Formally Verified Rounding Errors of the Logarithm-Sum-Exponential Function

We study the numerical accuracy of some specific computer program performing numerical computations. Such a numerical accuracy is expressed in terms of a bound on the difference between the floating-point computation and the corresponding rounding-free computation using mathematical real numbers. We do not only seek to discover such a bound “on paper” but we aim at obtaining computer-assisted formal proofs that this bound is correct for any possible inputs. The function we study comes from the domain of machine learning: a function computing the logarithm of the sum of exponentials of a sequence. The bound obtained is an original result, parameterized by the error bounds of the underlying implementations of the logarithm and exponential functions. The methodology we follow to conduct our formal proofs is also original, using a combination of the Why3 environment for deductive verification, an original modelling of floating-point computations using unbounded numbers, and the J³ environment for proving properties on C source code.