The rst important observation is that none of the functionals gives a satisfactory density for liquid water, which should be around 1 g/cm 3 . Also, the experimental density dierence between liquid water and ice is not well reproduced; experimentally, ice is about 90 % less dense than water. We start the discussion with the semilocal revised Perdew-Burke-Ernzerhof (RPBE and revPBE ) functionals, all of which consistently give too low densities. The results also depend very much on the choice of damping, i.e. how the attractive van der Waals (vdW) terms are "damped" to zero as the constituents approach each other. The damping is necessary because the approximate semi-local DFT functionals already account for some of the interactions associated with correlation eects. Observing such a strong dependence on the choice of damping is very disturbing. In particular, only the zero damping (zd), i.e. the original functional form proposed by Grimme et al., yields a higher density for water than for ice, while the Becke-Johnson damping (BJ) incorrectly predicts ice to be denser than water so that the slope of the melting line is likely to be inverted. The baseline DFT functionals also have some inuence on the results. The RPBE and revPBE functionals are closely related. The revPBE functional is a revised version of the original PBE functional. It includes mod-ications to the exchange part of the functional aimed at improving its accuracy in describing properties such as reaction energies of molecules. RPBE was primarily aimed at improving adsorption energies at surfaces and as opposed to revPBE locally observes the Lieb-Oxford criterion. Overall, revPBED3-zd gives the best agreement with the experiment, although it underestimates the density of ice and, even more so, water. This underestimation is also true for the dierence in densities between water and ice, as illustrated in Fig. . The ice melting points are presented in Table . They were determined by performing anisotropic NpT calculations on 864 molecules at standard pressure and at various temperatures varying by 2 K. The temperatures reported are usually the average of the lowest temperature at which melting occurred and the highest temperature at which freezing occurred (or, in a few cases, the temperature at which neither freezing nor melting occurred). We expect the errors to be smaller than ±2 K, and note that agree-ment with our previously reported melting temperature for larger ensembles for RPBED3-zd is excellent. Table shows that most semi-local functionals overestimate the experimental value of 273.15 K. Among these, the -zd ones align reasonably well, whereas the -BJ versions notably deviate, even exceeding the temperature of maximum density. Now we discuss the results for hybrid functionals, as well as SCAN and optPBE. For optPBE and revPBE0-D3/BJ we illustrate results for the original database (open circles), as well as on-the-y results for optPBE and results using the hybrid database covering a larger density variation for revPBE0-D3/BJ (lled circles). The predicted densities are somewhat too small using the databases constructed for RPBE-D3 structures, and the error increases as the density dierence to RPBE-D3 increases. The error becomes even larger for revPBE0-D3/zd (not shown). This is a sign of the somewhat limited extrapolation capabilities of the MLPs applied in the present work, but we note that despite a density dierence of 10-20% density errors remain below 1%.