On the Computation of Correctly-Rounded Sums

This paper presents a study of some basic blocks needed in the design of floating-point summation algorithms. In particular, we show that among the set of the algorithms with no comparisons performing only floating-point additions/subtractions, the 2Sum algorithm introduced by Knuth is minimal, both in terms of number of operations and depth of the dependency graph. Under reasonable conditions, we also prove that no algorithms performing only round-to-nearest additions/subtractions exist to compute the round-to-nearest sum of at least three floating-point numbers. Starting from an algorithm due to Boldo and Melquiond, we also present new results about the computation of the correctly-rounded sum of three floating-point numbers.