Side note—finite differencing (which, you’re right, typically throws away half of your precision) isn’t the only way to get a computer to take a derivative. Automatic differentiation packages will typically get you the derivative of an explicitly defined function to roughly the accuracy with which you can evaluate the function itself.
I’m not familiar with the n-body problem series solution, though; there’s lots of other ways that could turn out to be impractical to evaluate.
Side note—finite differencing (which, you’re right, typically throws away half of your precision) isn’t the only way to get a computer to take a derivative. Automatic differentiation packages will typically get you the derivative of an explicitly defined function to roughly the accuracy with which you can evaluate the function itself.
I’m not familiar with the n-body problem series solution, though; there’s lots of other ways that could turn out to be impractical to evaluate.