The floating-point representation provides widely-used data types (such as “float” and “double”) for modern numerical software. Numerical errors are inherent due to floating-point’s approximate nature, and pose an important, well-known challenge. It is nontrivial to fix/repair numerical code to reduce numerical errors — it requires eithernumerical expertise(for manual fixing) or high-precisionoracles(for automatic repair); both are difficult requirements. To tackle this challenge, this paper introduces aprincipled dynamic approachthat isfully automatedandoracle-freefor effectively repairing floating-point errors. The key of our approach is the novel notion ofmicro-structurethat characterizes structural patterns of floating-point errors. We leverage micro-structures’ statistical information on floating-point errors to effectively guide repair synthesis and validation. Compared with existing state-of-the-art repair approaches, our work is fully automatic and has the distinctive benefit of not relying on the difficult to obtain high-precision oracles. Evaluation results on 36 commonly-used numerical programs show that our approach is highly efficient and effective: (1) it is able to synthesize repairs instantaneously, and (2) versus the original programs, the repaired programs have orders of magnitude smaller floating-point errors, while having faster runtime performance.

The performance of fault localization techniques is critical to their adoption in practice. This paper reports on an empirical study of a wide range of fault localization techniques on real-world faults. Different from previous studies, this paper (1) considers a wide range of techniques from different families, (2) combines different techniques, and (3) considers the execution time of different techniques. Our results reveal that a combined technique significantly outperforms any individual technique (200 percent increase in faults localized in Top 1), suggesting that combination may be a desirable way to apply fault localization techniques and that future techniques should also be evaluated in the combined setting. Our implementation is publicly available for evaluating and combining fault localization techniques.

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.

The accuracy of the floating-point calculation is critical to many applications and different methods have been proposed around floating-point accuracies, such as detecting the errors in the program, verifying the accuracy of the program, and optimizing the program to produce more accurate results. These approaches need a specification of the program to understand the ideal calculation performed by the program, which is usually approached by interpreting the program in a precision-unspecific way.

It is well-known that using floating-point numbers may inevitably result in inaccurate results and sometimes even cause serious software failures. Safety-critical software often has strict requirements on the upper bound of inaccuracy, and a crucial task in testing is to check whether significant inaccuracies may be produced. The main existing approach to the floating-point inaccuracy problem is error analysis, which produces an upper bound of inaccuracies that may occur. However, a high upper bound does not guarantee the existence of inaccuracy defects, nor does it give developers any concrete test inputs for debugging. In this paper, we propose the first metaheuristic search-based approach to automatically generating test inputs that aim to trigger significant inaccuracies in floating-point programs. Our approach is based on the following two insights: (1) with FPDebug, a recently proposed dynamic analysis approach, we can build a reliable fitness function to guide the search; (2) two main factors -- the scales of exponents and the bit formations of significands -- may have significant impact on the accuracy of the output, but in largely different ways. We have implemented and evaluated our approach over 154 real-world floating-point functions. The results show that our approach can detect significant inaccuracies in the subjects.