The IceCube Neutrino Observatory, deployed inside the deep glacial ice at the South Pole, is the largest neutrino telescope in the world. While eight years have passed since IceCube discovered a diffuse flux of high-energy astrophysical neutrinos, the sources of the vast majority of these neutrinos remain unknown. Here, we present a new search for neutrino point sources that improves the accuracy of the statistical analysis, especially in the low energy regime. We replaced the usual Gaussian approximations of IceCube's point spread function with precise numerical representations, obtained from simulations, and combined them with new machine learning-based estimates of event energies and angular errors. Depending on the source properties, the new analysis provides improved source localization, flux characterization and thereby discovery potential (by up to 30%) over previous works. The analysis will be applied to IceCube data that has been recorded with the full 86-string detector configuration from 2011 to 2020 and includes improved detector calibration.