Gauss-Kummer and Euler methods for ellipse circumference calculation can be used for extremely high eccentricities:
Note for example in the Gauss-Kummer:
the fractions can be pulled out:
Note for this case, the 1+1/4+1/64+... sums to 4/pi.
You can then work with:
and when h is VERY near 1 you can obtain a workable number of significant digits in short order. This can be done similarly for Euler's method. Do note, however, the rate of convergence is nothing like that of Cayley.
With only a couple hundred terms of the series, at least 6 significant digits can be produced over all eccentricities, and with exact endpoints using this method. Since I see some people throw their own names about to gain popularity, I suppose could rename this the Blankenhorn ellipse circumference modification to the Gauss-Kummer method, haha.
Use the popular method when the ratio a/b>=0.0005 and the modified method for a/b<0.0005.