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 CTN - Combinatoire, théorie des nombres

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.

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

Domaines: Arithmétique des ordinateurs

Nicolas Brisebarre, Guillaume Hanrot, Olivier Robert. Exponential sums and correctly-rounded functions. IEEE Transactions on Computers, 2017, 66 (12), pp.2044-2057. ⟨10.1109/TC.2017.2690850⟩. ⟨hal-01396027v3⟩