Compact Numerical Function Generators Based on Quadratic Approximation: Architecture and Synthesis Method

Abstract

SUMMARYThis paper presents an architecture and a synthesis method for compact numerical function generators (NFGs) for trigonometric, logarithmic, square root, reciprocal, and combinations of these functions. Our NFG partitions a given domain of the function into non-uniform segments using an LUT cascade, and approximates the given function by a quadratic polynomial for each segment. Thus, we can implement fast and compact NFGs for a wide range of functions. Experimental results show that: 1) our NFGs require, on average, only 4% of the memory needed by NFGs based on the linear approximation with non-uniform segmentation; 2) our NFG for 2 x − 1 requires only 22% of the memory needed by the NFG based on a 5th-order approximation with uniform segmentation; and 3) our NFGs achieve about 70% of the throughput of the existing table-based NFGs using only a few percent of the memory. Thus, our NFGs can be implemented with more compact FPGAs than needed for the existing NFGs. Our automatic synthesis system generates such compact NFGs quickly.

Programmable Architectures and Design Methods for Two-Variable Numeric Function Generators

“…For one-variable functions, we have proposed linear segmentation algorithms 14), 19) to find an optimum segmentation of a linear domain (an approximation with the fewest segments) efficiently. However, for two-variable functions, a planar segmentation algorithm is now required to find an optimum segmentation of a planar domain.…”

Design Method for Numerical Function Generators Using Recursive Segmentation and EVBDDs

“…The size of all segments is the same as the smallest segment size needed to achieve the desired accuracy. Therefore, depending on functions, uniform segmentation can yield too many segments even if a higher order polynomial is used [15], [17].…”

“…In this method, segments are chosen to be as wide as possible while still achieving the desired accuracy. Such an optimum nonuniform segmentation yields the fewest segments for the given function, and so reduces memory size to store all the coefficients [15], [22], [24].…”

Design Method for Numerical Function Generators Based on Polynomial Approximation for FPGA Implementation

“…For elementary functions, such as sin(x) and e x , by using higher-order polynomial approximations, the number of uniform segments can be reduced, and therefore the memory size can be reduced. However, for some numerical functions, such as − ln(x), methods based on uniform segmentation yield large memory size for implementation on conventional FPGAs even if second-order polynomials are used [21]. On the other hand, since our NFG is based on non-uniform segmentation, for a wide range of functions, the memory size can be reduced by using secondorder polynomials [21].…”

“…However, for some numerical functions, such as − ln(x), methods based on uniform segmentation yield large memory size for implementation on conventional FPGAs even if second-order polynomials are used [21]. On the other hand, since our NFG is based on non-uniform segmentation, for a wide range of functions, the memory size can be reduced by using secondorder polynomials [21]. However, although second-order polynomial approximations reduce memory size, more multipliers and adders are required.…”