# bibtab

bibtype: book
filename: fparith.bib
label: Kastner:2010:AOT
author: Ryan Kastner and Anup Hosangadi and Farzan Fallah
title: Arithmetic optimization techniques for hardware and software design
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
pages: vii + 187
year: 2010
ISBN: 0-521-88099-8 OR 0521880998
ISBN13: 978-0-521-88099-2 OR 9780521880992
LCCN: QA76.9.C62 K37 2010; QA76.9.C62 KAS 2010
bibdate: Mon Jul 12 14:32:42 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; library.ox.ac.uk:210/ADVANCE
bibtimestamp: 2010.07.12 14:32:42 MDT
URL: http://assets.cambridge.org/97805218/80992/cover/9780521880992.jpg
acknowledgement: ack-nhfb
rawauthor: Ryan Kastner and Anup Hosangadi and Farzan Fallah
rawtitle: Arithmetic optimization techniques for hardware and software design
abstract:
Obtain better system performance, lower energy consumption, and avoid hand-coding arithmetic functions with this concise guide to automated optimization techniques for hardware and software design. High-level compiler optimizations and high-speed architectures for implementing FIR filters are covered, which can improve performance in communications, signal processing, computer graphics, and cryptography. Clearly explained algorithms and illustrative examples throughout make it easy to understand the techniques and write software for their implementation. Background information on the synthesis of arithmetic expressions and computer arithmetic is also included, making the book ideal for newcomers to the subject. This is an invaluable resource for researchers, professionals, and graduate students working in system level design and automation, compilers, and VLSI CAD.

subject:
computer arithmetic; electronic digital computers; design and construction; computer software; development; mathematical optimization

TOC:
1. Introduction; 2. Use of arithmetic expressions 3. Software compilation 4. Hardware synthesis 5. Fundamentals of digital arithmetic 6. Polynomial expressions 7. Linear systems

rawabstract:
Obtain better system performance, lower energy consumption, and avoid hand-coding arithmetic functions with this concise guide to automated optimization techniques for hardware and software design. High-level compiler optimizations and high-speed architectures for implementing FIR filters are covered, which can improve performance in communications, signal processing, computer graphics, and cryptography. Clearly explained algorithms and illustrative examples throughout make it easy to understand the techniques and write software for their implementation. Background information on the synthesis of arithmetic expressions and computer arithmetic is also included, making the book ideal for newcomers to the subject. This is an invaluable resource for researchers, professionals, and graduate students working in system level design and automation, compilers, and VLSI CAD.

rawTOC:
1. Introduction; \\ 2. Use of arithmetic expressions\\ 3. Software compilation\\ 4. Hardware synthesis\\ 5. Fundamentals of digital arithmetic\\ 6. Polynomial expressions\\ 7. Linear systems

entry:
@Book{Kastner:2010:AOT,
  author =       "Ryan Kastner and Anup Hosangadi and Farzan Fallah",
  title =        "Arithmetic optimization techniques for hardware and
                 software design",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  pages =        "vii + 187",
  year =         "2010",
  ISBN =         "0-521-88099-8",
  ISBN-13 =      "978-0-521-88099-2",
  LCCN =         "QA76.9.C62 K37 2010; QA76.9.C62 KAS 2010",
  bibdate =      "Mon Jul 12 14:32:42 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 library.ox.ac.uk:210/ADVANCE",
  URL =          "http://assets.cambridge.org/97805218/80992/cover/9780521880992.jpg",
  abstract =     "Obtain better system performance, lower energy
                 consumption, and avoid hand-coding arithmetic functions
                 with this concise guide to automated optimization
                 techniques for hardware and software design. High-level
                 compiler optimizations and high-speed architectures for
                 implementing FIR filters are covered, which can improve
                 performance in communications, signal processing,
                 computer graphics, and cryptography. Clearly explained
                 algorithms and illustrative examples throughout make it
                 easy to understand the techniques and write software
                 for their implementation. Background information on the
                 synthesis of arithmetic expressions and computer
                 arithmetic is also included, making the book ideal for
                 newcomers to the subject. This is an invaluable
                 resource for researchers, professionals, and graduate
                 students working in system level design and automation,
                 compilers, and VLSI CAD.",
  acknowledgement = ack-nhfb,
  subject =      "computer arithmetic; electronic digital computers;
                 design and construction; computer software;
                 development; mathematical optimization",
  tableofcontents = "1. Introduction; \\
                 2. Use of arithmetic expressions\\
                 3. Software compilation\\
                 4. Hardware synthesis\\
                 5. Fundamentals of digital arithmetic\\
                 6. Polynomial expressions\\
                 7. Linear systems",
}


bibtype: book
filename: fparith.bib
label: Kirk:2010:PMP
author: David B. Kirk and Wen-mei W. Hwu
title: Programming Massively Parallel Processors: a Hands-on Approach
publisher: pub-MORGAN-KAUFMANN
address: pub-MORGAN-KAUFMANN:adr
pages: xviii + 258
year: 2010
ISBN: 0-12-381472-3 OR 0123814723
ISBN13: 978-0-12-381472-2 OR 9780123814722
LCCN: QA76.642 .K57 2010
bibdate: Thu Jul 29 13:33:50 MDT 2010
bibtimestamp: 2010.07.29 13:33:50 MDT
note: Chapter 7 (pages 125--140) discusses GPU floating-point considerations.
keywords: CUDA; nVIDIA
acknowledgement: ack-nhfb
rawauthor: David B. Kirk and Wen-mei W. Hwu
rawnote: Chapter 7 (pages 125--140) discusses GPU floating-point considerations.
rawtitle: Programming Massively Parallel Processors: a Hands-on Approach
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/pvm.bib; https://www.math.utah.edu/pub/tex/bib/scpe.bib; https://www.math.utah.edu/pub/tex/bib/unix.bib; z3950.bibsys.no:2100/BIBSYS; z3950.loc.gov:7090/Voyager

subject:
parallel programming (computer science); parallel processing (electronic computers); multiprocessors; computer architecture

TOC:
1: Introduction 1.1 GPUs as Parallel Computers 1.2 Architecture of a Modern GPU 1.3 Why More Speed or Parallelism? 1.4 Parallel Programming Languages and Models 1.5 Overarching Goals 1.6 Organization of the Book 2: History of GPU Computing 2.1. Evolution of Graphics Pipelines The Era of Fixed Function Graphics Pipeline Evolution of Programmable Real-Time Graphics Unified Graphics and Computing Processors 2.2. GPGPU: an Intermediate Step Scalable GPUs Recent Developments Future Trends 3: Introduction to CUDA 3.1. Data Parallelism 3.2. CUDA Program Structure 3.3. A Matrix--Matrix Multiplication Example 3.4. Device Memories and Data Transfer 3.5. Kernel Functions and Threading 3.6. Summary Function Declarations Kernel Launch Predefined Variables Runtime API 4: CUDA Threads 4.1. CUDA Thread Organization 4.2. More on BlockIdx and ThreadIdx 4.3. Synchronization and Transparent Scalability 4.4. Thread Assignment 4.5. Thread Scheduling and Latency Tolerance 4.6. Summary 5: CUDA Memories 5.1. Importance of Memory Access Efficiency 5.2. CUDA Device Memory Types 5.3. A Strategy for Reducing Global Memory Traffic 5.4. Memory as a Limiting Factor to Parallelism 5.5. Summary 6: Performance Considerations 6.1. More on Thread Execution 6.2. Global Memory Bandwidth 6.3. Dynamic Partitioning of SM Resources 6.4. Data Prefetching 6.5. Instruction Mix 6.6. Thread Granularity 6.7. Measured Performance and Summary 7: Floating-Point Considerations 7.1. Floating-Point Format Normalized representation of M Excess encoding of E 7.2. Representable Numbers 7.3. Special Bit Patterns and Precision 7.4. Arithmetic Accuracy and Rounding 7.5. Algorithm Considerations 7.6. Summary 8: Application Case Study I Advanced MRI Reconstruction 8.1. Application Background 8.2. Iterative Reconstruction 8.3. Computing FHd Step 1: Determine the Kernel Parallelism Structure Step 2: Getting Around the Memory Bandwidth Limitation Step 3: Use Hardware Trigonometry Functions Step 4: Experimental Performance Testing 8.4. Final Evaluation 9: Application Case Study II Molecular Visualization and Analysis 9.1. Application Background 9.2. A Simple Kernel Implementation 9.3. Instruction Execution Efficiency 9.4. Memory Coalescing 9.5. Additional Performance Comparisons 9.6. Using Multiple GPUs 10: Parallel Programming and Computational Thinking 10.1. Goals of Parallel Programming 10.2. Problem Decomposition 10.3. Algorithm Selection 10.4. Computational Thinking 11: A Brief Introduction to OpenCL? 11.1. Background 11.2. Data Parallelism Model 11.3. Device Architecture 11.4. Kernel Functions 11.5. Device Management and Kernel Launch 11.6. Electrostatic Potential Map in OpenCL 11.7. Summary 12: Conclusion and Future Outlook 12.1. Goals Revisited 12.2. Memory Architecture Evolution 12.3. Kernel Execution Control Evolution 12.4. Core Performance 12.5. Programming Environment 12.6. A Bright Outlook Appendix A: Matrix Multiplication Example Code Appendix B: Speed and feed of current generation CUDA devices

rawTOC:
1: Introduction \\ 1.1 GPUs as Parallel Computers \\ 1.2 Architecture of a Modern GPU \\ 1.3 Why More Speed or Parallelism? \\ 1.4 Parallel Programming Languages and Models \\ 1.5 Overarching Goals \\ 1.6 Organization of the Book \\ 2: History of GPU Computing \\ 2.1. Evolution of Graphics Pipelines The Era of Fixed Function Graphics Pipeline Evolution of Programmable Real-Time Graphics Unified Graphics and Computing Processors \\ 2.2. GPGPU: an Intermediate Step Scalable GPUs Recent Developments Future Trends \\ 3: Introduction to CUDA \\ 3.1. Data Parallelism \\ 3.2. CUDA Program Structure \\ 3.3. A Matrix--Matrix Multiplication Example \\ 3.4. Device Memories and Data Transfer \\ 3.5. Kernel Functions and Threading \\ 3.6. Summary Function Declarations Kernel Launch Predefined Variables Runtime API \\ 4: CUDA Threads \\ 4.1. CUDA Thread Organization \\ 4.2. More on BlockIdx and ThreadIdx \\ 4.3. Synchronization and Transparent Scalability \\ 4.4. Thread Assignment \\ 4.5. Thread Scheduling and Latency Tolerance \\ 4.6. Summary \\ 5: CUDA Memories \\ 5.1. Importance of Memory Access Efficiency \\ 5.2. CUDA Device Memory Types \\ 5.3. A Strategy for Reducing Global Memory Traffic \\ 5.4. Memory as a Limiting Factor to Parallelism \\ 5.5. Summary \\ 6: Performance Considerations \\ 6.1. More on Thread Execution \\ 6.2. Global Memory Bandwidth \\ 6.3. Dynamic Partitioning of SM Resources \\ 6.4. Data Prefetching \\ 6.5. Instruction Mix \\ 6.6. Thread Granularity \\ 6.7. Measured Performance and Summary \\ \\ 7: Floating-Point Considerations \\ 7.1. Floating-Point Format Normalized representation of M Excess encoding of E \\ 7.2. Representable Numbers \\ 7.3. Special Bit Patterns and Precision \\ 7.4. Arithmetic Accuracy and Rounding \\ 7.5. Algorithm Considerations \\ 7.6. Summary \\ 8: Application Case Study I \\ Advanced MRI Reconstruction \\ 8.1. Application Background \\ 8.2. Iterative Reconstruction \\ 8.3. Computing FHd \\ Step 1: Determine the Kernel Parallelism Structure \\ Step 2: Getting Around the Memory Bandwidth Limitation \\ Step 3: Use Hardware Trigonometry Functions \\ Step 4: Experimental Performance Testing \\ 8.4. Final Evaluation \\ 9: Application Case Study II \\ Molecular Visualization and Analysis \\ 9.1. Application Background \\ 9.2. A Simple Kernel Implementation \\ 9.3. Instruction Execution Efficiency \\ 9.4. Memory Coalescing \\ 9.5. Additional Performance Comparisons \\ 9.6. Using Multiple GPUs \\ 10: Parallel Programming and Computational Thinking \\ 10.1. Goals of Parallel Programming \\ 10.2. Problem Decomposition \\ 10.3. Algorithm Selection \\ 10.4. Computational Thinking \\ 11: A Brief Introduction to OpenCL? \\ 11.1. Background \\ 11.2. Data Parallelism Model \\ 11.3. Device Architecture \\ 11.4. Kernel Functions \\ 11.5. Device Management and Kernel Launch \\ 11.6. Electrostatic Potential Map in OpenCL \\ 11.7. Summary \\ 12: Conclusion and Future Outlook \\ 12.1. Goals Revisited \\ 12.2. Memory Architecture Evolution \\ 12.3. Kernel Execution Control Evolution \\ 12.4. Core Performance \\ 12.5. Programming Environment \\ 12.6. A Bright Outlook \\ Appendix A: Matrix Multiplication Example Code \\ Appendix B: Speed and feed of current generation CUDA devices

entry:
@Book{Kirk:2010:PMP,
  author =       "David B. Kirk and Wen-mei W. Hwu",
  title =        "Programming Massively Parallel Processors: a Hands-on
                 Approach",
  publisher =    pub-MORGAN-KAUFMANN,
  address =      pub-MORGAN-KAUFMANN:adr,
  pages =        "xviii + 258",
  year =         "2010",
  ISBN =         "0-12-381472-3",
  ISBN-13 =      "978-0-12-381472-2",
  LCCN =         "QA76.642 .K57 2010",
  bibdate =      "Thu Jul 29 13:33:50 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/pvm.bib;
                 https://www.math.utah.edu/pub/tex/bib/scpe.bib;
                 https://www.math.utah.edu/pub/tex/bib/unix.bib;
                 z3950.bibsys.no:2100/BIBSYS;
                 z3950.loc.gov:7090/Voyager",
  note =         "Chapter 7 (pages 125--140) discusses GPU
                 floating-point considerations.",
  acknowledgement = ack-nhfb,
  keywords =     "CUDA; nVIDIA",
  libnote =      "Not yet in my library.",
  subject =      "parallel programming (computer science); parallel
                 processing (electronic computers); multiprocessors;
                 computer architecture",
  tableofcontents = "1: Introduction \\
                 1.1 GPUs as Parallel Computers \\
                 1.2 Architecture of a Modern GPU \\
                 1.3 Why More Speed or Parallelism? \\
                 1.4 Parallel Programming Languages and Models \\
                 1.5 Overarching Goals \\
                 1.6 Organization of the Book \\
                 2: History of GPU Computing \\
                 2.1. Evolution of Graphics Pipelines The Era of Fixed
                 Function Graphics Pipeline Evolution of Programmable
                 Real-Time Graphics Unified Graphics and Computing
                 Processors \\
                 2.2. GPGPU: an Intermediate Step Scalable GPUs Recent
                 Developments Future Trends \\
                 3: Introduction to CUDA \\
                 3.1. Data Parallelism \\
                 3.2. CUDA Program Structure \\
                 3.3. A Matrix--Matrix Multiplication Example \\
                 3.4. Device Memories and Data Transfer \\
                 3.5. Kernel Functions and Threading \\
                 3.6. Summary Function Declarations Kernel Launch
                 Predefined Variables Runtime API \\
                 4: CUDA Threads \\
                 4.1. CUDA Thread Organization \\
                 4.2. More on BlockIdx and ThreadIdx \\
                 4.3. Synchronization and Transparent Scalability \\
                 4.4. Thread Assignment \\
                 4.5. Thread Scheduling and Latency Tolerance \\
                 4.6. Summary \\
                 5: CUDA Memories \\
                 5.1. Importance of Memory Access Efficiency \\
                 5.2. CUDA Device Memory Types \\
                 5.3. A Strategy for Reducing Global Memory Traffic \\
                 5.4. Memory as a Limiting Factor to Parallelism \\
                 5.5. Summary \\
                 6: Performance Considerations \\
                 6.1. More on Thread Execution \\
                 6.2. Global Memory Bandwidth \\
                 6.3. Dynamic Partitioning of SM Resources \\
                 6.4. Data Prefetching \\
                 6.5. Instruction Mix \\
                 6.6. Thread Granularity \\
                 6.7. Measured Performance and Summary \\
                 \\
                 7: Floating-Point Considerations \\
                 7.1. Floating-Point Format Normalized representation of
                 M Excess encoding of E \\
                 7.2. Representable Numbers \\
                 7.3. Special Bit Patterns and Precision \\
                 7.4. Arithmetic Accuracy and Rounding \\
                 7.5. Algorithm Considerations \\
                 7.6. Summary \\
                 8: Application Case Study I \\
                 Advanced MRI Reconstruction \\
                 8.1. Application Background \\
                 8.2. Iterative Reconstruction \\
                 8.3. Computing FHd \\
                 Step 1: Determine the Kernel Parallelism Structure \\
                 Step 2: Getting Around the Memory Bandwidth Limitation
                 \\
                 Step 3: Use Hardware Trigonometry Functions \\
                 Step 4: Experimental Performance Testing \\
                 8.4. Final Evaluation \\
                 9: Application Case Study II \\
                 Molecular Visualization and Analysis \\
                 9.1. Application Background \\
                 9.2. A Simple Kernel Implementation \\
                 9.3. Instruction Execution Efficiency \\
                 9.4. Memory Coalescing \\
                 9.5. Additional Performance Comparisons \\
                 9.6. Using Multiple GPUs \\
                 10: Parallel Programming and Computational Thinking \\
                 10.1. Goals of Parallel Programming \\
                 10.2. Problem Decomposition \\
                 10.3. Algorithm Selection \\
                 10.4. Computational Thinking \\
                 11: A Brief Introduction to OpenCL? \\
                 11.1. Background \\
                 11.2. Data Parallelism Model \\
                 11.3. Device Architecture \\
                 11.4. Kernel Functions \\
                 11.5. Device Management and Kernel Launch \\
                 11.6. Electrostatic Potential Map in OpenCL \\
                 11.7. Summary \\
                 12: Conclusion and Future Outlook \\
                 12.1. Goals Revisited \\
                 12.2. Memory Architecture Evolution \\
                 12.3. Kernel Execution Control Evolution \\
                 12.4. Core Performance \\
                 12.5. Programming Environment \\
                 12.6. A Bright Outlook \\
                 Appendix A: Matrix Multiplication Example Code \\
                 Appendix B: Speed and feed of current generation CUDA
                 devices",
}


bibtype: article
filename: fparith.bib
label: Knezevic:2010:FIM
author: M. Knezevic and F. Vercauteren and I. Verbauwhede
title: Faster Interleaved Modular Multiplication Based on Barrett and Montgomery Reduction Methods
journal: j-IEEE-TRANS-COMPUT
volume: 59
number: 12
pages: 1715--1721
year: 2010
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.93
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Jul 3 11:52:34 MDT 2011
bibtimestamp: 2011.07.03 11:52:34 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5453352
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: M. Knezevic and F. Vercauteren and I. Verbauwhede
rawtitle: Faster Interleaved Modular Multiplication Based on {Barrett} and {Montgomery} Reduction Methods
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib

entry:
@Article{Knezevic:2010:FIM,
  author =       "M. Knezevic and F. Vercauteren and I. Verbauwhede",
  title =        "Faster Interleaved Modular Multiplication Based on
                 {Barrett} and {Montgomery} Reduction Methods",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "59",
  number =       "12",
  pages =        "1715--1721",
  month =        "????",
  year =         "2010",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.93",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Jul 3 11:52:34 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5453352",
  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: Kong:2010:RMR
author: Inwook Kong and E. E. Swartzlander
title: A Rounding Method to Reduce the Required Multiplier Precision for Goldschmidt Division
journal: j-IEEE-TRANS-COMPUT
volume: 59
number: 12
pages: 1703--1708
month: dec
monthnumber: 12
year: 2010
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.86
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Jul 3 11:52:34 MDT 2011
bibtimestamp: 2011.07.03 11:52:34 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5453345
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Inwook Kong and E. E. Swartzlander
rawtitle: A Rounding Method to Reduce the Required Multiplier Precision for {Goldschmidt} Division
entry:
@Article{Kong:2010:RMR,
  author =       "Inwook Kong and E. E. Swartzlander",
  title =        "A Rounding Method to Reduce the Required Multiplier
                 Precision for {Goldschmidt} Division",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "59",
  number =       "12",
  pages =        "1703--1708",
  month =        dec,
  year =         "2010",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.86",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Jul 3 11:52:34 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5453345",
  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: Kornerup:2010:CCRa
author: Peter Kornerup and Vincent Lefevre and Nicholas Louvet and Jean-Michel Muller
title: On the computation of correctly-rounded sums
type: Research report
number: RR-7262
institution: INRIA
address: Lyon, France
pages: 28
month: apr
monthnumber: 4
year: 2010
bibdate: Tue Dec 26 14:47:39 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.12.26 14:47:39 ???
URL: http://hal.inria.fr/inria-00475279
acknowledgement: ack-nhfb
rawauthor: Peter Kornerup and Vincent Lef{\`e}vre and Nicholas Louvet and Jean-Michel Muller
rawtitle: On the computation of correctly-rounded sums
abstract:
This paper presents a study of some basic blocks needed in the design of floating-point summation algorithms. In particular, we show that among the set of the algorithms with no comparisons performing only floating-point additions / subtractions, the 2Sum algorithm introduced by Knuth is minimal, both in terms of number of operations and depth of the dependency graph. We investigate the possible use of another algorithm, Dekker's Fast2Sum algorithm, in radix-10 arithmetic. We give methods for computing, in radix 10, the floating-point number nearest the average value of two floating-point numbers. Under reasonable conditions, we also prove that no algorithms performing only round-to-nearest additions / subtractions exist to compute the round-to-nearest sum of at least three floating-point numbers. Starting from an algorithm due to Boldo and Melquiond, we also present new results about the computation of the correctly-rounded sum of three floating-point numbers.

rawabstract:
This paper presents a study of some basic blocks needed in the design of floating-point summation algorithms. In particular, we show that among the set of the algorithms with no comparisons performing only floating-point additions\slash subtractions, the 2Sum algorithm introduced by Knuth is minimal, both in terms of number of operations and depth of the dependency graph. We investigate the possible use of another algorithm, Dekker's Fast2Sum algorithm, in radix-10 arithmetic. We give methods for computing, in radix 10, the floating-point number nearest the average value of two floating-point numbers. Under reasonable conditions, we also prove that no algorithms performing only round-to-nearest additions\slash subtractions exist to compute the round-to-nearest sum of at least three floating-point numbers. Starting from an algorithm due to Boldo and Melquiond, we also present new results about the computation of the correctly-rounded sum of three floating-point numbers.

entry:
@TechReport{Kornerup:2010:CCRa,
  author =       "Peter Kornerup and Vincent Lef{\`e}vre and Nicholas
                 Louvet and Jean-Michel Muller",
  title =        "On the computation of correctly-rounded sums",
  type =         "Research report",
  number =       "RR-7262",
  institution =  "INRIA",
  address =      "Lyon, France",
  pages =        "28",
  month =        apr,
  year =         "2010",
  bibdate =      "Tue Dec 26 14:47:39 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://hal.inria.fr/inria-00475279",
  abstract =     "This paper presents a study of some basic blocks
                 needed in the design of floating-point summation
                 algorithms. In particular, we show that among the set
                 of the algorithms with no comparisons performing only
                 floating-point additions\slash subtractions, the 2Sum
                 algorithm introduced by Knuth is minimal, both in terms
                 of number of operations and depth of the dependency
                 graph. We investigate the possible use of another
                 algorithm, Dekker's Fast2Sum algorithm, in radix-10
                 arithmetic. We give methods for computing, in radix 10,
                 the floating-point number nearest the average value of
                 two floating-point numbers. Under reasonable
                 conditions, we also prove that no algorithms performing
                 only round-to-nearest additions\slash subtractions
                 exist to compute the round-to-nearest sum of at least
                 three floating-point numbers. Starting from an
                 algorithm due to Boldo and Melquiond, we also present
                 new results about the computation of the
                 correctly-rounded sum of three floating-point
                 numbers.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Kornerup:2010:CCRb
author: Peter Kornerup and Christoph Lauter and Vincent Lefevre and Nicolas Louvet and Jean-Michel Muller
title: Computing correctly rounded integer powers in floating-point arithmetic
journal: j-TOMS
volume: 37
number: 1
pages: 4:1--4:23
month: jan
monthnumber: 1
year: 2010
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1644001.1644005
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Mar 15 10:45:33 MDT 2010
bibsource: http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.03.15 10:45:33 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
keywords: correct rounding; floating-point arithmetic; integer power function
acknowledgement: ack-nhfb
articleno: 4
journalURL: https://dl.acm.org/loi/toms
rawtitle: Computing correctly rounded integer powers in floating-point arithmetic
abstract:
We introduce several algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a em fused multiply-add (fma) instruction is available. For bounded, yet very large values of the exponent, we aim at obtaining correctly rounded results in round-to-nearest mode, that is, our algorithms return the floating-point number that is nearest the exact value.

rawabstract:
We introduce several algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a {\em fused multiply-add\/} (fma) instruction is available. For bounded, yet very large values of the exponent, we aim at obtaining correctly rounded results in round-to-nearest mode, that is, our algorithms return the floating-point number that is nearest the exact value.

rawauthor:
Peter Kornerup and Christoph Lauter and Vincent Lef{\`e}vre and Nicolas Louvet and Jean-Michel Muller

entry:
@Article{Kornerup:2010:CCRb,
  author =       "Peter Kornerup and Christoph Lauter and Vincent
                 Lef{\`e}vre and Nicolas Louvet and Jean-Michel Muller",
  title =        "Computing correctly rounded integer powers in
                 floating-point arithmetic",
  journal =      j-TOMS,
  volume =       "37",
  number =       "1",
  pages =        "4:1--4:23",
  month =        jan,
  year =         "2010",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1644001.1644005",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Mar 15 10:45:33 MDT 2010",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We introduce several algorithms for accurately
                 evaluating powers to a positive integer in
                 floating-point arithmetic, assuming a {\em fused
                 multiply-add\/} (fma) instruction is available. For
                 bounded, yet very large values of the exponent, we aim
                 at obtaining correctly rounded results in
                 round-to-nearest mode, that is, our algorithms return
                 the floating-point number that is nearest the exact
                 value.",
  acknowledgement = ack-nhfb,
  articleno =    "4",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  keywords =     "correct rounding; floating-point arithmetic; integer
                 power function",
}


bibtype: book
filename: fparith.bib
label: Kornerup:2010:FPN
author: Peter Kornerup and David W. Matula
title: Finite Precision Number Systems and Arithmetic
volume: 133
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
pages: xv + 699
year: 2010
ISBN: 0-521-76135-2 (hardcover) OR 0521761352 (hardcover)
ISBN13: 978-0-521-76135-2 (hardcover) OR 9780521761352 (hardcover)
LCCN: QA248 .K627 2010
bibdate: Sun Jun 19 14:21:37 MDT 2016
bibtimestamp: 2016.06.19 14:21:37 MDT
series: Encyclopedia of mathematics and its applications
subject: Arithmetic; Foundations
acknowledgement: ack-nhfb
rawauthor: Peter Kornerup and David W. Matula
rawtitle: Finite Precision Number Systems and Arithmetic
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager

URL:
http://assets.cambridge.org/97805217/61352/cover/9780521761352.jpg; http://catdir.loc.gov/catdir/enhancements/fy1011/2010030521-b.html; http://catdir.loc.gov/catdir/enhancements/fy1011/2010030521-d.html; http://catdir.loc.gov/catdir/enhancements/fy1011/2010030521-t.html

abstract:
Fundamental arithmetic operations support virtually all of the engineering, scientific, and financial computations required for practical applications, from cryptography, to financial planning, to rocket science. This comprehensive reference provides researchers with the thorough understanding of number representations that is a necessary foundation for designing efficient arithmetic algorithms. Using the elementary foundations of radix number systems as a basis for arithmetic, the authors develop and compare alternative algorithms for the fundamental operations of addition, multiplication, division, and square root with precisely defined roundings. Various finite precision number systems are investigated, with the focus on comparative analysis of practically efficient algorithms for closed arithmetic operations over these systems. Each chapter begins with an introduction to its contents and ends with bibliographic notes and an extensive bibliography. The book may also be used for graduate teaching: problems and exercises are scattered throughout the text and a solutions manual is available for instructors.

TOC:
Preface / xi 1. Radix polynomial representations / 1 2. Base and digit set conversion / 59 3. Addition / 4. Multiplication / 5. Division / 275 6. Square root / 398 7. Floating-point number systems / 447 8. Modular arithmetic and residue number systems / 528 9. Rational arithmetic / 63 Author index / 691 Index / 693

rawabstract:
Fundamental arithmetic operations support virtually all of the engineering, scientific, and financial computations required for practical applications, from cryptography, to financial planning, to rocket science. This comprehensive reference provides researchers with the thorough understanding of number representations that is a necessary foundation for designing efficient arithmetic algorithms. Using the elementary foundations of radix number systems as a basis for arithmetic, the authors develop and compare alternative algorithms for the fundamental operations of addition, multiplication, division, and square root with precisely defined roundings. Various finite precision number systems are investigated, with the focus on comparative analysis of practically efficient algorithms for closed arithmetic operations over these systems. Each chapter begins with an introduction to its contents and ends with bibliographic notes and an extensive bibliography. The book may also be used for graduate teaching: problems and exercises are scattered throughout the text and a solutions manual is available for instructors.

rawTOC:
Preface / xi \\ 1. Radix polynomial representations / 1 \\ 2. Base and digit set conversion / 59 \\ 3. Addition / \\ 4. Multiplication / \\ 5. Division / 275 \\ 6. Square root / 398 \\ 7. Floating-point number systems / 447 \\ 8. Modular arithmetic and residue number systems / 528 \\ 9. Rational arithmetic / 63 \\ Author index / 691 \\ Index / 693

entry:
@Book{Kornerup:2010:FPN,
  author =       "Peter Kornerup and David W. Matula",
  title =        "Finite Precision Number Systems and Arithmetic",
  volume =       "133",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  pages =        "xv + 699",
  year =         "2010",
  ISBN =         "0-521-76135-2 (hardcover)",
  ISBN-13 =      "978-0-521-76135-2 (hardcover)",
  LCCN =         "QA248 .K627 2010",
  bibdate =      "Sun Jun 19 14:21:37 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 z3950.loc.gov:7090/Voyager",
  series =       "Encyclopedia of mathematics and its applications",
  URL =          "http://assets.cambridge.org/97805217/61352/cover/9780521761352.jpg;
                 http://catdir.loc.gov/catdir/enhancements/fy1011/2010030521-b.html;
                 http://catdir.loc.gov/catdir/enhancements/fy1011/2010030521-d.html;
                 http://catdir.loc.gov/catdir/enhancements/fy1011/2010030521-t.html",
  abstract =     "Fundamental arithmetic operations support virtually
                 all of the engineering, scientific, and financial
                 computations required for practical applications, from
                 cryptography, to financial planning, to rocket science.
                 This comprehensive reference provides researchers with
                 the thorough understanding of number representations
                 that is a necessary foundation for designing efficient
                 arithmetic algorithms. Using the elementary foundations
                 of radix number systems as a basis for arithmetic, the
                 authors develop and compare alternative algorithms for
                 the fundamental operations of addition, multiplication,
                 division, and square root with precisely defined
                 roundings. Various finite precision number systems are
                 investigated, with the focus on comparative analysis of
                 practically efficient algorithms for closed arithmetic
                 operations over these systems. Each chapter begins with
                 an introduction to its contents and ends with
                 bibliographic notes and an extensive bibliography. The
                 book may also be used for graduate teaching: problems
                 and exercises are scattered throughout the text and a
                 solutions manual is available for instructors.",
  acknowledgement = ack-nhfb,
  subject =      "Arithmetic; Foundations",
  tableofcontents = "Preface / xi \\
                 1. Radix polynomial representations / 1 \\
                 2. Base and digit set conversion / 59 \\
                 3. Addition / \\
                 4. Multiplication / \\
                 5. Division / 275 \\
                 6. Square root / 398 \\
                 7. Floating-point number systems / 447 \\
                 8. Modular arithmetic and residue number systems / 528
                 \\
                 9. Rational arithmetic / 63 \\
                 Author index / 691 \\
                 Index / 693",
}


bibtype: techreport
filename: fparith.bib
label: KrusemanAretz:2010:DCP
author: F. E. J. Kruseman Aretz
type: Computer Science Report
number: 1002
institution: Technische Universiteit Eindhoven
address: Eindhoven, The Netherlands
pages: 58
day: 30
month: mar
monthnumber: 3
year: 2010
bibdate: Mon Oct 17 16:37:56 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.10.17 16:37:56 ???
URL: http://repository.tue.nl/674735
acknowledgement: ack-nhfb
rawauthor: F. E. J. {Kruseman Aretz}
title:
Design and correctness proof of an emulation of the floating-point operations of the Electrologica X8: a case study

abstract:
Some time ago I decided to write an emulator for a Dutch computer from the sixties of the previous century, the Electrologica X8, in order to be able to run its ALGOL 60 implementation and to do some measurements with it. That emulator was written in (standard ISO) Pascal.

rawabstract:
Some time ago I decided to write an emulator for a Dutch computer from the sixties of the previous century, the Electrologica X8, in order to be able to run its ALGOL 60 implementation and to do some measurements with it. That emulator was written in (standard ISO) Pascal.

rawtitle:
Design and correctness proof of an emulation of the floating-point operations of the {Electrologica X8}: a case study

entry:
@TechReport{KrusemanAretz:2010:DCP,
  author =       "F. E. J. {Kruseman Aretz}",
  title =        "Design and correctness proof of an emulation of the
                 floating-point operations of the {Electrologica X8}: a
                 case study",
  type =         "Computer Science Report",
  number =       "1002",
  institution =  "Technische Universiteit Eindhoven",
  address =      "Eindhoven, The Netherlands",
  pages =        "58",
  day =          "30",
  month =        mar,
  year =         "2010",
  bibdate =      "Mon Oct 17 16:37:56 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://repository.tue.nl/674735",
  abstract =     "Some time ago I decided to write an emulator for a
                 Dutch computer from the sixties of the previous
                 century, the Electrologica X8, in order to be able to
                 run its ALGOL 60 implementation and to do some
                 measurements with it. That emulator was written in
                 (standard ISO) Pascal.",
  acknowledgement = ack-nhfb,
  remark-1 =     "From page 2: ``Electrologica was a Dutch computer
                 factory, founded in 1956. It produced the Electrologica
                 X1 (from 1958) and its successor, the Electrologica X8
                 (from 1965). The latter was more or less upwards
                 compatible with the former, about a factor of 12
                 faster, and in addition it had floating-point hardware:
                 an additional register F and instructions for
                 floating-point addition, subtraction, multiplication,
                 and division.''",
  remark-2 =     "From page 3: ``the floating-point operations `+', `-',
                 `*', and `/' all delivered the best possible result,
                 i.e. that floating-point number in standard form whose
                 value is closest to the exact result. In case that the
                 exact result of the operation was precisely midway two
                 consecutive floating-point numbers, the result was
                 rounded upwards for positive results and downwards for
                 negative results.''",
  remark-3 =     "From page 29: ``The number representation of the EL X8
                 was in one-complement, with preference for $-0$ over
                 $+0$ (I have not analyzed whether an implementation
                 directly in the one-complement representation needs
                 more, or perhaps less guarding bits).",
  remark-4 =     "Page 56 presents the four-step Newton--Raphson
                 algorithm used to implement the square-root operation
                 on the EL8, producing results with a maximum error of
                 0.90 bit.",
}


bibtype: article
filename: fparith.bib
label: Lamotte:2010:CVC
author: Jean-Luc Lamotte and Jean-Marie Chesneaux and Fabienne Jezequel
title: CADNA_C: a version of CADNA for use with C or C++ programs
journal: j-COMP-PHYS-COMM
volume: 181
number: 11
pages: 1925--1926
month: nov
monthnumber: 11
year: 2010
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2010.07.006
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Sat Feb 11 09:54:31 MST 2012
bibtimestamp: 2012.02.11 09:54:31 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465510002353
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Jean-Luc Lamotte and Jean-Marie Chesneaux and Fabienne J{\'e}z{\'e}quel
rawtitle: {CADNA\_C}: a version of {CADNA} for use with {C} or {C++} programs
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Lamotte:2010:CVC,
  author =       "Jean-Luc Lamotte and Jean-Marie Chesneaux and Fabienne
                 J{\'e}z{\'e}quel",
  title =        "{CADNA\_C}: a version of {CADNA} for use with {C} or
                 {C++} programs",
  journal =      j-COMP-PHYS-COMM,
  volume =       "181",
  number =       "11",
  pages =        "1925--1926",
  month =        nov,
  year =         "2010",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2010.07.006",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Sat Feb 11 09:54:31 MST 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465510002353",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: fparith.bib
label: Lefevre:2010:LTL
title: LEMA: towards a language for reliable arithmetic
journal: j-ACM-COMM-COMP-ALGEBRA
volume: 44
number: 2
pages: 41--52
month: jun
monthnumber: 6
year: 2010
DOI: https://doi.org/10.1145/1838599.1838622
ISSN: 1932-2232 (print), 1932-2240 (electronic) OR 19322232 (print), 19322240 (electronic)
ISSNL: 1932-2232 OR 19322232
bibdate: Mon Aug 2 13:47:24 MDT 2010
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.08.02 13:47:24 MDT
fjournal: ACM Communications in Computer Algebra
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
issue: 172
rawtitle: {LEMA}: towards a language for reliable arithmetic
author:
Vincent Lefevre and Philippe Theveny and Florent de Dinechin and Claude-Pierre Jeannerod and Christophe Mouilleron and David Pfannholzer and Nathalie Revol

abstract:
Generating certified and efficient numerical codes requires information ranging from the mathematical level to the representation of numbers. Even though the mathematical semantics can be expressed using the content part of MathML, this language does not encompass the implementation on computers. Indeed various arithmetics may be involved, like floating-point or fixed-point, in fixed precision or arbitrary precision, and current tools do not handle all of these.par Therefore we propose in this paper LEMA (Langage pour les Expressions Mathematiques Annotees), a descriptive language based on MathML with additional expressiveness. LEMA will be used during the automatic generation of certified numerical codes. Such a generation process typically involves several steps, and LEMA would thus act as a glue to represent and store the information at every stage.par First, we specify in the language the characteristics of the arithmetic as described in the IEEE 754 floating-point standard: formats, exceptions, rounding modes. This can be generalized to other arithmetics. Then, we use annotations to attach a specific arithmetic context to an expression tree. Finally, considering the evaluation of the expression in this context allows us to deduce several properties on the result, like being exact or being an exception. Other useful properties include numerical ranges and error bounds.

rawabstract:
Generating certified and efficient numerical codes requires information ranging from the mathematical level to the representation of numbers. Even though the mathematical semantics can be expressed using the content part of MathML, this language does not encompass the implementation on computers. Indeed various arithmetics may be involved, like floating-point or fixed-point, in fixed precision or arbitrary precision, and current tools do not handle all of these.\par Therefore we propose in this paper LEMA (Langage pour les Expressions Math{\'e}matiques Annot{\'e}es), a descriptive language based on MathML with additional expressiveness. LEMA will be used during the automatic generation of certified numerical codes. Such a generation process typically involves several steps, and LEMA would thus act as a glue to represent and store the information at every stage.\par First, we specify in the language the characteristics of the arithmetic as described in the IEEE 754 floating-point standard: formats, exceptions, rounding modes. This can be generalized to other arithmetics. Then, we use annotations to attach a specific arithmetic context to an expression tree. Finally, considering the evaluation of the expression in this context allows us to deduce several properties on the result, like being exact or being an exception. Other useful properties include numerical ranges and error bounds.

