How can I know in advance which real numbers would have an imprecise representation usinf float variables em C?

I know that the number 159.95 cannot be precisely represented using float variables in C.

For example, considering the following piece of code:

#include <stdio.h>
int main()
{
    float x = 159.95;
    printf("%f\n",x);
    return 0;
}

It outputs 159.949997.

I would like to know if there is some way to know in advance which real value (in decimal system) would be represented in an imprecise way like the 159.95 number.

Best regards.

>Solution :

Usually, a float is an IEEE754 binary32 float (this is not guaranteed by spec and may be different on some compilers/systems). This data type specifies a 24-bit significand; this means that if you write the number in binary, it should require no more than 24 bits excluding trailing zeros.

159.95’s binary representation is 10011111.11110011001100110011… with repeating 0011 forever, so it requires an infinite number of bits to represent precisely with a binary format.

Other examples:

1073741760 has a binary representation of 111111111111111111111111000000. It has 30 bits in that representation, but only 24 significant bits (since the remainder are trailing zero bits). It has an exact float representation.

1073741761 has a binary representation of 111111111111111111111111000001. It has 30 significant bits and cannot be represented exactly as a float.

0.000000059604644775390625 has a binary representation of 0.000000000000000000000001. It has one significant bit and can be represented exactly.

0.750000059604644775390625 has a binary representation of 0.110000000000000000000001, which is 24 significant bits. It can be represented exactly as a float.

1.000000059604644775390625 has a binary representation of 1.000000000000000000000001, which is 25 significant bits. It cannot be represented exactly as a float.

Another factor (which applies to very large and very small numbers) is that the exponent is limited to the -126 to +127 range. With some handwaving around denormal values and other special cases, this generally allows values ranging from roughly 2^-126 to slightly under 2¹²⁸.