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

Jean-Michel Muller (1, 2)

1 ARIC - Arithmetic and Computing
2 LIP - Laboratoire de l'Informatique du Parallélisme

In their book Scientific Computing on The Itanium, Cornea, Harrison and Tang 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 (FP) 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.

Mots clés: Floating-Point arithmetic; Complex multiplication; 2D determinants

Domaines: Autre [cs.OH]

Jean-Michel Muller. On the error of Computing ab + cd using Cornea, Harrison and Tang's method. 2013. ⟨ensl-00862910v1⟩