In Numerical Analysis by Timothy Sauer (Pearson, 2nd Edition) it says that $\tilde{f'}(x+h) = f(x+h) + \epsilon_{\text{mach}}$, where $\tilde{f'}(x)$ is the floating-point representation of the given derivative, $f'(x)$ is the exact value of the derivative and $\epsilon_{\text{mach}}$ is the machine epsilon. I, however, believe that it should say $\tilde{f'}(x+h) = f(x+h) + \frac{\epsilon_{\text{mach}}}{2}$.
To my understanding, the machine epsilon is the difference between any two consecutive floating-point numbers in a given floating-point system. Accordingly, the distance between any real number and the nearest floating point number should, in the worst case, be $\frac{\epsilon_{\text{mach}}}{2}$. For this reason, I fail to see how loss of significance in the floating-point representation of the derivative can cause an absolute error on the size of $\epsilon_{\text{mach}}$. Have I missed something?
Update 1
As has been pointed out in the comments, my understanding of machine epsilon is incorrect and this is the main source of my confusion. If I was to understand the true definition of machine epsilon, the above formula would probably make sense to me.
Furhtermore, an important detail from the book that I did not include is that it says "we will assume that the function errors are on the order of 1 for the present discussion, so that relative and absolute errors are about equal".
