I spent three days in coffee shops with a pen and paper taking first and then second derivatives of the Winkel Tripel formula

I felt bad for you reading that. There are loads of tools to do it automatically (matlab, wolfram, etc).

The stuff coming out of those computer algebra things is not always exactly the form you want to be stuffing in your code. Working it out manually focuses the mind.

They also tend to do it quite badly in some non-trivial cases. I've seen Wolfram spewing a three-line result which could be further simplified to a simple expression. It also doesn't always help to simply know the result; a lot of insight about numerical problems can be gained from how the result is derived.

I also spent several days churning formulae in bars. Got a lot of weird looks.

Great read! All of these bug stories are making me want bugs now.

Don't worry, you have them. :)

Bugs? There are no bugs. Just undocumented features. And solutions for things that I wasn't actually trying to solve.