How do I calculate retained interest on a partial payout out of a saving account?

Not quite sure I'm interpreting that right, but if you're just saying you're removed money from the account and what interest did you earn in the meantime, the interest was earned on what you had in there and has nothing to do with what you took out.

So if on Jan 10 you deposted $1000, and you are now taking out $500 of it, you are getting interest from Jan 10 until now on the initial $1000. Taking out $500 won't affect you now - but rather will affect what interest you earn from this point onwards.

And how you calculate it depends on how the bank is doing interest and when it compounds.