The Pool Calculator uses a table-based approximation for a limited range of pH and TA though it does account for Borates (CYA turns out doesn't matter that much if one looks at TA, at least for typical pool pH ranges). The accurate calculation is complicated and is done in my Pool Equations spreadsheet. I didn't provide a simple enough formula to be used online in a timely manner. This post gives a formula where one can calculate the amount of acid or base needed for a given pH change without too much difficulty (but I found that after The Pool Calculator was already done), but the inverse calculation of predicting the pH change from a quantity of acid or base addition requires iterative searching (i.e. no closed-form formula).