Table-based polynomials for fast hardware function evaluation

Abstract

Many general table-based methods for the evaluation in hardware of elementary functions have been published. The bipartite and multipartite methods implement a first-order approximation of the function using only table lookups and additions. Recently, a single-multiplier second-order method of similar inspiration has also been published. This paper presents a general framework extending such methods to approximations of arbitrary order, using adders, small multipliers, and very small ad-hoc powering units. We obtain implementations that are both smaller and faster than all previously published approaches. This paper also deals with the FPGA implementation of such methods. Previous work have consistently shown that the more complex methods were also faster: The reduction of the table size meant a reduction of its lookup time, which compensated for the addition and multiplication time. A second contribution is therefore to finally create a tradeoff between space and time among table-based methods.

A Systematic Design Method for Two-Variable Numeric Function Generators Using Multiple-Valued Decision Diagrams

“…3 from the EVMDD. Since our NFG directly realizes the function table, it is more accurate than existing NFGs using polynomial approximation [7], [16], [25], [30], [31].…”

“…Various design methods for numeric function generators (NFGs) have been developed [18]. However, most existing methods are intended for one-variable numeric functions [7], [16], [21], [25], [29]- [31], and only a few methods have been reported for specific multi-variable numeric functions [9], [10], [34]. Thus, different numeric functions require different methods.…”

Numerical function generators using bilinear interpolation

“…sin(x) and log(x)) has been extensively studied [2,6,8,[10][11][12]. However, significantly less work has been done on the high-speed implementation of multi-variable functions (e.g.…”

Design Method for Numerical Function Generators Based on Polynomial Approximation for FPGA Implementation

“…To reduce memory size, polynomial approximations have been used [3,5,6,7,12,14,22,27,28,29]. These methods approximate the given numerical functions by piecewise polynomials, and realize the polynomials with hardware.…”