Threaded Mode | Linear Mode

Thomas Klemm · 03-16-2019, 06:14 PM

(03-15-2019 12:45 AM)Valentin Albillo Wrote: … trivially easy to convert to RPN, RPL or whatever:

Here's a program for the HP-42S:

Code:

00 { 36-Byte Prgm }

01▸LBL "A002491"

02 1ᴇ-3

03 RCL+ ST Y

04 X<>Y

05▸LBL 00

06 DSE ST Y

07 GTO 01

08 RTN

09▸LBL 01

10 ENTER

11 +/-

12 RCL ST Z

13 IP

14 MOD

15 +

16 GTO 00

Quote:FNP(1000000) -> 3.14159

Example:

1E6
XEQ "A002491"
318,310,503,562

1E12
X<>Y
÷
3.14158656032

Cheers
Thomas

Valentin Albillo · 03-16-2019, 10:43 PM

(03-16-2019 05:11 PM)Thomas Klemm Wrote:

Excellent ! It's a "The Far Side" one, isn't it ?

I was presented with an original "The Far Side" 1992 Calendar back then and every day of the year had one such (366 in all) and I loved it, the humour and art were truly amazing !

Have a nice weekend.
V.
.

Thomas Okken · (This post was last modified: 03-17-2019 12:39 AM by Thomas Okken.)

(03-16-2019 06:27 AM)Carsen Wrote:
(03-15-2019 05:07 AM)Valentin Albillo Wrote: To reciprocate, I'd recommend the song "PI" (Greek character, actually) from Kate Bush's 2005 double album "Aerial", in which she sings up to the 78^th decimal place of Pi, and after that from the 101^st to the 137 ^th.

Interesting! (I like exploring new styles). After listening to the entire song, your description describes it perfectly. Especially the word "atmospheric.

I like how there's another song about Pi. Pretty cool.

If you like that, she took the whole laid back and atmospheric thing to another level on "50 Words For Snow." The title alludes to the many words for snow the Inuit languages are supposed to have, but it's actually a lot more poetic than that. Smile

ttw · 03-17-2019, 12:56 AM

For songs, there's always, "By By Miss American Pi."

Gerson W. Barbosa · (This post was last modified: 03-22-2019 04:39 PM by Gerson W. Barbosa.)

(03-14-2019 08:58 PM)Gerson W. Barbosa Wrote: Optimization and conversion to HP calculators are left as an exercise for the reader.

The following is a very readable RPL version of the wp34s program in post #6:

« '3/2*√3' '2*√3' → n a b
  « 1 n
    START '√(a*b)' →NUM 'a' STO '2*a*b/(a+b)' →NUM 'b' STO
    NEXT '√(a*b)' →NUM 'a' 1 'n' STO+ STO '((2+2^((1/a-72*n-1)/36))*a+b)/(3+2^((1/a-72*n-1)/36))' EVAL
  »
»

The original algorithm (in modern terms) by Archimedes of Syracuse is followed by a simple empirical adjustment formula in function of a and b, his lower and upper bounds, respectively. Starting with inscribed and circumscribed hexagons to a circle of diameter 1, after four more iterations we obtain a = 3.14103195089 and b = 3.14271459965, which Archimedes rationalized to 223/71 and 22/7.

' (27*(a^2*b)^(1/3) - 4*(2*a + b))/15' is a better formula. Furthermore, it allows for a shorter wp34s program:

0001 **LBL A
0002 # 003
0003 [sqrt]
0004 RCL+ X
0005 # 1/2
0006 INC X
0007 RCL[times] L
0008 RCL[times] Y
0009 [sqrt]
0010 ||
0011 STO+ X
0012 RCL L
0013 DSE Z
0014 BACK 006
0015 RCL[times] Y
0016 [sqrt]
0017 [cmplx]ENTER
0018 x[^2]
0019 [times]
0020 [^3][sqrt]
0021 # 027
0022 [times]
0023 [<->] YZXT
0024 STO+ X
0025 +
0026 # 004
0027 [times]
0028 -
0029 # 015
0030 /
0031 END

