We demonstrate the high accuracy of the density splitting method to compute the gravitational potential and field in the plane of razor-thin, axisymmetric disks, as we preliminarily outlined in an earlier paper. Because residual kernels in Poisson integrals are not C^∞-class functions, we use a dynamical space mapping in order to increase the efficiency of advanced quadrature schemes. In terms of accuracy, results are better by orders of magnitude than for the classical fast Fourier transform (FFT) methods.