  SMC X-1 is a high-mass X-ray binary with an orbital period of 3.9 days. The
mass of the neutron star is as low as $\sim 1 M_{\sun}$, suggesting that it was
likely to be formed through an electron-capture supernova rather an iron-core
collapse supernova. From the present system configurations, we argue that the
orbital period at the supernova was $\lesssim 10$ days. Since the mass transfer
process between the neutron star's progenitor and the companion star before the
supernova should have increased the orbital period to tens of days, a mechanism
with efficient orbit angular momentum loss and relatively small mass loss is
required to account for its current orbital period. We have calculated the
evolution of the progenitor binary systems from zero-age main-sequence to the
pre-supernova stage with different initial parameters and various mass and
angular momentum loss mechanisms. Our results show that the outflow from the
outer Langrangian point or a circumbinary disk formed during the mass transfer
phase may be qualified for this purpose. We point out that these mechanisms may
be popular in binary evolution and significantly affect the formation of
compact star binaries.
