Understanding the binary representation of a float

Question

I am using the following code to print out a "binary representation" of a floating point number:

template<class F>
void printBinary(F value)
{
    std::cout <<
        std::bitset<sizeof(F) * 8>(*reinterpret_cast<unsigned long*>(&value)).to_string()
    << std::endl;
}

int main()
{
    float f = 1;
    printBinary(f);
    f = 2;
    printBinary(f);
    f = 3;
    printBinary(f);
    f = 4;
    printBinary(f);
    f = 16;
    printBinary(f);
    f = 0.2;
    printBinary(f);
}

It outputs:

00111111100000000000000000000000
01000000000000000000000000000000
01000000010000000000000000000000
01000000100000000000000000000000
01000001100000000000000000000000
00111110010011001100110011001101

Can someone explain what parts of the output binary numbers correspond to what parts of the float? I would expect the first one just to be 10000... . The second makes sense. I am confused about every output after that, especially the last.

Thanks in advance.

Answer 1

Assuming you're referring to IEEE754 binary floating point format, a 32-bit float consists of 1 sign bit, 8 exponent bits, and 23 significand (aka fraction) bits. Here's how one of your examples 0.2 get represented, as an example:

                  3  2          1         0
                  1 09876543 21098765432109876543210
                  S ---E8--- ----------F23----------
          Binary: 0 01111100 10011001100110011001101
             Hex: 3E4C CCCD
       Precision: SP
            Sign: Positive
        Exponent: -3 (Stored: 124, Bias: 127)
       Hex-float: +0x1.99999ap-3
           Value: +0.2 (NORMAL)

You can read more about the format itself on the wikipedia page: https://en.wikipedia.org/wiki/IEEE_754#Basic_and_interchange_formats and also the specifics of the single-precision format in: https://en.wikipedia.org/wiki/Single-precision_floating-point_format