We describe an optimal procedure, as well as its efficient software implementation, for exact and approximate synthesis of 4x4 unitaries into any discrete set of XX–type gates (a family that describes all 2-qubit controlled gates). This gateset is well-motivated on contemporary hardware where fractionally-entangling gates are often faster/higher-fidelity than fully-entangling gates. Our method arises from analysis and manipulation of certain polyhedral subsets of the space of canonical gates. Using this, we analyze which small sets of XX–type interactions cause the greatest improvement in expected fidelity under experimentally-motivated error models. For the exact circuit synthesis of Haar-randomly selected two-qubit operations, we find an improvement in estimated infidelity by ≈31.4% when including CX^1/2 and CX^1/3 alongside the standard gate CX, near to the optimal limit of ≈36.9% obtained by including all fractional CX^α, α ∈ [0, 1]. This motivates the calibration of a (slightly) over-complete gateset for quantum computers to reduce circuit costs, while maintaining efficient compilation.