Detecting floating-point errors via atomic conditions

Abstract

This paper tackles the important, difficult problem of detecting program inputs that trigger large floating-point errors in numerical code. It introduces a novel, principled dynamic analysis that leverages the mathematically rigorously analyzed condition numbers for atomic numerical operations, which we call atomic conditions, to effectively guide the search for large floating-point errors. Compared with existing approaches, our work based on atomic conditions has several distinctive benefits: (1) it does not rely on high-precision implementations to act as approximate oracles, which are difficult to obtain in general and computationally costly; and (2) atomic conditions provide accurate, modular search guidance. These benefits in combination lead to a highly effective approach that detects more significant errors in real-world code (e.g., widely-used numerical library functions) and achieves several orders of speedups over the state-of-the-art, thus making error analysis significantly more practical. We expect the methodology and principles behind our approach to benefit other floating-point program analysis tasks such as debugging, repair and synthesis. To facilitate the reproduction of our work, we have made our implementation, evaluation data and results publicly available on GitHub at https://github.com/FP-Analysis/atomic-condition. CCS Concepts: • Software and its engineering → General programming languages; Software testing and debugging.

An approach to generate correctly rounded math libraries for new floating point variants

“…Mathematical rewriting tools are other alternatives to create correctly rounded functions. If the rounding error in the implementation is the root cause of an incorrect result, we can use tools that detect numerical errors to diagnose them [Benz et al 2012;Chowdhary et al 2020;Goubault 2001;Sanchez-Stern et al 2018;Yi et al 2019;Zou et al 2019]. Subsequently, we can rewrite them using tools such as Herbie [Panchekha et al 2015] or Salsa [Damouche and Martel 2018].…”

High performance correctly rounded math libraries for 32-bit floating point representations

“…Further, seminal research on range reduction has made such approximation feasible [2, 12, 43ś 46]. Simultaneously, there are verification efforts to prove bounds for math libraries [21ś23, 27, 41], identify numerical errors with expressions that can be used in the implementation of math libraries [1,11,16,18,40], and repair individual outputs of math libraries [38,48,50].…”

A Two-Phase Approach for Conditional Floating-Point Verification

“…None of the state-ofthe-art static roundoff-error analysis tools [43,33,31,30,60,65,72] work on the whole applications in our benchmark set. Available dynamic analyses for finding large roundoff errors [10,21,77,21,78,44] or special values [38,57,9] also work only on smaller programs (often restricted to kernels). Only the dynamic-analysis tool FPDebug [10] has been shown to scale beyond numerical kernels, but unfortunately the code has not been actively maintained over the years.…”

“…Blackbox testing has been explored to find large roundoff errors by executing a higher-precision version of the program side-by-side [10,21,77]. Recently, whitebox testing has been used for detecting overflows [38], by phrasing the search as a mathematical optimization problem, and large roundoff errors [21,78], by adapting the notion of condition numbers. KLEE-Float [57], FPGen [44] and Ariadne [9] use symbolic execution for finding bugs in floating-point code, including overflows and large precision loss and cancellation.…”

“…On the other hand, there exist dynamic analyses that search for concrete inputs proving the presence of Infinities [38], NaNs or cancellation errors [10,21,78]. We could not apply any of these tools on our example, to a large part because they, too, have been designed for relatively small programs.…”