  The one-point distribution of the height for the continuum
Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping
to the directed polymer in a random potential at high temperature. Using an
importance sampling approach, the distribution is obtained over a large range
of values, down to a probability density as small as $10^{-1000}$ in the tails.
The short time behavior is investigated and compared with recent analytical
predictions for the large-deviation forms of the probability of rare
fluctuations, showing a spectacular agreement with the analytical expressions.
The flat and stationary initial conditions are studied in the full space,
together with the droplet initial condition in the half-space.
