1) Numerator having an outer multiple of pi and denominator having pi as a multiple of a constant (or of a term such as (1-x));

2) Numerator having an outer multiple of pi excepting on a constant;

3) Numerator having an outer multiple of pi raised to the power of x (or of (1-x) depending on the variable chosen).

In going this route, there would certainly be some condensing of the form presently shown here.

Alternatively, the formula could be written as:

A most interesting form for the calculation of the circumference of an ellipse as there is no π used in the formula. This is one of the most accurate fairly simple solitary formulas for the circumference of an ellipse over the range from the degenerate collapsed ellipse to circle. Being that π is not used, there can only be one exact endpoint, although with rounding to 8 significant digits, practically speaking this formula has two exact endpoints - the relative error for the circumference of a circle is exceedingly small, on the order of just greater than 1 part per quadrillion. The maximum relative error is about 4.2 parts per billion, permitting 8 significant digits.

Graph of Error Function:

A note on the development: The fractional approximation for pi, 80143857/25510582, was utilized. Note the sum of the coefficients of the denominator is 25510582 and 4 times this quantity plus the sum of the coefficients in the numerator is twice 80143857. One could parse out the pi from the large fraction in the formula leaving coefficients enabling the same 8 digits of accuracy. One such example for the coefficients, respectively, for the numerator: 955, 398483, 4717940, 5908033, 3134181, and for the denominator: 65872, 1469981, 5050067, 5568734, 4916104, 2412404.

Math Forum Discussion: 8 significant digit ellipse circumference formula, simplistic

Thomas Blankenhorn