A Branch-and-bound Algorithm to Rigorously Enclose the Round-Off Errors

Abstract

Round-off errors occur each time a program makes use of floating-point computations. They denote the existence of a distance between the actual floating-point computations and the intended computations over the reals. Therefore, analyzing round-off errors is a key issue in the verification of programs with floating-point computations. Most existing tools for round-off error analysis provide an over-approximation of the error. The point is that these approximations are often too coarse to evaluate the effective consequences of the error on the behaviour of a program. Some other tools compute an under-approximation of the maximal error. But these under-approximations are either not rigorous or not reachable. In this paper, we introduce a branch-and-bound algorithm to rigorously enclose the maximal error. Thanks to the use of rational arithmetic, our branch-and-bound algorithm provides a tight upper bound of the maximal error and a lower bound that can be exercised by input values. We outline the advantages and limits of our framework and compare it with state-of-the-art methods. Preliminary experiments on standard benchmarks show promising results.