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How to read floating point numbers accurately

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William D. ClingerAuthors Info & Claims

PLDI '90: Proceedings of the ACM SIGPLAN 1990 conference on Programming language design and implementation

Pages 92 - 101

https://doi.org/10.1145/93542.93557

Published: 01 June 1990 Publication History

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Abstract

Consider the problem of converting decimal scientific notation for a number into the best binary floating point approximation to that number, for some fixed precision. This problem cannot be solved using arithmetic of any fixed precision. Hence the IEEE Standard for Binary Floating-Point Arithmetic does not require the result of such a conversion to be the best approximation.

This paper presents an efficient algorithm that always finds the best approximation. The algorithm uses a few extra bits of precision to compute an IEEE-conforming approximation while testing an intermediate result to determine whether the approximation could be other than the best. If the approximation might not be the best, then the best approximation is determined by a few simple operations on multiple-precision integers, where the precision is determined by the input. When using 64 bits of precision to compute IEEE double precision results, the algorithm avoids higher-precision arithmetic over 99% of the time.

The input problem considered by this paper is the inverse of an output problem considered by Steele and White: Given a binary floating point number, print a correctly rounded decimal representation of it using the smallest number of digits that will allow the number to be read without loss of accuracy. The Steele and White algorithm assumes that the input problem is solved; an imperfect solution to the input problem, as allowed by the IEEE standard and ubiquitous in current practice, defeats the purpose of their algorithm.

References

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Clinger, William, and Jonathan Rees {editors}. Revisedn report on the algorithmic language Scheme. Technical Report CIS-TR-90-02, Department of Computer and Information Science, University of Oregon, 1990.

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Cited By

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Watson DWatson D(2017)Lexical AnalysisA Practical Approach to Compiler Construction10.1007/978-3-319-52789-5_3(37-73)Online publication date: 23-Mar-2017
https://doi.org/10.1007/978-3-319-52789-5_3
Boldo SJourdan JLeroy XMelquiond G(2015)Verified Compilation of Floating-Point ComputationsJournal of Automated Reasoning10.1007/s10817-014-9317-x54:2(135-163)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1007/s10817-014-9317-x
Boldo SJourdan JLeroy XMelquiond G(2013)A Formally-Verified C Compiler Supporting Floating-Point ArithmeticProceedings of the 2013 IEEE 21st Symposium on Computer Arithmetic10.1109/ARITH.2013.30(107-115)Online publication date: 7-Apr-2013
https://dl.acm.org/doi/10.1109/ARITH.2013.30
Show More Cited By

Index Terms

How to read floating point numbers accurately
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
      1. Approximation
    2. Numerical analysis
      1. Arbitrary-precision arithmetic
      2. Interval arithmetic
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

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Published In

PLDI '90: Proceedings of the ACM SIGPLAN 1990 conference on Programming language design and implementation

June 1990

351 pages

ISBN:0897913647

DOI:10.1145/93542

Chairman:
Bernard N. Fischer

ACM SIGPLAN Notices Volume 25, Issue 6
Jun. 1990
343 pages
ISSN:0362-1340
EISSN:1558-1160
DOI:10.1145/93548
Editor:
Marina Chen
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: 01 June 1990

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Cited By

View all

Watson DWatson D(2017)Lexical AnalysisA Practical Approach to Compiler Construction10.1007/978-3-319-52789-5_3(37-73)Online publication date: 23-Mar-2017
https://doi.org/10.1007/978-3-319-52789-5_3
Boldo SJourdan JLeroy XMelquiond G(2015)Verified Compilation of Floating-Point ComputationsJournal of Automated Reasoning10.1007/s10817-014-9317-x54:2(135-163)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1007/s10817-014-9317-x
Boldo SJourdan JLeroy XMelquiond G(2013)A Formally-Verified C Compiler Supporting Floating-Point ArithmeticProceedings of the 2013 IEEE 21st Symposium on Computer Arithmetic10.1109/ARITH.2013.30(107-115)Online publication date: 7-Apr-2013
https://dl.acm.org/doi/10.1109/ARITH.2013.30
Sperber MDybvig RFlatt MVan straaten AFindler RMatthews J(2009)Revised6 report on the algorithmic language schemeJournal of Functional Programming10.1017/S095679680999007419:S1(1-301)Online publication date: 1-Aug-2009
https://dl.acm.org/doi/10.1017/S0956796809990074
Monniaux D(2008)The pitfalls of verifying floating-point computationsACM Transactions on Programming Languages and Systems10.1145/1353445.135344630:3(1-41)Online publication date: 21-May-2008
https://dl.acm.org/doi/10.1145/1353445.1353446
Fousse LHanrot GLefèvre VPélissier PZimmermann P(2007)MPFRACM Transactions on Mathematical Software10.1145/1236463.123646833:2(13-es)Online publication date: 1-Jun-2007
https://dl.acm.org/doi/10.1145/1236463.1236468
Nikmehr HPhillips BCheng-Chew Lim (2006)Fast Decimal Floating-Point DivisionIEEE Transactions on Very Large Scale Integration (VLSI) Systems10.1109/TVLSI.2006.88404714:9(951-961)Online publication date: Sep-2006
https://doi.org/10.1109/TVLSI.2006.884047
Jacobi CBerg C(2005)Formal Verification of the VAMP Floating Point UnitFormal Methods in System Design10.1007/s10703-005-1613-y26:3(227-266)Online publication date: 1-May-2005
https://dl.acm.org/doi/10.1007/s10703-005-1613-y
Steele GWhite J(2004)How to print floating-point numbers accuratelyACM SIGPLAN Notices10.1145/989393.98943139:4(372-389)Online publication date: 1-Apr-2004
https://dl.acm.org/doi/10.1145/989393.989431
Quach NTakagi NFlynn M(2004)Systematic IEEE rounding method for high-speed floating-point multipliersIEEE Transactions on Very Large Scale Integration (VLSI) Systems10.1109/TVLSI.2004.82586012:5(511-521)Online publication date: 1-May-2004
https://dl.acm.org/doi/10.1109/TVLSI.2004.825860
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