Properties such as provable security and correctness for randomized programs are naturally expressed relationally as approximate equivalences. As a result, a number of relational program logics have been developed to reason about such approximate equivalences of probabilistic programs. However, existing approximate relational logics are mostly restricted to first-order programs without general state.

In this paper we develop Approxis, a \emph{higher-order approximate relational separation logic} for reasoning about approximate equivalence of programs written in an expressive ML-like language with discrete probabilistic sampling, higher-order functions, and higher-order state. The Approxis logic recasts the concept of \emph{error credits} in the relational setting to reason about relational approximation, which allows for expressive notions of modularity and composition, a range of new approximate relational rules, and an internalization of a standard limiting argument for showing exact probabilistic equivalences by approximation. We also use Approxis to develop a logical relation model that quantifies over error credits, which can be used to prove \emph{exact contextual equivalence}. We demonstrate the flexibility of our approach on a range of examples, including the PRP/PRF switching lemma, IND$-CPA security of an encryption scheme, and a collection of rejection samplers. All of the results have been mechanized in the Coq proof assistant and the Iris separation logic framework.