What is the fastest way to get the value of π?

Question

I\'m looking for the fastest way to obtain the value of π, as a personal challenge. More specifically, I\'m using ways that don\'t involve using #define constan

Answer 1

Calculate PI at compile-time with D.

( Copied from DSource.org )

/** Calculate pi at compile time
 *
 * Compile with dmd -c pi.d
 */
module calcpi;

import meta.math;
import meta.conv;

/** real evaluateSeries!(real x, real metafunction!(real y, int n) term)
 *
 * Evaluate a power series at compile time.
 *
 * Given a metafunction of the form
 *  real term!(real y, int n),
 * which gives the nth term of a convergent series at the point y
 * (where the first term is n==1), and a real number x,
 * this metafunction calculates the infinite sum at the point x
 * by adding terms until the sum doesn't change any more.
 */
template evaluateSeries(real x, alias term, int n=1, real sumsofar=0.0)
{
  static if (n>1 && sumsofar == sumsofar + term!(x, n+1)) {
     const real evaluateSeries = sumsofar;
  } else {
     const real evaluateSeries = evaluateSeries!(x, term, n+1, sumsofar + term!(x, n));
  }
}

/*** Calculate atan(x) at compile time.
 *
 * Uses the Maclaurin formula
 *  atan(z) = z - z^3/3 + Z^5/5 - Z^7/7 + ...
 */
template atan(real z)
{
    const real atan = evaluateSeries!(z, atanTerm);
}

template atanTerm(real x, int n)
{
    const real atanTerm =  (n & 1 ? 1 : -1) * pow!(x, 2*n-1)/(2*n-1);
}

/// Machin's formula for pi
/// pi/4 = 4 atan(1/5) - atan(1/239).
pragma(msg, "PI = " ~ fcvt!(4.0 * (4*atan!(1/5.0) - atan!(1/239.0))) );

Answer 2

This version (in Delphi) is nothing special, but it is at least faster than the version Nick Hodge posted on his blog :). On my machine, it takes about 16 seconds to do a billion iterations, giving a value of 3.1415926525879 (the accurate part is in bold).

program calcpi;

{$APPTYPE CONSOLE}

uses
  SysUtils;

var
  start, finish: TDateTime;

function CalculatePi(iterations: integer): double;
var
  numerator, denominator, i: integer;
  sum: double;
begin
  {
  PI may be approximated with this formula:
  4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 .......)
  //}
  numerator := 1;
  denominator := 1;
  sum := 0;
  for i := 1 to iterations do begin
    sum := sum + (numerator/denominator);
    denominator := denominator + 2;
    numerator := -numerator;
  end;
  Result := 4 * sum;
end;

begin
  try
    start := Now;
    WriteLn(FloatToStr(CalculatePi(StrToInt(ParamStr(1)))));
    finish := Now;
    WriteLn('Seconds:' + FormatDateTime('hh:mm:ss.zz',finish-start));
  except
    on E:Exception do
      Writeln(E.Classname, ': ', E.Message);
  end;
end.

Answer 3

Instead of defining pi as a constant, I always use acos(-1).

Answer 4

If by fastest you mean fastest to type in the code, here's the golfscript solution:

;''6666,-2%{2+.2/@*\/10.3??2*+}*`1000<~\;

Answer 5

The Monte Carlo method, as mentioned, applies some great concepts but it is, clearly, not the fastest, not by a long shot, not by any reasonable measure. Also, it all depends on what kind of accuracy you are looking for. The fastest π I know of is the one with the digits hard coded. Looking at Pi and Pi[PDF], there are a lot of formulae.

Here is a method that converges quickly — about 14 digits per iteration. PiFast, the current fastest application, uses this formula with the FFT. I'll just write the formula, since the code is straightforward. This formula was almost found by Ramanujan and discovered by Chudnovsky. It is actually how he calculated several billion digits of the number — so it isn't a method to disregard. The formula will overflow quickly and, since we are dividing factorials, it would be advantageous then to delay such calculations to remove terms.

where,

Below is the Brent–Salamin algorithm. Wikipedia mentions that when a and b are "close enough" then (a + b)² / 4t will be an approximation of π. I'm not sure what "close enough" means, but from my tests, one iteration got 2 digits, two got 7, and three had 15, of course this is with doubles, so it might have an error based on its representation and the true calculation could be more accurate.

let pi_2 iters =
    let rec loop_ a b t p i =
        if i = 0 then a,b,t,p
        else
            let a_n = (a +. b) /. 2.0 
            and b_n = sqrt (a*.b)
            and p_n = 2.0 *. p in
            let t_n = t -. (p *. (a -. a_n) *. (a -. a_n)) in
            loop_ a_n b_n t_n p_n (i - 1)
    in 
    let a,b,t,p = loop_ (1.0) (1.0 /. (sqrt 2.0)) (1.0/.4.0) (1.0) iters in
    (a +. b) *. (a +. b) /. (4.0 *. t)

Lastly, how about some pi golf (800 digits)? 160 characters!

int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,f[b]=d%--g,d/=g--,--b;d*=b);}

Answer 6

Brent's method posted above by Chris is very good; Brent generally is a giant in the field of arbitrary-precision arithmetic.

If all you want is the Nth digit, the famous BBP formula is useful in hex