I work at NASA, supporting the person who wrote the codes that is now used for planetary dynamics simulation, like this. It is an extraordinarily complex problem, and a specialized academic discipline in its own right.

n-body dynamics is a complicated, chaotic system. Getting any precision at all over any reasonable period of time is a feat of itself.

Not entirely trivial. Because it is chaotic, it's not terribly well behaved. People who try to integrate the solar system forward in time have to put some effort into accuracy, and ultimately accuracy is limited by our limited knowledge of the initial state of the system.