In Gray and Wang [Gray,H. L.,Wang,S. (1991). A general method for approximating tail probabilities. J. Am. Stat. Assoc. 86(413):159–166] the deterministic version of the generalized jackknife,referred to as the transform,was shown to be a powerful tool for obtaining simple approximation functions for tail probabilities of most pdfs. They demonstrated that in many cases m=1 is quite adequate to produce approximation functions for tail probabilities that are highly accurate. Moreover all of these approximations are of the convenient form f(x)R(x), where f is the pdf and R is a rational function. Gray and Wang were only able to give R(x) for for n = 1,2,3 due to the extensive algebra required. Even so these approximations yielded relative errors typically in the 10−5 range. In this paper we make use of the computer algebra program Mathematica to extend these approximation functions up to n = 8 (up to n = 10 in the normal case). The resulting approximation functions have relative errors typically in the 10−10 range and in some cases 10−20 or better.