Perimeter of Ellipse

[Albert Chan]

Improvement to my previous EK(m)
When m is small, the term E(m) - K(m) is losing significant digits.
Worse, catastrophic cancellation occurs at the base of recursion.

EKmc(m) returns E(m)-c, K(m)-c, where c=pi/2, similarly to expm1(x) = exp(x) - 1
EK(m) is now a simple wrapper for EKmc(m), adding back the c's.

Code:

from cmath import sqrt, pi
c = pi/2

def EKmc(m):
    'return E(m)-c, K(m)-c, where c=pi/2, m!=1'
    if 1+abs(m) == 1: m *= 0.25*c; return -m, m
    b = sqrt(1-m)
    h = m / ((b+2)*b+1)     # = (1-b)/(1+b)
    e, k = EKmc(h*h)
    hk, hc = h*k, h*c
    return (hk-k) + (e+b*e) - b*hc, (hk+k) + hc

def EK(m):
    'return E(m), K(m), assumed K(1)=99 (should be inf)'
    if abs(m-1)==0: return 1, 99
    e, k = EKmc(m)
    return e+c, k+c

>>> e, k = EK(0.96)
>>> 4 * 50 * abs(e)                       # ellipse_perimeter(10,50), error = 0 ULP
210.10044539689002