Automated Synthesis of Target-Dependent Programs for Polynomial Evaluation in Fixed-Point Arithmetic

Abstract—The design of both fast and numerically accurate programs is a real challenge. Thus, the CGPE tool was introduced to assist programmers in synthesizing fast and numerically certified codes in fixed-point arithmetic for the particular case of polynomial evaluation. For performance purposes, this tool produces programs using exclusively unsigned arithmetic and addition/subtraction or multiplication operations, thus requiring some constraints on the fixed-point operands. These choices are well-suited when dealing with the implementation of certain mathematical functions, however they prevent from tackling a broader class of polynomial evaluation problems. In this paper, we first extend the arithmetic model of CGPE to handle signed arithmetic and alignment shifts. Then, in order to make the most out of advanced instructions, we propose an enhancement of this tool based on instruction selection. This allows us to optimize the generated codes according to different criteria, like operation count, evaluation latency, or accuracy. Finally, we illustrate this technique on operation count, and we show that it yields an average reduction of up to 22.3 % of the number of operations in the synthesized codes of some functions. Keywords—fixed-point arithmetic, automated code synthesis, error analysis, polynomial evaluation I