IEEE
title: 1788-2015 --- IEEE Standard for Interval Arithmetic
publisher: pub-IEEE-STD
address: pub-IEEE-STD:adr
pages: xiv + 79
day: 30
month: jun
monthnumber: 6
year: 2015
DOI: https://doi.org/10.1109/IEEESTD.2015.7140721
ISBN: 0-7381-9721-1 (PDF), 0-7381-9720-3 (electronic) OR 0738197211 (PDF), 0738197203 (electronic)
bibdate: Mon Jan 16 18:28:35 2017
bibtimestamp: 2017.01.16 18:28:35 ???
note: Approved 11 June 2015 by IEEE-SA Standards Board.
URL: http://ieeexplore.ieee.org/servlet/opac?punumber=7140719
keywords: arithmetic; computing; decoration; enclosure; hull; IEEE 1788; interval; operation; verified
acknowledgement: ack-nhfb
rawauthor: {IEEE}
rawnote: Approved 11 June 2015 by IEEE-SA Standards Board.
rawtitle: 1788-2015 --- {IEEE} Standard for Interval Arithmetic
ISBN13:
978-0-7381-9721-0 (PDF), 978-0-7381-9720-3 (electronic) OR 9780738197210 (PDF), 9780738197203 (electronic)

bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeestd.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib

abstract:
Abstract: This standard species basic interval arithmetic (IA) operations selecting and following one of the commonly used mathematical interval models. This standard supports the IEEE 755 floating-point formats of practical use in interval computations. Exception conditions are defined, and standard handling of these conditions is specified. Consistency with the interval model is tempered with practical considerations based on input from representatives of vendors, developers and maintainers of existing systems. The standard provides a layer between the hardware and the programming language levels. It does not mandate that any operations be implemented in hardware. It does not define any realization of the basic operations as functions in a programming language.

rawabstract:
Abstract: This standard species basic interval arithmetic (IA) operations selecting and following one of the commonly used mathematical interval models. This standard supports the IEEE 755 floating-point formats of practical use in interval computations. Exception conditions are defined, and standard handling of these conditions is specified. Consistency with the interval model is tempered with practical considerations based on input from representatives of vendors, developers and maintainers of existing systems. The standard provides a layer between the hardware and the programming language levels. It does not mandate that any operations be implemented in hardware. It does not define any realization of the basic operations as functions in a programming language.

entry:
@Book{IEEE:2015:ISI,
  author =       "{IEEE}",
  title =        "1788-2015 --- {IEEE} Standard for Interval
                 Arithmetic",
  publisher =    pub-IEEE-STD,
  address =      pub-IEEE-STD:adr,
  pages =        "xiv + 79",
  day =          "30",
  month =        jun,
  year =         "2015",
  DOI =          "https://doi.org/10.1109/IEEESTD.2015.7140721",
  ISBN =         "0-7381-9721-1 (PDF), 0-7381-9720-3 (electronic)",
  ISBN-13 =      "978-0-7381-9721-0 (PDF), 978-0-7381-9720-3
                 (electronic)",
  LCCN =         "????",
  bibdate =      "Mon Jan 16 18:28:35 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeestd.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  note =         "Approved 11 June 2015 by IEEE-SA Standards Board.",
  URL =          "http://ieeexplore.ieee.org/servlet/opac?punumber=7140719",
  abstract =     "Abstract: This standard species basic interval
                 arithmetic (IA) operations selecting and following one
                 of the commonly used mathematical interval models. This
                 standard supports the IEEE 755 floating-point formats
                 of practical use in interval computations. Exception
                 conditions are defined, and standard handling of these
                 conditions is specified. Consistency with the interval
                 model is tempered with practical considerations based
                 on input from representatives of vendors, developers
                 and maintainers of existing systems. The standard
                 provides a layer between the hardware and the
                 programming language levels. It does not mandate that
                 any operations be implemented in hardware. It does not
                 define any realization of the basic operations as
                 functions in a programming language.",
  acknowledgement = ack-nhfb,
  keywords =     "arithmetic; computing; decoration; enclosure; hull;
                 IEEE 1788; interval; operation; verified",
}


bibtype: article
filename: fparith.bib
label: Jacobsen:2015:PFP
author: Charles Jacobsen and Alexey Solovyev and Ganesh Gopalakrishnan
title: A Parameterized Floating-Point Formalizaton in HOL Light
journal: j-ELECT-NOTES-THEOR-COMP-SCI
volume: 317
pages: 1--6
day: 18
month: nov
monthnumber: 11
year: 2015
DOI: https://doi.org/10.1016/j.entcs.2015.10.010
ISSN: 1571-0661 OR 15710661
ISSNL: 1571-0661 OR 15710661
bibdate: Fri Apr 17 18:06:24 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.04.17 18:06:24 ???
fjournal: Electronic Notes in Theoretical Computer Science
remark: Not yet listed at journal Web site (up to volume 311 5-Jan-2015).
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/15710661
rawauthor: Charles Jacobsen and Alexey Solovyev and Ganesh Gopalakrishnan
rawtitle: A Parameterized Floating-Point Formalizaton in {HOL Light}
note:
Also presented at NSV 2015: 8th International Workshop on Numerical Software Verification 2015, Seattle, WA, USA.

URL:
http://formalverification.cs.utah.edu/papers/nsv15-fp-hol-light.pdf; http://nsv2015.informatik.uni-freiburg.de/

entry:
@Article{Jacobsen:2015:PFP,
  author =       "Charles Jacobsen and Alexey Solovyev and Ganesh
                 Gopalakrishnan",
  title =        "A Parameterized Floating-Point Formalizaton in {HOL
                 Light}",
  journal =      j-ELECT-NOTES-THEOR-COMP-SCI,
  volume =       "317",
  number =       "??",
  pages =        "1--6",
  day =          "18",
  month =        nov,
  year =         "2015",
  DOI =          "https://doi.org/10.1016/j.entcs.2015.10.010",
  ISSN =         "1571-0661",
  ISSN-L =       "1571-0661",
  bibdate =      "Fri Apr 17 18:06:24 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  note =         "Also presented at NSV 2015: 8th International Workshop
                 on Numerical Software Verification 2015, Seattle, WA,
                 USA.",
  URL =          "http://formalverification.cs.utah.edu/papers/nsv15-fp-hol-light.pdf;
                 http://nsv2015.informatik.uni-freiburg.de/",
  acknowledgement = ack-nhfb,
  fjournal =     "Electronic Notes in Theoretical Computer Science",
  journal-URL =  "http://www.sciencedirect.com/science/journal/15710661",
  remark =       "Not yet listed at journal Web site (up to volume 311
                 5-Jan-2015).",
  xxjournal-url = "http://www.elsevier.nl/locate/entcs",
}


bibtype: techreport
filename: fparith.bib
label: Johansson:2015:ADR
author: Fredrik Johansson
title: Arb Documentation Release 2.6.0
type: Report
address: Softwarepark 23, Austria 4113
pages: 1--6
day: 19
month: apr
monthnumber: 4
year: 2015
bibdate: Wed Apr 22 09:04:21 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.04.22 09:04:21 ???
URL: http://fredrikj.net/arb/
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: {Arb} Documentation Release 2.6.0
abstract:
Arb is a C library for arbitrary-precision floating-point ball arithmetic, developed by Fredrik Johansson (tt fredrik.johansson@gmail.com). It supports real and complex numbers, polynomials, power series, matrices, and evaluation of many transcendental functions. All is done with automatic, rigorous error bounds.

remark:
From the Feature Overview: ``Ball arithmetic, also known as mid-rad interval arithmetic, is an extension of floating-point arithmetic in which an error bound is attached to each variable. This allows doing rigorous computations over the real numbers, while avoiding the overhead of traditional (inf-sup) interval arithmetic at high precision, and eliminating much of the need for time-consuming and bug-prone manual error analysis associated with standard floating-point arithmetic.''

rawabstract:
Arb is a C library for arbitrary-precision floating-point ball arithmetic, developed by Fredrik Johansson ({\tt fredrik.johansson@gmail.com}). It supports real and complex numbers, polynomials, power series, matrices, and evaluation of many transcendental functions. All is done with automatic, rigorous error bounds.

entry:
@TechReport{Johansson:2015:ADR,
  author =       "Fredrik Johansson",
  title =        "{Arb} Documentation Release 2.6.0",
  type =         "Report",
  institution =  "????",
  address =      "Softwarepark 23, Austria 4113",
  day =          "19",
  month =        apr,
  year =         "2015",
  bibdate =      "Wed Apr 22 09:04:21 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://fredrikj.net/arb/",
  abstract =     "Arb is a C library for arbitrary-precision
                 floating-point ball arithmetic, developed by Fredrik
                 Johansson ({\tt fredrik.johansson@gmail.com}). It
                 supports real and complex numbers, polynomials, power
                 series, matrices, and evaluation of many transcendental
                 functions. All is done with automatic, rigorous error
                 bounds.",
  acknowledgement = ack-nhfb,
  remark =       "From the Feature Overview: ``Ball arithmetic, also
                 known as mid-rad interval arithmetic, is an extension
                 of floating-point arithmetic in which an error bound is
                 attached to each variable. This allows doing rigorous
                 computations over the real numbers, while avoiding the
                 overhead of traditional (inf-sup) interval arithmetic
                 at high precision, and eliminating much of the need for
                 time-consuming and bug-prone manual error analysis
                 associated with standard floating-point arithmetic.''",
}


bibtype: inproceedings
filename: fparith.bib
label: Johansson:2015:EIE
author: Fredrik Johansson
title: Efficient Implementation of Elementary Functions in the Medium-Precision Range
crossref: Muller:2015:ISC
pages: 83--89
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.16
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: Efficient Implementation of Elementary Functions in the Medium-Precision Range
entry:
@InProceedings{Johansson:2015:EIE,
  author =       "Fredrik Johansson",
  title =        "Efficient Implementation of Elementary Functions in
                 the Medium-Precision Range",
  crossref =     "Muller:2015:ISC",
  pages =        "83--89",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.16",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: fparith.bib
label: Kamm:2015:SFP
author: Liina Kamm and Jan Willemson
title: Secure floating point arithmetic and private satellite collision analysis
journal: j-INT-J-INFO-SEC
volume: 14
number: 6
pages: 531--548
month: nov
monthnumber: 11
year: 2015
DOI: https://doi.org/10.1007/s10207-014-0271-8
ISSN: 1615-5262 (print), 1615-5270 (electronic) OR 16155262 (print), 16155270 (electronic)
ISSNL: 1615-5262 OR 16155262
bibdate: Tue Jan 23 16:01:44 MST 2018
bibtimestamp: 2018.01.23 16:01:44 MST
fjournal: International Journal of Information Security
acknowledgement: ack-nhfb
journalURL: https://link.springer.com/journal/10207
rawauthor: Liina Kamm and Jan Willemson
rawtitle: Secure floating point arithmetic and private satellite collision analysis
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/intjinfosec.bib

