Machine epsilon is fine for doing error analysis. For example, I used machine epsilon instead of ULP for an article I recently wrote which walks through impleme

Machine epsilon is fine for doing error analysis. For example, I used machine epsilon instead of ULP for an article I recently wrote which walks through implementation and (numerical) analysis of various algorithms for e^x: https://www.pseudorandom.com/implementing-exp https://www.pseudorandom.com/implementing-exp So you don't need ULP (and if you did, you could convert to ULP from machine epsilon). But you are right that machine epsilon varies according to where you are in the interval of floating point numbers. Some authors don't mention that, but yes, since floating point values are not evenly distributed the machine epsilon for any particular x will vary. This is good to know, because even if error analysis demonstrates that your approximation function is bounded by machine epsilon, it's still going to exhibit less accuracy as you move away from the origin.