Online HP-35 Emulator - First scientific pocket calculator from 1972

[Gerson W. Barbosa]

(12-30-2023 04:46 PM)EdS2 Wrote:  

(12-25-2023 01:50 PM)Gerson W. Barbosa Wrote:  8 7 8 ln 1 6 × ENTER↑ ln ÷ ln

This is a completely unexpected and remarkable pi approximation - how did you find it? Is there any significance - surely getting 10 digits of closeness out of 5 digits of input can't just be a numerical coincidence?

In the thread linked by Thomas Klemm above, there's a link to this paper. At page 23 therein there's this approximation by Simon Plouffe:

\(\pi \approx \) 689/396/ln(689/396)

When I saw it I thought of a more slightly complex approximation and tried it on the HP-32S Solver:

LN(A×LN(B)÷LN(A×LN(B)))=\(\pi \)

When making A=16 and solving for B I got B=877.999998596 , which I found a remarkable result (I'm doing this on my remaining HP-33s because my HP-32S is not working properly anymore). I don't remember whether I used Markovitch's method on it, but I suspect it gets a better score than Plouffe's approximation. It was about 14 years ago, judging by this thread.

Regards,

Gerson.