Ebook: Handbook of Floating-Point Arithmetic

Floating-point arithmetic is by far the most widely used way of implementing real-number arithmetic on modern computers. Although the basic principles of floating-point arithmetic can be explained in a short amount of time, making such an arithmetic reliable and portable, yet fast, is a very difficult task. From the 1960s to the early 1980s, many different arithmetics were developed, but their implementation varied widely from one machine to another, making it difficult for nonexperts to design, learn, and use the required algorithms. As a result, floating-point arithmetic is far from being exploited to its full potential.

This handbook aims to provide a complete overview of modern floating-point arithmetic, including a detailed treatment of the newly revised (IEEE 754-2008) standard for floating-point arithmetic. Presented throughout are algorithms for implementing floating-point arithmetic as well as algorithms that use floating-point arithmetic. So that the techniques presented can be put directly into practice in actual coding or design, they are illustrated, whenever possible, by a corresponding program.

Key topics and features include:

* Presentation of the history and basic concepts of floating-point arithmetic and various aspects of the past and current standards

* Development of smart and nontrivial algorithms, and algorithmic possibilities induced by the availability of a fused multiply-add (fma) instruction, e.g., correctly rounded software division and square roots

* Implementation of floating-point arithmetic, either in software—on an integer processor—or hardware, and a discussion of issues related to compilers and languages

* Coverage of several recent advances related to elementary functions: correct rounding of these functions and computation of very accurate approximations under constraints

* Extensions of floating-point arithmetic such as certification, verification, and big precision

Handbook of Floating-Point Arithmetic is designed for programmers of numerical applications, compiler designers, programmers of floating-point algorithms, designers of arithmetic operators, and more generally, students and researchers in numerical analysis who wish to better understand a tool used in their daily work and research.

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Floating-point arithmetic is by far the most widely used way of implementing real-number arithmetic on modern computers. Although the basic principles of floating-point arithmetic can be explained in a short amount of time, making such an arithmetic reliable and portable, yet fast, is a very difficult task. From the 1960s to the early 1980s, many different arithmetics were developed, but their implementation varied widely from one machine to another, making it difficult for nonexperts to design, learn, and use the required algorithms. As a result, floating-point arithmetic is far from being exploited to its full potential.

This handbook aims to provide a complete overview of modern floating-point arithmetic, including a detailed treatment of the newly revised (IEEE 754-2008) standard for floating-point arithmetic. Presented throughout are algorithms for implementing floating-point arithmetic as well as algorithms that use floating-point arithmetic. So that the techniques presented can be put directly into practice in actual coding or design, they are illustrated, whenever possible, by a corresponding program.

Key topics and features include:

* Presentation of the history and basic concepts of floating-point arithmetic and various aspects of the past and current standards

* Development of smart and nontrivial algorithms, and algorithmic possibilities induced by the availability of a fused multiply-add (fma) instruction, e.g., correctly rounded software division and square roots

* Implementation of floating-point arithmetic, either in software—on an integer processor—or hardware, and a discussion of issues related to compilers and languages

* Coverage of several recent advances related to elementary functions: correct rounding of these functions and computation of very accurate approximations under constraints

* Extensions of floating-point arithmetic such as certification, verification, and big precision

Handbook of Floating-Point Arithmetic is designed for programmers of numerical applications, compiler designers, programmers of floating-point algorithms, designers of arithmetic operators, and more generally, students and researchers in numerical analysis who wish to better understand a tool used in their daily work and research.
 

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Floating-point arithmetic is by far the most widely used way of implementing real-number arithmetic on modern computers. Although the basic principles of floating-point arithmetic can be explained in a short amount of time, making such an arithmetic reliable and portable, yet fast, is a very difficult task. From the 1960s to the early 1980s, many different arithmetics were developed, but their implementation varied widely from one machine to another, making it difficult for nonexperts to design, learn, and use the required algorithms. As a result, floating-point arithmetic is far from being exploited to its full potential.

This handbook aims to provide a complete overview of modern floating-point arithmetic, including a detailed treatment of the newly revised (IEEE 754-2008) standard for floating-point arithmetic. Presented throughout are algorithms for implementing floating-point arithmetic as well as algorithms that use floating-point arithmetic. So that the techniques presented can be put directly into practice in actual coding or design, they are illustrated, whenever possible, by a corresponding program.

