  We present a novel method for approximating the probability density function
(PDF) of the first-passage times in the Ratcliff diffusion decision model
(DDM). We implemented this approximation method in $\texttt{C++}$ using the
$\texttt{R}$ package $\texttt{Rcpp}$ to utilize the faster $\texttt{C++}$
language while maintaining the $\texttt{R}$ language interface. In addition to
our novel approximation method, we also compiled all known approximation
methods for the DDM density function (with fixed and variable drift rate),
including previously unused combinations of techniques found in the relevant
literature. We ported these approximation methods to $\texttt{C++}$ and
optimized them to run in this new language. Given an acceptable error tolerance
in the value of the PDF approximation, we benchmarked all of these
approximation methods to compare their speed against each other and also
against commonly used $\texttt{R}$ functions from the literature. The results
of these tests show that our novel approximation method is not only orders of
magnitude faster than the current standards, but it is also faster than all of
the other approximation methods available even after translation and
optimization to the faster $\texttt{C++}$ language. All of these approximation
methods are bundled in the $\texttt{fddm}$ package for the $\texttt{R}$
statistical computing language; this package is available via CRAN, and the
source code is available on GitHub.
