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**    A very easy program to compute Pi with many digits.
**    No optimisations, no tricks, just a basic program to learn how
**    to compute in multiprecision.
**
** Formulae:
**
**    Pi/4 =    arctan(1/2)+arctan(1/3)                     (Hutton 1)
**    Pi/4 =  2*arctan(1/3)+arctan(1/7)                     (Hutton 2)
**    Pi/4 =  4*arctan(1/5)-arctan(1/239)                   (Machin)
**    Pi/4 = 12*arctan(1/18)+8*arctan(1/57)-5*arctan(1/239) (Gauss)
**
**      with arctan(x) =  x - x^3/3 + x^5/5 - ...
**
**    The Lehmer's measure is the sum of the inverse of the decimal
**    logarithm of the pk in the arctan(1/pk). The more the measure
**    is small, the more the formula is efficient.
**    For example, with Machin's formula:
**
**      E = 1/log10(5)+1/log10(239) = 1.852
**
** Data:
**
**    A big real (or multiprecision real) is defined in base B as:
**      X = x(0) + x(1)/B^1 + ... + x(n-1)/B^(n-1)
**      where 0<=x(i)<B
**
** Results: (PentiumII, 450Mhz)
**
**   Formula      :    Hutton 1  Hutton 2   Machin   Gauss
**   Lehmer's measure:   5.418     3.280      1.852    1.786
**
**  1000   decimals:     0.2s      0.1s       0.06s    0.06s
**  10000  decimals:    19.0s     11.4s       6.7s     6.4s
**  100000 decimals:  1891.0s   1144.0s     785.0s   622.0s
**
** With a little work it's possible to reduce those computation
** times by a factor 3 and more:
**
**     => Work with double instead of long and the base B can
**        be choosen as 10^8
**     => During the iterations the numbers you add are smaller
**        and smaller, take this in account in the +, *, /
**     => In the division of y=x/d, you may precompute 1/d and
**        avoid multiplications in the loop (only with doubles)
**     => MaxDiv may be increased to more than 3000 with doubles
**     => ...
*/
#include <time.h>
#include <stdio.h>
#include <malloc.h>
#include <math.h>
long B=10000; /* Working base */
long LB=4;    /* Log10(base)  */
long MaxDiv=450;  /* about sqrt(2^31/B) */
/*
** Set the big real x to the small integer Integer
*/
void SetToInteger (long n, long *x, long Integer) {
  long i;
  for (i=1; i<n; i++) x[i] = 0;
  x[0] = Integer;
}
/*
** Is the big real x equal to zero ?
*/
long IsZero (long n, long *x) {
  long i;
  for (i=0; i<n; i++)
    if (x[i]) return 0;
return 1;
}
/*
** Addition of big reals : x += y
**  Like school addition with carry management
*/
void Add (long n, long *x, long *y) {
  long carry=0, i;
  for (i=n-1; i>=0; i--) {
    x[i] += y[i]+carry;
    if (x[i]<B) carry = 0;
    else {
      carry = 1;
      x[i] -= B;
    }
  }
}
/*
** Substraction of big reals : x -= y
**  Like school substraction with carry management
**  x must be greater than y
*/
void Sub (long n, long *x, long *y) {
  long i;
  for (i=n-1; i>=0; i--) {
    x[i] -= y[i];
  if (x[i]<0) {
    if (i) {
        x[i] += B;
        x[i-1]--;
      }
  }
  }
}
/*
** Multiplication of the big real x by the integer q
** x = x*q.
**  Like school multiplication with carry management
*/
void Mul (long n, long *x, long q) {
  long carry=0, xi, i;
  for (i=n-1; i>=0; i--) {
    xi  = x[i]*q;
    xi += carry;
    if (xi>=B) {
      carry = xi/B;
      xi -= (carry*B);
    }
    else
      carry = 0;
    x[i] = xi;
}
}
/*
** Division of the big real x by the integer d
** The result is y=x/d.
**  Like school division with carry management
**  d is limited to MaxDiv*MaxDiv.
*/
void Div (long n, long *x, long d, long *y) {
  long carry=0, xi, q, i;
  for (i=0; i<n; i++) {
    xi    = x[i]+carry*B;
    q     = xi/d;
    carry = xi-q*d;
    y[i]  = q;
  }
}
/*
** Find the arc cotangent of the integer p = arctan (1/p)
**  Result in the big real x (size n)
**  buf1 and buf2 are two buffers of size n
*/
void arccot (long p, long n, long *x, long *buf1, long *buf2) {
  long p2=p*p, k=3, sign=0;
  long *uk=buf1, *vk=buf2;
  SetToInteger (n, x, 0);
  SetToInteger (n, uk, 1); /* uk = 1/p */
  Div (n, uk, p, uk);
  Add (n, x, uk);           /* x  = uk */

  while (!IsZero(n, uk)) {
    if (p<MaxDiv)
      Div (n, uk, p2, uk);  /* One step for small p */
    else {
      Div (n, uk, p, uk);   /* Two steps for large p (see division) */
      Div (n, uk, p, uk);
    }
    /* uk = u(k-1)/(p^2) */
    Div (n, uk, k, vk);       /* vk = uk/k  */
    if (sign) Add (n, x, vk); /* x = x+vk   */
    else Sub (n, x, vk);      /* x = x-vk   */
    k+=2;
    sign = 1-sign;
  }
}
/*
** Print the big real x
*/
void Print (long n, long *x) {
  long i;
  printf ("%d.", x[0]);
  for (i=1; i<n; i++) {
    printf ("%.4d", x[i]);
    if (i%25==0) printf ("%8d\n", i*4);
  }
  printf ("\n");
}
/*
** Computation of the constant Pi with arctan relations
*/
void main () {
  clock_t endclock, startclock;
  long NbDigits=10000, NbArctan;
  long p[10], m[10];
  long size=1+NbDigits/LB, i;
  long *Pi      = (long *)malloc(size*sizeof(long));
  long *arctan  = (long *)malloc(size*sizeof(long));
  long *buffer1 = (long *)malloc(size*sizeof(long));
  long *buffer2 = (long *)malloc(size*sizeof(long));
  startclock = clock();
  /*
  ** Formula used:
  **
  **   Pi/4 = 12*arctan(1/18)+8*arctan(1/57)-5*arctan(1/239) (Gauss)
  */
  NbArctan = 3;
  m[0] = 12; m[1] = 8;  m[2] = -5;
  p[0] = 18; p[1] = 57; p[2] = 239;
  SetToInteger (size, Pi, 0);
  /*
  ** Computation of Pi/4 = Sum(i) [m[i]*arctan(1/p[i])]
  */
  for (i=0; i<NbArctan; i++) {
    arccot (p[i], size, arctan, buffer1, buffer2);
    Mul (size, arctan, abs(m[i]));
    if (m[i]>0) Add (size, Pi, arctan);
    else        Sub (size, Pi, arctan);
  }
  Mul (size, Pi, 4);
  endclock = clock ();
  Print (size, Pi);  /* Print out of Pi */
  printf ("Computation time is : %9.2f seconds\n",
         (float)(endclock-startclock)/(float)CLOCKS_PER_SEC );
  free (Pi);
  free (arctan);
free (buffer1);
free (buffer2);
}


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