Variable Latency Goldschmidt Algorithm Based on a New Rounding Method and a Remainder Estimate

A new variable latency Goldschmidt algorithm is presented. The algorithm is based on a new rounding method for division, square root, and their reciprocals that avoids the conventional remainder calculation in most of cases and improves previous proposals. The rounding decision is taken by checking the least significant bits of the output of the last Goldschmidt iteration without any other transformation. This helps to reduce the number of cases which need the calculation of the remainder. Additionally, we avoid the calculation of the remainder for most of those cases by using a remainder estimate that can be easily obtained from the Goldschmidt iteration. The calculation of the estimate is much simpler and less time consuming than the calculation of the remainder and this contributes to reducing the number of cases which need a large latency. The combination of both techniques allows us to define a variable latency algorithm which needs to compute the remainder in just nine percent of the total number of cases for reciprocal and division and in 12 percent for square root and square root reciprocal.

keywords: Goldschmidt algorithm, division, square root, reciprocal, square root reciprocal, rounding, variable latency, computer arithmetic