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The Classical Relative Error Bounds for Computing $\sqrt(a^2 + b^2)$ and $c/\sqrt(a^2 + b^2)$ in Binary Floating-Point Arithmetic are Asymptotically Optimal

Claude-Pierre Jeannerod 1 Jean-Michel Muller 1 Antoine Plet 1
1 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : We study the accuracy of classical algorithms for evaluating expressions of the form √a2 + b2 and c/ √a2 + b2 in radix-2, precision-p floating-point arithmetic, assuming that the elementary arithmetic operations ±, ×, /, √ are rounded to nearest, and assuming an unbounded exponent range. Classical analyses show that the relative error is bounded by 2u + O(u2 ) for √a2 + b2 , and by 3u + O(u2 ) for c/ √a2 + b2 , where u = 2−p is the unit round off. Recently, it was observed that for √a2 + b2 the O(u2 ) term is in fact not needed [1]. We show here that it is not needed either for c/√a2 + b2 . Furthermore, we show that these error bounds are asymptotically optimal. Finally, we show that both the bounds and their asymptotic optimality remain valid when an FMA instruction is used to evaluate a2 + b2 .
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Claude-Pierre Jeannerod, Jean-Michel Muller, Antoine Plet. The Classical Relative Error Bounds for Computing $\sqrt(a^2 + b^2)$ and $c/\sqrt(a^2 + b^2)$ in Binary Floating-Point Arithmetic are Asymptotically Optimal. ARITH-24 2017 - 24th IEEE Symposium on Computer Arithmetic, Jul 2017, London, United Kingdom. pp.8. ⟨ensl-01527202⟩

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