On the relative error of computing complex square roots in floating-point arithmetic

We study the accuracy of a classical approach to computing complex square-roots in floating-point arithmetic. Our analyses are done in binary floating-point arithmetic in precision p, and we assume that the (real) arithmetic operations +, —, x, ÷, √ are rounded to nearest, so the unit roundoff is u = 2−p. We show that in the absence of underflow and overflow, the componentwise and normwise relative errors of this approach are at most 7/2u and √37/2u, respectively, and this without having to neglect terms of higher order in u. We then provide some input examples showing that these bounds are reasonably sharp for the three basic binary interchange formats (binary32, binary64, and binary128) of the IEEE 754 standard for floating-point arithmetic.