Firstly, looking at the discussion https://math.stackexchange.com/questions/241315/three-ideas-of-perfect-sampling, the term "perfect sampling" does not seem adequate, since there are several aspects that don't sound perfect at all. The sampling becomes perfect from some point in the chain, but establishing a criteria may use different approaches. I don't know if you agree, but, isn't Perfect sampling achieved only through the Inverse transform sampling?

Another and most important question concerns the Inverse transform sampling (ITS). Let $X\sim F(x)$, where $F(x)$ is the cdf of $X$. According to the ITS, we have that $U=F(x)\sim Unif(0,1)$, then we sample $u$ from $Unif(0,1)$ and obtain sampled value $x$ of $X$ from $x=F^{-1}(u)$. Although this look more "perfect" to me, we have a problem, that an expression for the inverse of $F$ mostly is not available, that is why we have all the approximated methods.

So, I was wondering, what is wrong with the following strategy?

Suppose that $F^{-1}(x)$ is not available, but evaluating $F(x)$ is (computationally) easy, so I could set a range $a<x_1<x_2<...x_n<b$ where $[a,b]\subset \mathbb{R}$, the number of points $n$ is chosen suitably, then by computing $u_1=F(x_1),...,u_n=F(x_n)$, we have table of values from which we can draw $x_i$ promptly by drawing $u_i\sim Unif(0,1)$. If we choose a large number of $x_i$ within the range, can we get a very good random number generator?

A problem is when $X$ is defined in ranges such as $(-\infty,\infty)$ and $(0,\infty)$, because we can't be sure of what range to sample from.

Has anybody came across with some similar algorithm?

I hope to have passed the main idea, sorry, I was not formal because I would like to hear from your experiences. I would appreciate any comments. thanks in advance.