关于浮点运算的可靠相对误差界
On Sound Relative Error Bounds for Floating-Point Arithmetic
摘要：
最先进的静态分析工具用于验证有限精度代码计算数值误差的最坏情况绝对误差范围。然而，这些通常不是准确性的良好估计，因为它们没有考虑计算值的大小。因此，计算相对于值的大小的误差的相对误差是优选的。虽然今天的工具确实报告了相对误差范围，但这些只是通过绝对误差计算的，因此不一定紧密或更有信息性。此外，每当计算值在域的一部分上接近零时，工具根本不报告任何相对误差估计。令人惊讶的是，由今天的工具计算的相对误差范围的质量还没有被系统地研究或报告。在本文中，我们研究如何国家的最先进的静态技术计算声音绝对误差界可以使用，扩展和组合的相对误差的计算。我们在标准基准集上的实验表明，与通过绝对误差直接计算相对误差相比，直接计算相对误差通常是有益的，并且可以提供高达六个数量级的误差估计，即更准确。我们还表明，区间细分，另一种常用的技术，以减少过度近似，有较少的好处时，直接计算相对误差，但它可以帮助减轻相对误差估计接近零的固有问题的影响。
英文摘要：
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.
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