Pi Approximation Day

[Valentin Albillo]

.
Hi, Jean-François,

(07-23-2022 11:44 AM)J-F Garnier Wrote:  

(07-23-2022 01:20 AM)Valentin Albillo Wrote:  Set FIX 6 and then run on your HP-42S (physical or virtual) this little piece of mystery code I've hastily concocted and see what you get:

Wonderful little program!

Thank you very much for your appreciation.

Quote:It uses the specific HP-42S stack capability, running it on my trusty 32S would require a few more steps. Of course the interest is in this 42S version.

The little code I gave was intentionally obfuscated and particularized to add just 10 terms, to try and hide the "10" as if it were only the loop counter. My general non-obfuscated version which will add N terms is this 14-step, 26-byte HP-42S program:

01 LBL "?"
02 10
03 LBL 00
04 RCL ST Y
05 X↑2
06 RCL+ ST L
07 X↑2
08 RCL× ST L
09 1/X
10 -
11 DSE ST Y
12 GTO 00
13 SQRT
14 END

Key in N and XEQ "?". Running it on Free42 Decimal produces the following results for N=100, 1000, 10000 and 100000 terms:

      PI      3.14159 26535 89793 23846 26433 83279 502
      -------------------------------------------------
      100     3.14159 26535 92821 63577 23401 91552 596      (13 digits save 3 ulp)
      1000    3.14159 26535 89793 27013 49308 33666 846      (18 digits save 4 ulp)
      10000   3.14159 26535 89793 23846 29615 34056 207      (23 digits save 3 ulp)
      100000  3.14159 26535 89793 23846 26433 86462 443      (28 digits save 3 ulp)
      -------------------------------------------------
      PI      3.14159 26535 89793 23846 26433 83279 502

I like this series for Pi very much: it converges very fast (order 6), the general term is extremely simple to program needing just 6 steps, and this makes for speedy looping and very short running times, e.g. using as little as 100 terms gets you better than 12-digit accuracy and on Free42 it runs instantly.

Quote:Thank you Valentin, it's a pleasure to read you again.

Thank you, J-F, your feedback is most appreciated and if more people would do likewise that would surely motivate me to post again.

Best regards.
V.