On the Componentwise Accuracy of Complex Floating-Point Division with an FMA

Abstract

This paper deals with the accuracy of complex division in radix-two floating-point arithmetic. Assuming that a fused multiply-add (FMA) instruction is available and that no underflow/overflow occurs, we study how to ensure high relative accuracy in the component wise sense. Since this essentially reduces to evaluating accurately three expressions of the form ac+bd, an obvious approach would be to perform three calls to Kahan's compensated algorithm for 2 by 2 determinants. However, in the context of complex division, two of those expressions are such that ac and bd have the same sign, suggesting that cheaper schemes should be used here (since cancellation cannot occur). We first give a detailed accuracy analysis of such schemes for the sum of two nonnegative products, providing not only sharp bounds on both their absolute and relative errors, but also sufficient conditions for the output of one of them to coincide with the output of Kahan's algorithm. By combining Kahan's algorithm with this particular scheme, we then deduce two new division algorithms. Our first algorithm is a straight-line program whose component wise relative error is always at most 5u+13u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> with u the unit round off, we also provide examples of inputs for which the error of this algorithm approaches 5u, thus showing that our upper bound is essentially the best possible. When tests are allowed we show with a second algorithm that the bound above can be further reduced to 4.5u+9u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , and that this improved bound is reasonably sharp.