At the present time, IEEE 64-bit floating-point arithmetic is sufficiently accurate for most sci-entific applications. However, for a rapidly growing body of important scientific computing ap-plications, a higher level of numeric precision is required. Such calculations are facilitated by high-precision software packages that include high-level language translation modules to min-imize the conversion effort. This paper presents a survey of recent applications of these tech-niques and provides some analysis of their numerical requirements. These applications include supernova simulations, climate modeling, planetary orbit calculations, Coulomb n-body atomic systems, scattering amplitudes of quarks, gluons and bosons, nonlinear oscillator theory, Ising theory, quantum field theory and experimental mathematics. We conclude that high-precision arithmetic facilities are now an indispensable component of a modern large-scale scientific com-puting environment.

Generating certified and efficient numerical codes requires information ranging from the mathematical level to the representation of numbers. Even though the mathematical semantics can be expressed using the content part of MathML, this language does not encompass the implementation on computers. Indeed various arithmetics may be involved, like floating-point or fixed-point, in fixed precision or arbitrary precision, and current tools do not handle all of these. Therefore we propose in this paper LEMA (Langage pour les Expressions Mathématiques Annotées), a descriptive language based on MathML with additional expressiveness. LEMA will be used during the automatic generation of certified numerical codes. Such a generation process typically involves several steps, and LEMA would thus act as a glue

c t i v it y e p o r t 2009 Table of contents

Abstract. We state and analyze a generalization of the “truncation trick ” suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of D-finite functions (i.e., functions described as solutions of linear differential equations with polynomial coefficients) may be computed with error bounded by 2 −p in timeO(p(lgp) 3+o(1) ) and spaceO(p). The standard fast algorithm for this task, due to Chudnovsky and Chudnovsky, achieves the same time complexity bound but requires Θ(p lgp) bits of memory. 1.