Convergence guarantee and improvements for a fast hardware exponential and logarithm evaluation scheme

Abstract

In one iteration, Chen's algorithm for evaluating exponentials and logarithms advances by 2 bits on the average, yet may not advance at all. Analysis reveals that the no-advance situation actually paves the way for sizable advance in the next iteration, and the guaranteed advance, after a one iteration overhead, is one bit per iteration. Two new schemes raise the guaranteed advance to 1.5 bits per iteration, after a two-iteration overhead, while maintaining the original requirement of one stored constant per operand bit. Adopting as a figure of merit the following quantity Q = advance per iteration/memory words per operand bit for the steady-state iterations, the new schemes appears to be better than other methods heretofore proposed.