Not-Quite Transcendental Functions and their Applications

TL;DR

This paper introduces new non-transcendental functions that serve as efficient, accurate replacements for exponential and logarithmic functions in various computational applications, significantly improving performance without sacrificing accuracy.

Contribution

The authors propose novel functions that mimic transcendental functions' properties, enabling faster computations with no accuracy loss in many scenarios.

Findings

New functions provide computational speedups

No accuracy loss in key applications

Effective as drop-in replacements for exponentials and logarithms

Abstract

Transcendental functions, such as exponentials and logarithms, appear in a broad array of computational domains: from simulations in curvilinear coordinates, to interpolation, to machine learning. Unfortunately they are typically expensive to compute accurately. In this note, we argue that in many cases, the properties of the function matters more than the exact functional form. We present new functions, which are not transcendental, that can be used as drop-in replacements for the exponential and logarithm in many settings for a significant performance boost. We show that for certain applications using these functions result in no drop in the accuracy at all, as they are perfectly accurate representations of themselves, if not the original transcendental functions.