5.1 — Operator precedence and associativity

Chapter introduction

This chapter builds on top of the concepts from lesson 1.9 -- Introduction to literals and operators. A quick review follows:

In mathematics, an operation is a mathematical calculation involving zero or more input values (called operands) that produces a new value (called an output value). The specific operation to be performed is denoted by a construct (typically a symbol or pair of symbols) called an operator.

For example, as children we all learn that 2 + 3 equals 5. In this case, the literals 2 and 3 are the operands, and the symbol + is the operator that tells us to apply mathematical addition on the operands to produce the new value 5.

In this chapter, we’ll discuss topics related to operators, and explore many of the common operators that C++ supports.

Operator precedence

Now, let’s consider a more complicated expression, such as 4 + 2 * 3. An expression that has multiple operators is called a compound expression. In order to evaluate this compound expression, we must understand both what the operators do, and the correct order to apply them. The order in which operators are evaluated in a compound expression is determined by an operator’s precedence. Using normal mathematical precedence rules (which state that multiplication is resolved before addition), we know that the above expression should evaluate as 4 + (2 * 3) to produce the value 10.

In C++, when the compiler encounters an expression, it must similarly analyze the expression and determine how it should be evaluated. To assist with this, all operators are assigned a level of precedence. Operators with the highest level of precedence are evaluated first.

You can see in the table below that multiplication and division (precedence level 5) have more precedence than addition and subtraction (precedence level 6). Thus, 4 + 2 * 3 evaluates as 4 + (2 * 3) because multiplication has a higher level of precedence than addition.

Operator associativity

What happens if two operators in the same expression have the same precedence level? For example, in the expression 3 * 4 / 2, the multiplication and division operators are both precedence level 5. In this case, the compiler can’t rely upon precedence alone to determine how to evaluate the result.

If two operators with the same precedence level are adjacent to each other in an expression, the operator’s associativity tells the compiler whether to evaluate the operators from left to right or from right to left. The operators in precedence level 5 have an associativity of left to right, so the expression is resolved from left to right: (3 * 4) / 2 = 6.

Table of operators

The below table is primarily meant to be a reference chart that you can refer back to in the future to resolve any precedence or associativity questions you have.

Notes:

• Precedence level 1 is the highest precedence level, and level 17 is the lowest. Operators with a higher precedence level get evaluated first.
• L->R means left to right associativity.
• R->L means right to left associativity.

You should already recognize a few of these operators, such as +, -, *, /, (), and sizeof. However, unless you have experience with another programming language, the majority of the operators in this table will probably be incomprehensible to you right now. That’s expected at this point. We’ll cover many of them in this chapter, and the rest will be introduced as there is a need for them.

Parenthesization

In normal arithmetic, you learned that you can use parentheses to change the order of application of operations. For example, we know that 4 + 2 * 3 evaluates as 4 + (2 * 3), but if you want it to evaluate as (4 + 2) * 3 instead, you can explicitly parenthesize the expression to make it evaluate the way you want. This works in C++ because parentheses have one of the highest precedence levels, so parentheses generally evaluate before whatever is inside them.

Now consider an expression like x && y || z. Does this evaluate as (x && y) || z or x && (y || z)? You could look up in the table and see that && takes precedence over ||. But there are so many operators and precedence levels that it’s hard to remember them all.

In order to reduce mistakes and make your code easier to understand without referencing a precedence table, it’s a good idea to parenthesize any non-trivial compound expression, so it’s clear what your intent is.

There is one notable exception to the above best practice: Expressions that have a single assignment operator do not need to have the right operand of the assignment wrapped in parenthesis.

For example:

x = (y + z + w);   // instead of this
x = y + z + w;     // it's okay to do this

x = ((y || z) && w); // instead of this
x = (y || z) && w;   // it's okay to do this

x = (y *= z); // expressions with multiple assignments still benefit from parenthesis

The assignment operators have the second lowest precedence (only the comma operator is lower, and it’s rarely used). Therefore, so long as there is only one assignment (and no commas), we know the right operand will fully evaluate before the assignment.

The order of evaluation of expressions and function arguments is mostly unspecified

Consider the following expression:

a + b * c

We know from the precedence and associativity rules above that this expression will evaluate as if we had typed:

a + (b * c)

If a is 1, b is 2, and c is 3, this expression will evaluate to the answer 7.

However, the precedence and associativity rules only tell us how operators evaluate in relation to other operators. It does not tell us anything about the order in which the rest of the expression evaluates. For example, does variable a, b, or c get evaluated first?

Perhaps surprisingly, in many cases, the order of evaluation of any part of a compound expression (including function calls and argument evaluation) is unspecified. In such cases, the compiler is free to choose any evaluation order it believes is optimal.

For most expressions, this is irrelevant. In our sample expression above, it doesn’t matter whether in which order variables a, b, or c are evaluated for their values: the answer will always be 7. There is no ambiguity here.

But it is possible to write expressions where the order of evaluation does matter. Consider this program, which contains a mistake often made by new C++ programmers:

#include <iostream>

int getValue()
{
    std::cout << "Enter an integer: ";

    int x{};
    std::cin >> x;
    return x;
}

int main()
{
    std::cout << getValue() + (getValue() * getValue()); // a + (b * c)
    return 0;
}

If you run this program and enter inputs 1, 2, and 3, you might assume that this program would print 7. But that is making the assumption that the calls to getValue() will evaluate in left-to-right order. The compiler may choose a different order. For example, if the compiler chose a right-to-left order instead, the program would print 5 for the same set of inputs.

The above program can be made unambiguous by making each function call a separate statement:

#include <iostream>

int getValue()
{
    std::cout << "Enter an integer: ";

    int x{};
    std::cin >> x;
    return x;
}

int main()
{
    int a{ getValue() }; // will execute first
    int b{ getValue() }; // will execute second
    int c{ getValue() }; // will execute third
    
    std::cout << a + (b * c); // order of eval doesn't matter now

    return 0;
}

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