Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 40 / 1 / b / ii / Solution

Integration of the Laplace distribution probability density function givesSolving these two equations for $x$ givesGenerate one uniform $U\in (0,1)$ and return this piecewise expression. The branches agree at $U=1/2$ and the inverse transform sampling proof establishes the target law without any rejection step.