I don't have the exact formula, I'm not sure anyone does.
One thing to note is there is also an underpayment penalty of 0.5% per month or partial month. So if the IRS determined that you owed $1000 as of June 15, and you paid June 16, you would be assessed one day of interest but the full penalty. And the interest is compounded on the penalty as well as the balance.
Also, I think the interest rate is variable.
I think the challenge is not the exact formula but how the IRS applies it. For example, suppose you owe $5000 when you file your return and pay in full on April 15, 2024. The IRS wants taxes to be paid in quarterly installments or by withholding, so you technically owed $1250 on the previous April 15 2023, June 15, September 15, and January 15. So you need to take that amount owed from April 16, add the monthly penalty and the daily interest through June 15, then add the new amount owed and keep calculating monthly penalties and daily interest through September 15, then add the new amount for Sept 16, and so on. It's a complicated moving target.
