Astrophysicists rely on solution for the $N$-body problem for a myriad of dynamics problems, from those in dark matter dynamics to those in planetary dynamics.  Over 50 years ago, Miller found that errors in our simulations of the N-body problem grow exponentially in time.  So why should one trust any such simulation over long times?  In this talk, I explore a three-body planetary system.  The allowed motion over long times can be estimated analytically using recent results from Hadden & Lithwich.  I find conventional integration methods yield statistics that are as accurate as higher order symplectic methods.  In one test, a symplectic corrector method gave unreliable statistics.    
(Joint work with Sam Hadden and Junichiro Makino; Reference: arXiv:1910.08667)

