Fast and correctly rounded logarithms in double-precision

Abstract

This article is a case study in the implementation of a portable, proven and efficient correctly rounded elementary function in double-precision. We describe the methodology used to achieve these goals in the crlibm library. There are two novel aspects to this approach. The first is the proof framework, and in general the techniques used to balance performance and provability. The second is the introduction of processor-specific optimizations to get performance equivalent to the best current mathematical libraries, while trying to minimize the proof work. The implementation of the natural logarithm is detailed to illustrate these questions.Keywords: floating-point, elementary functions, logarithm, correct rounding Résumé Cet article montre comment implémenter une fonctionélémentaire efficace avec arrondi correct prouvé en double-précision. La méthodologie employée dans ce but par la bibliothèque crlibm présente deux aspects novateurs. Le premier concerne la preuve de l'arrondi correct, et plus généralement les techniques employées pour gérer les compromis entre performance et facilité de preuve. Le second est l'utilisation d'optimisations utilisant des caractéristiques les plus avancées des processeurs, ce qui permet d'obtenir une performanceéquivalente aux meilleures implémentations existantes. L'implémentation du logarithme népérien est décrite en détailà titre d'illustration. Mots-clés: virgule flottante, fonctionsélémentaires, logarithme, arrondi correctFast and correctly rounded logarithms in double-precision 1

A new open-source SIMD vector libm fully implemented with high-level scalar C

“…The uncommon range for the significand m between 0.75 and 1.5 requires extra work but allows a catastrophic cancellation to be avoided [7]. It is possible to perform that split of the input x into the exponent E and the significand m without using branches, merely with integer operations on the IEEE754-2008 memory representation, even when targeting the uncommon significand range between 0.75 and 1.5.…”

“…The function log x is defined for all inputs x > 0. On output, it may not produce any underflow nor overflow [7].…”

“…Most approaches for argument reduction found in the literature for the implementation of mathematical functions do use tables [4], [7]. We have to revise the argument reduction techniques.…”

“…The implementation of a mathematical function, such as exp(x), sin(x) or atan(x), in the floating-point environment provided by IEEE754-2008 has been extensively studied and described in the literature [4], [5], [6], [7], [8], [9], [10]. Classically, a mathematical function on a floating-point argument is computed in four or five steps.…”

Generating function approximations at compile time

“…• Gappa (Melquiond): a tool that computes error bounds on FP calculations, and generates formal proofs of these bounds [9]; These tools have been heavily used for building our CR-LIBM library of correctly-rounded elementary functions [4], [6], [7].…”