Perturbing numerical calculations for statistical analysis of floating-point program (in)stability

  • Authors:

  • Enyi Tang;Earl Barr;Xuandong Li;Zhendong Su

  • Affiliations:

  • Nanjing University, Nanjing, China;University of California, Davis, Davis, CA, USA;Nanjing University, Nanjing, China;University of California, Davis, Davis, CA, USA

  • Venue:
  • Proceedings of the 19th international symposium on Software testing and analysis
  • Year:
  • 2010

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Abstract

Writing reliable software is difficult. It becomes even more difficult when writing scientific software involving floating-point numbers. Computers provide numbers with limited precision; when confronted with a real whose precision exceeds that limit, they introduce approximation and error. Numerical analysts have developed sophisticated mathematical techniques for performing error and stability analysis of numerical algorithms. However, these are generally not accessible to application programmers or scientists who often do not have in-depth training in numerical analysis and who thus need more automated techniques to analyze their code. In this paper, we develop a novel, practical technique to help application programmers (or even numerical experts) obtain high-level information regarding the numerical stability and accuracy of their code. Our main insight is that by systematically altering (or perturbing) the underlying numerical calculation, we can uncover potential pitfalls in the numerical code. We propose two complementary perturbations to statistically measure numerical stability: value perturbation and expression perturbation. Value perturbation dynamically replaces the least significant bits of each floating-point value, including intermediate values, with random bits to statistically induce numerical error in the code. Expression perturbation statically changes the numerical expressions in the user program to mathematically equivalent (in the reals, likely not in floating-point numbers), but syntactically different forms. We then compare the executions of these "equivalent" forms to help discover and remedy potential instabilities. Value perturbation can overstate error, while expression perturbation is relatively conservative, so we use value perturbation to generate candidates for expression perturbation. We have implemented our technique, and evaluation results on various programs from the literature and the GNU Scientific Library (GSL) show that our technique is effective and offers a practical alternative for understanding numerical stability in scientific software.