Gap between numbers in fixed-point vs. floating point arithmetic
Yes, the difference is constant.
It is not really constant, but approximately, yes.  With exceptions.


With binary floating point numbers, the expression $(f(r)−r)/r$ is constant within a factor of $2$.  It is between $1 \over 2^m$ and $1 \over 2^{m-1}$ where m is the number of bits in the mantissa.  For rounding error calculations, you can assume it is constant.

A note on 0:
Obviously, you cannot apply the formula when $r=0$.  But it is important to know that while $f(r)-r$ decreases when $r$ becomes small, $f(0)-0$ is much larger than, for instance, $f(f(r))-f(r)$.  There is a huge non-representable gap around $0$.