  We prove sharp, computable error estimates for the propagation of errors in
the numerical solution of ordinary differential equations. The new estimates
extend previous estimates of the influence of data errors and discretisation
errors with a new term accounting for the propagation of numerical round-off
errors, showing that the accumulated round-off error is inversely proportional
to the square root of the step size. As a consequence, the numeric precision
eventually sets the limit for the pointwise computability of accurate solutions
of any ODE. The theoretical results are supported by numerically computed
solutions and error estimates for the Lorenz system and the van der Pol
oscillator.
