Several years ago I had an intern investigate this very problem. A good intern task is an interesting but nonessential problem. This qualified as such; he even managed to turn that internship work as the basis of his masters thesis. Note: I obtained the author's permission to link to the thesis, so I've added a link to his thesis at the end of this answer.
What he found was that
- Using single precision floating point arithmetic is a bad idea for modeling an Earth to Moon trajectory or a Moon to Earth trajectory.
- Using double precision floating point arithmetic can be a good idea if one switches from geocentric to Moon-centric (or vice versa) at a "reasonable" place.
- Using extended precision extends the concept of "reasonableness".
A "reasonable" place to switch from Earth-centered to Moon-centered (or vice versa) was a place that resulted in an acceptably small loss in accuracy. Switching from Earth-centered to Moon-centered while the vehicle was still in low Earth orbit was not "reasonable", nor was waiting to switch until after performing insertion into a low lunar orbit. (Exception: Almost any place, or no place at all qualified as "reasonable" with 100 decimal place precision arithmetic.) With doubles, there was a fairly broad band of what qualified as "reasonable".
The Apollo program used the Laplace sphere of influence:
$$r_{\text{SOI}} \approx a \left(\frac m M\right)^{2/5}$$
where $a$ is the semi-major axis length, $m$ is the mass of the smaller object (the Moon in the case of a transfer from the Earth to the Moon), and $M$ is the mass of the larger object (the Earth in this case). This turned out to be "reasonable". Then again, so did the Hill sphere:
$$r_{\text{Hill}} \approx a \left(\frac m {3M}\right)^{1/3}$$
For the Earth-Moon system, with $a\approx 385000\,\text{km}$ and $m/M\approx 0.0123$ (a handy number to remember), the Laplace sphere of influence radius is about 66300 km while the Hill sphere radius is about 61600 km.
The optimal place to switch was in between the sphere of influence and the Hill sphere, but a bit closer to the sphere of influence. With double precision floating point arithmetic, the error in using either the sphere of influence or the Hill sphere was rather small compared to the error that resulted from using this optimal transition point. You won't get fired for choosing either the sphere of influence or the Hill sphere as the transition point.
Reference:
M. Vautier, Effect of Coordinate Switching on Simulation Accuracy of Translunar Trajectories, August 2008