Why not both standardize and normalize features for machine learning? If one has data that's assumed to be normal distributed and want to use it as input in a machine learning model, why not first standardize the data and then normalize (min max scale it between zero and one)?
So first transform as follows
$$
S = \frac{X - \mu}{\sigma}
$$
...and then transform it one more time to
$$
X_{standaardizedAndNormalized} = \frac{S - S_{min}}{S_{max}-S_{min}}
$$

That is equivalent to normalizing only $X$ since the standardization step does not change the min/max values. Besides, these transformations are not associated with normality assumption.
