This square root function is not compressed in the way that we normally talk about dynamic range compression (though the signal strength certainly increases). That kind of compression is a dynamically changing amplification factor that multiplies smaller portions of the audio to make it “look” about the same size as the louder portions. A Compressor doesn’t react to individual samples, it reacts to the “recent average.”
You asked if any Compressed sound would be distorted. Technically it is, but think of it this way: At it’s most basic, a Compressor is simply a fancy version of amplification. While it’s not technically a linear operation, the non-linearities occur at such a slow rate that only incredibly low frequencies (that we can’t hear) will be distorted. The only frequencies added to the signal will be subaudible.
But this square root stuff isn’t just multiplying the signal, it’s performing a more complicated operation to it. And it’s doing this in a constantly non-linear fashion. This is because it’s affecting each sample differently based on how big the individual sample is, NOT on how big the “recent average” is (like a Compressor would).
I guess the easy way to look at it would be this: If you take a sine wave and apply several iterations of this square root function, it will end up looking more and more like a square wave.
To steal your original listing (and add to it):
x x^.5 x^.25 ... x^.031
0 0 0 0
0.2 0.45 0.66 0.95
0.4 0.63 0.79 0.97
0.6 0.77 0.88 0.98
0.8 0.89 0.94 0.99
1 1 1 1
From this, the ‘squareness’ of the distortion should be pretty evident. But what does that mean for the spectral content? It means that we’re adding odd harmonics to the signal, and we’re adding quite a few of them.
Take a look at the wikipedia article on distortion:
Note forms 4 and 5 they have at that list right there. The transfer function for a square root (or whatever it is we should call this effect) will fall somewhere between those two extremes. It will look like number 4, but will have more of a “shelf” at the top, and the upward swing (near the zero crossing) will be much sharper.
What we audio people call “distortion” is basically the addition of musically related harmonics to a signal. You don’t need abrupt clipping to add harmonics to a sound. It’s true that that’s exactly what number 5 is doing, but number 4 is still a form of distortion despite making no abrupt changes.
The reason this might not sound as nice your run of the mill distortion pedal is probably due to the lack of any equalization to the signal. I’m sure with some tweaking it can sound great.
Does that make sense?