A radix-independent error analysis of the Cornea-Harrison-Tang method

Claude-Pierre Jeannerod 1, 2, *
* Auteur correspondant
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Assuming floating-point arithmetic with a fused multiply-add operation and rounding to nearest, the Cornea-Harrison-Tang method aims to evaluate expressions of the form $ab+cd$ with high relative accuracy. In this paper we provide a rounding error analysis of this method, which unlike previous studies is not restricted to binary floating-point arithmetic but holds for any radix $\beta$. We show first that an asymptotically optimal bound on the relative error of this method is $2u + O(u^2)$, where $u= \frac{1}{2}\beta^{1-p}$ is the unit roundoff in radix $\beta$ and precision $p$. Then we show that the possibility of removing the $O(u^2)$ term from this bound is governed by the radix parity and the tie-breaking strategy used for rounding: if $\beta$ is odd or rounding is \emph{to nearest even}, then the simpler bound $2u$ is obtained, while if $\beta$ is even and rounding is \emph{to nearest away}, then there exist floating-point inputs $a,b,c,d$ that lead to a relative error larger than $2u + \frac{2}{\beta} u^2 - 4u^3$. All these results hold provided underflows and overflows do not occur and under some mild assumptions on $p$ satisfied by IEEE 754-2008 formats.
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ACM Transactions on Mathematical Software, Association for Computing Machinery, 2016, 〈10.1145/2824252〉
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Claude-Pierre Jeannerod. A radix-independent error analysis of the Cornea-Harrison-Tang method. ACM Transactions on Mathematical Software, Association for Computing Machinery, 2016, 〈10.1145/2824252〉. 〈hal-01050021v2〉

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