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

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 article, 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 β. We show first that an asymptotically optimal bound on the relative error of this method is 2β u + 2 u 2 /β - 2 u 2 = 2 u + 2/β u 2 + O ( u 3 ), where u = 1/2β 1- p is the unit roundoff in radix β 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 β is odd or rounding is to nearest even , then the simpler bound 2 u is obtained, while if β is even and rounding is to nearest away , then there exist floating-point inputs a , b , c , d that lead to a relative error larger than 2 u + 2/β u 2 — 4 u 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.