  For random variables produced through the inverse transform method,
approximate random variables are introduced, which are produced by
approximations to a distribution's inverse cumulative distribution function.
These approximations are designed to be computationally inexpensive, and much
cheaper than exact library functions, and thus highly suitable for use in Monte
Carlo simulations. Two approximations are presented for the Gaussian
distribution: a piecewise constant on equally spaced intervals, and a piecewise
linear using geometrically decaying intervals. The error of the approximations
are bounded and the convergence demonstrated, and the computational savings
measured for C and C++ implementations. Implementations tailored for Intel and
Arm hardwares are inspected, alongside hardware agnostic implementations built
using OpenMP. The savings are incorporated into a nested multilevel Monte Carlo
framework with the Euler-Maruyama scheme to exploit the speed ups without
losing accuracy, offering speed ups by a factor of 5--7. These ideas are
empirically extended to the Milstein scheme, and the Cox-Ingersoll-Ross
process' non central chi-squared distribution, which offer speed ups by a
factor of 250 or more.
