Hardware operators for function evaluation using sparse-coefficient polynomials

This article presents dedicated hardware arithmetic operators for function evaluation. The proposed solution uses polynomial approximations with sparse coefficients which leads to efficient hardware implementations. Up to 2× faster and 8× smaller operators are reported compared to standard implementations. Introduction: Polynomial approximations are widely used in digital systems for function evaluation (e.g. 1/x, √ x, sin, cos, exp, log). In some digital signal processing applications, such as frequency demodulation, low-degree polynomials are often used for evaluating reciprocals. In hardware implementation of polynomial approximations, the size of the multipliers is a major concern. Several solutions have been investigated to limit their size: argument reduction and series expansions in [1], small table and a modified multiplication in [2], or the multipartite tables method [3, 4]. In this work we focus on polynomial approximations with sparse coefficients (i.e. the multiplier operand can be written so that it contains predefined strings of bits stuck at 0). Sparse coefficients allow us to replace the complete reduction tree of the multipliers by smaller ones. This letter is an improved and extended version of the paper presented in [5] and uses a new recursive coefficient filtering. Background: When evaluating a function f on a real interval [a, b], one usually uses polynomial approximations, such as minimax approximations which minimize the distance ||p− f ||∞,[a,b] = sup a≤x≤b |p(x)− f(x)|, where p ∈ Rd[X], the set of polynomials with real coefficients and degree at most d. Minimax approximations, that can be computed thanks to an algorithm due 1 to Remez have a major drawback: in most cases, their coefficients are not exactly representable using a finite number of bits. In [6], the authors propose an efficient method for computing a polynomial which minimizes the distance ||p − f ||∞,[a,b] among the polynomials p ∈ Rd[X] that fulfill some given constraints on the format of the coefficients. The result polynomials q are of the form: q(x) = q0 2m0 + q1 2m1 x+ q2 2m2 x + . . .+ qd 2md x. The degree d and the integer sequence m0, . . . ,md are input parameters of the method. The coefficients are such that qi ∈ Z. The method presented in [6] provides result polynomials q such that ||q− f ||∞,[a,b] is minimal among the polynomials that fulfill the constraints. Those polynomials can be represented as the integer points of a polytope (cf. [6]) and efficient scanning of all the polytope points is performed using linear programming tools. Here, we look for polynomials with sparse coefficients, i.e. there are several bits predefined to 0 as illustrated in Figure 1. Each coefficient qi/2 mi is decomposed into k chunks ci,j with j ∈ {1, . . . , k}. The chunks are small signed sj-bit integers. The number of useful bits in coefficients is S = ∑k j=1 sj. The weight of the chunk ci,j is the value 2j as illustrated on Figure 1. Notice that the size and the weight of the chunks are the same for all the coefficients. The value of a coefficient numerator is then qi = ∑k j=1 ci,j × 2j . For a given number of chunks k, there are several possible sparse decompositions (the values sj and wj) with non-overlapping chunks. Polynomials with sparse coefficients are interesting for circuit implementation as soon as the number of nonzero bits is much smaller than the total format width. This corresponds to: