  We revisit the study of the mass functions and the bias of dark matter halos.
Focusing on the limit of rare massive halos, we point out that exact analytical
results can be obtained for the large-mass tail of the halo mass function. This
is most easily seen from a steepest-descent approach, that becomes
asymptotically exact for rare events. We also revisit the traditional
derivation of the bias of massive halos, associated with overdense regions in
the primordial density field. We check that the theoretical large-mass cutoff
agrees with the mass functions measured in numerical simulations. For halos
defined by a nonlinear threshold $\delta=200$ this corresponds to using a
linear threshold $\delta_L\simeq 1.59$ instead of the traditional value $\simeq
1.686$. We also provide a fitting formula that matches simulations over all
mass scales and obeys the exact large-mass tail. Next, paying attention to the
Lagrangian-Eulerian mapping (i.e. corrections associated with the motions of
halos), we improve the standard analytical formula for the bias of massive
halos. We check that our prediction, which contains no free parameter, agrees
reasonably well with numerical simulations. In particular, it recovers the
steepening of the dependence on scale of the bias that is observed at higher
redshifts, which published fitting formulae did not capture. This behavior
mostly arises from nonlinear biasing.