rawauthor:
Vincent Lef{\'e}vre and Philippe Th{\'e}veny and Florent de Dinechin and Claude-Pierre Jeannerod and Christophe Mouilleron and David Pfannholzer and Nathalie Revol

entry:
@Article{Lefevre:2010:LTL,
  author =       "Vincent Lef{\'e}vre and Philippe Th{\'e}veny and
                 Florent de Dinechin and Claude-Pierre Jeannerod and
                 Christophe Mouilleron and David Pfannholzer and
                 Nathalie Revol",
  title =        "{LEMA}: towards a language for reliable arithmetic",
  journal =      j-ACM-COMM-COMP-ALGEBRA,
  volume =       "44",
  number =       "2",
  pages =        "41--52",
  month =        jun,
  year =         "2010",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/1838599.1838622",
  ISSN =         "1932-2232 (print), 1932-2240 (electronic)",
  ISSN-L =       "1932-2232",
  bibdate =      "Mon Aug 2 13:47:24 MDT 2010",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Generating certified and efficient numerical codes
                 requires information ranging from the mathematical
                 level to the representation of numbers. Even though the
                 mathematical semantics can be expressed using the
                 content part of MathML, this language does not
                 encompass the implementation on computers. Indeed
                 various arithmetics may be involved, like
                 floating-point or fixed-point, in fixed precision or
                 arbitrary precision, and current tools do not handle
                 all of these.\par

                 Therefore we propose in this paper LEMA (Langage pour
                 les Expressions Math{\'e}matiques Annot{\'e}es), a
                 descriptive language based on MathML with additional
                 expressiveness. LEMA will be used during the automatic
                 generation of certified numerical codes. Such a
                 generation process typically involves several steps,
                 and LEMA would thus act as a glue to represent and
                 store the information at every stage.\par

                 First, we specify in the language the characteristics
                 of the arithmetic as described in the IEEE 754
                 floating-point standard: formats, exceptions, rounding
                 modes. This can be generalized to other arithmetics.
                 Then, we use annotations to attach a specific
                 arithmetic context to an expression tree. Finally,
                 considering the evaluation of the expression in this
                 context allows us to deduce several properties on the
                 result, like being exact or being an exception. Other
                 useful properties include numerical ranges and error
                 bounds.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Communications in Computer Algebra",
  issue =        "172",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
}


bibtype: article
filename: fparith.bib
label: Lima:2010:KBA
author: J. B. Lima and D. Panario and Qiang Wang
title: A Karatsuba-Based Algorithm for Polynomial Multiplication in Chebyshev Form
journal: j-IEEE-TRANS-COMPUT
volume: 59
number: 6
pages: 835--841
year: 2010
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.45
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Jul 3 11:52:29 MDT 2011
bibtimestamp: 2011.07.03 11:52:29 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5416688
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J. B. Lima and D. Panario and Qiang Wang
rawtitle: A {Karatsuba}-Based Algorithm for Polynomial Multiplication in {Chebyshev} Form
entry:
@Article{Lima:2010:KBA,
  author =       "J. B. Lima and D. Panario and Qiang Wang",
  title =        "A {Karatsuba}-Based Algorithm for Polynomial
                 Multiplication in {Chebyshev} Form",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "59",
  number =       "6",
  pages =        "835--841",
  month =        "????",
  year =         "2010",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.45",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Jul 3 11:52:29 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5416688",
  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: Loitsch:2010:PFP
author: Florian Loitsch
title: Printing floating-point numbers quickly and accurately with integers
journal: j-SIGPLAN
volume: 45
number: 6
pages: 233--243
month: jun
monthnumber: 6
year: 2010
CODEN: SINODQ
DOI: https://doi.org/10.1145/1809028.1806623
ISSN: 0362-1340 (print), 1523-2867 (print), 1558-1160 OR 03621340 (print), 15232867 (print), 15581160
ISSNL: 0362-1340 OR 03621340
bibdate: Fri Oct 8 17:53:18 MDT 2010
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.10.08 17:53:18 MDT
fjournal: ACM SIGPLAN Notices
keywords: correct rounding; dtoa; floating-point printing
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J706
rawauthor: Florian Loitsch
rawtitle: Printing floating-point numbers quickly and accurately with integers
abstract:
We present algorithms for accurately converting floating-point numbers to decimal representation. They are fast (up to 4 times faster than commonly used algorithms that use high-precision integers) and correct: any printed number will evaluate to the same number, when read again.par Our algorithms are fast, because they require only fixed-size integer arithmetic. The sole requirement for the integer type is that it has at least two more bits than the significand of the floating-point number. Hence, for IEEE 754 double-precision numbers (having a 53-bit significand) an integer type with 55 bits is sufficient. Moreover we show how to exploit additional bits to improve the generated output.par We present three algorithms with different properties: the first algorithm is the most basic one, and does not take advantage of any extra bits. It simply shows how to perform the binary-to-decimal transformation with the minimal number of bits. Our second algorithm improves on the first one by using the additional bits to produce a shorter (often the shortest) result.par Finally we propose a third version that can be used when the shortest output is a requirement. The last algorithm either produces optimal decimal representations (with respect to shortness and rounding) or rejects its input. For IEEE 754 double-precision numbers and 64-bit integers roughly 99.4% of all numbers can be processed efficiently. The remaining 0.6% are rejected and need to be printed by a slower complete algorithm.

rawabstract:
We present algorithms for accurately converting floating-point numbers to decimal representation. They are fast (up to 4 times faster than commonly used algorithms that use high-precision integers) and correct: any printed number will evaluate to the same number, when read again.\par Our algorithms are fast, because they require only fixed-size integer arithmetic. The sole requirement for the integer type is that it has at least two more bits than the significand of the floating-point number. Hence, for IEEE 754 double-precision numbers (having a 53-bit significand) an integer type with 55 bits is sufficient. Moreover we show how to exploit additional bits to improve the generated output.\par We present three algorithms with different properties: the first algorithm is the most basic one, and does not take advantage of any extra bits. It simply shows how to perform the binary-to-decimal transformation with the minimal number of bits. Our second algorithm improves on the first one by using the additional bits to produce a shorter (often the shortest) result.\par Finally we propose a third version that can be used when the shortest output is a requirement. The last algorithm either produces optimal decimal representations (with respect to shortness and rounding) or rejects its input. For IEEE 754 double-precision numbers and 64-bit integers roughly 99.4\% of all numbers can be processed efficiently. The remaining 0.6\% are rejected and need to be printed by a slower complete algorithm.

entry:
@Article{Loitsch:2010:PFP,
  author =       "Florian Loitsch",
  title =        "Printing floating-point numbers quickly and accurately
                 with integers",
  journal =      j-SIGPLAN,
  volume =       "45",
  number =       "6",
  pages =        "233--243",
  month =        jun,
  year =         "2010",
  CODEN =        "SINODQ",
  DOI =          "https://doi.org/10.1145/1809028.1806623",
  ISSN =         "0362-1340 (print), 1523-2867 (print), 1558-1160
                 (electronic)",
  ISSN-L =       "0362-1340",
  bibdate =      "Fri Oct 8 17:53:18 MDT 2010",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We present algorithms for accurately converting
                 floating-point numbers to decimal representation. They
                 are fast (up to 4 times faster than commonly used
                 algorithms that use high-precision integers) and
                 correct: any printed number will evaluate to the same
                 number, when read again.\par

                 Our algorithms are fast, because they require only
                 fixed-size integer arithmetic. The sole requirement for
                 the integer type is that it has at least two more bits
                 than the significand of the floating-point number.
                 Hence, for IEEE 754 double-precision numbers (having a
                 53-bit significand) an integer type with 55 bits is
                 sufficient. Moreover we show how to exploit additional
                 bits to improve the generated output.\par

                 We present three algorithms with different properties:
                 the first algorithm is the most basic one, and does not
                 take advantage of any extra bits. It simply shows how
                 to perform the binary-to-decimal transformation with
                 the minimal number of bits. Our second algorithm
                 improves on the first one by using the additional bits
                 to produce a shorter (often the shortest)
                 result.\par

                 Finally we propose a third version that can be used
                 when the shortest output is a requirement. The last
                 algorithm either produces optimal decimal
                 representations (with respect to shortness and
                 rounding) or rejects its input. For IEEE 754
                 double-precision numbers and 64-bit integers roughly
                 99.4\% of all numbers can be processed efficiently. The
                 remaining 0.6\% are rejected and need to be printed by
                 a slower complete algorithm.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGPLAN Notices",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J706",
  keywords =     "correct rounding; dtoa; floating-point printing",
}


bibtype: inproceedings
filename: fparith.bib
label: Louvet:2010:NRA
author: Nicolas Louvet and Jean-Michel Muller and Adrien Panhaleux
title: Newton--Raphson algorithms for floating-point division using an FMA
crossref: Charot:2010:API
pages: 200--207
year: 2010
DOI: https://doi.org/10.1109/ASAP.2010.5540948
bibdate: Thu Feb 17 08:21:17 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.02.17 08:21:17 ???
acknowledgement: ack-nhfb
rawauthor: Nicolas Louvet and Jean-Michel Muller and Adrien Panhaleux
rawtitle: {Newton--Raphson} algorithms for floating-point division using an {FMA}
abstract:
When implementing a function $f$ in floating-point arithmetic, if we wish correct rounding and good performance, it is important to know if there are input floating-point values $x$ such that $ f(x) $ is either the middle of two consecutive floating-point numbers (assuming rounded-to-nearest arithmetic), or a floating-point number (assuming rounded toward $ pm infty $ or toward $0$ arithmetic). In the first case, we say that $ f(x) $ is a midpoint, and in the second case, we say that $ f(x) $ is an exact point. For some usual algebraic functions and various floating-point formats, we prove whether or not there exist midpoints or exact points. When there exist midpoints or exact points, we characterize them or list all of them (if there are not too many). The results and the techniques presented in this paper can be used in particular to deal with both the binary and the decimal formats defined in the IEEE 754-2008 standard for floating-point arithmetic.

rawabstract:
When implementing a function $f$ in floating-point arithmetic, if we wish correct rounding and good performance, it is important to know if there are input floating-point values $x$ such that $ f(x) $ is either the middle of two consecutive floating-point numbers (assuming rounded-to-nearest arithmetic), or a floating-point number (assuming rounded toward $ \pm \infty $ or toward $0$ arithmetic). In the first case, we say that $ f(x) $ is a midpoint, and in the second case, we say that $ f(x) $ is an exact point. For some usual algebraic functions and various floating-point formats, we prove whether or not there exist midpoints or exact points. When there exist midpoints or exact points, we characterize them or list all of them (if there are not too many). The results and the techniques presented in this paper can be used in particular to deal with both the binary and the decimal formats defined in the IEEE 754-2008 standard for floating-point arithmetic.

entry:
@InProceedings{Louvet:2010:NRA,
  author =       "Nicolas Louvet and Jean-Michel Muller and Adrien
                 Panhaleux",
  title =        "{Newton--Raphson} algorithms for floating-point
                 division using an {FMA}",
  crossref =     "Charot:2010:API",
  pages =        "200--207",
  year =         "2010",
  DOI =          "https://doi.org/10.1109/ASAP.2010.5540948",
  bibdate =      "Thu Feb 17 08:21:17 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "When implementing a function $f$ in floating-point
                 arithmetic, if we wish correct rounding and good
                 performance, it is important to know if there are input
                 floating-point values $x$ such that $ f(x) $ is either
                 the middle of two consecutive floating-point numbers
                 (assuming rounded-to-nearest arithmetic), or a
                 floating-point number (assuming rounded toward $ \pm
                 \infty $ or toward $0$ arithmetic). In the first case,
                 we say that $ f(x) $ is a midpoint, and in the second
                 case, we say that $ f(x) $ is an exact point. For some
                 usual algebraic functions and various floating-point
                 formats, we prove whether or not there exist midpoints
                 or exact points. When there exist midpoints or exact
                 points, we characterize them or list all of them (if
                 there are not too many). The results and the techniques
                 presented in this paper can be used in particular to
                 deal with both the binary and the decimal formats
                 defined in the IEEE 754-2008 standard for
                 floating-point arithmetic.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Maruyama:2010:SVN
title: Sparc64 VIIIfx: a New-Generation Octocore Processor for Petascale Computing
journal: j-IEEE-MICRO
volume: 30
number: 2
pages: 30--40
month: mar
monthnumber: 3
year: 2010
CODEN: IEMIDZ
DOI: https://doi.org/10.1109/MM.2010.40
ISSN: 0272-1732 (print), 1937-4143 (electronic) OR 02721732 (print), 19374143 (electronic)
ISSNL: 0272-1732 OR 02721732
bibdate: Fri Apr 23 06:37:18 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.04.23 06:37:18 MDT
fjournal: IEEE Micro
keywords: fused multiply-add (FMA); IEEE 754-2008 arithmetic
acknowledgement: ack-nhfb
journalURL: http://www.computer.org/csdl/mags/mi/index.html
rawtitle: {Sparc64 VIIIfx}: a New-Generation Octocore Processor for Petascale Computing
author:
Takumi Maruyama and Toshio Yoshida and Ryuji Kan and Iwao Yamazaki and Shuji Yamamura and Noriyuki Takahashi and Mikio Hondou and Hiroshi Okano

rawauthor:
Takumi Maruyama and Toshio Yoshida and Ryuji Kan and Iwao Yamazaki and Shuji Yamamura and Noriyuki Takahashi and Mikio Hondou and Hiroshi Okano

entry:
@Article{Maruyama:2010:SVN,
  author =       "Takumi Maruyama and Toshio Yoshida and Ryuji Kan and
                 Iwao Yamazaki and Shuji Yamamura and Noriyuki Takahashi
                 and Mikio Hondou and Hiroshi Okano",
  title =        "{Sparc64 VIIIfx}: a New-Generation Octocore Processor
                 for Petascale Computing",
  journal =      j-IEEE-MICRO,
  volume =       "30",
  number =       "2",
  pages =        "30--40",
  month =        mar # "\slash " # apr,
  year =         "2010",
  CODEN =        "IEMIDZ",
  DOI =          "https://doi.org/10.1109/MM.2010.40",
  ISSN =         "0272-1732 (print), 1937-4143 (electronic)",
  ISSN-L =       "0272-1732",
  bibdate =      "Fri Apr 23 06:37:18 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Micro",
  journal-URL =  "http://www.computer.org/csdl/mags/mi/index.html",
  keywords =     "fused multiply-add (FMA); IEEE 754-2008 arithmetic",
}


bibtype: article
filename: fparith.bib
label: Mathews:2010:AOE
author: Deborah Mathews
title: Abstract only: an empirical study of parallel big number arithmetic
journal: j-ACM-COMM-COMP-ALGEBRA
volume: 44
number: 2
pages: 25--25
month: jun
monthnumber: 6
year: 2010
DOI: https://doi.org/10.1145/1838599.1838615
ISSN: 1932-2232 (print), 1932-2240 (electronic) OR 19322232 (print), 19322240 (electronic)
ISSNL: 1932-2232 OR 19322232
bibdate: Mon Aug 2 13:47:24 MDT 2010
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.08.02 13:47:24 MDT
fjournal: ACM Communications in Computer Algebra
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
issue: 172
rawauthor: Deborah Mathews
rawtitle: Abstract only: an empirical study of parallel big number arithmetic
abstract:
To capitalize on multi-core processing, it would be good to perform big number arithmetic in parallel. While a performance increase equal to the number of processors is theoretically possible, our experiments show that in practice the likelihood of gaining any performance increase for big number arithmetic through parallel processing is low. A speed-up approaching the number of processors was not achieved for multiplication until the operands had at least 215 bits. No performance gain was realized for addition. The base algorithm used for multiplication was O(n2) and, therefore, suboptimal. The expected performance gain achieved when parallelizing a more efficient base algorithm should be even smaller.

rawabstract:
To capitalize on multi-core processing, it would be good to perform big number arithmetic in parallel. While a performance increase equal to the number of processors is theoretically possible, our experiments show that in practice the likelihood of gaining any performance increase for big number arithmetic through parallel processing is low. A speed-up approaching the number of processors was not achieved for multiplication until the operands had at least 215 bits. No performance gain was realized for addition. The base algorithm used for multiplication was O(n2) and, therefore, suboptimal. The expected performance gain achieved when parallelizing a more efficient base algorithm should be even smaller.

entry:
@Article{Mathews:2010:AOE,
  author =       "Deborah Mathews",
  title =        "Abstract only: an empirical study of parallel big
                 number arithmetic",
  journal =      j-ACM-COMM-COMP-ALGEBRA,
  volume =       "44",
  number =       "2",
  pages =        "25--25",
  month =        jun,
  year =         "2010",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/1838599.1838615",
  ISSN =         "1932-2232 (print), 1932-2240 (electronic)",
  ISSN-L =       "1932-2232",
  bibdate =      "Mon Aug 2 13:47:24 MDT 2010",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "To capitalize on multi-core processing, it would be
                 good to perform big number arithmetic in parallel.
                 While a performance increase equal to the number of
                 processors is theoretically possible, our experiments
                 show that in practice the likelihood of gaining any
                 performance increase for big number arithmetic through
                 parallel processing is low. A speed-up approaching the
                 number of processors was not achieved for
                 multiplication until the operands had at least 215
                 bits. No performance gain was realized for addition.
                 The base algorithm used for multiplication was O(n2)
                 and, therefore, suboptimal. The expected performance
                 gain achieved when parallelizing a more efficient base
                 algorithm should be even smaller.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Communications in Computer Algebra",
  issue =        "172",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
}


bibtype: article
filename: fparith.bib
label: Mehrotra:2010:SLR
author: Sanjay Mehrotra and Zhifeng Li
title: Segment LLL Reduction of Lattice Bases Using Modular Arithmetic
journal: j-ALGORITHMS-BASEL
volume: 3
number: 3
pages: 224--243
month: sep
monthnumber: 9
year: 2010
CODEN: ALGOCH
DOI: https://doi.org/10.3390/a3030224
ISSN: 1999-4893 (electronic) OR 19994893 (electronic)
ISSNL: 1999-4893 OR 19994893
bibdate: Fri May 3 13:50:12 MDT 2019
bibtimestamp: 2019.05.03 13:50:12 MDT
URL: https://www.mdpi.com/1999-4893/3/3/224
fjournal: Algorithms (Basel)
acknowledgement: ack-nhfb
journalURL: https://www.mdpi.com/journal/algorithms
rawauthor: Sanjay Mehrotra and Zhifeng Li
rawtitle: Segment {LLL} Reduction of Lattice Bases Using Modular Arithmetic
bibsource:
https://www.math.utah.edu/pub/tex/bib/algorithms.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Mehrotra:2010:SLR,
  author =       "Sanjay Mehrotra and Zhifeng Li",
  title =        "Segment {LLL} Reduction of Lattice Bases Using Modular
                 Arithmetic",
  journal =      j-ALGORITHMS-BASEL,
  volume =       "3",
  number =       "3",
  pages =        "224--243",
  month =        sep,
  year =         "2010",
  CODEN =        "ALGOCH",
  DOI =          "https://doi.org/10.3390/a3030224",
  ISSN =         "1999-4893 (electronic)",
  ISSN-L =       "1999-4893",
  bibdate =      "Fri May 3 13:50:12 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/algorithms.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.mdpi.com/1999-4893/3/3/224",
  acknowledgement = ack-nhfb,
  fjournal =     "Algorithms (Basel)",
  journal-URL =  "https://www.mdpi.com/journal/algorithms",
  pubdates =     "Received: 28 May 2010 / Accepted: 29 June 2010 /
                 Published: 12 July 2010",
}


bibtype: article
filename: fparith.bib
label: Meyer:2010:CGT
author: Quirin Meyer and Jochen Suss muth and Gerd Suss ner and Marc Stamminger and Gunther Greiner
title: Computer Graphics Theory: On Floating-Point Normal Vectors
journal: j-CGF
volume: 29
number: 4
pages: 1405--1409
month: jun
monthnumber: 6
year: 2010
CODEN: CGFODY
DOI: https://doi.org/10.1111/j.1467-8659.2010.01737.x
ISSN: 0167-7055 (print), 1467-8659 (electronic) OR 01677055 (print), 14678659 (electronic)
ISSNL: 0167-7055 OR 01677055
bibdate: Sat May 11 13:28:01 MDT 2013
bibsource: https://www.math.utah.edu/pub/tex/bib/cgf.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2013.05.11 13:28:01 MDT
fjournal: Computer Graphics Forum
acknowledgement: ack-nhfb
journalURL: http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-8659/
onlinedate: 26 Aug 2010
rawtitle: Computer Graphics Theory: On Floating-Point Normal Vectors
rawauthor:
Quirin Meyer and Jochen S{\"u}{\ss}muth and Gerd Su{\ss}ner and Marc Stamminger and G{\"u}nther Greiner

entry:
@Article{Meyer:2010:CGT,
  author =       "Quirin Meyer and Jochen S{\"u}{\ss}muth and Gerd
                 Su{\ss}ner and Marc Stamminger and G{\"u}nther
                 Greiner",
  title =        "Computer Graphics Theory: On Floating-Point Normal
                 Vectors",
  journal =      j-CGF,
  volume =       "29",
  number =       "4",
  pages =        "1405--1409",
  month =        jun,
  year =         "2010",
  CODEN =        "CGFODY",
  DOI =          "https://doi.org/10.1111/j.1467-8659.2010.01737.x",
  ISSN =         "0167-7055 (print), 1467-8659 (electronic)",
  ISSN-L =       "0167-7055",
  bibdate =      "Sat May 11 13:28:01 MDT 2013",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cgf.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Com{\-}pu{\-}ter Graphics Forum",
  journal-URL =  "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-8659/",
  onlinedate =   "26 Aug 2010",
}


bibtype: techreport
filename: fparith.bib
label: Moller:2010:IDI
author: Niels Moller and Torbjorn Granlund
title: Improved division by invariant integers
type: Report
institution: Centre for Industrial and Applied Mathematics, KTH
address: Stockholm, Sweden
pages: 10
day: 13
month: apr
monthnumber: 4
year: 2010
bibdate: Mon Oct 14 15:57:19 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2019.10.14 15:57:19 ???
URL: https://gmplib.org/~tege/division-paper.pdf
acknowledgement: ack-nhfb
rawauthor: Niels M{\"o}ller and Torbj{\"o}rn Granlund
rawtitle: Improved division by invariant integers
abstract:
This paper considers the problem of dividing a two-word integer by a single-word integer, together with a few extensions and applications. Due to lack of efficient division instructions in current processors, the division is performed as a multiplication using a precomputed single-word approximation of the reciprocal of the divisor, followed by a couple of adjustment steps. There are three common types of unsigned multiplication instructions; we define full word multiplication (umul) which produces the two-word product of two single-word integers, low multiplication (umullo) which produces only the least significant word of the product, and high multiplication (umulhi), which produces only the most significant word. We describe an algorithm which produces a quotient and remainder using one umul and one umullo. This is an improvement over earlier methods, since the new method uses cheaper multiplication operations. It turns out we also get some additional savings from simpler adjustment conditions. The algorithm has been implemented in version 4.3 of the GMP library. When applied to the problem of dividing a large integer by a single word, the new algorithm gives a speedup of roughly 30%, benchmarked on AMD and Intel processors in the x86-64 family.

rawabstract:
This paper considers the problem of dividing a two-word integer by a single-word integer, together with a few extensions and applications. Due to lack of efficient division instructions in current processors, the division is performed as a multiplication using a precomputed single-word approximation of the reciprocal of the divisor, followed by a couple of adjustment steps. There are three common types of unsigned multiplication instructions; we define full word multiplication (umul) which produces the two-word product of two single-word integers, low multiplication (umullo) which produces only the least significant word of the product, and high multiplication (umulhi), which produces only the most significant word. We describe an algorithm which produces a quotient and remainder using one umul and one umullo. This is an improvement over earlier methods, since the new method uses cheaper multiplication operations. It turns out we also get some additional savings from simpler adjustment conditions. The algorithm has been implemented in version 4.3 of the GMP library. When applied to the problem of dividing a large integer by a single word, the new algorithm gives a speedup of roughly 30\%, benchmarked on AMD and Intel processors in the x86-64 family.

entry:
@TechReport{Moller:2010:IDI,
  author =       "Niels M{\"o}ller and Torbj{\"o}rn Granlund",
  title =        "Improved division by invariant integers",
  type =         "Report",
  institution =  "Centre for Industrial and Applied Mathematics, KTH",
  address =      "Stockholm, Sweden",
  pages =        "10",
  day =          "13",
  month =        apr,
  year =         "2010",
  bibdate =      "Mon Oct 14 15:57:19 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://gmplib.org/~tege/division-paper.pdf",
  abstract =     "This paper considers the problem of dividing a
                 two-word integer by a single-word integer, together
                 with a few extensions and applications. Due to lack of
                 efficient division instructions in current processors,
                 the division is performed as a multiplication using a
                 precomputed single-word approximation of the reciprocal
                 of the divisor, followed by a couple of adjustment
                 steps. There are three common types of unsigned
                 multiplication instructions; we define full word
                 multiplication (umul) which produces the two-word
                 product of two single-word integers, low multiplication
                 (umullo) which produces only the least significant word
                 of the product, and high multiplication (umulhi), which
                 produces only the most significant word. We describe an
                 algorithm which produces a quotient and remainder using
                 one umul and one umullo. This is an improvement over
                 earlier methods, since the new method uses cheaper
                 multiplication operations. It turns out we also get
                 some additional savings from simpler adjustment
                 conditions. The algorithm has been implemented in
                 version 4.3 of the GMP library. When applied to the
                 problem of dividing a large integer by a single word,
                 the new algorithm gives a speedup of roughly 30\%,
                 benchmarked on AMD and Intel processors in the x86-64
                 family.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Morisita:2010:IEA
author: Hirokazu Morisita and Kenta Inakagata and Yasunori Osana and Naoyuki Fujita and Hideharu Amano
title: Implementation and evaluation of an arithmetic pipeline on FLOPS-$2$D: multi-FPGA system
journal: j-COMP-ARCH-NEWS
volume: 38
number: 4
pages: 8--13
month: sep
monthnumber: 9
year: 2010
CODEN: CANED2
DOI: https://doi.org/10.1145/1926367.1926370
ISSN: 0163-5964 (print), 1943-5851 (electronic) OR 01635964 (print), 19435851 (electronic)
ISSNL: 0163-5964 OR 01635964
bibdate: Thu Jan 20 14:27:03 MST 2011
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.01.20 14:27:03 MST
fjournal: ACM SIGARCH Computer Architecture News
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J89
rawauthor: Hirokazu Morisita and Kenta Inakagata and Yasunori Osana and Naoyuki Fujita and Hideharu Amano
rawtitle: Implementation and evaluation of an arithmetic pipeline on {FLOPS-$2$D}: multi-{FPGA} system
abstract:
UPACS (Unified Platform for Aerospace Computational Simulation) is one of the practical CFD (Computational Fluid Dynamics) packages supporting various selectability. A custom machine for efficient execution of MUSCL; a core functions of UPACS is implemented on FLOPS-2D (Flexibly Linkable Object for Programmable System); multi-FPGA reconfigurable system. The deep and complicated pipeline structure generated from MUSCL dataflow is divided and optimized into two FPGA boards by using a tuning tool called RER. With optimization of the order of operations and pipeline structure, about 60% utilization of the pipeline is achieved even by using serial links between two boards.

rawabstract:
UPACS (Unified Platform for Aerospace Computational Simulation) is one of the practical CFD (Computational Fluid Dynamics) packages supporting various selectability. A custom machine for efficient execution of MUSCL; a core functions of UPACS is implemented on FLOPS-2D (Flexibly Linkable Object for Programmable System); multi-FPGA reconfigurable system. The deep and complicated pipeline structure generated from MUSCL dataflow is divided and optimized into two FPGA boards by using a tuning tool called RER. With optimization of the order of operations and pipeline structure, about 60\% utilization of the pipeline is achieved even by using serial links between two boards.

entry:
@Article{Morisita:2010:IEA,
  author =       "Hirokazu Morisita and Kenta Inakagata and Yasunori
                 Osana and Naoyuki Fujita and Hideharu Amano",
  title =        "Implementation and evaluation of an arithmetic
                 pipeline on {FLOPS-$2$D}: multi-{FPGA} system",
  journal =      j-COMP-ARCH-NEWS,
  volume =       "38",
  number =       "4",
  pages =        "8--13",
  month =        sep,
  year =         "2010",
  CODEN =        "CANED2",
  DOI =          "https://doi.org/10.1145/1926367.1926370",
  ISSN =         "0163-5964 (print), 1943-5851 (electronic)",
  ISSN-L =       "0163-5964",
  bibdate =      "Thu Jan 20 14:27:03 MST 2011",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "UPACS (Unified Platform for Aerospace Computational
                 Simulation) is one of the practical CFD (Computational
                 Fluid Dynamics) packages supporting various
                 selectability. A custom machine for efficient execution
                 of MUSCL; a core functions of UPACS is implemented on
                 FLOPS-2D (Flexibly Linkable Object for Programmable
                 System); multi-FPGA reconfigurable system. The deep and
                 complicated pipeline structure generated from MUSCL
                 dataflow is divided and optimized into two FPGA boards
                 by using a tuning tool called RER. With optimization of
                 the order of operations and pipeline structure, about
                 60\% utilization of the pipeline is achieved even by
                 using serial links between two boards.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGARCH Computer Architecture News",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J89",
}


bibtype: article
filename: fparith.bib
label: Mukherjee:2010:NAC
author: Manideepa Mukherjee and Amitabha Sinha
title: A novel architecture for conversion of binary to single digit double base numbers
journal: j-COMP-ARCH-NEWS
volume: 38
number: 5
pages: 1--6
month: dec
monthnumber: 12
year: 2010
CODEN: CANED2
DOI: https://doi.org/10.1145/1978907.1978909
ISSN: 0163-5964 (print), 1943-5851 (electronic) OR 01635964 (print), 19435851 (electronic)
ISSNL: 0163-5964 OR 01635964
bibdate: Fri May 13 11:25:46 MDT 2011
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.05.13 11:25:46 MDT
fjournal: ACM SIGARCH Computer Architecture News
keywords: Double-Based Number System (DBNS)
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J89
rawauthor: Manideepa Mukherjee and Amitabha Sinha
rawtitle: A novel architecture for conversion of binary to single digit double base numbers
abstract:
Double base number systems are increasingly attractive for many compute intensive applications especially in signal processing because of their capabilities of handling arithmetic operations efficiently. However, the complexity involved in converting binary to DBNS becomes a major bottleneck and the efficiency of performance goes down drastically due to the complexity involved in conversion. Since complexity of multi digit DBNS multiplications and additions increases with the number of digits (index i,j), in this paper a novel conversion scheme has been proposed where a given binary number will be converted to a single digit (index i,j) double base number. The proposed scheme not only reduces the hardware complexity of the arithmetic operations but also reduces the time of execution.

rawabstract:
Double base number systems are increasingly attractive for many compute intensive applications especially in signal processing because of their capabilities of handling arithmetic operations efficiently. However, the complexity involved in converting binary to DBNS becomes a major bottleneck and the efficiency of performance goes down drastically due to the complexity involved in conversion. Since complexity of multi digit DBNS multiplications and additions increases with the number of digits (index i,j), in this paper a novel conversion scheme has been proposed where a given binary number will be converted to a single digit (index i,j) double base number. The proposed scheme not only reduces the hardware complexity of the arithmetic operations but also reduces the time of execution.

entry:
@Article{Mukherjee:2010:NAC,
  author =       "Manideepa Mukherjee and Amitabha Sinha",
  title =        "A novel architecture for conversion of binary to
                 single digit double base numbers",
  journal =      j-COMP-ARCH-NEWS,
  volume =       "38",
  number =       "5",
  pages =        "1--6",
  month =        dec,
  year =         "2010",
  CODEN =        "CANED2",
  DOI =          "https://doi.org/10.1145/1978907.1978909",
  ISSN =         "0163-5964 (print), 1943-5851 (electronic)",
  ISSN-L =       "0163-5964",
  bibdate =      "Fri May 13 11:25:46 MDT 2011",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Double base number systems are increasingly attractive
                 for many compute intensive applications especially in
                 signal processing because of their capabilities of
                 handling arithmetic operations efficiently. However,
                 the complexity involved in converting binary to DBNS
                 becomes a major bottleneck and the efficiency of
                 performance goes down drastically due to the complexity
                 involved in conversion. Since complexity of multi digit
                 DBNS multiplications and additions increases with the
                 number of digits (index i,j), in this paper a novel
                 conversion scheme has been proposed where a given
                 binary number will be converted to a single digit
                 (index i,j) double base number. The proposed scheme not
                 only reduces the hardware complexity of the arithmetic
                 operations but also reduces the time of execution.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGARCH Computer Architecture News",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J89",
  keywords =     "Double-Based Number System (DBNS)",
}


bibtype: book
filename: fparith.bib
label: Muller:2010:HFP
title: Handbook of Floating-Point Arithmetic
publisher: pub-BIRKHAUSER-BOSTON
address: pub-BIRKHAUSER-BOSTON:adr
pages: xxiii + 572
year: 2010
DOI: https://doi.org/10.1007/978-0-8176-4704-9; https://doi.org/10.1007/978-0-8176-4705-6
ISBN: 0-8176-4704-X (hardcover), 0-8176-4705-8 (e-book) OR 081764704X (hardcover), 0817647058 (ebook)
LCCN: QA76.9.C62 H36 2010
MRnumber: MR2568265
bibdate: Thu Jan 27 16:18:58 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib
bibtimestamp: 2011.01.27 16:18:58 ???
acknowledgement: ack-nhfb
rawtitle: Handbook of Floating-Point Arithmetic
author:
Jean-Michel Muller and Nicolas Brisebarre and Florent de Dinechin and Claude-Pierre Jeannerod and Vincent Lefevre and Guillaume Melquiond and Nathalie Revol and Damien Stehle and Serge Torres

ISBN13:
978-0-8176-4704-9 (hardcover), 978-0-8176-4705-6 (e-book) OR 9780817647049 (hardcover), 9780817647056 (ebook)

rawauthor:
Jean-Michel Muller and Nicolas Brisebarre and Florent de Dinechin and Claude-Pierre Jeannerod and Vincent Lef{\`e}vre and Guillaume Melquiond and Nathalie Revol and Damien Stehl{\'e} and Serge Torres

entry:
@Book{Muller:2010:HFP,
  author =       "Jean-Michel Muller and Nicolas Brisebarre and Florent
                 de Dinechin and Claude-Pierre Jeannerod and Vincent
                 Lef{\`e}vre and Guillaume Melquiond and Nathalie Revol
                 and Damien Stehl{\'e} and Serge Torres",
  title =        "Handbook of Floating-Point Arithmetic",
  publisher =    pub-BIRKHAUSER-BOSTON,
  address =      pub-BIRKHAUSER-BOSTON:adr,
  pages =        "xxiii + 572",
  year =         "2010",
  DOI =          "https://doi.org/10.1007/978-0-8176-4704-9;
                 https://doi.org/10.1007/978-0-8176-4705-6",
  ISBN =         "0-8176-4704-X (hardcover), 0-8176-4705-8 (e-book)",
  ISBN-13 =      "978-0-8176-4704-9 (hardcover), 978-0-8176-4705-6
                 (e-book)",
  LCCN =         "QA76.9.C62 H36 2010",
  MRnumber =     "MR2568265",
  bibdate =      "Thu Jan 27 16:18:58 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  price =        "US\$90 (est.)",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Nickolls:2010:GCE
author: John Nickolls and William J. Dally
title: The GPU Computing Era
journal: j-IEEE-MICRO
volume: 30
number: 2
pages: 56--69
month: mar
monthnumber: 3
year: 2010
CODEN: IEMIDZ
DOI: https://doi.org/10.1109/MM.2010.41
ISSN: 0272-1732 (print), 1937-4143 (electronic) OR 02721732 (print), 19374143 (electronic)
ISSNL: 0272-1732 OR 02721732
bibdate: Fri Apr 23 06:37:18 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.04.23 06:37:18 MDT
fjournal: IEEE Micro
keywords: fused multiply-add (FMA); IEEE 754-2008 arithmetic; subnormal numbers in hardware; underflow
acknowledgement: ack-nhfb
journalURL: http://www.computer.org/csdl/mags/mi/index.html
rawauthor: John Nickolls and William J. Dally
rawtitle: The {GPU} Computing Era
remark:
From page 64 of the article:par IEEE 754-2008 floating-point arithmeticpar The Fermi CUDA core floating-point unit implements the IEEE 754-2008 floating-point arithmetic standard for 32-bit single-precision and 64-bit double-precision results, including fused multiply-add (FMA) instructions. FMA computes $ D = A times B + C $ with no loss of precision by retaining full precision in the intermediate product and addition, then rounding the final sum to form the result. Using FMA enables fast division and square-root operations with exactly rounded results.par Fermi raises the throughput of 64-bit double-precision operations to half that of single-precision operations, a dramatic improvement over the T10 GPU. This performance level enables broader deployment of GPUs in high-performance computing. The floating-point instructions handle subnormal numbers at full speed in hardware, allowing small values to retain partial precision rather than flushing them to zero or calculating subnormal values in multicycle software exception handlers as most CPUs do.par The SFUs execute 32-bit floating-point instructions for fast approximations of reciprocal, reciprocal square root, sin, cos, exp, and log functions. The approximations are precise to better than 22 mantissa bits.

entry:
@Article{Nickolls:2010:GCE,
  author =       "John Nickolls and William J. Dally",
  title =        "The {GPU} Computing Era",
  journal =      j-IEEE-MICRO,
  volume =       "30",
  number =       "2",
  pages =        "56--69",
  month =        mar # "\slash " # apr,
  year =         "2010",
  CODEN =        "IEMIDZ",
  DOI =          "https://doi.org/10.1109/MM.2010.41",
  ISSN =         "0272-1732 (print), 1937-4143 (electronic)",
  ISSN-L =       "0272-1732",
  bibdate =      "Fri Apr 23 06:37:18 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Micro",
  journal-URL =  "http://www.computer.org/csdl/mags/mi/index.html",
  keywords =     "fused multiply-add (FMA); IEEE 754-2008 arithmetic;
                 subnormal numbers in hardware; underflow",
  remark =       "From page 64 of the article:\par

                 IEEE 754-2008 floating-point arithmetic\par

                 The Fermi CUDA core floating-point unit implements the
                 IEEE 754-2008 floating-point arithmetic standard for
                 32-bit single-precision and 64-bit double-precision
                 results, including fused multiply-add (FMA)
                 instructions. FMA computes $ D = A \times B + C $ with
                 no loss of precision by retaining full precision in the
                 intermediate product and addition, then rounding the
                 final sum to form the result. Using FMA enables fast
                 division and square-root operations with exactly
                 rounded results.\par

                 Fermi raises the throughput of 64-bit double-precision
                 operations to half that of single-precision operations,
                 a dramatic improvement over the T10 GPU. This
                 performance level enables broader deployment of GPUs in
                 high-performance computing. The floating-point
                 instructions handle subnormal numbers at full speed in
                 hardware, allowing small values to retain partial
                 precision rather than flushing them to zero or
                 calculating subnormal values in multicycle software
                 exception handlers as most CPUs do.\par

                 The SFUs execute 32-bit floating-point instructions for
                 fast approximations of reciprocal, reciprocal square
                 root, sin, cos, exp, and log functions. The
                 approximations are precise to better than 22 mantissa
                 bits.",
}


bibtype: misc
filename: fparith.bib
label: NURCL:2010:VNV
author: Northeastern University Reconfigurable Computing Laboratory
title: Vfloat: The Northeastern Variable precision FLOATing point library
pages: 56--69
year: 2010
bibdate: Sat Oct 9 12:56:05 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.10.09 12:56:05 ???
URL: http://www.ece.neu.edu/groups/rcl/projects/floatingpoint/
acknowledgement: ack-nhfb
howpublished: Web site.
rawauthor: {Northeastern University Reconfigurable Computing Laboratory}
rawtitle: {Vfloat}: The {Northeastern Variable precision FLOATing point library}
entry:
@Misc{NURCL:2010:VNV,
  author =       "{Northeastern University Reconfigurable Computing
                 Laboratory}",
  title =        "{Vfloat}: The {Northeastern Variable precision
                 FLOATing point library}",
  howpublished = "Web site.",
  year =         "2010",
  bibdate =      "Sat Oct 9 12:56:05 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.ece.neu.edu/groups/rcl/projects/floatingpoint/",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: fparith.bib
label: Parhami:2010:CAA
author: Behrooz Parhami
title: Computer arithmetic: algorithms and hardware designs
publisher: pub-OXFORD
address: pub-OXFORD:adr
edition: Second
pages: xxv + 641
year: 2010
ISBN: 0-19-532848-5 (hardcover) OR 0195328485 (hardcover)
ISBN13: 978-0-19-532848-6 (hardcover) OR 9780195328486 (hardcover)
LCCN: QA76.9.C62 P37 2010
bibdate: Sat Sep 20 16:57:27 MDT 2014
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager
bibtimestamp: 2014.09.20 16:57:27 MDT
series: The Oxford series in electrical and computer engineering
subject: Computer arithmetic; Computer algorithms; Computer algorithms.; Computer arithmetic.
acknowledgement: ack-nhfb
rawauthor: Behrooz Parhami
rawtitle: Computer arithmetic: algorithms and hardware designs
URL:
http://www.loc.gov/catdir/enhancements/fy1006/2009034155-d.html; http://www.loc.gov/catdir/enhancements/fy1006/2009034155-t.html

abstract:
Ideal for graduate and senior undergraduate courses in computer arithmetic and advanced digital design, Computer Arithmetic: Algorithms and Hardware Designs, Second Edition, provides a balanced, comprehensive treatment of computer arithmetic. It covers topics in arithmetic unit design and circuit implementation that complement the architectural and algorithmic speedup techniques used in high-performance computer architecture and parallel processing. Using a unified and consistent framework, the text begins with number representation and proceeds through basic arithmetic operations, floating-point arithmetic, and function evaluation methods. Later chapters cover broad design and implementation topics --- including techniques for high-throughput, low-power, fault-tolerant, and reconfigurable arithmetic. An appendix provides a historical view of the field and speculates on its future. An indispensable resource for instruction, professional development, and research, Computer Arithmetic: Algorithms and Hardware Designs, Second Edition, combines broad coverage of the underlying theories of computer arithmetic with numerous examples of practical designs, worked-out examples, and a large collection of meaningful problems. This second edition includes a new chapter on reconfigurable arithmetic, in order to address the fact that arithmetic functions are increasingly being implemented on field-programmable gate arrays (FPGAs) and FPGA-like configurable devices. Updated and thoroughly revised, the book offers new and expanded coverage of saturating adders and multipliers, truncated multipliers, fused multiply-add units, overlapped quotient digit selection, bipartite and multipartite tables, reversible logic, dot notation, modular arithmetic, Montgomery modular reduction, division by constants, IEEE floating-point standard formats, and interval arithmetic.

