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pinkman · (This post was last modified: 06-17-2020 08:55 PM by pinkman.)

Hi,
Once again, I had another discussion with my cousin about PI (it's either PI or prime numbers...).
He told me about the beauty of the Viète's formula :

History here: link

This video shows the formula entered in CAS, and yes! I find it beautiful: https://youtu.be/BEYDBl94UpM
Regards,
Thibault

ramon_ea1gth · 06-17-2020, 09:37 PM

Cool! Really beautiful to see in the CAS the pi approximation in action! :-)

pinkman · (This post was last modified: 06-18-2020 02:49 PM by pinkman.)

And the convergence speed is acceptable : for p(18) we have 10 digits.
p(18) means Σ(x,x,1,18) = 171 iterations.
Compared to Wallis formula (https://www.hpmuseum.org/forum/thread-14...ght=wallis), it's absolutely fast!

cdmackay · 06-19-2020, 09:00 PM

very nice, thank you!

tiny tip: in CAS, for algebraic or fractional input, the a.b/c key also runs approx(); one less key-stroke than shift-enter.

Gerson W. Barbosa · 06-19-2020, 11:12 PM

(06-18-2020 12:51 PM)pinkman Wrote: Compared to Wallis formula (https://www.hpmuseum.org/forum/thread-14...ght=wallis), it's absolutely fast!

Yes, but it pales in comparison to the Wallis-Wasicki formula :-)

+---+---------------------+---------------------+
| N | 2*W | 2*W*C |
+---+---------------------+---------------------+
| 2 | 2.84444444444444444 | 3.14385964912280701 |
| 4 | 2.97215419501133786 | 3.14158816337302932 |
| 6 | 3.02317019200136082 | 3.14159266276745771 |
| 8 | 3.05058999605551092 | 3.14159265357083669 |
|10 | 3.06770380664349896 | 3.14159265358983256 |
|12 | 3.07940134316788626 | 3.14159265358979314 |
|14 | 3.08790206983111306 | 3.14159265358979321 |
+---+---------------------+---------------------+

Only 7 iterations (or 14, depending on how you implement the algorithm) for 18 correct decimal digits
(21697209162666264236130304/6906436179074198667175275 = 3.141592653589793238[633]...).
The Pascal code is available here. It should translate easily into the Prime programming language, but probably no joy with only 12 significant digits...

Gerson W. Barbosa · 06-23-2020, 06:39 PM

(06-18-2020 12:51 PM)pinkman Wrote: And the convergence speed is acceptable : for p(18) we have 10 digits.
p(18) means Σ(x,x,1,18) = 171 iterations.

For faster convergence, you might want to try this algorithm:

Code:

Program  Viete;                                                                                 

Uses Crt;                                                                                                            

var         k:  Byte;                                                                                                

            n: LongInt;                                                                                             

            c: Char;                                                                                                

   d, p, t, v: Extended;                                                                                             

begin                                                                                                               

  ClrScr;                                                                                                           

  WriteLn('+---+----------------------+----------------------+');                                                    

  WriteLn('| k |         2*v          |         ~ pi         |');                                                    

  WriteLn('+---+----------------------+----------------------+');                                                    

  v := 1;                                                                                                            

  n := 1;                                                                                                           

  d := 0;                                                                                                           

  for k := 1 to 15 do                                                                                                

    begin                                                                                                           

      n := n + n;                                                                                                   

      d := Sqrt(d + 2);                                                                                             

      v := v*d;                                                                                                      

      t := 6*v*v - 2/3;                                                                                             

      p := 2*n/v*(t + 1)/t;                                                                                         

      WriteLn('|',k:2,' |',2*n/v:21:18,' |',p:21:18,' |')

    end;                                                                                                             

  WriteLn('+---+----------------------+----------------------+');                                                   

  c := ReadKey                                                                                                       

end.

