A Radix-Independent Error Analysis of the Cornea-Harrison-Tang Method

International audienceAssuming 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 studiesis 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 bythe radix parity andthe 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 occurand under some mild assumptions on $p$ satisfied by IEEE 754-2008 formats