TOC:
Part I: Number representation Numbers and Arithmetic Representing Signed Numbers Redundant Number Systems Residue Number Systems Part II: Addition / Subtraction. Basic Addition and Counting Carry-Lookahead Adders Variations in Fast Adders Multioperand Addition Part III: Multiplication. Basic Multiplication Schemes High-Radix Multipliers Tree and Array Multipliers Variations in Multipliers Part IV: Division. Basic Division Schemes High-Radix Dividers Variations in Dividers Division by Convergence Part V: Real arithmetic. Floating-Point Representations Floating-Point Operations Errors and Error Control Precise and Certifiable Arithmetic Part VI: Function evaluation. Square-Rooting Methods The CORDIC Algorithms Variations in Function Evaluation Arithmetic by Table Lookup High-Throughput Arithmetic Part VII: Implementation topics Low-Power Arithmetic Fault-Tolerant Arithmetic Reconfigurable Arithmetic

rawabstract:
Ideal for graduate and senior undergraduate courses in computer arithmetic and advanced digital design, Computer Arithmetic: Algorithms and Hardware Designs, Second Edition, provides a balanced, comprehensive treatment of computer arithmetic. It covers topics in arithmetic unit design and circuit implementation that complement the architectural and algorithmic speedup techniques used in high-performance computer architecture and parallel processing. Using a unified and consistent framework, the text begins with number representation and proceeds through basic arithmetic operations, floating-point arithmetic, and function evaluation methods. Later chapters cover broad design and implementation topics --- including techniques for high-throughput, low-power, fault-tolerant, and reconfigurable arithmetic. An appendix provides a historical view of the field and speculates on its future. An indispensable resource for instruction, professional development, and research, Computer Arithmetic: Algorithms and Hardware Designs, Second Edition, combines broad coverage of the underlying theories of computer arithmetic with numerous examples of practical designs, worked-out examples, and a large collection of meaningful problems. This second edition includes a new chapter on reconfigurable arithmetic, in order to address the fact that arithmetic functions are increasingly being implemented on field-programmable gate arrays (FPGAs) and FPGA-like configurable devices. Updated and thoroughly revised, the book offers new and expanded coverage of saturating adders and multipliers, truncated multipliers, fused multiply-add units, overlapped quotient digit selection, bipartite and multipartite tables, reversible logic, dot notation, modular arithmetic, Montgomery modular reduction, division by constants, IEEE floating-point standard formats, and interval arithmetic.

rawTOC:
Part I: Number representation \\ Numbers and Arithmetic \\ Representing Signed Numbers \\ Redundant Number Systems \\ Residue Number Systems \\ Part II: Addition\slash Subtraction. Basic Addition and Counting \\ Carry-Lookahead Adders \\ Variations in Fast Adders \\ Multioperand Addition \\ Part III: Multiplication. Basic Multiplication Schemes \\ High-Radix Multipliers \\ Tree and Array Multipliers \\ Variations in Multipliers \\ Part IV: Division. Basic Division Schemes \\ High-Radix Dividers \\ Variations in Dividers \\ Division by Convergence \\ Part V: Real arithmetic. Floating-Point Representations \\ Floating-Point Operations \\ Errors and Error Control \\ Precise and Certifiable Arithmetic \\ Part VI: Function evaluation. Square-Rooting Methods \\ The CORDIC Algorithms \\ Variations in Function Evaluation \\ Arithmetic by Table Lookup \\ High-Throughput Arithmetic \\ Part VII: Implementation topics \\ Low-Power Arithmetic \\ Fault-Tolerant Arithmetic \\ Reconfigurable Arithmetic

entry:
@Book{Parhami:2010:CAA,
  author =       "Behrooz Parhami",
  title =        "Computer arithmetic: algorithms and hardware designs",
  publisher =    pub-OXFORD,
  address =      pub-OXFORD:adr,
  edition =      "Second",
  pages =        "xxv + 641",
  year =         "2010",
  ISBN =         "0-19-532848-5 (hardcover)",
  ISBN-13 =      "978-0-19-532848-6 (hardcover)",
  LCCN =         "QA76.9.C62 P37 2010",
  bibdate =      "Sat Sep 20 16:57:27 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 z3950.loc.gov:7090/Voyager",
  series =       "The Oxford series in electrical and computer
                 engineering",
  URL =          "http://www.loc.gov/catdir/enhancements/fy1006/2009034155-d.html;
                 http://www.loc.gov/catdir/enhancements/fy1006/2009034155-t.html",
  abstract =     "Ideal for graduate and senior undergraduate courses in
                 computer arithmetic and advanced digital design,
                 Computer Arithmetic: Algorithms and Hardware Designs,
                 Second Edition, provides a balanced, comprehensive
                 treatment of computer arithmetic. It covers topics in
                 arithmetic unit design and circuit implementation that
                 complement the architectural and algorithmic speedup
                 techniques used in high-performance computer
                 architecture and parallel processing. Using a unified
                 and consistent framework, the text begins with number
                 representation and proceeds through basic arithmetic
                 operations, floating-point arithmetic, and function
                 evaluation methods. Later chapters cover broad design
                 and implementation topics --- including techniques for
                 high-throughput, low-power, fault-tolerant, and
                 reconfigurable arithmetic. An appendix provides a
                 historical view of the field and speculates on its
                 future. An indispensable resource for instruction,
                 professional development, and research, Computer
                 Arithmetic: Algorithms and Hardware Designs, Second
                 Edition, combines broad coverage of the underlying
                 theories of computer arithmetic with numerous examples
                 of practical designs, worked-out examples, and a large
                 collection of meaningful problems. This second edition
                 includes a new chapter on reconfigurable arithmetic, in
                 order to address the fact that arithmetic functions are
                 increasingly being implemented on field-programmable
                 gate arrays (FPGAs) and FPGA-like configurable devices.
                 Updated and thoroughly revised, the book offers new and
                 expanded coverage of saturating adders and multipliers,
                 truncated multipliers, fused multiply-add units,
                 overlapped quotient digit selection, bipartite and
                 multipartite tables, reversible logic, dot notation,
                 modular arithmetic, Montgomery modular reduction,
                 division by constants, IEEE floating-point standard
                 formats, and interval arithmetic.",
  acknowledgement = ack-nhfb,
  subject =      "Computer arithmetic; Computer algorithms; Computer
                 algorithms.; Computer arithmetic.",
  tableofcontents = "Part I: Number representation \\
                 Numbers and Arithmetic \\
                 Representing Signed Numbers \\
                 Redundant Number Systems \\
                 Residue Number Systems \\
                 Part II: Addition\slash Subtraction. Basic Addition and
                 Counting \\
                 Carry-Lookahead Adders \\
                 Variations in Fast Adders \\
                 Multioperand Addition \\
                 Part III: Multiplication. Basic Multiplication Schemes
                 \\
                 High-Radix Multipliers \\
                 Tree and Array Multipliers \\
                 Variations in Multipliers \\
                 Part IV: Division. Basic Division Schemes \\
                 High-Radix Dividers \\
                 Variations in Dividers \\
                 Division by Convergence \\
                 Part V: Real arithmetic. Floating-Point Representations
                 \\
                 Floating-Point Operations \\
                 Errors and Error Control \\
                 Precise and Certifiable Arithmetic \\
                 Part VI: Function evaluation. Square-Rooting Methods
                 \\
                 The CORDIC Algorithms \\
                 Variations in Function Evaluation \\
                 Arithmetic by Table Lookup \\
                 High-Throughput Arithmetic \\
                 Part VII: Implementation topics \\
                 Low-Power Arithmetic \\
                 Fault-Tolerant Arithmetic \\
                 Reconfigurable Arithmetic",
}


bibtype: article
filename: fparith.bib
label: Pence:2010:OCF
author: W. D. Pence and R. L. White and R. Seaman
title: Optimal Compression of Floating-point Astronomical Images Without Significant Loss of Information
journal: arXiv.org
pages: ??--??
day: 7
month: jul
monthnumber: 7
year: 2010
bibdate: Wed Sep 30 12:43:49 MDT 2015
bibtimestamp: 2015.09.30 12:43:49 MDT
note: Published in PASP bf 122, 1096 (2010).
URL: http://arxiv.org/abs/1007.1179
subject: Instrumentation and Methods for Astrophysics (astro-ph.IM)
acknowledgement: ack-nhfb
rawauthor: W. D. Pence and R. L. White and R. Seaman
rawnote: Published in PASP {\bf 122}, 1096 (2010).
rawtitle: Optimal Compression of Floating-point Astronomical Images Without Significant Loss of Information
bibsource:
https://www.math.utah.edu/pub/tex/bib/datacompression.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

abstract:
We describe a compression method for floating-point astronomical images that gives compression ratios of $6$--$ 10$ while still preserving the scientifically important information in the image. The pixel values are first preprocessed by quantizing them into scaled integer intensity levels, which removes some of the uncompressible noise in the image. The integers are then losslessly compressed using the fast and efficient Rice algorithm and stored in a portable FITS format file. Quantizing an image more coarsely gives greater image compression, but it also increases the noise and degrades the precision of the photometric and astrometric measurements in the quantized image. Dithering the pixel values during the quantization process can greatly improve the precision of measurements in the images. This is especially important if the analysis algorithm relies on the mode or the median which would be similarly quantized if the pixel values are not dithered. We perform a series of experiments on both synthetic and real astronomical CCD images to quantitatively demonstrate that the magnitudes and positions of stars in the quantized images can be measured with the predicted amount of precision. In order to encourage wider use of these image compression methods, we have made available a pair of general-purpose image compression programs, called fpack and funpack, which can be used to compress any FITS format image.

rawabstract:
We describe a compression method for floating-point astronomical images that gives compression ratios of $6$--$ 10$ while still preserving the scientifically important information in the image. The pixel values are first preprocessed by quantizing them into scaled integer intensity levels, which removes some of the uncompressible noise in the image. The integers are then losslessly compressed using the fast and efficient Rice algorithm and stored in a portable FITS format file. Quantizing an image more coarsely gives greater image compression, but it also increases the noise and degrades the precision of the photometric and astrometric measurements in the quantized image. Dithering the pixel values during the quantization process can greatly improve the precision of measurements in the images. This is especially important if the analysis algorithm relies on the mode or the median which would be similarly quantized if the pixel values are not dithered. We perform a series of experiments on both synthetic and real astronomical CCD images to quantitatively demonstrate that the magnitudes and positions of stars in the quantized images can be measured with the predicted amount of precision. In order to encourage wider use of these image compression methods, we have made available a pair of general-purpose image compression programs, called fpack and funpack, which can be used to compress any FITS format image.

entry:
@Article{Pence:2010:OCF,
  author =       "W. D. Pence and R. L. White and R. Seaman",
  title =        "Optimal Compression of Floating-point Astronomical
                 Images Without Significant Loss of Information",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "??--??",
  day =          "7",
  month =        jul,
  year =         "2010",
  CODEN =        "????",
  ISSN =         "????",
  ISSN-L =       "????",
  bibdate =      "Wed Sep 30 12:43:49 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/datacompression.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  note =         "Published in PASP {\bf 122}, 1096 (2010).",
  URL =          "http://arxiv.org/abs/1007.1179",
  abstract =     "We describe a compression method for floating-point
                 astronomical images that gives compression ratios of
                 $6$--$ 10$ while still preserving the scientifically
                 important information in the image. The pixel values
                 are first preprocessed by quantizing them into scaled
                 integer intensity levels, which removes some of the
                 uncompressible noise in the image. The integers are
                 then losslessly compressed using the fast and efficient
                 Rice algorithm and stored in a portable FITS format
                 file. Quantizing an image more coarsely gives greater
                 image compression, but it also increases the noise and
                 degrades the precision of the photometric and
                 astrometric measurements in the quantized image.
                 Dithering the pixel values during the quantization
                 process can greatly improve the precision of
                 measurements in the images. This is especially
                 important if the analysis algorithm relies on the mode
                 or the median which would be similarly quantized if the
                 pixel values are not dithered. We perform a series of
                 experiments on both synthetic and real astronomical CCD
                 images to quantitatively demonstrate that the
                 magnitudes and positions of stars in the quantized
                 images can be measured with the predicted amount of
                 precision. In order to encourage wider use of these
                 image compression methods, we have made available a
                 pair of general-purpose image compression programs,
                 called fpack and funpack, which can be used to compress
                 any FITS format image.",
  acknowledgement = ack-nhfb,
  subject =      "Instrumentation and Methods for Astrophysics
                 (astro-ph.IM)",
}


bibtype: inproceedings
filename: fparith.bib
label: Qi:2010:DLC
author: Zichu Qi and Qi Guo and Ge Zhang and Xiangku Li and Weiwu Hu
title: Design of Low-Cost High-Performance Floating-Point Fused Multiply-Add with Reduced Power
crossref: IEEE:2010:ICV
pages: 206--211
year: 2010
DOI: https://doi.org/10.1109/VLSI.Design.2010.41
bibdate: Sun Feb 20 10:01:52 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.02.20 10:01:52 ???
acknowledgement: ack-nhfb
rawauthor: Zichu Qi and Qi Guo and Ge Zhang and Xiangku Li and Weiwu Hu
rawtitle: Design of Low-Cost High-Performance Floating-Point Fused Multiply-Add with Reduced Power
abstract:
This paper presents a floating-point fused multiply-add (FMA) unit with low-cost and low power techniques. To improve the performance, two single-precision operations can be performed concurrently with one double-precision datapath, which is very useful in multimedia and even scientific applications. Moreover, to reduce the additional area costs for supporting two single-precision operations in parallel, multiple double-precision units, i.e., the multiplier, shifter and adder, are reused as much as possible. A modified dual-path algorithm is proposed by classifying the exponent difference into three cases and implementing them with close and far paths, which can reduce latency and facilitate lowering power consumption by enabling only one of the two paths. In addition, in case of FADD instructions, the multiplier in the first stage is bypassed and kept in stable mode, which can significantly improve FADD instruction performance and lower power consumption. The overall FMA unit has a latency of 4 cycles while the FADD operation has 3 cycles. Each cycle has a time delay of about 0.66 ns in the ST 65 nm CMOS technology. Compared with the conventional double-precision FMA, about 13% delay is reduced and about 22% area is increased, which is acceptable since two single-precision results can be generated simultaneously.

rawabstract:
This paper presents a floating-point fused multiply-add (FMA) unit with low-cost and low power techniques. To improve the performance, two single-precision operations can be performed concurrently with one double-precision datapath, which is very useful in multimedia and even scientific applications. Moreover, to reduce the additional area costs for supporting two single-precision operations in parallel, multiple double-precision units, i.e., the multiplier, shifter and adder, are reused as much as possible. A modified dual-path algorithm is proposed by classifying the exponent difference into three cases and implementing them with close and far paths, which can reduce latency and facilitate lowering power consumption by enabling only one of the two paths. In addition, in case of FADD instructions, the multiplier in the first stage is bypassed and kept in stable mode, which can significantly improve FADD instruction performance and lower power consumption. The overall FMA unit has a latency of 4 cycles while the FADD operation has 3 cycles. Each cycle has a time delay of about 0.66 ns in the ST 65 nm CMOS technology. Compared with the conventional double-precision FMA, about 13\% delay is reduced and about 22\% area is increased, which is acceptable since two single-precision results can be generated simultaneously.

entry:
@InProceedings{Qi:2010:DLC,
  author =       "Zichu Qi and Qi Guo and Ge Zhang and Xiangku Li and
                 Weiwu Hu",
  title =        "Design of Low-Cost High-Performance Floating-Point
                 Fused Multiply-Add with Reduced Power",
  crossref =     "IEEE:2010:ICV",
  pages =        "206--211",
  year =         "2010",
  DOI =          "https://doi.org/10.1109/VLSI.Design.2010.41",
  bibdate =      "Sun Feb 20 10:01:52 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "This paper presents a floating-point fused
                 multiply-add (FMA) unit with low-cost and low power
                 techniques. To improve the performance, two
                 single-precision operations can be performed
                 concurrently with one double-precision datapath, which
                 is very useful in multimedia and even scientific
                 applications. Moreover, to reduce the additional area
                 costs for supporting two single-precision operations in
                 parallel, multiple double-precision units, i.e., the
                 multiplier, shifter and adder, are reused as much as
                 possible. A modified dual-path algorithm is proposed by
                 classifying the exponent difference into three cases
                 and implementing them with close and far paths, which
                 can reduce latency and facilitate lowering power
                 consumption by enabling only one of the two paths. In
                 addition, in case of FADD instructions, the multiplier
                 in the first stage is bypassed and kept in stable mode,
                 which can significantly improve FADD instruction
                 performance and lower power consumption. The overall
                 FMA unit has a latency of 4 cycles while the FADD
                 operation has 3 cycles. Each cycle has a time delay of
                 about 0.66 ns in the ST 65 nm CMOS technology. Compared
                 with the conventional double-precision FMA, about 13\%
                 delay is reduced and about 22\% area is increased,
                 which is acceptable since two single-precision results
                 can be generated simultaneously.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Roldao:2010:HTF
author: Antonio Roldao and George A. Constantinides
title: A High Throughput FPGA-Based Floating Point Conjugate Gradient Implementation for Dense Matrices
journal: j-TRETS
volume: 3
number: 1
pages: 1:1--1:??
month: jan
monthnumber: 1
year: 2010
DOI: https://doi.org/10.1145/1661438.1661439
ISSN: 1936-7406 (print), 1936-7414 (electronic) OR 19367406 (print), 19367414 (electronic)
ISSNL: 1936-7406 OR 19367406
bibdate: Tue Mar 16 09:47:03 MDT 2010
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.03.16 09:47:03 MDT
fjournal: ACM Transactions on Reconfigurable Technology and Systems (TRETS)
acknowledgement: ack-nhfb
articleno: 1
journalURL: http://portal.acm.org/toc.cfm?id=J1151
rawauthor: Antonio Roldao and George A. Constantinides
rawtitle: A High Throughput {FPGA}-Based Floating Point Conjugate Gradient Implementation for Dense Matrices
entry:
@Article{Roldao:2010:HTF,
  author =       "Antonio Roldao and George A. Constantinides",
  title =        "A High Throughput {FPGA}-Based Floating Point
                 Conjugate Gradient Implementation for Dense Matrices",
  journal =      j-TRETS,
  volume =       "3",
  number =       "1",
  pages =        "1:1--1:??",
  month =        jan,
  year =         "2010",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/1661438.1661439",
  ISSN =         "1936-7406 (print), 1936-7414 (electronic)",
  ISSN-L =       "1936-7406",
  bibdate =      "Tue Mar 16 09:47:03 MDT 2010",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  articleno =    "1",
  fjournal =     "ACM Transactions on Reconfigurable Technology and
                 Systems (TRETS)",
  journal-URL =  "http://portal.acm.org/toc.cfm?id=J1151",
}


bibtype: inproceedings
filename: fparith.bib
label: Rummer:2010:IPI
author: Philipp Rummer and Thomas Wahl
editor: An smt-lib theory of binary floating-point arithmetic
pages: ??--??
year: 2010
bibdate: Sat Apr 02 09:57:20 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.04.02 09:57:20 ???
acknowledgement: ack-nhfb
rawauthor: Philipp R{\"u}mmer and Thomas Wahl
raweditor: An smt-lib theory of binary floating-point arithmetic
title:
Informal proceedings of 8th International Workshop on Satisfiability Modulo Theories (SMT) at FLoC, Edinburgh, Scotland

rawtitle:
Informal proceedings of {8th International Workshop on Satisfiability Modulo Theories (SMT) at FLoC, Edinburgh, Scotland}

entry:
@InProceedings{Rummer:2010:IPI,
  author =       "Philipp R{\"u}mmer and Thomas Wahl",
  editor =       "An smt-lib theory of binary floating-point
                 arithmetic",
  booktitle =    "????",
  title =        "Informal proceedings of {8th International Workshop on
                 Satisfiability Modulo Theories (SMT) at FLoC,
                 Edinburgh, Scotland}",
  publisher =    "????",
  address =      "??",
  pages =        "??--??",
  year =         "2010",
  DOI =          "",
  ISBN =         "",
  ISBN-13 =      "",
  LCCN =         "",
  bibdate =      "Sat Apr 02 09:57:20 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Rump:2010:ARC
author: Siegfried M. Rump
title: Accurate and Reliable Computing in Floating-Point Arithmetic
crossref: Fukuda:2010:MSI
pages: 105--108
year: 2010
DOI: https://doi.org/10.1007/978-3-642-15582-6_22
bibdate: Sat Sep 23 06:20:46 2023
bibtimestamp: 2023.09.23 06:20:46 ???
URL: http://link.springer.com/content/pdf/10.1007/978-3-642-15582-6_22.pdf
acknowledgement: ack-nhfb
rawauthor: Siegfried M. Rump
rawtitle: Accurate and Reliable Computing in Floating-Point Arithmetic
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/lncs2010a.bib

entry:
@InProceedings{Rump:2010:ARC,
  author =       "Siegfried M. Rump",
  title =        "Accurate and Reliable Computing in Floating-Point
                 Arithmetic",
  crossref =     "Fukuda:2010:MSI",
  pages =        "105--108",
  year =         "2010",
  DOI =          "https://doi.org/10.1007/978-3-642-15582-6_22",
  bibdate =      "Sat Sep 23 06:20:46 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/lncs2010a.bib",
  URL =          "http://link.springer.com/content/pdf/10.1007/978-3-642-15582-6_22.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Rump:2010:FHP
author: Siegfried M. Rump and T. Ogita and S. Oishi
title: Fast high precision summation
journal: Nonlinear Theory and Its Applications (NOLTA)
volume: 1
number: 1
pages: 2--24
year: 2010
DOI: https://doi.org/10.1587/nolta.1.2
bibdate: Wed Oct 12 18:34:53 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.10.12 18:34:53 ???
keywords: accurate dot product; accurate summation
acknowledgement: ack-nhfb
rawauthor: Siegfried M. Rump and T. Ogita and S. Oishi
rawtitle: Fast high precision summation
entry:
@Article{Rump:2010:FHP,
  author =       "Siegfried M. Rump and T. Ogita and S. Oishi",
  title =        "Fast high precision summation",
  journal =      "Nonlinear Theory and Its Applications ({NOLTA})",
  volume =       "1",
  number =       "1",
  pages =        "2--24",
  month =        "????",
  year =         "2010",
  DOI =          "https://doi.org/10.1587/nolta.1.2",
  bibdate =      "Wed Oct 12 18:34:53 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "accurate dot product; accurate summation",
}


bibtype: inproceedings
filename: fparith.bib
label: Rump:2010:VMRa
author: Siegfried M. Rump
title: Verification methods: rigorous results using floating-point arithmetic
crossref: Watt:2010:IPI
pages: 3--4
year: 2010
DOI: https://doi.org/10.1145/1837934.1837937
bibdate: Fri Jun 17 08:06:37 MDT 2011
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.06.17 08:06:37 MDT
acknowledgement: ack-nhfb
rawauthor: Siegfried M. Rump
rawtitle: Verification methods: rigorous results using floating-point arithmetic
abstract:
The classical mathematical proof is performed by pencil and paper. However, there are many ways in which computers may be used in a mathematical proof. But `proofs by computers' or even the use of computers in the course of a proof are not so readily accepted (the December 2008 issue of the Notices of the American Mathematical Society is devoted to formal proofs by computers).par In this talk we discuss how verification methods may assist in achieving a mathematically rigorous result. In particular we emphasize how floating-point arithmetic is used.par The goal of verification methods is ambitious: For a given problem it is proved, with the aid of a computer, that there exists a (unique) solution within computed bounds. The methods are constructive, and the results are rigorous in every respect. Verification methods apply to data with tolerances as well.par Rigorous results are the main goal in computer algebra. However, verification methods use solely floating-point arithmetic, so that the total computational effort is not too far from that of a purely (approximate) numerical method. Nontrivial problems have been solved using verification methods. For example:par Tucker (1999) received the 2004 EMS prize awarded by the European Mathematical Society for (citation) `giving a rigorous proof that the Lorenz attractor exists for the parameter values provided by Lorenz. This was a long standing challenge to the dynamical system community, and was included by Smale in his list of problems for the new millennium. The proof uses computer estimates with rigorous bounds based on higher dimensional interval arithmetics.'par Sahinidis and Tawaralani (2005) received the 2006 Beale-Orchard-Hays Prize for their package BARON which (citation) `incorporates techniques from automatic differentiation, interval arithmetic, and other areas to yield an automatic, modular, and relatively efficient solver for the very difficult area of global optimization'.par A main goal of this talk is to introduce the principles of how to design verification algorithms, and how these principles differ from those for traditional numerical algorithms.par We begin with a brief discussion of the working tools of verification methods, in particular floating-point and interval arithmetic. In particular the development and limits of verification methods for finite dimensional problems are discussed in some detail; problems include dense systems of linear equations, sparse linear systems, systems of nonlinear equations, semi-definite programming and other special linear and nonlinear problems including M-matrices, simple and multiple roots of polynomials, bounds for simple and multiple eigenvalues or clusters, and quadrature. We mention that automatic differentiation tools to compute the range of gradients, Hessians, Taylor coefficients, and slopes are necessary. If time permits, verification methods for continuous problems, namely two-point boundary value problems and semilinear elliptic boundary value problems are presented.par Throughout the talk, a number of examples of the wrong use of interval operations are given. In the past such examples contributed to the dubious reputation of interval arithmetic, whereas they are, in fact, just a misuse.par Some algorithms are presented in executable Matlab/INTLAB-code. INTLAB, the Matlab toolbox for reliable computing and free for academic use, is developed and written by Rump (1999). It was, for example, used by Bornemann, Laurie, Wagon, and Waldvogel (2004) in the solution of half of the problems of the $ 10 times 10 $-digit challenge by Trefethen (2002).

rawabstract:
The classical mathematical proof is performed by pencil and paper. However, there are many ways in which computers may be used in a mathematical proof. But `proofs by computers' or even the use of computers in the course of a proof are not so readily accepted (the December 2008 issue of the Notices of the American Mathematical Society is devoted to formal proofs by computers).\par In this talk we discuss how verification methods may assist in achieving a mathematically rigorous result. In particular we emphasize how floating-point arithmetic is used.\par The goal of verification methods is ambitious: For a given problem it is proved, with the aid of a computer, that there exists a (unique) solution within computed bounds. The methods are constructive, and the results are rigorous in every respect. Verification methods apply to data with tolerances as well.\par Rigorous results are the main goal in computer algebra. However, verification methods use solely floating-point arithmetic, so that the total computational effort is not too far from that of a purely (approximate) numerical method. Nontrivial problems have been solved using verification methods. For example:\par Tucker (1999) received the 2004 EMS prize awarded by the European Mathematical Society for (citation) `giving a rigorous proof that the Lorenz attractor exists for the parameter values provided by Lorenz. This was a long standing challenge to the dynamical system community, and was included by Smale in his list of problems for the new millennium. The proof uses computer estimates with rigorous bounds based on higher dimensional interval arithmetics.'\par Sahinidis and Tawaralani (2005) received the 2006 Beale-Orchard-Hays Prize for their package BARON which (citation) `incorporates techniques from automatic differentiation, interval arithmetic, and other areas to yield an automatic, modular, and relatively efficient solver for the very difficult area of global optimization'.\par A main goal of this talk is to introduce the principles of how to design verification algorithms, and how these principles differ from those for traditional numerical algorithms.\par We begin with a brief discussion of the working tools of verification methods, in particular floating-point and interval arithmetic. In particular the development and limits of verification methods for finite dimensional problems are discussed in some detail; problems include dense systems of linear equations, sparse linear systems, systems of nonlinear equations, semi-definite programming and other special linear and nonlinear problems including M-matrices, simple and multiple roots of polynomials, bounds for simple and multiple eigenvalues or clusters, and quadrature. We mention that automatic differentiation tools to compute the range of gradients, Hessians, Taylor coefficients, and slopes are necessary. If time permits, verification methods for continuous problems, namely two-point boundary value problems and semilinear elliptic boundary value problems are presented.\par Throughout the talk, a number of examples of the wrong use of interval operations are given. In the past such examples contributed to the dubious reputation of interval arithmetic, whereas they are, in fact, just a misuse.\par Some algorithms are presented in executable Matlab/INTLAB-code. INTLAB, the Matlab toolbox for reliable computing and free for academic use, is developed and written by Rump (1999). It was, for example, used by Bornemann, Laurie, Wagon, and Waldvogel (2004) in the solution of half of the problems of the $ 10 \times 10 $-digit challenge by Trefethen (2002).

entry:
@InProceedings{Rump:2010:VMRa,
  author =       "Siegfried M. Rump",
  title =        "Verification methods: rigorous results using
                 floating-point arithmetic",
  crossref =     "Watt:2010:IPI",
  pages =        "3--4",
  year =         "2010",
  DOI =          "https://doi.org/10.1145/1837934.1837937",
  bibdate =      "Fri Jun 17 08:06:37 MDT 2011",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The classical mathematical proof is performed by
                 pencil and paper. However, there are many ways in which
                 computers may be used in a mathematical proof. But
                 `proofs by computers' or even the use of computers in
                 the course of a proof are not so readily accepted (the
                 December 2008 issue of the Notices of the American
                 Mathematical Society is devoted to formal proofs by
                 computers).\par

                 In this talk we discuss how verification methods may
                 assist in achieving a mathematically rigorous result.
                 In particular we emphasize how floating-point
                 arithmetic is used.\par

                 The goal of verification methods is ambitious: For a
                 given problem it is proved, with the aid of a computer,
                 that there exists a (unique) solution within computed
                 bounds. The methods are constructive, and the results
                 are rigorous in every respect. Verification methods
                 apply to data with tolerances as well.\par

                 Rigorous results are the main goal in computer algebra.
                 However, verification methods use solely floating-point
                 arithmetic, so that the total computational effort is
                 not too far from that of a purely (approximate)
                 numerical method. Nontrivial problems have been solved
                 using verification methods. For example:\par

                 Tucker (1999) received the 2004 EMS prize awarded by
                 the European Mathematical Society for (citation)
                 `giving a rigorous proof that the Lorenz attractor
                 exists for the parameter values provided by Lorenz.
                 This was a long standing challenge to the dynamical
                 system community, and was included by Smale in his list
                 of problems for the new millennium. The proof uses
                 computer estimates with rigorous bounds based on higher
                 dimensional interval arithmetics.'\par

                 Sahinidis and Tawaralani (2005) received the 2006
                 Beale-Orchard-Hays Prize for their package BARON which
                 (citation) `incorporates techniques from automatic
                 differentiation, interval arithmetic, and other areas
                 to yield an automatic, modular, and relatively
                 efficient solver for the very difficult area of global
                 optimization'.\par

                 A main goal of this talk is to introduce the principles
                 of how to design verification algorithms, and how these
                 principles differ from those for traditional numerical
                 algorithms.\par

                 We begin with a brief discussion of the working tools
                 of verification methods, in particular floating-point
                 and interval arithmetic. In particular the development
                 and limits of verification methods for finite
                 dimensional problems are discussed in some detail;
                 problems include dense systems of linear equations,
                 sparse linear systems, systems of nonlinear equations,
                 semi-definite programming and other special linear and
                 nonlinear problems including M-matrices, simple and
                 multiple roots of polynomials, bounds for simple and
                 multiple eigenvalues or clusters, and quadrature. We
                 mention that automatic differentiation tools to compute
                 the range of gradients, Hessians, Taylor coefficients,
                 and slopes are necessary. If time permits, verification
                 methods for continuous problems, namely two-point
                 boundary value problems and semilinear elliptic
                 boundary value problems are presented.\par

                 Throughout the talk, a number of examples of the wrong
                 use of interval operations are given. In the past such
                 examples contributed to the dubious reputation of
                 interval arithmetic, whereas they are, in fact, just a
                 misuse.\par

                 Some algorithms are presented in executable
                 Matlab/INTLAB-code. INTLAB, the Matlab toolbox for
                 reliable computing and free for academic use, is
                 developed and written by Rump (1999). It was, for
                 example, used by Bornemann, Laurie, Wagon, and
                 Waldvogel (2004) in the solution of half of the
                 problems of the $ 10 \times 10 $-digit challenge by
                 Trefethen (2002).",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Rump:2010:VMRb
author: Siegfried M. Rump
title: Verification methods: Rigorous results using floating-point arithmetic
journal: j-ACTA-NUMERICA
volume: 19
pages: 287--449
year: 2010
CODEN: ANUMFU
DOI: https://doi.org/10.1017/S096249291000005X
ISBN: 0-521-19284-6 OR 0521192846
ISBN13: 978-0-521-19284-2 OR 9780521192842
ISSN: 0962-4929 (print), 1474-0508 (electronic) OR 09624929 (print), 14740508 (electronic)
ISSNL: 0962-4929 OR 09624929
MRclass: 65G20 (03B35 68T05)
MRnumber: 2652784 (2011j:65093)
MRreviewer: Ljiljana Petkovi{\'c}
bibdate: Sat Sep 24 10:35:17 2011
bibtimestamp: 2011.09.24 10:35:17 ???
URL: http://www.ti3.tu-harburg.de/paper/rump/Ru10.pdf
fjournal: Acta Numerica
acknowledgement: ack-nhfb
ajournal: Acta Numer.
journalURL: http://journals.cambridge.org/action/displayJournal?jid=ANU
onlinedate: 10 May 2010
rawauthor: Siegfried M. Rump
rawtitle: Verification methods: Rigorous results using floating-point arithmetic
bibsource:
https://www.math.utah.edu/pub/tex/bib/actanumerica.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

abstract:
This review article is devoted to verification methods and consists of three parts of similar length. In Part 1 the working tools of verification methods are discussed, in particular floating-point and interval arithmetic; my findings in Section 1.5 (Historical remarks) seem new, even to experts in the field.par In Part 2, the development and limits of verification methods for finite-dimensional problems are discussed in some detail. In particular, we discuss how verification is _not_ working. For example, we give a probabilistic argument that the so-called interval Gaussian elimination (IGA) does not work even for (well-conditioned) random matrices of small size. Verification methods are discussed for problems such as dense systems of linear equations, sparse linear systems, systems of nonlinear equations, semi-definite programming and other special linear and nonlinear problems, including M-matrices, finding simple and multiple roots of polynomials, bounds for simple and multiple eigenvalues or clusters, and quadrature. The necessary automatic differentiation tools to compute the range of gradients, Hessians, Taylor coefficients, and slopes are also introduced.par Concerning the important area of optimization, Neumaier (2004) gave in his Acta Numerica article an overview on global optimization and constraint satisfaction methods. In view of the thorough treatment there, showing the essential role of interval methods in this area, we restrict our discussion to a few recent, complementary issues.par Finally, in Part 3, verification methods for infinite-dimensional problems are presented, namely two-point boundary value problems and semilinear elliptic boundary value problems.par Throughout the article, many examples of the inappropriate use of interval operations are given. In the past such examples contributed to the dubious reputation of interval arithmetic (see Section 1.3), whereas they are, in fact, simply a misuse.par One main goal of this review article is to introduce the principles of the design of verification algorithms, and how these principles differ from those for traditional numerical algorithms (see Section 1.4).par Many algorithms are presented in executable MATLAB/INTLAB code, providing the opportunity to test the methods directly. INTLAB, the MATLAB toolbox for reliable computing, was, for example, used by Bornemann, Laurie, Wagon and Waldvogel (2004) in the solution of half of the problems of the SIAM $ 10 times 10 $-digit challenge by Trefethen (2002).

rawabstract:
This review article is devoted to verification methods and consists of three parts of similar length. In Part 1 the working tools of verification methods are discussed, in particular floating-point and interval arithmetic; my findings in Section 1.5 (Historical remarks) seem new, even to experts in the field.\par In Part 2, the development and limits of verification methods for finite-dimensional problems are discussed in some detail. In particular, we discuss how verification is _not_ working. For example, we give a probabilistic argument that the so-called interval Gaussian elimination (IGA) does not work even for (well-conditioned) random matrices of small size. Verification methods are discussed for problems such as dense systems of linear equations, sparse linear systems, systems of nonlinear equations, semi-definite programming and other special linear and nonlinear problems, including M-matrices, finding simple and multiple roots of polynomials, bounds for simple and multiple eigenvalues or clusters, and quadrature. The necessary automatic differentiation tools to compute the range of gradients, Hessians, Taylor coefficients, and slopes are also introduced.\par Concerning the important area of optimization, Neumaier (2004) gave in his Acta Numerica article an overview on global optimization and constraint satisfaction methods. In view of the thorough treatment there, showing the essential role of interval methods in this area, we restrict our discussion to a few recent, complementary issues.\par Finally, in Part 3, verification methods for infinite-dimensional problems are presented, namely two-point boundary value problems and semilinear elliptic boundary value problems.\par Throughout the article, many examples of the inappropriate use of interval operations are given. In the past such examples contributed to the dubious reputation of interval arithmetic (see Section 1.3), whereas they are, in fact, simply a misuse.\par One main goal of this review article is to introduce the principles of the design of verification algorithms, and how these principles differ from those for traditional numerical algorithms (see Section 1.4).\par Many algorithms are presented in executable MATLAB/INTLAB code, providing the opportunity to test the methods directly. INTLAB, the MATLAB toolbox for reliable computing, was, for example, used by Bornemann, Laurie, Wagon and Waldvogel (2004) in the solution of half of the problems of the SIAM $ 10 \times 10 $-digit challenge by Trefethen (2002).