URL:
http://link.springer.com/article/10.1007/s10207-014-0271-8; http://link.springer.com/content/pdf/10.1007/s10207-014-0271-8.pdf

entry:
@Article{Kamm:2015:SFP,
  author =       "Liina Kamm and Jan Willemson",
  title =        "Secure floating point arithmetic and private satellite
                 collision analysis",
  journal =      j-INT-J-INFO-SEC,
  volume =       "14",
  number =       "6",
  pages =        "531--548",
  month =        nov,
  year =         "2015",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s10207-014-0271-8",
  ISSN =         "1615-5262 (print), 1615-5270 (electronic)",
  ISSN-L =       "1615-5262",
  bibdate =      "Tue Jan 23 16:01:44 MST 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/intjinfosec.bib",
  URL =          "http://link.springer.com/article/10.1007/s10207-014-0271-8;
                 http://link.springer.com/content/pdf/10.1007/s10207-014-0271-8.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "International Journal of Information Security",
  journal-URL =  "https://link.springer.com/journal/10207",
}


bibtype: article
filename: fparith.bib
label: Kornerup:2015:RHR
author: P. Kornerup
title: Reviewing High-Radix Signed-Digit Adders
journal: j-IEEE-TRANS-COMPUT
volume: 64
number: 5
pages: 1502--1505
month: may
monthnumber: 5
year: 2015
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2329678
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Jun 4 19:46:44 MDT 2015
bibtimestamp: 2015.06.04 19:46:44 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: P. Kornerup
rawtitle: Reviewing High-Radix Signed-Digit Adders
entry:
@Article{Kornerup:2015:RHR,
  author =       "P. Kornerup",
  title =        "Reviewing High-Radix Signed-Digit Adders",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "64",
  number =       "5",
  pages =        "1502--1505",
  month =        may,
  year =         "2015",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2329678",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Jun 4 19:46:44 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: techreport
filename: fparith.bib
label: Kulisch:2015:HSA
author: Ulrich Kulisch and Gerd Bohlender
title: High Speed Associative Accumulation of Floating-point Numbers and Floating-point Intervals
type: Report
institution: Institut fur Angewandte und Numerische Mathematik, Karlsruher Institut fur Technologie
address: D-76128 Karlsruhe, Germany
pages: 8
day: 21
month: dec
monthnumber: 12
year: 2015
bibdate: Sat Jan 02 12:26:31 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.01.02 12:26:31 ???
remark: Distributed to stds-1788 mailing list on Fri, 1 Jan 2016 09:22:19 +0100; as yet unpublished.
acknowledgement: ack-nhfb
rawauthor: Ulrich Kulisch and Gerd Bohlender
rawtitle: High Speed Associative Accumulation of Floating-point Numbers and Floating-point Intervals
entry:
@TechReport{Kulisch:2015:HSA,
  author =       "Ulrich Kulisch and Gerd Bohlender",
  title =        "High Speed Associative Accumulation of Floating-point
                 Numbers and Floating-point Intervals",
  type =         "Report",
  institution =  "Institut f{\"u}r Angewandte und Numerische Mathematik,
                 Karlsruher Institut f{\"u}r Technologie",
  address =      "D-76128 Karlsruhe, Germany",
  pages =        "8",
  day =          "21",
  month =        dec,
  year =         "2015",
  bibdate =      "Sat Jan 02 12:26:31 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  remark =       "Distributed to stds-1788 mailing list on Fri, 1 Jan
                 2016 09:22:19 +0100; as yet unpublished.",
}


bibtype: inproceedings
filename: fparith.bib
label: Kumm:2015:ESM
author: Martin Kumm and Shahid Abbas and Peter Zipf
title: An Efficient Softcore Multiplier Architecture for Xilinx FPGAs
crossref: Muller:2015:ISC
pages: 18--25
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.17
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Martin Kumm and Shahid Abbas and Peter Zipf
rawtitle: An Efficient Softcore Multiplier Architecture for {Xilinx FPGAs}
entry:
@InProceedings{Kumm:2015:ESM,
  author =       "Martin Kumm and Shahid Abbas and Peter Zipf",
  title =        "An Efficient Softcore Multiplier Architecture for
                 {Xilinx FPGAs}",
  crossref =     "Muller:2015:ISC",
  pages =        "18--25",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.17",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: phdthesis
filename: fparith.bib
label: Kupriianova:2015:TMF
author: Olga Kupriianova
title: Towards a modern floating-point environment
type: These de doctorat
school: Universite Pierre et Marie Curie --- Paris VI
address: Paris, France
pages: xi + 130
year: 2015
bibdate: Fri Jan 31 14:59:32 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.01.31 14:59:32 ???
URL: https://pdfs.semanticscholar.org/94ca/770b22739bdcf22fb9f131d9d4665dc28031.pdf
acknowledgement: ack-nhfb
rawauthor: Olga Kupriianova
rawtitle: Towards a modern floating-point environment
abstract:
This work investigates two ways of enlarging the current floating-point environment. The first is to support several implementation versions of each mathematical function (elementary such as $ exp $ or $ log $ and special such as $ erf $ or $ Gamma $), the second one is to provide IEEE754 operations that mix the inputs and the output of different radixes. As the number of various implementations for each mathematical function is large, this work is focused on code generation. Our code generator supports the huge variety of functions: it generates parametrized implementations for the user-specified functions. So it may be considered as a black-box function generator. This work contains a novel algorithm for domain splitting and an approach to replace branching on reconstruction by a polynomial. This new domain splitting algorithm produces less subdomains and the polynomial degrees on adjacent subdomains do not change much. To produce vectorizable implementations, if-else statements on the reconstruction step have to be avoided. Since the revision of the IEEE754 Standard in 2008 it is possible to mix numbers of different precisions in one operation. However, there is no mechanism that allows users to mix numbers of different radices in one operation. This research starts an examination of mixed-radix arithmetic with the worst cases search for FMA. A novel algorithm to convert a decimal character sequence of arbitrary length to a binary floating-point number is presented. It is independent of currently-set rounding mode and produces correctly-rounded results.

rawabstract:
This work investigates two ways of enlarging the current floating-point environment. The first is to support several implementation versions of each mathematical function (elementary such as $ \exp $ or $ \log $ and special such as $ \erf $ or $ \Gamma $), the second one is to provide IEEE754 operations that mix the inputs and the output of different radixes. As the number of various implementations for each mathematical function is large, this work is focused on code generation. Our code generator supports the huge variety of functions: it generates parametrized implementations for the user-specified functions. So it may be considered as a black-box function generator. This work contains a novel algorithm for domain splitting and an approach to replace branching on reconstruction by a polynomial. This new domain splitting algorithm produces less subdomains and the polynomial degrees on adjacent subdomains do not change much. To produce vectorizable implementations, if-else statements on the reconstruction step have to be avoided. Since the revision of the IEEE754 Standard in 2008 it is possible to mix numbers of different precisions in one operation. However, there is no mechanism that allows users to mix numbers of different radices in one operation. This research starts an examination of mixed-radix arithmetic with the worst cases search for FMA. A novel algorithm to convert a decimal character sequence of arbitrary length to a binary floating-point number is presented. It is independent of currently-set rounding mode and produces correctly-rounded results.

entry:
@PhdThesis{Kupriianova:2015:TMF,
  author =       "Olga Kupriianova",
  title =        "Towards a modern floating-point environment",
  type =         "Th{\`e}se de doctorat",
  school =       "Universit{\'e} Pierre et Marie Curie --- Paris VI",
  address =      "Paris, France",
  pages =        "xi + 130",
  year =         "2015",
  bibdate =      "Fri Jan 31 14:59:32 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://pdfs.semanticscholar.org/94ca/770b22739bdcf22fb9f131d9d4665dc28031.pdf",
  abstract =     "This work investigates two ways of enlarging the
                 current floating-point environment. The first is to
                 support several implementation versions of each
                 mathematical function (elementary such as $ \exp $ or $
                 \log $ and special such as $ \erf $ or $ \Gamma $), the
                 second one is to provide IEEE754 operations that mix
                 the inputs and the output of different radixes. As the
                 number of various implementations for each mathematical
                 function is large, this work is focused on code
                 generation. Our code generator supports the huge
                 variety of functions: it generates parametrized
                 implementations for the user-specified functions. So it
                 may be considered as a black-box function generator.
                 This work contains a novel algorithm for domain
                 splitting and an approach to replace branching on
                 reconstruction by a polynomial. This new domain
                 splitting algorithm produces less subdomains and the
                 polynomial degrees on adjacent subdomains do not change
                 much. To produce vectorizable implementations, if-else
                 statements on the reconstruction step have to be
                 avoided. Since the revision of the IEEE754 Standard in
                 2008 it is possible to mix numbers of different
                 precisions in one operation. However, there is no
                 mechanism that allows users to mix numbers of different
                 radices in one operation. This research starts an
                 examination of mixed-radix arithmetic with the worst
                 cases search for FMA. A novel algorithm to convert a
                 decimal character sequence of arbitrary length to a
                 binary floating-point number is presented. It is
                 independent of currently-set rounding mode and produces
                 correctly-rounded results.",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Kurka:2015:ERA
author: Petr Kurka
title: The Exact Real Arithmetical Algorithm in Binary Continued Fractions
crossref: Muller:2015:ISC
pages: 168--175
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.20
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Petr K{\r{u}}rka
rawtitle: The Exact Real Arithmetical Algorithm in Binary Continued Fractions
entry:
@InProceedings{Kurka:2015:ERA,
  author =       "Petr K{\r{u}}rka",
  title =        "The Exact Real Arithmetical Algorithm in Binary
                 Continued Fractions",
  crossref =     "Muller:2015:ISC",
  pages =        "168--175",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.20",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: inproceedings
filename: fparith.bib
label: Langhammer:2015:DIE
author: Martin Langhammer and Bogdan Pasca
title: Design and Implementation of an Embedded FPGA Floating Point DSP Block
crossref: Muller:2015:ISC
pages: 26--33
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.18
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
acknowledgement: ack-nhfb
rawauthor: Martin Langhammer and Bogdan Pasca
rawtitle: Design and Implementation of an Embedded {FPGA} Floating Point {DSP} Block
entry:
@InProceedings{Langhammer:2015:DIE,
  author =       "Martin Langhammer and Bogdan Pasca",
  title =        "Design and Implementation of an Embedded {FPGA}
                 Floating Point {DSP} Block",
  crossref =     "Muller:2015:ISC",
  pages =        "26--33",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.18",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Langroudi:2015:MPP
author: Seyed Hamed Fatemi Langroudi and Ghassem Jaberipur
title: Modulo-$ (2^n 2^q 1) $ Parallel Prefix Addition via Excess-Modulo Encoding of Residues
crossref: Muller:2015:ISC
pages: 121--128
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.9
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Seyed Hamed Fatemi Langroudi and Ghassem Jaberipur
rawtitle: Modulo-$ (2^n 2^q 1) $ Parallel Prefix Addition via Excess-Modulo Encoding of Residues
entry:
@InProceedings{Langroudi:2015:MPP,
  author =       "Seyed Hamed Fatemi Langroudi and Ghassem Jaberipur",
  title =        "Modulo-$ (2^n 2^q 1) $ Parallel Prefix Addition via
                 Excess-Modulo Encoding of Residues",
  crossref =     "Muller:2015:ISC",
  pages =        "121--128",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.9",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: inproceedings
filename: fparith.bib
label: Laskar:2015:KTN
author: Jacques Laskar
title: Keynote Talk: Numerical Challenges in Long Term Integrations of the Solar System
crossref: Muller:2015:ISC
pages: 104--104
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.35
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
note: Abstract only.
keywords: ARITH-22; multiple-precision arithmetic
acknowledgement: ack-nhfb
rawauthor: Jacques Laskar
rawnote: Abstract only.
rawtitle: Keynote Talk: Numerical Challenges in Long Term Integrations of the Solar System
entry:
@InProceedings{Laskar:2015:KTN,
  author =       "Jacques Laskar",
  title =        "Keynote Talk: Numerical Challenges in Long Term
                 Integrations of the Solar System",
  crossref =     "Muller:2015:ISC",
  pages =        "104--104",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.35",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  note =         "Abstract only.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22; multiple-precision arithmetic",
}


bibtype: inproceedings
filename: fparith.bib
label: Lauter:2015:SAF
author: Christoph Lauter and Marc Mezzarobba
title: Semi-Automatic Floating-Point Implementation of Special Functions
crossref: Muller:2015:ISC
pages: 58--65
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.12
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Christoph Lauter and Marc Mezzarobba
rawtitle: Semi-Automatic Floating-Point Implementation of Special Functions
entry:
@InProceedings{Lauter:2015:SAF,
  author =       "Christoph Lauter and Marc Mezzarobba",
  title =        "Semi-Automatic Floating-Point Implementation of
                 Special Functions",
  crossref =     "Muller:2015:ISC",
  pages =        "58--65",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.12",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: fparith.bib
label: Lee:2015:RRA
author: Wen-Chuan Lee and Tao Bao and Yunhui Zheng and Xiangyu Zhang and Keval Vora and Rajiv Gupta
title: RAIVE: runtime assessment of floating-point instability by vectorization
journal: j-SIGPLAN
volume: 50
number: 10
pages: 623--638
month: oct
monthnumber: 10
year: 2015
CODEN: SINODQ
DOI: https://doi.org/10.1145/2858965.2814299
ISSN: 0362-1340 (print), 1523-2867 (print), 1558-1160 OR 03621340 (print), 15232867 (print), 15581160
ISSNL: 0362-1340 OR 03621340
bibdate: Tue Feb 16 12:01:43 MST 2016
bibtimestamp: 2016.02.16 12:01:43 MST
fjournal: ACM SIGPLAN Notices
remark: OOPSLA '15 conference proceedings.
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J706
rawauthor: Wen-Chuan Lee and Tao Bao and Yunhui Zheng and Xiangyu Zhang and Keval Vora and Rajiv Gupta
rawtitle: {RAIVE}: runtime assessment of floating-point instability by vectorization
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigplan2010.bib

abstract:
Floating point representation has limited precision and inputs to floating point programs may also have errors. Consequently, during execution, errors are introduced, propagated, and accumulated, leading to unreliable outputs. We call this the instability problem. We propose RAIVE, a technique that identifies output variations of a floating point execution in the presence of instability. RAIVE transforms every floating point value to a vector of multiple values --- the values added to create the vector are obtained by introducing artificial errors that are upper bounds of actual errors. The propagation of artificial errors models the propagation of actual errors. When values in vectors result in discrete execution differences (e.g., following different paths), the execution is forked to capture the resulting output variations. Our evaluation shows that RAIVE can precisely capture output variations. Its overhead (340%) is 2.43 times lower than the state of the art

rawabstract:
Floating point representation has limited precision and inputs to floating point programs may also have errors. Consequently, during execution, errors are introduced, propagated, and accumulated, leading to unreliable outputs. We call this the instability problem. We propose RAIVE, a technique that identifies output variations of a floating point execution in the presence of instability. RAIVE transforms every floating point value to a vector of multiple values --- the values added to create the vector are obtained by introducing artificial errors that are upper bounds of actual errors. The propagation of artificial errors models the propagation of actual errors. When values in vectors result in discrete execution differences (e.g., following different paths), the execution is forked to capture the resulting output variations. Our evaluation shows that RAIVE can precisely capture output variations. Its overhead (340\%) is 2.43 times lower than the state of the art

entry:
@Article{Lee:2015:RRA,
  author =       "Wen-Chuan Lee and Tao Bao and Yunhui Zheng and Xiangyu
                 Zhang and Keval Vora and Rajiv Gupta",
  title =        "{RAIVE}: runtime assessment of floating-point
                 instability by vectorization",
  journal =      j-SIGPLAN,
  volume =       "50",
  number =       "10",
  pages =        "623--638",
  month =        oct,
  year =         "2015",
  CODEN =        "SINODQ",
  DOI =          "https://doi.org/10.1145/2858965.2814299",
  ISSN =         "0362-1340 (print), 1523-2867 (print), 1558-1160
                 (electronic)",
  ISSN-L =       "0362-1340",
  bibdate =      "Tue Feb 16 12:01:43 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigplan2010.bib",
  abstract =     "Floating point representation has limited precision
                 and inputs to floating point programs may also have
                 errors. Consequently, during execution, errors are
                 introduced, propagated, and accumulated, leading to
                 unreliable outputs. We call this the instability
                 problem. We propose RAIVE, a technique that identifies
                 output variations of a floating point execution in the
                 presence of instability. RAIVE transforms every
                 floating point value to a vector of multiple values ---
                 the values added to create the vector are obtained by
                 introducing artificial errors that are upper bounds of
                 actual errors. The propagation of artificial errors
                 models the propagation of actual errors. When values in
                 vectors result in discrete execution differences (e.g.,
                 following different paths), the execution is forked to
                 capture the resulting output variations. Our evaluation
                 shows that RAIVE can precisely capture output
                 variations. Its overhead (340\%) is 2.43 times lower
                 than the state of the art",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGPLAN Notices",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J706",
  remark =       "OOPSLA '15 conference proceedings.",
}


bibtype: article
filename: fparith.bib
label: Liu:2015:IBI
title: Importance of bitwise identical reproducibility in earth system modeling and status report
journal: Geoscientific Model Development Discussions
volume: 8
number: 6
pages: 4375--4400
month: jun
monthnumber: 6
year: 2015
DOI: https://doi.org/10.5194/gmdd-8-4375-2015
ISSN: 1991-959X (print), 1991-9603 (electronic) OR 1991959X (print), 19919603 (electronic)
ISSNL: 1991-959X OR 1991959X
bibdate: Mon Feb 10 17:57:24 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.02.10 17:57:24 ???
URL: https://www.geosci-model-dev-discuss.net/gmd-2015-83/
acknowledgement: ack-nhfb
journalURL: https://www.geosci-model-dev.net/volumes.html
rawtitle: Importance of bitwise identical reproducibility in earth system modeling and status report
author:
L. Liu and S. Peng and C. Zhang and R. Li and B. Wang and C. Sun and Q. Liu and L. Dong and L. Li and Y. Shi and Y. He and W. Zhao and G. Yang

abstract:
Reproducibility is a fundamental principle of scientific research. Bitwise identical reproducibility, i.e., bitwise computational results can be reproduced, guarantees the reproduction of exactly the same results. Here we show the importance of bitwise identical reproducibility to Earth system modeling but the importance has not yet been widely recognized. Modeled mean climate states, variability and trends at different scales may be significantly changed or even lead to opposing results due to a slight change in the original simulation setting during a reproduction. Out of the large body of Earth system modeling publications, few thoroughly describe the whole original simulation setting. As a result, the reproduction of a particular simulation experiment by fellow scientists heavily depends on the interaction with the original authors, which is often inconvenient or even impossible. We anticipate bitwise identical reproducibility to be promoted as a worldwide standard, to guarantee the independent reproduction of simulation results and to further improve model development and scientific research.

rawabstract:
Reproducibility is a fundamental principle of scientific research. Bitwise identical reproducibility, i.e., bitwise computational results can be reproduced, guarantees the reproduction of exactly the same results. Here we show the importance of bitwise identical reproducibility to Earth system modeling but the importance has not yet been widely recognized. Modeled mean climate states, variability and trends at different scales may be significantly changed or even lead to opposing results due to a slight change in the original simulation setting during a reproduction. Out of the large body of Earth system modeling publications, few thoroughly describe the whole original simulation setting. As a result, the reproduction of a particular simulation experiment by fellow scientists heavily depends on the interaction with the original authors, which is often inconvenient or even impossible. We anticipate bitwise identical reproducibility to be promoted as a worldwide standard, to guarantee the independent reproduction of simulation results and to further improve model development and scientific research.

rawauthor:
L. Liu and S. Peng and C. Zhang and R. Li and B. Wang and C. Sun and Q. Liu and L. Dong and L. Li and Y. Shi and Y. He and W. Zhao and G. Yang

entry:
@Article{Liu:2015:IBI,
  author =       "L. Liu and S. Peng and C. Zhang and R. Li and B. Wang
                 and C. Sun and Q. Liu and L. Dong and L. Li and Y. Shi
                 and Y. He and W. Zhao and G. Yang",
  title =        "Importance of bitwise identical reproducibility in
                 earth system modeling and status report",
  journal =      "Geoscientific Model Development Discussions",
  volume =       "8",
  number =       "6",
  pages =        "4375--4400",
  month =        jun,
  year =         "2015",
  DOI =          "https://doi.org/10.5194/gmdd-8-4375-2015",
  ISSN =         "1991-959X (print), 1991-9603 (electronic)",
  ISSN-L =       "1991-959X",
  bibdate =      "Mon Feb 10 17:57:24 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.geosci-model-dev-discuss.net/gmd-2015-83/",
  abstract =     "Reproducibility is a fundamental principle of
                 scientific research. Bitwise identical reproducibility,
                 i.e., bitwise computational results can be reproduced,
                 guarantees the reproduction of exactly the same
                 results. Here we show the importance of bitwise
                 identical reproducibility to Earth system modeling but
                 the importance has not yet been widely recognized.
                 Modeled mean climate states, variability and trends at
                 different scales may be significantly changed or even
                 lead to opposing results due to a slight change in the
                 original simulation setting during a reproduction. Out
                 of the large body of Earth system modeling
                 publications, few thoroughly describe the whole
                 original simulation setting. As a result, the
                 reproduction of a particular simulation experiment by
                 fellow scientists heavily depends on the interaction
                 with the original authors, which is often inconvenient
                 or even impossible. We anticipate bitwise identical
                 reproducibility to be promoted as a worldwide standard,
                 to guarantee the independent reproduction of simulation
                 results and to further improve model development and
                 scientific research.",
  acknowledgement = ack-nhfb,
  journal-URL =  "https://www.geosci-model-dev.net/volumes.html",
}


bibtype: article
filename: fparith.bib
label: Liu:2015:SSS
author: Weifeng Liu and Brian Vinter
title: Speculative segmented sum for sparse matrix-vector multiplication on heterogeneous processors
journal: j-PARALLEL-COMPUTING
volume: 49
pages: 179--193
month: nov
monthnumber: 11
year: 2015
CODEN: PACOEJ
DOI: https://doi.org/10.1016/j.parco.2015.04.004
ISSN: 0167-8191 (print), 1872-7336 (electronic) OR 01678191 (print), 18727336 (electronic)
ISSNL: 0167-8191 OR 01678191
bibdate: Thu Oct 29 17:53:44 MDT 2015
bibtimestamp: 2015.10.29 17:53:44 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0167819115000770
fjournal: Parallel Computing
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01678191/
rawauthor: Weifeng Liu and Brian Vinter
rawtitle: Speculative segmented sum for sparse matrix-vector multiplication on heterogeneous processors
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/parallelcomputing.bib

entry:
@Article{Liu:2015:SSS,
  author =       "Weifeng Liu and Brian Vinter",
  title =        "Speculative segmented sum for sparse matrix-vector
                 multiplication on heterogeneous processors",
  journal =      j-PARALLEL-COMPUTING,
  volume =       "49",
  number =       "??",
  pages =        "179--193",
  month =        nov,
  year =         "2015",
  CODEN =        "PACOEJ",
  DOI =          "https://doi.org/10.1016/j.parco.2015.04.004",
  ISSN =         "0167-8191 (print), 1872-7336 (electronic)",
  ISSN-L =       "0167-8191",
  bibdate =      "Thu Oct 29 17:53:44 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/parallelcomputing.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0167819115000770",
  acknowledgement = ack-nhfb,
  fjournal =     "Parallel Computing",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01678191/",
}


bibtype: article
filename: fparith.bib
label: Lu:2015:REP
author: Xin Lu and Shufang Xu
title: Rounding errors of partial derivatives of simple eigenvalues of the quadratic eigenvalue problem
journal: j-J-COMPUT-APPL-MATH
volume: 282
pages: 268--277
month: jul
monthnumber: 7
year: 2015
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:34:50 MST 2017
bibtimestamp: 2017.02.25 13:34:50 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042715000151
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Xin Lu and Shufang Xu
rawtitle: Rounding errors of partial derivatives of simple eigenvalues of the quadratic eigenvalue problem
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib

entry:
@Article{Lu:2015:REP,
  author =       "Xin Lu and Shufang Xu",
  title =        "Rounding errors of partial derivatives of simple
                 eigenvalues of the quadratic eigenvalue problem",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "282",
  number =       "??",
  pages =        "268--277",
  month =        jul,
  year =         "2015",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:34:50 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042715000151",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: inproceedings
filename: fparith.bib
label: Lutz:2015:OLZ
author: David Raymond Lutz
title: Optimized leading zero anticipators for faster fused multiply-adds
crossref: Matthews:2017:CRF
pages: 741--744
year: 2015
DOI: https://doi.org/10.1109/ACSSC.2017.8335443
bibdate: Mon Feb 10 08:39:02 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.02.10 08:39:02 ???
URL: https://ieeexplore.ieee.org/document/8335443/
acknowledgement: ack-nhfb
rawauthor: David Raymond Lutz
rawtitle: Optimized leading zero anticipators for faster fused multiply-adds
entry:
@InProceedings{Lutz:2015:OLZ,
  author =       "David Raymond Lutz",
  title =        "Optimized leading zero anticipators for faster fused
                 multiply-adds",
  crossref =     "Matthews:2017:CRF",
  pages =        "741--744",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ACSSC.2017.8335443",
  bibdate =      "Mon Feb 10 08:39:02 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://ieeexplore.ieee.org/document/8335443/",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Martin-Dorel:2015:FVC
author: Erik Martin-Dorel and Guillaume Hanrot and Micaela Mayero and Laurent Thery
title: Formally Verified Certificate Checkers for Hardest-to-Round Computation
journal: j-J-AUTOM-REASON
volume: 54
number: 1
pages: 1--29
month: jan
monthnumber: 1
year: 2015
CODEN: JAREEW
DOI: https://doi.org/10.1007/s10817-014-9312-2
ISSN: 0168-7433 (print), 1573-0670 (electronic) OR 01687433 (print), 15730670 (electronic)
ISSNL: 0168-7433 OR 01687433
bibdate: Sat Apr 2 10:51:12 MDT 2016
bibtimestamp: 2016.04.02 10:51:12 MDT
URL: http://link.springer.com/article/10.1007/s10817-014-9312-2
fjournal: Journal of Automated Reasoning
acknowledgement: ack-nhfb
ajournal: J. Autom. Reason.
journalURL: http://link.springer.com/journal/10817
rawauthor: {\'E}rik Martin-Dorel and Guillaume Hanrot and Micaela Mayero and Laurent Th{\'e}ry
rawtitle: Formally Verified Certificate Checkers for Hardest-to-Round Computation
entry:
@Article{Martin-Dorel:2015:FVC,
  author =       "{\'E}rik Martin-Dorel and Guillaume Hanrot and Micaela
                 Mayero and Laurent Th{\'e}ry",
  title =        "Formally Verified Certificate Checkers for
                 Hardest-to-Round Computation",
  journal =      j-J-AUTOM-REASON,
  volume =       "54",
  number =       "1",
  pages =        "1--29",
  month =        jan,
  year =         "2015",
  CODEN =        "JAREEW",
  DOI =          "https://doi.org/10.1007/s10817-014-9312-2",
  ISSN =         "0168-7433 (print), 1573-0670 (electronic)",
  ISSN-L =       "0168-7433",
  bibdate =      "Sat Apr 2 10:51:12 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jautomreason.bib",
  URL =          "http://link.springer.com/article/10.1007/s10817-014-9312-2",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Autom. Reason.",
  fjournal =     "Journal of Automated Reasoning",
  journal-URL =  "http://link.springer.com/journal/10817",
}


