  It is well known that the computation of accurate trajectories of the Lorenz
system is a difficult problem. Computed solutions are very sensitive to the
discretization error determined by the time step size and polynomial order of
the method, as well as round-off errors.
  In this work, we show how round-off errors limit the computability of the
Lorenz system and quantify exactly the length of intervals over which solutions
can be computed, expressed in terms of the floating point precision. Using
adjoint-based \emph{a posteriori} error analysis techniques, we estimate the
stability of computations with respect to initial data, discretization, and
round-off errors, respectively.
  The analysis is verified by computing an accurate solution on the time
interval $[0, 1000]$ using a very high order (order 200) finite element method
and very high floating point precision 400 digits).
