Thread: binary representation of a float

I was curious to see exactly how a float (and a double) is stored in memory so I wrote this little program to let me see the actual bits for the numbers 0-9 stored as floats:

Code:

#include <iostream>
#include <string>
using namespace std;

int main( int argc, char* argv[] ) {
  float num; // = atof( argv[1] );
  int *ip = (int*)&num;
  int size, digit;
  string ch;
  string binstr;
  size = sizeof(num);
  cout << "sizeof int = " << sizeof(size) << endl;
  cout << "sizeof float = " << sizeof(num) << endl;

  for( int k = 0; k < 10; ++k ) {
    num = k;
    for( int j = 0; j < size; ++j ) {
      for( int i = 0; i < 8; ++i ) {
        digit = 1 & (*ip);
        ch = string( 1, (char)(48 + digit) );
        binstr.insert( 0, ch );
        *ip >>= 1;
      }
      binstr.insert( 0, " ");
    }
    cout << binstr << endl;
    binstr = "";
  }

  return 0;
}

I thought I understood that a 32-bit float is stored as an 8-bit exponent and 24-bit significand, so (assuming the first byte would be the exponent, 0, and the next 3 bytes would be the significand) I was expecting as output:

Code:

sizeof int = 4
sizeof float = 4
 00000000 00000000 00000000 00000000
 00000000 00000001 00000000 00000000
 00000000 00000010 00000000 00000000
 00000000 00000011 00000000 00000000
 00000000 00000100 00000000 00000000
 00000000 00000101 00000000 00000000
 00000000 00000110 00000000 00000000
 00000000 00000111 00000000 00000000
 00000000 00001000 00000000 00000000
 00000000 00001001 00000000 00000000

but instead, this is the surprising output:

Code:

sizeof int = 4
sizeof float = 4
 00000000 00000000 00000000 00000000
 00111111 10000000 00000000 00000000
 01000000 00000000 00000000 00000000
 01000000 01000000 00000000 00000000
 01000000 10000000 00000000 00000000
 01000000 10100000 00000000 00000000
 01000000 11000000 00000000 00000000
 01000000 11100000 00000000 00000000
 01000001 00000000 00000000 00000000
 01000001 00010000 00000000 00000000

Can you explain this?

(My code may not be the most elegant way to do this, but I don't see anything actually wrong in it, do you?)

Thanks. I had looked at a couple of articles in Wikipedia but didn't find that one.

That's quite a system. I'd love to hear Bob Newhart explain it.

Yes, the bias is 127 as explained in the Wikipedia article. All of the encodings in my output example bear that out.