#include <math.h>
#include <stdio.h>
#include <string.h>
#define DIAMETER 6875.0
#define CIRCLE_PI 3.1415926536
typedef
  struct
  {
    int sign;
    int deg;
    int min;
    int sec;
  }
  degree;
typedef
  struct
  {
    degree latitude;
    degree longitude;
  }
  coordinate;
typedef
  struct
  {
    double x;
    double y;
    double z;
  }
  vector;
coordinate ship;
coordinate iceberg;
double dist;
void str_to_coord(degree *ptr,char *p,char positive)
{
  while(*p<'0'||*p>'9')
    p++;
  ptr->deg=0;
  while(*p>='0'&&*p<='9')
    ptr->deg=ptr->deg*10+(*(p++)-'0');
  while(*p<'0'||*p>'9')
    p++;
  ptr->min=0;
  while(*p>='0'&&*p<='9')
    ptr->min=ptr->min*10+(*(p++)-'0');
  while(*p<'0'||*p>'9')
    p++;
  ptr->sec=0;
  while(*p>='0'&&*p<='9')
    ptr->sec=ptr->sec*10+(*(p++)-'0');
  while(*p<'A'||*p>'Z')
    p++;
  ptr->sign=(*p==positive?1:-1);
}
void init()
{
  char buf[80];
  gets(buf);
  gets(buf);
  gets(buf);
  gets(buf);
  str_to_coord(&(ship.latitude),buf,'N');
  gets(buf);
  str_to_coord(&(ship.longitude),buf,'E');
  gets(buf);
  gets(buf);
  str_to_coord(&(iceberg.latitude),buf,'N');
  gets(buf);
  str_to_coord(&(iceberg.longitude),buf,'E');
}
double deg_to_rad(degree angle)
{
  double res;
  res=(double)angle.deg+(double)angle.min/60.0+(double)
angle.sec/3600.0;
  res=res/180.0*CIRCLE_PI;
  if(angle.sign==-1)
    res=-res;
  return res;
}
void coord_to_vec(vector *vec,coordinate coord)
{
  double phi,theta;
  phi=deg_to_rad(coord.latitude);
  theta=deg_to_rad(coord.longitude);
  vec->x=cos(phi)*cos(theta)*DIAMETER/2.0;
  vec->y=cos(phi)*sin(theta)*DIAMETER/2.0;
  vec->z=sin(phi)*DIAMETER/2.0;
}
double distance(vector a,vector b)
{
  double t1=a.x-b.x;
  double t2=a.y-b.y;
  double t3=a.z-b.z;
  t1=t1*t1;
  t2=t2*t2;
  t3=t3*t3;
  return sqrt(t1+t2+t3);
}
void solve()
{
  vector a,b;
  double t;
  coord_to_vec(&a,ship);
  coord_to_vec(&b,iceberg);
  t=distance(a,b);
  dist=acos(1.0-(t*t)/(2.0*(DIAMETER/2.0)*(DIAMETER/2.0)))*
(DIAMETER/2.0);
}
void done()
{
  printf("The distance to the iceberg: %0.2lf miles.\n",dist);
  if(floor(dist*100.0+0.5)<10000.0)
    printf("DANGER!\n");
}
main()
{
  init();
  solve();
  done();
  return 0;
}

I think the problem is with this;

if(floor(dist*100.0+0.5)<10000.0)
    printf("DANGER!\n");

i can make a test in which the answer is much nearer to 100 than you
expect; one way to correct this:

#include <string.h>

...

void done()
{
  if (
   printf("The distance to the iceberg: %0.2lf miles.\n",dist) ==
   strlen("The distance to the iceberg: 100.00 miles.\n") )
   printf("DANGER!\n");
}

Good luck.

sorry, a mistake -- not == but <