entry:
@Article{Rump:2010:VMRb,
  author =       "Siegfried M. Rump",
  title =        "Verification methods: Rigorous results using
                 floating-point arithmetic",
  journal =      j-ACTA-NUMERICA,
  volume =       "19",
  pages =        "287--449",
  year =         "2010",
  CODEN =        "ANUMFU",
  DOI =          "https://doi.org/10.1017/S096249291000005X",
  ISBN =         "0-521-19284-6",
  ISBN-13 =      "978-0-521-19284-2",
  ISSN =         "0962-4929 (print), 1474-0508 (electronic)",
  ISSN-L =       "0962-4929",
  MRclass =      "65G20 (03B35 68T05)",
  MRnumber =     "2652784 (2011j:65093)",
  MRreviewer =   "Ljiljana Petkovi{\'c}",
  bibdate =      "Sat Sep 24 10:35:17 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/actanumerica.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.ti3.tu-harburg.de/paper/rump/Ru10.pdf",
  abstract =     "This review article is devoted to verification methods
                 and consists of three parts of similar length. In Part
                 1 the working tools of verification methods are
                 discussed, in particular floating-point and interval
                 arithmetic; my findings in Section 1.5 (Historical
                 remarks) seem new, even to experts in the field.\par

                 In Part 2, the development and limits of verification
                 methods for finite-dimensional problems are discussed
                 in some detail. In particular, we discuss how
                 verification is _not_ working. For example, we give a
                 probabilistic argument that the so-called interval
                 Gaussian elimination (IGA) does not work even for
                 (well-conditioned) random matrices of small size.
                 Verification methods are discussed for problems such as
                 dense systems of linear equations, sparse linear
                 systems, systems of nonlinear equations, semi-definite
                 programming and other special linear and nonlinear
                 problems, including M-matrices, finding simple and
                 multiple roots of polynomials, bounds for simple and
                 multiple eigenvalues or clusters, and quadrature. The
                 necessary automatic differentiation tools to compute
                 the range of gradients, Hessians, Taylor coefficients,
                 and slopes are also introduced.\par

                 Concerning the important area of optimization, Neumaier
                 (2004) gave in his Acta Numerica article an overview on
                 global optimization and constraint satisfaction
                 methods. In view of the thorough treatment there,
                 showing the essential role of interval methods in this
                 area, we restrict our discussion to a few recent,
                 complementary issues.\par

                 Finally, in Part 3, verification methods for
                 infinite-dimensional problems are presented, namely
                 two-point boundary value problems and semilinear
                 elliptic boundary value problems.\par

                 Throughout the article, many examples of the
                 inappropriate use of interval operations are given. In
                 the past such examples contributed to the dubious
                 reputation of interval arithmetic (see Section 1.3),
                 whereas they are, in fact, simply a misuse.\par

                 One main goal of this review article is to introduce
                 the principles of the design of verification
                 algorithms, and how these principles differ from those
                 for traditional numerical algorithms (see Section
                 1.4).\par

                 Many algorithms are presented in executable
                 MATLAB/INTLAB code, providing the opportunity to test
                 the methods directly. INTLAB, the MATLAB toolbox for
                 reliable computing, was, for example, used by
                 Bornemann, Laurie, Wagon and Waldvogel (2004) in the
                 solution of half of the problems of the SIAM $ 10
                 \times 10 $-digit challenge by Trefethen (2002).",
  acknowledgement = ack-nhfb,
  ajournal =     "Acta Numer.",
  fjournal =     "Acta Numerica",
  journal-URL =  "http://journals.cambridge.org/action/displayJournal?jid=ANU",
  onlinedate =   "10 May 2010",
}


bibtype: article
filename: fparith.bib
label: Ryden:2010:EIR
author: Jesper Ryden and Sven Erick Alm
journal: j-J-APPL-STAT
volume: 37
number: 10
pages: 1695--1701
month: oct
monthnumber: 10
year: 2010
DOI: https://doi.org/10.1080/02664760903143925
ISSN: 0266-4763 (print), 1360-0532 (electronic) OR 02664763 (print), 13600532 (electronic)
ISSNL: 0266-4763 OR 02664763
bibdate: Mon Sep 5 18:53:08 MDT 2011
bibtimestamp: 2011.09.05 18:53:08 MDT
fjournal: Journal of Applied Statistics
acknowledgement: ack-nhfb
journalURL: http://www.tandfonline.com/loi/cjas20
onlinedate: 21 Sep 2010
rawauthor: Jesper Ryd{\'e}n and Sven Erick Alm
title:
The effect of interaction and rounding error in two-way ANOVA: example of impact on testing for normality

rawtitle:
The effect of interaction and rounding error in two-way {ANOVA}: example of impact on testing for normality

entry:
@Article{Ryden:2010:EIR,
  author =       "Jesper Ryd{\'e}n and Sven Erick Alm",
  title =        "The effect of interaction and rounding error in
                 two-way {ANOVA}: example of impact on testing for
                 normality",
  journal =      j-J-APPL-STAT,
  volume =       "37",
  number =       "10",
  pages =        "1695--1701",
  month =        oct,
  year =         "2010",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1080/02664760903143925",
  ISSN =         "0266-4763 (print), 1360-0532 (electronic)",
  ISSN-L =       "0266-4763",
  bibdate =      "Mon Sep 5 18:53:08 MDT 2011",
  bibsource =    "http://www.tandf.co.uk/journals/routledge/02664763.html;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Applied Statistics",
  journal-URL =  "http://www.tandfonline.com/loi/cjas20",
  onlinedate =   "21 Sep 2010",
}


bibtype: article
filename: fparith.bib
label: Saito:2010:DQP
author: Tsubasa Saito and Emiko Ishiwata and Hidehiko Hasegawa
title: Development of quadruple precision arithmetic toolbox QuPAT on Scilab
journal: j-LECT-NOTES-COMP-SCI
volume: 6017
pages: 60--70
year: 2010
CODEN: LNCSD9
DOI: https://doi.org/10.1007/978-3-642-12165-4_5
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
bibdate: Sat Apr 01 07:57:10 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.04.01 07:57:10 ???
URL: http://link.springer.com/chapter/10.1007/978-3-642-12165-4_5
fjournal: Lecture Notes in Computer Science
keywords: floating-point arithmetic; QuPAT (Quadruple Precision Arithmetic Toolbox)
remark: International Conference on Computational Science and Its Applications, ICCSA 2010
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/bookseries/558
rawauthor: Tsubasa Saito and Emiko Ishiwata and Hidehiko Hasegawa
rawtitle: Development of quadruple precision arithmetic toolbox {QuPAT} on {Scilab}
entry:
@Article{Saito:2010:DQP,
  author =       "Tsubasa Saito and Emiko Ishiwata and Hidehiko
                 Hasegawa",
  title =        "Development of quadruple precision arithmetic toolbox
                 {QuPAT} on {Scilab}",
  journal =      j-LECT-NOTES-COMP-SCI,
  volume =       "6017",
  pages =        "60--70",
  year =         "2010",
  CODEN =        "LNCSD9",
  DOI =          "https://doi.org/10.1007/978-3-642-12165-4_5",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  bibdate =      "Sat Apr 01 07:57:10 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://link.springer.com/chapter/10.1007/978-3-642-12165-4_5",
  acknowledgement = ack-nhfb,
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
  keywords =     "floating-point arithmetic; QuPAT (Quadruple Precision
                 Arithmetic Toolbox)",
  remark =       "International Conference on Computational Science and
                 Its Applications, ICCSA 2010",
}


bibtype: inproceedings
filename: fparith.bib
label: Samy:2010:DFP
author: R. Samy and H. A. H. Fahmy and R. Raafat and A. Mohamed and T. ElDeeb and Y. Farouk
title: A decimal floating-point fused-multiply-add unit
crossref: Delgado-Frias:2010:IIM
pages: 529--532
year: 2010
DOI: https://doi.org/10.1109/MWSCAS.2010.5548746
bibdate: Thu Feb 17 08:38:22 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.02.17 08:38:22 ???
acknowledgement: ack-nhfb
rawauthor: R. Samy and H. A. H. Fahmy and R. Raafat and A. Mohamed and T. ElDeeb and Y. Farouk
rawtitle: A decimal floating-point fused-multiply-add unit
abstract:
This paper presents the first hardware implementation of a fully parallel decimal floating-point fused-multiply-add unit performing the operation $ pm (A times B) pm C $ on decimal floating-point operands. The proposed design is fully compliant with the IEEE 754-2008 standard and supports the two standard formats decimal64 and decimal128. Furthermore, the proposed design may be controlled to perform the multiplication or the addition / subtraction as standalone operations. Our decimal floating-point FMA may be pipelined so that a complete resultant decimal floating-point is available each clock cycle.

rawabstract:
This paper presents the first hardware implementation of a fully parallel decimal floating-point fused-multiply-add unit performing the operation $ \pm (A \times B) \pm C $ on decimal floating-point operands. The proposed design is fully compliant with the IEEE 754-2008 standard and supports the two standard formats decimal64 and decimal128. Furthermore, the proposed design may be controlled to perform the multiplication or the addition\slash subtraction as standalone operations. Our decimal floating-point FMA may be pipelined so that a complete resultant decimal floating-point is available each clock cycle.

entry:
@InProceedings{Samy:2010:DFP,
  author =       "R. Samy and H. A. H. Fahmy and R. Raafat and A.
                 Mohamed and T. ElDeeb and Y. Farouk",
  title =        "A decimal floating-point fused-multiply-add unit",
  crossref =     "Delgado-Frias:2010:IIM",
  pages =        "529--532",
  year =         "2010",
  DOI =          "https://doi.org/10.1109/MWSCAS.2010.5548746",
  bibdate =      "Thu Feb 17 08:38:22 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "This paper presents the first hardware implementation
                 of a fully parallel decimal floating-point
                 fused-multiply-add unit performing the operation $ \pm
                 (A \times B) \pm C $ on decimal floating-point
                 operands. The proposed design is fully compliant with
                 the IEEE 754-2008 standard and supports the two
                 standard formats decimal64 and decimal128. Furthermore,
                 the proposed design may be controlled to perform the
                 multiplication or the addition\slash subtraction as
                 standalone operations. Our decimal floating-point FMA
                 may be pipelined so that a complete resultant decimal
                 floating-point is available each clock cycle.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Sasaki:2010:CFP
author: Tateaki Sasaki and Fujio Kako
title: Computing floating-point Grobner bases accurately
journal: j-ACM-COMM-COMP-ALGEBRA
volume: 44
number: 3
pages: 142--143
month: sep
monthnumber: 9
year: 2010
DOI: https://doi.org/10.1145/1940475.1940504
ISSN: 1932-2232 (print), 1932-2240 (electronic) OR 19322232 (print), 19322240 (electronic)
ISSNL: 1932-2232 OR 19322232
bibdate: Thu Mar 31 10:24:16 MDT 2011
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.03.31 10:24:16 MDT
fjournal: ACM Communications in Computer Algebra
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
issue: 173
rawauthor: Tateaki Sasaki and Fujio Kako
rawtitle: Computing floating-point {Gr{\"o}bner} bases accurately
abstract:
We propose a new method for floating-point Grobner basis computation. The idea is to improve the accuracies of the coefficients of polynomials of intermediate bases computed by our previous method, by reducing coefficient matrices by Gaussian elimination.

rawabstract:
We propose a new method for floating-point Gr{\"o}bner basis computation. The idea is to improve the accuracies of the coefficients of polynomials of intermediate bases computed by our previous method, by reducing coefficient matrices by Gaussian elimination.

entry:
@Article{Sasaki:2010:CFP,
  author =       "Tateaki Sasaki and Fujio Kako",
  title =        "Computing floating-point {Gr{\"o}bner} bases
                 accurately",
  journal =      j-ACM-COMM-COMP-ALGEBRA,
  volume =       "44",
  number =       "3",
  pages =        "142--143",
  month =        sep,
  year =         "2010",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/1940475.1940504",
  ISSN =         "1932-2232 (print), 1932-2240 (electronic)",
  ISSN-L =       "1932-2232",
  bibdate =      "Thu Mar 31 10:24:16 MDT 2011",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We propose a new method for floating-point Gr{\"o}bner
                 basis computation. The idea is to improve the
                 accuracies of the coefficients of polynomials of
                 intermediate bases computed by our previous method, by
                 reducing coefficient matrices by Gaussian
                 elimination.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Communications in Computer Algebra",
  issue =        "173",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
}


bibtype: book
filename: fparith.bib
label: Schertz:2010:CM
author: Reinhard Schertz
title: Complex Multiplication
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
pages: xiii + 361
year: 2010
ISBN: 0-521-76668-0 OR 0521766680
ISBN13: 978-0-521-76668-5 OR 9780521766685
LCCN: QA564 .S294 2010
bibdate: Thu Mar 04 18:34:14 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.03.04 18:34:14 ???
subject: Multiplication, Complex
acknowledgement: ack-nhfb
rawauthor: Reinhard Schertz
rawtitle: Complex Multiplication
remark:
This is a self-contained account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant $j$ and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass $p$-function; relations between elliptic units and class numbers.

TOC:
Preface 1. Elliptic functions 2. Modular functions 3. Basic facts from number theory 4. Factorisation of singular values 5. The reciprocity law 6. Generation of ring class fields and ray class fields 7. Integral basis in ray class fields 8. Galois module structure 9. Berwick's congruences 10. Cryptographically relevant elliptic curves 11. The class number formulas of Curt Meyer 12. Arithmetic interpretation of class number formulas References Index of notation Index

rawTOC:
Preface \\ 1. Elliptic functions \\ 2. Modular functions \\ 3. Basic facts from number theory \\ 4. Factorisation of singular values \\ 5. The reciprocity law \\ 6. Generation of ring class fields and ray class fields \\ 7. Integral basis in ray class fields \\ 8. Galois module structure \\ 9. Berwick's congruences \\ 10. Cryptographically relevant elliptic curves \\ 11. The class number formulas of Curt Meyer \\ 12. Arithmetic interpretation of class number formulas \\ References \\ Index of notation \\ Index

entry:
@Book{Schertz:2010:CM,
  author =       "Reinhard Schertz",
  title =        "Complex Multiplication",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  pages =        "xiii + 361",
  year =         "2010",
  ISBN =         "0-521-76668-0",
  ISBN-13 =      "978-0-521-76668-5",
  LCCN =         "QA564 .S294 2010",
  bibdate =      "Thu Mar 04 18:34:14 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  remark =       "This is a self-contained account of the state of the
                 art in classical complex multiplication that includes
                 recent results on rings of integers and applications to
                 cryptography using elliptic curves. The author is
                 exhaustive in his treatment, giving a thorough
                 development of the theory of elliptic functions,
                 modular functions and quadratic number fields and
                 providing a concise summary of the results from class
                 field theory. The main results are accompanied by
                 numerical examples, equipping any reader with all the
                 tools and formulas they need. Topics covered include:
                 the construction of class fields over quadratic
                 imaginary number fields by singular values of the
                 modular invariant $j$ and Weber's tau-function;
                 explicit construction of rings of integers in ray class
                 fields and Galois module structure; the construction of
                 cryptographically relevant elliptic curves over finite
                 fields; proof of Berwick's congruences using division
                 values of the Weierstrass $p$-function; relations
                 between elliptic units and class numbers.",
  subject =      "Multiplication, Complex",
  tableofcontents = "Preface \\
                 1. Elliptic functions \\
                 2. Modular functions \\
                 3. Basic facts from number theory \\
                 4. Factorisation of singular values \\
                 5. The reciprocity law \\
                 6. Generation of ring class fields and ray class fields
                 \\
                 7. Integral basis in ray class fields \\
                 8. Galois module structure \\
                 9. Berwick's congruences \\
                 10. Cryptographically relevant elliptic curves \\
                 11. The class number formulas of Curt Meyer \\
                 12. Arithmetic interpretation of class number formulas
                 \\
                 References \\
                 Index of notation \\
                 Index",
}


bibtype: article
filename: fparith.bib
label: Schneeweiss:2010:SAR
author: H. Schneeweiss and J. Komlos and A. S. Ahmad
title: Symmetric and asymmetric rounding: a review and some new results
journal: j-ASTA-ADV-STAT-ANAL
volume: 94
number: 3
pages: 247--271
month: sep
monthnumber: 9
year: 2010
DOI: https://doi.org/10.1007/s10182-010-0125-2
ISSN: 1863-8171 (print), 1863-818X (electronic) OR 18638171 (print), 1863818X (electronic)
ISSNL: 1863-8171 OR 18638171
bibdate: Thu Feb 5 09:35:46 MST 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/asta.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.02.05 09:35:46 MST
URL: http://link.springer.com/article/10.1007/s10182-010-0125-2
fjournal: AStA. Advances in Statistical Analysis
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10182
rawauthor: H. Schneeweiss and J. Komlos and A. S. Ahmad
rawtitle: Symmetric and asymmetric rounding: a review and some new results
entry:
@Article{Schneeweiss:2010:SAR,
  author =       "H. Schneeweiss and J. Komlos and A. S. Ahmad",
  title =        "Symmetric and asymmetric rounding: a review and some
                 new results",
  journal =      j-ASTA-ADV-STAT-ANAL,
  volume =       "94",
  number =       "3",
  pages =        "247--271",
  month =        sep,
  year =         "2010",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s10182-010-0125-2",
  ISSN =         "1863-8171 (print), 1863-818X (electronic)",
  ISSN-L =       "1863-8171",
  bibdate =      "Thu Feb 5 09:35:46 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/asta.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://link.springer.com/article/10.1007/s10182-010-0125-2",
  acknowledgement = ack-nhfb,
  fjournal =     "AStA. Advances in Statistical Analysis",
  journal-URL =  "http://link.springer.com/journal/10182",
}


bibtype: inproceedings
filename: fparith.bib
label: Sheikh:2010:OOA
author: Basit Riaz Sheikh and Rajit Manohar
editor: IEEE
title: An Operand-Optimized Asynchronous IEEE 754 Double-Precision Floating-Point Adder
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 151--162
year: 2010
DOI: https://doi.org/10.1109/ASYNC.2010.24
ISBN: 0-7695-4032-5 OR 0769540325
ISBN13: 978-0-7695-4032-0 OR 9780769540320
bibdate: Sat Apr 02 16:34:52 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.04.02 16:34:52 ???
acknowledgement: ack-nhfb
bookpages: xviii + 197
rawauthor: Basit Riaz Sheikh and Rajit Manohar
raweditor: IEEE
rawtitle: An Operand-Optimized Asynchronous {IEEE 754} Double-Precision Floating-Point Adder
booktitle:
ASYNC 2010: 16th IEEE Symposium on Asynchronous Circuits and Systems, 3--6 May 2010, Grenoble, France, Proceedings

URL:
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5476966; https://www.semanticscholar.org/paper/An-Operand-Optimized-Asynchronous-IEEE-754-Double-Sheikh-Manohar/3e30463d11ba059f19c5959e0acc66709390475e

rawbooktitle:
{ASYNC 2010: 16th IEEE Symposium on Asynchronous Circuits and Systems, 3--6 May 2010, Grenoble, France, Proceedings}

entry:
@InProceedings{Sheikh:2010:OOA,
  author =       "Basit Riaz Sheikh and Rajit Manohar",
  editor =       "IEEE",
  booktitle =    "{ASYNC 2010: 16th IEEE Symposium on Asynchronous
                 Circuits and Systems, 3--6 May 2010, Grenoble, France,
                 Proceedings}",
  title =        "An Operand-Optimized Asynchronous {IEEE 754}
                 Double-Precision Floating-Point Adder",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  bookpages =    "xviii + 197",
  pages =        "151--162",
  year =         "2010",
  DOI =          "https://doi.org/10.1109/ASYNC.2010.24",
  ISBN =         "0-7695-4032-5",
  ISBN-13 =      "978-0-7695-4032-0",
  LCCN =         "????",
  bibdate =      "Sat Apr 02 16:34:52 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5476966;
                 https://www.semanticscholar.org/paper/An-Operand-Optimized-Asynchronous-IEEE-754-Double-Sheikh-Manohar/3e30463d11ba059f19c5959e0acc66709390475e",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Shieh:2010:WBM
author: Ming-Der Shieh and Wen-Ching Lin
title: Word-Based Montgomery Modular Multiplication Algorithm for Low-Latency Scalable Architectures
journal: j-IEEE-TRANS-COMPUT
volume: 59
number: 8
pages: 1145--1151
year: 2010
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.72
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Jul 3 11:52:31 MDT 2011
bibtimestamp: 2011.07.03 11:52:31 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5441286
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Ming-Der Shieh and Wen-Ching Lin
rawtitle: Word-Based {Montgomery} Modular Multiplication Algorithm for Low-Latency Scalable Architectures
entry:
@Article{Shieh:2010:WBM,
  author =       "Ming-Der Shieh and Wen-Ching Lin",
  title =        "Word-Based {Montgomery} Modular Multiplication
                 Algorithm for Low-Latency Scalable Architectures",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "59",
  number =       "8",
  pages =        "1145--1151",
  month =        "????",
  year =         "2010",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.72",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Jul 3 11:52:31 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5441286",
  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: Shuster:2010:ECN
author: John A. Shuster and Jens Koplinger
title: Elliptic complex numbers with dual multiplication
journal: j-APPL-MATH-COMP
volume: 216
number: 12
pages: 3497--3514
day: 15
month: aug
monthnumber: 8
year: 2010
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Sep 3 10:53:27 MDT 2010
bibtimestamp: 2010.09.03 10:53:27 MDT
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: John A. Shuster and Jens K{\"o}plinger
rawtitle: Elliptic complex numbers with dual multiplication
entry:
@Article{Shuster:2010:ECN,
  author =       "John A. Shuster and Jens K{\"o}plinger",
  title =        "Elliptic complex numbers with dual multiplication",
  journal =      j-APPL-MATH-COMP,
  volume =       "216",
  number =       "12",
  pages =        "3497--3514",
  day =          "15",
  month =        aug,
  year =         "2010",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Sep 3 10:53:27 MDT 2010",
  bibsource =    "http://www.sciencedirect.com/science/journal/00963003;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: book
filename: fparith.bib
label: Smiley:2010:MWI
author: Jane Smiley
title: The man who invented the computer: the biography of John Atanasoff, digital pioneer
publisher: pub-DOUBLEDAY
address: pub-DOUBLEDAY:adr
pages: 246 + 8
year: 2010
LCCN: QA76.2.A75 S64 2010
bibdate: Mon Oct 25 10:49:42 MDT 2010
bibtimestamp: 2010.10.25 10:49:42 MDT
acknowledgement: ack-nhfb
rawauthor: Jane Smiley
rawtitle: The man who invented the computer: the biography of {John Atanasoff}, digital pioneer
ISBN:
0-385-52713-6, 0-385-53372-1 (e-book), 1-299-11995-6 (e-book) OR 0385527136, 0385533721 (ebook), 1299119956 (ebook)

ISBN13:
978-0-385-52713-2, 978-0-385-53372-0 (e-book), 978-1-299-11995-6 (e-book) OR 9780385527132, 9780385533720 (ebook), 9781299119956 (ebook)

bibsource:
fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/t/teller-edward.bib; https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/annhistcomput.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib

abstract:
One night in the late 1930s, in a bar on the Illinois--Iowa border, John Vincent Atanasoff, a professor of physics at Iowa State University, after a frustrating day performing tedious mathematical calculations in his lab, hit on the idea that the binary number system and electronic switches, combined with an array of capacitors on a moving drum to serve as memory, could yield a computing machine that would make his life easier. Then he went back and built the machine. It worked, but he never patented the device, and the developers of the far-better-known ENIAC almost certainly stole critical ideas from him. But in 1973 a court declared that the patent on that Sperry Rand device was invalid, opening the gates to the computer revolution. Biographer Jane Smiley makes the race to develop digital computing as gripping as a real-life techno-thriller.

keywords:
ABC computer; Alan Turing; Charles Babbage; Clifford Berry; J. Presper Eckert; John V (John Vincent) Atanasoff; John von Neumann; John W. Mauchly; Konrad Zuse; Max Newman

subject:
Atanasoff, John V (John Vincent); Computer scientists; United States; Biography; Inventors; Physicists; Iowa; College teachers; Electronic digital computers; History; 20th century; Patents; Intellectual property

rawabstract:
One night in the late 1930s, in a bar on the Illinois--Iowa border, John Vincent Atanasoff, a professor of physics at Iowa State University, after a frustrating day performing tedious mathematical calculations in his lab, hit on the idea that the binary number system and electronic switches, combined with an array of capacitors on a moving drum to serve as memory, could yield a computing machine that would make his life easier. Then he went back and built the machine. It worked, but he never patented the device, and the developers of the far-better-known ENIAC almost certainly stole critical ideas from him. But in 1973 a court declared that the patent on that Sperry Rand device was invalid, opening the gates to the computer revolution. Biographer Jane Smiley makes the race to develop digital computing as gripping as a real-life techno-thriller.

entry:
@Book{Smiley:2010:MWI,
  author =       "Jane Smiley",
  title =        "The man who invented the computer: the biography of
                 {John Atanasoff}, digital pioneer",
  publisher =    pub-DOUBLEDAY,
  address =      pub-DOUBLEDAY:adr,
  pages =        "246 + 8",
  year =         "2010",
  ISBN =         "0-385-52713-6, 0-385-53372-1 (e-book), 1-299-11995-6
                 (e-book)",
  ISBN-13 =      "978-0-385-52713-2, 978-0-385-53372-0 (e-book),
                 978-1-299-11995-6 (e-book)",
  LCCN =         "QA76.2.A75 S64 2010",
  bibdate =      "Mon Oct 25 10:49:42 MDT 2010",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/bibnet/authors/t/teller-edward.bib;
                 https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib;
                 https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib;
                 https://www.math.utah.edu/pub/tex/bib/annhistcomput.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib",
  price =        "US\$25.95",
  abstract =     "One night in the late 1930s, in a bar on the
                 Illinois--Iowa border, John Vincent Atanasoff, a
                 professor of physics at Iowa State University, after a
                 frustrating day performing tedious mathematical
                 calculations in his lab, hit on the idea that the
                 binary number system and electronic switches, combined
                 with an array of capacitors on a moving drum to serve
                 as memory, could yield a computing machine that would
                 make his life easier. Then he went back and built the
                 machine. It worked, but he never patented the device,
                 and the developers of the far-better-known ENIAC almost
                 certainly stole critical ideas from him. But in 1973 a
                 court declared that the patent on that Sperry Rand
                 device was invalid, opening the gates to the computer
                 revolution. Biographer Jane Smiley makes the race to
                 develop digital computing as gripping as a real-life
                 techno-thriller.",
  acknowledgement = ack-nhfb,
  keywords =     "ABC computer; Alan Turing; Charles Babbage; Clifford
                 Berry; J. Presper Eckert; John V (John Vincent)
                 Atanasoff; John von Neumann; John W. Mauchly; Konrad
                 Zuse; Max Newman",
  remark-1 =     "This book contains discussions of the patent battles
                 over early computer designs (pp 131--133 and Chapter
                 9), and some limited information about arithmetic
                 choices, such as on pp. 162--163 ``Howard Aiken, who
                 was still advocating for decimal numbers for
                 computers''. It also contains statements of how Konrad
                 Zuse's early work on electromechanical computers was
                 finally made known in the 1950s in the US (see p.
                 159).",
  remark-2 =     "The chapters are numbered, but without titles, so
                 there is no tableofcontents field in this entry.",
  remark-3 =     "Chapters 10--12 on the Honeywell vs Sperry lawsuit
                 (1968--1971) over who discovered the computer present a
                 strong rebuttal to the views in
                 \cite{McCartney:2001:ETT}.",
  remark-4 =     "Pages 188--189 mention Edward Teller's testimony in
                 August 1971 in the Honeywell vs Sperry lawsuit that the
                 ENIAC had been used for calculations by Los Alamos
                 scientists in 1945--1945, almost two years before the
                 Mauchly and Eckert (i.e., Sperry) patent claims on
                 ENIAC technology. That is `prior art' that led to
                 Sperry losing its patent claims for ENIAC.",
  subject =      "Atanasoff, John V (John Vincent); Computer scientists;
                 United States; Biography; Inventors; Physicists; Iowa;
                 College teachers; Electronic digital computers;
                 History; 20th century; Patents; Intellectual property",
}


bibtype: article
filename: fparith.bib
label: Smith:2010:AFA
author: Alastair M. Smith and George A. Constantinides and Peter Y. K. Cheung
title: An Automated Flow for Arithmetic Component Generation in Field-Programmable Gate Arrays
journal: j-TRETS
volume: 3
number: 3
pages: 13:1--13:??
month: sep
monthnumber: 9
year: 2010
DOI: https://doi.org/10.1145/1839480.1839483
ISSN: 1936-7406 (print), 1936-7414 (electronic) OR 19367406 (print), 19367414 (electronic)
ISSNL: 1936-7406 OR 19367406
bibdate: Fri Oct 8 18:26:34 MDT 2010
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.10.08 18:26:34 MDT
fjournal: ACM Transactions on Reconfigurable Technology and Systems (TRETS)
keywords: common subgraph; FPGA; reconfigurable logic
acknowledgement: ack-nhfb
articleno: 13
journalURL: http://portal.acm.org/toc.cfm?id=J1151
rawauthor: Alastair M. Smith and George A. Constantinides and Peter Y. K. Cheung
rawtitle: An Automated Flow for Arithmetic Component Generation in Field-Programmable Gate Arrays
abstract:
State-of-the-art configurable logic platforms, such as Field-Programmable Gate Arrays (FPGAs), consist of a heterogeneous mixture of different component types. Compared to traditional homogeneous configurable platforms, heterogeneity provides speed and density advantages. This is due to the replacement of inefficient programmable logic and routing with specialized logic and fixed interconnect in components such as memories, embedded processor units, and fused arithmetic units. Given the increasing complexity of these components, this article introduces a method to automatically propose and explore the benefits of different types of fused arithmetic units. The methods are based on common subgraph extraction techniques, meaning that it is possible to explore different subcircuits that occur frequently across a set of benchmarks. A quantitative analysis is performed of the various fused arithmetic circuits identified by our tool, which are then automatically synthesized to an ASIC process, providing a study of the speed and area benefits of the components. The results of this study provide bounds on the performance of heterogeneous FPGAs: by incorporating coarse-grain components which match the specific needs of a set of benchmarks we show that significant improvements in circuit speed and area can be made.

rawabstract:
State-of-the-art configurable logic platforms, such as Field-Programmable Gate Arrays (FPGAs), consist of a heterogeneous mixture of different component types. Compared to traditional homogeneous configurable platforms, heterogeneity provides speed and density advantages. This is due to the replacement of inefficient programmable logic and routing with specialized logic and fixed interconnect in components such as memories, embedded processor units, and fused arithmetic units. Given the increasing complexity of these components, this article introduces a method to automatically propose and explore the benefits of different types of fused arithmetic units. The methods are based on common subgraph extraction techniques, meaning that it is possible to explore different subcircuits that occur frequently across a set of benchmarks. A quantitative analysis is performed of the various fused arithmetic circuits identified by our tool, which are then automatically synthesized to an ASIC process, providing a study of the speed and area benefits of the components. The results of this study provide bounds on the performance of heterogeneous FPGAs: by incorporating coarse-grain components which match the specific needs of a set of benchmarks we show that significant improvements in circuit speed and area can be made.

entry:
@Article{Smith:2010:AFA,
  author =       "Alastair M. Smith and George A. Constantinides and
                 Peter Y. K. Cheung",
  title =        "An Automated Flow for Arithmetic Component Generation
                 in Field-Programmable Gate Arrays",
  journal =      j-TRETS,
  volume =       "3",
  number =       "3",
  pages =        "13:1--13:??",
  month =        sep,
  year =         "2010",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/1839480.1839483",
  ISSN =         "1936-7406 (print), 1936-7414 (electronic)",
  ISSN-L =       "1936-7406",
  bibdate =      "Fri Oct 8 18:26:34 MDT 2010",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "State-of-the-art configurable logic platforms, such as
                 Field-Programmable Gate Arrays (FPGAs), consist of a
                 heterogeneous mixture of different component types.
                 Compared to traditional homogeneous configurable
                 platforms, heterogeneity provides speed and density
                 advantages. This is due to the replacement of
                 inefficient programmable logic and routing with
                 specialized logic and fixed interconnect in components
                 such as memories, embedded processor units, and fused
                 arithmetic units. Given the increasing complexity of
                 these components, this article introduces a method to
                 automatically propose and explore the benefits of
                 different types of fused arithmetic units. The methods
                 are based on common subgraph extraction techniques,
                 meaning that it is possible to explore different
                 subcircuits that occur frequently across a set of
                 benchmarks. A quantitative analysis is performed of the
                 various fused arithmetic circuits identified by our
                 tool, which are then automatically synthesized to an
                 ASIC process, providing a study of the speed and area
                 benefits of the components. The results of this study
                 provide bounds on the performance of heterogeneous
                 FPGAs: by incorporating coarse-grain components which
                 match the specific needs of a set of benchmarks we show
                 that significant improvements in circuit speed and area
                 can be made.",
  acknowledgement = ack-nhfb,
  articleno =    "13",
  fjournal =     "ACM Transactions on Reconfigurable Technology and
                 Systems (TRETS)",
  journal-URL =  "http://portal.acm.org/toc.cfm?id=J1151",
  keywords =     "common subgraph; FPGA; reconfigurable logic",
}


bibtype: inproceedings
filename: fparith.bib
label: Sze:2010:TQB
author: Tsz-Wo Sze
editor: IEEE
booktitle: 2010 IEEE Second International Conference on Cloud Computing Technology and Science (CloudCom)
title: The Two Quadrillionth Bit of Pi is $0$ ! Distributed Computation of Pi with Apache Hadoop
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 727
year: 2010
DOI: https://doi.org/10.1109/CloudCom.2010.57
ISBN: 1-4244-9405-2 OR 1424494052
ISBN13: 978-1-4244-9405-7 OR 9781424494057
bibdate: Mon Apr 25 18:16:05 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.04.25 18:16:05 ???
acknowledgement: ack-nhfb
rawauthor: Tsz-Wo Sze
rawbooktitle: {2010 IEEE Second International Conference on Cloud Computing Technology and Science (CloudCom)}
raweditor: {IEEE}
rawtitle: The Two Quadrillionth Bit of Pi is $0$ ! Distributed Computation of Pi with {Apache Hadoop}
abstract:
We present a new record on computing specific bits of Pi, the mathematical constant, and discuss performing such computations on Apache Hadoop clusters. The specific bits represented in hexadecimal are 0E6C1294 AED40403 F56D2D76 4026265B CA98511D 0FCFFAA1 0F4D28B1 BB5392B8. These 256 bits end at the 2,000,000,000,000,252nd bit position, which doubles the position and quadruples the precision of the previous known record. The position of the first bit is 1,999,999,999,999,997 and the value of the two quadrillionth bit is 0. The computation is carried out by a MapReduce program called DistBbp. To effectively utilize available cluster resources without monopolizing the whole cluster, we develop an elastic computation framework that automatically schedules computation slices, each a DistBbp job, as either map-side or reduce-side computation based on changing cluster load condition. We have calculated Pi at varying bit positions and precisions, and one of the largest computations took 23 days of wall clock time and 503 years of CPU time on a 1000-node cluster.

remark:
This paper contains a good discussion of floating-point rounding errors in the BBP algorithm, and of the optimal way to distribute computations over multiple independent systems sharing a common filesystem (needed to permit restart after node failure).

rawabstract:
We present a new record on computing specific bits of Pi, the mathematical constant, and discuss performing such computations on Apache Hadoop clusters. The specific bits represented in hexadecimal are 0E6C1294 AED40403 F56D2D76 4026265B CA98511D 0FCFFAA1 0F4D28B1 BB5392B8. These 256 bits end at the 2,000,000,000,000,252nd bit position, which doubles the position and quadruples the precision of the previous known record. The position of the first bit is 1,999,999,999,999,997 and the value of the two quadrillionth bit is 0. The computation is carried out by a MapReduce program called DistBbp. To effectively utilize available cluster resources without monopolizing the whole cluster, we develop an elastic computation framework that automatically schedules computation slices, each a DistBbp job, as either map-side or reduce-side computation based on changing cluster load condition. We have calculated Pi at varying bit positions and precisions, and one of the largest computations took 23 days of wall clock time and 503 years of CPU time on a 1000-node cluster.

entry:
@InProceedings{Sze:2010:TQB,
  author =       "Tsz-Wo Sze",
  editor =       "{IEEE}",
  booktitle =    "{2010 IEEE Second International Conference on Cloud
                 Computing Technology and Science (CloudCom)}",
  title =        "The Two Quadrillionth Bit of Pi is $0$ ! Distributed
                 Computation of Pi with {Apache Hadoop}",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "727",
  year =         "2010",
  DOI =          "https://doi.org/10.1109/CloudCom.2010.57",
  ISBN =         "1-4244-9405-2",
  ISBN-13 =      "978-1-4244-9405-7",
  LCCN =         "????",
  bibdate =      "Mon Apr 25 18:16:05 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We present a new record on computing specific bits of
                 Pi, the mathematical constant, and discuss performing
                 such computations on Apache Hadoop clusters. The
                 specific bits represented in hexadecimal are 0E6C1294
                 AED40403 F56D2D76 4026265B CA98511D 0FCFFAA1 0F4D28B1
                 BB5392B8. These 256 bits end at the
                 2,000,000,000,000,252nd bit position, which doubles the
                 position and quadruples the precision of the previous
                 known record. The position of the first bit is
                 1,999,999,999,999,997 and the value of the two
                 quadrillionth bit is 0. The computation is carried out
                 by a MapReduce program called DistBbp. To effectively
                 utilize available cluster resources without
                 monopolizing the whole cluster, we develop an elastic
                 computation framework that automatically schedules
                 computation slices, each a DistBbp job, as either
                 map-side or reduce-side computation based on changing
                 cluster load condition. We have calculated Pi at
                 varying bit positions and precisions, and one of the
                 largest computations took 23 days of wall clock time
                 and 503 years of CPU time on a 1000-node cluster.",
  acknowledgement = ack-nhfb,
  remark =       "This paper contains a good discussion of
                 floating-point rounding errors in the BBP algorithm,
                 and of the optimal way to distribute computations over
                 multiple independent systems sharing a common
                 filesystem (needed to permit restart after node
                 failure).",
}


bibtype: article
filename: fparith.bib
label: Szewczak:2010:LTR
author: Zbigniew S. Szewczak
title: A limit theorem for random sums modulo $1$
journal: j-STAT-PROB-LETT
volume: 80
number: 9 10
pages: 747--751
year: 2010
CODEN: SPLTDC
DOI: https://doi.org/10.1016/j.spl.2010.01.005
ISSN: 0167-7152 (print), 1879-2103 (electronic) OR 01677152 (print), 18792103 (electronic)
ISSNL: 0167-7152 OR 01677152
bibdate: Wed Jan 28 10:08:12 2015
bibtimestamp: 2015.01.28 10:08:12 ???
URL: http://www.sciencedirect.com/science/article/pii/S0167715210000106
fjournal: Statistics & Probability Letters
keywords: Benford's Law
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01677152
rawauthor: Zbigniew S. Szewczak
rawtitle: A limit theorem for random sums modulo $1$
abstract:
Residues of partial sums in a class of dependent random variables, including functionals of uniformly recurrent Markov chains, are in the domain of attraction of the uniform distribution. These types of limit theorems arise for example in the multiplication of floating-point numbers.

rawabstract:
Residues of partial sums in a class of dependent random variables, including functionals of uniformly recurrent Markov chains, are in the domain of attraction of the uniform distribution. These types of limit theorems arise for example in the multiplication of floating-point numbers.

