  Quadrature by Expansion (QBX) is a quadrature method for approximating the
value of the singular integrals encountered in the evaluation of layer
potentials. It exploits the smoothness of the layer potential by forming
locally-valid expansion which are then evaluated to compute the near or
on-surface value of the integral. Recent work towards coupling of a Fast
Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation
of such integrals (and the solution of related integral equations), albeit with
only empirically understood error behavior. In this paper, we improve upon this
approach with a modified algorithm for which we give a comprehensive analysis
of error and cost in the case of the Laplace equation in two dimensions. For
the same levels of (user-specified) accuracy, the new algorithm empirically has
cost-per-accuracy comparable to prior approaches. We provide experimental
results to demonstrate scalability and numerical accuracy.
