Normal form for mathematical expressions I have a somewhat general question regarding the notation of mathematical expressions. I am interested in further information, that is if you know a book, a pdf, or just a wikipedia page about my question that would help a lot. Of course, if you know a specific answer feel free to share.
My question is: Is there a normal form / convention for the notation of mathematical expressions? As an example:

*

*$\frac{\sqrt{x^2+2xy+y^2}}{z}$

*$\frac{x+y}{z}$

*$\frac{x}{z}+\frac{y}{\sqrt{z^2}}$
Those three expressions are mathematically equal but semantically different. Is there a normal form i.e. a rule set that - when given a those three expressions - defines one "normalized" way to write them down? And if there is, is there an associated normalization algorithm that would, with absolute certainty, convert these three examples (that are mathematically equal) into the same (syntactically equal) expression? The resulting expression does not necessarily have to be the "simplest form" that contains the least amount of symbols or whatever. I am also aware that there are probably many different "normal forms". I am just searching for one of them.
Normal forms exist in expression logic (disjunctive, conjunctive,...). I am searching for something analogous for math expressions.
Thanks for the help in advance :)
Edit:
I realized that my question is a bit to general. Maybe it is easier to first limit the expressions to algebraic expressions, that is expressions involving integer constants, variables, and the operations

*

*addition

*subtraction

*multiplication

*division

*exponentiation by a rational

 A: Edit: The question was changed substantially after I posted this answer. It does not address the current form of the question.
Mathematical expression is far to vague and wide to ask for normal forms, it is not even clear how to define what exactly makes for a mathematical expression. You also have the syntax vs semantics issue: You don't just care about the sequence of symbols, but rather their meaning.
Your examples all denote rational functions, they are built from variables, integers, addition, substraction, multiplication and division. These do have normal forms:
Going from outside to inside, we first have a single quotient. Both numerator and denominator are sums, and each summand is a product of variables and integers.
To really standardize things, we put an order on the variables, and can then order both the factors inside the products and the summands inside the products in a canonic way.
We can compute the normal form of a given term, essentially following school arithmetic. However, the normal form will not necessarily be the most concise way to write a given rational function.
