The relative accuracy of (x plus y) * (x-y)

We consider the relative accuracy of evaluating (x + y)(x - y) in IEEE floating-point arithmetic, when x and 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 infinity). 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 such that the error of the associated result is 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) exceeds that of x(2). (C) 2019 Elsevier B.V. All rights reserved.