Preface to the Third Edition

Since the publication of the second edition of this book, in 2006, several important changes have occurred in the domain of computer arithmetic. First, a new version of the IEEE-754 Standard for Floating-Point Arithmetic was adopted in June 2008. This new version was merged with the previous binary (754) and “radix independent” (854) standards, resolved some ambiguities of the previous release, standardized the fused multiply– add (FMA) instruction, and included new formats (among them, the binary128 format, previously called “quad precision”). An appendix to this new IEEE 754-2008 standard also makes some important recommendations concerning the elementary functions. New tools have been released that make much easier the work of a programmer eager to implement very accurate functions. A typical example is Sollya. Sollya offers, among many interesting features, a certified supremum norm of the difference between a polynomial and a function. It also computes very good polynomial approximations with constraints (such as requiring the coefficients to be exactly representable in a given format). Another example is Gappa, which simplifies the calculation of error bounds for small “straight-line” numerical programs (such as those designed for evaluating elementary functions), and makes it possible to use proof checkers such as Coq for certifying these bounds. FloPoCo is a wonderful tool for implementing floating-point functions on FPGAs. Research in this area is still very active. To cite a few examples: Harrison designed clever techniques for implementing decimal transcendental functions using the binary functions; Chevillard, Harrison, Joldes, and Lauter introduced a new algorithm for computing certified supremum norms of approximation errors; Johansson designed new algorithms for implementing functions in the “medium precision” range; Brunie et al. designed code generators for mathematical functions; several authors (especially de Dinechin) introduced hardware-oriented techniques targeted at FPGA implementation; Brisebarre and colleagues introduced methods for rigorous polynomial approximation.