  The recoil associated with photon emission is key to the dynamics of
ultrarelativistic electrons in strong electromagnetic fields, as are found in
high-intensity laser-matter interactions and astrophysical environments such as
neutron star magnetospheres. When the energy of the photon becomes comparable
to that of the electron, it is necessary to use quantum electrodynamics (QED)
to describe the dynamics accurately. However, computing the appropriate
scattering matrix element from strong-field QED is not generally possible due
to multiparticle effects and the complex structure of the electromagnetic
fields. Therefore these interactions are treated semiclassically, coupling
probabilistic emission events to classical electrodynamics using rates
calculated in the locally constant field approximation. Here we provide
comprehensive benchmarking of this approach against the exact QED calculation
for nonlinear Compton scattering of electrons in an intense laser pulse. We
find agreement at the percentage level between the photon spectra, as well as
between the models' predictions of absorption from the background field, for
normalized amplitudes $a_0 > 5$. We discuss possible routes towards improved
numerical methods and the implications of our results for the study of QED
cascades.
