Exponential sums and correctly-rounded functions
Résumé
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.
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