Numerical validation of compensated algorithms with stochastic arithmetic

Compensated algorithms consist in computing the rounding errors of individual operations and then adding them later on to the computed result. This makes it possible to increase the accuracy of the computed result efficiently. Computing the rounding error of an individual operation is possible through the use of a so-called error-free transformation. In this article, we show that it is possible to validate the result of compensated algorithms using stochastic arithmetic. We study compensated algorithms for summation, dot product and polynomial evaluation. We prove that the use of the random rounding mode inherent to stochastic arithmetic does not change much the accuracy of compensated methods. This is due to the fact that error-free transformations are no more exact but still sufficiently accurate to improve the numerical quality of results.