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)

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Recensione

"There a few classic books on algorithms for computing elementary functions.... These books focused on software implementation using polynomial approximations. Perhaps Muller's book is destined to become a new classic in this subject, but only time will tell.... Muller's book contains few theorems and even fewer proofs. It does contain many numerical examples, complete with Maple code.... In summary, this 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)

"This book is devoted to the computation of elementary functions (such as sine, cosine, tan, exponentials and logarithms) and it is intended for specialists and inquiring minds as the author says in his preface. I also think that the book will be very valuable to students both in numerical analysis and in computer science. The author is well known among people working on computer arithmetic. I found the book well written and containing much interesting material, most of the time disseminated in specialized papers published in specialized journals difficult to find. Moreover, there are very few books on these topics and they are not recent." ―Numerical Algorithms (Review of First Edition)

"This book is intended for two different audiences: specialists, who have to design floating-point systems...or to do research on algorithms, and inquiring minds, who just want to know what kind of methods are used to compute mathematical functions in current computers or pocket calculators. Because of this, it will be helpful for postgraduate and advanced undergraduate students in computer science or applied mathematics as well 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. Much of the book can be understood with only a basic grounding in computer science and mathematics." ―Mathematica Bohemia (Review of First Edition)

"The author presents a state-of-the-art review of techniques used to compute the values of common elementary functions. Chapter 1 introduces the goals of techniques that produce good approximations. Chapter 2 reviews topics in computer arithmetic, including number representation (redundant and nonredundant) and the IEEE standard for binary floating-point arithmetic. Chapters 3 and 4 review the techniques (polynomial, rational, and table-based) used in some current microprocessors. Chapters 5, 6, and 7 review shift-and-add techniques, including the CORDIC method frequently used by calculator designers. Chapter 8 discusses range reduction. Chapter 9 discusses techniques that help produce correctly rounded results." ―Mathematical Reviews (Review of First Edition)

"A must for those involved with designing numerical processors or mathematical software, the book should also interest calculus students for the new perspectives it offers on topics they might think they know very well. Suitable for upper-division undergraduates through faculty." ―Choice (Review of First Edition)

"This fascinating book describes the techniques used by high-level compilers and by pocket book calculators to generate values of the common elementary mathematical functions." ―ASLIB Book Guide (Review of First Edition)

"The author fully accomplishes his aim of giving the necessary theoretical background in order to both understand and build algorithms for the computation of elementary functions (such as sine, cosine, exponential, logarithms), that are the most commonly used mathematical functions. Hardware- as well as software-oriented algorithms are presented, together with a pertinent analysis of accurate floating-point implementations...Good examples are always chosen in order to introduce or to illustrate the methods, following the given cases. The book is very well structured..." ―Analele Stiintifice ale Universitatii “Al. I. Cuza” din Iasi

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Elementary Functions:: Algorithms and Implementation

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