Algorithms for fast and precise computation of the normal integral

Five highly accurate algorithms that evaluate the normal distribution function are presented. The algorithms, which are based on either numerical integration or series expansions, are explicated, and applications of Simpson’s rule are discussed. Computer programs that implement the algorithms were compared with respect to accuracy and speed. The programs attain from 11- to 15-decimal-place accuracy in finding one-tailed areas of the normal curve, and most execute very quickly. Recommendations are made for selection of algorithms and programs to approximate the normal distribution.