**abstract:**
The one-point distribution of the height for the continuumKardar-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 ex- pression 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.

† A. Hartmann, S. N. Majumdar, P. Le Doussal, A. Rosso, G. Schehr, Europhys. Lett. 121, 67004 (2018)

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