Typically the error due to your numerical integrator that runs Newtons law backward in time (or forward for predictions) is larger than the floating point error

Typically the error due to your numerical integrator that runs Newtons law backward in time (or forward for predictions) is larger than the floating point error after a few time steps and it is not uncommon to run for millions of time steps. This is not just true for the trivial but terrible Euler integrator, but also for high-order symplectic integrators. I know that my office mate has run simulations of planet systems with different codes and they tend to diverge at around 100 million years. They still look like the same system, but orbital phases disagree. The uncertainty in the orbital elements (current orbital position and velocities) from which you start is not a major problem as long as all bodies are well separated. But our model would for example completely miss the body that crashed into the young Earth and left us with a moon.