Algebraic techniques to enhance common sub-expression elimination for polynomial system synthesis

Abstract

Common sub-expression elimination (CSE) serves as a useful optimization technique in the synthesis of arithmetic datapaths described at RTL. However, CSE has a limited potential for optimization when many common sub-expressions are not exposed. Given a suitable transformation of the polynomial system representation, which exposes many common sub-expressions, subsequent CSE can offer a higher degree of optimization. The objective of this paper is to develop algebraic techniques that perform such a transformation, and present a methodology to integrate it with CSE to further enhance the potential for optimization. In our experiments, we show that this integrated approach outperforms conventional methods in deriving areaefficient hardware implementations of polynomial systems. I. IntroductionHigh-level descriptions of arithmetic datapaths that perform polynomial computations over bit-vectors are found in many practical applications, such as in Digital Signal Processing (DSP) for multi-media applications and embedded systems. These polynomial designs are initially specified using behavioural or Register-Transfer-Level (RTL) descriptions, which are subsequently synthesized into hardware using high-level/logic synthesis tools [1]. The growing market for such applications requires sophisticated CAD support for their design, optimization and synthesis.The general area of high-level synthesis has seen extensive research over the years. Various algorithmic techniques have been devised, and CAD tools have been developed that are quite adept at capturing hardware description language (HDL) models and mapping them into control/data-flow graphs (CDFGs), performing scheduling, resource allocation and sharing, binding, retiming, etc., [2]. However, these tools lack the mathematical wherewithal to perform sophisticated algebraic manipulation for arithmetic datapath-intensive designs. Such designs implement a sequence of add, mult type of algebraic computations over bit-vectors; they are generally modeled at RTL or behavioral-level as systems of multi-variate polynomials of finite degree [3] [4]. Hence, there has been increasing interest in exploring the use of algebraic manipulation of polynomial expressions, for RTL synthesis of arithmetic datapaths. Several techniques such as Horner decomposition, factoring with common sub-expression elimination [5], term-rewriting [6], etc., have been proposed. Symbolic computer algebra [3] [4] [7] has also been employed for

Polynomial datapath optimization using constraint solving and formal modelling

“…On the other hand we are able to reduce the area on instance 20 by 30.93% and its delay by 15.17% compared to [13]. Furthermore on instance 5 we achieve a reduction of 46.14% in delay and 16.87% in area compared to [10]. Overall we obtained average savings of 12.15% in the area and 3.05% in the delay over [13].…”

“…In addition to the results from [13] we also included results from [10]. Table II summarizes the results for 22 configurations.…”

“…In the first experiment, we have employed phase-shift keying (PSK) that is used in digital communication from [4], digital image rejection unit (DIRU), Degree-5 Filter (PFLT), multivariate cosine wavelet polynomial (MVCS), an anti-aliasing function (ANTI) [4], a Savitzky Golay filter (SG2) and Quadratic polynomial (QUAD) benchmarks and applied our method to the benchmarks listed in Table I as singleoutput polynomials. These benchmarks have been used in previous work on polynomial datapath optimization such as [13], [10]. We have synthesized the polynomials using a traditional logic synthesis tool in 0.25µm CMOS technology.…”

“…The technique in [10] first extracts coefficient multiplications. Then using the kernel/co-kernel extraction techniques from [7] and [11], common cubes are extracted.…”

“…Algebraic techniques in [9], [10] employed various optimization techniques to manipulate the polynomials and extract common subexpressions. The technique in [10] first extracts coefficient multiplications.…”