bibtype: article
filename: fparith.bib
label: Matula:2015:MDE
author: D. W. Matula and M. T. Panu and J. Y. Zhang
title: Multiplicative Division Employing Independent Factors
journal: j-IEEE-TRANS-COMPUT
volume: 64
number: 7
pages: 2012--2019
year: 2015
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2346206
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sat Aug 1 08:53:41 MDT 2015
bibtimestamp: 2015.08.01 08:53:41 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: D. W. Matula and M. T. Panu and J. Y. Zhang
rawtitle: Multiplicative Division Employing Independent Factors
entry:
@Article{Matula:2015:MDE,
  author =       "D. W. Matula and M. T. Panu and J. Y. Zhang",
  title =        "Multiplicative Division Employing Independent
                 Factors",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "64",
  number =       "7",
  pages =        "2012--2019",
  month =        "????",
  year =         "2015",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2346206",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sat Aug 1 08:53:41 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: McCleeary:2015:LAA
author: Ryan McCleeary and Martin Brain and Aaron Stump
title: A lazy approach to adaptive exact real arithmetic using floating-point operations
journal: j-ACM-COMM-COMP-ALGEBRA
volume: 49
number: 3
pages: 83--86
month: sep
monthnumber: 9
year: 2015
DOI: https://doi.org/10.1145/2850449.2850456
ISSN: 1932-2232 (print), 1932-2240 (electronic) OR 19322232 (print), 19322240 (electronic)
ISSNL: 1932-2232 OR 19322232
bibdate: Wed Dec 2 14:16:56 MST 2015
bibtimestamp: 2015.12.02 14:16:56 MST
fjournal: ACM Communications in Computer Algebra
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
rawauthor: Ryan McCleeary and Martin Brain and Aaron Stump
rawtitle: A lazy approach to adaptive exact real arithmetic using floating-point operations
bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib

abstract:
Arithmetic operations with high degrees of precision are needed for an increasing number of applications. We propose an exact real arithmetic system that achieves adaptive precision using lazy infinite lists of floating-point values.

rawabstract:
Arithmetic operations with high degrees of precision are needed for an increasing number of applications. We propose an exact real arithmetic system that achieves adaptive precision using lazy infinite lists of floating-point values.

entry:
@Article{McCleeary:2015:LAA,
  author =       "Ryan McCleeary and Martin Brain and Aaron Stump",
  title =        "A lazy approach to adaptive exact real arithmetic
                 using floating-point operations",
  journal =      j-ACM-COMM-COMP-ALGEBRA,
  volume =       "49",
  number =       "3",
  pages =        "83--86",
  month =        sep,
  year =         "2015",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2850449.2850456",
  ISSN =         "1932-2232 (print), 1932-2240 (electronic)",
  ISSN-L =       "1932-2232",
  bibdate =      "Wed Dec 2 14:16:56 MST 2015",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigsam.bib",
  abstract =     "Arithmetic operations with high degrees of precision
                 are needed for an increasing number of applications. We
                 propose an exact real arithmetic system that achieves
                 adaptive precision using lazy infinite lists of
                 floating-point values.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Communications in Computer Algebra",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
}


bibtype: article
filename: fparith.bib
label: Meloni:2015:EDB
author: N. Meloni and M. A. Hasan
title: Efficient Double Bases for Scalar Multiplication
journal: j-IEEE-TRANS-COMPUT
volume: 64
number: 8
pages: 2204--2212
month: aug
monthnumber: 8
year: 2015
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2360539
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Aug 11 08:24:23 MDT 2015
bibtimestamp: 2015.08.11 08:24:23 MDT
fjournal: IEEE Transactions on Computers
keywords: double-base number system; elliptic curve
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: N. Meloni and M. A. Hasan
rawtitle: Efficient Double Bases for Scalar Multiplication
entry:
@Article{Meloni:2015:EDB,
  author =       "N. Meloni and M. A. Hasan",
  title =        "Efficient Double Bases for Scalar Multiplication",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "64",
  number =       "8",
  pages =        "2204--2212",
  month =        aug,
  year =         "2015",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2360539",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Aug 11 08:24:23 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "double-base number system; elliptic curve",
}


bibtype: article
filename: fparith.bib
label: Michelogiannakis:2015:ESP
author: George Michelogiannakis and Xiaoye S. Li
title: Extending Summation Precision for Network Reduction Operations
journal: j-INT-J-PARALLEL-PROG
volume: 43
number: 6
pages: 1218--1243
month: dec
monthnumber: 12
year: 2015
CODEN: IJPPE5
DOI: https://doi.org/10.1007/s10766-014-0326-5
ISSN: 0885-7458 (print), 1573-7640 (electronic) OR 08857458 (print), 15737640 (electronic)
ISSNL: 0885-7458 OR 08857458
bibdate: Tue Sep 29 10:13:48 MDT 2015
bibtimestamp: 2015.09.29 10:13:48 MDT
URL: http://link.springer.com/article/10.1007/s10766-014-0326-5
fjournal: International Journal of Parallel Programming
keywords: accurate floating-point summation
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10766
rawauthor: George Michelogiannakis and Xiaoye S. Li
rawtitle: Extending Summation Precision for Network Reduction Operations
bibsource:
http://link.springer.com/journal/10766/43/6; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/intjparallelprogram.bib

entry:
@Article{Michelogiannakis:2015:ESP,
  author =       "George Michelogiannakis and Xiaoye S. Li",
  title =        "Extending Summation Precision for Network Reduction
                 Operations",
  journal =      j-INT-J-PARALLEL-PROG,
  volume =       "43",
  number =       "6",
  pages =        "1218--1243",
  month =        dec,
  year =         "2015",
  CODEN =        "IJPPE5",
  DOI =          "https://doi.org/10.1007/s10766-014-0326-5",
  ISSN =         "0885-7458 (print), 1573-7640 (electronic)",
  ISSN-L =       "0885-7458",
  bibdate =      "Tue Sep 29 10:13:48 MDT 2015",
  bibsource =    "http://link.springer.com/journal/10766/43/6;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/intjparallelprogram.bib",
  URL =          "http://link.springer.com/article/10.1007/s10766-014-0326-5",
  acknowledgement = ack-nhfb,
  fjournal =     "International Journal of Parallel Programming",
  journal-URL =  "http://link.springer.com/journal/10766",
  keywords =     "accurate floating-point summation",
}


bibtype: article
filename: fparith.bib
label: Momeni:2015:DAA
author: A. Momeni and Jie Han and P. Montuschi and F. Lombardi
title: Design and Analysis of Approximate Compressors for Multiplication
journal: j-IEEE-TRANS-COMPUT
volume: 64
number: 4
pages: 984--994
month: apr
monthnumber: 4
year: 2015
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2014.2308214
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Jun 4 19:46:38 MDT 2015
bibtimestamp: 2015.06.04 19:46:38 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: A. Momeni and Jie Han and P. Montuschi and F. Lombardi
rawtitle: Design and Analysis of Approximate Compressors for Multiplication
entry:
@Article{Momeni:2015:DAA,
  author =       "A. Momeni and Jie Han and P. Montuschi and F.
                 Lombardi",
  title =        "Design and Analysis of Approximate Compressors for
                 Multiplication",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "64",
  number =       "4",
  pages =        "984--994",
  month =        apr,
  year =         "2015",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2014.2308214",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Jun 4 19:46:38 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Morar:2015:RMT
author: Florin-Stefan Morar
title: Reinventing machines: the transmission history of the Leibniz calculator
journal: j-BRITISH-J-HIST-SCI
volume: 48
number: 1
pages: 123--146
month: mar
monthnumber: 3
year: 2015
CODEN: BJHSAT
DOI: https://doi.org/10.1017/S0007087414000429
ISSN: 0007-0874 (print), 1474-001X (electronic) OR 00070874 (print), 1474001X (electronic)
ISSNL: 0007-0874 OR 00070874
bibdate: Wed Feb 4 10:40:51 MST 2015
bibtimestamp: 2015.02.04 10:40:51 MST
fjournal: British Journal for the History of Science
acknowledgement: ack-nhfb
ajournal: British J. Hist. Sci.
journalURL: http://journals.cambridge.org/action/displayJournal?jid=BJH
onlinedate: Mon Jul 14 00:00:00 BST 2014
rawauthor: Florin-Stefan Morar
rawtitle: Reinventing machines: the transmission history of the {Leibniz} calculator
bibsource:
https://www.math.utah.edu/pub/tex/bib/bjhs2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Morar:2015:RMT,
  author =       "Florin-Stefan Morar",
  title =        "Reinventing machines: the transmission history of the
                 {Leibniz} calculator",
  journal =      j-BRITISH-J-HIST-SCI,
  volume =       "48",
  number =       "1",
  pages =        "123--146",
  month =        mar,
  year =         "2015",
  CODEN =        "BJHSAT",
  DOI =          "https://doi.org/10.1017/S0007087414000429",
  ISSN =         "0007-0874 (print), 1474-001X (electronic)",
  ISSN-L =       "0007-0874",
  bibdate =      "Wed Feb 4 10:40:51 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/bjhs2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "British J. Hist. Sci.",
  fjournal =     "British Journal for the History of Science",
  journal-URL =  "http://journals.cambridge.org/action/displayJournal?jid=BJH",
  onlinedate =   "Mon Jul 14 00:00:00 BST 2014",
}


bibtype: article
filename: fparith.bib
label: Muller:2015:ECC
author: Jean-Michel Muller
title: On the Error of Computing $ a b + c d $ using Cornea, Harrison and Tang's Method
journal: j-TOMS
volume: 41
number: 2
pages: 7:1--7:8
month: jan
monthnumber: 1
year: 2015
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2629615
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Wed Feb 4 17:49:11 MST 2015
bibtimestamp: 2015.02.04 17:49:11 MST
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 7
journalURL: https://dl.acm.org/loi/toms
rawauthor: Jean-Michel Muller
rawtitle: On the Error of Computing $ a b + c d $ using {Cornea}, {Harrison} and {Tang}'s Method
bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
In their book, booktitle Scientific Computing on the Itanium, Cornea et al. [2002] introduce an accurate algorithm for evaluating expressions of the form $ a b + c d $ in binary floating-point arithmetic, assuming an FMA instruction is available. They show that if $p$ is the precision of the floating-point format and if $ u = 2^-p $, the relative error of the result is of order $u$. We improve their proof to show that the relative error is bounded by $ 2 u + 7 u^2 + 6 u^3 $. Furthermore, by building an example for which the relative error is asymptotically (as $ p to infty $ or, equivalently, as $ u to 0 $) equivalent to $ 2 u $, we show that our error bound is asymptotically optimal.

remark:
This article compares two algorithms (Kahan's and Cornea / Harrison / Tang's) for computing $ a b + c d $. It shows that the worst-case error with FMA and round-to-nearest arithmetic is $ 2 u $ for the first, and $ 2 u + 7 u^2 + 6 u^3 $ for the second, suggesting that Kahan's is preferred. However, the second guarantees that $ a b + c d = = c d + a b $, whereas the first does not, so it may be preferred for applications like complex multiplication and division, in order to guarantee commutative arithmetic

rawabstract:
In their book, \booktitle{Scientific Computing on the Itanium}, Cornea et al. [2002] introduce an accurate algorithm for evaluating expressions of the form $ a b + c d $ in binary floating-point arithmetic, assuming an FMA instruction is available. They show that if $p$ is the precision of the floating-point format and if $ u = 2^{-p} $, the relative error of the result is of order $u$. We improve their proof to show that the relative error is bounded by $ 2 u + 7 u^2 + 6 u^3 $. Furthermore, by building an example for which the relative error is asymptotically (as $ p \to \infty $ or, equivalently, as $ u \to 0 $) equivalent to $ 2 u $, we show that our error bound is asymptotically optimal.

