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 univariate Kolmogorov-Smirnof test)
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.