entry:
@Article{Szewczak:2010:LTR,
  author =       "Zbigniew S. Szewczak",
  title =        "A limit theorem for random sums modulo $1$",
  journal =      j-STAT-PROB-LETT,
  volume =       "80",
  number =       "9 10",
  pages =        "747--751",
  year =         "2010",
  CODEN =        "SPLTDC",
  DOI =          "https://doi.org/10.1016/j.spl.2010.01.005",
  ISSN =         "0167-7152 (print), 1879-2103 (electronic)",
  ISSN-L =       "0167-7152",
  bibdate =      "Wed Jan 28 10:08:12 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/benfords-law.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0167715210000106",
  abstract =     "Residues of partial sums in a class of dependent
                 random variables, including functionals of uniformly
                 recurrent Markov chains, are in the domain of
                 attraction of the uniform distribution. These types of
                 limit theorems arise for example in the multiplication
                 of floating-point numbers.",
  acknowledgement = ack-nhfb,
  fjournal =     "Statistics \& Probability Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01677152",
  keywords =     "Benford's Law",
}


bibtype: article
filename: fparith.bib
label: Takahashi:2010:PIM
author: Daisuke Takahashi
journal: j-PARALLEL-COMPUTING
volume: 36
number: 8
pages: 439--448
month: aug
monthnumber: 8
year: 2010
CODEN: PACOEJ
DOI: https://doi.org/10.1016/j.parco.2010.02.007
ISSN: 0167-8191 (print), 1872-7336 (electronic) OR 01678191 (print), 18727336 (electronic)
ISSNL: 0167-8191 OR 01678191
bibdate: Thu Sep 2 17:51:13 MDT 2010
bibtimestamp: 2010.09.02 17:51:13 MDT
fjournal: Parallel Computing
keywords: distributed-memory parallel computer; Fast Fourier transform; multiple-precision arithmetic
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01678191
rawauthor: Daisuke Takahashi
title:
Parallel implementation of multiple-precision arithmetic and $ 2, 576, 980, 370, 000 $ decimal digits of $ pi $ calculation

bibsource:
http://www.sciencedirect.com/science/journal/01678191; https://www.math.utah.edu/pub/tex/bib/fparith.bib

abstract:
We present efficient parallel algorithms for multiple-precision arithmetic operations of more than several million decimal digits on distributed-memory parallel computers. A parallel implementation of floating-point real FFT-based multiplication is used, since the key operation for fast multiple-precision arithmetic is multiplication. The operation for releasing propagated carries and borrows in multiple-precision addition, subtraction and multiplication was also parallelized. More than 2.576 trillion decimal digits of $ pi $ were computed on 640 nodes of Appro Xtreme-X3 (648 nodes, 147.2 GFlops/node, 95.4 TFlops peak performance) with a computing elapsed time of 73 h 36 min which includes the time required for verification.

rawabstract:
We present efficient parallel algorithms for multiple-precision arithmetic operations of more than several million decimal digits on distributed-memory parallel computers. A parallel implementation of floating-point real FFT-based multiplication is used, since the key operation for fast multiple-precision arithmetic is multiplication. The operation for releasing propagated carries and borrows in multiple-precision addition, subtraction and multiplication was also parallelized. More than 2.576 trillion decimal digits of $ \pi $ were computed on 640 nodes of Appro Xtreme-X3 (648 nodes, 147.2 GFlops/node, 95.4 TFlops peak performance) with a computing elapsed time of 73 h 36 min which includes the time required for verification.

rawtitle:
Parallel implementation of multiple-precision arithmetic and $ 2, 576, 980, 370, 000 $ decimal digits of $ \pi $ calculation

entry:
@Article{Takahashi:2010:PIM,
  author =       "Daisuke Takahashi",
  title =        "Parallel implementation of multiple-precision
                 arithmetic and $ 2, 576, 980, 370, 000 $ decimal digits
                 of $ \pi $ calculation",
  journal =      j-PARALLEL-COMPUTING,
  volume =       "36",
  number =       "8",
  pages =        "439--448",
  month =        aug,
  year =         "2010",
  CODEN =        "PACOEJ",
  DOI =          "https://doi.org/10.1016/j.parco.2010.02.007",
  ISSN =         "0167-8191 (print), 1872-7336 (electronic)",
  ISSN-L =       "0167-8191",
  bibdate =      "Thu Sep 2 17:51:13 MDT 2010",
  bibsource =    "http://www.sciencedirect.com/science/journal/01678191;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We present efficient parallel algorithms for
                 multiple-precision arithmetic operations of more than
                 several million decimal digits on distributed-memory
                 parallel computers. A parallel implementation of
                 floating-point real FFT-based multiplication is used,
                 since the key operation for fast multiple-precision
                 arithmetic is multiplication. The operation for
                 releasing propagated carries and borrows in
                 multiple-precision addition, subtraction and
                 multiplication was also parallelized. More than 2.576
                 trillion decimal digits of $ \pi $ were computed on 640
                 nodes of Appro Xtreme-X3 (648 nodes, 147.2 GFlops/node,
                 95.4 TFlops peak performance) with a computing elapsed
                 time of 73 h 36 min which includes the time required
                 for verification.",
  acknowledgement = ack-nhfb,
  fjournal =     "Parallel Computing",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01678191",
  keywords =     "distributed-memory parallel computer; Fast Fourier
                 transform; multiple-precision arithmetic",
}


bibtype: article
filename: fparith.bib
label: Tichy:2010:GAF
author: Milan Tichy and Jan Schier and David Gregg
title: GSFAP adaptive filtering using log arithmetic for resource-constrained embedded systems
journal: j-TECS
volume: 9
number: 3
pages: 29:1--29:??
month: feb
monthnumber: 2
year: 2010
ISSN: 1539-9087 (print), 1558-3465 (electronic) OR 15399087 (print), 15583465 (electronic)
ISSNL: 1539-9087 OR 15399087
bibdate: Mon Mar 15 18:41:02 MDT 2010
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.03.15 18:41:02 MDT
fjournal: ACM Transactions on Embedded Computing Systems
acknowledgement: ack-nhfb
articleno: 29
journalURL: http://portal.acm.org/browse_dl.cfm?&idx=J840
rawauthor: Milan Tichy and Jan Schier and David Gregg
rawtitle: {GSFAP} adaptive filtering using log arithmetic for resource-constrained embedded systems
entry:
@Article{Tichy:2010:GAF,
  author =       "Milan Tichy and Jan Schier and David Gregg",
  title =        "{GSFAP} adaptive filtering using log arithmetic for
                 resource-constrained embedded systems",
  journal =      j-TECS,
  volume =       "9",
  number =       "3",
  pages =        "29:1--29:??",
  month =        feb,
  year =         "2010",
  CODEN =        "????",
  ISSN =         "1539-9087 (print), 1558-3465 (electronic)",
  ISSN-L =       "1539-9087",
  bibdate =      "Mon Mar 15 18:41:02 MDT 2010",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  articleno =    "29",
  fjournal =     "ACM Transactions on Embedded Computing Systems",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?&idx=J840",
}


bibtype: article
filename: fparith.bib
label: Vazquez:2010:IDH
author: A. Vazquez and E. Antelo and P. Montuschi
title: Improved Design of High-Performance Parallel Decimal Multipliers
journal: j-IEEE-TRANS-COMPUT
volume: 59
number: 5
pages: 679--693
year: 2010
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2009.167
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Jul 3 11:52:28 MDT 2011
bibtimestamp: 2011.07.03 11:52:28 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5313798
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: A. Vazquez and E. Antelo and P. Montuschi
rawtitle: Improved Design of High-Performance Parallel Decimal Multipliers
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib

entry:
@Article{Vazquez:2010:IDH,
  author =       "A. Vazquez and E. Antelo and P. Montuschi",
  title =        "Improved Design of High-Performance Parallel Decimal
                 Multipliers",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "59",
  number =       "5",
  pages =        "679--693",
  month =        "????",
  year =         "2010",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2009.167",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Jul 3 11:52:28 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5313798",
  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: Vestias:2010:PDM
author: M. P. Vestias and H. C. Neto
title: Parallel decimal multipliers using binary multipliers
crossref: Santos:2010:PVS
pages: 73--78
year: 2010
DOI: https://doi.org/10.1109/SPL.2010.5483001
bibdate: Thu Feb 17 08:10:02 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.02.17 08:10:02 ???
acknowledgement: ack-nhfb
rawauthor: M. P. V{\'e}stias and H. C. Neto
rawtitle: Parallel decimal multipliers using binary multipliers
abstract:
Human-centric applications, like financial and commercial, depend on decimal arithmetic since the results must match exactly those obtained by human calculations. The IEEE-754 2008 standard for floating point arithmetic has definitely recognized the importance of decimal for computer arithmetic. A number of hardware approaches have already been proposed for decimal arithmetic operations, including addition, subtraction, multiplication and division. However, few efforts have been done to develop decimal IP cores able to take advantage of the binary multipliers available in most reconfigurable computing architectures. In this paper, we analyze the tradeoffs involved in the design of a parallel decimal multiplier, for decimal operands with 8 and 16 digits, using existent coarse-grained embedded binary arithmetic blocks. The proposed circuits were implemented in a Xilinx Virtex 4 FPGA. The results indicate that the proposed parallel multipliers are very competitive when compared to decimal multipliers implemented with direct manipulation of BCD numbers.

rawabstract:
Human-centric applications, like financial and commercial, depend on decimal arithmetic since the results must match exactly those obtained by human calculations. The IEEE-754 2008 standard for floating point arithmetic has definitely recognized the importance of decimal for computer arithmetic. A number of hardware approaches have already been proposed for decimal arithmetic operations, including addition, subtraction, multiplication and division. However, few efforts have been done to develop decimal IP cores able to take advantage of the binary multipliers available in most reconfigurable computing architectures. In this paper, we analyze the tradeoffs involved in the design of a parallel decimal multiplier, for decimal operands with 8 and 16 digits, using existent coarse-grained embedded binary arithmetic blocks. The proposed circuits were implemented in a Xilinx Virtex 4 FPGA. The results indicate that the proposed parallel multipliers are very competitive when compared to decimal multipliers implemented with direct manipulation of BCD numbers.

entry:
@InProceedings{Vestias:2010:PDM,
  author =       "M. P. V{\'e}stias and H. C. Neto",
  title =        "Parallel decimal multipliers using binary
                 multipliers",
  crossref =     "Santos:2010:PVS",
  pages =        "73--78",
  year =         "2010",
  DOI =          "https://doi.org/10.1109/SPL.2010.5483001",
  bibdate =      "Thu Feb 17 08:10:02 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Human-centric applications, like financial and
                 commercial, depend on decimal arithmetic since the
                 results must match exactly those obtained by human
                 calculations. The IEEE-754 2008 standard for floating
                 point arithmetic has definitely recognized the
                 importance of decimal for computer arithmetic. A number
                 of hardware approaches have already been proposed for
                 decimal arithmetic operations, including addition,
                 subtraction, multiplication and division. However, few
                 efforts have been done to develop decimal IP cores able
                 to take advantage of the binary multipliers available
                 in most reconfigurable computing architectures. In this
                 paper, we analyze the tradeoffs involved in the design
                 of a parallel decimal multiplier, for decimal operands
                 with 8 and 16 digits, using existent coarse-grained
                 embedded binary arithmetic blocks. The proposed
                 circuits were implemented in a Xilinx Virtex 4 FPGA.
                 The results indicate that the proposed parallel
                 multipliers are very competitive when compared to
                 decimal multipliers implemented with direct
                 manipulation of BCD numbers.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Wang:2010:AOB
author: Chih-Yueh Wang and Chen-Yang Yin and Hong-Yu Chen and Yung-Ko Chen
title: Arithmetic Operations Beyond Floating Point Number Precision
journal: arXiv.org
pages: ??--??
day: 29
month: sep
monthnumber: 9
year: 2010
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/1009.5911
subject: Computational Physics (physics.comp-ph); Physics Education (physics.ed-ph)
acknowledgement: ack-nhfb
rawauthor: Chih-Yueh Wang and Chen-Yang Yin and Hong-Yu Chen and Yung-Ko Chen
rawtitle: Arithmetic Operations Beyond Floating Point Number Precision
abstract:
In basic computational physics classes, students often raise the question of how to compute a number that exceeds the numerical limit of the machine. While technique of avoiding overflow/underflow has practical application in the electrical and electronics engineering industries, it is not commonly utilized in scientific computing, because scientific notation is adequate in most cases. We present an undergraduate project that deals with such calculations beyond a machine's numerical limit, known as arbitrary precision arithmetic. The assignment asks students to investigate the approach of calculating the exact value of a large number beyond the floating point number precision, using the basic scientific programming language Fortran. The basic concept is to utilize arrays to decompose the number and allocate finite memory. Examples of the successive multiplication of even number and the multiplication and division of two overflowing floats are presented. The multiple precision scheme has been applied to hardware and firmware design for digital signal processing (DSP) systems, and is gaining importance to scientific computing. Such basic arithmetic operations can be integrated to solve advanced mathematical problems to almost arbitrarily-high precision that is limited by the memory of the host machine.

rawabstract:
In basic computational physics classes, students often raise the question of how to compute a number that exceeds the numerical limit of the machine. While technique of avoiding overflow/underflow has practical application in the electrical and electronics engineering industries, it is not commonly utilized in scientific computing, because scientific notation is adequate in most cases. We present an undergraduate project that deals with such calculations beyond a machine's numerical limit, known as arbitrary precision arithmetic. The assignment asks students to investigate the approach of calculating the exact value of a large number beyond the floating point number precision, using the basic scientific programming language Fortran. The basic concept is to utilize arrays to decompose the number and allocate finite memory. Examples of the successive multiplication of even number and the multiplication and division of two overflowing floats are presented. The multiple precision scheme has been applied to hardware and firmware design for digital signal processing (DSP) systems, and is gaining importance to scientific computing. Such basic arithmetic operations can be integrated to solve advanced mathematical problems to almost arbitrarily-high precision that is limited by the memory of the host machine.

entry:
@Article{Wang:2010:AOB,
  author =       "Chih-Yueh Wang and Chen-Yang Yin and Hong-Yu Chen and
                 Yung-Ko Chen",
  title =        "Arithmetic Operations Beyond Floating Point Number
                 Precision",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "??--??",
  day =          "29",
  month =        sep,
  year =         "2010",
  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/1009.5911",
  abstract =     "In basic computational physics classes, students often
                 raise the question of how to compute a number that
                 exceeds the numerical limit of the machine. While
                 technique of avoiding overflow/underflow has practical
                 application in the electrical and electronics
                 engineering industries, it is not commonly utilized in
                 scientific computing, because scientific notation is
                 adequate in most cases. We present an undergraduate
                 project that deals with such calculations beyond a
                 machine's numerical limit, known as arbitrary precision
                 arithmetic. The assignment asks students to investigate
                 the approach of calculating the exact value of a large
                 number beyond the floating point number precision,
                 using the basic scientific programming language
                 Fortran. The basic concept is to utilize arrays to
                 decompose the number and allocate finite memory.
                 Examples of the successive multiplication of even
                 number and the multiplication and division of two
                 overflowing floats are presented. The multiple
                 precision scheme has been applied to hardware and
                 firmware design for digital signal processing (DSP)
                 systems, and is gaining importance to scientific
                 computing. Such basic arithmetic operations can be
                 integrated to solve advanced mathematical problems to
                 almost arbitrarily-high precision that is limited by
                 the memory of the host machine.",
  acknowledgement = ack-nhfb,
  subject =      "Computational Physics (physics.comp-ph); Physics
                 Education (physics.ed-ph)",
}


bibtype: inproceedings
filename: fparith.bib
label: Wang:2010:DAH
author: Dong Wang and M. D. Ercegovac and Nanning Zheng
title: Design and analysis of high radix complex dividers
crossref: IEEE:2010:ICC
pages: V1--84--V1--88
year: 2010
DOI: https://doi.org/10.1109/ICCET.2010.5486287
bibdate: Fri Dec 03 15:18:03 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.12.03 15:18:03 ???
acknowledgement: ack-nhfb
rawauthor: Dong Wang and M. D. Ercegovac and Nanning Zheng
rawtitle: Design and analysis of high radix complex dividers
abstract:
This paper evaluates FPGA-based high radix hardware architecture for complex division. The architecture uses the digit-recurrence algorithm with prescaling of complex operands. It efficiently executes the prescaling and recurrence procedures in shared logic resources. Thirty independent designs of different radices from 4 to 64 and input precisions from 16 to 64 are implemented in Stratix-II FPGA and results on cost and performance provide a broad space of design parameters. Moreover, methods for estimating logic resource consumption and timing performance are also given so that one could make quick evaluations on the design before any actual implementations. The proposed architecture and design can be used as standalone arithmetic units for systems-on-chip implementations, in embedded processors, or as IP for VLSI implementation in general.

keywords:
design parameter; digit-recurrence algorithm; digital arithmetic; embedded processor; field programmable gate arrays; high radix complex dividers; IP implementation; logic design; shared logic resource consumption; standalone arithmetic unit; Stratix-II FPGA-based high radix hardware architecture; system-on-chip; systems-on-chip implementation; VLSI implementation

rawabstract:
This paper evaluates FPGA-based high radix hardware architecture for complex division. The architecture uses the digit-recurrence algorithm with prescaling of complex operands. It efficiently executes the prescaling and recurrence procedures in shared logic resources. Thirty independent designs of different radices from 4 to 64 and input precisions from 16 to 64 are implemented in Stratix-II FPGA and results on cost and performance provide a broad space of design parameters. Moreover, methods for estimating logic resource consumption and timing performance are also given so that one could make quick evaluations on the design before any actual implementations. The proposed architecture and design can be used as standalone arithmetic units for systems-on-chip implementations, in embedded processors, or as IP for VLSI implementation in general.

entry:
@InProceedings{Wang:2010:DAH,
  author =       "Dong Wang and M. D. Ercegovac and Nanning Zheng",
  title =        "Design and analysis of high radix complex dividers",
  crossref =     "IEEE:2010:ICC",
  pages =        "V1--84--V1--88",
  year =         "2010",
  DOI =          "https://doi.org/10.1109/ICCET.2010.5486287",
  bibdate =      "Fri Dec 03 15:18:03 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "This paper evaluates FPGA-based high radix hardware
                 architecture for complex division. The architecture
                 uses the digit-recurrence algorithm with prescaling of
                 complex operands. It efficiently executes the
                 prescaling and recurrence procedures in shared logic
                 resources. Thirty independent designs of different
                 radices from 4 to 64 and input precisions from 16 to 64
                 are implemented in Stratix-II FPGA and results on cost
                 and performance provide a broad space of design
                 parameters. Moreover, methods for estimating logic
                 resource consumption and timing performance are also
                 given so that one could make quick evaluations on the
                 design before any actual implementations. The proposed
                 architecture and design can be used as standalone
                 arithmetic units for systems-on-chip implementations,
                 in embedded processors, or as IP for VLSI
                 implementation in general.",
  acknowledgement = ack-nhfb,
  keywords =     "design parameter; digit-recurrence algorithm; digital
                 arithmetic; embedded processor; field programmable gate
                 arrays; high radix complex dividers; IP implementation;
                 logic design; shared logic resource consumption;
                 standalone arithmetic unit; Stratix-II FPGA-based high
                 radix hardware architecture; system-on-chip;
                 systems-on-chip implementation; VLSI implementation",
}


bibtype: article
filename: fparith.bib
label: Wang:2010:SHD
author: Liang-Kai Wang and Mark A. Erle and Charles Tsen and Eric M. Schwarz and Michael J. Schulte
title: A survey of hardware designs for decimal arithmetic
journal: j-IBM-JRD
volume: 54
number: 2
pages: 8:1--8:15
year: 2010
CODEN: IBMJAE
DOI: https://doi.org/10.1147/JRD.2010.2040930
ISSN: 0018-8646 (print), 2151-8556 (electronic) OR 00188646 (print), 21518556 (electronic)
ISSNL: 0018-8646 OR 00188646
bibdate: Sat May 1 17:44:14 MDT 2010
bibsource: http://www.research.ibm.com/journal/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.05.01 17:44:14 MDT
URL: http://www.research.ibm.com/journal/abstracts/rd/542/wang-schwarz.html
fjournal: IBM Journal of Research and Development
keywords: decimal floating-point arithmetic
acknowledgement: ack-nhfb
articleno: 8
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520
rawauthor: Liang-Kai Wang and Mark A. Erle and Charles Tsen and Eric M. Schwarz and Michael J. Schulte
rawtitle: A survey of hardware designs for decimal arithmetic
abstract:
Decimal data and decimal arithmetic operations are ubiquitous in daily life. Although microprocessors normally use binary arithmetic for computations, decimal arithmetic is often required in financial and commercial applications. Due to the increasing importance of and demand for decimal arithmetic, decimal floating-point (DFP) formats and operations are specified in the revised IEEE Standard for Floating-Point Arithmetic (IEEE 754-2008). This paper provides a survey of hardware designs for decimal arithmetic. It gives an overview of DFP arithmetic in IEEE 754-2008, describes processors that provide hardware and instruction set support for decimal arithmetic, and provides a survey of hardware designs for decimal addition, subtraction, multiplication, and division. Finally, it describes potential areas for future research.

rawabstract:
Decimal data and decimal arithmetic operations are ubiquitous in daily life. Although microprocessors normally use binary arithmetic for computations, decimal arithmetic is often required in financial and commercial applications. Due to the increasing importance of and demand for decimal arithmetic, decimal floating-point (DFP) formats and operations are specified in the revised IEEE Standard for Floating-Point Arithmetic (IEEE 754-2008). This paper provides a survey of hardware designs for decimal arithmetic. It gives an overview of DFP arithmetic in IEEE 754-2008, describes processors that provide hardware and instruction set support for decimal arithmetic, and provides a survey of hardware designs for decimal addition, subtraction, multiplication, and division. Finally, it describes potential areas for future research.

entry:
@Article{Wang:2010:SHD,
  author =       "Liang-Kai Wang and Mark A. Erle and Charles Tsen and
                 Eric M. Schwarz and Michael J. Schulte",
  title =        "A survey of hardware designs for decimal arithmetic",
  journal =      j-IBM-JRD,
  volume =       "54",
  number =       "2",
  pages =        "8:1--8:15",
  month =        "????",
  year =         "2010",
  CODEN =        "IBMJAE",
  DOI =          "https://doi.org/10.1147/JRD.2010.2040930",
  ISSN =         "0018-8646 (print), 2151-8556 (electronic)",
  ISSN-L =       "0018-8646",
  bibdate =      "Sat May 1 17:44:14 MDT 2010",
  bibsource =    "http://www.research.ibm.com/journal/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.research.ibm.com/journal/abstracts/rd/542/wang-schwarz.html",
  abstract =     "Decimal data and decimal arithmetic operations are
                 ubiquitous in daily life. Although microprocessors
                 normally use binary arithmetic for computations,
                 decimal arithmetic is often required in financial and
                 commercial applications. Due to the increasing
                 importance of and demand for decimal arithmetic,
                 decimal floating-point (DFP) formats and operations are
                 specified in the revised IEEE Standard for
                 Floating-Point Arithmetic (IEEE 754-2008). This paper
                 provides a survey of hardware designs for decimal
                 arithmetic. It gives an overview of DFP arithmetic in
                 IEEE 754-2008, describes processors that provide
                 hardware and instruction set support for decimal
                 arithmetic, and provides a survey of hardware designs
                 for decimal addition, subtraction, multiplication, and
                 division. Finally, it describes potential areas for
                 future research.",
  acknowledgement = ack-nhfb,
  articleno =    "8",
  fjournal =     "IBM Journal of Research and Development",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520",
  keywords =     "decimal floating-point arithmetic",
}


bibtype: article
filename: fparith.bib
label: Wang:2010:VVP
author: Xiaojun Wang and Miriam Leeser
title: VFloat: a Variable Precision Fixed- and Floating-Point Library for Reconfigurable Hardware
journal: j-TRETS
volume: 3
number: 3
pages: 16:1--16:??
month: sep
monthnumber: 9
year: 2010
DOI: https://doi.org/10.1145/1839480.1839486
ISSN: 1936-7406 (print), 1936-7414 (electronic) OR 19367406 (print), 19367414 (electronic)
ISSNL: 1936-7406 OR 19367406
bibdate: Fri Oct 8 18:26:34 MDT 2010
bibtimestamp: 2010.10.08 18:26:34 MDT
fjournal: ACM Transactions on Reconfigurable Technology and Systems (TRETS)
keywords: clustering; floating-point; Reconfigurable hardware
acknowledgement: ack-nhfb
articleno: 16
journalURL: http://portal.acm.org/toc.cfm?id=J1151
rawauthor: Xiaojun Wang and Miriam Leeser
rawtitle: {VFloat}: a Variable Precision Fixed- and Floating-Point Library for Reconfigurable Hardware
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:
Optimal reconfigurable hardware implementations may require the use of arbitrary floating-point formats that do not necessarily conform to IEEE specified sizes. We present a variable precision floating-point library (VFloat) that supports general floating-point formats including IEEE standard formats. Most previously published floating-point formats for use with reconfigurable hardware are subsets of our format. Custom datapaths with optimal bitwidths for each operation can be built using the variable precision hardware modules in the VFloat library, enabling a higher level of parallelism. The VFloat library includes three types of hardware modules for format control, arithmetic operations, and conversions between fixed-point and floating-point formats. The format conversions allow for hybrid fixed- and floating-point operations in a single design. This gives the designer control over a large number of design possibilities including format as well as number range within the same application. In this article, we give an overview of the components in the VFloat library and demonstrate their use in an implementation of the K-means clustering algorithm applied to multispectral satellite images.

rawabstract:
Optimal reconfigurable hardware implementations may require the use of arbitrary floating-point formats that do not necessarily conform to IEEE specified sizes. We present a variable precision floating-point library (VFloat) that supports general floating-point formats including IEEE standard formats. Most previously published floating-point formats for use with reconfigurable hardware are subsets of our format. Custom datapaths with optimal bitwidths for each operation can be built using the variable precision hardware modules in the VFloat library, enabling a higher level of parallelism. The VFloat library includes three types of hardware modules for format control, arithmetic operations, and conversions between fixed-point and floating-point formats. The format conversions allow for hybrid fixed- and floating-point operations in a single design. This gives the designer control over a large number of design possibilities including format as well as number range within the same application. In this article, we give an overview of the components in the VFloat library and demonstrate their use in an implementation of the K-means clustering algorithm applied to multispectral satellite images.

entry:
@Article{Wang:2010:VVP,
  author =       "Xiaojun Wang and Miriam Leeser",
  title =        "{VFloat}: a Variable Precision Fixed- and
                 Floating-Point Library for Reconfigurable Hardware",
  journal =      j-TRETS,
  volume =       "3",
  number =       "3",
  pages =        "16:1--16:??",
  month =        sep,
  year =         "2010",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/1839480.1839486",
  ISSN =         "1936-7406 (print), 1936-7414 (electronic)",
  ISSN-L =       "1936-7406",
  bibdate =      "Fri Oct 8 18:26:34 MDT 2010",
  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 =     "Optimal reconfigurable hardware implementations may
                 require the use of arbitrary floating-point formats
                 that do not necessarily conform to IEEE specified
                 sizes. We present a variable precision floating-point
                 library (VFloat) that supports general floating-point
                 formats including IEEE standard formats. Most
                 previously published floating-point formats for use
                 with reconfigurable hardware are subsets of our format.
                 Custom datapaths with optimal bitwidths for each
                 operation can be built using the variable precision
                 hardware modules in the VFloat library, enabling a
                 higher level of parallelism. The VFloat library
                 includes three types of hardware modules for format
                 control, arithmetic operations, and conversions between
                 fixed-point and floating-point formats. The format
                 conversions allow for hybrid fixed- and floating-point
                 operations in a single design. This gives the designer
                 control over a large number of design possibilities
                 including format as well as number range within the
                 same application. In this article, we give an overview
                 of the components in the VFloat library and demonstrate
                 their use in an implementation of the K-means
                 clustering algorithm applied to multispectral satellite
                 images.",
  acknowledgement = ack-nhfb,
  articleno =    "16",
  fjournal =     "ACM Transactions on Reconfigurable Technology and
                 Systems (TRETS)",
  journal-URL =  "http://portal.acm.org/toc.cfm?id=J1151",
  keywords =     "clustering; floating-point; Reconfigurable hardware",
}


bibtype: article
filename: fparith.bib
label: Waters:2010:RCW
author: Ron S. Waters and Earl E. Swartzlander
title: A Reduced Complexity Wallace Multiplier Reduction
journal: j-IEEE-TRANS-COMPUT
volume: 59
number: 8
pages: 1134--1137
year: 2010
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.103
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Jul 3 11:52:31 MDT 2011
bibtimestamp: 2011.07.03 11:52:31 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5467045
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Ron S. Waters and Earl E. Swartzlander
rawtitle: A Reduced Complexity {Wallace} Multiplier Reduction
entry:
@Article{Waters:2010:RCW,
  author =       "Ron S. Waters and Earl E. Swartzlander",
  title =        "A Reduced Complexity {Wallace} Multiplier Reduction",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "59",
  number =       "8",
  pages =        "1134--1137",
  month =        "????",
  year =         "2010",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.103",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Jul 3 11:52:31 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5467045",
  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: Zanoni:2010:ITC
author: Alberto Zanoni
title: Iterative Toom--Cook methods for very unbalanced long integer multiplication
crossref: Watt:2010:IPI
pages: 319--323
year: 2010
DOI: https://doi.org/10.1145/1837934.1837995
bibdate: Fri Jun 17 08:06:37 MDT 2011
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.06.17 08:06:37 MDT
acknowledgement: ack-nhfb
rawauthor: Alberto Zanoni
rawtitle: Iterative {Toom--Cook} methods for very unbalanced long integer multiplication
abstract:
We consider the multiplication of long integers when one factor is much larger than the other one. We describe an iterative approach using Toom--Cook unbalanced methods, which results in the evaluation of the smaller integer only once. The particular case of Toom-2.5 is considered in full detail. A further optimization depending on the parity of the shortest operand evaluation in 1 is also described. A comparison with GMP library is also presented.

rawabstract:
We consider the multiplication of long integers when one factor is much larger than the other one. We describe an iterative approach using Toom--Cook unbalanced methods, which results in the evaluation of the smaller integer only once. The particular case of Toom-2.5 is considered in full detail. A further optimization depending on the parity of the shortest operand evaluation in 1 is also described. A comparison with GMP library is also presented.

entry:
@InProceedings{Zanoni:2010:ITC,
  author =       "Alberto Zanoni",
  title =        "Iterative {Toom--Cook} methods for very unbalanced
                 long integer multiplication",
  crossref =     "Watt:2010:IPI",
  pages =        "319--323",
  year =         "2010",
  DOI =          "https://doi.org/10.1145/1837934.1837995",
  bibdate =      "Fri Jun 17 08:06:37 MDT 2011",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We consider the multiplication of long integers when
                 one factor is much larger than the other one. We
                 describe an iterative approach using Toom--Cook
                 unbalanced methods, which results in the evaluation of
                 the smaller integer only once. The particular case of
                 Toom-2.5 is considered in full detail. A further
                 optimization depending on the parity of the shortest
                 operand evaluation in 1 is also described. A comparison
                 with GMP library is also presented.",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Zhao:2010:GMP
author: Kaiyong Zhao and Xiaowen Chu
editor: IEEE
title: GPUMP: a Multiple-Precision Integer Library for GPUs
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 1164--1168
year: 2010
DOI: https://doi.org/10.1109/CIT.2010.211
ISBN: 0-7695-4108-9 (print), 1-4244-7547-3 OR 0769541089 (print), 1424475473
ISBN13: 978-0-7695-4108-2 (print), 978-1-4244-7547-6 OR 9780769541082 (print), 9781424475476
bibdate: Thu Jan 16 10:33:01 2014
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/pvm.bib
bibtimestamp: 2014.01.16 10:33:01 ???
note: IEEE Computer Society Order Number E4108. BMS Part Number: CFP10355-CDR
acknowledgement: ack-nhfb
bookURL: http://ieeexplore.ieee.org/servlet/opac?punumber=5575291
bookpages: xcix + 2987 (est.)
rawauthor: Kaiyong Zhao and Xiaowen Chu
raweditor: {IEEE}
rawnote: IEEE Computer Society Order Number E4108. BMS Part Number: CFP10355-CDR
rawtitle: {GPUMP}: a Multiple-Precision Integer Library for {GPUs}
booktitle:
IEEE 10th International Conference on Computer and Information Technology (CIT), 2010: June 29, 2010--July 1, 2010, Bradford, West Yorkshire, UK

keywords:
CUDA; GPU; multiple-precision algorithm; multiple-precision comparison; multiple-precision division; multiple-precision exponentiation; multiple-precision modular addition; multiple-precision modular multiplication; multiple-precision Montgomery exponentiation; multiple-precision Montgomery multiplication; multiple-precision Montgomery reduction; multiple-precision multiplication; nVidia GT200 GPU

rawbooktitle:
{IEEE 10th International Conference on Computer and Information Technology (CIT), 2010: June 29, 2010--July 1, 2010, Bradford, West Yorkshire, UK}

entry:
@InProceedings{Zhao:2010:GMP,
  author =       "Kaiyong Zhao and Xiaowen Chu",
  editor =       "{IEEE}",
  booktitle =    "{IEEE 10th International Conference on Computer and
                 Information Technology (CIT), 2010: June 29, 2010--July
                 1, 2010, Bradford, West Yorkshire, UK}",
  title =        "{GPUMP}: a Multiple-Precision Integer Library for
                 {GPUs}",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  bookpages =    "xcix + 2987 (est.)",
  pages =        "1164--1168",
  year =         "2010",
  DOI =          "https://doi.org/10.1109/CIT.2010.211",
  ISBN =         "0-7695-4108-9 (print), 1-4244-7547-3",
  ISBN-13 =      "978-0-7695-4108-2 (print), 978-1-4244-7547-6",
  LCCN =         "????",
  bibdate =      "Thu Jan 16 10:33:01 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/pvm.bib",
  note =         "IEEE Computer Society Order Number E4108. BMS Part
                 Number: CFP10355-CDR",
  acknowledgement = ack-nhfb,
  book-URL =     "http://ieeexplore.ieee.org/servlet/opac?punumber=5575291",
  keywords =     "CUDA; GPU; multiple-precision algorithm;
                 multiple-precision comparison; multiple-precision
                 division; multiple-precision exponentiation;
                 multiple-precision modular addition; multiple-precision
                 modular multiplication; multiple-precision Montgomery
                 exponentiation; multiple-precision Montgomery
                 multiplication; multiple-precision Montgomery
                 reduction; multiple-precision multiplication; nVidia
                 GT200 GPU",
}


bibtype: article
filename: fparith.bib
label: Zhu:2010:AOE
author: Yong-Kang Zhu and Wayne B. Hayes
title: Algorithm 908: Online Exact Summation of Floating-Point Streams
journal: j-TOMS
volume: 37
number: 3
pages: 37:1--37:13
year: 2010
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1824801.1824815
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Sep 27 10:15:50 MDT 2010
bibsource: http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2010.09.27 10:15:50 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
keywords: accurate floating-point summation; correct rounding; floating-point arithmetic; rounding error
acknowledgement: ack-nhfb
articleno: 37
journalURL: https://dl.acm.org/loi/toms
rawauthor: Yong-Kang Zhu and Wayne B. Hayes
rawtitle: {Algorithm 908}: Online Exact Summation of Floating-Point Streams
abstract:
We present a novel, online algorithm for exact summation of a stream of floating-point numbers. By ``online'' we mean that the algorithm needs to see only one input at a time, and can take an arbitrary length input stream of such inputs while requiring only constant memory. By ``exact'' we mean that the sum of the internal array of our algorithm is exactly equal to the sum of all the inputs, and the returned result is the correctly-rounded sum. The proof of correctness is valid for all inputs (including nonnormalized numbers but modulo intermediate overflow), and is independent of the number of summands or the condition number of the sum. The algorithm asymptotically needs only 5 FLOPs per summand, and due to instruction-level parallelism runs only about 2--3 times slower than the obvious, fast-but-dumb ``ordinary recursive summation'' loop when the number of summands is greater than 10,000. Thus, to our knowledge, it is the fastest, most accurate, and most memory efficient among known algorithms. Indeed, it is difficult to see how a faster algorithm or one requiring significantly fewer FLOPs could exist without hardware improvements. An application for a large number of summands is provided.

rawabstract:
We present a novel, online algorithm for exact summation of a stream of floating-point numbers. By ``online'' we mean that the algorithm needs to see only one input at a time, and can take an arbitrary length input stream of such inputs while requiring only constant memory. By ``exact'' we mean that the sum of the internal array of our algorithm is exactly equal to the sum of all the inputs, and the returned result is the correctly-rounded sum. The proof of correctness is valid for all inputs (including nonnormalized numbers but modulo intermediate overflow), and is independent of the number of summands or the condition number of the sum. The algorithm asymptotically needs only 5 FLOPs per summand, and due to instruction-level parallelism runs only about 2--3 times slower than the obvious, fast-but-dumb ``ordinary recursive summation'' loop when the number of summands is greater than 10,000. Thus, to our knowledge, it is the fastest, most accurate, and most memory efficient among known algorithms. Indeed, it is difficult to see how a faster algorithm or one requiring significantly fewer FLOPs could exist without hardware improvements. An application for a large number of summands is provided.

