On the error of Computing ab + cd using Cornea, Harrison and Tang's method

Jean-Michel Muller 1, 2
1 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : In their book, Scientific Computing on the Itanium, Cornea et al. [2002] introduce an accurate algorithm for evaluating expressions of the form ab + cd in binary floating-point arithmetic, assuming an FMA instruction is available. They show that if p is the precision of the floating-point format and if u = 2^{−p}, the relative error of the result is of order u. We improve their proof to show that the relative error is bounded by 2u + 7u^2 + 6u^3. Furthermore, by building an example for which the relative error is asymptotically (as p → ∞ or, equivalently, as u → 0) equivalent to 2u, we show that our error bound is asymptotically optimal.
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Jean-Michel Muller. On the error of Computing ab + cd using Cornea, Harrison and Tang's method. ACM Transactions on Mathematical Software, Association for Computing Machinery, 2015, 41 (2), pp.8. ⟨ensl-00862910v2⟩

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