  We provide a non-deterministic quantum protocol that approximates the single
qubit rotations R_x(2a^2 b^2)$ using R_x(2a) and R_x(2b) and a constant number
of Clifford and T operations. We then use this method to construct a "floating
point" implementation of a small rotation wherein we use the aforementioned
method to construct the exponent part of the rotation and also to combine it
with a mantissa. This causes the cost of the synthesis to depend more strongly
on the relative (rather than absolute) precision required. We analyze the mean
and variance of the \Tcount required to use our techniques and provide new
lower bounds for the T-count for ancilla free synthesis of small single-qubit
axial rotations. We further show that our techniques can use ancillas to beat
these lower bounds with high probability. We also discuss the T-depth of our
method and see that the vast majority of the cost of the resultant circuits can
be shifted to parallel computation paths.
