A new model for on-line arithmetic with an application to the reciprocal calculation

Abstract

A new model for on-line arithmetic functions is presented. It differs from the current models that in each iteration step the exact function value is used to generate the on-line digits. From this model the on-line properties of a large class of arithmetic functions can be determined. In general, the on-line properties of an arithmetic function derived from the model cannot be improved by any iterative approximation algorithm for that arithmetic function. The on-line properties of a number of standard arithmetic functions are given. It is shown that the above model of on-line computation can be easily implemented by means of a table look-up system. Furthermore, a table implementation can be used to start the on-line computation of an iterative approximation method. This is shown by an example, the reciprocal calculation, where the combination of a seed table and an adapted Newton-Raphson iteration method leads to a system with a low on-line delay and fast cycle times. The algorithm works for normalized, quasi-normalized, and pseudo-normalized numbers and can therefore be applied to chained on-line computations.