printing a float has unexpected value

Question

Why has '32' been added to the value of v in the output below?

int main()
{
    float v = 9.999e8;

    std::cout << "v --> " << v << std::endl;

    std::cout << std::fixed;
    std::cout << "v --> " << v << std::endl;

    return 0;
}

output:

v --> 9.999e+08
v --> 999900032.000000
             ^^ 

Is it an artefact of printing the value, or that 9.999e+08 cannot be accurately represented in a float?

Changing v to a double resolves the issue but I want to understand the actual cause.

strong text

Answer 1

With the common desktop implementation of float, the IEEE 754 standard, there are only 24 significand bits (and 8 exponent bits). This means that the biggest consecutive value that can be represented, without precision loss, is 2²⁴ which is approximately 1.6E7 (1.6•10⁷). For example, an odd value above this threshold can't be represented accurately by such a float, since the least significant 1-bit "falls off" the representation.

Your value is approximately 1E9 (10⁹), which is 60 times larger. It means that the value can be about 30 off (after rounding to the nearest representable value). Note that the exponent bits contribute 2^exp (sans the nitpicking about the offset thing), and not 10^exp.

With double there are 53 bits, which are more than enough for 1E9 (10⁹).