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On Ziv's rounding test

Authors:

Florent de Dinechin,

Christoph Lauter,

Jean-Michel Muller,

Serge TorresAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 39, Issue 4

Article No.: 25, Pages 1 - 19

https://doi.org/10.1145/2491491.2491495

Published: 23 July 2013 Publication History

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Abstract

A very simple test, introduced by Ziv, allows one to determine if an approximation to the value f(x) of an elementary function at a given point x suffices to return the floating-point number nearest f(x). The same test may be used when implementing floating-point operations with input and output operands of different formats, using arithmetic operators tailored for manipulating operands of the same format. That test depends on a “magic constant” e. We show how to choose that constant e to make the test reliable and efficient. Various cases are considered, depending on the availability of an fma instruction, and on the range of f(x).

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Cited By

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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
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
Jeangoudoux CLauter C(2018)A Correctly Rounded Mixed-Radix Fused-Multiply-Add2018 IEEE 25th Symposium on Computer Arithmetic (ARITH)10.1109/ARITH.2018.8464818(21-28)Online publication date: Jun-2018
https://doi.org/10.1109/ARITH.2018.8464818
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Index Terms

On Ziv's rounding test
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Arbitrary-precision arithmetic
      2. Interval arithmetic

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Reviewer: Kai Diethelm

When working in classical finite precision arithmetic, one is often faced with the following problem: the result of a computation cannot be represented exactly. In such cases, one usually has the goal of constructing the computation algorithm so that the output is the floating-point number nearest to the exact result, where "nearest" is determined according to the rounding mode selected by the user. Ziv developed a now-classical test that can be used to find out whether or not a given approximate solution to a calculation has this desirable property [1]. An interesting extension of Ziv's approach arises when the algorithm's input values are assumed to be of a higher precision than the output data. A crucial parameter in all variants of Ziv's test is a so-called "magic constant" that needs to be chosen by the user in a very careful way. If it is too small, then the test will be worthless because it can produce falsely positive results. In other words, incorrectly rounded values will not be recognized as wrong and thus will be accepted. This is an outcome that should certainly be avoided. On the other hand, a too-large value of the magic constant will lead to a high probability of false negatives, meaning correct results will not be recognized as such. In those cases, the user will be unnecessarily forced to repeat his or her computation with smaller tolerances, thus increasing the computational cost. In this paper, de Dinechin et al. provide a detailed investigation of the properties of the magic constant and derive a method for choosing this parameter in a suitable way. Moreover, they demonstrate that their ideas lead to an almost optimal choice of the magic constant. Online Computing Reviews Service

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cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 39, Issue 4

July 2013

163 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/2491491

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

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Publication History

Published: 23 July 2013

Accepted: 01 January 2013

Received: 01 May 2012

Published in TOMS Volume 39, Issue 4

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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
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
Jeangoudoux CLauter C(2018)A Correctly Rounded Mixed-Radix Fused-Multiply-Add2018 IEEE 25th Symposium on Computer Arithmetic (ARITH)10.1109/ARITH.2018.8464818(21-28)Online publication date: Jun-2018
https://doi.org/10.1109/ARITH.2018.8464818
Muller JBrunie Nde Dinechin FJeannerod CJoldes MLefèvre VMelquiond GRevol NTorres SMuller JBrunie Nde Dinechin FJeannerod CJoldes MLefèvre VMelquiond GRevol NTorres S(2018)Evaluating Floating-Point Elementary FunctionsHandbook of Floating-Point Arithmetic10.1007/978-3-319-76526-6_10(375-433)Online publication date: 3-May-2018
https://doi.org/10.1007/978-3-319-76526-6_10
Maire JBrunie NDinechin FMuller J(2016)Computing floating-point logarithms with fixed-point operations2016 IEEE 23nd Symposium on Computer Arithmetic (ARITH)10.1109/ARITH.2016.24(156-163)Online publication date: Jul-2016
https://doi.org/10.1109/ARITH.2016.24
Lauter C(2016)A new open-source SIMD vector libm fully implemented with high-level scalar C2016 50th Asilomar Conference on Signals, Systems and Computers10.1109/ACSSC.2016.7869070(407-411)Online publication date: Nov-2016
https://doi.org/10.1109/ACSSC.2016.7869070

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MPFR: A multiple-precision binary floating-point library with correct rounding

A Software Implementation of the IEEE 754R Decimal Floating-Point Arithmetic Using the Binary Encoding Format

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MPFR: A multiple-precision binary floating-point library with correct rounding

A Software Implementation of the IEEE 754R Decimal Floating-Point Arithmetic Using the Binary Encoding Format

On the Computation of Correctly Rounded Sums

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