Report Number: CSL-TR-00-791
Institution: Stanford University, Computer Systems Laboratory
Title: Precision of Semi-Exact Redundant Continued Fraction
Author: Mencer, Oskar
Author: Morf, Martin
Author: Flynn, Michael J.
Date: February 2000
Abstract: Continued fractions (CFs) enable straightforward representation of elementary functions and rational approximations. We improve the positional algebraic algorithm, which computes homographic functions. The improved algorithm for the linear fractional transformation produces exact results, given regular continued fraction input. In case the input is in redundant continued fraction form, our improved linear algorithm increases the percentage of exact results with 12-bit state registers from 78% to 98%. The maximal error of non-exact results is improved. Indeed, by detecting a small number of cases, we can add a final correction step to improve the guaranteed accuracy of non-exact results. We refer to the fact that a few results may not be exact as "Semi-Exact" arithmetic. We detail the adjustments to the positional algebraic algorithm concerning register overflow, the virtual singularities that occur during the computation, and the errors due to non-regular, redundant CF inputs.