Three-dimensional phase-field simulations are performed to follow the formation of facets at the surface of a silicon crystal in contact with its melt. We focus on the anisotropies of both the surface energy and the kinetic attachment coefficient. Analytical expressions are proposed for the corresponding anisotropy functions on the basis of simple physical considerations and of available experimental results for the equilibrium and the early growth shapes (Yang et al., 2014). Direct comparisons of these experimental results with the results of our simulations allow us to validate the proposed anisotropy functions. Numerical tests to cancel one of the anisotropies while keeping the other one show that both effects must be included to obtain agreement with the experiment.