A parametric error analysis of Goldschmidt's square-root algorithm

Back in the 1960s Goldschmidt presented a variation of Newton-Raphson iterations that are the basis for a division and a square root algorithm that are well suited for pipelining. The problem in using Goldschmidt's algorithms is to present an error analysis that enables one to save hardware by using just the right amount of precision for intermediate calculations while still providing correct rounding. Previous implementations relied on combining formal proof methods (that span thousands of lines) with millions of test vectors. These techniques yield correct designs but the analysis is hard to follow and is not quite tight. We have previously presented a simple parametric error analysis of Goldschmidt's division algorithm to allow for improved division implementations and parameter optimizations for the choice of the intermediate precisions. In this work we extend our analysis to Goldschmidt's square root algorithm. This analysis sheds more light on the effect of the different parameters on the error of the square root implementations. In addition, we derive error formulae that help determine optimal parameter choices in practical implementation settings.