4 A -> 3.141592653583662847516781540551053

10 A -> 3.141592653589793238462554257894971

16 A -> 3.141592653589793238462643383279503

(03-17-2019, 01:48 PM) PS: In order to keep both a and b on the stack, let’s change lines 0023 through 0027 to

0023 RCL Y
0024 RCL+ X
0025 RCL+ T
0026 RCL+ X
0027 RCL+ X

4 A -> 3.14159265358 ; (27*(a^2*b)^(1/3) - 4*(2*a + b))/15
Rv -> 3.14103195089 ; a
Rv -> 3.14271459965 ; b

(03-19-2019, 06:42 PM) PPS: 1 iteration should return the values for the hexagon, not the dodecagon. Fixed.

--------

WP 34S:

0001 **LBL A
0002 # 004
0003 # 027
0004 [sqrt]
0005 RCL/ Y
0006 y[<->] L
0007 RCL[times] Y
0008 [sqrt]
0009 ||
0010 STO+ X
0011 RCL L
0012 DSE Z
0013 BACK 006
0014 RCL[times] Y
0015 [sqrt]
0016 [cmplx]ENTER
0017 x[^2]
0018 [times]
0019 [^3][sqrt]
0020 # 027
0021 [times]
0022 RCL Y
0023 STO+ X
0024 RCL+ T
0025 STO+ X
0026 STO+ X
0027 -
0028 # 015
0029 /
0030 END

1 A -> 3.141460911773498978633977995517088 ; (27*(a^2*b)^(1/3) - 4*(2*a + b))/15
  Rv -> 2.999999999999999999999999999999999 ; a, perimeter of the inscribed hexagon ( 3 )
  Rv -> 3.464101615137754587054892683011744 ; b, perimeter of the circumscribed hexagon ( √12 )

5 A -> 3.141592653583662847516781540551053 ; (27*(a^2*b)^(1/3) - 4*(2*a + b))/15
  Rv -> 3.141031950890509638111352926459658 ; a, perimeter of the inscribed 96-gon
  Rv -> 3.142714599645368298168859093772122 ; b, perimeter of the circumscribed 96-gon

17 A -> 3.141592653589793238462643383279503 ; (27*(a^2*b)^(1/3) - 4*(2*a + b))/15
  Rv -> 3.141592653556370963662823316554114 ; a, perimeter of the inscribed 196608-gon
  Rv -> 3.141592653656637788064203581586042 ; b, perimeter of the circumscribed 196608-gon

--------

HP-42S/Free42:

00 { 72-Byte Prgm }
01▸LBL "PI"
02 27
03 SQRT
04 1.6875
05 SQRT
06▸LBL 00
07 RCL× ST Y
08 SQRT
09 RCL× ST Y
10 X<>Y
11 LASTX
12 +
13 LASTX
14 R↓
15 ÷
16 STO+ ST X
17 R↑
18 DSE ST Z
19 GTO 00
20 RCL× ST Y
21 SQRT
22 STO ST Z
23 X↑2
24 RCL× ST Y
25 3
26 1/X
27 Y↑X
28 27
29 ×
30 R↑
31 STO+ ST X
32 RCL+ ST Z
33 STO+ ST X
34 STO+ ST X
35 -
36 15
37 ÷
38 .END.

1 XEQ PI -> X: 3.141460911773498978633977995517088

R↓       -> Y: 3
          -> X: 3.464101615137754587054892683011746

17 XEQ PI -> X: 3.141592653589793238462643383279505

R↓       -> Y: 3.141592653556370963662823316554116
          -> X: 3.141592653656637788064203581586044

--------

HP 50g:

5

« '3/4*√3' '3*√3' → n a b
  « 1 n
    START '√(a*b)' →NUM 'a' STO
      '2*a*b/(a+b)' →NUM 'b' STO
    NEXT
    '√(a*b)' →NUM 'a' STO a b
    '(27*(a^2*b)^(1/3)-4*(2*a+b))/15' EVAL
  »
»

EVAL ->

3:    3.14103195089
2:    3.14271459965
1:    3.14159265358

--------

Finally, a stack-only wp34s version that preserves the original X-register content:

