Exponential sums and correctly-rounded functions

Nicolas Brisebarre (1, 2) , Guillaume Hanrot (1, 2) , Olivier Robert (3)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 ARIC - Arithmetic and Computing
3 ICJ - Institut Camille Jordan

The 2008 revision of the IEEE-754 standard, which rules floating-point arithmetic, recommends that a certain set of elementary functions should be correctly rounded. Successful attempts for solving the Table Maker's Dilemma in binary64 made it possible to design CRlibm, a library which offers correctly rounded evaluation in binary64 of some functions of the usual libm. It evaluates functions using a two step strategy, which relies on a heuristic that is well spread in the community of mathematical functions designers. This heuristic associates a probability to the occurrence of some runs of zeros or ones after the rounding bit of the value of the function at a given floating-point number. The goal of this paper is to change, when possible, this heuristic into a rigorous statement. We use so-called exponential sums techniques, a tool from analytic number theory, to achieve this goal.

Mots clés: Computer arithmetic; Correct rounding; Elementary function; Exponential sums; Function evaluation; Ziv strategy

Domaines: Arithmétique des ordinateurs

Nicolas Brisebarre, Guillaume Hanrot, Olivier Robert. Exponential sums and correctly-rounded functions. 2016. ⟨hal-01396027v1⟩