Quantitative precipitation estimation (QPE) is a difficult task, particularly in complex topography, and requires the adjustment of empirical relations between radar observables and precipitation quantities, as well as methods to transform observations aloft to estimations at the ground level. In this work, we tackle this classical problem with a new twist, by training a random forest (RF) regression to learn a QPE model directly from a large database comprising 4 years of combined gauge and polarimetric radar observations. This algorithm is carefully fine-tuned by optimizing its hyperparameters and then compared with MeteoSwiss' current operational non-polarimetric QPE method. The evaluation shows that the RF algorithm is able to significantly reduce the error and the bias of the predicted precipitation intensities, especially for large and solid or mixed precipitation. In weak precipitation, however, and despite a posteriori bias correction, the RF method has a tendency to overestimate. The trained RF is then adapted to run in a quasi-operational setup providing 5 min QPE estimates on a Cartesian grid, using a simple temporal disaggregation scheme. A series of six case studies reveal that the RF method creates realistic precipitation fields, with no visible radar artifacts, that appear less smooth than the original non-polarimetric QPE and offers an improved performance for five out of six events.