0001 **LBL A
0002 # 004
0003 # 027
0004 [sqrt]
0005 RCL/ Y
0006 y[<->] L
0007 RCL[times] Y
0008 [sqrt]
0009 ||
0010 STO+ X
0011 RCL L
0012 DSE Z
0013 BACK 006
0014 RCL[times] Y
0015 [sqrt]
0016 STO Z
0017 x[^2]
0018 RCL[times] Y
0019 [^3][sqrt]
0020 STO L
0021 CLx
0022 # 027
0023 STO[times] L
0024 x[<->] Z
0025 z[<->] L
0026 STO L
0027 STO+ L
0028 x[<->] Y
0029 STO+ L
0030 x[<->] L
0031 STO+ X
0032 STO+ X
0033 STO- Z
0034 CLx
0035 # 015
0036 STO/ Z
0037 x[<->] L
0038 x[<->] Z
0039 END

314 ENTER 5 A -> 3.141592653583662847516781540551053
           Rv -> 3.141031950890509638111352926459658
           Rv -> 3.142714599645368298168859093772122
           Rv -> 314

--------

(03-20-2019, 02:17 PM) PPPS: It ain't over till it's over (which doesn't mean further optimizations are not possible).

--------

WP 34S:

0001 **LBL A
0002 # 027
0003 [sqrt]
0004 # 1/2
0005 RCL[times] Y
0006 ||
0007 STO+ X
0008 RCL L
0009 RCL[times] Y
0010 [sqrt]
0011 DSE Z
0012 BACK 006
0013 [cmplx]ENTER
0014 x[^2]
0015 [times]
0016 [^3][sqrt]
0017 # 027
0018 [times]
0019 RCL Y
0020 STO+ X
0021 RCL+ T
0022 STO+ X
0023 STO+ X
0024 -
0025 # 015
0026 /
0027 END

1 A -> 3.141460911773498978633977995517088 ; (27*(a^2*b)^(1/3) - 4*(2*a + b))/15
  Rv -> 2.999999999999999999999999999999999 ; a, perimeter of the inscribed hexagon ( 3 )
  Rv -> 3.464101615137754587054892683011744 ; b, perimeter of the circumscribed hexagon ( √12 )

5 A -> 3.141592653583662847516781540551053 ; (27*(a^2*b)^(1/3) - 4*(2*a + b))/15
  Rv -> 3.141031950890509638111352926459658 ; a, perimeter of the inscribed 96-gon
  Rv -> 3.142714599645368298168859093772122 ; b, perimeter of the circumscribed 96-gon

17 A -> 3.141592653589793238462643383279503 ; (27*(a^2*b)^(1/3) - 4*(2*a + b))/15
  Rv -> 3.141592653556370963662823316554114 ; a, perimeter of the inscribed 196608-gon
  Rv -> 3.141592653656637788064203581586042 ; b, perimeter of the circumscribed 196608-gon

--------

HP-42S/Free42:

00 { 68-Byte Prgm }
01▸LBL "PI"
02 27
03 SQRT
04 6.75
05 SQRT
06▸LBL 00
07 R↓
08 RCL× ST T
09 STO+ ST X
10 R↑
11 RCL+ ST L
12 STO÷ ST Y
13 X<> ST L
14 RCL× ST Y
15 SQRT
16 DSE ST Z
17 GTO 00
18 STO ST Z
19 X↑2
20 RCL× ST Y
21 3
22 1/X
23 Y↑X
24 27
25 ×
26 R↑
27 STO+ ST X
28 RCL+ ST Z
29 STO+ ST X
30 STO+ ST X
31 -
32 15
33 ÷
34 .END.

