/* pi_compute.c there are many ways to compute pi     */
/*              based on 4*atan(1) = pi               */
/*              based on tan(a/2) = (1-cos(a))/sin(a) */
/*              further reduction for tan(a/4)        */
/*              sin(a/2)=sqrt((1-cos(a))/2)           */
/*              cos(a/2)=sqrt((1+cos(a))/2)           */
/* atan(1/x)=Pi/2-atan(x)   atan(-x)=-atan(x)         */

#include <stdio.h>
#include <math.h>

int main(int argc, char *argv[])
{
  double Pi = 3.1415926535897932384626433832795028841971 ;
  double b1, b1a, b2, b3;

  printf("pi_compute.c running (all values are approximate) \n");
  printf("string pi =  3.1415926535897932384626433832795028841971 \n");
  printf("21.17f pi = %20.17f \n",Pi);
  printf("g      pi =  %g \n", Pi);
  printf("4*atan(1) = %20.17f \n", 4.0*atan(1.0));
  printf("4*(5*atan(1/7) + 2*atan(3/79))\n");
  printf("          = %20.17f \n", 4.0*(5.0*atan(1.0/7.0)+2.0*atan(3.0/79.0)));
  /* 30 degrees, pi/6, sine=1/2, cos=sqrt(3)/2, tan=sin/cos */
  b1 = 1.0/sqrt(3.0);
  printf("6*atan(1/sqrt3)  b1=1/sqrt3=%g\n", b1);
  printf("          = %20.17f \n", 6.0*atan(b1));
  /* 22.5 degrees, pi/8, tan(pi/8) = (1-cos(pi/4))/sin(pi/4) or sqrt2-1 */
  b1a = sqrt(2.0)-1.0;
  printf("8*atan(sqrt2-1)  b1a=sqrt2-1=%g\n", b1a);
  printf("          = %20.17f \n", 8.0*atan(b1a));
  /* 15 degrees, pi/12, tan=((1-cos(pi/6))/sin(pi/6)) */
  b2 = 2.0-sqrt(3.0);
  printf("12*atan(2-sqrt3)  b2=2-sqrt3=%g\n", b2);
  printf("          = %20.17f \n", 12.0*atan(b2));
  /* 7.5 degrees, pi/24, tan=((2sqrt(b2)-1)/b2) */
  b3 = (2.0*sqrt(b2)-1.0)/b2;
  printf("24*atan(b3)  b3=(2*sqrt(b2)-1)/b2=%g\n", b3);
  printf("          = %20.17f \n", 24.0*atan(b3));
  printf("\nseries for atan(x) shows need for x<1/2 for fast convergence\n");
  printf("x - x^3/3 + x^5/5 - x^7/7 + x^9/9 ...\n");
  return 0;
} /* end pi_compute */