entry:
@Article{Muller:2015:ECC,
  author =       "Jean-Michel Muller",
  title =        "On the Error of Computing $ a b + c d $ using
                 {Cornea}, {Harrison} and {Tang}'s Method",
  journal =      j-TOMS,
  volume =       "41",
  number =       "2",
  pages =        "7:1--7:8",
  month =        jan,
  year =         "2015",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2629615",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Wed Feb 4 17:49:11 MST 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "In their book, \booktitle{Scientific Computing on the
                 Itanium}, Cornea et al. [2002] introduce an accurate
                 algorithm for evaluating expressions of the form $ a b
                 + c d $ in binary floating-point arithmetic, assuming
                 an FMA instruction is available. They show that if $p$
                 is the precision of the floating-point format and if $
                 u = 2^{-p} $, the relative error of the result is of
                 order $u$. We improve their proof to show that the
                 relative error is bounded by $ 2 u + 7 u^2 + 6 u^3 $.
                 Furthermore, by building an example for which the
                 relative error is asymptotically (as $ p \to \infty $
                 or, equivalently, as $ u \to 0 $) equivalent to $ 2 u
                 $, we show that our error bound is asymptotically
                 optimal.",
  acknowledgement = ack-nhfb,
  articleno =    "7",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  remark =       "This article compares two algorithms (Kahan's and
                 Cornea / Harrison / Tang's) for computing $ a b + c d
                 $. It shows that the worst-case error with FMA and
                 round-to-nearest arithmetic is $ 2 u $ for the first,
                 and $ 2 u + 7 u^2 + 6 u^3 $ for the second, suggesting
                 that Kahan's is preferred. However, the second
                 guarantees that $ a b + c d = = c d + a b $, whereas
                 the first does not, so it may be preferred for
                 applications like complex multiplication and division,
                 in order to guarantee commutative arithmetic",
}


bibtype: techreport
filename: fparith.bib
label: Neal:2015:FESa
author: Radford M. Neal
title: Fast exact summation using small and large superaccumulators
type: Report
institution: Department of Statistical Sciences and Department of Computer Science, University of Toronto
address: Toronto, ON, Canada
pages: 22
year: 2015
bibdate: Sat Oct 31 07:19:27 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib
bibtimestamp: 2015.10.31 07:19:27 ???
URL: http://www.cs.toronto.edu/~radford/ftp/xsum.pdf
acknowledgement: ack-nhfb
rawauthor: Radford M. Neal
rawtitle: Fast exact summation using small and large superaccumulators
entry:
@TechReport{Neal:2015:FESa,
  author =       "Radford M. Neal",
  title =        "Fast exact summation using small and large
                 superaccumulators",
  type =         "Report",
  institution =  "Department of Statistical Sciences and Department of
                 Computer Science, University of Toronto",
  address =      "Toronto, ON, Canada",
  pages =        "22",
  year =         "2015",
  bibdate =      "Sat Oct 31 07:19:27 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  URL =          "http://www.cs.toronto.edu/~radford/ftp/xsum.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Neal:2015:FESb
author: Radford M. Neal
title: Fast exact summation using small and large superaccumulators
journal: arxiv.org
pages: 22
year: 2015
bibdate: Sat Oct 31 07:19:27 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib
bibtimestamp: 2015.10.31 07:19:27 ???
URL: https://arxiv.org/pdf/1505.05571v1.pdf
acknowledgement: ack-nhfb
rawauthor: Radford M. Neal
rawtitle: Fast exact summation using small and large superaccumulators
abstract:
I present two new methods for exactly summing a set of floating-point numbers, and then correctly rounding to the nearest floating-point number. Higher accuracy than simple summation (rounding after each addition) is important in many applications, such as finding the sample mean of data. Exact summation also guarantees identical results with parallel and serial implementations, since the exact sum is independent of order. The new methods use variations on the concept of a ``superaccumulator'' --- a large fixed-point number that can exactly represent the sum of any reasonable number of floating-point values. One method uses a ``small'' superaccumulator with sixty-seven 64-bit chunks, each with 32-bit overlap with the next chunk, allowing carry propagation to be done infrequently. The small superaccumulator is used alone when summing a small number of terms. For big summations, a ``large'' superaccumulator is used as well. It consists of 4096 64-bit chunks, one for every possible combination of exponent bits and sign bit, plus counts of when each chunk needs to be transferred to the small superaccumulator. To add a term to the large superaccumulator, only a single chunk and its associated count need to be updated, which takes very few instructions if carefully implemented. On modern 64-bit processors, exactly summing a large array using this combination of large and small superaccumulators takes less than twice the time of simple, inexact, ordered summation, with a serial implementation. A parallel implementation using a small number of processor cores can be expected to perform exact summation of large arrays at a speed that reaches the limit imposed by memory bandwidth. Some common methods that attempt to improve accuracy without being exact may therefore be pointless, at least for large summations, since they are slower than computing the sum exactly.

rawabstract:
I present two new methods for exactly summing a set of floating-point numbers, and then correctly rounding to the nearest floating-point number. Higher accuracy than simple summation (rounding after each addition) is important in many applications, such as finding the sample mean of data. Exact summation also guarantees identical results with parallel and serial implementations, since the exact sum is independent of order. The new methods use variations on the concept of a ``superaccumulator'' --- a large fixed-point number that can exactly represent the sum of any reasonable number of floating-point values. One method uses a ``small'' superaccumulator with sixty-seven 64-bit chunks, each with 32-bit overlap with the next chunk, allowing carry propagation to be done infrequently. The small superaccumulator is used alone when summing a small number of terms. For big summations, a ``large'' superaccumulator is used as well. It consists of 4096 64-bit chunks, one for every possible combination of exponent bits and sign bit, plus counts of when each chunk needs to be transferred to the small superaccumulator. To add a term to the large superaccumulator, only a single chunk and its associated count need to be updated, which takes very few instructions if carefully implemented. On modern 64-bit processors, exactly summing a large array using this combination of large and small superaccumulators takes less than twice the time of simple, inexact, ordered summation, with a serial implementation. A parallel implementation using a small number of processor cores can be expected to perform exact summation of large arrays at a speed that reaches the limit imposed by memory bandwidth. Some common methods that attempt to improve accuracy without being exact may therefore be pointless, at least for large summations, since they are slower than computing the sum exactly.

entry:
@Article{Neal:2015:FESb,
  author =       "Radford M. Neal",
  title =        "Fast exact summation using small and large
                 superaccumulators",
  journal =      "arxiv.org",
  pages =        "22",
  year =         "2015",
  bibdate =      "Sat Oct 31 07:19:27 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  URL =          "https://arxiv.org/pdf/1505.05571v1.pdf",
  abstract =     "I present two new methods for exactly summing a set of
                 floating-point numbers, and then correctly rounding to
                 the nearest floating-point number. Higher accuracy than
                 simple summation (rounding after each addition) is
                 important in many applications, such as finding the
                 sample mean of data. Exact summation also guarantees
                 identical results with parallel and serial
                 implementations, since the exact sum is independent of
                 order. The new methods use variations on the concept of
                 a ``superaccumulator'' --- a large fixed-point number
                 that can exactly represent the sum of any reasonable
                 number of floating-point values. One method uses a
                 ``small'' superaccumulator with sixty-seven 64-bit
                 chunks, each with 32-bit overlap with the next chunk,
                 allowing carry propagation to be done infrequently. The
                 small superaccumulator is used alone when summing a
                 small number of terms. For big summations, a ``large''
                 superaccumulator is used as well. It consists of 4096
                 64-bit chunks, one for every possible combination of
                 exponent bits and sign bit, plus counts of when each
                 chunk needs to be transferred to the small
                 superaccumulator. To add a term to the large
                 superaccumulator, only a single chunk and its
                 associated count need to be updated, which takes very
                 few instructions if carefully implemented. On modern
                 64-bit processors, exactly summing a large array using
                 this combination of large and small superaccumulators
                 takes less than twice the time of simple, inexact,
                 ordered summation, with a serial implementation. A
                 parallel implementation using a small number of
                 processor cores can be expected to perform exact
                 summation of large arrays at a speed that reaches the
                 limit imposed by memory bandwidth. Some common methods
                 that attempt to improve accuracy without being exact
                 may therefore be pointless, at least for large
                 summations, since they are slower than computing the
                 sum exactly.",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Negre:2015:EME
author: Christophe Negre and Thomas Plantard and Jean-Marc Robert
title: Efficient Modular Exponentiation Based on Multiple Multiplications by a Common Operand
crossref: Muller:2015:ISC
pages: 144--151
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.24
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Christophe Negre and Thomas Plantard and Jean-Marc Robert
rawtitle: Efficient Modular Exponentiation Based on Multiple Multiplications by a Common Operand
entry:
@InProceedings{Negre:2015:EME,
  author =       "Christophe Negre and Thomas Plantard and Jean-Marc
                 Robert",
  title =        "Efficient Modular Exponentiation Based on Multiple
                 Multiplications by a Common Operand",
  crossref =     "Muller:2015:ISC",
  pages =        "144--151",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.24",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: inproceedings
filename: fparith.bib
label: Nguyen:2015:RTS
author: Hong Diep Nguyen and James Demmel
title: Reproducible Tall-Skinny QR
crossref: Muller:2015:ISC
pages: 152--159
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.28
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
URL: http://bebop.cs.berkeley.edu/reproblas/docs/papers/RepTSQR.pdf
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Hong Diep Nguyen and James Demmel
rawtitle: Reproducible Tall-Skinny {QR}
entry:
@InProceedings{Nguyen:2015:RTS,
  author =       "Hong Diep Nguyen and James Demmel",
  title =        "Reproducible Tall-Skinny {QR}",
  crossref =     "Muller:2015:ISC",
  pages =        "152--159",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.28",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://bebop.cs.berkeley.edu/reproblas/docs/papers/RepTSQR.pdf",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: fparith.bib
label: Ozaki:2015:IEF
author: Katsuhisa Ozaki and Takeshi Ogita and Shin'ichi Oishi
title: Improvement of error-free splitting for accurate matrix multiplication
journal: j-J-COMPUT-APPL-MATH
volume: 288
pages: 127--140
month: nov
monthnumber: 11
year: 2015
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:34:52 MST 2017
bibtimestamp: 2017.02.25 13:34:52 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042715002289
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Katsuhisa Ozaki and Takeshi Ogita and Shin'ichi Oishi
rawtitle: Improvement of error-free splitting for accurate matrix multiplication
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib

entry:
@Article{Ozaki:2015:IEF,
  author =       "Katsuhisa Ozaki and Takeshi Ogita and Shin'ichi
                 Oishi",
  title =        "Improvement of error-free splitting for accurate
                 matrix multiplication",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "288",
  number =       "??",
  pages =        "127--140",
  month =        nov,
  year =         "2015",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:34:52 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042715002289",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: fparith.bib
label: Palmer:2015:MBI
author: Tim Palmer
title: Modelling: Build imprecise supercomputers
journal: j-NATURE
volume: 526
number: 7571
pages: 32--33
day: 29
month: sep
monthnumber: 9
year: 2015
CODEN: NATUAS
DOI: https://doi.org/10.1038/526032a
ISSN: 0028-0836 (print), 1476-4687 (electronic) OR 00280836 (print), 14764687 (electronic)
ISSNL: 0028-0836 OR 00280836
bibdate: Sat Mar 25 18:29:10 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/super.bib
bibtimestamp: 2017.03.25 18:29:10 ???
fjournal: Nature
acknowledgement: ack-nhfb
journalURL: http://www.nature.com/nature/archive/
rawauthor: Tim Palmer
rawtitle: Modelling: Build imprecise supercomputers
remark:
This provocative paper argues that future supercomputers should be built with a small number of processors that can do accurate high-precision floating-point arithmetic, with the bulk of computations carried out, to save energy, at lower precision, and possibly with occasional arithmetic errors. I [NHFB] find that a perilous approach that would likely lead to entirely unrepeatable, and thus, almost certainly useless, computations to which it is impossible to apply sensible numerical analysis.!

