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Rigorous Estimation of Floating-Point Round-Off Errors with Symbolic Taylor Expansions

Authors:

Alexey Solovyev,

Marek S. Baranowski,

Charles Jacobsen,

Zvonimir Rakamarić,

Ganesh GopalakrishnanAuthors Info & Claims

ACM Transactions on Programming Languages and Systems (TOPLAS), Volume 41, Issue 1

Article No.: 2, Pages 1 - 39

https://doi.org/10.1145/3230733

Published: 11 December 2018 Publication History

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Abstract

Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide very pessimistic overestimates, causing unnecessary verification failure. We have developed a new approach called Symbolic Taylor Expansions that avoids these problems, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. FPTaylor emits per-instance analysis certificates in the form of HOL Light proofs that can be machine checked.

In this article, we present the basic ideas behind Symbolic Taylor Expansions in detail. We also survey as well as thoroughly evaluate six tool families, namely, Gappa (two tool options studied), Fluctuat, PRECiSA, Real2Float, Rosa, and FPTaylor (two tool options studied) on 24 examples, running on the same machine, and taking care to find the best options for running each of these tools. This study demonstrates that FPTaylor estimates round-off errors within much tighter bounds compared to other tools on a significant number of case studies. We also release FPTaylor along with our benchmarks, thus contributing to future studies and tool development in this area.

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Xu JCui MLi FZhang ZYang HZhou BZhao JChristakis MPradel M(2024)Arfa: An Agile Regime-Based Floating-Point Optimization Approach for Rounding ErrorsProceedings of the 33rd ACM SIGSOFT International Symposium on Software Testing and Analysis10.1145/3650212.3680378(1516-1528)Online publication date: 11-Sep-2024
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Index Terms

Rigorous Estimation of Floating-Point Round-Off Errors with Symbolic Taylor Expansions
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
    2. Numerical analysis
2. Software and its engineering
  1. Software notations and tools
    1. Compilers
  2. Software organization and properties
    1. Software functional properties
      1. Formal methods
        Software verification

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cover image ACM Transactions on Programming Languages and Systems

ACM Transactions on Programming Languages and Systems Volume 41, Issue 1

March 2019

235 pages

ISSN:0164-0925

EISSN:1558-4593

DOI:10.1145/3299867

Editor:
Andrew Myers
Cornell University, USA

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Publication History

Published: 11 December 2018

Accepted: 01 June 2018

Revised: 01 March 2018

Received: 01 January 2017

Published in TOPLAS Volume 41, Issue 1

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Rivera JFranchetti FPüschel M(2024)Floating-Point TVPI Abstract DomainProceedings of the ACM on Programming Languages10.1145/36563958:PLDI(442-466)Online publication date: 20-Jun-2024
https://dl.acm.org/doi/10.1145/3656395
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