While I'm glad to see the OP got a good minimax solution at the end, it seems like the article missed clarifying one of the key points: error waveforms over a specified interval are critical, and if you don't see the characteristic minimax-like wiggle, you're wasting easy opportunity for improvement.

Taylor series in general are a poor choice, and Pade approximants of Taylor series are equally poor. If you're going to use Pade approximants, they should be of the original function.

I prefer Chebyshev approximation: https://www.embeddedrelated.com/showarticle/152.php which is often close enough to the more complicated Remez algorithm.