Error Bounds on Complex Floating-Point Multiplication

Richard Brent Colin Percival Paul Zimmermann 1
1 CACAO - Curves, Algebra, Computer Arithmetic, and so On
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
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 $\abs{z_0} \abs{z_1} \frac{1}{2} \beta^{1 - t} \sqrt{5}$. 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.
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Richard Brent, Colin Percival, Paul Zimmermann. Error Bounds on Complex Floating-Point Multiplication. Mathematics of Computation, American Mathematical Society, 2007, 76, pp.1469-1481. ⟨inria-00120352v2⟩

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