We provide a non-deterministic quantum protocol that approximates Rx(a2 b2) using Rx(a) and Rx(b) 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 T-count required to use our techniques and show that, with high probability, the required T-count will be lower than lower bounds for the T-count required to do ancilla-free circuit synthesis. We also discuss the T-depth of our method and show that the vast majority of the cost of the resultant circuits can be shifted offline.