  An algorithm of Ross and Selinger for the factorization of diagonal elements
of PU(2) to within distance $\varepsilon$ was adapted by Parzanchevski and
Sarnak into an efficient probabilistic algorithm for any element of PU(2) using
at most effective $3\log_p\frac{1}{\varepsilon^{3}}$ factors from certain
well-chosen sets associated to a number field and a prime $p$. The icosahedral
super golden gates are one such set associated to $\mathbb{Q}(\sqrt{5})$. We
leverage recent work of Carvalho Pinto, Petit, and Stier to reduce this bound
to $\frac{7}{3}\log_{59}\frac{1}{\varepsilon^3}$, and we implement the
algorithm in Python. This represents an improvement by a multiplicative factor
of $\log_259\approx5.9$ over the analogous result for the Clifford+$T$ gates.
This is of interest because the icosahedral gates have shortest factorization
lengths among all super golden gates.
