ERL: an algorithm for fast evaluation of exponential, reciprocal, and logarithmic functions

Abstract

A fast algorithm (ERL) is proposed for waluating Exponential, Reciprocal, and Logarithmic functions. The algorithm requires two to three iterations to complete using simple operations such as multiply, accumulate, and table lookup. The algorithm is independent of the number format used by the machine. Thus it can be implemented using the IEEE 754 ?oating-point standard or any other special format used by special-purpose processors. The dynamic range of the algorithm is limited only by the dynamic range of the machine on which it is implemented Numerical simulations are performed which veri& the speed and accuracy of the algorithm. I . INTRODUCTION Iterative techniques are used in computer arithmetic for the evaluation of elementary functions using minimal hardware. CORDIC is a prime example of such techniques where it has been used to evaluate trigonometric and hyperbolic functions as well as logarithms and division [l], [2], [3], [43. However, CORDIC is a slow algorithm that suffers fiom limited dynamic range and several authors attempted to remedy that with limited [14]. The main drawbacks of CORDIC are discussed in detail in [14]. Basically CORDIC is an algorithm with slow convergence and limited range. This paper is organized as follows. Section I1 discusses the new algorithm. Section V discusses the main operations that can be performed on an input vector. Section VI1 shows some numerical simulations to verify the algorithm. Section VI11 summarizes the main conclusions. 11. THE ERL ALGORITHM Consider the vector whose end point (X,Y) moves along the curve success PI, ~41, ~61, 171, PI, ~91, [io], [111, [121, [131, X x Y = R 2 (1) Differentiate the above equation Starting from an initial pint (Xo,Yo), we can move to another point (X,Y) on the curve such that Using (2), the corresponding change in Y is given by X = & + A X (3) Y = Yo+AY (4) Choosing A X = a x , we can ?nd AY fiom (2). Thus X and Y are given by x = XO(l+(Y) ( 5 ) Y = Yo(1-a) (6) 111. THE SCALE FACTOR The radius of the vector after the updates is given by R2 = X X = xo x Yo x (1 -2) = Ri x (1-(r2) For R to remain real, LY must lie within the range