Fixed-Point Implementations of the Reciprocal, Square Root and Reciprocal Square Root Functions

Implementations of the reciprocal, square root and reciprocal square root often share a common structure. This article is a survey and comparison of methods (with only slight variations for the three cases) for computing these functions. The comparisons are made in the context of the same accuracy target (faithful rounding) and of an arbitrary fixed-point format for the inputs and outputs (precisions of up to 32 bits). Some of the methods discussed might require some form of range reduction, depending on the input's range. The objective of the article is to optimize the use of fixed-size FPGA resources (block multipliers and block RAMs). The discussions and conclusions are based on synthesis results for FPGAs. They try to suggest the best method to compute the previously mentioned fixed-point functions on a FPGA, given the input precision. This work compares classical methods (direct tabulation, multipartite tables, piecewise polynomials, Taylor-based polynomials, Newton-Raphson iterations). It also studies methods that are novel in this context: the Halley method and, more generally, the Householder method