  Let $\Gamma$ be a group and $(\Gamma_n)_{n=1} ^{\infty}$ be a descending
sequence of finite-index normal subgroups. We establish explicit upper bounds
on the diameters of the directed Cayley graphs of the $\Gamma/\Gamma_n$ , under
some natural hypotheses on the behaviour of power and commutator words in
$\Gamma$. The bounds we obtain do not depend on a choice of generating set.
Moreover under reasonable conditions our method provides a fast algorithm for
constructing directed Cayley graphs of diameter satisfying our bounds. The
proof is closely analogous to the the Solovay-Kitaev procedure, which only uses
commutator words, but also only constructs small-diameter undirected Cayley
graphs. As an application we give directed diameter bounds on finite quotients
of two very different groups: $SL_2 (\mathbb{F}_q [[t]])$ (for $q$ even) and a
group of automorphisms of the ternary rooted tree introduced by Fabrykowski and
Gupta.
