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Implementation of the reciprocal square root in MPFR

Authors: Paul Zimmermann

Published in: Dagstuhl Seminar Proceedings, Volume 8021, Numerical Validation in Current Hardware Architectures (2008)

Abstract

We describe the implementation of the reciprocal square root --- also called inverse square root --- as a native function in the MPFR library. The difficulty is to implement Newton's iteration for the reciprocal square root on top's of GNU MP's extsc{mpn} layer, while guaranteeing a rigorous $1/2$ ulp bound on the roundoff error.

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Paul Zimmermann. Implementation of the reciprocal square root in MPFR. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{zimmermann:DagSemProc.08021.12,
  author =	{Zimmermann, Paul},
  title =	{{Implementation of the reciprocal square root in MPFR}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--3},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8021},
  editor =	{Annie Cuyt and Walter Kr\"{a}mer and Wolfram Luther and Peter Markstein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.08021.12},
  URN =		{urn:nbn:de:0030-drops-14357},
  doi =		{10.4230/DagSemProc.08021.12},
  annote =	{Keywords: Multiple precision, floating-point, inverse square root, correct rounding, MPFR library}
}

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DOI: 10.4230/DagSemProc.06021.11

Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format

Authors: Vincent Lefèvre, Damien Stehlé, and Paul Zimmermann

Published in: Dagstuhl Seminar Proceedings, Volume 6021, Reliable Implementation of Real Number Algorithms: Theory and Practice (2006)

Abstract

We searched for the worst cases for correct rounding of the exponential function in the IEEE 754r decimal64 format, and computed all the bad cases whose distance from a breakpoint (for all rounding modes) is less than $10^{-15}$,ulp, and we give the worst ones. In particular, the worst case for $|x| geq 3 imes 10^{-11}$ is $exp(9.407822313572878 imes 10^{-2}) = 1.098645682066338,5,0000000000000000,278ldots$. This work can be extended to other elementary functions in the decimal64 format and allows the design of reasonably fast routines that will evaluate these functions with correct rounding, at least in some domains.

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Vincent Lefèvre, Damien Stehlé, and Paul Zimmermann. Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format. In Reliable Implementation of Real Number Algorithms: Theory and Practice. Dagstuhl Seminar Proceedings, Volume 6021, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{lefevre_et_al:DagSemProc.06021.11,
  author =	{Lef\`{e}vre, Vincent and Stehl\'{e}, Damien and Zimmermann, Paul},
  title =	{{Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format}},
  booktitle =	{Reliable Implementation of Real Number Algorithms: Theory and Practice},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6021},
  editor =	{Peter Hertling and Christoph M. Hoffmann and Wolfram Luther and Nathalie Revol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06021.11},
  URN =		{urn:nbn:de:0030-drops-7483},
  doi =		{10.4230/DagSemProc.06021.11},
  annote =	{Keywords: Floating-point arithmetic, decimal arithmetic, table maker's dilemma, correct rounding, elementary functions}
}

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Documents authored by Zimmermann, Paul

Implementation of the reciprocal square root in MPFR

Abstract

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Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format

Abstract

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