Evaluation of elementary functions without range reduction

Abstract

The evaluation of elementary functions can be performed by approximations using minimax polynomials requiring simple hardware resources. The general method to calculate an elementary function is composed by three steps: range reduction, computation of the polynomial in the reduced argument and range reconstruction. This approach allows a low-degree polynomial approximation but range reduction and reconstruction introduce a computation overhead. This work proposes an evaluation methodology without range reduction and range reconstruction steps. Applications that need to compute elementary functions may benefit from avoiding these steps if the argument belongs to a sub-domain of the function. Particularly in the context of embedded systems, applications related to digital signal processing most of the times require function evaluation within a specific interval. As a consequence of not doing range reduction, the degree of the approximant polynomials increases to maintain the required precision. Interval segmentation is an effective way to overcome this issue because the approximations are computed in smaller intervals. The proposed methodology uses non-uniform segmentation as a way to mitigate the problem arising from not carrying out range reduction. The benefits that come from applying interval segmentation to the general evaluation technique are limited by the range reduction and reconstruction steps because the segmentation only applies to the approximation step. However, when used in the proposed methodology it reveals more effective. Some elementary functions were implemented using the proposed methodology in a FPGA device. The metric used to characterize the proposed technique are the area occupation and the corresponding latency. The results of each implementation without range reduction were compared with the corresponding ones of the general method using range reduction. The results show that latency can be significantly reduced while the area is approximately the same.