Automated backward error analysis for numerical code

Fu, Zhoulai; Bai, Zhaojun; Su, Zhendong

doi:10.1145/2814270.2814317

Cited by 14 publications

(3 citation statements)

References 33 publications

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“…Worst-case analysis of roundoff errors has been an active research area with numerous published approaches [12][13][14][15][16]18,22,33,35,37,38,46,47,50]. Our symbolic affine arithmetic used in PAF (Sect.…”

Section: Related Workmentioning

confidence: 99%

Rigorous Roundoff Error Analysis of Probabilistic Floating-Point Computations

Constantinides

Dahlqvist

Rakamarić

et al. 2021

Computer Aided Verification

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We present a detailed study of roundoff errors in probabilistic floating-point computations. We derive closed-form expressions for the distribution of roundoff errors associated with a random variable, and we prove that roundoff errors are generally close to being uncorrelated with their generating distribution. Based on these theoretical advances, we propose a model of IEEE floating-point arithmetic for numerical expressions with probabilistic inputs and an algorithm for evaluating this model. Our algorithm provides rigorous bounds to the output and error distributions of arithmetic expressions over random variables, evaluated in the presence of roundoff errors. It keeps track of complex dependencies between random variables using an SMT solver, and is capable of providing sound but tight probabilistic bounds to roundoff errors using symbolic affine arithmetic. We implemented the algorithm in the PAF tool, and evaluated it on FPBench, a standard benchmark suite for the analysis of roundoff errors. Our evaluation shows that PAF computes tighter bounds than current state-of-the-art on almost all benchmarks.

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Section: Related Workmentioning

confidence: 99%

Rigorous Roundoff Error Analysis of Probabilistic Floating-Point Computations

Constantinides

Dahlqvist

Rakamarić

et al. 2021

Computer Aided Verification

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“…We call △x the backward error B, and let δ = △x/x. We follow the assumption of [Fu et al 2015] that the mathematical function f is smooth in a neighborhood of x and the backward error is small. Then by the Taylor expansion, we have…”

Section: Preliminariesmentioning

confidence: 99%

“…The MCMC algorithm is a sampling method that draws samples from the target (usually unknown) distribution. MCMC has been used to search the maximum backward error [Fu et al 2015] and to achieve high coverage for floating-point code [Fu and Su 2017], and has also been applied in STOKE [Schkufza et al 2014] for stochastic search of floating-point optimization. We configure the MCMC sampling such that it tends to attain the inputs that may trigger maximum floating-point errors with higher probability than the other points.…”

Section: Detecting High Floating-point Errorsmentioning

confidence: 99%

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

Chen

Mao

et al. 2019

Proc. ACM Program. Lang.

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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;

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