The paper considers approximate distribution theory as a way to deliver practical improvements over asymptotic methods. Trials of the adequacy of asymptotic series based approximations are conducted and the results exhibit the need for further improvements. An alternative approach is suggested which utilizes a family of extended rational approximants (ERA’s). ERA’s build on the strength of primitive exact theory or asymptotic series; they allow us to blend information from diverse analytic numerical and experimental sources. The ultimate objective of the approach is to incorporate directly into software constructive functional approximants relevant to statistical practice. An illustration is provided.