Pi Day MMXVIII

Today it is the fourteenth of March 2018. Today’s date — when written in the M/D/Y format –, 3/14/18, looks close enough to Archimedes’ constant’s decimal representation for it to be the constant’s celebratory day.
As always on Pi Day, I have implemented an algorithm to generate \pi, albeit this year’s accuracy is not the greatest (Try it online).

                        typedef double d;typedef long l;l f(l n          
                   ){l f=1;while(n>1)f*=n--;return f;}d ne(d v,          
                 l p){d r=1;for(l k=0;k<p;k++)r*=v;return r;}d           
                ps(d(*c)(l),l i,d x){d s=0;for(l k=0;k<i;k++)s           
               +=c(k)*       ne(x,        k);return                      
              s;}           d exc         (     l                        
             n){            return       1./f (n)                        
                           ; } d         exp(d x                         
                          )   {         return                           
                         ps(exc        ,20,x);}                          
                        d G( d         x){return                         
                        exp(-x        *x);}d I                           
                       (d a,d         b,d g,d                            
                     (* f)(d         )){d cs=                            
                    0;for( d         x=a;x<=                             
                   b;x +=g)         cs+=f(x)                             
                 *g;return          cs ;  }          int                 
               main( ) { d          pi_root         =I(                  
              -2.5, 2.5 ,           1e-4,G);      d pi                   
             = pi_root *            pi_root+(0xf&0xf0                    
             ) ; printf(             "%c%c%c%c%c%f%c"                    
             ,'p','i',                ' ','=',' ',pi                     
               ,'\n'                     ) ; }                           

I use various methods of generating \pi throughout the Pi Days; this time I chose to use an improper integral paired with a power series. \pi is calculated using a famous identity involving infinite continuous sums, roots, e, statistics and — of course — \pi.

\int\limits_{-\infty}^\infty e^{-x^2}\mathrm{d}x = \sqrt{\pi}

Furthermore, to compute e, the following identity is used.

\exp{x} = \sum\limits_{n=0}^\infty\frac{x^n}{n!}

Both formulae are combined, the approximated value of \sqrt{\pi} is squared and \pi is printed to stdout.

You can download this program’s prettified (some call it obfuscated, see above) source code pi.c and also the (nearly, as #include is not missing so that the compiler does not need to guess my dependencies) equivalent code in a more traditional source layout tpi.c.

Happy Pi Day!

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