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

Appendix A provides supplementary material that does not appear in the journal version.International audienceWe 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 $u$ denoting 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 (as the precision tends to $\infty$).In contrast, if the tie-breaking rule is to even, we show that asymptotically optimal bounds are now $2.25u$ for base twoand $2u$ for larger bases, such as base ten. In each case, asymptotic optimality is obtained bythe explicit construction of a certificate, that is, some floating-point input $(x,y)$ parametrized by $u$and such that the error of the associated resultis equivalent to the error bound as $u$ tends to zero.We conclude with comments on how $(x+y)(x-y)$ compares with $x^2$in the presence of floating-point arithmetic,in particular showing cases where the computed value of $(x+y)(x-y)$ exceedsthat of $x^2$