Title: Numerical reproducibility in HPC: issues in interval arithmetic Authors:

Abstract. The problem of numerical reproducibility is the problem of getting the same result when a numerical computation is run several times, whether on the same machine (and then the word repeatability is often preferred) or on different machines. This phenomenon is best exemplified by the sum of n floating-point numbers: as the addition of floating-point numbers is not associa-tive, the result of the sum depends on the order used to perform the additions. For instance, let us consider the computation of −L + L + 1, where L is so large that L + 1 is rounded as L, then (−L + L) + 1 gives 1 in floating-point arithmetic, whereas −L + (L + 1) returns 0. If several cores are available, the addends can be split among the cores and the sum is split into sub-sums which are computed in parallel and then added together. Depending on the number of cores available at run time, the sum can be split into a variable number of sub-sums. Furthermore, the order used for the final reduction (the final sum of the sub-sums) is not deterministic. Thus the result may depend on the run. Numerical reproducibility thus means getting exactly the same result, bit fo