Antithetic and control variates for E[e^U]

You want to estimate $\theta = E[e^U] = e - 1$ by simulation, with $U \sim U(0,1)$.

(a) What is the variance of one plain draw $e^U$?

(b) Antithetic variates: what is the variance of $\frac12(e^U + e^{1-U})$? Compare it with the average of two independent draws.

(c) Control variates: use $U$ as the control, with known mean $\frac12$. Find the optimal coefficient $c^*$ and the variance of $e^U - c^*(U - \frac12)$.

(d) Per evaluation of $e^x$, which method wins?