Error bounds on complex floating-point multiplication

Abstract:

Given floating-point arithmetic with $t$-digit base-$\beta$ significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values $z_0$ and $z_1$ can be computed with maximum absolute error $|z_0\|z_1| \frac {1}{2} \beta ^{1 - t} \sqrt {5}$. In particular, this provides relative error bounds of $2^{-24} \sqrt {5}$ and $2^{-53} \sqrt {5}$ for IEEE 754 single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for IEEE 754 single and double precision arithmetic.