Computing Pi Via New Polynomial Approximations to Arctangent

Rational functions after integration can produce arctangent, and therefore can be used to approximate pi. Using rational functions we produce different families of efficient polynomial approximations to arctangent on the interval [0,a], and hence, provide approximations to pi via known arctangent values. The polynomials produce approximations to pi that require only the computation of a single square root (square root of 3); moreover, on the interval [0,a] they are orders of magnitude more accurate than Maclaurin polynomials and other approximations to arctangent recently studied. We analyze the efficiency of the approximations and provide algebraic and analytic properties of the sequences of polynomials. Finally, we turn the most efficient approximation of pi into a series that gives about 21 more decimal digits of accuracy with each successive term; this is significantly more digits per term than other well-known series.