  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.

Numerical Analysis;Numerical Analysis