Jean-Michel Muller – författare

Floating-point arithmetic is the most widely used way of implementing real-number arithmetic on modern computers. However, making such an arithmetic reliable and portable, yet fast, is a very difficult task. 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. So that the techniques presented can directly be put into practice in actual coding or design, they are illustrated, whenever possible, by a corresponding program in C.The handbook 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.

The elementary functions (sine, cosine, exponentials, logarithms . . . ) are the most commonly used mathematical functions. Computing them quickly and accurately is a major goal in computer arithmetic. This book gives the theoretical background necessary to understand and/or build algorithms for computing these functions, presents algorithms (hardware-oriented as well as software-oriented), and discusses issues related to the accurate floating-point implementation of these functions. My purpose was not to give "cooking recipes" that allow to implement some given functions on some given floating-point systems, but to provide the reader with the knowledge that is necessary to build, or adapt algorithms to his or her computing environment. When writing this book, I have had in mind two different audiences: specialists, who will have to design floating-point systems (hardware or soft­ ware parts) orto do research on algorithms, and inquiring minds, who just want to know what kind of methods are used to compute the math functions in current computers or pocket calculators. Because of this, the book is in­ tended to be helpful as well for postgraduate and advanced undergraduate students in computer science or applied mathematics as for professionals engaged in the design of algorithms, programs or circuits that implement floating-point arithmetic, or simply for engineers or scientists who want to improve their culture in that domain.

This textbook presents the concepts and tools necessary to understand, build, and implement algorithms for computing elementary functions (e.g., logarithms, exponentials, and the trigonometric functions).  Both hardware- and software-oriented algorithms are included, along with issues related to accurate floating-point implementation.  This third edition has been updated and expanded to incorporate the most recent advances in the field, new elementary function algorithms, and function software.After a preliminary chapter that briefly introduces some fundamental concepts of computer arithmetic, such as floating-point arithmetic and redundant number systems, the text is divided into three main parts.  Part I considers the computation of elementary functions using algorithms based on polynomial or rational approximations and using table-based methods;  the final chapter in this section deals with basic principles of multiple-precision arithmetic.  Part II is devoted to a presentation of “shift-and-add” algorithms (hardware-oriented algorithms that use additions and shifts only).  Issues related to accuracy, including range reduction, preservation of monotonicity, and correct rounding, as well as some examples of implementation are explored in Part III.  Numerous examples of command lines and full programs are provided throughout for various software packages, including Maple, Sollya, and Gappa.  New to this edition are an in-depth overview of the IEEE-754-2008 standard for floating-point arithmetic; a section on using double- and triple-word numbers; a presentation of new tools for designing accurate function software; and a section on the Toom-Cook family of multiplication algorithms.The techniques presented in this book will be of interest to implementers of elementary function libraries or circuits and programmers of numerical applications.  Additionally, graduate and advanced undergraduate students, professionals, and researchers in scientific computing, numerical analysis, software engineering, and computer engineering will find this a useful reference and resource.PRAISE FOR PREVIOUS EDITIONS“[T]his book seems like an essential reference for the experts (which I'm not).  More importantly, this is an interesting book for the curious (which I am).  In this case, you'll probably learn many interesting things from this book.  If you teach numerical analysis or approximation theory, then this book will give you some good examples to discuss in class." — MAA Reviews (Review of Second Edition)"The rich content of ideas sketched or presented in some detail in this book is supplemented by a list of over three hundred references, most of them of 1980 or more recent.  The book also contains some relevant typical programs." — Zentralblatt MATH (Review of Second Edition)“I think that the book will be very valuable to students both in numerical analysis and in computer science.  I found [it to be] well written and containing much interesting material, most of the time disseminated in specialized papers published in specialized journals difficult to find." — Numerical Algorithms (Review of First Edition)

