| This dissertation presents two significant contributions for computing elementary function. The first contribution is a method for computing elementary function using the optimized memory requirements along with truncated multipliers, squarers and cubers for designing linear, quadratic and cubic interpolators. The proposed method optimizes the initial coefficient values found using a Chebyshev series approximation and minimizes the maximum absolute error of the interpolator output. The resulting designs can be utilized for any approximation for functions up to and beyond 53-bits (IEEE double precision significant) of precision with a reduced requirement for table lookup sizes. Designs for linear, quadratic and cubic interpolators that implement reciprocal, square root, reciprocal square root and sine are presented and analyzed, and the method can be extended easily to other functions. Overall, the first part of the dissertation demonstrates a method to optimize a given accuracy for computation that employ hardware units that may have different precision limitations. Area, delay and power estimates are given for 16, 24 and 32-bit interpolators that compute the reciprocal function targeting a 65nm CMOS technology from IBM. Results indicate the proposed method uses smaller arithmetic units and has reduced lookup table sizes than previously proposed methods. This method can be employed within any system that has similar truncation and rounding effects within multiple logic units.;The second contribution is an optimization method for computing an optimum lookup table size for two well-known look up table elementary function approximation methods: Symmetric Table Additional Method (STAM) and Multipartite Table Method (MTM). Using a discrete optimization algorithm called Leapfrogging, this part utilizes a method to find the best decomposition of the coefficients to optimize look up table sizes. The resulting designs can easily be utilized for any approximation for functions up 24-bits of precision with significantly smaller requirements for lookup table sizes. Results show that the proposed optimized method is able to achieve higher memory efficiency than the best existing MTM. |