entry:
@Article{Zhu:2010:AOE,
  author =       "Yong-Kang Zhu and Wayne B. Hayes",
  title =        "{Algorithm 908}: Online Exact Summation of
                 Floating-Point Streams",
  journal =      j-TOMS,
  volume =       "37",
  number =       "3",
  pages =        "37:1--37:13",
  year =         "2010",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1824801.1824815",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Sep 27 10:15:50 MDT 2010",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We present a novel, online algorithm for exact
                 summation of a stream of floating-point numbers. By
                 ``online'' we mean that the algorithm needs to see only
                 one input at a time, and can take an arbitrary length
                 input stream of such inputs while requiring only
                 constant memory. By ``exact'' we mean that the sum of
                 the internal array of our algorithm is exactly equal to
                 the sum of all the inputs, and the returned result is
                 the correctly-rounded sum. The proof of correctness is
                 valid for all inputs (including nonnormalized numbers
                 but modulo intermediate overflow), and is independent
                 of the number of summands or the condition number of
                 the sum. The algorithm asymptotically needs only 5
                 FLOPs per summand, and due to instruction-level
                 parallelism runs only about 2--3 times slower than the
                 obvious, fast-but-dumb ``ordinary recursive summation''
                 loop when the number of summands is greater than
                 10,000. Thus, to our knowledge, it is the fastest, most
                 accurate, and most memory efficient among known
                 algorithms. Indeed, it is difficult to see how a faster
                 algorithm or one requiring significantly fewer FLOPs
                 could exist without hardware improvements. An
                 application for a large number of summands is
                 provided.",
  acknowledgement = ack-nhfb,
  articleno =    "37",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  keywords =     "accurate floating-point summation; correct rounding;
                 floating-point arithmetic; rounding error",
}


bibtype: inproceedings
filename: fparith.bib
label: Zimmermann:2010:RCG
author: Paul Zimmermann
title: Reliable Computing with GNU MPFR
crossref: Fukuda:2010:MSI
pages: 42--45
year: 2010
DOI: https://doi.org/10.1007/978-3-642-15582-6_8
bibdate: Sat Sep 23 06:20:46 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.09.23 06:20:46 ???
acknowledgement: ack-nhfb
rawauthor: Paul Zimmermann
rawtitle: Reliable Computing with {GNU MPFR}
entry:
@InProceedings{Zimmermann:2010:RCG,
  author =       "Paul Zimmermann",
  title =        "Reliable Computing with {GNU MPFR}",
  crossref =     "Fukuda:2010:MSI",
  pages =        "42--45",
  year =         "2010",
  DOI =          "https://doi.org/10.1007/978-3-642-15582-6_8",
  bibdate =      "Sat Sep 23 06:20:46 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Adikari:2011:HBT
author: Jithra Adikari and Vassil S. Dimitrov and Laurent Imbert
title: Hybrid Binary-Ternary Number System for Elliptic Curve Cryptosystems
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 2
pages: 254--265
month: feb
monthnumber: 2
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.138
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Feb 20 19:15:33 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.02.20 19:15:33 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Jithra Adikari and Vassil S. Dimitrov and Laurent Imbert
rawtitle: Hybrid Binary-Ternary Number System for Elliptic Curve Cryptosystems
entry:
@Article{Adikari:2011:HBT,
  author =       "Jithra Adikari and Vassil S. Dimitrov and Laurent
                 Imbert",
  title =        "Hybrid Binary-Ternary Number System for Elliptic Curve
                 Cryptosystems",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "60",
  number =       "2",
  pages =        "254--265",
  month =        feb,
  year =         "2011",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.138",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Feb 20 19:15:33 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.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: Al-Ashrafy:2011:EIF
author: M. Al-Ashrafy and A. Salem and W. Anis
booktitle: 2011 Saudi International Electronics, Communications and Photonics Conference (SIECPC)
title: An efficient implementation of floating point multiplier
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 1--5
year: 2011
DOI: https://doi.org/10.1109/SIECPC.2011.5876905
bibdate: Tue Sep 27 08:11:02 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.09.27 08:11:02 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5876905
acknowledgement: ack-nhfb
bookURL: http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5783968
rawauthor: M. Al-Ashrafy and A. Salem and W. Anis
rawbooktitle: {2011 Saudi International Electronics, Communications and Photonics Conference (SIECPC)}
rawtitle: An efficient implementation of floating point multiplier
entry:
@InProceedings{Al-Ashrafy:2011:EIF,
  author =       "M. Al-Ashrafy and A. Salem and W. Anis",
  booktitle =    "{2011 Saudi International Electronics, Communications
                 and Photonics Conference (SIECPC)}",
  title =        "An efficient implementation of floating point
                 multiplier",
  publisher =    "pub-IEEE",
  address =      "pub-IEEE:adr",
  pages =        "1--5",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/SIECPC.2011.5876905",
  bibdate =      "Tue Sep 27 08:11:02 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5876905",
  acknowledgement = ack-nhfb,
  book-URL =     "http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5783968",
}


bibtype: inproceedings
filename: fparith.bib
label: Anderson:2011:GVD
title: A 1.5 Ghz VLIW DSP CPU with Integrated Floating Point and Fixed Point Instructions in 40 nm CMOS
crossref: Schwarz:2011:PIS
pages: 82--86
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.20
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992112
keywords: ARITH-20
acknowledgement: ack-nhfb
author:
Timothy Anderson and Duc Bui and Shriram Moharil and Soujanya Narnur and Mujibur Rahman and Anthony Lell and Eric Biscondi and Ashish Shrivastava and Peter Dent and Mingjian Yan and Hasan Mahmood

rawauthor:
Timothy Anderson and Duc Bui and Shriram Moharil and Soujanya Narnur and Mujibur Rahman and Anthony Lell and Eric Biscondi and Ashish Shrivastava and Peter Dent and Mingjian Yan and Hasan Mahmood

rawtitle:
A {1.5 Ghz VLIW DSP CPU} with Integrated Floating Point and Fixed Point Instructions in 40 nm {CMOS}

entry:
@InProceedings{Anderson:2011:GVD,
  author =       "Timothy Anderson and Duc Bui and Shriram Moharil and
                 Soujanya Narnur and Mujibur Rahman and Anthony Lell and
                 Eric Biscondi and Ashish Shrivastava and Peter Dent and
                 Mingjian Yan and Hasan Mahmood",
  title =        "A {1.5 Ghz VLIW DSP CPU} with Integrated Floating
                 Point and Fixed Point Instructions in 40 nm {CMOS}",
  crossref =     "Schwarz:2011:PIS",
  pages =        "82--86",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.20",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992112",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: article
filename: fparith.bib
label: Anonymous:2011:CPSa
author: Anonymous
title: Call for Papers: Special Section on Computer Arithmetic
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 6
pages: 910--910
month: jun
monthnumber: 6
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2011.88
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Wed Jun 29 10:26:18 2011
bibtimestamp: 2011.06.29 10:26:18 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Anonymous
rawtitle: Call for Papers: Special Section on Computer Arithmetic
entry:
@Article{Anonymous:2011:CPSa,
  author =       "Anonymous",
  title =        "Call for Papers: Special Section on Computer
                 Arithmetic",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "60",
  number =       "6",
  pages =        "910--910",
  month =        jun,
  year =         "2011",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2011.88",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Wed Jun 29 10:26:18 2011",
  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: book
filename: fparith.bib
label: Antelo:2011:IIFa
editor: Elisardo Antelo
title: Industrial Implementations of Floating-Point Units
volume: 1
publisher: pub-IEEE
address: pub-IEEE:adr
year: 2011
bibdate: Thu Aug 18 08:02:12 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.18 08:02:12 ???
remark: Filename ES0000033.
acknowledgement: ack-nhfb
raweditor: Elisardo Antelo
rawtitle: Industrial Implementations of Floating-Point Units
TOC:
Editor's Introduction Related Videos ``P6 Binary Floating-Point Unit,'' by Son Dao Trong, Martin Schmookler, Eric M. Schwarz, and Michael Kroener ``FPU Implementations with Denormalized Numbers,'' by Eric M. Schwarz, Martin Schmookler, and Son Dao Trong ``1-GHz HAL SPARC64 Dual Floating Point Unit with RAS Features,'' by Ajay Naini, Atul Dhablania, Warren James, and Debjit Das Sarma ``Low-Power Multiple-Precision Iterative Floating-Point Multiplier with SIMD Support,'' by Dimitri Tan, Carl E. Lemonds, and Michael J. Schulte ``The Vector Floating-Point Unit in a Synergistic Processor Element of a CELL Processor,'' by Silvia M. Mueller, Christian Jacobi, Hwa-Joon Oh, Kevin D. Tran, Scott R. Cottier, Brad W. Michael, Hiroo Nishikawa, Yonetaro Totsuka, Tatsuya Namatame, Naoka Yano, Takashi Machida, and Sang H. Dhong ``A High-Performance SIMD Floating Point Unit for BlueGene/L: Architecture, Compilation, and Algorithm Design,'' by Leonardo Bachega, Siddhartha Chatterjee, Kenneth A. Dockser, John A. Gunnels, Manish Gupta, Fred G. Gustavson, Christopher A. Lapkowski, Gary K. Liu, Mark P. Mendell, Charles D. Wait, and T.J. Chris Ward ``Design of the ARM VFP11 Divide and Square Root Synthesisable Macrocell,'' by Neil Burgess and Chris N. Hinds ``Correctness Proofs Outline for Newton-Raphson Based Floating-Point Divide and Square Root Algorithms,'' by Marius A. Cornea-Hasegan, Roger A. Golliver, and Peter Markstein Recommended Resources About the Editor

rawTOC:
Editor's Introduction \\ Related Videos \\ ``P6 Binary Floating-Point Unit,'' by Son Dao Trong, Martin Schmookler, Eric M. Schwarz, and Michael Kroener \\ ``FPU Implementations with Denormalized Numbers,'' by Eric M. Schwarz, Martin Schmookler, and Son Dao Trong \\ ``1-GHz HAL SPARC64 Dual Floating Point Unit with RAS Features,'' by Ajay Naini, Atul Dhablania, Warren James, and Debjit Das Sarma \\ ``Low-Power Multiple-Precision Iterative Floating-Point Multiplier with SIMD Support,'' by Dimitri Tan, Carl E. Lemonds, and Michael J. Schulte \\ ``The Vector Floating-Point Unit in a Synergistic Processor Element of a CELL Processor,'' by Silvia M. Mueller, Christian Jacobi, Hwa-Joon Oh, Kevin D. Tran, Scott R. Cottier, Brad W. Michael, Hiroo Nishikawa, Yonetaro Totsuka, Tatsuya Namatame, Naoka Yano, Takashi Machida, and Sang H. Dhong \\ ``A High-Performance SIMD Floating Point Unit for BlueGene/L: Architecture, Compilation, and Algorithm Design,'' by Leonardo Bachega, Siddhartha Chatterjee, Kenneth A. Dockser, John A. Gunnels, Manish Gupta, Fred G. Gustavson, Christopher A. Lapkowski, Gary K. Liu, Mark P. Mendell, Charles D. Wait, and T.J. Chris Ward \\ ``Design of the ARM VFP11 Divide and Square Root Synthesisable Macrocell,'' by Neil Burgess and Chris N. Hinds \\ ``Correctness Proofs Outline for Newton-Raphson Based Floating-Point Divide and Square Root Algorithms,'' by Marius A. Cornea-Hasegan, Roger A. Golliver, and Peter Markstein \\ Recommended Resources \\ About the Editor

entry:
@Book{Antelo:2011:IIFa,
  editor =       "Elisardo Antelo",
  title =        "Industrial Implementations of Floating-Point Units",
  volume =       "1",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "????",
  year =         "2011",
  ISBN =         "????",
  ISBN-13 =      "????",
  LCCN =         "????",
  bibdate =      "Thu Aug 18 08:02:12 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  remark =       "Filename ES0000033.",
  tableofcontents = "Editor's Introduction \\
                 Related Videos \\
                 ``P6 Binary Floating-Point Unit,'' by Son Dao Trong,
                 Martin Schmookler, Eric M. Schwarz, and Michael Kroener
                 \\
                 ``FPU Implementations with Denormalized Numbers,'' by
                 Eric M. Schwarz, Martin Schmookler, and Son Dao Trong
                 \\
                 ``1-GHz HAL SPARC64 Dual Floating Point Unit with RAS
                 Features,'' by Ajay Naini, Atul Dhablania, Warren
                 James, and Debjit Das Sarma \\
                 ``Low-Power Multiple-Precision Iterative Floating-Point
                 Multiplier with SIMD Support,'' by Dimitri Tan, Carl E.
                 Lemonds, and Michael J. Schulte \\
                 ``The Vector Floating-Point Unit in a Synergistic
                 Processor Element of a CELL Processor,'' by Silvia M.
                 Mueller, Christian Jacobi, Hwa-Joon Oh, Kevin D. Tran,
                 Scott R. Cottier, Brad W. Michael, Hiroo Nishikawa,
                 Yonetaro Totsuka, Tatsuya Namatame, Naoka Yano, Takashi
                 Machida, and Sang H. Dhong \\
                 ``A High-Performance SIMD Floating Point Unit for
                 BlueGene/L: Architecture, Compilation, and Algorithm
                 Design,'' by Leonardo Bachega, Siddhartha Chatterjee,
                 Kenneth A. Dockser, John A. Gunnels, Manish Gupta, Fred
                 G. Gustavson, Christopher A. Lapkowski, Gary K. Liu,
                 Mark P. Mendell, Charles D. Wait, and T.J. Chris Ward
                 \\
                 ``Design of the ARM VFP11 Divide and Square Root
                 Synthesisable Macrocell,'' by Neil Burgess and Chris N.
                 Hinds \\
                 ``Correctness Proofs Outline for Newton-Raphson Based
                 Floating-Point Divide and Square Root Algorithms,'' by
                 Marius A. Cornea-Hasegan, Roger A. Golliver, and Peter
                 Markstein \\
                 Recommended Resources \\
                 About the Editor",
}


bibtype: book
filename: fparith.bib
label: Antelo:2011:IIFb
editor: Elisardo Antelo
title: Industrial Implementations of Floating-Point Units
volume: 2
publisher: pub-IEEE
address: pub-IEEE:adr
year: 2011
bibdate: Thu Aug 18 08:02:12 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.18 08:02:12 ???
remark: Filename ES0000034.
acknowledgement: ack-nhfb
raweditor: Elisardo Antelo
rawtitle: Industrial Implementations of Floating-Point Units
TOC:
Editor's Introduction Related Videos ``A High-Performance Area-Efficient Multifunction Interpolator,'' by Stuart F. Oberman and Michael Y. Siu ``New Algorithms for Improved Transcendental Functions on IA-64,'' by Shane Story and Ping Tak Peter Tang ``Energy-Delay Estimation Technique for High-Performance Microprocessor VLSI Adders,'' by Vojin G. Oklobdzija, Bart R. Zeydel, Hoang Dao, Sanu Mathew, and Ram Krishnamurthy ``Advanced Clockgating Schemes for Fused-Multiply-Add-Type Floating-Point Units,'' by Jochen Preiss, Maarten Boersma, and Silvia Melitta Mueller ``Design of Low-Cost High-Performance Floating-Point Fused Multiply-Add with Reduced Power,'' by Zichu Qi, Qi Guo, Ge Zhang, Xiangku Li, and Weiwu Hu ``Decimal Floating-Point: Algorism for Computers,'' by Michael F. Cowlishaw ``A Software Implementation of the IEEE 754R Decimal Floating-Point Arithmetic Using the Binary Encoding Format,'' by Marius Cornea, John Harrison, Cristina Anderson, Ping Tak Peter Tang, Eric Schneider, and Evgeny Gvozdev Recommended Resources About the Editor

rawTOC:
Editor's Introduction \\ Related Videos \\ ``A High-Performance Area-Efficient Multifunction Interpolator,'' by Stuart F. Oberman and Michael Y. Siu \\ ``New Algorithms for Improved Transcendental Functions on IA-64,'' by Shane Story and Ping Tak Peter Tang \\ ``Energy-Delay Estimation Technique for High-Performance Microprocessor VLSI Adders,'' by Vojin G. Oklobdzija, Bart R. Zeydel, Hoang Dao, Sanu Mathew, and Ram Krishnamurthy \\ ``Advanced Clockgating Schemes for Fused-Multiply-Add-Type Floating-Point Units,'' by Jochen Preiss, Maarten Boersma, and Silvia Melitta Mueller \\ ``Design of Low-Cost High-Performance Floating-Point Fused Multiply-Add with Reduced Power,'' by Zichu Qi, Qi Guo, Ge Zhang, Xiangku Li, and Weiwu Hu \\ ``Decimal Floating-Point: Algorism for Computers,'' by Michael F. Cowlishaw \\ ``A Software Implementation of the IEEE 754R Decimal Floating-Point Arithmetic Using the Binary Encoding Format,'' by Marius Cornea, John Harrison, Cristina Anderson, Ping Tak Peter Tang, Eric Schneider, and Evgeny Gvozdev \\ Recommended Resources \\ About the Editor

entry:
@Book{Antelo:2011:IIFb,
  editor =       "Elisardo Antelo",
  title =        "Industrial Implementations of Floating-Point Units",
  volume =       "2",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "????",
  year =         "2011",
  ISBN =         "????",
  ISBN-13 =      "????",
  LCCN =         "????",
  bibdate =      "Thu Aug 18 08:02:12 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  remark =       "Filename ES0000034.",
  tableofcontents = "Editor's Introduction \\
                 Related Videos \\
                 ``A High-Performance Area-Efficient Multifunction
                 Interpolator,'' by Stuart F. Oberman and Michael Y. Siu
                 \\
                 ``New Algorithms for Improved Transcendental Functions
                 on IA-64,'' by Shane Story and Ping Tak Peter Tang \\
                 ``Energy-Delay Estimation Technique for
                 High-Performance Microprocessor VLSI Adders,'' by Vojin
                 G. Oklobdzija, Bart R. Zeydel, Hoang Dao, Sanu Mathew,
                 and Ram Krishnamurthy \\
                 ``Advanced Clockgating Schemes for
                 Fused-Multiply-Add-Type Floating-Point Units,'' by
                 Jochen Preiss, Maarten Boersma, and Silvia Melitta
                 Mueller \\
                 ``Design of Low-Cost High-Performance Floating-Point
                 Fused Multiply-Add with Reduced Power,'' by Zichu Qi,
                 Qi Guo, Ge Zhang, Xiangku Li, and Weiwu Hu \\
                 ``Decimal Floating-Point: Algorism for Computers,'' by
                 Michael F. Cowlishaw \\
                 ``A Software Implementation of the IEEE 754R Decimal
                 Floating-Point Arithmetic Using the Binary Encoding
                 Format,'' by Marius Cornea, John Harrison, Cristina
                 Anderson, Ping Tak Peter Tang, Eric Schneider, and
                 Evgeny Gvozdev \\
                 Recommended Resources \\
                 About the Editor",
}


bibtype: inproceedings
filename: fparith.bib
label: Arias-Garcia:2011:SFI
author: J. Arias-Garcia and R. Pezzuol Jacobi and C. H. Llanos and M. Ayala-Rincon
booktitle: 2011 VII Southern Conference on Programmable Logic (SPL)
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 263--268
year: 2011
DOI: https://doi.org/10.1109/SPL.2011.5782659
bibdate: Tue Sep 27 08:11:02 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.09.27 08:11:02 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5782659
acknowledgement: ack-nhfb
bookURL: http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5771182
rawauthor: J. Arias-Garcia and R. Pezzuol Jacobi and C. H. Llanos and M. Ayala-Rincon
rawbooktitle: {2011 VII Southern Conference on Programmable Logic (SPL)}
title:
A suitable FPGA implementation of floating-point matrix inversion based on Gauss--Jordan elimination

rawtitle:
A suitable {FPGA} implementation of floating-point matrix inversion based on {Gauss--Jordan} elimination

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  booktitle =    "{2011 VII Southern Conference on Programmable Logic
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  title =        "A suitable {FPGA} implementation of floating-point
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  publisher =    "pub-IEEE",
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  pages =        "263--268",
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  acknowledgement = ack-nhfb,
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bibtype: article
filename: fparith.bib
label: Arnold:2011:RCL
author: Mark G. Arnold and Sylvain Collange
title: A Real / Complex Logarithmic Number System ALU
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 2
pages: 202--213
month: feb
monthnumber: 2
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.154
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Feb 20 19:15:33 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.02.20 19:15:33 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Mark G. Arnold and Sylvain Collange
rawtitle: A Real\slash Complex Logarithmic Number System {ALU}
entry:
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  title =        "A Real\slash Complex Logarithmic Number System {ALU}",
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  number =       "2",
  pages =        "202--213",
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  year =         "2011",
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  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
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bibtype: inproceedings
filename: fparith.bib
label: Arnold:2011:TQC
author: Mark G. Arnold and John Cowles and Vassilis Paliouras and Ioannis Kouretas
title: Towards a Quaternion Complex Logarithmic Number System
crossref: Schwarz:2011:PIS
pages: 33--42
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.14
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992106
keywords: ARITH-20; logarithmic number system (LNS)
acknowledgement: ack-nhfb
rawauthor: Mark G. Arnold and John Cowles and Vassilis Paliouras and Ioannis Kouretas
rawtitle: Towards a Quaternion Complex Logarithmic Number System
entry:
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  author =       "Mark G. Arnold and John Cowles and Vassilis Paliouras
                 and Ioannis Kouretas",
  title =        "Towards a Quaternion Complex Logarithmic Number
                 System",
  crossref =     "Schwarz:2011:PIS",
  pages =        "33--42",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.14",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992106",
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bibtype: article
filename: fparith.bib
label: Badin:2011:IAM
author: Matthew Badin and Lubomir Bic and Michael Dillencourt and Alexandru Nicolau
title: Improving accuracy for matrix multiplications on GPUs
journal: j-SCI-PROG
volume: 19
number: 1
pages: 3--11
year: 2011
CODEN: SCIPEV
DOI: https://doi.org/10.3233/SPR-2011-0315
ISSN: 1058-9244 (print), 1875-919X (electronic) OR 10589244 (print), 1875919X (electronic)
ISSNL: 1058-9244 OR 10589244
bibdate: Tue Dec 13 19:01:37 MST 2011
bibtimestamp: 2011.12.13 19:01:37 MST
fjournal: Scientific Programming
acknowledgement: ack-nhfb
journalURL: http://iospress.metapress.com/content/1058-9244
rawauthor: Matthew Badin and Lubomir Bic and Michael Dillencourt and Alexandru Nicolau
rawtitle: Improving accuracy for matrix multiplications on {GPUs}
bibsource:
http://www.iospress.nl/journal/scientific-programming/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sciprogram.bib

entry:
@Article{Badin:2011:IAM,
  author =       "Matthew Badin and Lubomir Bic and Michael Dillencourt
                 and Alexandru Nicolau",
  title =        "Improving accuracy for matrix multiplications on
                 {GPUs}",
  journal =      j-SCI-PROG,
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  number =       "1",
  pages =        "3--11",
  month =        "????",
  year =         "2011",
  CODEN =        "SCIPEV",
  DOI =          "https://doi.org/10.3233/SPR-2011-0315",
  ISSN =         "1058-9244 (print), 1875-919X (electronic)",
  ISSN-L =       "1058-9244",
  bibdate =      "Tue Dec 13 19:01:37 MST 2011",
  bibsource =    "http://www.iospress.nl/journal/scientific-programming/;
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  acknowledgement = ack-nhfb,
  fjournal =     "Scientific Programming",
  journal-URL =  "http://iospress.metapress.com/content/1058-9244",
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bibtype: techreport
filename: fparith.bib
label: Bailey:2011:GMD
author: David H. Bailey and Jonathan M. Borwein
title: The Greatest Mathematical Discovery?
type: Report
institution: Lawrence Berkeley National Laboratory and Centre for Computer Assisted RMA, University of Newcastle
address: Berkeley, CA 94720, USA and Callaghan, NSW 2308, Australia
pages: 10
day: 8
month: may
monthnumber: 5
year: 2011
bibdate: Fri Aug 12 18:29:28 2016
bibtimestamp: 2016.08.12 18:29:28 ???
URL: http://www.davidhbailey.com/dhbpapers/decimal.pdf
keywords: decimal arithmetic history
acknowledgement: ack-nhfb
authordates: Jonathan Michael Borwein (20 May 1951--2 August 2016)
rawauthor: David H. Bailey and Jonathan M. Borwein
rawtitle: The Greatest Mathematical Discovery?
ORCID: Bailey, David H./0000-0002-7574-8342; Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@TechReport{Bailey:2011:GMD,
  author =       "David H. Bailey and Jonathan M. Borwein",
  title =        "The Greatest Mathematical Discovery?",
  type =         "Report",
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  bibdate =      "Fri Aug 12 18:29:28 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.davidhbailey.com/dhbpapers/decimal.pdf",
  acknowledgement = ack-nhfb,
  author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
                 2016)",
  keywords =     "decimal arithmetic history",
  ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
                 Jonathan/0000-0002-1263-0646",
}


bibtype: techreport
filename: fparith.bib
label: Baudin:2011:EBC
author: M. Baudin
title: Error bounds of complex arithmetic
type: Report
pages: 10
month: jun
monthnumber: 6
year: 2011
bibdate: Sun Feb 19 07:29:34 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.02.19 07:29:34 ???
URL: http://forge.scilab.org/upload/compdiv/files/complexerrorbounds_v0.2.pdf
acknowledgement: ack-nhfb
rawauthor: M. Baudin
rawtitle: Error bounds of complex arithmetic
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  bibdate =      "Sun Feb 19 07:29:34 2017",
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  URL =          "http://forge.scilab.org/upload/compdiv/files/complexerrorbounds_v0.2.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: techreport
filename: fparith.bib
label: Beebe:2011:BPAb
author: Nelson H. F. Beebe
title: A Bibliography of Publications about Benford's Law, Heaps' Law, and Zipf's Law
institution: inst-CSC
address: inst-CSC:adr
pages: 62
day: 12
month: nov
monthnumber: 11
year: 2011
bibdate: Thu Nov 10 12:15:06 2011
bibtimestamp: 2011.11.10 12:15:06 ???
URL: https://www.math.utah.edu/pub/tex/bib/index-table-b.html#benfords-law
acknowledgement: ack-nhfb
rawauthor: Nelson H. F. Beebe
rawtitle: A Bibliography of Publications about {Benford's Law}, {Heaps' Law}, and {Zipf's Law}
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https://www.math.utah.edu/pub/bibnet/authors/b/beebe-nelson-h-f.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
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  title =        "A Bibliography of Publications about {Benford's Law},
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bibtype: article
filename: fparith.bib
label: Berger:2011:FSM
author: Arno Berger and Theodore P. Hill and Bahar Kaynar and Ad Ridder
title: Finite-state Markov Chains Obey Benford's Law
journal: j-SIAM-J-MAT-ANA-APPL
volume: 32
number: 3
pages: 665--684
year: 2011
CODEN: SJMAEL
DOI: https://doi.org/10.1137/100789890
ISSN: 0895-4798 (print), 1095-7162 (electronic) OR 08954798 (print), 10957162 (electronic)
ISSNL: 0895-4798 OR 08954798
bibdate: Wed Aug 24 11:18:18 MDT 2011
bibtimestamp: 2011.08.24 11:18:18 MDT
URL: http://epubs.siam.org/simax/resource/1/sjmael/v32/i3/p665_s1
fjournal: SIAM Journal on Matrix Analysis and Applications
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/simax
onlinedate: July 21, 2011
rawauthor: Arno Berger and Theodore P. Hill and Bahar Kaynar and Ad Ridder
rawtitle: Finite-state {Markov} Chains Obey {Benford's Law}
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bibtype: article
filename: fparith.bib
label: Beuchat:2011:FAP
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 2
pages: 266--281
month: feb
monthnumber: 2
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.163
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Feb 20 19:15:33 2011
bibtimestamp: 2011.02.20 19:15:33 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
author:
Jean-Luc Beuchat and Jeremie Detrey and Nicolas Estibals and Eiji Okamoto and Francisco Rodriguez-Henriquez

title:
Fast Architectures for the $ eta _T $ Pairing over Small-Characteristic Supersingular Elliptic Curves

rawauthor:
Jean-Luc Beuchat and Jeremie Detrey and Nicolas Estibals and Eiji Okamoto and Francisco Rodriguez-Henriquez

rawtitle:
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entry:
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  author =       "Jean-Luc Beuchat and Jeremie Detrey and Nicolas
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  bibdate =      "Sun Feb 20 19:15:33 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
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bibtype: inproceedings
filename: fparith.bib
label: Bodrato:2011:HDT
author: Marco Bodrato
title: High Degree Toom'n'Half for Balanced and Unbalanced Multiplication
crossref: Schwarz:2011:PIS
pages: 15--22
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.12
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992104
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Marco Bodrato
rawtitle: High Degree {Toom'n'Half} for Balanced and Unbalanced Multiplication
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  crossref =     "Schwarz:2011:PIS",
  pages =        "15--22",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.12",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992104",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
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bibtype: inproceedings
filename: fparith.bib
label: Boersma:2011:PBF
title: The POWER7 Binary Floating-Point Unit
crossref: Schwarz:2011:PIS
pages: 87--91
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.21
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992113
keywords: ARITH-20
acknowledgement: ack-nhfb
rawtitle: The {POWER7} Binary Floating-Point Unit
author:
Maarten Boersma and Michael Kroner and Christophe Layer and Petra Leber and Silvia M. Muller and Kerstin Schelm

rawauthor:
Maarten Boersma and Michael Kroner and Christophe Layer and Petra Leber and Silvia M. Muller and Kerstin Schelm

entry:
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  author =       "Maarten Boersma and Michael Kroner and Christophe
                 Layer and Petra Leber and Silvia M. Muller and Kerstin
                 Schelm",
  title =        "The {POWER7} Binary Floating-Point Unit",
  crossref =     "Schwarz:2011:PIS",
  pages =        "87--91",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.21",
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  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992113",
  acknowledgement = ack-nhfb,
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}


bibtype: article
filename: fparith.bib
label: Boldo:2011:EAE
author: Sylvie Boldo and Jean-Michel Muller
title: Exact and Approximated Error of the FMA
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 2
pages: 157--164
month: feb
monthnumber: 2
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.139
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Feb 20 19:15:33 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.02.20 19:15:33 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Sylvie Boldo and Jean-Michel Muller
rawtitle: Exact and Approximated Error of the {FMA}
abstract:
The fused multiply accumulate-add (FMA) instruction, specified by the IEEE 754-2008 Standard for Floating-Point Arithmetic, eases some calculations, and is already available on some current processors such as the Power PC or the Itanium. We first extend an earlier work on the computation of the exact error of an FMA (by giving more general conditions and providing a formal proof). Then, we present a new algorithm that computes an approximation to the error of an FMA, and provide error bounds and a formal proof for that algorithm.

rawabstract:
The fused multiply accumulate-add (FMA) instruction, specified by the IEEE 754-2008 Standard for Floating-Point Arithmetic, eases some calculations, and is already available on some current processors such as the Power PC or the Itanium. We first extend an earlier work on the computation of the exact error of an FMA (by giving more general conditions and providing a formal proof). Then, we present a new algorithm that computes an approximation to the error of an FMA, and provide error bounds and a formal proof for that algorithm.

entry:
@Article{Boldo:2011:EAE,
  author =       "Sylvie Boldo and Jean-Michel Muller",
  title =        "Exact and Approximated Error of the {FMA}",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "60",
  number =       "2",
  pages =        "157--164",
  month =        feb,
  year =         "2011",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.139",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Feb 20 19:15:33 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The fused multiply accumulate-add (FMA) instruction,
                 specified by the IEEE 754-2008 Standard for
                 Floating-Point Arithmetic, eases some calculations, and
                 is already available on some current processors such as
                 the Power PC or the Itanium. We first extend an earlier
                 work on the computation of the exact error of an FMA
                 (by giving more general conditions and providing a
                 formal proof). Then, we present a new algorithm that
                 computes an approximation to the error of an FMA, and
                 provide error bounds and a formal proof for that
                 algorithm.",
  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: Boldo:2011:FUL
author: Sylvie Boldo and Guillaume Melquiond
title: Flocq: a Unified Library for Proving Floating-Point Algorithms in Coq
crossref: Schwarz:2011:PIS
pages: 243--252
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.40
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992132
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Sylvie Boldo and Guillaume Melquiond
rawtitle: {Flocq}: a Unified Library for Proving Floating-Point Algorithms in {Coq}
entry:
@InProceedings{Boldo:2011:FUL,
  author =       "Sylvie Boldo and Guillaume Melquiond",
  title =        "{Flocq}: a Unified Library for Proving Floating-Point
                 Algorithms in {Coq}",
  crossref =     "Schwarz:2011:PIS",
  pages =        "243--252",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.40",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992132",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: article
filename: fparith.bib
label: Boldo:2011:FVN
author: Sylvie Boldo and Claude Marche
title: Formal Verification of Numerical Programs: From C Annotated Programs to Mechanical Proofs
journal: j-MATH-COMPUT-SCI
volume: 5
number: 4
pages: 377--393
month: dec
monthnumber: 12
year: 2011
DOI: https://doi.org/10.1007/s11786-011-0099-9
ISSN: 1661-8270 (print), 1661-8289 (electronic) OR 16618270 (print), 16618289 (electronic)
ISSNL: 1661-8270 OR 16618270
bibdate: Tue Nov 6 10:16:23 MST 2012
bibtimestamp: 2012.11.06 10:16:23 MST
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1661-8270&volume=5&issue=4&spage=377
fjournal: Mathematics in Computer Science
acknowledgement: ack-nhfb
journalURL: http://www.springerlink.com/content/1661-8270/
rawauthor: Sylvie Boldo and Claude March{\'e}
rawtitle: Formal Verification of Numerical Programs: From {C} Annotated Programs to Mechanical Proofs
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1661-8270&volume=5&issue=4; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-sci.bib

keywords:
Alt-Ergo; automated reasoning; C program; Coq; CVC3; floating-point arithmetic; formal specification; Frama-C; Gappa; Jessie plugin; Why; Z3

entry:
@Article{Boldo:2011:FVN,
  author =       "Sylvie Boldo and Claude March{\'e}",
  title =        "Formal Verification of Numerical Programs: From {C}
                 Annotated Programs to Mechanical Proofs",
  journal =      j-MATH-COMPUT-SCI,
  volume =       "5",
  number =       "4",
  pages =        "377--393",
  month =        dec,
  year =         "2011",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s11786-011-0099-9",
  ISSN =         "1661-8270 (print), 1661-8289 (electronic)",
  ISSN-L =       "1661-8270",
  bibdate =      "Tue Nov 6 10:16:23 MST 2012",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1661-8270&volume=5&issue=4;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/math-comput-sci.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1661-8270&volume=5&issue=4&spage=377",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics in Computer Science",
  journal-URL =  "http://www.springerlink.com/content/1661-8270/",
  keywords =     "Alt-Ergo; automated reasoning; C program; Coq; CVC3;
                 floating-point arithmetic; formal specification;
                 Frama-C; Gappa; Jessie plugin; Why; Z3",
}


bibtype: inproceedings
filename: fparith.bib
label: Bos:2011:ESA
author: Joppe W. Bos and Thorsten Kleinjung and Arjen K. Lenstra and Peter L. Montgomery
title: Efficient SIMD Arithmetic Modulo a Mersenne Number
crossref: Schwarz:2011:PIS
pages: 213--221
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.37
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992129
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Joppe W. Bos and Thorsten Kleinjung and Arjen K. Lenstra and Peter L. Montgomery
rawtitle: Efficient {SIMD} Arithmetic Modulo a {Mersenne} Number
entry:
@InProceedings{Bos:2011:ESA,
  author =       "Joppe W. Bos and Thorsten Kleinjung and Arjen K.
                 Lenstra and Peter L. Montgomery",
  title =        "Efficient {SIMD} Arithmetic Modulo a {Mersenne}
                 Number",
  crossref =     "Schwarz:2011:PIS",
  pages =        "213--221",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.37",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992129",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: book
filename: fparith.bib
label: Brent:2011:MCA
author: Richard P. Brent and Paul Zimmermann
title: Modern Computer Arithmetic
volume: 18
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
pages: xvi + 221
year: 2011
DOI: https://doi.org/10.1017/CBO9780511921698
ISBN: 0-521-19469-5 (hardcover) OR 0521194695 (hardcover)
ISBN13: 978-0-521-19469-3 (hardcover) OR 9780521194693 (hardcover)
LCCN: QA76.9.C62 BRE 2011
bibdate: Sat Jan 15 12:25:22 MST 2011
bibtimestamp: 2011.01.15 12:25:22 MST
series: Cambridge monographs on applied and computational mathematics
URL: http://www.loria.fr/~zimmerma/mca/pub226.html
subject: Computer arithmetic
acknowledgement: ack-nhfb
rawauthor: Richard P. Brent and Paul Zimmermann
rawtitle: Modern Computer Arithmetic
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/sigact.bib; library.ox.ac.uk:210/ADVANCE

TOC:
Preface / ix Acknowledgements / xi Notation / xiii Integer arithmetic / 1 1.1 Representation and notations / 1 1.2 Addition and subtraction / 2 1.3 Multiplication / J i.3.1 Naive multiplication / 4 1.3.2 Karatsuba's algorithm / 5 1.3.3 Toom--Cook multiplication / 6 1.3.4 Use of the fast Fourier transform (FFT) / 8 1.3.5 Unbalanced multiplication / 8 1.3.6 Squaring / 11 1.3.7 Multiplication by a constant / 13 1.4 Division / 14 1.4.1 Naive division / 14 1.4.2 Divisor preconditioning / 16 1.4.3 Divide and conquer division / 18 1.4.4 Newton's method / 21 1.4.5 Exact division / 21 1.4.6 Only quotient or remainder wanted / 22 1.4.7 Division by a single word / 23 1.4.8 Hensel's division / 24 1.5 Roots / 25 1.5.1 Square root / 25 1.5.2 $k$th root / 27 1.5.3 Exact root / 28 1.6 Greatest common divisor / 29 1.6.1 Naive GCD / 29 1.6.2 Extended GCD / 32 1.6.3 Half binary GCD, divide and conquer GCD / 33 1.7 Base conversion / 37 1.7.1 Quadratic algorithms / 37 1.7.2 Subquadratic algorithms / 38 1.8 Exercises / 39 1.9 Notes and references / 44 Modular arithmetic and the FFT / 47 2.1 Representation / 47 2.1.1 Classical representation / 47 2.1.2 Montgomery's form / 48 2.1.3 Residue number systems / 48 2.1.4 MSB vs LSB algorithms / 49 2.1.5 Link with Polynomials / 49 2.2 Modular addition and subtraction / 50 2.3 The Fourier transform / 50 2.3.1 Theoretical setting / 50 2.3.2 The fast Fourier transform / 51 2.3.3 The Schonhage--Strassen algorithm / 55 2.4 Modular multiplication / 58 2.4.1 Barrett's algorithm / 58 2.4.2 Montgomery's multiplication / 60 2.4.3 McLaughlin's algorithm / 63 2.4.4 Special moduli / 65 2.5 Modular division and inversion / 65 2.5.1 Several inversions at once / 61 2.6 Modular exponentiation / 68 2.6.1 Binary exponentiation / 70 2.6.2 Exponentiation with a larger base / 70 2.6.3 Sliding window and redundant representation / 72 2.7 Chinese remainder theorem / 75 2.8 Exercises / 75 2.9 Notes and references / 77 Floating-point arithmetic / 79 3.1 Representation / 79 3.1.1 Radix choice / 80 3.1.2 Exponent range / 81 3.1.3 Special values / 82 3.1.4 Subnormal numbers / 82 3.1.5 Encoding / 83 3.1.6 Precision: local, global, operation, operand / 84 3.1.7 Link to integers / 86 3.1.8 Ziv's algorithm and error analysis / 86 3.1.9 Rounding / 87 3.1.10 Strategies / 90 3.2 Addition, subtraction, comparison / 91 3.2.1 Floating-point addition / 92 3.2.2 Floating-point subtraction / 93 3.3 Multiplication / 95 3.3.1 Integer multiplication via complex FFT / 98 3.3.2 The middle product / 99 3.4 Reciprocal and division / 101 3.4.1 Reciprocal / 102 3.4.2 Division / 106 3.5 Square root / 111 3.5.1 Reciprocal square root / 112 3.6 Conversion / 114 3.6.1 Floating-point output / 115 3.6.2 Floating-point input / 117 3.7 Exercises / 118 3.8 Notes and references / 120 Elementary and special function evaluation / 125 4.1 Introduction / 125 4.2 Newton's method / 126 4.2.1 Newton's method for inverse roots / 127 4.2.2 Newton's method for reciprocals / 128 4.2.3 Newton's method for (reciprocal) square roots / 129 4.2.4 Newton's method for formal power series / 129 4.2.5 Newton's method for functional inverses / 130 4.2.6 Higher-order Newton-like methods / 131 4.3 Argument reduction / 132 4.3.1 Repeated use of a doubling formula / 134 4.3.2 Loss of precision / 134 4.3.3 Guard digits / 135 4.3.4 Doubling versus tripling / 136 4.4 Power series / 136 4.4.1 Direct power series evaluation / 140 4.4.2 Power series with argument reduction / 140 4.4.3 Rectangular series splitting / 141 4.5 Asymptotic expansions / 144 4.6 Continued fractions / 150 4.7 Recurrence relations / 152 4.7.1 Evaluation of Bessel functions / 153 4.7.2 Evaluation of Bernoulli and tangent numbers / 154 4.8 Arithmetic--geometric mean / 158 4.8.1 Elliptic integrals / 158 4.8.2 First AGM algorithm for the logarithm / 159 4.8.3 Theta functions / 160 4.8.4 Second AGM algorithm for the logarithm / 162 4.8.5 The complex AGM / 163 4.9 Binary splitting / 163 4.9.1 A binary splitting algorithm for sin, cos / 166 4.9.2 The bit-burst algorithm / 161 4.10 Contour integration / 169 4.11 Exercises / 171 4.12 Notes and references / 179 Implementations and pointers / 185 5.1 Software tools / 185 5.1.1 CLN / 185 5.1.2 GNUMP (GMP) / 185 5.1.3 MPFQ / 186 5.1.4 GNU MPFR / 187 5.1.5 Other multiple-precision packages / 187 5.1.6 Computational algebra packages / 188 5.2 Mailing lists / 189 5.2.1 The GMP lists / 189 5.2.2 The MPFR list / 190 5.3 Online documents / 190 References / 191 Index / 207

