# Handling full expressions

Now we are ready to extend our calculator to cover the full range of arithmetic expressions (well, at least the ones you learned in elementary school). Here is the next calculator example, calculator3:

use std::str::FromStr;

grammar;

pub Expr: i32 = {
<l:Expr> "+" <r:Factor> => l + r,
<l:Expr> "-" <r:Factor> => l - r,
Factor,
};

Factor: i32 = {
<l:Factor> "*" <r:Term> => l * r,
<l:Factor> "/" <r:Term> => l / r,
Term,
};

Term: i32 = {
Num,
"(" <Expr> ")",
};

Num: i32 = {
r"[0-9]+" => i32::from_str(<>).unwrap(),
};


Perhaps the most interesting thing about this example is the way it encodes precedence. The idea of precedence of course is that in an expression like 2+3*4, we want to do the multiplication first, and then the addition. LALRPOP doesn't have any built-in features for giving precedence to operators, mostly because I consider those to be creepy, but it's pretty straightforward to express precedence in your grammar by structuring it in tiers -- for example, here we have the nonterminal Expr, which covers all expressions. It consists of a series of factors that are added or subtracted from one another. A Factor is then a series of terms that are multiplied or divided. Finally, a Term is either a single number or, using parenthesis, an entire expr.

Abstracting from this example, the typical pattern for encoding precedence is to have one nonterminal per precedence level, where you begin with the operators of lowest precedence (+, -), add in the next highest precedence level (*, /), and finish with the bare "atomic" expressions like Num. Finally, you add in a parenthesized version of your top-level as an atomic expression, which lets people reset.

To see why this works, consider the two possible parse trees for something like 2+3*4:

2 + 3   *    4          2   +  3   *    4
| | |   |    |          |   |  |   |    |
| | +-Factor-+    OR    +-Expr-+   |    |
| |     |                   |      |    |
+-Expr -+                   +----Factor-+


In the first one, we give multiplication higher precedence, and in the second one, we (incorrectly) give addition higher precedence. If you look at the grammar now, you can see that the second one is impossible: a Factor cannot have an Expr as its left-hand side. This is the purpose of the tiers: to force the parser into the precedence you want.