High-order Taylor series approximation for efficient computation of elementary functions

Abstract

A new piecewise polynomial method is proposed to compute elementary functions by using high-order Taylor approximation. The high-order power terms of the series are proposed to be approximated by using simple and fast table lookup. Furthermore, the similarity and regularity among the Taylor coefficients can make possible the sharing of the lookup tables. The authors have developed an error analysis method to estimate the maximum error of the proposed approximation approach, and formulated the procedure for determining the hardware parameters in the approximation unit. Finally, the authors have designed a single-precision approximation unit for computing six common elementary functions. The coefficient sharing approach can result in at least 30.5% reduction in the coefficient lookup tables. Compared with a previous work by Piñeiro et al., the authors can save 27.91% of the lookup tables with some extra cost in computation hardware. Compared with the work by Alimohammad et al., 34.85% of the lookup tables can be saved with the same computation hardware cost. The authors conclude that the proposed approaches can effectively reduce the lookup tables required in the piecewise polynomial approximation for efficient elementary function computation.