Quantum circuit optimisation aims to reduce the gate count, gate depth, or other cost metrics of a quantum circuit without changing the overall computation. Most automated approaches focus on exact optimisations, i.e. transformations that preserve the exact unitary map of the quantum computation. However, in most applications of quantum computing, it suffices to merely approximate a target unitary up to some given precision. In this paper, we introduce a simple approximate reduction, which we call “phase squashing”, into an existing quantum circuit optimisation routine based on the ZX calculus. We show that we can rigorously bound the overall error introduced by phase squashing when it is applied to ZX diagrams with generalised flow, allowing one to perform automated circuit optimisation up to a fixed error budget. We then introduce several optimisation heuristics that incorporate phase squashing and benchmark their performance on two families of circuits.