Ellipse circumference calculation method?

Question

Actually I know how to calculate the circumference of an ellipse using two methods and each one of them giving me different result.

The first method is using the formula: $E_c=2\pi\sqrt{\dfrac{a^2+b^2}{2}}$

The second method is determining the arc length of the first quart in the ellipse using elliptic integral multiplied by 4 (Look at picture below):

I want to know wath is the best method to get the exact circumference of an ellipse ?

Answer 1

The first one is an approximation, but it fails badly for $a \ll b $. Overall, I would recommend the exact formula of $4a E(e) $, which can be calculated quickly using the Arithmetic-Geometric Mean Method.

Answer 2

The formula $E_c=2\pi\sqrt{\frac{a^2+b^2}{2}}$ is not a bad approximation when $a$ is not far from $b$, but it is not a correct expression for the perimeter.

For example, when $a=1000$ and $b=1$, the true perimeter is not far from $4000$, and the formula predicts about $4440$.