Roll Twice Take Highest

The generalizable way to get the mean value of a rerolled N-sided die (taking the highest result):

(1)

\begin{align} \sum_{n=1}^{N} \frac{2n^{2}-n}{N^{2}} = \frac{N(N + 1)(4N - 1)}{6N^{2}} \end{align}

… and the mean value 'R' of a re-rolled 'n' on a N-sided die:

(2)

\begin{align} R(n) = \frac{n^{2} - n + N^{2} + N}{2N} = \frac{n^{2} - n}{2N} + \frac{N + 1}{2} \end{align}

Examples:

dX	mean reroll	mean 2x	reroll >2x for
d3	2.444	4	1
d4	3.125	5	1
d6	4.472	7	1-2
d8	5.8125	9	1-2
d10	7.15	11	1-2
d12	8.486	13	1-3

What About the Lowest of Two Rolls?

The generalizable way to get the mean value of a rerolled N-sided die (taking the lowest result):

(3)

\begin{align} \sum_{n=1}^{N} \frac{Nn}{N^{2}} - \frac{n(n - 1)}{2N^{2}} = \frac{N(N + 1)(2N + 1)}{6N^{2}} \end{align}