printing the integral part of a floating point num

Question

I am trying to figure out how to print floating point numbers without using library functions. Printing the decimal part of a floating point number turned out to be quite easy. Printing the integral part is harder:

static const int base = 2;
static const char hex[] = "0123456789abcdef";

void print_integral_part(float value)
{
    assert(value >= 0);
    char a[129]; // worst case is 128 digits for base 2 plus NUL
    char * p = a + 128;
    *p = 0;
    do
    {
        int digit = fmod(value, base);
        value /= base;
        assert(p > a);
        *--p = hex[digit];
    } while (value >= 1);
    printf("%s", p);
}

Printing the integral part of FLT_MAX works flawlessly with base 2 and base 16:

11111111111111111111111100000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000 (base 2)

ffffff00000000000000000000000000 (base 16)

However, printing in base 10 results in errors after the first 7 digits:

340282368002860660002286082464244022240 (my own function)
340282346638528859811704183484516925440 (printf)

I assume this is a result of the division by 10. It gets better if I use double instead of float:

340282346638528986604286022844204804240 (my own function)
340282346638528859811704183484516925440 (printf)

(If you don't believe printf, enter 2^128-2^104 into Wolfram Alpha. It is correct.)

Now, how does printf manage to print the correct result? Does it use some bigint facilities internally? Or is there some floating point trick I am missing?

Answer 1

According to IEEE single precision float implementation, only 24 bits of data is stored at any time in a float variable. This means only maximum 7 decimal digits are stored in the floating number.

Rest of the hugeness of the number is stored in the exponent. FLT_MAX is initialized as 3.402823466e+38F. So, after the 10th precision, which digit should get printed is not defined anywhere.

From Visual C++ 2010 compiler, I get this output 340282346638528860000000000000000000000.000000, which is the only vaild output.

So, initially we have these many valid digits 3402823466 So after the 1st division we have only 0402823466 So, the system need to get rid of the left 0 and introduce a new digit at the right. In ideal integer division, it is 0. Because you are doing floating division (value /= base;) , system is getting some other digit to fill in that location.

So, in my opinion, the printf could be assigning the above available significant digits to an integer and working with this.