On the computation of the reciprocal of floating point expansions using an adapted Newton-Raphson iteration

Abstract

Many numerical problems require a higher computing precision than that offered by common floating point (FP) formats. One common way of extending the precision is to represent numbers in a \emph{multiple component} format. With so-called \emph{floating point expansions}, numbers are represented as the unevaluated sum of standard machine precision FP numbers. This format offers the simplicity of using directly available and highly optimized FP operations and is used by multiple-precisions libraries such as Bailey's QD or the analogue Graphics Processing Units tuned version, GQD. In this article we present a new algorithm for computing the reciprocal FP expansion ${a}^{-1}$ of a FP expansion $a$. Our algorithm is based on an adapted Newton-Raphson iteration where we use truncated operations (additions, multiplications) involving FP expansions. The thorough error analysis given shows that our algorithm allows for computations of very accurate quotients. Precisely, after $i\geq0$ iterations, the computed FP expansion $x=x_0+\ldots+x_{2^i-1}$ satisfies the relative error bound: $\abs{\frac{x-a^{-1}}{a^{-1}}}\leq 2^{-2^i(p-3)-1}$, where $p>2$ is the precision of the FP representation used ($p=24$ for single precision and $p=53$ for double precision).