  From the beginning of KAM theory, it was realized that its applicability to
realistic problems depended on developing quantitative estimates on the sizes
of the perturbations allowed. In this paper we present results on the existence
of quasi-periodic solutions for conformally symplectic systems in
non-perturbative regimes. We recall that, for conformally symplectic systems,
finding the solution requires also to find a "drift parameter". We present a
proof on the existence of solutions for values of the parameters which agree
with more than three figures with the numerically conjectured optimal values.
  The verification of the estimates presented here is not completely rigorous
since we do not control the round-off error. Nevertheless, running with
different precision shows very little difference in the results. Given the high
precision of the calculation and the simplicity of the estimates, this does not
seem to affect the results. A full verification should be done implementing
interval arithmetic.
  We make available the approximate solutions, the highly efficient algorithms
to generate them (incorporating high precision based on the MPFR library) and
the routines used to verify the applicability of the theorem.