rawTOC:
Preface / ix \\ Acknowledgements / xi \\ Notation / xiii \\ Integer arithmetic / 1 \\ 1.1 Representation and notations / 1 \\ 1.2 Addition and subtraction / 2 \\ 1.3 Multiplication / J \\ i.3.1 Naive multiplication / 4 \\ 1.3.2 Karatsuba's algorithm / 5 \\ 1.3.3 Toom--Cook multiplication / 6 \\ 1.3.4 Use of the fast Fourier transform (FFT) / 8 \\ 1.3.5 Unbalanced multiplication / 8 \\ 1.3.6 Squaring / 11 \\ 1.3.7 Multiplication by a constant / 13 \\ 1.4 Division / 14 \\ 1.4.1 Naive division / 14 \\ 1.4.2 Divisor preconditioning / 16 \\ 1.4.3 Divide and conquer division / 18 \\ 1.4.4 Newton's method / 21 \\ 1.4.5 Exact division / 21 \\ 1.4.6 Only quotient or remainder wanted / 22 \\ 1.4.7 Division by a single word / 23 \\ 1.4.8 Hensel's division / 24 \\ 1.5 Roots / 25 \\ 1.5.1 Square root / 25 \\ 1.5.2 $k$th root / 27 \\ 1.5.3 Exact root / 28 \\ 1.6 Greatest common divisor / 29 \\ 1.6.1 Naive GCD / 29 \\ 1.6.2 Extended GCD / 32 \\ 1.6.3 Half binary GCD, divide and conquer GCD / 33 \\ 1.7 Base conversion / 37 \\ 1.7.1 Quadratic algorithms / 37 \\ 1.7.2 Subquadratic algorithms / 38 \\ 1.8 Exercises / 39 \\ 1.9 Notes and references / 44 \\ Modular arithmetic and the FFT / 47 \\ 2.1 Representation / 47 \\ 2.1.1 Classical representation / 47 \\ 2.1.2 Montgomery's form / 48 \\ 2.1.3 Residue number systems / 48 \\ 2.1.4 MSB vs LSB algorithms / 49 \\ 2.1.5 Link with Polynomials / 49 \\ 2.2 Modular addition and subtraction / 50 \\ 2.3 The Fourier transform / 50 \\ 2.3.1 Theoretical setting / 50 \\ 2.3.2 The fast Fourier transform / 51 \\ 2.3.3 The Sch{\"o}nhage--Strassen algorithm / 55 \\ 2.4 Modular multiplication / 58 \\ 2.4.1 Barrett's algorithm / 58 \\ 2.4.2 Montgomery's multiplication / 60 \\ 2.4.3 McLaughlin's algorithm / 63 \\ 2.4.4 Special moduli / 65 \\ 2.5 Modular division and inversion / 65 \\ 2.5.1 Several inversions at once / 61 \\ 2.6 Modular exponentiation / 68 \\ 2.6.1 Binary exponentiation / 70 \\ 2.6.2 Exponentiation with a larger base / 70 \\ 2.6.3 Sliding window and redundant representation / 72 \\ 2.7 Chinese remainder theorem / 75 \\ 2.8 Exercises / 75 \\ 2.9 Notes and references / 77 \\ Floating-point arithmetic / 79 \\ 3.1 Representation / 79 \\ 3.1.1 Radix choice / 80 \\ 3.1.2 Exponent range / 81 \\ 3.1.3 Special values / 82 \\ 3.1.4 Subnormal numbers / 82 \\ 3.1.5 Encoding / 83 \\ 3.1.6 Precision: local, global, operation, operand / 84 \\ 3.1.7 Link to integers / 86 \\ 3.1.8 Ziv's algorithm and error analysis / 86 \\ 3.1.9 Rounding / 87 \\ 3.1.10 Strategies / 90 \\ 3.2 Addition, subtraction, comparison / 91 \\ 3.2.1 Floating-point addition / 92 \\ 3.2.2 Floating-point subtraction / 93 \\ 3.3 Multiplication / 95 \\ 3.3.1 Integer multiplication via complex FFT / 98 \\ 3.3.2 The middle product / 99 \\ 3.4 Reciprocal and division / 101 \\ 3.4.1 Reciprocal / 102 \\ 3.4.2 Division / 106 \\ 3.5 Square root / 111 \\ 3.5.1 Reciprocal square root / 112 \\ 3.6 Conversion / 114 \\ 3.6.1 Floating-point output / 115 \\ 3.6.2 Floating-point input / 117 \\ 3.7 Exercises / 118 \\ 3.8 Notes and references / 120 \\ Elementary and special function evaluation / 125 \\ 4.1 Introduction / 125 \\ 4.2 Newton's method / 126 \\ 4.2.1 Newton's method for inverse roots / 127 \\ 4.2.2 Newton's method for reciprocals / 128 \\ 4.2.3 Newton's method for (reciprocal) square roots / 129 \\ 4.2.4 Newton's method for formal power series / 129 \\ 4.2.5 Newton's method for functional inverses / 130 \\ 4.2.6 Higher-order Newton-like methods / 131 \\ 4.3 Argument reduction / 132 \\ 4.3.1 Repeated use of a doubling formula / 134 \\ 4.3.2 Loss of precision / 134 \\ 4.3.3 Guard digits / 135 \\ 4.3.4 Doubling versus tripling / 136 \\ 4.4 Power series / 136 \\ 4.4.1 Direct power series evaluation / 140 \\ 4.4.2 Power series with argument reduction / 140 \\ 4.4.3 Rectangular series splitting / 141 \\ 4.5 Asymptotic expansions / 144 \\ 4.6 Continued fractions / 150 \\ 4.7 Recurrence relations / 152 \\ 4.7.1 Evaluation of Bessel functions / 153 \\ 4.7.2 Evaluation of Bernoulli and tangent numbers / 154 \\ 4.8 Arithmetic--geometric mean / 158 \\ 4.8.1 Elliptic integrals / 158 \\ 4.8.2 First AGM algorithm for the logarithm / 159 \\ 4.8.3 Theta functions / 160 \\ 4.8.4 Second AGM algorithm for the logarithm / 162 \\ 4.8.5 The complex AGM / 163 \\ 4.9 Binary splitting / 163 \\ 4.9.1 A binary splitting algorithm for sin, cos / 166 \\ 4.9.2 The bit-burst algorithm / 161 \\ 4.10 Contour integration / 169 \\ 4.11 Exercises / 171 \\ 4.12 Notes and references / 179 \\ Implementations and pointers / 185 \\ 5.1 Software tools / 185 \\ 5.1.1 CLN / 185 \\ 5.1.2 GNUMP (GMP) / 185 \\ 5.1.3 MPFQ / 186 \\ 5.1.4 GNU MPFR / 187 \\ 5.1.5 Other multiple-precision packages / 187 \\ 5.1.6 Computational algebra packages / 188 \\ 5.2 Mailing lists / 189 \\ 5.2.1 The GMP lists / 189 \\ 5.2.2 The MPFR list / 190 \\ 5.3 Online documents / 190 \\ References / 191 \\ Index / 207

entry:
@Book{Brent:2011:MCA,
  author =       "Richard P. Brent and Paul Zimmermann",
  title =        "Modern Computer Arithmetic",
  volume =       "18",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  pages =        "xvi + 221",
  year =         "2011",
  DOI =          "https://doi.org/10.1017/CBO9780511921698",
  ISBN =         "0-521-19469-5 (hardcover)",
  ISBN-13 =      "978-0-521-19469-3 (hardcover)",
  LCCN =         "QA76.9.C62 BRE 2011",
  bibdate =      "Sat Jan 15 12:25:22 MST 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigact.bib;
                 library.ox.ac.uk:210/ADVANCE",
  series =       "Cambridge monographs on applied and computational
                 mathematics",
  URL =          "http://www.loria.fr/~zimmerma/mca/pub226.html",
  acknowledgement = ack-nhfb,
  subject =      "Computer arithmetic",
  tableofcontents = "Preface / ix \\
                 Acknowledgements / xi \\
                 Notation / xiii \\
                 Integer arithmetic / 1 \\
                 1.1 Representation and notations / 1 \\
                 1.2 Addition and subtraction / 2 \\
                 1.3 Multiplication / J \\
                 i.3.1 Naive multiplication / 4 \\
                 1.3.2 Karatsuba's algorithm / 5 \\
                 1.3.3 Toom--Cook multiplication / 6 \\
                 1.3.4 Use of the fast Fourier transform (FFT) / 8 \\
                 1.3.5 Unbalanced multiplication / 8 \\
                 1.3.6 Squaring / 11 \\
                 1.3.7 Multiplication by a constant / 13 \\
                 1.4 Division / 14 \\
                 1.4.1 Naive division / 14 \\
                 1.4.2 Divisor preconditioning / 16 \\
                 1.4.3 Divide and conquer division / 18 \\
                 1.4.4 Newton's method / 21 \\
                 1.4.5 Exact division / 21 \\
                 1.4.6 Only quotient or remainder wanted / 22 \\
                 1.4.7 Division by a single word / 23 \\
                 1.4.8 Hensel's division / 24 \\
                 1.5 Roots / 25 \\
                 1.5.1 Square root / 25 \\
                 1.5.2 $k$th root / 27 \\
                 1.5.3 Exact root / 28 \\
                 1.6 Greatest common divisor / 29 \\
                 1.6.1 Naive GCD / 29 \\
                 1.6.2 Extended GCD / 32 \\
                 1.6.3 Half binary GCD, divide and conquer GCD / 33 \\
                 1.7 Base conversion / 37 \\
                 1.7.1 Quadratic algorithms / 37 \\
                 1.7.2 Subquadratic algorithms / 38 \\
                 1.8 Exercises / 39 \\
                 1.9 Notes and references / 44 \\
                 Modular arithmetic and the FFT / 47 \\
                 2.1 Representation / 47 \\
                 2.1.1 Classical representation / 47 \\
                 2.1.2 Montgomery's form / 48 \\
                 2.1.3 Residue number systems / 48 \\
                 2.1.4 MSB vs LSB algorithms / 49 \\
                 2.1.5 Link with Polynomials / 49 \\
                 2.2 Modular addition and subtraction / 50 \\
                 2.3 The Fourier transform / 50 \\
                 2.3.1 Theoretical setting / 50 \\
                 2.3.2 The fast Fourier transform / 51 \\
                 2.3.3 The Sch{\"o}nhage--Strassen algorithm / 55 \\
                 2.4 Modular multiplication / 58 \\
                 2.4.1 Barrett's algorithm / 58 \\
                 2.4.2 Montgomery's multiplication / 60 \\
                 2.4.3 McLaughlin's algorithm / 63 \\
                 2.4.4 Special moduli / 65 \\
                 2.5 Modular division and inversion / 65 \\
                 2.5.1 Several inversions at once / 61 \\
                 2.6 Modular exponentiation / 68 \\
                 2.6.1 Binary exponentiation / 70 \\
                 2.6.2 Exponentiation with a larger base / 70 \\
                 2.6.3 Sliding window and redundant representation / 72
                 \\
                 2.7 Chinese remainder theorem / 75 \\
                 2.8 Exercises / 75 \\
                 2.9 Notes and references / 77 \\
                 Floating-point arithmetic / 79 \\
                 3.1 Representation / 79 \\
                 3.1.1 Radix choice / 80 \\
                 3.1.2 Exponent range / 81 \\
                 3.1.3 Special values / 82 \\
                 3.1.4 Subnormal numbers / 82 \\
                 3.1.5 Encoding / 83 \\
                 3.1.6 Precision: local, global, operation, operand / 84
                 \\
                 3.1.7 Link to integers / 86 \\
                 3.1.8 Ziv's algorithm and error analysis / 86 \\
                 3.1.9 Rounding / 87 \\
                 3.1.10 Strategies / 90 \\
                 3.2 Addition, subtraction, comparison / 91 \\
                 3.2.1 Floating-point addition / 92 \\
                 3.2.2 Floating-point subtraction / 93 \\
                 3.3 Multiplication / 95 \\
                 3.3.1 Integer multiplication via complex FFT / 98 \\
                 3.3.2 The middle product / 99 \\
                 3.4 Reciprocal and division / 101 \\
                 3.4.1 Reciprocal / 102 \\
                 3.4.2 Division / 106 \\
                 3.5 Square root / 111 \\
                 3.5.1 Reciprocal square root / 112 \\
                 3.6 Conversion / 114 \\
                 3.6.1 Floating-point output / 115 \\
                 3.6.2 Floating-point input / 117 \\
                 3.7 Exercises / 118 \\
                 3.8 Notes and references / 120 \\
                 Elementary and special function evaluation / 125 \\
                 4.1 Introduction / 125 \\
                 4.2 Newton's method / 126 \\
                 4.2.1 Newton's method for inverse roots / 127 \\
                 4.2.2 Newton's method for reciprocals / 128 \\
                 4.2.3 Newton's method for (reciprocal) square roots /
                 129 \\
                 4.2.4 Newton's method for formal power series / 129 \\
                 4.2.5 Newton's method for functional inverses / 130 \\
                 4.2.6 Higher-order Newton-like methods / 131 \\
                 4.3 Argument reduction / 132 \\
                 4.3.1 Repeated use of a doubling formula / 134 \\
                 4.3.2 Loss of precision / 134 \\
                 4.3.3 Guard digits / 135 \\
                 4.3.4 Doubling versus tripling / 136 \\
                 4.4 Power series / 136 \\
                 4.4.1 Direct power series evaluation / 140 \\
                 4.4.2 Power series with argument reduction / 140 \\
                 4.4.3 Rectangular series splitting / 141 \\
                 4.5 Asymptotic expansions / 144 \\
                 4.6 Continued fractions / 150 \\
                 4.7 Recurrence relations / 152 \\
                 4.7.1 Evaluation of Bessel functions / 153 \\
                 4.7.2 Evaluation of Bernoulli and tangent numbers / 154
                 \\
                 4.8 Arithmetic--geometric mean / 158 \\
                 4.8.1 Elliptic integrals / 158 \\
                 4.8.2 First AGM algorithm for the logarithm / 159 \\
                 4.8.3 Theta functions / 160 \\
                 4.8.4 Second AGM algorithm for the logarithm / 162 \\
                 4.8.5 The complex AGM / 163 \\
                 4.9 Binary splitting / 163 \\
                 4.9.1 A binary splitting algorithm for sin, cos / 166
                 \\
                 4.9.2 The bit-burst algorithm / 161 \\
                 4.10 Contour integration / 169 \\
                 4.11 Exercises / 171 \\
                 4.12 Notes and references / 179 \\
                 Implementations and pointers / 185 \\
                 5.1 Software tools / 185 \\
                 5.1.1 CLN / 185 \\
                 5.1.2 GNUMP (GMP) / 185 \\
                 5.1.3 MPFQ / 186 \\
                 5.1.4 GNU MPFR / 187 \\
                 5.1.5 Other multiple-precision packages / 187 \\
                 5.1.6 Computational algebra packages / 188 \\
                 5.2 Mailing lists / 189 \\
                 5.2.1 The GMP lists / 189 \\
                 5.2.2 The MPFR list / 190 \\
                 5.3 Online documents / 190 \\
                 References / 191 \\
                 Index / 207",
}


bibtype: inproceedings
filename: fparith.bib
label: Brisebarre:2011:APS
crossref: Schwarz:2011:PIS
pages: 23--30
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.13
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992105
keywords: ARITH-20; correct rounding; floating-point arithmetic; hypotenuse
acknowledgement: ack-nhfb
author:
Nicolas Brisebarre and Mioara Joldes and Peter Kornerup and Erik Martin-Dorel and Jean-Michel Muller

title:
Augmented Precision Square Roots and $2$-D Norms, and Discussion on Correctly Rounding $ sqrt x^2 + y^2 $

rawauthor:
Nicolas Brisebarre and Mioara Joldes and Peter Kornerup and {\'E}rik Martin-Dorel and Jean-Michel Muller

rawtitle:
Augmented Precision Square Roots and {$2$-D} Norms, and Discussion on Correctly Rounding $ \sqrt {x^2 + y^2} $

entry:
@InProceedings{Brisebarre:2011:APS,
  author =       "Nicolas Brisebarre and Mioara Joldes and Peter
                 Kornerup and {\'E}rik Martin-Dorel and Jean-Michel
                 Muller",
  title =        "Augmented Precision Square Roots and {$2$-D} Norms,
                 and Discussion on Correctly Rounding $ \sqrt {x^2 +
                 y^2} $",
  crossref =     "Schwarz:2011:PIS",
  pages =        "23--30",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.13",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
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  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992105",
  acknowledgement = ack-nhfb,
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bibtype: article
filename: fparith.bib
label: Bruguera:2011:GEI
author: Javier Bruguera and Marius Cornea and Debjit Das Sarma
title: Guest Editors' Introduction: Special Section on Computer Arithmetic
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 2
pages: 145--147
month: feb
monthnumber: 2
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2011.15
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Feb 20 19:15:33 2011
bibtimestamp: 2011.02.20 19:15:33 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Javier Bruguera and Marius Cornea and Debjit Das Sarma
rawtitle: {Guest Editors}' Introduction: Special Section on Computer Arithmetic
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  author =       "Javier Bruguera and Marius Cornea and Debjit Das
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  ajournal =     "IEEE Trans. Comput.",
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bibtype: inproceedings
filename: fparith.bib
label: Brumley:2011:BSB
author: Billy Bob Brumley and Dan Page
title: Bit-Sliced Binary Normal Basis Multiplication
crossref: Schwarz:2011:PIS
pages: 205--212
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.36
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992128
keywords: ARITH-20; correct rounding; floating-point arithmetic
acknowledgement: ack-nhfb
rawauthor: Billy Bob Brumley and Dan Page
rawtitle: Bit-Sliced Binary Normal Basis Multiplication
entry:
@InProceedings{Brumley:2011:BSB,
  author =       "Billy Bob Brumley and Dan Page",
  title =        "Bit-Sliced Binary Normal Basis Multiplication",
  crossref =     "Schwarz:2011:PIS",
  pages =        "205--212",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.36",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992128",
  acknowledgement = ack-nhfb,
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                 arithmetic",
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bibtype: inproceedings
filename: fparith.bib
label: Brunie:2011:MPF
author: Nicolas Brunie and Florent de Dinechin and Benoit de Dinechin
title: A mixed-precision fused multiply and add
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 205--212
month: nov
monthnumber: 11
year: 2011
DOI: https://doi.org/10.1109/ACSSC.2011.6189977
bibdate: Wed Aug 7 14:09:53 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2024.08.07 14:09:53 ???
acknowledgement: ack-nhfb
rawauthor: Nicolas Brunie and Florent de Dinechin and Benoit de Dinechin
rawtitle: A mixed-precision fused multiply and add
booktitle:
2011 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR)

rawbooktitle:
2011 Conference Record of the {Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR)}

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  author =       "Nicolas Brunie and Florent de Dinechin and Benoit de
                 Dinechin",
  booktitle =    "2011 Conference Record of the {Forty Fifth Asilomar
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  bibdate =      "Wed Aug 7 14:09:53 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
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bibtype: inproceedings
filename: fparith.bib
label: Brusentsov:2011:TCS
author: Nikolay Petrovich Brusentsov and Jose Ramil Alvarez
title: Ternary Computers: The Setun and the Setun 70
crossref: Impagliazzo:2011:PSR
pages: 74--80
year: 2011
DOI: https://doi.org/10.1007/978-3-642-22816-2_10
bibdate: Sun Aug 16 19:31:45 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.08.16 19:31:45 ???
URL: http://link.springer.com/openurl?genre=book%26isbn=978-3-642-22815-5
acknowledgement: ack-nhfb
bookDOI: https://doi.org/10.1007/978-3-642-22816-2
rawauthor: Nikolay Petrovich Brusentsov and Jos{\'e} Ramil Alvarez
rawtitle: Ternary Computers: The {Setun} and the {Setun 70}
remark:
The Setun computer series was unusual in being proposed to use base-3 (ternary) floating-point arithmetic.

entry:
@InProceedings{Brusentsov:2011:TCS,
  author =       "Nikolay Petrovich Brusentsov and Jos{\'e} Ramil
                 Alvarez",
  title =        "Ternary Computers: The {Setun} and the {Setun 70}",
  crossref =     "Impagliazzo:2011:PSR",
  pages =        "74--80",
  year =         "2011",
  DOI =          "https://doi.org/10.1007/978-3-642-22816-2_10",
  bibdate =      "Sun Aug 16 19:31:45 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://link.springer.com/openurl?genre=book%26isbn=978-3-642-22815-5",
  acknowledgement = ack-nhfb,
  book-DOI =     "https://doi.org/10.1007/978-3-642-22816-2",
  remark =       "The Setun computer series was unusual in being
                 proposed to use base-3 (ternary) floating-point
                 arithmetic.",
}


bibtype: inproceedings
filename: fparith.bib
label: Burgess:2011:FRC
author: Neil Burgess
title: Fast Ripple-Carry Adders in Standard-Cell CMOS VLSI
crossref: Schwarz:2011:PIS
pages: 103--111
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.23
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992115
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Neil Burgess
rawtitle: Fast Ripple-Carry Adders in Standard-Cell {CMOS VLSI}
entry:
@InProceedings{Burgess:2011:FRC,
  author =       "Neil Burgess",
  title =        "Fast Ripple-Carry Adders in Standard-Cell {CMOS
                 VLSI}",
  crossref =     "Schwarz:2011:PIS",
  pages =        "103--111",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.23",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992115",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: inproceedings
filename: fparith.bib
label: Butts:2011:RDR
author: J. Adam Butts and Ping Tak Peter Tang and Ron O. Dror and David E. Shaw
crossref: Schwarz:2011:PIS
pages: 149--158
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.28
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992120
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: J. Adam Butts and Ping Tak Peter Tang and Ron O. Dror and David E. Shaw
title:
Radix-8 Digit-by-Rounding: Achieving High-Performance Reciprocals, Square Roots, and Reciprocal Square Roots

rawtitle:
Radix-8 Digit-by-Rounding: Achieving High-Performance Reciprocals, Square Roots, and Reciprocal Square Roots

entry:
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  author =       "J. Adam Butts and Ping Tak Peter Tang and Ron O. Dror
                 and David E. Shaw",
  title =        "Radix-8 Digit-by-Rounding: Achieving High-Performance
                 Reciprocals, Square Roots, and Reciprocal Square
                 Roots",
  crossref =     "Schwarz:2011:PIS",
  pages =        "149--158",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.28",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992120",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: article
filename: fparith.bib
label: Calamia:2011:CGG
author: J. Calamia
title: China's Godson gamble
journal: j-IEEE-SPECTRUM
volume: 48
number: 5
pages: 14--16
month: may
monthnumber: 5
year: 2011
CODEN: IEESAM
DOI: https://doi.org/10.1109/MSPEC.2011.5753229
ISSN: 0018-9235 (print), 1939-9340 (electronic) OR 00189235 (print), 19399340 (electronic)
ISSNL: 0018-9235 OR 00189235
bibdate: Fri Jan 17 18:54:05 2020
bibtimestamp: 2020.01.17 18:54:05 ???
fjournal: IEEE Spectrum
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6
rawauthor: J. Calamia
rawtitle: {China}'s {Godson} gamble
bibsource:
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keywords:
American born chip; China; Chinese researcher; country homegrown high end processor; energy conservation; energy efficiency; floating point operation; Godson-3B; mainframes; Mesh networks; multiprocessing systems; supercomputer; Supercomputers; The Dawning

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@Article{Calamia:2011:CGG,
  author =       "J. Calamia",
  title =        "{China}'s {Godson} gamble",
  journal =      j-IEEE-SPECTRUM,
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  number =       "5",
  pages =        "14--16",
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  bibdate =      "Fri Jan 17 18:54:05 2020",
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  acknowledgement = ack-nhfb,
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  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6",
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                 energy efficiency; floating point operation; Godson-3B;
                 mainframes; Mesh networks; multiprocessing systems;
                 supercomputer; Supercomputers; The Dawning",
}


bibtype: inproceedings
filename: fparith.bib
label: Carlough:2011:IZD
author: Steven Carlough and Adam Collura and Silvia Mueller and Michael Kroener
title: The IBM zEnterprise-196 Decimal Floating-Point Accelerator
crossref: Schwarz:2011:PIS
pages: 139--146
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.27
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992119
keywords: ARITH-20; decimal floating-point arithmetic
acknowledgement: ack-nhfb
rawauthor: Steven Carlough and Adam Collura and Silvia Mueller and Michael Kroener
rawtitle: The {IBM zEnterprise-196} Decimal Floating-Point Accelerator
entry:
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  author =       "Steven Carlough and Adam Collura and Silvia Mueller
                 and Michael Kroener",
  title =        "The {IBM zEnterprise-196} Decimal Floating-Point
                 Accelerator",
  crossref =     "Schwarz:2011:PIS",
  pages =        "139--146",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.27",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992119",
  acknowledgement = ack-nhfb,
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}


bibtype: article
filename: fparith.bib
label: Cavagnino:2011:AAD
author: D. Cavagnino and A. E. Werbrouck
journal: j-COMP-J
volume: 54
number: 1
pages: 148--156
month: jan
monthnumber: 1
year: 2011
CODEN: CMPJA6
DOI: https://doi.org/10.1093/comjnl/bxp117
ISSN: 0010-4620 (print), 1460-2067 (electronic) OR 00104620 (print), 14602067 (electronic)
ISSNL: 0010-4620 OR 00104620
bibdate: Tue Dec 21 19:26:47 MST 2010
bibtimestamp: 2010.12.21 19:26:47 MST
URL: http://comjnl.oxfordjournals.org/content/54/1/148.full.pdf+html
fjournal: Computer Journal
keywords: dividend distribution; division by multiplication (DBM); integer division; multiplicative division
remark: See cite Cavagnino:2008:EAI.
acknowledgement: ack-nhfb
journalURL: http://comjnl.oxfordjournals.org/
onlinedate: December 30, 2009
rawauthor: D. Cavagnino and A. E. Werbrouck
title:
An Analysis of Associated Dividends in the DBM Algorithm for Division by Constants Using Multiplication

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abstract:
When a compiler encounters a fixed integer divisor, which is not a power of 2, it can calculate its inverse to be multiplied by the run-time integer dividends to obtain the quotients, using our very efficient, recently published [Cavagnino, D. and Werbrouck, A.E. (2008) em Efficient algorithms for integer division by constants using multiplication. Comp. J., 51, 470--480] division by multiplication algorithms. Essentially our algorithms permit a complete partition of a defined number space into non-adverse and adverse divisors on the basis of whether a dividend associated with each divisor is, or is not, greater than the maximum dividend size. In this paper, we demonstrate useful relations between the dividends associated with all divisors and with their multiples by positive powers

rawabstract:
When a compiler encounters a fixed integer divisor, which is not a power of 2, it can calculate its inverse to be multiplied by the run-time integer dividends to obtain the quotients, using our very efficient, recently published [Cavagnino, D. and Werbrouck, A.E. (2008) {\em Efficient algorithms for integer division by constants using multiplication}. Comp. J., 51, 470--480] division by multiplication algorithms. Essentially our algorithms permit a complete partition of a defined number space into non-adverse and adverse divisors on the basis of whether a dividend associated with each divisor is, or is not, greater than the maximum dividend size. In this paper, we demonstrate useful relations between the dividends associated with all divisors and with their multiples by positive powers

rawtitle:
An Analysis of Associated Dividends in the {DBM} Algorithm for Division by Constants Using Multiplication

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@Article{Cavagnino:2011:AAD,
  author =       "D. Cavagnino and A. E. Werbrouck",
  title =        "An Analysis of Associated Dividends in the {DBM}
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  journal =      j-COMP-J,
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  number =       "1",
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  ISSN-L =       "0010-4620",
  bibdate =      "Tue Dec 21 19:26:47 MST 2010",
  bibsource =    "http://comjnl.oxfordjournals.org/content/54/1.toc;
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  URL =          "http://comjnl.oxfordjournals.org/content/54/1/148.full.pdf+html",
  abstract =     "When a compiler encounters a fixed integer divisor,
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                 to be multiplied by the run-time integer dividends to
                 obtain the quotients, using our very efficient,
                 recently published [Cavagnino, D. and Werbrouck, A.E.
                 (2008) {\em Efficient algorithms for integer division
                 by constants using multiplication}. Comp. J., 51,
                 470--480] division by multiplication algorithms.
                 Essentially our algorithms permit a complete partition
                 of a defined number space into non-adverse and adverse
                 divisors on the basis of whether a dividend associated
                 with each divisor is, or is not, greater than the
                 maximum dividend size. In this paper, we demonstrate
                 useful relations between the dividends associated with
                 all divisors and with their multiples by positive
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  acknowledgement = ack-nhfb,
  fjournal =     "Computer Journal",
  journal-URL =  "http://comjnl.oxfordjournals.org/",
  keywords =     "dividend distribution; division by multiplication
                 (DBM); integer division; multiplicative division",
  onlinedate =   "December 30, 2009",
  remark =       "See \cite{Cavagnino:2008:EAI}.",
}


bibtype: article
filename: fparith.bib
label: Cenk:2011:EM
author: Murat Cenk and Ferruh Ozbudak
title: Efficient multiplications in $ mathbb F_5^5n $ and $ mathbb F_7^7n $
journal: j-J-COMPUT-APPL-MATH
volume: 236
number: 2
pages: 177--183
day: 15
month: aug
monthnumber: 8
year: 2011
CODEN: JCAMDI
DOI: https://doi.org/10.1016/j.cam.2011.06.016
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
MRclass: 11Y16 (11T99 12E20)
MRnumber: 2827399
bibdate: Sat Feb 25 13:24:37 MST 2017
bibtimestamp: 2017.02.25 13:24:37 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042711003396
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Murat Cenk and Ferruh {\"O}zbudak
rawtitle: Efficient multiplications in {$ \mathbb {F}_{5^{5n}} $} and {$ \mathbb {F}_{7^{7n}} $}
bibsource:
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  number =       "2",
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  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  MRclass =      "11Y16 (11T99 12E20)",
  MRnumber =     "2827399",
  bibdate =      "Sat Feb 25 13:24:37 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042711003396",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
  xxtitle =      "Efficient multiplications in {[??IMAGE??]} and
                 {[??IMAGE??]}",
}


bibtype: article
filename: fparith.bib
label: Chakraborty:2011:CBS
author: Anindita Chakraborty and Amitabha Sinha
title: Conversion of binary to single-term triple base numbers for DSP applications
journal: j-COMP-ARCH-NEWS
volume: 39
number: 5
pages: 5--11
month: dec
monthnumber: 12
year: 2011
CODEN: CANED2
DOI: https://doi.org/10.1145/2093339.2093342
ISSN: 0163-5964 (print), 1943-5851 (electronic) OR 01635964 (print), 19435851 (electronic)
ISSNL: 0163-5964 OR 01635964
bibdate: Thu Mar 15 14:07:10 MDT 2012
bibtimestamp: 2012.03.15 14:07:10 MDT
fjournal: ACM SIGARCH Computer Architecture News
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J89
rawauthor: Anindita Chakraborty and Amitabha Sinha
rawtitle: Conversion of binary to single-term triple base numbers for {DSP} applications
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:
In this paper high speed Residue Number System (RNS) based FIR filter using Distributed Arithmetic (DA) is proposed. The proposed architecture uses the module set having the value of numbers as small as possible. In case of using Distributed Arithmetic in FIR filter; the size of LUTs gets increased exponentially with the increase of tap of the filter. Here care has been taken so that sizes of LUTs do not get increased. The proposed architecture is designed using verilog HDL; a popular hardware description language [9]. The design is synthesized with ISE 10.1 and implemented on Xilinx's Virtex-4. The proposed architecture is also compared with conventional RNS-DA FIR filter. The results show that the proposed architecture can implement FIR filter with high speed.

rawabstract:
In this paper high speed Residue Number System (RNS) based FIR filter using Distributed Arithmetic (DA) is proposed. The proposed architecture uses the module set having the value of numbers as small as possible. In case of using Distributed Arithmetic in FIR filter; the size of LUTs gets increased exponentially with the increase of tap of the filter. Here care has been taken so that sizes of LUTs do not get increased. The proposed architecture is designed using verilog HDL; a popular hardware description language [9]. The design is synthesized with ISE 10.1 and implemented on Xilinx's Virtex-4. The proposed architecture is also compared with conventional RNS-DA FIR filter. The results show that the proposed architecture can implement FIR filter with high speed.

entry:
@Article{Chakraborty:2011:CBS,
  author =       "Anindita Chakraborty and Amitabha Sinha",
  title =        "Conversion of binary to single-term triple base
                 numbers for {DSP} applications",
  journal =      j-COMP-ARCH-NEWS,
  volume =       "39",
  number =       "5",
  pages =        "5--11",
  month =        dec,
  year =         "2011",
  CODEN =        "CANED2",
  DOI =          "https://doi.org/10.1145/2093339.2093342",
  ISSN =         "0163-5964 (print), 1943-5851 (electronic)",
  ISSN-L =       "0163-5964",
  bibdate =      "Thu Mar 15 14:07:10 MDT 2012",
  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 =     "In this paper high speed Residue Number System (RNS)
                 based FIR filter using Distributed Arithmetic (DA) is
                 proposed. The proposed architecture uses the module set
                 having the value of numbers as small as possible. In
                 case of using Distributed Arithmetic in FIR filter; the
                 size of LUTs gets increased exponentially with the
                 increase of tap of the filter. Here care has been taken
                 so that sizes of LUTs do not get increased. The
                 proposed architecture is designed using verilog HDL; a
                 popular hardware description language [9]. The design
                 is synthesized with ISE 10.1 and implemented on
                 Xilinx's Virtex-4. The proposed architecture is also
                 compared with conventional RNS-DA FIR filter. The
                 results show that the proposed architecture can
                 implement FIR filter with high speed.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGARCH Computer Architecture News",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J89",
}


bibtype: article
filename: fparith.bib
label: Chang:2011:CGR
author: Harry M. Chang
journal: j-COMPUTING
volume: 91
number: 3
pages: 241--264
month: mar
monthnumber: 3
year: 2011
CODEN: CMPTA2
ISSN: 0010-485X (print), 1436-5057 (electronic) OR 0010485X (print), 14365057 (electronic)
ISSNL: 0010-485X OR 0010485X
bibdate: Tue Sep 6 19:14:24 MDT 2011
bibtimestamp: 2011.09.06 19:14:24 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=0010-485X&volume=91&issue=3&spage=241
fjournal: Computing
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/607
rawauthor: Harry M. Chang
title:
Constructing $n$-gram rules for natural language models through exploring the limitation of the Zipf--Mandelbrot law

bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0010-485X&volume=91&issue=3; https://www.math.utah.edu/pub/tex/bib/fparith.bib

rawtitle:
Constructing $n$-gram rules for natural language models through exploring the limitation of the {Zipf--Mandelbrot} law

entry:
@Article{Chang:2011:CGR,
  author =       "Harry M. Chang",
  title =        "Constructing $n$-gram rules for natural language
                 models through exploring the limitation of the
                 {Zipf--Mandelbrot} law",
  journal =      j-COMPUTING,
  volume =       "91",
  number =       "3",
  pages =        "241--264",
  month =        mar,
  year =         "2011",
  CODEN =        "CMPTA2",
  ISSN =         "0010-485X (print), 1436-5057 (electronic)",
  ISSN-L =       "0010-485X",
  bibdate =      "Tue Sep 6 19:14:24 MDT 2011",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0010-485X&volume=91&issue=3;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=0010-485X&volume=91&issue=3&spage=241",
  acknowledgement = ack-nhfb,
  fjournal =     "Computing",
  journal-URL =  "http://link.springer.com/journal/607",
}


bibtype: article
filename: fparith.bib
label: Chen:2011:PIM
author: Zhimin Chen and Patrick Schaumont
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 12
pages: 1692--1703
month: dec
monthnumber: 12
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.256
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Nov 6 07:52:27 MST 2011
bibtimestamp: 2011.11.06 07:52:27 MST
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5669278
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Zhimin Chen and Patrick Schaumont
title:
A Parallel Implementation of Montgomery Multiplication on Multicore Systems: Algorithm, Analysis, and Prototype

rawtitle:
A Parallel Implementation of {Montgomery} Multiplication on Multicore Systems: Algorithm, Analysis, and Prototype

entry:
@Article{Chen:2011:PIM,
  author =       "Zhimin Chen and Patrick Schaumont",
  title =        "A Parallel Implementation of {Montgomery}
                 Multiplication on Multicore Systems: Algorithm,
                 Analysis, and Prototype",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "60",
  number =       "12",
  pages =        "1692--1703",
  month =        dec,
  year =         "2011",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.256",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Nov 6 07:52:27 MST 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5669278",
  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: Chen:2011:TSA
author: Jianxun Chen and Yongzhong Huang and Shaozhong Guo and Shimiao Chen and Wei Wang
booktitle: 2011 Third International Conference on Measuring Technology and Mechatronics Automation (ICMTMA)
title: Test Standardization and Analyse Model of Mathematical Functions for Precision
volume: 3
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 652--655
year: 2011
DOI: https://doi.org/10.1109/ICMTMA.2011.734
ISBN: 0-7695-4296-4 OR 0769542964
ISBN13: 978-0-7695-4296-6 OR 9780769542966
bibdate: Tue Sep 27 08:11:02 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.09.27 08:11:02 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5721571
acknowledgement: ack-nhfb
bookURL: http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5720445
rawauthor: Jianxun Chen and Yongzhong Huang and Shaozhong Guo and Shimiao Chen and Wei Wang
rawbooktitle: {2011 Third International Conference on Measuring Technology and Mechatronics Automation (ICMTMA)}
rawtitle: Test Standardization and Analyse Model of Mathematical Functions for Precision
abstract:
This article describes problems of meet the requirements to implementations of mathematical functions working with floating-point numbers, and so facilitate the comprehensive testing of mathematical functions. Inconsistency and incompleteness of available standards in the domain is demonstrated. Correct rounding requirement is suggested to guarantee preservation of all important properties of functions and to support high level of interoperability between different mathematical libraries and software using them. The article also concerns precision analyse of mathematical functions. Conformance test construction method is proposed based on different sources of test data.

remark:
This paper contains unattributed plagiaristic copying of material from url https://www.math.utah.edu/~beebe/software/ieee/index.html.

rawabstract:
This article describes problems of meet the requirements to implementations of mathematical functions working with floating-point numbers, and so facilitate the comprehensive testing of mathematical functions. Inconsistency and incompleteness of available standards in the domain is demonstrated. Correct rounding requirement is suggested to guarantee preservation of all important properties of functions and to support high level of interoperability between different mathematical libraries and software using them. The article also concerns precision analyse of mathematical functions. Conformance test construction method is proposed based on different sources of test data.

entry:
@InProceedings{Chen:2011:TSA,
  author =       "Jianxun Chen and Yongzhong Huang and Shaozhong Guo and
                 Shimiao Chen and Wei Wang",
  booktitle =    "{2011 Third International Conference on Measuring
                 Technology and Mechatronics Automation (ICMTMA)}",
  title =        "Test Standardization and Analyse Model of Mathematical
                 Functions for Precision",
  volume =       "3",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "652--655",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ICMTMA.2011.734",
  ISBN =         "0-7695-4296-4",
  ISBN-13 =      "978-0-7695-4296-6",
  bibdate =      "Tue Sep 27 08:11:02 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5721571",
  abstract =     "This article describes problems of meet the
                 requirements to implementations of mathematical
                 functions working with floating-point numbers, and so
                 facilitate the comprehensive testing of mathematical
                 functions. Inconsistency and incompleteness of
                 available standards in the domain is demonstrated.
                 Correct rounding requirement is suggested to guarantee
                 preservation of all important properties of functions
                 and to support high level of interoperability between
                 different mathematical libraries and software using
                 them. The article also concerns precision analyse of
                 mathematical functions. Conformance test construction
                 method is proposed based on different sources of test
                 data.",
  acknowledgement = ack-nhfb,
  book-URL =     "http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5720445",
  remark =       "This paper contains unattributed plagiaristic copying
                 of material from
                 \url{https://www.math.utah.edu/~beebe/software/ieee/index.html}.",
}


bibtype: inproceedings
filename: fparith.bib
label: Chevillard:2011:AGC
author: Sylvain Chevillard
crossref: Schwarz:2011:PIS
pages: 225--232
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.38
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992130
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Sylvain Chevillard
title:
Automatic Generation of Code for the Evaluation of Constant Expressions at Any Precision with a Guaranteed Error Bound

rawtitle:
Automatic Generation of Code for the Evaluation of Constant Expressions at Any Precision with a Guaranteed Error Bound

entry:
@InProceedings{Chevillard:2011:AGC,
  author =       "Sylvain Chevillard",
  title =        "Automatic Generation of Code for the Evaluation of
                 Constant Expressions at Any Precision with a Guaranteed
                 Error Bound",
  crossref =     "Schwarz:2011:PIS",
  pages =        "225--232",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.38",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992130",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: article
filename: fparith.bib
label: Colberg:2011:HAS
author: Peter H. Colberg and Felix Hofling
journal: j-COMP-PHYS-COMM
volume: 182
number: 5
pages: 1120--1129
month: may
monthnumber: 5
year: 2011
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2011.01.009
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Sat Feb 11 10:10:57 MST 2012
bibtimestamp: 2012.02.11 10:10:57 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465511000294
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Peter H. Colberg and Felix H{\"o}fling
title:
Highly accelerated simulations of glassy dynamics using GPUs: Caveats on limited floating-point precision

bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

rawtitle:
Highly accelerated simulations of glassy dynamics using {GPUs}: Caveats on limited floating-point precision

entry:
@Article{Colberg:2011:HAS,
  author =       "Peter H. Colberg and Felix H{\"o}fling",
  title =        "Highly accelerated simulations of glassy dynamics
                 using {GPUs}: Caveats on limited floating-point
                 precision",
  journal =      j-COMP-PHYS-COMM,
  volume =       "182",
  number =       "5",
  pages =        "1120--1129",
  month =        may,
  year =         "2011",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2011.01.009",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Sat Feb 11 10:10:57 MST 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465511000294",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: fparith.bib
label: Corless:2011:RCA
author: Robert Corless and Erik Postma and David R. Stoutemyer
title: Rounding coefficients and artificially underflowing terms in non-numeric expressions
journal: j-ACM-COMM-COMP-ALGEBRA
volume: 45
number: 1
pages: 17--48
month: mar
monthnumber: 3
year: 2011
DOI: https://doi.org/10.1145/2016567.2016570
ISSN: 1932-2232 (print), 1932-2240 (electronic) OR 19322232 (print), 19322240 (electronic)
ISSNL: 1932-2232 OR 19322232
bibdate: Wed Aug 17 08:55:07 MDT 2011
bibsource: http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.17 08:55:07 MDT
fjournal: ACM Communications in Computer Algebra
keywords: floating-point arithmetic; rounding; symbolic algebra
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
issue: 175
rawauthor: Robert Corless and Erik Postma and David R. Stoutemyer
rawtitle: Rounding coefficients and artificially underflowing terms in non-numeric expressions
abstract:
This article takes an analytical viewpoint to address the following questions: 1. How can we justifiably beautify an input or result sum of non-numeric terms that has some approximate coefficients by deleting some terms and/or rounding some coefficients to simpler floating-point or rational numbers? 2. When we add two expressions, how can we justifiably delete more non-zero result terms and/or round some result coefficients to even simpler floating-point, rational or irrational numbers? The methods considered in this paper provide a justifiable scale-invariant way to attack these problems for subexpressions that are multivariate sums of monomials with real exponents.

