Section: New Results
Tools for numerically guaranteed computations
Participant : Sylvain Chevillard.
This work has been performed in collaboration with members of the teams Arénaire in Lyon and Caramel in Nancy. The overall and longterm goal is to enhance the quality of numerical computations. Several aspects are studied:

A first topic is the development of software code for the multiprecision evaluation of elementary and special functions. Developing such codes is a long and errorprone task. It is hence relevant to automatically generate such codes whenever possible. A first step has been to design an algorithm that automatically generates multiprecision code for the evaluation of constant expressions with an a priori guaranteed error [25] . This is usually necessary for the evaluation, e.g., of the first terms of a Taylor series.

Another topic consists in the design of algorithms that allow developers of double precision mathematical libraries (socalled libm) to certify their library. In the process of developing a libm, one usually replaces the function $f$ to be evaluated by a good polynomial approximation $p$. In order to certify the quality of the library, it is then necessary to give a rigorous mathematical proof that the relative error $\epsilon =(pf)/f$ between $p$ and $f$ is bounded by a small constant. This turns out to be equivalent to the problem of computing a sharp yet certified upper bound of the supremum norm of $\epsilon $. An efficient algorithm has been designed for this purpose [20] (this work is the publication of a work initially begun in the Arénaire team and continued in the Caramel team).

Finally, a more general endeavor is to develop a tool that helps developers of libms in their task. This is performed by the software Sollya, which has originally been developed in the Arénaire team, in collaboration with C. Lauter and M. Joldeş. A new release has been performed this year [32] .