I read your note and can tell you I did come up with some formulas that are not so complicated yet with [subjectively] high precision.
I commented on your other thread re: ellipse circumference. If I have time I may play around with your formula more. It's surprising that what you came up wasn't known long ago - it's an example of experts being so much into theory that they overlook some approaches. In the sciences, this sort of inventive ability can be welcoming, unless there is pressure from jealous people to thwart advancements as I sadly had to endure. **No paper certificate measures the creative mind** yet human resource managers don't seem to comprehend this so well. I welcome you to view some of my websites such as http://freeideas2society.blogspot.com where I know my efforts are futile unless having a tremendous backing so I just give everything away for free.
Anyway, after Mr. Michon's conversion, your formula has great eye appeal. You should be proud of your ingenuity.