Linear and Quadratic Interpolators Using Truncated-Matrix Multipliers and Squarers

Abstract

This paper presents a technique for designing linear and quadratic interpolators for function approximation using truncated multipliers and squarers. Initial coefficient values are found using a Chebyshev-series approximation and then adjusted through exhaustive simulation to minimize the maximum absolute error of the interpolator output. This technique is suitable for any function and any precision up to 24 bits (IEEE single precision). Designs for linear and quadratic interpolators that implement the 1/x, 1/ √ x, log 2 (1 + 2 x ), log 2 (x) and 2 x functions are presented and analyzed as examples. Results show that a proposed 24-bit interpolator computing 1/x with a design specification of ±1 unit in the last place of the product (ulp) error uses 16.4% less area and 15.3% less power than a comparable standard interpolator with the same error specification. Sixteen-bit linear interpolators for other functions are shown to use up to 17.3% less area and 12.1% less power, and 16-bit quadratic interpolators are shown to use up to 25.8% less area and 24.7% less power.

Optimized Linear, Quadratic and Cubic Interpolators for Elementary Function Hardware Implementations

“…One method to reduce the hardware needed to accurately approximate elementary functions using polynomials is to use truncated multipliers and squarers within the hardware [26,27]. Unfortunately, incorporating truncated arithmetic units into the architecture complicates the error analysis.…”

“…With the approach presented in this paper, Chebyshev series approximations are used to select the initial coefficient values for each subinterval [26,27]. First, the number of subintervals, 2 m , is determined.…”

“…For a cubic interpolator, the coefficients for the interval [x m , x m + 2 −m ) are given by Equations (24)- (27) where…”

“…Truncated multiplication and squaring units can be used to reduce the area, delay, and power requirements. However, the error associated with the reduced hardware complicates the error analysis required to ensure accurate evaluation of functions [26,27]. This paper presents an optimization method to localize and find a closed-form solution for interpolators that allows for smaller architectures to be realized.…”

“…In-between methods can be subdivided into linear approximations [12,24] and quadratic interpolation methods [22,23,[25][26][27] based on the degree of the polynomial approximation employed. The intermediate size of the lookup tables makes them suitable for IEEE single-precision computations (24-bit significand), attaining fast execution with reasonable hardware requirements.…”