entry:
@Article{Palmer:2015:MBI,
  author =       "Tim Palmer",
  title =        "Modelling: Build imprecise supercomputers",
  journal =      j-NATURE,
  volume =       "526",
  number =       "7571",
  pages =        "32--33",
  day =          "29",
  month =        sep,
  year =         "2015",
  CODEN =        "NATUAS",
  DOI =          "https://doi.org/10.1038/526032a",
  ISSN =         "0028-0836 (print), 1476-4687 (electronic)",
  ISSN-L =       "0028-0836",
  bibdate =      "Sat Mar 25 18:29:10 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/super.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Nature",
  journal-URL =  "http://www.nature.com/nature/archive/",
  remark =       "This provocative paper argues that future
                 supercomputers should be built with a small number of
                 processors that can do accurate high-precision
                 floating-point arithmetic, with the bulk of
                 computations carried out, to save energy, at lower
                 precision, and possibly with occasional arithmetic
                 errors. I [NHFB] find that a perilous approach that
                 would likely lead to entirely unrepeatable, and thus,
                 almost certainly useless, computations to which it is
                 impossible to apply sensible numerical analysis.!",
}


bibtype: article
filename: fparith.bib
label: Panchekha:2015:AIA
author: Pavel Panchekha and Alex Sanchez-Stern and James R. Wilcox and Zachary Tatlock
title: Automatically improving accuracy for floating point expressions
journal: j-SIGPLAN
volume: 50
number: 6
pages: 1--11
month: jun
monthnumber: 6
year: 2015
CODEN: SINODQ
DOI: https://doi.org/10.1145/2813885.2737959
ISSN: 0362-1340 (print), 1523-2867 (print), 1558-1160 OR 03621340 (print), 15232867 (print), 15581160
ISSNL: 0362-1340 OR 03621340
bibdate: Tue Feb 16 12:01:41 MST 2016
bibtimestamp: 2016.02.16 12:01:41 MST
fjournal: ACM SIGPLAN Notices
remark: PLDI '15 conference proceedings.
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J706
rawauthor: Pavel Panchekha and Alex Sanchez-Stern and James R. Wilcox and Zachary Tatlock
rawtitle: Automatically improving accuracy for floating point expressions
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigplan2010.bib

abstract:
Scientific and engineering applications depend on floating point arithmetic to approximate real arithmetic. This approximation introduces rounding error, which can accumulate to produce unacceptable results. While the numerical methods literature provides techniques to mitigate rounding error, applying these techniques requires manually rearranging expressions and understanding the finer details of floating point arithmetic. We introduce Herbie, a tool which automatically discovers the rewrites experts perform to improve accuracy. Herbie's heuristic search estimates and localizes rounding error using sampled points (rather than static error analysis), applies a database of rules to generate improvements, takes series expansions, and combines improvements for different input regions. We evaluated Herbie on examples from a classic numerical methods textbook, and found that Herbie was able to improve accuracy on each example, some by up to 60 bits, while imposing a median performance overhead of 40%. Colleagues in machine learning have used Herbie to significantly improve the results of a clustering algorithm, and a mathematical library has accepted two patches generated using Herbie.

rawabstract:
Scientific and engineering applications depend on floating point arithmetic to approximate real arithmetic. This approximation introduces rounding error, which can accumulate to produce unacceptable results. While the numerical methods literature provides techniques to mitigate rounding error, applying these techniques requires manually rearranging expressions and understanding the finer details of floating point arithmetic. We introduce Herbie, a tool which automatically discovers the rewrites experts perform to improve accuracy. Herbie's heuristic search estimates and localizes rounding error using sampled points (rather than static error analysis), applies a database of rules to generate improvements, takes series expansions, and combines improvements for different input regions. We evaluated Herbie on examples from a classic numerical methods textbook, and found that Herbie was able to improve accuracy on each example, some by up to 60 bits, while imposing a median performance overhead of 40\%. Colleagues in machine learning have used Herbie to significantly improve the results of a clustering algorithm, and a mathematical library has accepted two patches generated using Herbie.

entry:
@Article{Panchekha:2015:AIA,
  author =       "Pavel Panchekha and Alex Sanchez-Stern and James R.
                 Wilcox and Zachary Tatlock",
  title =        "Automatically improving accuracy for floating point
                 expressions",
  journal =      j-SIGPLAN,
  volume =       "50",
  number =       "6",
  pages =        "1--11",
  month =        jun,
  year =         "2015",
  CODEN =        "SINODQ",
  DOI =          "https://doi.org/10.1145/2813885.2737959",
  ISSN =         "0362-1340 (print), 1523-2867 (print), 1558-1160
                 (electronic)",
  ISSN-L =       "0362-1340",
  bibdate =      "Tue Feb 16 12:01:41 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigplan2010.bib",
  abstract =     "Scientific and engineering applications depend on
                 floating point arithmetic to approximate real
                 arithmetic. This approximation introduces rounding
                 error, which can accumulate to produce unacceptable
                 results. While the numerical methods literature
                 provides techniques to mitigate rounding error,
                 applying these techniques requires manually rearranging
                 expressions and understanding the finer details of
                 floating point arithmetic. We introduce Herbie, a tool
                 which automatically discovers the rewrites experts
                 perform to improve accuracy. Herbie's heuristic search
                 estimates and localizes rounding error using sampled
                 points (rather than static error analysis), applies a
                 database of rules to generate improvements, takes
                 series expansions, and combines improvements for
                 different input regions. We evaluated Herbie on
                 examples from a classic numerical methods textbook, and
                 found that Herbie was able to improve accuracy on each
                 example, some by up to 60 bits, while imposing a median
                 performance overhead of 40\%. Colleagues in machine
                 learning have used Herbie to significantly improve the
                 results of a clustering algorithm, and a mathematical
                 library has accepted two patches generated using
                 Herbie.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGPLAN Notices",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J706",
  remark =       "PLDI '15 conference proceedings.",
}


bibtype: article
filename: fparith.bib
label: Parhami:2015:DAN
author: Behrooz Parhami
title: Digital Arithmetic in Nature: Continuous-Digit RNS
journal: j-COMP-J
volume: 58
number: 5
pages: 1214--1223
month: may
monthnumber: 5
year: 2015
CODEN: CMPJA6
DOI: https://doi.org/10.1093/comjnl/bxu060
ISSN: 0010-4620 (print), 1460-2067 (electronic) OR 00104620 (print), 14602067 (electronic)
ISSNL: 0010-4620 OR 00104620
bibdate: Sat May 2 08:05:35 MDT 2015
bibtimestamp: 2015.05.02 08:05:35 MDT
URL: http://comjnl.oxfordjournals.org/content/58/5/1214
fjournal: Computer Journal
keywords: Residue Number System (RNS)
acknowledgement: ack-nhfb
journalURL: http://comjnl.oxfordjournals.org/
onlinedate: July 17, 2014
rawauthor: Behrooz Parhami
rawtitle: Digital Arithmetic in Nature: Continuous-Digit {RNS}
bibsource:
http://comjnl.oxfordjournals.org/content/58/5.toc; https://www.math.utah.edu/pub/tex/bib/compj2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Parhami:2015:DAN,
  author =       "Behrooz Parhami",
  title =        "Digital Arithmetic in Nature: Continuous-Digit {RNS}",
  journal =      j-COMP-J,
  volume =       "58",
  number =       "5",
  pages =        "1214--1223",
  month =        may,
  year =         "2015",
  CODEN =        "CMPJA6",
  DOI =          "https://doi.org/10.1093/comjnl/bxu060",
  ISSN =         "0010-4620 (print), 1460-2067 (electronic)",
  ISSN-L =       "0010-4620",
  bibdate =      "Sat May 2 08:05:35 MDT 2015",
  bibsource =    "http://comjnl.oxfordjournals.org/content/58/5.toc;
                 https://www.math.utah.edu/pub/tex/bib/compj2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://comjnl.oxfordjournals.org/content/58/5/1214",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Journal",
  journal-URL =  "http://comjnl.oxfordjournals.org/",
  keywords =     "Residue Number System (RNS)",
  onlinedate =   "July 17, 2014",
}


bibtype: inproceedings
filename: fparith.bib
label: Patil:2015:OFP
author: Vinayak Patil and Aneesh Raveendran and P. M. Sobha and A. David Selvakumar and D. Vivian
editor: IEEE
booktitle: 2015 19th International Symposium on VLSI Design and Test
title: Out of order floating point coprocessor for RISC V ISA
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 1--7
year: 2015
DOI: https://doi.org/10.1109/ISVDAT.2015.7208116
bibdate: Sat Dec 16 15:51:40 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib
bibtimestamp: 2023.12.16 15:51:40 ???
acknowledgement: ack-nhfb
rawauthor: Vinayak Patil and Aneesh Raveendran and P. M. Sobha and A. David Selvakumar and D. Vivian
rawbooktitle: {2015 19th International Symposium on VLSI Design and Test}
raweditor: {IEEE}
rawtitle: Out of order floating point coprocessor for {RISC V ISA}
entry:
@InProceedings{Patil:2015:OFP,
  author =       "Vinayak Patil and Aneesh Raveendran and P. M. Sobha
                 and A. David Selvakumar and D. Vivian",
  editor =       "{IEEE}",
  booktitle =    "{2015 19th International Symposium on VLSI Design and
                 Test}",
  title =        "Out of order floating point coprocessor for {RISC V
                 ISA}",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "1--7",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ISVDAT.2015.7208116",
  bibdate =      "Sat Dec 16 15:51:40 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/risc-v.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Peeper:2015:DDP
author: D. Peeper
title: Deep-dish peeper [The Big Picture]
journal: j-IEEE-SPECTRUM
volume: 52
number: 11
pages: 16--17
month: nov
monthnumber: 11
year: 2015
CODEN: IEESAM
DOI: https://doi.org/10.1109/MSPEC.2015.7335888
ISSN: 0018-9235 (print), 1939-9340 (electronic) OR 00189235 (print), 19399340 (electronic)
ISSNL: 0018-9235 OR 00189235
bibdate: Sat Jan 18 07:02:09 2020
bibtimestamp: 2020.01.18 07:02:09 ???
fjournal: IEEE Spectrum
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6
rawauthor: D. Peeper
rawtitle: Deep-dish peeper [The Big Picture]
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeespectrum2010.bib

keywords:
FAST; five-hundred-meter aperture spherical radio telescope; floating point operations; Puerto Rico; radiotelescopes

entry:
@Article{Peeper:2015:DDP,
  author =       "D. Peeper",
  title =        "Deep-dish peeper [The Big Picture]",
  journal =      j-IEEE-SPECTRUM,
  volume =       "52",
  number =       "11",
  pages =        "16--17",
  month =        nov,
  year =         "2015",
  CODEN =        "IEESAM",
  DOI =          "https://doi.org/10.1109/MSPEC.2015.7335888",
  ISSN =         "0018-9235 (print), 1939-9340 (electronic)",
  ISSN-L =       "0018-9235",
  bibdate =      "Sat Jan 18 07:02:09 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeespectrum2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Spectrum",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6",
  keywords =     "FAST; five-hundred-meter aperture spherical radio
                 telescope; floating point operations; Puerto Rico;
                 radiotelescopes",
}


bibtype: inproceedings
filename: fparith.bib
label: Proust:2015:KTC
author: Christine Proust
title: Keynote Talk: Calculating in Floating Sexagesimal Place Value Notation, 4000 years ago
crossref: Muller:2015:ISC
pages: 1--1
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.33
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
note: Abstract only.
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Christine Proust
rawnote: Abstract only.
rawtitle: Keynote Talk: Calculating in Floating Sexagesimal Place Value Notation, 4000 years ago
entry:
@InProceedings{Proust:2015:KTC,
  author =       "Christine Proust",
  title =        "Keynote Talk: Calculating in Floating Sexagesimal
                 Place Value Notation, 4000 years ago",
  crossref =     "Muller:2015:ISC",
  pages =        "1--1",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.33",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  note =         "Abstract only.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: inproceedings
filename: fparith.bib
label: Renardy:2015:HIM
author: Antonius P. Renardy and Nur Ahmadi and Ashbir A. Fadila and Naufal Shidqi and Trio Adiono
editor: IEEE
title: Hardware Implementation of Montgomery Modular Multiplication Algorithm Using Iterative Architecture
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 99--102
year: 2015
DOI: https://doi.org/10.1109/ISITIA.2015.7219961
ISBN: 1-4799-7711-X OR 147997711X
ISBN13: 978-1-4799-7711-6 OR 9781479977116
LCCN: TA347.A78
bibdate: Tue Jul 06 18:27:11 2021
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2021.07.06 18:27:11 ???
keywords: FPGA; Iterative Architecture; Modular Multiplication; Montgomery's Algorithm; RSA Cryptosystem
acknowledgement: ack-nhfb
bookpages: 453
rawauthor: Antonius P. Renardy and Nur Ahmadi and Ashbir A. Fadila and Naufal Shidqi and Trio Adiono
raweditor: {IEEE}
booktitle:
2015 International Seminar on Intelligent Technology and Its Applications (ISITIA): proceeding: Surabaya, Indonesia, 20--21 May 2015

abstract:
Modular multiplication is an integral part of RSA cryptosystems and its performance heavily determines the performance of the encryption hardware. This paper provides a hardware implementation of Montgomery's modular multiplication algorithm using iterative architecture. The proposed design is implemented in Verilog HDL and simulated functionally using ModelSim Altera 10.1E. The synthesis is performed using Altera Quartus II 9.1 with target FPGA board Altera DE2-70. The proposed design consumes 17540 logic elements with 15480 LUT and takes 2048 clock cycles to perform multiplication process. Based on trade-off parameter AT$^2$ measure, the proposed design offers the best performance among other designs.

rawabstract:
Modular multiplication is an integral part of RSA cryptosystems and its performance heavily determines the performance of the encryption hardware. This paper provides a hardware implementation of Montgomery's modular multiplication algorithm using iterative architecture. The proposed design is implemented in Verilog HDL and simulated functionally using ModelSim Altera 10.1E. The synthesis is performed using Altera Quartus II 9.1 with target FPGA board Altera DE2-70. The proposed design consumes 17540 logic elements with 15480 LUT and takes 2048 clock cycles to perform multiplication process. Based on trade-off parameter AT$^2$ measure, the proposed design offers the best performance among other designs.

