research-article

Formalization of Double-Word Arithmetic, and Comments on “Tight and Rigorous Error Bounds for Basic Building Blocks of Double-Word Arithmetic”

Authors:

Jean-Michel Muller,

Laurence RideauAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 48, Issue 1

Article No.: 9, Pages 1 - 24

https://doi.org/10.1145/3484514

Published: 16 February 2022 Publication History

Abstract

Recently, a complete set of algorithms for manipulating double-word numbers (some classical, some new) was analyzed [16]. We have formally proven all the theorems given in that article, using the Coq proof assistant. The formal proof work led us to: (i) locate mistakes in some of the original paper proofs (mistakes that, however, do not hinder the validity of the algorithms), (ii) significantly improve some error bounds, and (iii) generalize some results by showing that they are still valid if we slightly change the rounding mode. The consequence is that the algorithms presented in [16] can be used with high confidence, and that some of them are even more accurate than what was believed before. This illustrates what formal proof can bring to computer arithmetic: beyond mere (yet extremely useful) verification, correction, and consolidation of already known results, it can help to find new properties. All our formal proofs are freely available.

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Cited By

Appel AKellison ATimany ATraytel DPientka BBlazy S(2024)VCFloat2: Floating-Point Error Analysis in CoqProceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3636501.3636953(14-29)Online publication date: 9-Jan-2024
https://dl.acm.org/doi/10.1145/3636501.3636953
Hubrecht TJeannerod CMuller J(2024)Useful applications of correctly-rounded operators of the form ab + cd + e2024 IEEE 31st Symposium on Computer Arithmetic (ARITH)10.1109/ARITH61463.2024.00015(32-39)Online publication date: 10-Jun-2024
https://doi.org/10.1109/ARITH61463.2024.00015
Lefèvre VLouvet NMuller JPicot JRideau L(2023)Accurate Calculation of Euclidean Norms Using Double-word ArithmeticACM Transactions on Mathematical Software10.1145/356867249:1(1-34)Online publication date: 21-Mar-2023
https://dl.acm.org/doi/10.1145/3568672
Show More Cited By

Index Terms

Formalization of Double-Word Arithmetic, and Comments on “Tight and Rigorous Error Bounds for Basic Building Blocks of Double-Word Arithmetic”
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Arbitrary-precision arithmetic
2. Theory of computation
  1. Design and analysis of algorithms

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 48, Issue 1

March 2022

320 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/3505199

Editors:
Zhaojun Bai
University of California at Davis, USA
,
Wolfgang Bangerth
Colorado State University, USA

Issue’s Table of Contents

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 16 February 2022

Accepted: 01 August 2021

Revised: 01 May 2021

Received: 01 October 2020

Published in TOMS Volume 48, Issue 1

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Cited By

Appel AKellison ATimany ATraytel DPientka BBlazy S(2024)VCFloat2: Floating-Point Error Analysis in CoqProceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3636501.3636953(14-29)Online publication date: 9-Jan-2024
https://dl.acm.org/doi/10.1145/3636501.3636953
Hubrecht TJeannerod CMuller J(2024)Useful applications of correctly-rounded operators of the form ab + cd + e2024 IEEE 31st Symposium on Computer Arithmetic (ARITH)10.1109/ARITH61463.2024.00015(32-39)Online publication date: 10-Jun-2024
https://doi.org/10.1109/ARITH61463.2024.00015
Lefèvre VLouvet NMuller JPicot JRideau L(2023)Accurate Calculation of Euclidean Norms Using Double-word ArithmeticACM Transactions on Mathematical Software10.1145/356867249:1(1-34)Online publication date: 21-Mar-2023
https://dl.acm.org/doi/10.1145/3568672
Qu YXu JZhou BHao JLi FZhang Z(2023)SCR-LIBM: A Correctly Rounded Elementary Function Library in Double-PrecisionInternational Journal of Software Engineering and Knowledge Engineering10.1142/S021819402350067534:04(675-703)Online publication date: 7-Dec-2023
https://doi.org/10.1142/S0218194023500675

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