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On relative errors of floating-point operations: optimal bounds and applications

Abstract : Rounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function fl and barring underflow and overflow, such models bound the relative errors E 1 (t) = |t − fl(t)|/|t| and E 2 (t) = |t − fl(t)|/|fl(t)| by the unit roundoff u. This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real t and in the case where t is the exact result of an arithmetic operation on some floating-point numbers. We show that E 1 (t) and E 2 (t) are optimally bounded by u/(1 + u) and u, respectively, when t is real or, under mild assumptions on the base and the precision, when t = x ± y or t = xy with x, y two floating-point numbers. We prove that while this remains true for division in base β > 2, smaller, attainable bounds can be derived for both division in base β = 2 and square root. This set of optimal bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain significantly shorter proofs of the best-known error bounds for such algorithms, and/or improvements on these bounds themselves.
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Preprints, Working Papers, ...
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Contributor : Claude-Pierre Jeannerod <>
Submitted on : Wednesday, April 20, 2016 - 6:38:16 PM
Last modification on : Friday, April 20, 2018 - 3:44:26 PM
Long-term archiving on: : Tuesday, November 15, 2016 - 8:00:13 AM


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  • HAL Id : hal-00934443, version 3


Claude-Pierre Jeannerod, Siegfried M. Rump. On relative errors of floating-point operations: optimal bounds and applications. 2016. ⟨hal-00934443v3⟩



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