This textbook presents the concepts and tools necessary to understand, build, and implement algorithms for computing elementary functions (e.g., logarithms, exponentials, and the trigonometric functions).  Both hardware- and software-oriented algorithms are included, along with issues related to accurate floating-point implementation.  This third edition has been updated and expanded to incorporate the most recent advances in the field, new elementary function algorithms, and function software.After a preliminary chapter that briefly introduces some fundamental concepts of computer arithmetic, such as floating-point arithmetic and redundant number systems, the text is divided into three main parts.  Part I considers the computation of elementary functions using algorithms based on polynomial or rational approximations and using table-based methods;  the final chapter in this section deals with basic principles of multiple-precision arithmetic.  Part II is devoted to a presentation of “shift-and-add” algorithms (hardware-oriented algorithms that use additions and shifts only).  Issues related to accuracy, including range reduction, preservation of monotonicity, and correct rounding, as well as some examples of implementation are explored in Part III.  Numerous examples of command lines and full programs are provided throughout for various software packages, including Maple, Sollya, and Gappa.  New to this edition are an in-depth overview of the IEEE-754-2008 standard for floating-point arithmetic; a section on using double- and triple-word numbers; a presentation of new tools for designing accurate function software; and a section on the Toom-Cook family of multiplication algorithms.The techniques presented in this book will be of interest to implementers of elementary function libraries or circuits and programmers of numerical applications.  Additionally, graduate and advanced undergraduate students, professionals, and researchers in scientific computing, numerical analysis, software engineering, and computer engineering will find this a useful reference and resource.PRAISE FOR PREVIOUS EDITIONS“[T]his book seems like an essential reference for the experts (which I''m not).  More importantly, this is an interesting book for the curious (which I am).  In this case, you''ll probably learn many interesting things from this book.  If you teach numerical analysis or approximation theory, then this book will give you some good examples to discuss in class." — MAA Reviews (Review of Second Edition)"The rich content of ideas sketched or presented in some detail in this book is supplemented by a list of over three hundred references, most of them of 1980 or more recent.  The book also contains some relevant typical programs." — Zentralblatt MATH (Review of Second Edition)“I think that the book will be very valuable to students both in numerical analysis and in computer science.  I found [it to be] well written and containing much interesting material, most of the time disseminated in specialized papers published in specialized journals difficult to find." — Numerical Algorithms (Review of First Edition)

This handbook is a definitive guide to the effective use of modern floating-point arithmetic, which has considerably evolved, from the frequently inconsistent floating-point number systems of early computing to the recent IEEE 754-2008 standard. Most of computational mathematics depends on floating-point numbers, and understanding their various implementations will allow readers to develop programs specifically tailored for the standard’s technical features. Algorithms for floating-point arithmetic are presented throughout the book and illustrated where possible by example programs which show how these techniques appear in actual coding and design.The volume itself breaks its core topic into four parts: the basic concepts and history of floating-point arithmetic; methods of analyzing floating-point algorithms and optimizing them; implementations of IEEE 754-2008 in hardware and software; and useful extensions to the standard floating-point system, such as interval arithmetic, double- and triple-word arithmetic, operations on complex numbers, and formal verification of floating-point algorithms. This new edition updates chapters to reflect recent changes to programming languages and compilers and the new prevalence of GPUs in recent years. The revisions also add material on fused multiply-add instruction, and methods of extending the floating-point precision. As supercomputing becomes more common, more numerical engineers will need to use number representation to account for trade-offs between various parameters, such as speed, accuracy, and energy consumption. The Handbook of Floating-Point Arithmetic is designed for students and researchers in numerical analysis, programmers of numerical algorithms, compiler designers, and designers of arithmetic operators.

This handbook is a definitive guide to the effective use of modern floating-point arithmetic, which has considerably evolved, from the frequently inconsistent floating-point number systems of early computing to the recent IEEE 754-2008 standard. Most of computational mathematics depends on floating-point numbers, and understanding their various implementations will allow readers to develop programs specifically tailored for the standard’s technical features. Algorithms for floating-point arithmetic are presented throughout the book and illustrated where possible by example programs which show how these techniques appear in actual coding and design.The volume itself breaks its core topic into four parts: the basic concepts and history of floating-point arithmetic; methods of analyzing floating-point algorithms and optimizing them; implementations of IEEE 754-2008 in hardware and software; and useful extensions to the standard floating-point system, such as interval arithmetic, double- and triple-word arithmetic, operations on complex numbers, and formal verification of floating-point algorithms. This new edition updates chapters to reflect recent changes to programming languages and compilers and the new prevalence of GPUs in recent years. The revisions also add material on fused multiply-add instruction, and methods of extending the floating-point precision. As supercomputing becomes more common, more numerical engineers will need to use number representation to account for trade-offs between various parameters, such as speed, accuracy, and energy consumption. The Handbook of Floating-Point Arithmetic is designed for students and researchers in numerical analysis, programmers of numerical algorithms, compiler designers, and designers of arithmetic operators.