Thread: Floating Point Underflow

Because DBL_MIN isn't the smallest value possible for double. DBL_MIN is the smallest normalized value.

2.225074e-309 is, indeed an underflow (what IEEE call it a "gracious underflow") because it is a subnormal double value (where E == 0). A little change in your program can show this:

Code:

#include <stdio.h>
#include <float.h>

struct fp_s {
  unsigned long long m:52;
  unsigned int e:11;
  unsigned int s:1;
} __attribute__((packed));

int main( void )
{
  double x, y;
  struct fp_s *px, *py;

  px = (struct fp_s *)&x;
  py = (struct fp_s *)&y;

  x = DBL_MIN;
  y = x / 10;

  // Remember: the structure for normalized values (0 < E < 2047) is:
  //    v = (-1)^S * ( 1 + M/2^52 ) * 2^(E-1023)
  // but for subnormal values (E=0) is:
  //    v = (-1)^S * ( M/2^52 ) * 2^-1022
  //
  printf( "%e (M=%#llx, E=%u, S=%u) %e (M=%#llx, E=%u, S=%u)\n",
     x, (unsigned long long)px->m, (unsigned int)px->e, (unsigned int)px->s,
     y, (unsigned long long)py->m, (unsigned int)py->e, (unsigned int)py->s );

  // THIS is the smallest subnormal value (not 0.0) possible for a double.
  // Yes... 0.0 is a subnormal special value (M=0, E=0).
  px->m=1;
  px->e=0;
  px->s=0;
  printf( "%e\n", x );

  return 0;
}

Compiling and running:

Code:

$ cc -O2 -o test test.c
$ ./test
2.225074e-308 (M=0, E=1, S=0) 2.225074e-309 (M=0x199999999999a, E=0, S=0)
4.940656e-324