Taylor’s Formula with Remainder

In this paper, we present a proof in ACL2(r) of Taylor’s formula with remainder. This important theorem allows a function f with n + 1 derivatives on the interval [a, b] to be approximated with a Taylor series of n terms centered at a. Moreover, the formula allows the error in the approximation to be bounded by a term involving the (n + 1)st derivative of f on (a, b). The results in this paper were motivated in part by Jun Sawada’s work with ACL2(r) verifying that the approximation used in the square root calculation of the IBM Power4 processor has the accuracy required. Sawada’s proof effort used a Taylor approximation to the square root function. However, the support for such development in ACL2(r) is lacking [17]. This paper shows how such results can be proved in ACL2(r). It also shines a spotlight on some limitations of ACL2(r) that complicate the proof. Future work will address these limitations.