  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.
Both short and long times are investigated and compared with recent analytical
predictions for the large-deviation forms of the probability of rare
fluctuations. At short times the agreement with the analytical expression is
spectacular. We observe that the far left and right tails, with exponents 5/2
and 3/2 respectively, are preserved until large time. We present some evidence
for the predicted non-trivial crossover in the left tail from the 5/2 tail
exponent to the cubic tail of Tracy-Widom, although the details of the full
scaling form remains beyond reach.
