Predicting the orbits of the sun-earth-moon system is easy to do accurately because of the relative sizes and proximity of the bodies.
You don't even need to treat it like a 3-body problem, because the size of the earth/moon have an inconsequential impact on the sun, the size of the moon had an inconsequential impact on the earth, and the proximity of the moon means the sun has an inconsequential impact on it too. So you use a traditional "sphere of influence" 2 body problems, and get very accurate results.

But when the bodies are all large enough to significantly affect eachother's orbits, like a trinary star system, then doing simulation (iterative calculations) quickly builds up errors, especially as the steps are far apart (which they must be with their biological computer, given how slow it is). I haven't run the exact calculations, but I've done interactive simulation in the past and it can pretty quickly fall apart

Edit:

You're gonna look at me and you're gonna tell me that I'm wrong? Am I wrong? She wore a crown and came down in a bubble, Doug. Grow up bro, grow up.