Here is an online renderer and the description: https://asciimath.org/

The rules are basically the same as LaTeX, with saner symbol names, support for fractions, \ is not needed before symbols and () can be used instead of {}.

> Supposing that the parentheses aren't necessary, as implied by your edit: how does AsciiMath determine that e^y isn't in the numerator in "e^y dy/dx"

It seems to me that dx,dy,dz,dt behave like numbers, single letter variables and symbols (probably they are symbols, but not listed for some reason). Just as LaTeX doesn't need {} parentheses for numbers, single letter variables and symbols, AsciiMath allows omitting them too.

So `/` captures a single number/symbol/variable left to it, and that is `dy`. But if there was `du` for example it would only capture u, and you would need to put du between parentheses.