  An analytical approximation of a nonlinear force-free magnetic field (NLFFF)
solution was developed in Paper I, while a numerical code that performs fast
forward-fitting of this NLFFF approximation to a line-of-sight magnetogram and
coronal 3D loops has been described and tested in Paper II. Here we calculate
the free magnetic energy $E_{\rm free}=E_{\rm N}-E_{\rm P}$, i.e. the
difference of the magnetic energies between the nonpotential field and the
potential field. A second method to estimate the free energy is obtained from
the mean misalignment angle change $\Delta\mu=\mu_{\rm P}-\mu_{\rm N}$ between
the potential and nonpotential field, which scales as $E_{\rm free}/E_{\rm P}
\approx \tan^2{(\Delta\mu)}$. For four active regions observed with STEREO in
2007 we find free energies in the range of $q_{\rm free}=(E_{\rm free}/E_{\rm
P}) \approx 1%-10%$, with an uncertainty of less than $\pm 2%$ between the two
methods, while the free energies obtained from 11 other NLFFF codes exhibit a
larger scatter of order $\approx\pm10%$. We find also a correlation between the
free magnetic energy and the GOES flux of the largest flare that occurred
during the observing period, which can be quantified by an exponential
relationship, $F_{\rm GOES} \propto \exp{(q_{\rm free}/0.015)}$, implying an
exponentiation of the dissipated currents.
