Generating function approximations at compile time

Abstract

Abstract-Usually, the mathematical functions used in a numerical programs are decomposed into elementary functions (such as sine, cosine, exponential, logarithm...), and for each of these functions, we use a program from a library. This may have some drawbacks: first in frequent cases, it is a compound function (e.g. log(1 + exp(−x))) that is needed, so that directly building a polynomial or rational approximation for that function (instead of decomposing it) would result in a faster and/or more accurate calculation. Also, at compile-time, we might have some information (e.g., on the range of the input value) that could help to simplify the program. We investigate the possibility of directly building accurate approximations at compile-time.

An optimization‐based approach to lookup table program transformations

“…In hardware, functions are often calculated using polynomial approximations such as the minimax , which provides fine‐grained control over accuracy . The polynomial approximation technique has also been applied to software in a paper by Muller that proposes a tool to generate code for polynomial approximations at compile‐time . One advantage of Muller's approach, as with the LUT methodology presented here, is that it supports compound functions that combine multiple, expensive elementary functions.…”