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Fast evaluation of elementary mathematical functions with correctly rounded last bit

Author:

Abraham ZivAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 17, Issue 3

Pages 410 - 423

https://doi.org/10.1145/114697.116813

Published: 01 September 1991 Publication History

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Cited By

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Wang XYi XYu HHuang CPeng L(2024)Parallel Optimization for Accelerating the Generation of Correctly Rounded Elementary FunctionsProceedings of the 53rd International Conference on Parallel Processing10.1145/3673038.3673125(21-31)Online publication date: 12-Aug-2024
https://dl.acm.org/doi/10.1145/3673038.3673125
Aanjaneya MNagarakatte SDubach CBruening DHardekopf B(2023)Fast Polynomial Evaluation for Correctly Rounded Elementary Functions using the RLIBM ApproachProceedings of the 21st ACM/IEEE International Symposium on Code Generation and Optimization10.1145/3579990.3580022(95-107)Online publication date: 17-Feb-2023
https://dl.acm.org/doi/10.1145/3579990.3580022
Hubrecht TJeannerod CZimmermann P(2023)Towards a correctly-rounded and fast power function in binary64 arithmetic2023 IEEE 30th Symposium on Computer Arithmetic (ARITH)10.1109/ARITH58626.2023.00028(111-118)Online publication date: 4-Sep-2023
https://doi.org/10.1109/ARITH58626.2023.00028
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Index Terms

Fast evaluation of elementary mathematical functions with correctly rounded last bit
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
      1. Approximation
    2. Numerical analysis
      1. Numerical differentiation
  2. Mathematical software
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

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Fast Polynomial Evaluation for Correctly Rounded Elementary Functions using the RLIBM Approach
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This paper proposes fast polynomial evaluation methods for correctly rounded elementary functions generated using our RLibm approach. The resulting functions produce correct results for all inputs with multiple representations and rounding modes. Given ...
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Reviews

Reviewer: Heinrich W. Guggenheimer

The variety of rounding rules used for elementary mathematical function routines makes it difficult to use different routines and obtain the same values up to the last bit. The technique proposed here to obtain correct function values (given a fixed rounding rule for the last digit) is to use a fast primary routine (such as Gal's look-up routines) and then to test for indeterminacy of the last digit, requiring that the number differ from any jump point of ROUND( ) by no more than a small e >0 . In the few cases where this condition is violated, a slow, high-accuracy routine can then be used (repeatedly, if necessary). If the last digits are random (a hypothesis certainly violated for functions such as cos x , x<<1 ), then a probabilistic argument shows that in most modern machine implementations, the secondary routine is almost never invoked, and therefore the slowness of the secondary routine will have no effect on expected CPU time.

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 17, Issue 3

Sept. 1991

142 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/114697

Editor:
John R. Rice
Purdue Univ., West Layfayette, IN

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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Publication History

Published: 01 September 1991

Published in TOMS Volume 17, Issue 3

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Wang XYi XYu HHuang CPeng L(2024)Parallel Optimization for Accelerating the Generation of Correctly Rounded Elementary FunctionsProceedings of the 53rd International Conference on Parallel Processing10.1145/3673038.3673125(21-31)Online publication date: 12-Aug-2024
https://dl.acm.org/doi/10.1145/3673038.3673125
Aanjaneya MNagarakatte SDubach CBruening DHardekopf B(2023)Fast Polynomial Evaluation for Correctly Rounded Elementary Functions using the RLIBM ApproachProceedings of the 21st ACM/IEEE International Symposium on Code Generation and Optimization10.1145/3579990.3580022(95-107)Online publication date: 17-Feb-2023
https://dl.acm.org/doi/10.1145/3579990.3580022
Hubrecht TJeannerod CZimmermann P(2023)Towards a correctly-rounded and fast power function in binary64 arithmetic2023 IEEE 30th Symposium on Computer Arithmetic (ARITH)10.1109/ARITH58626.2023.00028(111-118)Online publication date: 4-Sep-2023
https://doi.org/10.1109/ARITH58626.2023.00028
de Dinechin FKumm Mde Dinechin FKumm M(2023)Generalities on Fixed-Point Function ApproximationApplication-Specific Arithmetic10.1007/978-3-031-42808-1_16(477-495)Online publication date: 23-Aug-2023
https://doi.org/10.1007/978-3-031-42808-1_16
Aanjaneya MLim JNagarakatte SJhala RDillig I(2022)Progressive polynomial approximations for fast correctly rounded math librariesProceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation10.1145/3519939.3523447(552-565)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519939.3523447
Lim JNagarakatte S(2022)One polynomial approximation to produce correctly rounded results of an elementary function for multiple representations and rounding modesProceedings of the ACM on Programming Languages10.1145/34986646:POPL(1-28)Online publication date: 12-Jan-2022
https://dl.acm.org/doi/10.1145/3498664
Sibidanov AZimmermann PGlondu S(2022)The CORE-MATH Project2022 IEEE 29th Symposium on Computer Arithmetic (ARITH)10.1109/ARITH54963.2022.00014(26-34)Online publication date: Sep-2022
https://doi.org/10.1109/ARITH54963.2022.00014
Turner JNowotny T(2021)Arpra: An Arbitrary Precision Range Analysis LibraryFrontiers in Neuroinformatics10.3389/fninf.2021.63272915Online publication date: 25-Jun-2021
https://doi.org/10.3389/fninf.2021.632729
Lim JNagarakatte SFreund SYahav E(2021)High performance correctly rounded math libraries for 32-bit floating point representationsProceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation10.1145/3453483.3454049(359-374)Online publication date: 19-Jun-2021
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Lim JAanjaneya MGustafson JNagarakatte S(2021)An approach to generate correctly rounded math libraries for new floating point variantsProceedings of the ACM on Programming Languages10.1145/34343105:POPL(1-30)Online publication date: 4-Jan-2021
https://dl.acm.org/doi/10.1145/3434310
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Fast Polynomial Evaluation for Correctly Rounded Elementary Functions using the RLIBM Approach

Computing correctly rounded integer powers in floating-point arithmetic

On the Computation of Correctly Rounded Sums

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