Fast correct rounding of elementary functions in double precision using double-extended arithmetic

Florent de Dinechin 1 David Defour 1 Christoph Lauter 2
1 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : This article shows that IEEE-754 double-precision correct rounding of the most common elementary functions (exp/log, trigonometric and hyperbolic) is achievable on current processors using only double-double-extended arithmetic. This allows to improve by several orders of magnitude the worst case performance of a correctly-rounded mathematical library, compared to the current state of the art. This article builds up on previous work by Lefèvre and Muller, who have shown that an intermediate accuracy of up to 158 bits is required for the evaluation of some functions. We show that the practical accuracy required can always be reduced to less than 119 bits, which is easy to obtain using well-known and well-proven techniques of double-double-extended arithmetic. As an example, a prototype implementation of the exponential function on the Itanium has a worst-case time about twice that of the standard, highly optimized libm by Intel, which doesn't offer correct rounding. Such a small performance penalty should allow correct rounding of elementary functions to become the standard.
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Florent de Dinechin, David Defour, Christoph Lauter. Fast correct rounding of elementary functions in double precision using double-extended arithmetic. [Research Report] RR-5137, LIP RR-2004-10, INRIA, LIP. 2004. ⟨inria-00071446⟩

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