Numerical Validation of Compensated Summation Algorithms with Stochastic Arithmetic

Compensated summation algorithms are designed to improve the accuracy of ill-conditioned sums. They are based on algorithms, such as FastTwoSum, which are proved to provide, with rounding to nearest, the sum of two floating-point numbers and the associated rounding error. Discrete stochastic arithmetic enables one to estimate rounding error propagation in numerical codes. It requires a random rounding mode which consists in rounding each computed result toward −∞ or +∞ with the same probability. In this paper we analyse the impact of this random rounding mode on compensated summations based on the FastTwoSum algorithm. We show the accuracy improvement obtained using such compensated summations in numerical simulations controlled with discrete stochastic arithmetic.