The relative accuracy of $(x+y)*(x-y)$
Résumé
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.
Origine | Fichiers produits par l'(les) auteur(s) |
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