Normal rejection sampling with an exponential envelope ($M(\lambda )=\sqrt{2/\pi }{e}^{{\lambda }^2/2}/\lambda$)

For the half-normal distribution, the density ratio to $\mathrm{Exp}(\lambda )$ is maximized at $y=\lambda$. Completing the square gives the envelope constant above and acceptance probability $e^{-(y-\lambda {)}^2/2}$. Its logarithmic derivative is $\lambda -1/\lambda$, so the optimal rate is $\lambda =1$. Give an accepted nonnegative value an independent fair sign to obtain the standard normal distribution. The expected number of proposals is $M$, while the probability of rejecting one proposal is $1-1/M$; these are different quantities.