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Book Review
Digital Arithmetic by Miloš Ercegovac and Tomás Lang, Morgan Kaufmann (Elsevier) 2004, ISBN 1-55860-798-6
Reviewed by Vladimir Botchev (vladimir.botchev@analog.com)
Several years ago, while studying Amos Omondi’s Computer Arithmetic Systems, I thought that perhaps this is the book that would replace the monumental (albeit tiny in number of pages) 1979 work by Kai Hwang, Computer Arithmetic: Principles, Architecture, and Design. Well, despite some really good chapters—especially on elementary functions—I couldn’t decide if it were indeed such a replacement. Perhaps if it were coupled with the extreme clarity of Jean-Michel Muller’s Arithmetique des Ordinateurs, it could have “won”. Since then, many excellent texts on computer arithmetic have been published, but perhaps the best one to date is the present book, Digital Arithmetic, by Professors Miloš Ercegovac and Tomás Lang.
While in many respects this text goes much deeper than Hwang’s book, it is somewhat lacking in breadth. It is true that almost all topics are at least touched on, or extensive literature references are given for them. However a few more (tens of) pages about array dividers/function evaluators or residue-number arithmetic would definitely have enriched the text beyond the Hwang “threshold”. Adding to the positive side, each chapter is provided with excellent exercise sets, with selected ones having solutions that can be found among the on-line appendixes on the book’s very well organized website.
The first chapter introduces number representations and basic arithmetic algorithms, such as addition/subtraction, shifts, multiplication and division. What is immediately apparent is the extremely good notation and crystal clear illustrations that, fortunately, permeate the rest of the book and could almost be thought to be its trademark. Chapter Two is concerned with two-operand adders of about all flavors, including at the end redundant-digit-set adders at a similar level to the treatment in Hwang’s book. Multioperand addition, a very important topic from the point of view of signal-processing hardware, is considered at length in Chapter Three.
Chapter Four provides an extensive study of multiplier circuits, including the truncated multiplier, cherished-by signal-processing hardware designers. Chapters Five and Six develop the digit-recurrence methodology for division and square root; the latter chapter has a section on combining these two operations in one hardware block (a similar combination technique is also performed in universal cellular arrays for division/square root/multiplication, as described in Hwang’s book). Chapter Seven will again be much appreciated by signal-processing hardware designers: it presents division and square root methods that use iterative approximations (in hardware, it amounts to using multipliers that already exist in the design).
Chapter Eight, the longest one, gives extensive treatment to floating-point representations and hardware implementations. On-line arithmetic—very important in terms of tradeoffs between number of interconnections, latency and silicon—is given attention in Chapter Nine. Chapter Ten is about function evaluation by approximations or interpolations and includes a design using on-line arithmetic.
The book ends with an excellent chapter on CORDIC (coordinate-rotation digital-computer) arithmetic and a few flavors of its hardware implementation (perhaps I’m biased here, being a strong proponent of the CORDIC method—even for software implementations, where it is sometimes slower than regular methods). Floating-point CORDIC is also mentioned in this chapter, without further elaboration. Overall, this is an excellent textbook on computer arithmetic essentials, warmly recommended for students and practitioners alike.
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