FastTwoSum revisited

The FastTwoSum algorithm is a classical way to evaluate the rounding error that occurs when adding two numbers in finite precision arithmetic. Starting with Dekker in the early 1970s, numerous floating-point analyses have been made of this algorithm, that are aimed at identifying sufficient conditions for the error to be computed exactly and, otherwise, at quantifying the quality of the error estimate thus produced. In this paper we revisit these two aspects of FastTwoSum. We first provide new, less restrictive conditions for exactness, and show that FastTwoSum performs an error-free transform in more general situations than those found so far in the literature. Second, when exactness cannot be guaranteed we give several error analyses of the output of FastTwoSum and show that the bounds obtained are tight. In particular, this provides further insight into how the algorithm behaves when roundings other than ‘to nearest’ are used, or when the operands are reversed.