The author uses a transfer-matrix technique to simulate field electronic emission from a flat metal. He compares, in particular, the results provided by this numerical scheme with those predicted by the standard Fowler-Nordheim equation. He considers for this study electric fields between 1 and 10 V/nm as well as work functions between 1.5 and 5 eV. The results demonstrate that the Fowler-Nordheim theory and the transfer-matrix calculations are globally in good agreement. With the Fermi energy of 10 eV considered in this work, the results provided by the standard Fowler-Nordheim equation are, however, systematically larger than the quantum-mechanical result, especially for low values of the work function and for high electric fields. This is essentially due to the fact the standard Fowler-Nordheim theory relies on the simple Jeffreys-Wentzel-Kramers- Brillouin approximation for evaluating the electronic transmission through the surface barrier of the emitter. A correction factor is thus established that enables the temperature-dependent version of the standard Fowler-Nordheim equation to match the exact quantum-mechanical result.