Tight Error Analysis in Fixed-point Arithmetic

Abstract

We consider the problem of estimating the numerical accuracy of programs with operations in fixed-point arithmetic and variables of arbitrary, mixed precision, and possibly non-deterministic value. By applying a set of parameterised rewrite rules, we transform the relevant fragments of the program under consideration into sequences of operations in integer arithmetic over vectors of bits, thereby reducing the problem as to whether the error enclosures in the initial program can ever exceed a given order of magnitude to simple reachability queries on the transformed program. We describe a possible verification flow and a prototype analyser that implements our technique. We present an experimental evaluation on a particularly complex industrial case study, including a preliminary comparison between bit-level and word-level decision procedures.