Hardware Implementation Trade-Offs of Polynomial Approximations and Interpolations

This paper examines the hardware implementation tradeoffs when evaluating functions via piecewise polynomial approximations and interpolations for precisions up to 24 bits. In polynomial approximations, polynomials are evaluated using stored coefficients. Polynomial interpolations, however, require the coefficients to be computed on-the-fly using stored function values. Although it is known that interpolations require less memory than approximations at the expense of additional computation, the tradeoffs in memory, area, delay, and power consumption between the two approaches have not been examined in detail. This work quantitatively analyzes these tradeoffs for optimized approximations and interpolations across different functions and target precisions. Hardware architectures for degree-1 and degree-2 approximations and interpolations are described. The results show that the extent of memory savings realized by using interpolation is significantly lower than what is commonly believed. Furthermore, experimental results on a field-programmable gate array (FPGA) show that for high output precision, degree-1 interpolations offer considerable area and power savings over degree-1 approximations, but similar savings are not realized when degree-2 interpolations and approximations are compared. The availability of both interpolation-based and approximation-based designs offers a richer set of design tradeoffs than is available using either interpolation or approximation alone.