Formal Verification of Medina's Sequence of Polynomials for Approximating Arctangent

TL;DR

This paper presents a formal proof in ACL2(r) verifying Medina's polynomial sequence for efficiently approximating arctangent, demonstrating faster convergence than traditional Taylor series.

Contribution

The paper provides the first formal verification of Medina's polynomial sequence for arctangent approximation, including convergence rate proof using real analysis in ACL2(r).

Findings

Proves correctness of Medina's polynomial sequence

Shows faster convergence than Taylor series

Uses real analysis results in formal proof

Abstract

The verification of many algorithms for calculating transcendental functions is based on polynomial approximations to these functions, often Taylor series approximations. However, computing and verifying approximations to the arctangent function are very challenging problems, in large part because the Taylor series converges very slowly to arctangent-a 57th-degree polynomial is needed to get three decimal places for arctan(0.95). Medina proposed a series of polynomials that approximate arctangent with far faster convergence-a 7th-degree polynomial is all that is needed to get three decimal places for arctan(0.95). We present in this paper a proof in ACL2(r) of the correctness and convergence rate of this sequence of polynomials. The proof is particularly beautiful, in that it uses many results from real analysis. Some of these necessary results were proven in prior work, but some were proven as part of this effort.