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Some issues related to double rounding

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Abstract

Double rounding is a phenomenon that may occur when different floating-point precisions are available on the same system. Although double rounding is, in general, innocuous, it may change the behavior of some useful small floating-point algorithms. We analyze the potential influence of double rounding on the Fast2Sum and 2Sum algorithms, on some summation algorithms, and Veltkamp’s splitting.

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Notes

  1. The FMA instruction evaluates expressions of the form xy+z with one final rounding only.

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Acknowledgements

We are extremely grateful to the anonymous referees, whose suggestions have been very helpful for revising this paper. Especially, one of them suggested a drastic simplification of the proof of Theorem 4.1.

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Authors and Affiliations

  1. Inria Sophia Antipolis - Méditerranée, Marelle team, 2004 route des Lucioles, BP 93, 06902, Sophia Antipolis Cedex, France

    Érik Martin-Dorel

  2. Inria Saclay–Île-de-France, Toccata team, LRI Lab., CNRS, Bât. 650, Univ. Paris Sud, 91405, Orsay Cedex, France

    Guillaume Melquiond

  3. CNRS, lab. LIP, Inria Aric team, Université de Lyon, 46 Allée d’Italie, 69364, Lyon Cedex 07, France

    Jean-Michel Muller

Authors
  1. Érik Martin-Dorel
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  2. Guillaume Melquiond
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  3. Jean-Michel Muller
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Corresponding author

Correspondence to Jean-Michel Muller.

Additional information

Communicated by Axel Ruhe.

This work is partly supported by the TaMaDi project of the French Agence Nationale de la Recherche.

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Martin-Dorel, É., Melquiond, G. & Muller, JM. Some issues related to double rounding. Bit Numer Math 53, 897–924 (2013). https://doi.org/10.1007/s10543-013-0436-2

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