Arithmetic algorithms for extended precision using floating-point expansions

Mioara Joldes (1, 2) , Olivier Marty (3) , Jean-Michel Muller (4) , Valentina Popescu (4)

1 LAAS - Laboratoire d'analyse et d'architecture des systèmes
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
3 ENS Cachan - École normale supérieure - Cachan
4 ARIC - Arithmetic and Computing

Many numerical problems require a higher computing precision than the one offered by standard floating-point (FP) formats. One common way of extending the precision is to represent numbers in a multiple component format. By using the so-called floating-point expansions, real numbers are represented as the unevaluated sum of standard machine precision FP numbers. This representation offers the simplicity of using directly available, hardware implemented and highly optimized FP operations and is used by multiple-precision libraries such as Bailey's QD or the analogue Graphics Processing Units (GPU) tuned version, GQD. In this article we revisit algorithms for adding and multiplying FP expansions, then we introduce and prove new algorithms for normalizing, dividing and square rooting of FP expansions. The new method used for computing the reciprocal and the square root of a FP expansion is based on an adapted Newton-Raphson iteration where the intermediate calculations are done using "truncated" operations (additions, multiplications) involving FP expansions. We give here a thorough error analysis showing that it allows very accurate computations.

Mots clés: floating-point expansions; floating-point arithmetic; Newton-Raphson iteration; high precision arithmetic; division; multiple-precision arithmetic; reciprocal; square root

Domaines: Arithmétique des ordinateurs

Mioara Joldes, Olivier Marty, Jean-Michel Muller, Valentina Popescu. Arithmetic algorithms for extended precision using floating-point expansions. [Research Report] LIP - ENS Lyon; LAAS-CNRS; ENS Cachan. 2015. ⟨hal-01111551v1⟩