+---+----------------------+----------------------+
| k | 2*v | ~ pi |
+---+----------------------+----------------------+
| 1 | 2.828427124746190098 | 3.077994223988500989 |
| 2 | 3.061467458920718174 | 3.137427049842311165 |
| 3 | 3.121445152258052286 | 3.141329731135014592 |
| 4 | 3.136548490545939264 | 3.141576182507746574 |
| 5 | 3.140331156954752913 | 3.141591623553754287 |
| 6 | 3.141277250932772868 | 3.141592589203296386 |
| 7 | 3.141513801144301077 | 3.141592649565492850 |
| 8 | 3.141572940367091385 | 3.141592653338272210 |
| 9 | 3.141587725277159701 | 3.141592653574073140 |
|10 | 3.141591421511199975 | 3.141592653588810732 |
|11 | 3.141592345570117743 | 3.141592653589731833 |
|12 | 3.141592576584872667 | 3.141592653589789401 |
|13 | 3.141592634338562990 | 3.141592653589793000 |
|14 | 3.141592648776985670 | 3.141592653589793224 |
|15 | 3.141592652386591346 | 3.141592653589793238 |
+---+----------------------+----------------------+

pinkman · 06-23-2020, 09:58 PM

Quote:Yes, but it pales in comparison to the Wallis-Wasicki formula :-)

Here is a quick PPL port of your Wallis-Wasicki implementation:

Code:

EXPORT WallisWasicki()

BEGIN

LOCAL i, k, m, n;

LOCAL c, d, t, w;

  PRINT("+---+-------------+------------+");

  PRINT("| N |     2*W     |    2*W*C   |");

  PRINT("+---+-------------+------------+");

  FOR k := 1 TO 7 DO

      n := 2*k;

      w := 1;

      c := 0;

      d := 4*n;

      m := n + 1;

      FOR i := 1 TO k DO

          t := i*(16*i - 8);

          w := w*t*t/(t*(t - 2) - 3);

          c := (m - 4)/(d + m/(2 - c));

          m := m - 2

        END;

      c := 1 - c;

      PRINT("|" + n + " |" + 2*w + " |" + 2*w*c + " |");

    END;

  PRINT("+---+-------------+------------+");

  RETURN 2*w*c;

END;

The terminal output is not well formatted, but still easy to read.

I’m also porting your 314 pi digits algorithm, I had to stop (work...) but it will be ready in a few hours (if I find the time).

pinkman · 06-23-2020, 10:04 PM

(06-23-2020 06:39 PM)Gerson W. Barbosa Wrote:
(06-18-2020 12:51 PM)pinkman Wrote: And the convergence speed is acceptable : for p(18) we have 10 digits.
p(18) means Σ(x,x,1,18) = 171 iterations.

For faster convergence, you might want to try this algorithm:
[...]

Yes! But... I really love to see the CAS in action, even if it reaches its limit in terms of recursivity.

Gerson W. Barbosa · (This post was last modified: 07-15-2020 05:07 PM by Gerson W. Barbosa.)

(06-23-2020 06:39 PM)Gerson W. Barbosa Wrote: ...
For faster convergence, you might want to try this algorithm:
...

The following appears to be better (more tests required) . Only one line has been changed. Anyway, here is it again:

Code:

Program  Viete;                                                                                 

Uses Crt;                                                                                                            

var         k: Byte;                                                                                                

            n: LongInt;                                                                                             

            c: Char;                                                                                                

   d, p, t, v: Extended;                                                                                             

begin                                                                                                               

  ClrScr;                                                                                                           

  WriteLn('+---+----------------------+----------------------+');                                                    

  WriteLn('| k |         2*v          |         ~ pi         |');                                                    

  WriteLn('+---+----------------------+----------------------+');                                                    

  v := 1;                                                                                                            

  n := 1;                                                                                                           

  d := 0;                                                                                                           

  for k := 1 to 15 do                                                                                                

    begin                                                                                                           

      n := n + n;                                                                                                   

      d := Sqrt(d + 2);                                                                                             

      v := v*d;                                                                                                      

      t := 6*v*v - 27/10 - 1/(n*n);                                                                                            

      p := 2*n/v*(t + 1)/t;                                                                                         

      WriteLn('|',k:2,' |',2*n/v:21:18,' |',p:21:18,' |')

    end;                                                                                                             

  WriteLn('+---+----------------------+----------------------+');                                                   

  c := ReadKey                                                                                                       

end.

+---+----------------------+----------------------+
| k | 2*v | ~ pi |
+---+----------------------+----------------------+
| 1 | 2.828427124746190098 | 3.140960508696045357 |
| 2 | 3.061467458920718174 | 3.141593674140638978 |
| 3 | 3.121445152258052286 | 3.141592707164788547 |
| 4 | 3.136548490545939264 | 3.141592654569488290 |
| 5 | 3.140331156954752913 | 3.141592653605653769 |
| 6 | 3.141277250932772868 | 3.141592653590043215 |
| 7 | 3.141513801144301077 | 3.141592653589797153 |
| 8 | 3.141572940367091385 | 3.141592653589793300 |
| 9 | 3.141587725277159701 | 3.141592653589793240 |
|10 | 3.141591421511199975 | 3.141592653589793239 |
+---+----------------------+----------------------+

P.S.:

This yields 1.8 digits per iteration, three times as much when compared to the plain Viète's formula.

