(12C Platinum) Length of An Ellipse

[John Keith]

I have tried several of these methods and I believe that Ramanujan's approximation is the clear winner. It gives 6 - 9 digits of accuracy for reasonable ellipses and is small and fast. For exact solutions, Albert's AGM2 program is fastest by far, almost 200* as fast as numerical integration on the HP 50! The Gauss-Kummer infinite series (series 2 from the PDF in post #2) looks good but the terms are very slow to compute and the program is large and complicated.

At the risk of trolling this thread I am posting HP-48 versions of the programs for those who may be interested.

Ramanujan:

Code:


\<< DUP2 - ROT ROT + DUP \pi \->NUM * ROT ROT / SQ 3. * 4. OVER - \v/ 10. + / 1. + *
\>>

For the HP 50, ROT ROT can be replaced by UNROT.

AGM2:

Code:


\<< DUP2 SQ SWAP SQ + .5 * .25 0. 0. \-> a b s t k s1
  \<<
    WHILE b a - 'k' STO s t k SQ * - 's1' STO s s1 \=/
    REPEAT a b * \v/ 'b' STO 'a' k .5 * STO+ s1 's' STO t 't' STO+
    END \pi \->NUM 2. * s * k .5 * a + /
  \>>
\>>

This is just a straightforward translation of Albert's Lua code which could be made smaller and faster by moving local variables to the stack at the cost of readability (and effort!).