[VA] SRC #010 - Pi Day 2022 Special

[J-F Garnier]

Thanks Valentin for this interesting reading. Relations between pi and e always intrigued me.

(03-15-2022 03:11 PM)Ángel Martin Wrote:  As you guys know I'm "stuck" in the 41 world [...] and sure enough my FOCAL routines did not work at all.

Really? HP-41 user code can't do it?

Let's see:
Maybe it's better to transform Valentin's expression with log and then compute the exponential at the end.
Using pseudo algebraic language (I'm not comfortable with graphic equation editors), with ln as the natural log:

PN = exp(3/2) * Prod(n=2,N,e*(1-1/n²)^n²) / (1+1/(2*N)-1/(8*N²))
becomes
ln(PN) = 3/2 + Sum(n=2,N,1+n²*ln(1-1/n²)) - ln(1+1/(2*N)-1/(8*N²))

e doesn't appear explicitly any more, but of course it does at the end when computing exp(ln(PN)).

Here is the corresponding HP-41/42 program, using the here highly useful LN1+X function that preserves the accuracy for small 1/n² quantities to some extend:

01*LBL "PN2"
02 "RUNNING"
03 AVIEW
04 STO 00 ; N
05 1
06 -
07 STO 01 ; control loop 1..N-1
08 0
09*LBL 00 ; sum loop <---
10 RCL 01
11 1
12 + ; n=2..N
13 X^2
14 ENTER^
15 1/X
16 CHS
17 LN1+X
18 *
19 1
20 +
21 +
22 DSE 01
23 GTO 00 ; sum endloop --^
24 RCL 00
25 2
26 *
27 1/X
28 RCL 00
29 X^2
30 8
31 *
32 1/X
33 -
34 LN1+X ; correction factor
35 -
36*LBL 01 ; final result
37 1.5
38 +
39 E^X
40 CLD
41 END


and results for the HP-41:

10.00000000 RUN
RUNNING
3.141844397 ***

100.0000000 RUN
RUNNING
3.141592946 ***

200.0000000 RUN
RUNNING
3.141592701 ***

300.0000000 RUN
RUNNING
3.141592670 ***

400.0000000 RUN
RUNNING
3.141592651 *** best result

500.0000000 RUN
RUNNING
3.141592685 ***

Not bad for this ancient 10-digit machine, isn't it?

So thanks again Valentin for this contribution to the fascinating pi, and Ángel for giving me the opportunity to write a HP-41 code, something I'm rarely doing - I'm a HP-71B man - but the HP-41 language is so deeply buried in my mind since 40 years that it was very natural.

J-F