In 1914, Ramanujan presented two approximations for the perimeter of an
ellipse, the more accurate of which is often written as
[please view in a fixed-width font]

                         3h
  pi(a + b) ( 1 + ----------------- )
                  10 + Sqrt(4 - 3h)

where a and b are the lengths of the semiaxes of the ellipse and
h = ((a-b)/(a+b))^2. This approximation is of truly remarkable accuracy for
ellipses of low eccentricity; and its worst relative error, occurring when
eccentricity e = 1, is 7/22 pi -1, or about -4*10^(-4).

A natural question to ask is whether this approximation can be improved in
some attractive way. One possible way would be to add a corrective term c,
so that the approximation would look like

  pi(a + b) (1 + (original fraction) + c).

Indeed, just yesterday I received an email from one Edgar Erives specifying
an expression -- unfortunately, rather complicated -- for such a corrective
term. However, there is a simple corrective term which may be used:

  c = (4/pi - 14/11) h^12

which I designed to precisely correct the approximation when e = 1 and
reduce worst |relative error|, while being benign with respect to the
accuracy of the approximation when e is small.

We may then approximate the perimeter of an ellipse,
with |relative error| < 1.5*10^(-5), by

                       3h                            12
  pi(a + b) (1 + ----------------- + (4/pi - 14/11) h  ).
                 10 + Sqrt(4 - 3h)

[By the way, the exponent 12 is not "precise", in the sense that one could
numerically determine a nonintegral exponent which would give a slightly
smaller max|relative error|. Nonetheless, it would still exceed
1.4*10^(-5), and so, in the interest of computational efficiency, it seems
nicest to use exactly 12 for the exponent.]

Those who would like to have a simple approximation providing even smaller
worst |relative error| may be interested in item (2) at the end of my
recent article "Two New Approximations, in a Certain Form, for the
Perimeter of an Ellipse" at
<http://mathforum.org/discuss/sci.math/m/604793/604793>. However, the
accuracy of (2) for near-circular ellipses is far worse than that of the
approximation presented here.

Has anyone seen this "corrected" version of Ramanujan's approximation
before?