Depending on your needs, some characteristics per sample ($\times 10^n$ as appropriate) of simulated $f$ (plus the CDF, $F=\int f$, and quantile function $F^{-1}$) that may be of interest for procedure 1 vs procedure 2:

  * Computational time

  * Memory cost

  * Energy cost

  * Uniformity of $p$ (from $F^{-1}$)

  * Accuracy in the *tails* of $f$

  * Accuracy for the distribution overall from an analytic formulation of $f$ (e.g., using a [one-sample Kolmogorov-Smirnof test](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test) for each procedure for many samples of size $N$ and comparing the mean $p$ values).

The particular statistics tracked from simulating a univariate distribution will necessarily depend on the purposes envisioned for using it. For example, if the number of needed simulations is anticipated to be on the order of, say, $10^6$ per day or less, then you may not need to calculate compute time, memory, or energy costs. However, if anticipated need for simulated draws from $f$ is at a much greater rate, say $10^9$ per hour or more, then those costs per sample may be relevant.