Minimizing Coefficients Wordlength for Piecewise-Polynomial Hardware Function Evaluation With Exact or Faithful Rounding

Piecewise polynomial interpolation is a well-established technique for hardware function evaluation. The paper describes a novel technique to minimize polynomial coefficients wordlength with the aim of obtaining either exact or faithful rounding at a reduced hardware cost. The standard approaches employed in literature subdivide the design of piecewise-polynomial interpolators into three steps (coefficients calculation, coefficients quantization and arithmetic hardware optimization) and estimate conservatively the overall approximation error as the sum of the error components arising in each step. The proposed technique, using Integer Linear Programming (ILP), optimizes the polynomial coefficients taking into account all error components simultaneously. This gives two advantages. Firstly, we can obtain exactly rounded approximations; secondly, for faithfully rounded interpolators, we avoid any overdesign due to pessimistic assumptions on error components, optimizing in this way the resulting hardware. The proposed ILP based algorithm requires an acceptable CPU time (from few seconds to tens of minutes) and is suited for approximations up to, maximum, 24 input bits. The results compare favorably with previously published data. We present synthesis results in 28 nm and 90 nm CMOS technologies, to further assess the effectiveness of the proposed approach.