Improving the compensated Horner scheme with a fused multiply and add

Abstract

Several different techniques and softwares intend to improve the accuracy of results computed in a fixed finite precision. Here we focus on a method to improve the accuracy of the polynomial evaluation. It is well known that the use of the Fused Multiply and Add operation available on some microprocessors like Intel Itanium improves slightly the accuracy of the Horner scheme. In this paper, we propose an accurate compensated Horner scheme specially designed to take advantage of the Fused Multiply and Add. We prove that the computed result is as accurate as if computed in twice the working precision. The algorithm we present is fast since it only requires well optimizable floating point operations, performed in the same working precision as the given data.