1 XEQ PI -> X: 3.141460911773498978633977995517088

R↓       -> Y: 3
          -> X: 3.464101615137754587054892683011745

17 XEQ PI -> X: 3.141592653589793238462643383279505

R↓       -> Y: 3.141592653556370963662823316554116
          -> X: 3.141592653656637788064203581586044

--------

HP 50g:

5

« '3/2*√3' '3*√3' → n a b
  « 1 n
    START
      '2*a*b/(a+b)' →NUM 'b' STO
      '√(a*b)' →NUM 'a' STO
    NEXT
    a b '(27*(a^2*b)^(1/3)-4*(2*a+b))/15' EVAL
  »
»

EVAL ->

3:    3.14103195089
2:    3.14271459965
1:    3.14159265358

--------

Finally, a stack-only wp34s version that preserves the original X-register content:

0001 **LBL A
0002 # 027
0003 [sqrt]
0004 # 1/2
0005 RCL[times] Y
0006 ||
0007 STO+ X
0008 RCL L
0009 RCL[times] Y
0010 [sqrt]
0011 DSE Z
0012 BACK 006
0013 STO Z
0014 x[^2]
0015 RCL[times] Y
0016 [^3][sqrt]
0017 STO L
0018 CLx
0019 # 027
0020 STO[times] L
0021 x[<->] Z
0022 z[<->] L
0023 STO L
0024 STO+ L
0025 x[<->] Y
0026 STO+ L
0027 x[<->] L
0028 STO+ X
0029 STO+ X
0030 STO- Z
0031 CLx
0032 # 015
0033 STO/ Z
0034 x[<->] L
0035 x[<->] Z
0036 END

314 ENTER 5 A -> 3.141592653583662847516781540551053
           Rv -> 3.141031950890509638111352926459658
           Rv -> 3.142714599645368298168859093772122
           Rv -> 314

--------

(03-22-2019, 04:39 PM) PPPPS: At least one more step can be saved in the second wp34S program:

--------

0001 **LBL A
0002 # 027
0003 [sqrt]
0004 # 1/2
0005 RCL[times] Y
0006 ||
0007 STO+ X
0008 RCL L
0009 RCL[times] Y
0010 [sqrt]
0011 DSE Z
0012 BACK 006
0013 STO Z
0014 x[<->] L
0015 RCL[times] Y
0016 [^3][sqrt]
0017 STO L
0018 CLx
0019 # 027
0020 STO[times] L
0021 [<->] ZYZT
0022 STO+ Z
0023 [<->] YZXT
0024 STO+ Y
0025 x[<->] L
0026 [<->] YZXT
0027 STO+ X
0028 STO+ X
0029 STO- Z
0030 CLx
0031 # 015
0032 STO/ Z
0033 x[<->] L
0034 x[<->] Z
0035 END

314 ENTER 5 A -> 3.141592653583662847516781540551053
           Rv -> 3.141031950890509638111352926459658
           Rv -> 3.142714599645368298168859093772122
           Rv -> 314

--------

EdS2 · 03-18-2019, 08:07 AM

(03-16-2019 03:18 PM)Thomas Klemm Wrote:
(03-16-2019 07:50 AM)EdS2 Wrote: Indeed - a clue would be nice! Please?

A002491

Thanks! That led to Kevin Brown's Rounding Up To PI - MathPages.

Thomas Klemm · 03-18-2019, 05:40 PM

(03-18-2019 08:07 AM)EdS2 Wrote: Thanks! That led to Kevin Brown's Rounding Up To PI - MathPages.

Here are other links that I found interesting:

The Combinatorics of Mancala-Type Games:
Ayo, Tchoukaillon, and 1/π
Duane M. Broline, Daniel E. Loeb

My Favorite Integer Sequences
by N. J. A. Sloane
Chapter 12 Tchoukaillon solitaire (or Mancala, or Kalahari)

Quote:The sequence can be obtained by a sieving process: write 1, 2, . . . in a
column. To get the second column, cross off 1 and every second number. To
get the third column, cross off the first and every third number. Then cross off
the first and every fourth number, and so on (see Fig. 13). The top number in
column n is t(n).

1 1
2 2 2
3 3
4 4 4 4
5 5
6 6 6 6 6
7 7
8 8 8
9 9
10 10 10 10 10 10
11 11
12 12 12 12 12 12 12
13 13
14 14 14
15 15
16 16 16 16
17 17
18 18 18 18 18 18 18 18
19 19
20 20 20

Fig. 13. A sieve to generate the sequence t(1),t(2), . . . = 1, 2, 4, 6, 10, 12, 18, . . ..
At stage n, the first number and every nth are crossed off.

Cheers
Thomas