Low-order approximations for the normal probability integral and the error function

Abstract

Rational fractions of the form 0.5 / ( a + b x + … ) 2 q 0.5/{(a + bx + \ldots )^{2q}} are used to evaluate the function of interest. Polynomials of from third to sixth order are derived which achieve absolute errors ranging from 0.01 to 0.000001 for all (real) positive x, and relative errors of from 0.1 to 0.00001 for (real) positive x less than 3.1, 4.0, and 5.2. Denominator coefficients are calculated by linearizing the rational fraction about progressively improved nominal solutions and using linear programming to solve the resulting linear minimax problems.