The complexity of generating an exponentially distributed variate

We analyze in detail an almost optimal algorithm for generating an exponentially distributed variate. The algorithm is due to Knuth and Yao and relies on a method which goes back to J. von Neumann. It is shown here that it can generate k bits of an exponentially distributed variate using an average of about k + 5.67974692 coin flippings. This solves a problem left open by Knuth and Yao.