Tableless Calculation of Circular Functions on Dyadic Rationals

TL;DR

This paper introduces a novel algorithm and language framework in Scheme for calculating circular functions on dyadic rationals, emphasizing a purely functional, implementation-agnostic approach.

Contribution

It presents a new, mechanically-derived algorithm for trigonometric functions on dyadic rationals within an algorithmic language, addressing efficiency and accuracy.

Findings

Developed a scheme-based algorithm for circular functions on dyadic rationals

Provided a mechanically-derived inverse function algorithm

Addressed implementation concerns for hardware and software

Abstract

I would like to tell a story. A story about a beautiful mathematical relationship that elucidates the computational view on the classic subject of trigonometry. All stories need a language, and for this particular story an algorithmic language ought to do well. What makes a language algorithmic? From our perspective as the functional programming community, an algorithmic language provides means to express computation in terms of functions, with no implementation-imposed limitations. We develop a new algorithm for the computation of trigonometric functions on dyadic rationals, together with the language used to express it, in Scheme. We provide a mechanically-derived algorithm for the computation of the inverses of our target functions. We address efficiency and accuracy concerns that pertain to the implementation of the proposed algorithm either in hardware or software.