We use coupling into and from the past to sample perfectly in a simple and provably fast fashion from the Vervaat family of perpetuities. The family includes the Dickman distribution, which arises both in number theory and in the analysis of the Quickselect algorithm (the motivation for our work).
"Perfect Simulation of Vervaat Perpetuities." Electron. J. Probab. 15 96 - 109, 2010. https://doi.org/10.1214/EJP.v15-734