Key topics and features include:

* Presentation of the history and basic concepts of floating-point arithmetic and various aspects of the past and current standards

* Development of smart and nontrivial algorithms, and algorithmic possibilities induced by the availability of a fused multiply-add (fma) instruction, e.g., correctly rounded software division and square roots

* Implementation of floating-point arithmetic, either in software—on an integer processor—or hardware, and a discussion of issues related to compilers and languages

* Coverage of several recent advances related to elementary functions: correct rounding of these functions and computation of very accurate approximations under constraints

* Extensions of floating-point arithmetic such as certification, verification, and big precision

Handbook of Floating-Point Arithmetic is designed for programmers of numerical applications, compiler designers, programmers of floating-point algorithms, designers of arithmetic operators, and more generally, students and researchers in numerical analysis who wish to better understand a tool used in their daily work and research.
 

Content:
Front Matter....Pages i-xxiii
Front Matter....Pages 1-1
Introduction....Pages 3-12
Definitions and Basic Notions....Pages 13-53
Floating-Point Formats and Environment....Pages 55-116
Front Matter....Pages 117-117
Basic Properties and Algorithms....Pages 119-150
The Fused Multiply-Add Instruction....Pages 151-179
Enhanced Floating-Point Sums, Dot Products, and Polynomial Values....Pages 181-204
Languages and Compilers....Pages 205-235
Front Matter....Pages 237-237
Algorithms for the Five Basic Operations....Pages 239-267
Hardware Implementation of Floating-Point Arithmetic....Pages 269-320
Software Implementation of Floating-Point Arithmetic....Pages 321-372
Front Matter....Pages 373-373
Evaluating Floating-Point Elementary Functions....Pages 375-404
Solving the Table Maker’s Dilemma....Pages 405-459
Front Matter....Pages 461-461
Formalisms for Certifying Floating-Point Algorithms....Pages 463-491
Extending the Precision....Pages 493-516
Front Matter....Pages 517-517
Conclusion and Perspectives....Pages 519-519
Appendix: Number Theory Tools for Floating-Point Arithmetic....Pages 521-528
Back Matter....Pages 529-572

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Floating-point arithmetic is by far the most widely used way of implementing real-number arithmetic on modern computers. Although the basic principles of floating-point arithmetic can be explained in a short amount of time, making such an arithmetic reliable and portable, yet fast, is a very difficult task. From the 1960s to the early 1980s, many different arithmetics were developed, but their implementation varied widely from one machine to another, making it difficult for nonexperts to design, learn, and use the required algorithms. As a result, floating-point arithmetic is far from being exploited to its full potential.

This handbook aims to provide a complete overview of modern floating-point arithmetic, including a detailed treatment of the newly revised (IEEE 754-2008) standard for floating-point arithmetic. Presented throughout are algorithms for implementing floating-point arithmetic as well as algorithms that use floating-point arithmetic. So that the techniques presented can be put directly into practice in actual coding or design, they are illustrated, whenever possible, by a corresponding program.

Key topics and features include:

* Presentation of the history and basic concepts of floating-point arithmetic and various aspects of the past and current standards

* Development of smart and nontrivial algorithms, and algorithmic possibilities induced by the availability of a fused multiply-add (fma) instruction, e.g., correctly rounded software division and square roots

* Implementation of floating-point arithmetic, either in software—on an integer processor—or hardware, and a discussion of issues related to compilers and languages

* Coverage of several recent advances related to elementary functions: correct rounding of these functions and computation of very accurate approximations under constraints

* Extensions of floating-point arithmetic such as certification, verification, and big precision

Handbook of Floating-Point Arithmetic is designed for programmers of numerical applications, compiler designers, programmers of floating-point algorithms, designers of arithmetic operators, and more generally, students and researchers in numerical analysis who wish to better understand a tool used in their daily work and research.
 

Content:
Front Matter....Pages i-xxiii
Front Matter....Pages 1-1
Introduction....Pages 3-12
Definitions and Basic Notions....Pages 13-53
Floating-Point Formats and Environment....Pages 55-116
Front Matter....Pages 117-117
Basic Properties and Algorithms....Pages 119-150
The Fused Multiply-Add Instruction....Pages 151-179
Enhanced Floating-Point Sums, Dot Products, and Polynomial Values....Pages 181-204
Languages and Compilers....Pages 205-235
Front Matter....Pages 237-237
Algorithms for the Five Basic Operations....Pages 239-267
Hardware Implementation of Floating-Point Arithmetic....Pages 269-320
Software Implementation of Floating-Point Arithmetic....Pages 321-372
Front Matter....Pages 373-373
Evaluating Floating-Point Elementary Functions....Pages 375-404
Solving the Table Maker’s Dilemma....Pages 405-459
Front Matter....Pages 461-461
Formalisms for Certifying Floating-Point Algorithms....Pages 463-491
Extending the Precision....Pages 493-516
Front Matter....Pages 517-517
Conclusion and Perspectives....Pages 519-519
Appendix: Number Theory Tools for Floating-Point Arithmetic....Pages 521-528
Back Matter....Pages 529-572
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