Uniform-envelope rejection sampling for a beta distribution ($r(x)=x^{a-1}(1-x{)}^{b-1},a,b\ge 1$)

For $a,b>1$, the maximum of $r$ on $[0,1]$ occurs at $(a-1)/(a+b-2)$. If one parameter is one, the maximum is attained at the corresponding endpoint; if both are one, $r$ is constant. Let $M=\sup r$. Accept an independent uniform distribution proposal $X$ when another uniform draw is at most $r(X)/M$. The accepted probability density function is the beta distribution, and acceptance probability is $B(a,b)/M$.