For a specified hyperexponential function H of x, the (H1, H2) := ReduceHyperexp(H, x, newH) calling sequence constructs two hyperexponential functions H1 and H2 such that  and the certificate  has a differential rational normal form  with v of minimal degree.

The output from ReduceHyperexp is a sequence of two elements  each of which is either  or written in the form

(The form shown above is called a multiplicative decomposition of the hyperexponential function .)

ReduceHyperexp is a generalization of the reduction algorithm for rational functions by Hermite (recall that a rational function is also a hyperexponential function). It also covers the differential Gosper's algorithm.

Geddes, Keith; Le, Ha; and Li, Ziming. "Differential rational canonical forms and a reduction algorithm for hyperexponential functions." Proceedings of ISSAC 2004. ACM Press. (2004): 183-190.