P.P.S.:

This new formula is easier to program and will yield slightly more than two digits per term:

\(\pi \approx 2\left ( \frac{4}{3} \times \frac{16}{15}\times \frac{36}{35}\times\frac{64}{63} \times \cdots \times \frac{ 4n ^{2}}{ 4n ^{2}-1}\right )\cdot \left ( 1+\frac{2}{8n+3+\frac{3}{8n+4+\frac{15}{8n+4+ \frac{35}{8n+4 + \frac{63}{\dots\frac{\ddots }{8n+4+\frac{4n^{2}-1}{8n+4}}} }}} } \right )\)

TurboBCD program:

Code:

Program Wallis_Wasicki;

var d, n, i: Byte;

          t: Integer;

       c, w: Real;

begin

  ClrScr;

  WriteLn('+---+---------------------+---------------------+');

  WriteLn('| N |         2*W         |     2*W*C/(C-2)     |');

  WriteLn('+---+---------------------+---------------------+');

  for n := 1 to 8 do

    begin

      c := 0;

      d := 8*n + 4;

      w := 1;

      for i := n downto 1 do

        begin

          t := 4*i*i;

          w := w*t/(t - 1);

          c := (t - 1)/(c + d)

        end;

      c := c + d + 1;

      WriteLn('|',n:2,' |',2*w:20:17,' |',2*c*w/(c - 2):20:17,' |')

    end;

    WriteLn('+---+---------------------+---------------------+')

end.

+---+---------------------+---------------------+
| N | 2*W | 2*W*C/(C-2) |
+---+---------------------+---------------------+
| 1 | 2.66666666666666666 | 3.14074074074074073 |
| 2 | 2.84444444444444446 | 3.14159848961611078 |
| 3 | 2.92571428571428570 | 3.14159260997123044 |
| 4 | 2.97215419501133786 | 3.14159265392705764 |
| 5 | 3.00217595455690690 | 3.14159265358714120 |
| 6 | 3.02317019200136082 | 3.14159265358981426 |
| 7 | 3.03867362888341912 | 3.14159265358979309 |
| 8 | 3.05058999605551094 | 3.14159265358979325 |
+---+---------------------+---------------------+

In this table N is the number of terms, W is the Wallis Product evaluated to N terms and C/(C - 2) is the correction factor.

Gerson W. Barbosa · 06-23-2020, 11:00 PM

(06-23-2020 09:58 PM)pinkman Wrote:
Quote:Yes, but it pales in comparison to the Wallis-Wasicki formula :-)

Here is a quick PPL port of your Wallis-Wasicki implementation:

...

I’m also porting your 314 pi digits algorithm, I had to stop (work...) but it will be ready in a few hours (if I find the time).

Thank you very much for the PPL port!

Perhaps it's time I should get myself a Prime. But I think I will wait until a good arbitrary precision package is available, either built-in or third-party.

Gerson.

pinkman · 06-24-2020, 01:15 PM

The port of the 200-314-997 decimals will need the CAS, which has capabilities for manipulating big integers but not arbitrary precision decimal numbers.
I’ll check if I can avoid using decimals by calculating on a base of estimated_pi * 10^200 (or less, depending on CAS limit).
Need a few hours more!

CyberAngel · 06-29-2020, 05:52 AM

I wish the extremely fast HP Prime had the LongFloat Library integrated.
- -
VP

pinkman · 06-29-2020, 10:54 PM

Me too!

compsystems · (This post was last modified: 07-09-2020 04:07 AM by compsystems.)

There must be some open source LongFloat library, if so Xcas could include it.
[Image: 01-38.jpg]

https://keisan.casio.com/

pinkman · 07-16-2020, 04:42 PM

(06-23-2020 09:58 PM)pinkman Wrote: Here is a quick PPL port of your Wallis-Wasicki implementation:

[...]

Gerson sent me a PM to thank me for having posted this code to hpcalc.org, but in fact I did not post anything, I guess Eric did.
Funny, and good idea, but the credits come to Gerson and his continuous fraction quick convergence for Wallis product.