Microlensing with an advanced contour integration algorithm: Green's theorem to third order, error control, optimal sampling and limb darkening

Microlensing light curves are typically computed either by ray-shooting maps or by contour integration via Green's theorem. We present an improved version of the second method that includes a parabolic correction in Green's line integral. In addition, we present an accurate analytical estimate of the residual errors, which allows the implementation of an optimal strategy for the contour sampling. Finally, we give a prescription for dealing with limb-darkened sources, reaching arbitrary accuracy. These optimizations lead to a substantial speed-up of contour integration code along with a full mastery of the errors.