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

Ruben Gamboa
(University of Wyoming)
John Cowles
(University of Wyoming)

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.

In Freek Verbeek and Julien Schmaltz: Proceedings Twelfth International Workshop on the ACL2 Theorem Prover and its Applications (ACL2 2014), Vienna, Austria, 12-13th July 2014, Electronic Proceedings in Theoretical Computer Science 152, pp. 101–110.
Published: 4th June 2014.

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