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Ellipse circumference formulabetter than Ramanujan II
Posted:
Jan 18, 2013 12:50 AM


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*[((1sqrt(2)/2) + (sqrt(2)/2) * (b/a)^(0.454))^(22pi)](2pi4)4 Maximum absolute relative error: ~3.8E5 (~3.4E5 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(1x^2)):
C~4a[1+(pi/21)/(((1sqrt(2)/2)+(sqrt(2)/2)*(b/a)^(0.454))^(2pi2))]
I will continue on my quest to make even more dramatic improvements.



