binary - How do I convert from a decimal number to IEEE 754 single-precision floating-point format?

Question

How would I go about manually changing a decimal (base 10) number into IEEE 754 single-precision floating-point format? I understand that there is three parts to it, a sign, an exponent, and a mantissa. I just don't completely understand what the last two parts actually represent.

Answer 1

Find the largest power of 2 which is smaller than your number, e.g if you start with x = 10.0 then 2³ = 8, so the exponent is 3. The exponent is biased by 127 so this means the exponent will be represented as 127 + 3 = 130. The mantissa is then 10.0/8 = 1.25. The 1 is implicit so we just need to represent 0.25, which is 010 0000 0000 0000 0000 0000 when expressed as a 23 bit unsigned fractional quantity. The sign bit is 0 for positive. So we have:

s | exp [130]  | mantissa [(1).25]            |

0 | 100 0001 0 | 010 0000 0000 0000 0000 0000 |

0x41200000

You can test the representation with a simple C program, e.g.

#include <stdio.h>

typedef union
{
    int i;
    float f;
} U;

int main(void)
{
    U u;
    
    u.f = 10.0;
    
    printf("%g = %#x
", u.f, u.i);

    return 0;
}