Date: Jan 18, 2013 12:50 AM
Author: thomasinventions@yahoo.com
Subject: Ellipse circumference formula-better than Ramanujan II
A formula I just came up with today is simplistic, having a maximum relative error an entire magnitude lower than Ramanujan's second formula:

C~a*[((1-sqrt(2)/2) + (sqrt(2)/2) * (b/a)^(-0.454))^(2-2pi)](2pi-4)-4

Maximum absolute relative error: ~3.8E-5 (~3.4E-5 with -0.454012 as inner power) Note: b<>0

The above could be written as the following to see the direct decomposition into the elliptical integral of the second kind (C=4aE(1-x^2)):

C~4a[1+(pi/2-1)/(((1-sqrt(2)/2)+(sqrt(2)/2)*(b/a)^(-0.454))^(2pi-2))]

I will continue on my quest to make even more dramatic improvements.