 Normal ratio-of-uniforms envelope

For the unnormalized normal density $h(x)=e^{-x^2/(2{\sigma }^2)}$, the smallest axis-aligned envelope is $0<u<1$ and $|v|\le \sigma \sqrt{2/e}$. The boundary reaches that width at $u=e^{-1/2}$. Region area is $\sigma \sqrt{\pi /2}$, so the acceptance rate is $\sqrt{\pi e}/4$, independently of $\sigma$. Multiplying $h$ by a positive constant rescales both coordinates and their areas consistently.