rawbooktitle:
2015 International Seminar on Intelligent Technology and Its Applications {(ISITIA)}: proceeding: {Surabaya, Indonesia, 20--21 May 2015}

rawtitle:
Hardware Implementation of {Montgomery} Modular Multiplication Algorithm Using Iterative Architecture

entry:
@InProceedings{Renardy:2015:HIM,
  author =       "Antonius P. Renardy and Nur Ahmadi and Ashbir A.
                 Fadila and Naufal Shidqi and Trio Adiono",
  editor =       "{IEEE}",
  booktitle =    "2015 International Seminar on Intelligent Technology
                 and Its Applications {(ISITIA)}: proceeding: {Surabaya,
                 Indonesia, 20--21 May 2015}",
  title =        "Hardware Implementation of {Montgomery} Modular
                 Multiplication Algorithm Using Iterative Architecture",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  bookpages =    "453",
  pages =        "99--102",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ISITIA.2015.7219961",
  ISBN =         "1-4799-7711-X",
  ISBN-13 =      "978-1-4799-7711-6",
  LCCN =         "TA347.A78",
  bibdate =      "Tue Jul 06 18:27:11 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Modular multiplication is an integral part of RSA
                 cryptosystems and its performance heavily determines
                 the performance of the encryption hardware. This paper
                 provides a hardware implementation of Montgomery's
                 modular multiplication algorithm using iterative
                 architecture. The proposed design is implemented in
                 Verilog HDL and simulated functionally using ModelSim
                 Altera 10.1E. The synthesis is performed using Altera
                 Quartus II 9.1 with target FPGA board Altera DE2-70.
                 The proposed design consumes 17540 logic elements with
                 15480 LUT and takes 2048 clock cycles to perform
                 multiplication process. Based on trade-off parameter
                 AT$^2$ measure, the proposed design offers the best
                 performance among other designs.",
  acknowledgement = ack-nhfb,
  keywords =     "FPGA; Iterative Architecture; Modular Multiplication;
                 Montgomery's Algorithm; RSA Cryptosystem",
}


bibtype: article
filename: fparith.bib
label: Roegel:2015:MCA
author: Denis Roegel
journal: j-IEEE-ANN-HIST-COMPUT
volume: 37
number: 4
pages: 90--96
year: 2015
CODEN: IAHCEX
DOI: https://doi.org/10.1109/MAHC.2015.79
ISSN: 1058-6180 (print), 1934-1547 (electronic) OR 10586180 (print), 19341547 (electronic)
ISSNL: 1058-6180 OR 10586180
bibdate: Tue Dec 8 07:02:44 MST 2015
bibtimestamp: 2015.12.08 07:02:44 MST
fjournal: IEEE Annals of the History of Computing
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=85
rawauthor: Denis Roegel
title:
A Mechanical Calculator for Arithmetic Sequences (1844--1852): Part 1, Historical Context and Structure

rawtitle:
A Mechanical Calculator for Arithmetic Sequences (1844--1852): Part 1, Historical Context and Structure

entry:
@Article{Roegel:2015:MCA,
  author =       "Denis Roegel",
  title =        "A Mechanical Calculator for Arithmetic Sequences
                 (1844--1852): Part 1, Historical Context and
                 Structure",
  journal =      j-IEEE-ANN-HIST-COMPUT,
  volume =       "37",
  number =       "4",
  pages =        "90--96",
  month =        "????",
  year =         "2015",
  CODEN =        "IAHCEX",
  DOI =          "https://doi.org/10.1109/MAHC.2015.79",
  ISSN =         "1058-6180 (print), 1934-1547 (electronic)",
  ISSN-L =       "1058-6180",
  bibdate =      "Tue Dec 8 07:02:44 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeeannhistcomput.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Annals of the History of Computing",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=85",
}


bibtype: book
filename: fparith.bib
label: Ruckert:2015:MSS
author: Martin Ruckert
publisher: pub-AW
address: pub-AW:adr
pages: xxi + 193
year: 2015
ISBN: 0-13-399231-4 (paperback), 0-13-399289-6 OR 0133992314 (paperback), 0133992896
ISBN13: 978-0-13-399231-1 (paperback), 978-0-13-399289-2 OR 9780133992311 (paperback), 9780133992892
LCCN: QA76.6 .K64 2005 Suppl. 1
bibdate: Wed Feb 4 10:19:23 MST 2015
bibtimestamp: 2015.02.04 10:19:23 MST
URL: http://mmix.cs.hm.edu/
acknowledgement: ack-nhfb
rawauthor: Martin Ruckert
title:
The MMIX supplement: supplement to booktitle The Art of Computer Programming, volumes 1, 2, 3 by Donald E. Knuth

bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/hash.bib; https://www.math.utah.edu/pub/tex/bib/litprog.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/string-matching.bib; z3950.loc.gov:7090/Voyager

abstract:
In the first edition of Volume 1 of The Art of Computer Programming, Donald E. Knuth introduced the MIX computer and its machine language: a teaching tool that powerfully illuminated the inner workings of the algorithms he documents. Later, with the publication of his Fascicle 1, Knuth introduced MMIX: a modern, 64-bit RISC replacement to the now-obsolete MIX. Now, with Knuth's guidance and approval, Martin Ruckert has rewritten all MIX example programs from Knuth's Volumes 1--3 for MMIX, thus completing this MMIX update to the original classic.par From Donald E. Knuth's Foreword:par ``I am thrilled to see the present book by Martin Ruckert: It is jam-packed with goodies from which an extraordinary amount can be learned. Martin has not merely transcribed my early programs for MIX and recast them in a modern idiom. He has penetrated to their essence and rendered them anew with elegance and good taste. His carefully checked code represents a significant contribution to the art of pedagogy as well as to the art of programming.''

subject:
MMIX (Computer architecture); Assembly languages (Electronic computers); Microcomputers; Programming

TOC:
Foreword / iii Preface / v Style Guide / viii Programming Techniques / xii Basic Concepts / 1 Applications to Permutations / 1 Input and Output / 8 Information Structures / 15 Introduction / 15 Sequential Allocation / 17 Linked Allocation / 18 Circular Lists / 25 Doubly Linked Lists / 27 Arrays and Orthogonal Lists / 36 Traversing Binary Trees / 37 Binary Tree Representation of Trees / 39 Other Representations of Trees / 43 Lists and Garbage Collection / 44 Dynamic Storage Allocation / 45 Random Numbers / 48 Choice of modulus / 48 Potency / 49 Other Methods / 50 Numerical Distributions / 51 Summary / 52 Arithmetic / 53 Positional Number Systems / 53 Single-Precision Calculations / 53 Accuracy of Floating Point Arithmetic / 58 Double-Precision Calculations / 58 The Classical Algorithms / 62 Radix Conversion / 68 The Greatest Common Divisor / 70 Analysis of Euclid's Algorithm / 71 Factoring into Primes / 72 Evaluation of Powers / 72 Evaluation of Polynomials / 73 Sorting / 74 Internal Sorting / 74 Sorting by Insertion / 76 Sorting by Exchanging / 81 Sorting by Selection / 87 Sorting by Merging / 89 Sorting by Distribution / 93 Minimum-Comparison Sorting / 94 Summary, History and Bibliography / 95 Searching / 97 Sequential Searching / 97 Searching an Ordered Table / 99 Binary Tree Searching / 102 Balanced Trees / 103 Digital Searching / 106 Hashing / 108 Answers to Exercises / 117 The MMIX Assembly Language / 117 Applications to Permutations / 120 Input and Output / 120 Introduction / 122 Sequential Allocation / 123 Linked Allocation / 124 Circular Lists / 128 Doubly Linked Lists / 130 Arrays and Orthogonal Lists / 132 Traversing Binary Trees / 134 Binary Tree Representation of Trees / 136 Lists and Garbage Collection / 139 Dynamic Storage Allocation / 140 Choice of modulus / 147 Potency / 148 Other Methods / 148 Numerical Distributions / 149 Summary / 150 Positional Number Systems / 150 Single-Precision Calculations / 151 Accuracy of Floating Point Arithmetic / 152 Double-Precision Calculations / 153 The Classical Algorithms / 156 Radix Conversion / 158 The Greatest Common Divisor / 160 Analysis of Euclid's Algorithm / 160 Evaluation of Powers / 161 Evaluation of Polynomials / 161 Sorting / 162 Internal Sorting / 162 Sorting by Insertion / 165 Sorting by Exchanging / 169 Sorting by Selection / 174 Sorting by Distribution / 179 Minimum-Comparison Sorting / 180 Summary, History, and Bibliography / 183 Sequential Searching / 183 Searching an Ordered Table / 184 Binary Tree Searching / 185 Balanced Trees / 185 Digital Searching / 186 Hashing / 186 Acknowledgements / 188 Index / 189

rawabstract:
In the first edition of Volume 1 of The Art of Computer Programming, Donald E. Knuth introduced the MIX computer and its machine language: a teaching tool that powerfully illuminated the inner workings of the algorithms he documents. Later, with the publication of his Fascicle 1, Knuth introduced MMIX: a modern, 64-bit RISC replacement to the now-obsolete MIX. Now, with Knuth's guidance and approval, Martin Ruckert has rewritten all MIX example programs from Knuth's Volumes 1--3 for MMIX, thus completing this MMIX update to the original classic.\par From Donald E. Knuth's Foreword:\par ``I am thrilled to see the present book by Martin Ruckert: It is jam-packed with goodies from which an extraordinary amount can be learned. Martin has not merely transcribed my early programs for MIX and recast them in a modern idiom. He has penetrated to their essence and rendered them anew with elegance and good taste. His carefully checked code represents a significant contribution to the art of pedagogy as well as to the art of programming.''

rawtitle:
The {MMIX} supplement: supplement to {{\booktitle{The Art of Computer Programming, volumes 1, 2, 3}} by Donald E. Knuth}

rawTOC:
Foreword / iii \\ Preface / v \\ Style Guide / viii \\ Programming Techniques / xii \\ Basic Concepts / 1 \\ Applications to Permutations / 1 \\ Input and Output / 8 \\ Information Structures / 15 \\ Introduction / 15 \\ Sequential Allocation / 17 \\ Linked Allocation / 18 \\ Circular Lists / 25 \\ Doubly Linked Lists / 27 \\ Arrays and Orthogonal Lists / 36 \\ Traversing Binary Trees / 37 \\ Binary Tree Representation of Trees / 39 \\ Other Representations of Trees / 43 \\ Lists and Garbage Collection / 44 \\ Dynamic Storage Allocation / 45 \\ Random Numbers / 48 \\ Choice of modulus / 48 \\ Potency / 49 \\ Other Methods / 50 \\ Numerical Distributions / 51 \\ Summary / 52 \\ Arithmetic / 53 \\ Positional Number Systems / 53 \\ Single-Precision Calculations / 53 \\ Accuracy of Floating Point Arithmetic / 58 \\ Double-Precision Calculations / 58 \\ The Classical Algorithms / 62 \\ Radix Conversion / 68 \\ The Greatest Common Divisor / 70 \\ Analysis of Euclid's Algorithm / 71 \\ Factoring into Primes / 72 \\ Evaluation of Powers / 72 \\ Evaluation of Polynomials / 73 \\ Sorting / 74 \\ Internal Sorting / 74 \\ Sorting by Insertion / 76 \\ Sorting by Exchanging / 81 \\ Sorting by Selection / 87 \\ Sorting by Merging / 89 \\ Sorting by Distribution / 93 \\ Minimum-Comparison Sorting / 94 \\ Summary, History and Bibliography / 95 \\ Searching / 97 \\ Sequential Searching / 97 \\ Searching an Ordered Table / 99 \\ Binary Tree Searching / 102 \\ Balanced Trees / 103 \\ Digital Searching / 106 \\ Hashing / 108 \\ Answers to Exercises / 117 \\ The MMIX Assembly Language / 117 \\ Applications to Permutations / 120 \\ Input and Output / 120 \\ Introduction / 122 \\ Sequential Allocation / 123 \\ Linked Allocation / 124 \\ Circular Lists / 128 \\ Doubly Linked Lists / 130 \\ Arrays and Orthogonal Lists / 132 \\ Traversing Binary Trees / 134 \\ Binary Tree Representation of Trees / 136 \\ Lists and Garbage Collection / 139 \\ Dynamic Storage Allocation / 140 \\ Choice of modulus / 147 \\ Potency / 148 \\ Other Methods / 148 \\ Numerical Distributions / 149 \\ Summary / 150 \\ Positional Number Systems / 150 \\ Single-Precision Calculations / 151 \\ Accuracy of Floating Point Arithmetic / 152 \\ Double-Precision Calculations / 153 \\ The Classical Algorithms / 156 \\ Radix Conversion / 158 \\ The Greatest Common Divisor / 160 \\ Analysis of Euclid's Algorithm / 160 \\ Evaluation of Powers / 161 \\ Evaluation of Polynomials / 161 \\ Sorting / 162 \\ Internal Sorting / 162 \\ Sorting by Insertion / 165 \\ Sorting by Exchanging / 169 \\ Sorting by Selection / 174 \\ Sorting by Distribution / 179 \\ Minimum-Comparison Sorting / 180 \\ Summary, History, and Bibliography / 183 \\ Sequential Searching / 183 \\ Searching an Ordered Table / 184 \\ Binary Tree Searching / 185 \\ Balanced Trees / 185 \\ Digital Searching / 186 \\ Hashing / 186 \\ Acknowledgements / 188 \\ Index / 189

