  This article presents differential equations and solution methods for the
functions of the form $Q(x) = F^{-1}(G(x))$, where $F$ and $G$ are cumulative
distribution functions. Such functions allow the direct recycling of Monte
Carlo samples from one distribution into samples from another. The method may
be developed analytically for certain special cases, and illuminate the idea
that it is a more precise form of the traditional Cornish-Fisher expansion. In
this manner the model risk of distributional risk may be assessed free of the
Monte Carlo noise associated with resampling. Examples are given of equations
for converting normal samples to Student t, and converting exponential to
hyperbolic, variance gamma and normal. In the case of the normal distribution,
the change of variables employed allows the sampling to take place to good
accuracy based on a single rational approximation over a very wide range of the
sample space. The avoidance of any branching statement is of use in optimal GPU
computations as it avoids the effect of {\it warp divergence}, and we give
examples of branch-free normal quantiles that offer performance improvements in
a GPU environment, while retaining the best precision characteristics of
well-known methods. We also offer models based on a low-probability of warp
divergence. Comparisons of new and old forms are made on the Nvidia Quadro
4000, GTX 285 and 480, and Tesla C2050 GPUs. We argue that in single-precision
mode, the change-of-variables approach offers performance competitive with the
fastest existing scheme while substantially improving precision, and that in
double-precision mode, this approach offers the most GPU-optimal Gaussian
quantile yet, and without compromise on precision for Monte Carlo applications,
working twice as fast as the CUDA 4 library function with increased precision.
