Issue No. 12 - Dec. (2017 vol. 66)

ISSN: 0018-9340

pp: 2044-2057

Nicolas Brisebarre , Université de Lyon, CNRS, ENS de Lyon, Inria, Laboratoire LIP (UMR 5668), Université Claude-Bernard Lyon 1, Lyon, France

Guillaume Hanrot , Université de Lyon, CNRS, ENS de Lyon, Inria, Laboratoire LIP (UMR 5668), Université Claude-Bernard Lyon 1, Lyon, France

Olivier Robert , INSA Lyon, Centrale Lyon, Université de Lyon, CNRS, Université Claude-Bernard Lyon 1, Université de Saint-Étienne, Institut Camille Jordan (UMR 5208), Saint-Étienne, France

ABSTRACT

The 2008 revision of the IEEE-754 standard, which governs 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 folklore heuristic that is well spread in the community of mathematical functions designers. Under this heuristic, one can compute the distribution of the lengths of runs of zeros/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, whenever possible, this heuristic into a rigorous statement. The underlying mathematical problem amounts to counting integer points in the neighborhood of a curve, which we tackle using so-called exponential sums techniques, a tool from analytic number theory.

INDEX TERMS

Floating point arithmetic, Probabilistic logic, Function approximation, Memory management

CITATION

Nicolas Brisebarre, Guillaume Hanrot, Olivier Robert, "Exponential Sums and Correctly-Rounded Functions",

*IEEE Transactions on Computers*, vol. 66, no. , pp. 2044-2057, Dec. 2017, doi:10.1109/TC.2017.2690850