Automatic Computation of Exponentials, Logarithms, Ratios and Square Roots

It is shown how a relatively simple device can evaluate exponentials, logarithms, ratios and square roots for fraction arguments, employing only shifts, adds, high-speed table lookups, and bit counting. The scheme is based on the cotransformation of a number pair (x,y) such that the F(x,y) = f(x0) is invariant; when x is driven towards a known value xω, y is driven towards the result. For an N-bit fraction about N/4 iterations are required, each involving two or three adds; then a termination algorithm, based on an add and an abbreviated multiply, completes the process, for a total cost of about one conventional multiply time. Convergence, errors and simulation using APL are discussed.