Robustness Analysis of Finite Precision Implementations

Abstract

A desirable property of control systems is to be robust to inputs, that is small perturbations of the inputs of a system will cause only small perturbations on its outputs. But it is not clear whether this property is maintained at the implementation level, when two close inputs can lead to very different execution paths. The problem becomes particularly crucial when considering finite precision implementations, where any elementary computation can be affected by a small error. In this context, almost every test is potentially unstable, that is, for a given input, the computed (finite precision) path may differ from the ideal (same computation in real numbers) path. Still, state-of-the-art error analyses do not consider this possibility and rely on the stable test hypothesis, that control flows are identical. If there is a discontinuity between the treatments in the two branches, that is the conditional block is not robust to uncertainties, the error bounds can be unsound. We propose here a new abstract-interpretation based error analysis of finite precision implementations, relying on the analysis of [16] for rounding error propagation in a given path, but which is now made sound in presence of unstable tests. It automatically bounds the discontinuity error coming from the difference between the float and real values when there is a path divergence, and introduces a new error term labeled by the test that introduced this potential discontinuity. This gives a tractable error analysis, implemented in our static analyzer FLUCTUAT: we present results on representative extracts of control programs.

Generating test cases inside suspicious intervals for floating-point number programs

“…A new abstract-interpretation based robustness analysis of finite precision implementations has recently been proposed [9] for sound rounding error propagation in a given path in presence of unstable tests. Brain et al [10,2] have recently introduced a bit-precise decision procedure for the theory of point arithmetic.…”

“…Programs simple_interpolator and simple_square are two benchmarks extracted from [9]: the first benchmark computes an interpolator, affine by sub-intervals; the second is a rewrite of a square root function used in an industrial context. First column gives the name of the program while the second column gives the condition that is checked.…”

“…Abstract interpretation based error analysis [3,6,9] of finite precision implementations compute an over-approximation of the errors due to floating-point operations. Inappropriate behaviors of a program may thus be dreaded but we cannot know whether the predicted unstable behaviors will occur with actual data.…”

Daisy - Framework for Analysis and Optimization of Numerical Programs (Tool Paper)

“…Widening operators compute non-trivial bounds only for very special cases where roundoff errors decrease with each loop iteration. These challenges have been (partially) addressed [16,23], and we plan to include those techniques in Daisy in the future. Nonetheless, conditionals and loops remain open problems.…”

Combining Tools for Optimization and Analysis of Floating-Point Computations

“…Some of these tools use sound techniques to statically bound roundoff errors of straight-line floating-point programs [8,11,21,14,15,7] and partially automate complex analysis tasks [17,9]. Other such tools use dynamic techniques to find inputs that suffer from large rounding errors [3,22].…”