Quantum-mechanical calculations have become an indispensable tool for computational materials science due to their unprecedented versatility and reliability. Focusing specifically on the Density Functional Theory (DFT), the reliability of its numerous implementations was tested and verified mostly for pure elements. An extensive testing of binaries, ternaries and more-component phases is still rather rare due to a vast configurational space that is nearly infinite already for binaries. Importantly, there are well known cases of theoretical predictions contradicting experiments. In this paper, we analyze the failure of theory to reproduce correctly the ground state of the Fe3Al inter metallic compound. Namely, most exchange-correlation (xc) energies within the generalized gradient approximation (GGA) predict this material in the L1(2) structure instead of the experimentally found D0(3) structure. We test the performance of 36 combinations of 6 different GGA parametrizations and 6 different Fe and Al potentials. These combinations are evaluated employing a multi-dimensional multi-criteria descriptor {triangle E, a, {mu (Fe)}, {C-ij}} consisting of fundamental thermodynamic properties (energy difference triangle E between the D0(3) and L1(2) structures), a structural aspect (lattice parameter a), electronic structure related magnetic properties (local magnetic moments of Fe atoms {mu (Fe)}) and elastic properties (a complete set of second-order elastic constants {C-ij}). Considering the thermodynamic stability as the most critical aspect, we identify the Perdew-Wang (1991) GGA xc-functional parametrization as the optimum for describing the electronic structure of the Fe3Al compound.