Systems of particles with interactions given by the Jagla core-softened pair potential are known to exhibit water-like thermodynamic anomalies and a liquid-liquid phase transition. The drawback of the Jagla potential is that it is characterized by discontinuous forces acting between particles and thus is not suitable for standard molecular dynamics (MD) simulations. Here we introduce a smooth version of the Jagla potential based on two Fermi distributions and study the properties of a system of particles interacting via this new "Fermi-Jagla" pair potential by using standard MD simulations. We find that the liquid based on the Fermi-Jagla potential retains most of the properties of the liquid based on the original Jagla potential. Namely, it exhibits the following water-like anomalies: (i) decrease of density, (ii) increase of compressibility, kappa(T)(T,P), and (iii) increase of isobaric specific heat, C(P)(T,P), upon isobaric cooling, and (iv) increase of diffusivity upon isothermal compression. The Fermi-Jagla potential also exhibits (i') density minima, (ii') compressibility minima, (iii') isobaric specific heat minima upon isobaric cooling, and (iv') diffusivity minima upon isothermal compression. As in the Jagla model case, we find a liquid-liquid phase transition (LLPT) and a liquid-liquid critical point in the equilibrium liquid. Contrary to the case of the original Jagla model liquid, the LLPT line for the Fermi-Jagla potential has a negative slope in the P-T plane that extends well above the crystallization temperature. This feature makes the Fermi-Jagla potential a better candidate to reproduce the behavior of tetrahedral liquids including water, for which the LLPT line observed in simulations has also negative slope. In the glass state, the Fermi-Jagla pair potential results in reversible polyamorphism between low- and high-density amorphous solids (LDA and HDA, respectively). We also find that HDA results from pressure-induced amorphization of the model's low pressure crystal, as observed in water and other materials. The Fermi-Jagla pair potential, being a smooth function of the interparticle separation, can be easily implemented in standard MD simulation codes. Moreover, since spontaneous crystallization for the Fermi-Jagla potential can be avoided by fast cooling, it can be used to study the phenomenology of glasses.