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# 3.1 — Operator precedence and associativity

In mathematics, an operation is a mathematical calculation involving zero or more input values (called operands) that produces an output value. Common operations (such as addition) use special symbols (such as +) that denote the operation. These symbols are called operators. Operators in programming work the same way except the names may not always be a symbol. Operators work analogously to functions that take input parameters and return a value, except they are more concise. For example, 4 + 2 * 3 is much easier to read than add(4, mult(2, 3))!

In order to properly evaluate an expression such as `4 + 2 * 3`, 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 called operator 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. Those with the highest precedence are evaluated first. You can see in the table below that multiplication and division (precedence level 5) have a higher precedence than addition and subtraction (precedence level 6). The compiler uses these levels to determine how to evaluate expressions it encounters.

Thus, 4 + 2 * 3 evaluates as 4 + (2 * 3) because multiplication has a higher level of precedence than addition.

If two operators with the same precedence level are adjacent to each other in an expression, the operator associativity rules tell the compiler whether to evaluate the operators from left to right or from right to left. For example, in the expression `3 * 4 / 2`, the multiplication and division operators are both precedence level 5. Level 5 has an associativity of left to right, so the expression is resolved from left to right: `(3 * 4) / 2 = 6`.

Table of operators

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.
Prec/Ass Operator Description Pattern
1 None ::
::
Global scope (unary)
Class scope (binary)
::name
class_name::member_name
2 L->R ()
()
()
{}
type()
type{}
[]
.
->
++
––
typeid
const_cast
dynamic_cast
reinterpret_cast
static_cast
Parentheses
Function call
Initialization
Uniform initialization (C++11)
Functional cast
Functional cast (C++11)
Array subscript
Member access from object
Member access from object ptr
Post-increment
Post-decrement
Run-time type information
Cast away const
Run-time type-checked cast
Cast one type to another
Compile-time type-checked cast
(expression)
function_name(parameters)
type name(expression)
type name{expression}
new_type(expression)
new_type{expression}
pointer[expression]
object.member_name
object_pointer->member_name
lvalue++
lvalue––
typeid(type) or typeid(expression)
const_cast<type>(expression)
dynamic_cast<type>(expression)
reinterpret_cast<type>(expression)
static_cast<type>(expression)
3 R->L +
-
++
––
!
~
(type)
sizeof
&
*
new
new[]
delete
delete[]
Unary plus
Unary minus
Pre-increment
Pre-decrement
Logical NOT
Bitwise NOT
C-style cast
Size in bytes
Dereference
Dynamic memory allocation
Dynamic array allocation
Dynamic memory deletion
Dynamic array deletion
+expression
-expression
++lvalue
––lvalue
!expression
~expression
(new_type)expression
sizeof(type) or sizeof(expression)
&lvalue
*expression
new type
new type[expression]
delete pointer
delete[] pointer
4 L->R ->*
.*
Member pointer selector
Member object selector
object_pointer->*pointer_to_member
object.*pointer_to_member
5 L->R *
/
%
Multiplication
Division
Modulus
expression * expression
expression / expression
expression % expression
6 L->R +
-
Subtraction
expression + expression
expression - expression
7 L->R <<
>>
Bitwise shift left
Bitwise shift right
expression << expression
expression >> expression
8 L->R <
<=
>
>=
Comparison less than
Comparison less than or equals
Comparison greater than
Comparison greater than or equals
expression < expression
expression <= expression
expression > expression
expression >= expression
9 L->R ==
!=
Equality
Inequality
expression == expression
expression != expression
10 L->R & Bitwise AND expression & expression
11 L->R ^ Bitwise XOR expression ^ expression
12 L->R | Bitwise OR expression | expression
13 L->R && Logical AND expression && expression
14 L->R || Logical OR expression || expression
15 R->L ?:
=
*=
/=
%=
+=
-=
<<=
>>=
&=
|=
^=
Conditional (see note below)
Assignment
Multiplication assignment
Division assignment
Modulus assignment
Subtraction assignment
Bitwise shift left assignment
Bitwise shift right assignment
Bitwise AND assignment
Bitwise OR assignment
Bitwise XOR assignment
expression ? expression : expression
lvalue = expression
lvalue *= expression
lvalue /= expression
lvalue %= expression
lvalue += expression
lvalue -= expression
lvalue <<= expression
lvalue >>= expression
lvalue &= expression
lvalue |= expression
lvalue ^= expression
16 R->L throw Throw expression throw expression
17 L->R , Comma operator expression, expression

Note: The expression in the middle of the conditional operator ?: is evaluated as if it were parenthesized.

A few operators you should already recognize: +, -, *, /, (), =, <, >, <=, and >=. These arithmetic and relational operators have the same meaning in C++ as they do in every-day usage.

However, unless you have experience with another programming language, it’s likely the majority of the operators in this table will 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.

The above 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.

That said, if you have an expression that uses operators of different types, it is a best practice to use parenthesis to explicitly disambiguate the order of evaluation.

Rule: If your expression uses different operators, use parenthesis to make it clear how the expression should evaluate, even if they are technically unnecessary.

How do I do exponents?

You’ll note that the ^ operator (commonly used to denote exponentiation in standard mathematical nomenclature) is a Bitwise XOR operation in C++. C++ does not include an exponent operator. To do exponents in C++, #include the <cmath> header, and use the pow() function:

Note that the parameters and return value of pow are of type double. Note that due to rounding errors in floating point numbers, the results of pow() may not be precise (slightly smaller or larger than what you’d expect).

If you want to do integer exponents, you’re best off just using your own function to do so, like this one (that uses the “exponentiation by squaring” algorithm for efficiency):

Don’t worry if you don’t understand all of the parts of this function yet. Just beware of overflowing your integer result, which can happen very quickly if either argument is large.

Quiz

1) You know from everyday mathematics that expressions inside of parentheses get evaluated first. For example, in the expression `(2 + 3) * 4`, the `(2 + 3)` part is evaluated first.

For this exercise, you are given a set of expressions that have no parentheses. Using the operator precedence and associativity rules in the table above, add parentheses to each expression to make it clear how the compiler will evaluate the expression.

Hint: Use the pattern column in the table above to determine whether the operator is unary (has one operand) or binary (has two operands). Review section 1.5 -- A first look at operators [1] if you need a refresher on what unary and binary operators are.

 Sample problem: x = 2 + 3 % 4 Binary operator % has higher precedence than operator + or operator =, so it gets evaluated first: x = 2 + (3 % 4) Binary operator + has a higher precedence than operator =, so it gets evaluated next: Final answer: x = (2 + (3 % 4)) We now no longer need the table above to understand how this expression will evaluate.

a) x = 3 + 4 + 5;
b) x = y = z;
c) z *= ++y + 5;
d) a || b && c || d;

Solutions

1) Show Solution [2]

 3.2 -- Arithmetic operators [3] Index [4] 2.10 -- Chapter 2 comprehensive quiz [5]