New Algorithms for the Approximate Evaluation in Hardware of Binary Logarithms and Elementary Functions

After an analysis of the errors introduced in the approximate computation of the x
²
function (O ≤ x ≤ 1) and its distributions, we find that a parabolic rather than linear fit to log2 (1 + x), (O ≤ x ≤ 1) can be performed in hardware without increasing the number of necessary sums. An improvement, by a factor of about 2.5, in the absolute maximum error can be expected. Full simulation on a digital computer confirms the theoretical analysis. We used partitioning of the range in only two subranges; the resulting hardware is not harder than in piecewise linear approximation. Examples are also included to show the effectiveness of the method for approximation of different functions.