[edparcell]

It is simpler if you use the notation [x]_a^b (i.e. with a subscript and a superscript b) to mean x, clipped to the range a to b, and skip writing +/- infinity if you don't intend clipping on one side.

Then you get a bunch of obvious identities like [x]^b = min(x, b) = [b]^x (x capped by b is the same as the smaller of x and b which is the same as b capped by x), [x]_a^b = [b]_a^x, and [x]_a^b = [[x]_a]^b. Putting these together you get [x]_a^b = [[x]_a]^b = min(max(x, a), b). But honestly it's just easier to stick to the notation most of the time.

A better write-up, for everyone who doesn't like reading new math notations inline: https://imgur.com/gallery/593QEow (Imgur link with white background) https://quicklatex.com/cache3/71/ql_46c49ac709b3789482d0736d... (Original link - renders badly in Chrome due to PNG transparency)

[secondcoming]

Those are all valid C.