Toward mechanical verification of properties of roundoff error propagation

In this paper we will be concerned with portions of roundoff analysis which can be automated. Conditions are given under which proofs of numerical stability can be performed completely automatically and very economically (in particular, in polynomial time). We also discuss the use of “numerical heuristics” which apply “hill-climbing” methods to functionals measuring contamination from roundoff. In Section 2 we will relate this work to the extensive literature on roundoff error. Two properties of error propagation in straight-line programs are defined in Section 3, and their relationship demonstrated in Theorem 1. The properties are guaranteed to be effectively decidable since they can be formulated in the first-order theory of real-closed fields. In Section 4 we present sufficient conditions for the properties to hold; conditions which can be checked in time bounded by a polynomial in the size of the given straight-line program.