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I found out yesterday that the German government provides a flowchart for developers wanting to calculate income tax

And they even provide an API you can run tests against, to ensure that your implementation is correct:

Can you provide a link to the mentioned flowchart as well, please?

However, this is the calculation for the monthly (pay as you go) wage tax and other deductions and not the yearly income tax.

There's a pdf version as well:

The problem is in computing the input values; it's quite hard to figure out without examples, except the most simple cases.