  We show that crossing probabilities in 2D critical site percolation on the
triangular lattice in a piecewise analytic Jordan domain converge with power
law rate in the mesh size to their limit given by the Cardy-Smirnov formula. We
use this result to obtain new upper and lower bounds of exp(O(sqrt(log log R)))
R^(-1/3) for the probability that the cluster at the origin in the half-plane
has diameter R, improving the previously known estimate of R^(-1/3+o(1)).
