  A semi-relativistic formulation of light-matter interaction is derived using
the so called propagation gauge and the relativistic mass shift. We show that
relativistic effects induced by a super-intense laser field can, to a
surprisingly large extent, be accounted for by the Schr{\"o}dinger equation,
provided that we replace the rest mass in the propagation gauge Hamiltonian by
the corresponding time-dependent field-dressed mass. The validity of the
semi-relativistic approach is tested numerically on a hydrogen atom exposed to
an intense XUV laser pulse strong enough to accelerate the electron towards
relativistic velocities. It is found that while the results obtained from the
ordinary (non-relativistic) Schr{\"o}dinger equation generally differ from
those of the Dirac equation, merely demonstrating that relativistic effects are
significant, the semi-relativistic formulation provides results in quantitative
agreement with a fully relativistic treatment.