entry:
@Book{Ruckert:2015:MSS,
  author =       "Martin Ruckert",
  title =        "The {MMIX} supplement: supplement to {{\booktitle{The
                 Art of Computer Programming, volumes 1, 2, 3}} by
                 Donald E. Knuth}",
  publisher =    pub-AW,
  address =      pub-AW:adr,
  pages =        "xxi + 193",
  year =         "2015",
  ISBN =         "0-13-399231-4 (paperback), 0-13-399289-6",
  ISBN-13 =      "978-0-13-399231-1 (paperback), 978-0-13-399289-2",
  LCCN =         "QA76.6 .K64 2005 Suppl. 1",
  bibdate =      "Wed Feb 4 10:19:23 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/hash.bib;
                 https://www.math.utah.edu/pub/tex/bib/litprog.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/prng.bib;
                 https://www.math.utah.edu/pub/tex/bib/string-matching.bib;
                 z3950.loc.gov:7090/Voyager",
  URL =          "http://mmix.cs.hm.edu/",
  abstract =     "In the first edition of Volume 1 of The Art of
                 Computer Programming, Donald E. Knuth introduced the
                 MIX computer and its machine language: a teaching tool
                 that powerfully illuminated the inner workings of the
                 algorithms he documents. Later, with the publication of
                 his Fascicle 1, Knuth introduced MMIX: a modern, 64-bit
                 RISC replacement to the now-obsolete MIX. Now, with
                 Knuth's guidance and approval, Martin Ruckert has
                 rewritten all MIX example programs from Knuth's Volumes
                 1--3 for MMIX, thus completing this MMIX update to the
                 original classic.\par

                 From Donald E. Knuth's Foreword:\par

                 ``I am thrilled to see the present book by Martin
                 Ruckert: It is jam-packed with goodies from which an
                 extraordinary amount can be learned. Martin has not
                 merely transcribed my early programs for MIX and recast
                 them in a modern idiom. He has penetrated to their
                 essence and rendered them anew with elegance and good
                 taste. His carefully checked code represents a
                 significant contribution to the art of pedagogy as well
                 as to the art of programming.''",
  acknowledgement = ack-nhfb,
  libnote =      "Not yet in my library.",
  subject =      "MMIX (Computer architecture); Assembly languages
                 (Electronic computers); Microcomputers; Programming",
  tableofcontents = "Foreword / iii \\
                 Preface / v \\
                 Style Guide / viii \\
                 Programming Techniques / xii \\
                 Basic Concepts / 1 \\
                 Applications to Permutations / 1 \\
                 Input and Output / 8 \\
                 Information Structures / 15 \\
                 Introduction / 15 \\
                 Sequential Allocation / 17 \\
                 Linked Allocation / 18 \\
                 Circular Lists / 25 \\
                 Doubly Linked Lists / 27 \\
                 Arrays and Orthogonal Lists / 36 \\
                 Traversing Binary Trees / 37 \\
                 Binary Tree Representation of Trees / 39 \\
                 Other Representations of Trees / 43 \\
                 Lists and Garbage Collection / 44 \\
                 Dynamic Storage Allocation / 45 \\
                 Random Numbers / 48 \\
                 Choice of modulus / 48 \\
                 Potency / 49 \\
                 Other Methods / 50 \\
                 Numerical Distributions / 51 \\
                 Summary / 52 \\
                 Arithmetic / 53 \\
                 Positional Number Systems / 53 \\
                 Single-Precision Calculations / 53 \\
                 Accuracy of Floating Point Arithmetic / 58 \\
                 Double-Precision Calculations / 58 \\
                 The Classical Algorithms / 62 \\
                 Radix Conversion / 68 \\
                 The Greatest Common Divisor / 70 \\
                 Analysis of Euclid's Algorithm / 71 \\
                 Factoring into Primes / 72 \\
                 Evaluation of Powers / 72 \\
                 Evaluation of Polynomials / 73 \\
                 Sorting / 74 \\
                 Internal Sorting / 74 \\
                 Sorting by Insertion / 76 \\
                 Sorting by Exchanging / 81 \\
                 Sorting by Selection / 87 \\
                 Sorting by Merging / 89 \\
                 Sorting by Distribution / 93 \\
                 Minimum-Comparison Sorting / 94 \\
                 Summary, History and Bibliography / 95 \\
                 Searching / 97 \\
                 Sequential Searching / 97 \\
                 Searching an Ordered Table / 99 \\
                 Binary Tree Searching / 102 \\
                 Balanced Trees / 103 \\
                 Digital Searching / 106 \\
                 Hashing / 108 \\
                 Answers to Exercises / 117 \\
                 The MMIX Assembly Language / 117 \\
                 Applications to Permutations / 120 \\
                 Input and Output / 120 \\
                 Introduction / 122 \\
                 Sequential Allocation / 123 \\
                 Linked Allocation / 124 \\
                 Circular Lists / 128 \\
                 Doubly Linked Lists / 130 \\
                 Arrays and Orthogonal Lists / 132 \\
                 Traversing Binary Trees / 134 \\
                 Binary Tree Representation of Trees / 136 \\
                 Lists and Garbage Collection / 139 \\
                 Dynamic Storage Allocation / 140 \\
                 Choice of modulus / 147 \\
                 Potency / 148 \\
                 Other Methods / 148 \\
                 Numerical Distributions / 149 \\
                 Summary / 150 \\
                 Positional Number Systems / 150 \\
                 Single-Precision Calculations / 151 \\
                 Accuracy of Floating Point Arithmetic / 152 \\
                 Double-Precision Calculations / 153 \\
                 The Classical Algorithms / 156 \\
                 Radix Conversion / 158 \\
                 The Greatest Common Divisor / 160 \\
                 Analysis of Euclid's Algorithm / 160 \\
                 Evaluation of Powers / 161 \\
                 Evaluation of Polynomials / 161 \\
                 Sorting / 162 \\
                 Internal Sorting / 162 \\
                 Sorting by Insertion / 165 \\
                 Sorting by Exchanging / 169 \\
                 Sorting by Selection / 174 \\
                 Sorting by Distribution / 179 \\
                 Minimum-Comparison Sorting / 180 \\
                 Summary, History, and Bibliography / 183 \\
                 Sequential Searching / 183 \\
                 Searching an Ordered Table / 184 \\
                 Binary Tree Searching / 185 \\
                 Balanced Trees / 185 \\
                 Digital Searching / 186 \\
                 Hashing / 186 \\
                 Acknowledgements / 188 \\
                 Index / 189",
}


bibtype: article
filename: fparith.bib
label: Seo:2015:MMS
author: Hwajeong Seo and Zhe Liu and Yasuyuki Nogami and Jongseok Choi and Howon Kim
title: Montgomery multiplication and squaring for Optimal Prime Fields
journal: j-COMPUT-SECUR
volume: 52
pages: 276--291
month: jul
monthnumber: 7
year: 2015
CODEN: CPSEDU
ISSN: 0167-4048 (print), 1872-6208 (electronic) OR 01674048 (print), 18726208 (electronic)
ISSNL: 0167-4048 OR 01674048
bibdate: Mon Sep 23 09:46:42 MDT 2019
bibtimestamp: 2019.09.23 09:46:42 MDT
URL: https://www.sciencedirect.com/science/article/pii/S0167404815000371
fjournal: Computers & Security
acknowledgement: ack-nhfb
journalURL: https://www.sciencedirect.com/science/journal/01674048
rawauthor: Hwajeong Seo and Zhe Liu and Yasuyuki Nogami and Jongseok Choi and Howon Kim
rawtitle: {Montgomery} multiplication and squaring for Optimal Prime Fields
bibsource:
https://www.math.utah.edu/pub/tex/bib/computsecur2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Seo:2015:MMS,
  author =       "Hwajeong Seo and Zhe Liu and Yasuyuki Nogami and
                 Jongseok Choi and Howon Kim",
  title =        "{Montgomery} multiplication and squaring for Optimal
                 Prime Fields",
  journal =      j-COMPUT-SECUR,
  volume =       "52",
  number =       "??",
  pages =        "276--291",
  month =        jul,
  year =         "2015",
  CODEN =        "CPSEDU",
  ISSN =         "0167-4048 (print), 1872-6208 (electronic)",
  ISSN-L =       "0167-4048",
  bibdate =      "Mon Sep 23 09:46:42 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/computsecur2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.sciencedirect.com/science/article/pii/S0167404815000371",
  acknowledgement = ack-nhfb,
  fjournal =     "Computers \& Security",
  journal-URL =  "https://www.sciencedirect.com/science/journal/01674048",
}


bibtype: techreport
filename: fparith.bib
label: Solovyev:2015:REFa
author: Alexey Solovyev and Charles Jacobsen and Zvonimir Rakamaric and Ganesh Gopalakrishnan
title: Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions
type: Technical Report
number: UUCS-15-001
institution: School of Computing, University of Utah
address: Salt Lake City, UT 84112 USA
pages: 31
day: 6
month: apr
monthnumber: 4
year: 2015
bibdate: Fri Apr 17 18:02:17 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.04.17 18:02:17 ???
URL: http://formalverification.cs.utah.edu/papers/fptaylor_report.pdf
acknowledgement: ack-nhfb
rawauthor: Alexey Solovyev and Charles Jacobsen and Zvonimir Rakamari{\'c} and Ganesh Gopalakrishnan
rawtitle: Rigorous Estimation of Floating-Point Round-off Errors with Symbolic {Taylor} Expansions
entry:
@TechReport{Solovyev:2015:REFa,
  author =       "Alexey Solovyev and Charles Jacobsen and Zvonimir
                 Rakamari{\'c} and Ganesh Gopalakrishnan",
  title =        "Rigorous Estimation of Floating-Point Round-off Errors
                 with Symbolic {Taylor} Expansions",
  type =         "Technical Report",
  number =       "UUCS-15-001",
  institution =  "School of Computing, University of Utah",
  address =      "Salt Lake City, UT 84112 USA",
  pages =        "31",
  day =          "6",
  month =        apr,
  year =         "2015",
  bibdate =      "Fri Apr 17 18:02:17 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://formalverification.cs.utah.edu/papers/fptaylor_report.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Solovyev:2015:REFb
author: Alexey Solovyev and Charles Jacobsen and Zvonimir Rakamaric and Ganesh Gopalakrishnan
booktitle: 20th International Symposium on Formal Methods (FM 2015), Oslo, Norway
title: Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions
publisher: pub-SV
address: pub-SV:adr
pages: 532--550
day: 6
month: jun
monthnumber: 6
year: 2015
bibdate: Fri Apr 17 18:02:17 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.04.17 18:02:17 ???
note: Extended version available in cite Solovyev:2015:REFa.
URL: http://formalverification.cs.utah.edu/papers/fptaylor_report.pdf
acknowledgement: ack-nhfb
rawauthor: Alexey Solovyev and Charles Jacobsen and Zvonimir Rakamari{\'c} and Ganesh Gopalakrishnan
rawbooktitle: {20th International Symposium on Formal Methods (FM 2015), Oslo, Norway}
rawnote: Extended version available in \cite{Solovyev:2015:REFa}.
rawtitle: Rigorous Estimation of Floating-Point Round-off Errors with Symbolic {Taylor} Expansions
entry:
@InProceedings{Solovyev:2015:REFb,
  author =       "Alexey Solovyev and Charles Jacobsen and Zvonimir
                 Rakamari{\'c} and Ganesh Gopalakrishnan",
  booktitle =    "{20th International Symposium on Formal Methods (FM
                 2015), Oslo, Norway}",
  title =        "Rigorous Estimation of Floating-Point Round-off Errors
                 with Symbolic {Taylor} Expansions",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "532--550",
  day =          "6",
  month =        jun,
  year =         "2015",
  bibdate =      "Fri Apr 17 18:02:17 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  note =         "Extended version available in
                 \cite{Solovyev:2015:REFa}.",
  URL =          "http://formalverification.cs.utah.edu/papers/fptaylor_report.pdf",
  acknowledgement = ack-nhfb,
}


