lambdaway :: spigot

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_h1 computing π with spigot

_ul [[https://rosettacode.org/wiki/Pi|https://rosettacode.org/wiki/Pi]]
_ul [[https://stackoverflow.com/questions/30747235/javascript-pi-%CF%80-calculator|https://stackoverflow.com/questions/30747235/javascript-pi-%CF%80-calculator]]
_ul [[https://www.cs.ox.ac.uk/jeremy.gibbons/publications/spigot.pdf|https://www.cs.ox.ac.uk/jeremy.gibbons/publications/spigot.pdf]]   
_ul [[https://rosettacode.org/wiki/Pi|https://rosettacode.org/wiki/Pi]]
_ul [[https://fr.wikipedia.org/wiki/Approximation_de_%CF%80|https://fr.wikipedia.org/wiki/Approximation_de_%CF%80]]
_ul [[http://www.eveandersson.com/pi/digits/1000000|http://www.eveandersson.com/pi/digits/1000000]]
_ul [[https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/BigInt|https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/BigInt]]

_h2 lambdatalk code

_p We require the {b lib_BN} javascript library.

{require lib_BN}

{pre
'{def genpi
 {def genpi.loop
  {lambda {:n :pi :q :r :t :i :z}
   {if {> :z :n}
    then :pi
    else {let { {:n :n} {:pi :pi} {:q :q} {:r :r} {:t :t} {:i :i} {:z :z}
                {:digit {BN./ {BN.+ {BN.* {BN.- {BN.* :i 27} 12} :q} 
                                    {BN.* :r 5} }  
                              {BN.* :t 5} } }
                {:u {BN.* {BN.+ {BN.* :i 3} 1} 
                          {BN.* 3 {BN.+ {BN.* :i 3} 2} } } }
              } {genpi.loop :n
                            {BN.+ :pi :digit}
                            {BN.* :q {BN.* :i {BN.- {BN.* :i 2} 1} }} 
                            {BN.* :u {BN.+ {BN.* :q {BN.- {BN.* :i 5} 2} } 
                                     {BN.- :r {BN.* :t :digit} }}}
                            {BN.* :t :u} 
                            {BN.+ :i 1}
                            {+ :z 1}} }}}}
 {lambda {:n}
  {genpi.loop :n # 1 180 60 2 0} }}
-> {def genpi
 {def genpi.loop
  {lambda {:n :pi :q :r :t :i :z}
   {if {> :z :n}
    then :pi
    else 
     {let { {:n :n} {:pi :pi} {:q :q} {:r :r} {:t :t} {:i :i} {:z :z}
            {:digit {BN./ {BN.+ {BN.* {BN.- {BN.* :i 27} 12} :q} 
                                {BN.* :r 5} }  
                          {BN.* :t 5} } }
            {:u {BN.* {BN.+ {BN.* :i 3} 1} 
                      {BN.* 3 
                            {BN.+ {BN.* :i 3} 2} } } }
          } {genpi.loop :n
                        {BN.+ :pi :digit}
                        {BN.* :q 
                              {BN.* :i {BN.- {BN.* :i 2} 1} }} 
                        {BN.* :u
                              {BN.+ {BN.* :q {BN.- {BN.* :i 5} 2} } 
                                    {BN.- :r {BN.* :t :digit} }}}
                        {BN.* :t :u} 
                        {BN.+ :i 1}
                        {+ :z 1}}
 }}}}

{lambda {:n}
  {genpi.loop :n # 1 180 60 2 0}
}}
}

_p We can generate {b π} with a choosen number of digits.

{prewrap

'{BN.DEC 15} -> 15 digits

'{genpi 15} 
-> {PI}  // in about 50ms, equivalent to javascript

'{BN.DEC 72} -> {BN.DEC 72} 72 digits

'{genpi 60}
-> 3.141592653589793238462643383279502884197169399375105820974944592307816406  // in about 500ms
}

_h2 computing 1000 digits

_p To go further the best is to build a javascript primitive using {b bigInts}.

{pre
LAMBDATALK.DICT["spigot"] = function() '{
  function generateDigitsOfPi(max) {
    var pi = "";
    var z = 0;
    var q = 1n;
    var r = 180n;
    var t = 60n;
    var i = 2n;
    while (z < max) {
        var digit = ((i * 27n - 12n) * q + r * 5n) / (t * 5n);
        pi += digit;
        var  u = (i * 3n + 1n) * 3n * (i * 3n + 2n);
        r = u * 10n * (q * (i * 5n - 2n) + (r - t * digit));
        q = q * 10n * i * (i * 2n - 1n); 
        i = i + 1n;
        t = t * u; 
        z++;
    }
    return pi 
  }
  var args = arguments[0].trim();
  return generateDigitsOfPi( args ); 
};
}

_p We can generate 1000 digits in about 70ms

{prewrap
3.'{W.rest {spigot 100}}
-> 3.{W.rest {spigot 1000}}  // in about 75ms
}

_h2 control

_p Here is the value of {b π} with 1000 digits given by [[http://www.eveandersson.com|http://www.eveandersson.com/pi/digits/1000000]].
{prewrap 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198}

_p {i alain marty | 2023/01/22}

{script 
LAMBDATALK.DICT["spigot"] = function() {
  function generateDigitsOfPi(max) {
    var pi = "";
    var z = 0;
    var q = 1n;
    var r = 180n;
    var t = 60n;
    var i = 2n;
    while (z < max) {
        var digit = ((i * 27n - 12n) * q + r * 5n) / (t * 5n);
        pi += digit;
 
      //  var u = i * 3n;
      //  u = (u + 1n) * 3n * (u + 2n);
 
        var  u = (i * 3n + 1n) * 3n * (i * 3n + 2n);

r = u * 10n * (q * (i * 5n - 2n) + (r - t * digit));
        q = q * 10n * i * (i * 2n - 1n); 
        i = i + 1n;
        t = t * u;  
        z++;
    }
    return pi 
  }
  var args = arguments[0].trim();
  return generateDigitsOfPi( args ); 
};

}