Simplifying Probabilistic Programs Using Computer Algebra

We transform probabilistic programs to run more efficiently and read more easily, by composing three semantics-preserving transformations: (1) apply the denotational semantics; (2) improve the resulting integral; then (3) invert the denotational semantics. Whereas step 1 is a straightforward transformation from monadic to continuation-passing style, the rest builds on computer algebra: step 2 reorders and performs integrals, and step 3 represents density functions as differential operators.