  We improve the Solovay-Kitaev theorem and algorithm for a general finite,
inverse-closed generating set acting on a qudit. Prior versions of the
algorithm can efficiently find a word of length $O((\log
1/\epsilon)^{3+\delta})$ to approximate an arbitrary target gate to within
$\epsilon$. Using two new ideas, each of which reduces the exponent separately,
our new bound on the world length is $O((\log
1/\epsilon)^{1.44042\ldots+\delta})$. Our result holds more generally for any
finite set that densely generates any connected, semisimple real Lie group,
with an extra length term in the non-compact case to reach group elements far
away from the identity.