rawabstract:
This article takes an analytical viewpoint to address the following questions: 1. How can we justifiably beautify an input or result sum of non-numeric terms that has some approximate coefficients by deleting some terms and/or rounding some coefficients to simpler floating-point or rational numbers? 2. When we add two expressions, how can we justifiably delete more non-zero result terms and/or round some result coefficients to even simpler floating-point, rational or irrational numbers? The methods considered in this paper provide a justifiable scale-invariant way to attack these problems for subexpressions that are multivariate sums of monomials with real exponents.

entry:
@Article{Corless:2011:RCA,
  author =       "Robert Corless and Erik Postma and David R.
                 Stoutemyer",
  title =        "Rounding coefficients and artificially underflowing
                 terms in non-numeric expressions",
  journal =      j-ACM-COMM-COMP-ALGEBRA,
  volume =       "45",
  number =       "1",
  pages =        "17--48",
  month =        mar,
  year =         "2011",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2016567.2016570",
  ISSN =         "1932-2232 (print), 1932-2240 (electronic)",
  ISSN-L =       "1932-2232",
  bibdate =      "Wed Aug 17 08:55:07 MDT 2011",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "This article takes an analytical viewpoint to address
                 the following questions: 1. How can we justifiably
                 beautify an input or result sum of non-numeric terms
                 that has some approximate coefficients by deleting some
                 terms and/or rounding some coefficients to simpler
                 floating-point or rational numbers? 2. When we add two
                 expressions, how can we justifiably delete more
                 non-zero result terms and/or round some result
                 coefficients to even simpler floating-point, rational
                 or irrational numbers? The methods considered in this
                 paper provide a justifiable scale-invariant way to
                 attack these problems for subexpressions that are
                 multivariate sums of monomials with real exponents.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Communications in Computer Algebra",
  issue =        "175",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
  keywords =     "floating-point arithmetic; rounding; symbolic
                 algebra",
}


bibtype: inproceedings
filename: fparith.bib
label: Cui:2011:TDB
author: Mingyi Cui
title: A threshold denoising based floating point representation genetic algorithm
volume: 7
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 3305--3308
year: 2011
DOI: https://doi.org/10.1109/EMEIT.2011.6023063
bibdate: Tue Sep 27 08:11:02 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.09.27 08:11:02 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6023063
acknowledgement: ack-nhfb
bookURL: http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5996364
rawauthor: Mingyi Cui
rawtitle: A threshold denoising based floating point representation genetic algorithm
booktitle:
2011 International Conference on Electronic and Mechanical Engineering and Information Technology (EMEIT)

rawbooktitle:
{2011 International Conference on Electronic and Mechanical Engineering and Information Technology (EMEIT)}

entry:
@InProceedings{Cui:2011:TDB,
  author =       "Mingyi Cui",
  booktitle =    "{2011 International Conference on Electronic and
                 Mechanical Engineering and Information Technology
                 (EMEIT)}",
  title =        "A threshold denoising based floating point
                 representation genetic algorithm",
  volume =       "7",
  publisher =    "pub-IEEE",
  address =      "pub-IEEE:adr",
  pages =        "3305--3308",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/EMEIT.2011.6023063",
  bibdate =      "Tue Sep 27 08:11:02 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6023063",
  acknowledgement = ack-nhfb,
  book-URL =     "http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5996364",
}


bibtype: article
filename: fparith.bib
label: Curran:2011:ZSM
title: The zEnterprise 196 System and Microprocessor
journal: j-IEEE-MICRO
volume: 31
number: 2
pages: 26--40
month: mar
monthnumber: 3
year: 2011
CODEN: IEMIDZ
DOI: https://doi.org/10.1109/MM.2011.34
ISSN: 0272-1732 (print), 1937-4143 (electronic) OR 02721732 (print), 19374143 (electronic)
ISSNL: 0272-1732 OR 02721732
bibdate: Tue Apr 26 13:50:28 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.04.26 13:50:28 ???
fjournal: IEEE Micro
keywords: correct rounding; floating-point arithmetic; Hot Chips 22 conference proceedings
acknowledgement: ack-nhfb
journalURL: http://www.computer.org/csdl/mags/mi/index.html
rawtitle: The {zEnterprise 196} System and Microprocessor
author:
Brian W. Curran and Lee E. Eisen and Eric M. Schwarz and Pak-kin Mak and James Warnock and Patrick J. Meaney and Michael Fee

abstract:
The zEnterprise 196 is the latest IBM System zSeries mainframe computer, which builds on IBM's 46-year heritage of compatible enterprise-class machines. This design advances the prior z10 processor pipeline with out-of-order execution to achieve considerable performance gains in legacy online transaction processing and computationally intensive workloads. This article describes the system structure and details of this new high-frequency microprocessor.

remark:
From page 34: ``IEEE Std 754-2008 (IEEE Standard for Floating-Point Arithmetic) adds the requirement to support heterogeneous operations on operands with different precisions. In light of this standard, z196 added a new rounding mode called round to prepare for shorter precision to all binary floating-point instructions. Using this new rounding mode allows an addition with a double-precision addend and a single-precision augend to produce a single-precision sum with correct rounding.''

rawabstract:
The zEnterprise 196 is the latest IBM System zSeries mainframe computer, which builds on IBM's 46-year heritage of compatible enterprise-class machines. This design advances the prior z10 processor pipeline with out-of-order execution to achieve considerable performance gains in legacy online transaction processing and computationally intensive workloads. This article describes the system structure and details of this new high-frequency microprocessor.

rawauthor:
Brian W. Curran and Lee E. Eisen and Eric M. Schwarz and Pak-kin Mak and James Warnock and Patrick J. Meaney and Michael Fee

entry:
@Article{Curran:2011:ZSM,
  author =       "Brian W. Curran and Lee E. Eisen and Eric M. Schwarz
                 and Pak-kin Mak and James Warnock and Patrick J. Meaney
                 and Michael Fee",
  title =        "The {zEnterprise 196} System and Microprocessor",
  journal =      j-IEEE-MICRO,
  volume =       "31",
  number =       "2",
  pages =        "26--40",
  month =        mar # "\slash " # apr,
  year =         "2011",
  CODEN =        "IEMIDZ",
  DOI =          "https://doi.org/10.1109/MM.2011.34",
  ISSN =         "0272-1732 (print), 1937-4143 (electronic)",
  ISSN-L =       "0272-1732",
  bibdate =      "Tue Apr 26 13:50:28 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The zEnterprise 196 is the latest IBM System zSeries
                 mainframe computer, which builds on IBM's 46-year
                 heritage of compatible enterprise-class machines. This
                 design advances the prior z10 processor pipeline with
                 out-of-order execution to achieve considerable
                 performance gains in legacy online transaction
                 processing and computationally intensive workloads.
                 This article describes the system structure and details
                 of this new high-frequency microprocessor.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Micro",
  journal-URL =  "http://www.computer.org/csdl/mags/mi/index.html",
  keywords =     "correct rounding; floating-point arithmetic; Hot Chips
                 22 conference proceedings",
  remark =       "From page 34: ``IEEE Std 754-2008 (IEEE Standard for
                 Floating-Point Arithmetic) adds the requirement to
                 support heterogeneous operations on operands with
                 different precisions. In light of this standard, z196
                 added a new rounding mode called round to prepare for
                 shorter precision to all binary floating-point
                 instructions. Using this new rounding mode allows an
                 addition with a double-precision addend and a
                 single-precision augend to produce a single-precision
                 sum with correct rounding.''",
}


bibtype: article
filename: fparith.bib
label: Daisaka:2011:GMS
author: Hiroshi Daisaka and Naohito Nakasato and Junichiro Makino and Fukuko Yuasa and Tadashi Ishikawa
title: GRAPE-MP: An SIMD Accelerator Board for Multi-precision Arithmetic
journal: Procedia Computer Science
volume: 4
pages: 878--887
month: may
monthnumber: 5
year: 2011
DOI: https://doi.org/10.1016/j.procs.2011.04.093
ISSN: 1877-0509 OR 18770509
ISSNL: 1877-0509 OR 18770509
bibdate: Wed Jun 12 16:47:38 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2024.06.12 16:47:38 ???
acknowledgement: ack-nhfb
ajournal: Proc. Comput. Sci.
rawauthor: Hiroshi Daisaka and Naohito Nakasato and Junichiro Makino and Fukuko Yuasa and Tadashi Ishikawa
rawtitle: {GRAPE-MP}: An {SIMD} Accelerator Board for Multi-precision Arithmetic
entry:
@Article{Daisaka:2011:GMS,
  author =       "Hiroshi Daisaka and Naohito Nakasato and Junichiro
                 Makino and Fukuko Yuasa and Tadashi Ishikawa",
  title =        "{GRAPE-MP}: An {SIMD} Accelerator Board for
                 Multi-precision Arithmetic",
  journal =      "Procedia Computer Science",
  volume =       "4",
  pages =        "878--887",
  month =        may,
  year =         "2011",
  DOI =          "https://doi.org/10.1016/j.procs.2011.04.093",
  ISSN =         "1877-0509",
  ISSN-L =       "1877-0509",
  bibdate =      "Wed Jun 12 16:47:38 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "Proc. Comput. Sci.",
}


bibtype: article
filename: fparith.bib
label: Das:2011:HSR
author: Malay Das and Amitabha Sinha and Nishant Kumar Giri
title: High speed residue number system (RNS) based FIR filter using distributed arithmetic (DA)
journal: j-COMP-ARCH-NEWS
volume: 39
number: 5
pages: 1--4
month: dec
monthnumber: 12
year: 2011
CODEN: CANED2
DOI: https://doi.org/10.1145/2093339.2093341
ISSN: 0163-5964 (print), 1943-5851 (electronic) OR 01635964 (print), 19435851 (electronic)
ISSNL: 0163-5964 OR 01635964
bibdate: Thu Mar 15 14:07:10 MDT 2012
bibtimestamp: 2012.03.15 14:07:10 MDT
fjournal: ACM SIGARCH Computer Architecture News
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J89
rawauthor: Malay Das and Amitabha Sinha and Nishant Kumar Giri
rawtitle: High speed residue number system ({RNS}) based {FIR} filter using distributed arithmetic ({DA})
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

entry:
@Article{Das:2011:HSR,
  author =       "Malay Das and Amitabha Sinha and Nishant Kumar Giri",
  title =        "High speed residue number system ({RNS}) based {FIR}
                 filter using distributed arithmetic ({DA})",
  journal =      j-COMP-ARCH-NEWS,
  volume =       "39",
  number =       "5",
  pages =        "1--4",
  month =        dec,
  year =         "2011",
  CODEN =        "CANED2",
  DOI =          "https://doi.org/10.1145/2093339.2093341",
  ISSN =         "0163-5964 (print), 1943-5851 (electronic)",
  ISSN-L =       "0163-5964",
  bibdate =      "Thu Mar 15 14:07:10 MDT 2012",
  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",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGARCH Computer Architecture News",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J89",
}


bibtype: inproceedings
filename: fparith.bib
label: deDinechin:2011:AOY
author: Florent de Dinechin
title: The Arithmetic Operators You Will Never See in a Microprocessor
crossref: Schwarz:2011:PIS
pages: 189--190
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.33
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992125
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Florent de Dinechin
rawtitle: The Arithmetic Operators You Will Never See in a Microprocessor
entry:
@InProceedings{deDinechin:2011:AOY,
  author =       "Florent de Dinechin",
  title =        "The Arithmetic Operators You Will Never See in a
                 Microprocessor",
  crossref =     "Schwarz:2011:PIS",
  pages =        "189--190",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.33",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992125",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: article
filename: fparith.bib
label: deDinechin:2011:CFP
author: Florent de Dinechin and Christoph Lauter and Guillaume Melquiond
title: Certifying the Floating-Point Implementation of an Elementary Function Using Gappa
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 2
pages: 242--253
month: feb
monthnumber: 2
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.128
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Feb 20 19:15:33 2011
bibtimestamp: 2011.02.20 19:15:33 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Florent de Dinechin and Christoph Lauter and Guillaume Melquiond
rawtitle: Certifying the Floating-Point Implementation of an Elementary Function Using {Gappa}
abstract:
High confidence in floating-point programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. This certification may require a time-consuming proof for each line of code, and it is usually broken by the smallest change to the code, e.g., for maintenance or optimization purpose. Certifying floating-point programs by hand is, therefore, very tedious and error-prone. The Gappa proof assistant is designed to make this task both easier and more secure, due to the following novel features: It automates the evaluation and propagation of rounding errors using interval arithmetic. Its input format is very close to the actual code to validate. It can be used incrementally to prove complex mathematical properties pertaining to the code. It generates a formal proof of the results, which can be checked independently by a lower level proof assistant like Coq. Yet it does not require any specific knowledge about automatic theorem proving, and thus, is accessible to a wide community. This paper demonstrates the practical use of this tool for a widely used class of floating-point programs: implementations of elementary functions in a mathematical library.

rawabstract:
High confidence in floating-point programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. This certification may require a time-consuming proof for each line of code, and it is usually broken by the smallest change to the code, e.g., for maintenance or optimization purpose. Certifying floating-point programs by hand is, therefore, very tedious and error-prone. The Gappa proof assistant is designed to make this task both easier and more secure, due to the following novel features: It automates the evaluation and propagation of rounding errors using interval arithmetic. Its input format is very close to the actual code to validate. It can be used incrementally to prove complex mathematical properties pertaining to the code. It generates a formal proof of the results, which can be checked independently by a lower level proof assistant like Coq. Yet it does not require any specific knowledge about automatic theorem proving, and thus, is accessible to a wide community. This paper demonstrates the practical use of this tool for a widely used class of floating-point programs: implementations of elementary functions in a mathematical library.

entry:
@Article{deDinechin:2011:CFP,
  author =       "Florent de Dinechin and Christoph Lauter and Guillaume
                 Melquiond",
  title =        "Certifying the Floating-Point Implementation of an
                 Elementary Function Using {Gappa}",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "60",
  number =       "2",
  pages =        "242--253",
  month =        feb,
  year =         "2011",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.128",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Feb 20 19:15:33 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  abstract =     "High confidence in floating-point programs requires
                 proving numerical properties of final and intermediate
                 values. One may need to guarantee that a value stays
                 within some range, or that the error relative to some
                 ideal value is well bounded. This certification may
                 require a time-consuming proof for each line of code,
                 and it is usually broken by the smallest change to the
                 code, e.g., for maintenance or optimization purpose.
                 Certifying floating-point programs by hand is,
                 therefore, very tedious and error-prone. The Gappa
                 proof assistant is designed to make this task both
                 easier and more secure, due to the following novel
                 features: It automates the evaluation and propagation
                 of rounding errors using interval arithmetic. Its input
                 format is very close to the actual code to validate. It
                 can be used incrementally to prove complex mathematical
                 properties pertaining to the code. It generates a
                 formal proof of the results, which can be checked
                 independently by a lower level proof assistant like
                 Coq. Yet it does not require any specific knowledge
                 about automatic theorem proving, and thus, is
                 accessible to a wide community. This paper demonstrates
                 the practical use of this tool for a widely used class
                 of floating-point programs: implementations of
                 elementary functions in a mathematical library.",
  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: DelVento:2011:SLM
title: System-level monitoring of floating-point performance to improve effective system utilization
crossref: ACM:2011:SSP
pages: 5:1--5:6
year: 2011
DOI: https://doi.org/10.1145/2063348.2063355
bibdate: Fri Dec 16 11:19:26 MST 2011
bibtimestamp: 2011.12.16 11:19:26 MST
acknowledgement: ack-nhfb
articleno: 5
rawtitle: System-level monitoring of floating-point performance to improve effective system utilization
author:
Davide Del Vento and Thomas Engel and Siddhartha S. Ghosh and David L. Hart and Rory Kelly and Si Liu and Richard Valent

bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/supercomputing2011.bib

rawauthor:
Davide {Del Vento} and Thomas Engel and Siddhartha S. Ghosh and David L. Hart and Rory Kelly and Si Liu and Richard Valent

entry:
@InProceedings{DelVento:2011:SLM,
  author =       "Davide {Del Vento} and Thomas Engel and Siddhartha S.
                 Ghosh and David L. Hart and Rory Kelly and Si Liu and
                 Richard Valent",
  title =        "System-level monitoring of floating-point performance
                 to improve effective system utilization",
  crossref =     "ACM:2011:SSP",
  pages =        "5:1--5:6",
  year =         "2011",
  DOI =          "https://doi.org/10.1145/2063348.2063355",
  bibdate =      "Fri Dec 16 11:19:26 MST 2011",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/supercomputing2011.bib",
  acknowledgement = ack-nhfb,
  articleno =    "5",
}


bibtype: article
filename: fparith.bib
label: Dimitrov:2011:AEM
author: Vassil S. Dimitrov and Kimmo U. Jarvinen and Jithra Adikari
title: Area-Efficient Multipliers Based on Multiple-Radix Representations
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 2
pages: 189--201
month: feb
monthnumber: 2
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.200
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sun Feb 20 19:15:33 2011
bibtimestamp: 2011.02.20 19:15:33 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Vassil S. Dimitrov and Kimmo U. Jarvinen and Jithra Adikari
rawtitle: Area-Efficient Multipliers Based on Multiple-Radix Representations
entry:
@Article{Dimitrov:2011:AEM,
  author =       "Vassil S. Dimitrov and Kimmo U. Jarvinen and Jithra
                 Adikari",
  title =        "Area-Efficient Multipliers Based on Multiple-Radix
                 Representations",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "60",
  number =       "2",
  pages =        "189--201",
  month =        feb,
  year =         "2011",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.200",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sun Feb 20 19:15:33 2011",
  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: Dimond:2011:ALS
author: Rob Dimond and Sebastien Racaniere and Oliver Pell
title: Accelerating Large-Scale HPC Applications Using FPGAs
crossref: Schwarz:2011:PIS
pages: 191--192
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.34
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992126
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Rob Dimond and Sebastien Racani{\`e}re and Oliver Pell
rawtitle: Accelerating Large-Scale {HPC} Applications Using {FPGAs}
entry:
@InProceedings{Dimond:2011:ALS,
  author =       "Rob Dimond and Sebastien Racani{\`e}re and Oliver
                 Pell",
  title =        "Accelerating Large-Scale {HPC} Applications Using
                 {FPGAs}",
  crossref =     "Schwarz:2011:PIS",
  pages =        "191--192",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.34",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992126",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: article
filename: fparith.bib
label: Emmart:2011:HPI
author: Niall Emmart and Charles C. Weems
title: High Precision Integer Multiplication with a GPU Using Strassen's Algorithm with Multiple FFT Sizes
journal: j-PARALLEL-PROCESS-LETT
volume: 21
number: 3
pages: 359--375
month: sep
monthnumber: 9
year: 2011
CODEN: PPLTEE
DOI: https://doi.org/10.1142/S0129626411000266
ISSN: 0129-6264 (print), 1793-642X (electronic) OR 01296264 (print), 1793642X (electronic)
bibdate: Tue Feb 28 11:32:08 MST 2012
bibtimestamp: 2012.02.28 11:32:08 MST
note: See earlier work cite Emmart:2010:HPI.
fjournal: Parallel Processing Letters
acknowledgement: ack-nhfb
journalURL: http://www.worldscientific.com/loi/ppl
rawauthor: Niall Emmart and Charles C. Weems
rawnote: See earlier work \cite{Emmart:2010:HPI}.
bibsource:
http://ejournals.wspc.com.sg/ppl/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/parallelprocesslett.bib

remark:
The abstract reports an improvement (GPU compared to GMP) of 13.9x to 19x for multiplication with operands of 255Kbits to 16320Kbits, compared to earlier work cite Emmart:2010:HPI.

rawtitle:
High Precision Integer Multiplication with a {GPU} Using {Strassen}'s Algorithm with Multiple {FFT} Sizes

entry:
@Article{Emmart:2011:HPI,
  author =       "Niall Emmart and Charles C. Weems",
  title =        "High Precision Integer Multiplication with a {GPU}
                 Using {Strassen}'s Algorithm with Multiple {FFT}
                 Sizes",
  journal =      j-PARALLEL-PROCESS-LETT,
  volume =       "21",
  number =       "3",
  pages =        "359--375",
  month =        sep,
  year =         "2011",
  CODEN =        "PPLTEE",
  DOI =          "https://doi.org/10.1142/S0129626411000266",
  ISSN =         "0129-6264 (print), 1793-642X (electronic)",
  bibdate =      "Tue Feb 28 11:32:08 MST 2012",
  bibsource =    "http://ejournals.wspc.com.sg/ppl/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/parallelprocesslett.bib",
  note =         "See earlier work \cite{Emmart:2010:HPI}.",
  acknowledgement = ack-nhfb,
  fjournal =     "Parallel Processing Letters",
  journal-URL =  "http://www.worldscientific.com/loi/ppl",
  remark =       "The abstract reports an improvement (GPU compared to
                 GMP) of 13.9x to 19x for multiplication with operands
                 of 255Kbits to 16320Kbits, compared to earlier work
                 \cite{Emmart:2010:HPI}.",
}


bibtype: inproceedings
filename: fparith.bib
label: Fischer:2011:HIC
author: Ralf Fischer
title: High Intelligence Computing: The New Era of High Performance Computing
crossref: Schwarz:2011:PIS
pages: 3--3
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.42
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992102
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Ralf Fischer
rawtitle: High Intelligence Computing: The New Era of High Performance Computing
entry:
@InProceedings{Fischer:2011:HIC,
  author =       "Ralf Fischer",
  title =        "High Intelligence Computing: The New Era of High
                 Performance Computing",
  crossref =     "Schwarz:2011:PIS",
  pages =        "3--3",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.42",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992102",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: article
filename: fparith.bib
label: Galal:2011:EEF
author: Sameh Galal and Mark Horowitz
title: Energy-Efficient Floating-Point Unit Design
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 7
pages: 913--922
month: jul
monthnumber: 7
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.121
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Wed Jun 29 10:26:18 2011
bibtimestamp: 2011.06.29 10:26:18 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Sameh Galal and Mark Horowitz
rawtitle: Energy-Efficient Floating-Point Unit Design
entry:
@Article{Galal:2011:EEF,
  author =       "Sameh Galal and Mark Horowitz",
  title =        "Energy-Efficient Floating-Point Unit Design",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "60",
  number =       "7",
  pages =        "913--922",
  month =        jul,
  year =         "2011",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.121",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Wed Jun 29 10:26:18 2011",
  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: Galal:2011:LSF
author: Sameh Galal and Mark Horowitz
title: Latency Sensitive FMA Design
crossref: Schwarz:2011:PIS
pages: 129--138
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.26
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992118
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Sameh Galal and Mark Horowitz
rawtitle: Latency Sensitive {FMA} Design
entry:
@InProceedings{Galal:2011:LSF,
  author =       "Sameh Galal and Mark Horowitz",
  title =        "Latency Sensitive {FMA} Design",
  crossref =     "Schwarz:2011:PIS",
  pages =        "129--138",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.26",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992118",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: inproceedings
filename: fparith.bib
label: Gandino:2011:GAI
author: Filippo Gandino and Fabrizio Lamberti and Paolo Montuschi and Jean-Claude Bajard
title: A General Approach for Improving RNS Montgomery Exponentiation Using Pre-processing
crossref: Schwarz:2011:PIS
pages: 195--204
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.35
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992127
keywords: ARITH-20; residue number system (RNS)
acknowledgement: ack-nhfb
rawauthor: Filippo Gandino and Fabrizio Lamberti and Paolo Montuschi and Jean-Claude Bajard
rawtitle: A General Approach for Improving {RNS Montgomery} Exponentiation Using Pre-processing
entry:
@InProceedings{Gandino:2011:GAI,
  author =       "Filippo Gandino and Fabrizio Lamberti and Paolo
                 Montuschi and Jean-Claude Bajard",
  title =        "A General Approach for Improving {RNS Montgomery}
                 Exponentiation Using Pre-processing",
  crossref =     "Schwarz:2011:PIS",
  pages =        "195--204",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.35",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992127",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20; residue number system (RNS)",
}


bibtype: article
filename: fparith.bib
label: Garofalo:2011:ACM
author: Valeria Garofalo and Nicola Petra and Ettore Napoli
journal: j-IEEE-TRANS-COMPUT
volume: 60
number: 9
pages: 1366--1371
month: sep
monthnumber: 9
year: 2011
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2010.236
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Aug 22 09:18:06 MDT 2011
bibtimestamp: 2011.08.22 09:18:06 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5669274
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Valeria Garofalo and Nicola Petra and Ettore Napoli
title:
Analytical Calculation of the Maximum Error for a Family of Truncated Multipliers Providing Minimum Mean Square Error

rawtitle:
Analytical Calculation of the Maximum Error for a Family of Truncated Multipliers Providing Minimum Mean Square Error

entry:
@Article{Garofalo:2011:ACM,
  author =       "Valeria Garofalo and Nicola Petra and Ettore Napoli",
  title =        "Analytical Calculation of the Maximum Error for a
                 Family of Truncated Multipliers Providing Minimum Mean
                 Square Error",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "60",
  number =       "9",
  pages =        "1366--1371",
  month =        sep,
  year =         "2011",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2010.236",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Aug 22 09:18:06 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5669274",
  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: Gopikiran:2011:FIF
author: G. Gopikiran and R. Thilagavathy
title: FPGA implementation of floating-point rotation mode CORDIC algorithm
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 506--508
year: 2011
DOI: https://doi.org/10.1109/ICSCCN.2011.6024604
bibdate: Tue Sep 27 08:11:09 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.09.27 08:11:09 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6024604
acknowledgement: ack-nhfb
bookURL: http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6016617
rawauthor: G. Gopikiran and R. Thilagavathy
rawtitle: {FPGA} implementation of floating-point rotation mode {CORDIC} algorithm
booktitle:
2011 International Conference on Signal Processing, Communication, Computing and Networking Technologies (ICSCCN)

rawbooktitle:
{2011 International Conference on Signal Processing, Communication, Computing and Networking Technologies (ICSCCN)}

entry:
@InProceedings{Gopikiran:2011:FIF,
  author =       "G. Gopikiran and R. Thilagavathy",
  booktitle =    "{2011 International Conference on Signal Processing,
                 Communication, Computing and Networking Technologies
                 (ICSCCN)}",
  title =        "{FPGA} implementation of floating-point rotation mode
                 {CORDIC} algorithm",
  publisher =    "pub-IEEE",
  address =      "pub-IEEE:adr",
  pages =        "506--508",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ICSCCN.2011.6024604",
  bibdate =      "Tue Sep 27 08:11:09 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6024604",
  acknowledgement = ack-nhfb,
  book-URL =     "http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6016617",
}


bibtype: inproceedings
filename: fparith.bib
label: Gorgin:2011:FHR
author: Saeid Gorgin and Ghassem Jaberipur
title: A Family of High Radix Signed Digit Adders
crossref: Schwarz:2011:PIS
pages: 112--120
year: 2011
DOI: https://doi.org/10.1109/ARITH.2011.24
bibdate: Sat Aug 20 09:00:00 MDT 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2011.08.20 09:00:00 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992116
keywords: ARITH-20
acknowledgement: ack-nhfb
rawauthor: Saeid Gorgin and Ghassem Jaberipur
rawtitle: A Family of High Radix Signed Digit Adders
entry:
@InProceedings{Gorgin:2011:FHR,
  author =       "Saeid Gorgin and Ghassem Jaberipur",
  title =        "A Family of High Radix Signed Digit Adders",
  crossref =     "Schwarz:2011:PIS",
  pages =        "112--120",
  year =         "2011",
  DOI =          "https://doi.org/10.1109/ARITH.2011.24",
  bibdate =      "Sat Aug 20 09:00:00 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992116",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-20",
}


bibtype: inproceedings
filename: fparith.bib
label: Goubault:2011:SAF
author: Eric Goubault and Sylvie Putot
editor: Ranjit Jhala and David A. Schmidt
title: Static Analysis of Finite Precision Computations
publisher: pub-ACM
address: pub-ACM:adr
pages: 232--247
year: 2011
ISBN: 3-642-18274-7 OR 3642182747
ISBN13: 978-3-642-18274-7 OR 9783642182747
bibdate: Thu Oct 17 05:33:04 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2019.10.17 05:33:04 ???
URL: https://dl.acm.org/citation.cfm?id=1946284.1946301
acknowledgement: ack-nhfb
rawauthor: Eric Goubault and Sylvie Putot
raweditor: Ranjit Jhala and David A. Schmidt
rawtitle: Static Analysis of Finite Precision Computations
booktitle:
VMCAI'11: Proceedings of the 12th international conference on verification, model checking, and abstract interpretation. Austin, TX, USA --- January 23--25, 2011

abstract:
We define several abstract semantics for the static analysis of finite precision computations, that bound not only the ranges of values taken by numerical variables of a program, but also the difference with the result of the same sequence of operations in an idealized real number semantics. These domains point out with more or less detail (control point, block, function for instance) sources of numerical errors in the program and the way they were propagated by further computations, thus allowing to evaluate not only the rounding error, but also sensitivity to inputs or parameters of the program. We describe two classes of abstractions, a non relational one based on intervals, and a weakly relational one based on parametrized zonotopic abstract domains called affine sets, especially well suited for sensitivity analysis and test generation. These abstract domains are implemented in the Fluctuat static analyzer, and we finally present some experiments.

rawabstract:
We define several abstract semantics for the static analysis of finite precision computations, that bound not only the ranges of values taken by numerical variables of a program, but also the difference with the result of the same sequence of operations in an idealized real number semantics. These domains point out with more or less detail (control point, block, function for instance) sources of numerical errors in the program and the way they were propagated by further computations, thus allowing to evaluate not only the rounding error, but also sensitivity to inputs or parameters of the program. We describe two classes of abstractions, a non relational one based on intervals, and a weakly relational one based on parametrized zonotopic abstract domains called affine sets, especially well suited for sensitivity analysis and test generation. These abstract domains are implemented in the Fluctuat static analyzer, and we finally present some experiments.

rawbooktitle:
{VMCAI'11: Proceedings of the 12th international conference on verification, model checking, and abstract interpretation. Austin, TX, USA --- January 23--25, 2011}

entry:
@InProceedings{Goubault:2011:SAF,
  author =       "Eric Goubault and Sylvie Putot",
  editor =       "Ranjit Jhala and David A. Schmidt",
  booktitle =    "{VMCAI'11: Proceedings of the 12th international
                 conference on verification, model checking, and
                 abstract interpretation. Austin, TX, USA --- January
                 23--25, 2011}",
  title =        "Static Analysis of Finite Precision Computations",
  publisher =    pub-ACM,
  address =      pub-ACM:adr,
  pages =        "232--247",
  year =         "2011",
  ISBN =         "3-642-18274-7",
  ISBN-13 =      "978-3-642-18274-7",
  bibdate =      "Thu Oct 17 05:33:04 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://dl.acm.org/citation.cfm?id=1946284.1946301",
  abstract =     "We define several abstract semantics for the static
                 analysis of finite precision computations, that bound
                 not only the ranges of values taken by numerical
                 variables of a program, but also the difference with
                 the result of the same sequence of operations in an
                 idealized real number semantics. These domains point
                 out with more or less detail (control point, block,
                 function for instance) sources of numerical errors in
                 the program and the way they were propagated by further
                 computations, thus allowing to evaluate not only the
                 rounding error, but also sensitivity to inputs or
                 parameters of the program. We describe two classes of
                 abstractions, a non relational one based on intervals,
                 and a weakly relational one based on parametrized
                 zonotopic abstract domains called affine sets,
                 especially well suited for sensitivity analysis and
                 test generation. These abstract domains are implemented
                 in the Fluctuat static analyzer, and we finally present
                 some experiments.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Graillat:2011:SAM
author: Stef Graillat and Fabienne Jezequel and Shiyue Wang and Yuxiang Zhu
title: Stochastic Arithmetic in Multiprecision
journal: j-MATH-COMPUT-SCI
volume: 5
number: 4
pages: 359--375
month: dec
monthnumber: 12
year: 2011
DOI: https://doi.org/10.1007/s11786-011-0103-4
ISSN: 1661-8270 (print), 1661-8289 (electronic) OR 16618270 (print), 16618289 (electronic)
ISSNL: 1661-8270 OR 16618270
bibdate: Tue Nov 6 10:16:23 MST 2012
bibtimestamp: 2012.11.06 10:16:23 MST
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1661-8270&volume=5&issue=4&spage=359
fjournal: Mathematics in Computer Science
keywords: interval arithmetic; MPFI; MPFR; multiprecision; Stochastic Arithmetic in Multiprecision (SAM)
acknowledgement: ack-nhfb
journalURL: http://www.springerlink.com/content/1661-8270/
rawauthor: Stef Graillat and Fabienne J{\'e}z{\'e}quel and Shiyue Wang and Yuxiang Zhu
rawtitle: Stochastic Arithmetic in Multiprecision
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1661-8270&volume=5&issue=4; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-sci.bib

entry:
@Article{Graillat:2011:SAM,
  author =       "Stef Graillat and Fabienne J{\'e}z{\'e}quel and Shiyue
                 Wang and Yuxiang Zhu",
  title =        "Stochastic Arithmetic in Multiprecision",
  journal =      j-MATH-COMPUT-SCI,
  volume =       "5",
  number =       "4",
  pages =        "359--375",
  month =        dec,
  year =         "2011",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s11786-011-0103-4",
  ISSN =         "1661-8270 (print), 1661-8289 (electronic)",
  ISSN-L =       "1661-8270",
  bibdate =      "Tue Nov 6 10:16:23 MST 2012",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1661-8270&volume=5&issue=4;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/math-comput-sci.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1661-8270&volume=5&issue=4&spage=359",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics in Computer Science",
  journal-URL =  "http://www.springerlink.com/content/1661-8270/",
  keywords =     "interval arithmetic; MPFI; MPFR; multiprecision;
                 Stochastic Arithmetic in Multiprecision (SAM)",
}


bibtype: article
filename: fparith.bib
label: Grcar:2011:JNA
author: Joseph F. Grcar
title: John von Neumann's Analysis of Gaussian Elimination and the Origins of Modern Numerical Analysis
journal: j-SIAM-REVIEW
volume: 53
number: 4
pages: 607--682
year: 2011
CODEN: SIREAD
DOI: https://doi.org/10.1137/080734716
ISSN: 0036-1445 (print), 1095-7200 (electronic) OR 00361445 (print), 10957200 (electronic)
ISSNL: 0036-1445 OR 00361445
MRclass: 01-08, 65-03, 65F05, 65F35, 65G50, 65M12, 68-03, 65-03 (01A60 65F35 68-03)
MRnumber: 2861262 (2012m:65001)
MRreviewer: Ilse C. F. Ipsen
bibdate: Fri Jun 21 11:25:02 MDT 2013
bibtimestamp: 2013.06.21 11:25:02 MDT
URL: http://epubs.siam.org/sirev/resource/1/siread/v53/i4/p607_s1
fjournal: SIAM Review
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sirev
onlinedate: November 07, 2011
rawauthor: Joseph F. Grcar
bibsource:
http://epubs.siam.org/sam-bin/dbq/toc/SIREV/53/4; http://epubs.siam.org/toc/siread/53/4; https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib

abstract:
Just when modern computers (digital, electronic, and programmable) were being invented, John von Neumann and Herman Goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers. Their foresight and the congruence of historical events made their work the first modern paper in numerical analysis. Von Neumann once remarked that to found a mathematical theory one had to prove the first theorem, which he and Goldstine did for the accuracy of mechanized Gaussian elimination --- but their paper was about more than that. Von Neumann and Goldstine described what they surmized would be the significant questions once computers became available for computational science, and they suggested enduring ways to answer them.

keywords:
backward error, CFL condition, computer architecture, computer arithmetic, computer programming, condition number, decomposition paradigm, Gaussian elimination, history, matrix norms, numerical analysis, rounding error analysis, stability, stochastic linear algebra, von Neumann

rawabstract:
Just when modern computers (digital, electronic, and programmable) were being invented, John von Neumann and Herman Goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers. Their foresight and the congruence of historical events made their work the first modern paper in numerical analysis. Von Neumann once remarked that to found a mathematical theory one had to prove the first theorem, which he and Goldstine did for the accuracy of mechanized Gaussian elimination --- but their paper was about more than that. Von Neumann and Goldstine described what they surmized would be the significant questions once computers became available for computational science, and they suggested enduring ways to answer them.

rawtitle:
{John von Neumann}'s Analysis of {Gaussian} Elimination and the Origins of Modern Numerical Analysis

entry:
@Article{Grcar:2011:JNA,
  author =       "Joseph F. Grcar",
  title =        "{John von Neumann}'s Analysis of {Gaussian}
                 Elimination and the Origins of Modern Numerical
                 Analysis",
  journal =      j-SIAM-REVIEW,
  volume =       "53",
  number =       "4",
  pages =        "607--682",
  month =        "????",
  year =         "2011",
  CODEN =        "SIREAD",
  DOI =          "https://doi.org/10.1137/080734716",
  ISSN =         "0036-1445 (print), 1095-7200 (electronic)",
  ISSN-L =       "0036-1445",
  MRclass =      "01-08, 65-03, 65F05, 65F35, 65G50, 65M12, 68-03, 65-03
                 (01A60 65F35 68-03)",
  MRnumber =     "2861262 (2012m:65001)",
  MRreviewer =   "Ilse C. F. Ipsen",
  bibdate =      "Fri Jun 21 11:25:02 MDT 2013",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SIREV/53/4;
                 http://epubs.siam.org/toc/siread/53/4;
                 https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamreview.bib",
  URL =          "http://epubs.siam.org/sirev/resource/1/siread/v53/i4/p607_s1",
  abstract =     "Just when modern computers (digital, electronic, and
                 programmable) were being invented, John von Neumann and
                 Herman Goldstine wrote a paper to illustrate the
                 mathematical analyses that they believed would be
                 needed to use the new machines effectively and to guide
                 the development of still faster computers. Their
                 foresight and the congruence of historical events made
                 their work the first modern paper in numerical
                 analysis. Von Neumann once remarked that to found a
                 mathematical theory one had to prove the first theorem,
                 which he and Goldstine did for the accuracy of
                 mechanized Gaussian elimination --- but their paper was
                 about more than that. Von Neumann and Goldstine
                 described what they surmized would be the significant
                 questions once computers became available for
                 computational science, and they suggested enduring ways
                 to answer them.",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Review",
  journal-URL =  "http://epubs.siam.org/sirev",
  keywords =     "backward error, CFL condition, computer architecture,
                 computer arithmetic, computer programming, condition
                 number, decomposition paradigm, Gaussian elimination,
                 history, matrix norms, numerical analysis, rounding
                 error analysis, stability, stochastic linear algebra,
                 von Neumann",
  onlinedate =   "November 07, 2011",
}


