On sound relative error bounds for floating-point arithmetic

Abstract

Abstract-State-of-the-art static analysis tools for verifying finite-precision code compute worst-case absolute error bounds on numerical errors. These are, however, often not a good estimate of accuracy as they do not take into account the magnitude of the computed values. Relative errors, which compute errors relative to the value's magnitude, are thus preferable. While today's tools do report relative error bounds, these are merely computed via absolute errors and thus not necessarily tight or more informative. Furthermore, whenever the computed value is close to zero on part of the domain, the tools do not report any relative error estimate at all. Surprisingly, the quality of relative error bounds computed by today's tools has not been systematically studied or reported to date.In this paper, we investigate how state-of-the-art static techniques for computing sound absolute error bounds can be used, extended and combined for the computation of relative errors. Our experiments on a standard benchmark set show that computing relative errors directly, as opposed to via absolute errors, is often beneficial and can provide error estimates up to six orders of magnitude tighter, i.e. more accurate. We also show that interval subdivision, another commonly used technique to reduce over-approximations, has less benefit when computing relative errors directly, but it can help to alleviate the effects of the inherent issue of relative error estimates close to zero.

A Verified Certificate Checker for Finite-Precision Error Bounds in Coq and HOL4

“…FloVer furthermore focuses, like most tools, on certifying absolute error bounds. Bounding relative errors is challenging due to the increased complexity as well as due to the issue that often the error is not even well-defined due to an inherent division by zero [22]. Gappa does provide verified relative error support by optimizing a constraint based on Equation 3.…”

“…Gappa does provide verified relative error support by optimizing a constraint based on Equation 3. This approach has been shown to not provide tight bounds once input ranges and expressions become larger [22]. Finally, note that input ranges are also necessary for computing concrete relative error bounds.…”

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

“…These methods compute absolute error bounds, and whenever a relative error can be computed, it is also reported. Daisy also supports a dedicated relative error computation [26] which is often more accurate, but also more expensive. All methods can be combined with interval subdivision, which can provide tighter error bounds at the expense of larger running times.…”

“…For instance, when the range of f (x) includes zero, relative errors are not well defined and this is often the case in practice. For a more thorough discussion, we refer the reader to [26]; the technique is also implemented within Daisy.…”

“…The goal of this effort is not to develop the next even more accurate technique, rather to consolidate existing ones and to provide a solid basis for further research. Other efforts related to Daisy, which have been described elsewhere and which we do not focus on here are the generation and checking of formal certificates [4], relative error computation [26], and mixed-precision tuning [16].…”