  The nuclear symmetry energy is defined by the second derivative of the energy
per nucleon with respect to the proton-neutron asymmetry, and is sometimes
approximated by the energy difference between the neutron matter and the
symmetric matter. Accuracy of this approximation is assessed analytically and
numerically, within the Hartree-Fock theory using effective interactions. By
decomposing the nuclear-matter energy, the relative error of each term is
expressed analytically; it is constant or is a single-variable function
determined by the function type. The full errors are evaluated for several
effective interactions, by inserting values of the parameters. Although the
errors stay within $10\%$ up to twice of the normal density irrespective of the
interactions, at higher densities accuracy of the approximation significantly
depends on the interactions.
