  We prove that the function $P_\alpha(x) = \exp(1-x^{-\alpha})$ with $\alpha >
0$, is $1/\alpha$-mild. We apply this result to obtain a uniform
$1/\alpha$-mild parametrization of the family of curves $\{xy = \epsilon^2 \mid
(x,y) \in (0,1)^2\}$ for $\epsilon \in (0,1)$, which does not have a uniform
$0$-mild parametrization by work of Yomdin. More generally we can parametrize
families of power-subanalytic curves. This improves a result of Benjamini and
Novikov that gives a $2$-mild parametrization.
