The relative accuracy of $(x+y)*(x-y)$

Claude-Pierre Jeannerod 1, 2, 3
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : We consider the relative accuracy of evaluating $(x+y)(x-y)$ in IEEE floating-point arithmetic, when $x,y$ are two floating-point numbers and rounding is to nearest. This expression can be used, for example, as an efficient cancellation-free alternative to $x^2-y^2$ and (at least in the absence of underflow and overflow) is well known to have low relative error, namely, at most about $3u$ with $u$ the unit roundoff. In this paper we propose to complement this traditional analysis with a finer-grained one, aimed at improving and assessing the quality of that bound. Specifically, we show that if the tie-breaking rule is "to away" then the bound $3u$ is asymptotically optimal. In contrast, if the tie-breaking rule is "to even", we show that asymptotically optimal bounds are now $2.25u$ for base two and $2u$ for larger bases (such as base ten). In each case, asymptotic optimality is obtained by the explicit construction of a certificate, that is, some floating-point input $(x,y)$ parametrized by $u$ and for which the error of the result is equivalent to the error bound as $u$ tends to zero.
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Preprints, Working Papers, ...
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https://hal.inria.fr/hal-02100500
Contributor : Claude-Pierre Jeannerod <>
Submitted on : Tuesday, April 16, 2019 - 6:58:51 AM
Last modification on : Wednesday, April 17, 2019 - 1:32:25 AM

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Claude-Pierre Jeannerod. The relative accuracy of $(x+y)*(x-y)$. 2019. ⟨hal-02100500⟩

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