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Automatic differentiation (AD) aims to compute derivatives of user-definedfunctions, but in Turing-complete languages, this simple specification does notfully capture AD's behavior: AD sometimes disagrees with the true derivative ofa differentiable program, and when AD is applied to non-differentiable oreffectful programs, it is unclear what guarantees (if any) hold of theresulting code. We study an expressive differentiable programming language,with piecewise-analytic primitives, higher-order functions, and generalrecursion. Our main result is that even in this general setting, a version ofLee et al. [2020]'s correctness theorem (originally proven for a first-orderlanguage without partiality or recursion) holds: all programs denote so-called$\omega$PAP functions, and AD computes correct intensional derivatives of them.Mazza and Pagani [2021]'s recent theorem, that AD disagrees with the truederivative of a differentiable recursive program at a measure-zero set ofinputs, can be derived as a straightforward corollary of this fact. We alsoapply the framework to study probabilistic programs, and recover a recentresult from Mak et al. [2021] via a novel denotational argument.

Towards Denotational Semantics of AD for Higher-Order, Recursive, Probabilistic Languages