Demo (FX) Coding Tutorials

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);}

Here's how we do it. Note that there are probably more edge conditions than you realize at first glance.

This is the second version, unrolled for 5x more performance (and yes, I benchmarked it). I figured I'd optimize it because this function can be called hundreds of times per page.

/// <summary>
/// Produces optional, URL-friendly version of a title, "like-this-one". 
/// hand-tuned for speed, reflects performance refactoring contributed
/// by John Gietzen (user otac0n) 
/// </summary>
public static string URLFriendly(string title)
{
    if (title == null) return "";

    const int maxlen = 80;
    int len = title.Length;
    bool prevdash = false;
    var sb = new StringBuilder(len);
    char c;

    for (int i = 0; i < len; i++)
    {
        c = title[i];
        if ((c >= 'a' && c <= 'z') || (c >= '0' && c <= '9'))
        {
            sb.Append(c);
            prevdash = false;
        }
        else if (c >= 'A' && c <= 'Z')
        {
            // tricky way to convert to lowercase
            sb.Append((char)(c | 32));
            prevdash = false;
        }
        else if (c == ' ' || c == ',' || c == '.' || c == '/' || 
            c == '\\' || c == '-' || c == '_' || c == '=')
        {
            if (!prevdash && sb.Length > 0)
            {
                sb.Append('-');
                prevdash = true;
            }
        }
        else if ((int)c >= 128)
        {
            int prevlen = sb.Length;
            sb.Append(RemapInternationalCharToAscii(c));
            if (prevlen != sb.Length) prevdash = false;
        }
        if (i == maxlen) break;
    }

    if (prevdash)
        return sb.ToString().Substring(0, sb.Length - 1);
    else
        return sb.ToString();
}

To see the previous version of the code this replaced (but is functionally equivalent to, and 5x faster), view revision history of this post (click the date link).

Also, the RemapInternationalCharToAscii method source code can be found here.