Numerical validation in quadruple precision using stochastic arithmetic

International audienceDiscrete Stochastic Arithmetic (DSA) enables one to estimate rounding errors and to detect numerical instabilities in simulation programs. DSA is implemented in the CADNA library that can analyze the numerical quality of single and double precision programs. In this article, we show how the CADNA library has been improved to enable the estimation of rounding errors in programs using quadruple precision floating-point variables, i.e. having 113-bit mantissa length. Although an implementation of DSA called SAM exists for arbitrary precision programs, a significant performance improvement has been obtained with CADNA compared to SAM for the numerical validation of programs with 113-bit mantissa length variables. This new version of CADNA has been sucessfully used for the control of accuracy in quadruple precision applications, such as a chaotic sequence and the computation of multiple roots of polynomials. We also describe a new version of the PROMISE tool, based on CADNA, that aimed at reducing in numerical programs the number of double precision variable declarations in favor of single precision ones, taking into account a requested accuracy of the results. The new version of PROMISE can now provide type declarations mixing single, double and quadruple precision

Formalization of double-word arithmetic, and comments on "Tight and rigorous error bounds for basic building blocks of double-word arithmetic"

International audienceRecently, a complete set of algorithms for manipulating double-word numbers (some classical, some new) was analyzed [15]. We have formally proven all the theorems given in that paper, using the Coq proof assistant. The formal proof work led us to: i) locate mistakes in some of the original paper proofs (mistakes that, however, do not hinder the validity of the algorithms), ii) significantly improve some error bounds, and iii) generalize some results by showing that they are still valid if we slightly change the rounding mode. The consequence is that the algorithms presented in [15] can be used with high confidence, and that some of them are even more accurate than what was believed before. This illustrates what formal proof can bring to computer arithmetic: beyond mere (yet extremely useful) verification, correction and consolidation of already known results, it can help to find new properties. All our formal proofs are freely available