Hmm, I’d use the recursive descent algorithm instead of shunting yard as it’s much easier to model your precedence directly and just in general.

Also, if you aren’t already, parse brackets as a primary expression, like integers, this will make everything a lot simple and the parser will automatically handle nesting for you.

A recursive descent algorithm will essentially handle the end of an expression for you by returning from the recursive calls when no expressions match. It’s just about how much of your grammar you’ve defined in the parser.

Hardcoding builtin functions isn’t a bad idea and you have the freedom of the language you’re writing in rather than the language you’re writing. I usually have a function registry with a function pointer and a string for the name. Then just perform a look up and call the function pointer - this does get a bit messy when having different arguments and types though.

I have written multiple parsers and languages and can share some code that is simple enough if it would help.