  This work presents a faster method of simulating exactly from a distribution
known as a Vervaat perpetuity. A parameter of the Vervaat perpetuity is $\beta
\in (0,\infty)$. An earlier method for simulating from this distributon ran in
time $O((2.23\beta)^{\beta}).$ This earlier method utilized dominated coupling
from the past that bounded a stochastic process for perpetuities from above. By
extending to non-Markovian update functions, it is possible to create a new
method that bounds the perpetuities from both above and below. This new
approach is shown to run in $O(\beta \ln(\beta))$ time.
