Not-Quite Transcendental Functions and their Applications

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.