Piecewise Arithmetic Expressions of Numeric Functions and Their Application to Design of Numeric Function Generators
Shinobu Nagayama, Tsutomu Sasao and Jon T. Butler
In this paper, we propose a new representation of numeric functions using a piecewise arithmetic expression. To represent a numeric function compactly, we partition the domain of the function into uniform segments, and transform the sub-function in each segment into an arithmetic spectrum. From this arithmetic spectrum, we derive an arithmetic expression, and obtain a piecewise arithmetic expression for the function. By using the piecewise arithmetic expression, we can increase the number of zero arithmetic coefficients significantly, and represent a numeric function more compactly than using a conventional single arithmetic expression. We also present an application of the piecewise arithmetic expression to design of numeric function generators (NFGs). Since the piecewise arithmetic expression has many zero coefficients and repeated coefficients, by storing only distinct nonzero coefficients in a table, we can significantly reduce the table size needed to store arithmetic coefficients. Experimental results show that the table size can be reduced to only a small percent of the table size needed to store all the arithmetic coefficients.
Keywords: Piecewise arithmetic expressions, nonzero arithmetic coefficients, numeric function generators (NFGs), programmable architectures.