Efficient automated repair of high floating-point errors in numerical libraries

Abstract

Floating point computation is by nature inexact, and numerical libraries that intensively involve floating-point computations may encounter high floating-point errors. Due to the wide use of numerical libraries, it is highly desired to reduce high floating-point errors in them. Using higher precision will degrade performance and may also introduce extra errors for certain precision-specific operations in numerical libraries. Using mathematical rewriting that mostly focuses on rearranging floating-point expressions or taking Taylor expansions may not fit for reducing high floating-point errors evoked by ill-conditioned problems that are in the nature of the mathematical feature of many numerical programs in numerical libraries. In this paper, we propose a novel approach for efficient automated repair of high floating-point errors in numerical libraries. Our main idea is to make use of the mathematical feature of a numerical program for detecting and reducing high floating-point errors. The key components include a detecting method based on two algorithms for detecting high floating-point errors and a repair method for deriving an approximation of a mathematical function to generate patch to satisfy a given repair criterion. We implement our approach by constructing a new tool called AutoRNP. Our experiments are conducted on 20 numerical programs in GNU Scientific Library (GSL). Experimental results show that our approach can efficiently repair (with 100% accuracy over all randomly sampled points) high floating-point errors for 19 of the 20 numerical programs. CCS Concepts: • Mathematics of computing → Mathematical software; Computations in finite fields; • Theory of computation → Numeric approximation algorithms; • Software and its engineering → Search-based software engineering;

Detecting floating-point errors via atomic conditions

“…The state-of-the-art technique DEMC [Yi et al 2019] uses high-precision resultsf high to guide its search. The dataset for DEMC contains 49 GSL functions that are a subset of the 88 functions that we use as subjects.…”

“…We evaluate Atomu on 88 functions from the popular GNU Scientific Library (GSL) to demonstrate the effectiveness and runtime efficiency of Atomu. We find that Atomu is at least two orders of magnitude faster than state-of-the-art approaches [Yi et al 2019;Zou et al 2015]. When oracles are available (i.e., the same setting as all existing approaches), Atomu can detect significantly more (40%) buggy functions with significant errors with neither false positives nor false negatives on real-world evaluation subjects (see Section 5.3.2).…”

“…GSL is an open-source numerical library that provides a wide range of mathematical routines such as random number generators, special functions and least-squares fitting. GSL has been frequently used as test subjects in previous research [Barr et al [Yi et al 2019] for direct comparisons. Our approach is not limited to univariate functions and can be easily applied to multivariate functions by changing the search module's parameters.…”

“…For this comparison, we use the state-of-the-art DEMC [Yi et al 2019], whose search method consists of a partitioned global search and a fine-grained search guided by the high-precisionf high . We also compare with LSGA [Zou et al 2015], another approach for detecting floating-point errors.…”

“…As mentioned in Section 2.2, relative error is the prevalent measurement for floating-point errors. There is also the measurement of ErrBits, which we have mentioned in Section 5.3 when comparing our approach with DEMC [Yi et al 2019]. ErrCount is defined as the count of floating-point numbers (F) between the ideal f (x) and the floating-point resultf (x), and ErrBits is defined as the logarithm of ErrCount to base 2:…”