Single-qubit rotation algorithm with logarithmic Toffoli count and gate depth

We propose a direct (non-recursive) algorithm for applying a rotation Rθ^, ε-close to a desired rotation R_θ, to a single qubit using the Clifford+Toffoli gate set. Our algorithm does not rely on repeatedly applying a fixed rotation, but immediately applies Rθ^. It succeeds with probability strictly greater than $1/2$, has an expected number of repetitions strictly less than 2, expected Toffoli count logarithmic in 1/ε, and expected gate depth also logarithmic in 1/ε.