VERIFYING THE LENGTH OF ELLIPSES.

.Verifying the length of a Quarter of Ellipse.
The length L of the NE Quarter of an Ellipse (and so of any Quarter) is found here by means of asin and nacos functions, adding the values of two complementary arcs ending on the same point:
{ x, y }; the point is selected on RC:
{ x = √0.5, y = b√0.5 } when:  b ≥ c ;
it is selected between RC and FC:
( √0.5 ≤ x < c ), ( b√0.5 ≤ y < b2 ), when:  b < c .
So, you compute the length L this way:
 L = asin( y ) + nacos( x ) .
This value of L may be verified against methods of classic Calculus, such as the following.

.Series of MacLaurin.
This is not the famous Taylor-MacLaurin general Series, but a dedicated Series to compute le length of an Ellipse. It's an infinite Series on the powers of the Eccentricity. Since that vanishes for the Circle, the Series converges better on less eccentric Ellipse.
For the description of the Series see: numericana.

.Series of Gauss-Kummer.
It's an infinite Series on the powers of the parameter:
 h = (a - b)/(a + b) , which in the Implosive Model, where a = 1 always, becomes:  h = (1 - b)/(1 + b) .
This parameter vanishes for b = a (Circle), so this Series converges faster for less eccentric Ellipses.
For the description of the Series see: numericana (with language implementation), or: mathforum or: mathworld

.Series of Cayley.
It'a an infinite Series on the powers of the parameter:  (b/a)2 , which in the Implosive Model, where a = 1 always, becomes:  b2 .
This parameter becomes even smaller for growing eccentricity, so this Series converges faster for more eccentric Ellipses.
For the description of the Series see: numericana (with language implementation).

.Approximation formulas.
Besides the above exact Series, there is a lot of approximation formulas; near every famous mathematician made his own; Ramanujan made two formulas, may be the most famous of all; David W. Cantrell made the last, may be the first of current century. Many of these formulas are descripted in the references above.