Stoneham's series

[Gerson W. Barbosa]

(06-08-2024 05:02 AM)Johnh Wrote:  ....further on this, I put it into Free42, similar code that I posted above, with a 'View' line added to monitor the convergence towards Pi. I let it run 30 minutes, by which time it had completed 84 million loops, which is more than twice as many per minute as the Jovial hp15 sim!

Here’s an equivalent expression plus a correction factor which allows for 33 correct digits with only fifteen loops:

\[\pi \approx \left(4\prod _{k=1}^{n}{\frac {(2k+1)^2-1}{(2k+1)^2}}\right) \left({1-\frac{2}{8n+9+\frac{1\times3}{8n+8+\frac{3\times5}{8n+8+… +\frac{\left({2n}\right)^2-1}{8n+8}}}}}\right)\]

Code:


00 { 29-Byte Prgm }
01▸LBL "Sπ"
02 4
03▸LBL 00
04 RCL ST Y
05 STO+ ST X
06 SIGN
07 RCL+ ST L
08 X↑2
09 ABS
10 DSE ST X
11 RCL÷ ST L
12 ×
13 DSE ST Y
14 GTO 00
15 END

1332322 XEQ Sπ ->

3.141593243085161166771739970151(294) (~ 3 seconds on Free42)

Code:


00 { 68-Byte Prgm }
01▸LBL "SWπ"
02 4
03 STO 01
04 STO+ ST X
05 RCL× ST Y
06 RCL+ ST L
07 RCL ST X
08▸LBL 01
09 RCL ST Z
10 STO+ ST X
11 X↑2
12 DSE ST X
13 X<>Y
14 ÷
15 RCL+ ST Y
16 RCL ST Z
17 STO+ ST X
18 SIGN
19 RCL+ ST L
20 X↑2
21 ABS
22 DSE ST X
23 RCL÷ ST L
24 STO× 01
25 R↓
26 DSE ST Z
27 GTO 01
28 1
29 +
30 1/X
31 STO+ ST X
32 +/-
33 1
34 +
35 RCL× 01
36 END

15 XEQ SWπ ->

3.14159265358979323846264338327950(4)

5 XEQ SWπ ->

3.1415926535(8) (~ 2 seconds on the HP-42S)