Probabilistic Error Analysis For Sequential Summation of Real Floating Point Numbers

01/27/2021

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by   Johnathan Rhyne, et al.

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We derive two probabilistic bounds for the relative forward error in the floating point summation of n real numbers, by representing the roundoffs as independent, zero-mean, bounded random variables. The first probabilistic bound is based on Azuma's concentration inequality, and the second on the Azuma-Hoeffding Martingale. Our numerical experiments illustrate that the probabilistic bounds, with a stringent failure probability of 10^-16, can be 1-2 orders of magnitude tighter than deterministic bounds. We performed the numerical experiments in Julia by summing up to n=10^7 single precision (binary32) floating point numbers, and up to n=10^4 half precision (binary16) floating point numbers. We simulated exact computation with double precision (binary64). The bounds tend to be tighter when all summands have the same sign.