Laplace ratio-of-uniforms envelope ($a=1,b_1=-4e^{-5/4},b_2=4e^{-3/4}$)

For the unnormalized Laplace distribution probability density function $h(x)=e^{-|x-1|/2}$, the ratio-of-uniforms method uses $0<u\le \sqrt{h(v/u)}$. The maximum of $h$ is one. On $x\le 0$, the maximum of $x^{2}h(x)$ occurs at $x=-4$ and equals $16e^{-5/2}$. On $x\ge 0$, it occurs at $x=4$ and equals $16e^{-3/2}$. Thus the displayed rectangle contains the region. Its area is $4(e^{-3/4}+e^{-5/4})$, while the target region has area $\frac{1}{2}\int h=2$. Uniform proposals in the rectangle, accepted when $u^2\le h(v/u)$, therefore return the correct Laplace distribution with acceptance probability $[2(e^{-3/4}+e^{-5/4}){]}^{-1}$.