Arithmetic operators
Returns the result of specific arithmetic operation.
| Operator name | Syntax | Prototype examples (for class T) | ||
|---|---|---|---|---|
| Inside class definition | Outside class definition | |||
| Unary plus | +a | T T::operator+() const; | T operator+(const T& a); | |
| Unary minus | -a | T T::operator-() const; | T operator-(const T& a); | |
| Addition | a + b | T T::operator+(const T2& b) const; | T operator+(const T& a, const T2& b); | |
| Subtraction | a - b | T T::operator-(const T2& b) const; | T operator-(const T& a, const T2& b); | |
| Multiplication | a * b | T T::operator*(const T2& b) const; | T operator*(const T& a, const T2& b); | |
| Division | a / b | T T::operator/(const T2& b) const; | T operator/(const T& a, const T2& b); | |
| Remainder | a % b | T T::operator%(const T2& b) const; | T operator%(const T& a, const T2& b); | |
| Bitwise NOT | ~a | T T::operator~() const; | T operator~(const T& a); | |
| Bitwise AND | a & b | T T::operator&(const T2& b) const; | T operator&(const T& a, const T2& b); | |
| Bitwise OR | a | b | T T::operator|(const T2& b) const; | T operator|(const T& a, const T2& b); | |
| Bitwise XOR | a ^ b | T T::operator^(const T2& b) const; | T operator^(const T& a, const T2& b); | |
| Bitwise left shift | a << b | T T::operator<<(const T2& b) const; | T operator<<(const T& a, const T2& b); | |
| Bitwise right shift | a >> b | T T::operator>>(const T2& b) const; | T operator>>(const T& a, const T2& b); | |
| ||||
General explanation
All built-in arithmetic operators compute the result of specific arithmetic operation and returns its result. The arguments are not modified.
Conversions
If the operand passed to a built-in arithmetic operator is integral or unscoped enumeration type, then before any other action (but after lvalue-to-rvalue conversion, if applicable), the operand undergoes integral promotion. If an operand has array or function type, array-to-pointer and function-to-pointer conversions are applied.
For the binary operators (except shifts), if the promoted operands have different types, usual arithmetic conversions are applied.
Overflows
Unsigned integer arithmetic is always performed modulo 2n
where n is the number of bits in that particular integer. E.g. for unsigned int, adding one to UINT_MAX gives 0, and subtracting one from 0 gives UINT_MAX.
When signed integer arithmetic operation overflows (the result does not fit in the result type), the behavior is undefined, — the possible manifestations of such an operation include:
- it wraps around according to the rules of the representation (typically two's complement),
- it traps — on some platforms or due to compiler options (e.g.
-ftrapvin GCC and Clang), - it saturates to minimal or maximal value (on many DSPs),
- it is completely optimized out by the compiler.
Floating-point environment
If
#pragma STDC FENV_ACCESS is supported and set to ON, all floating-point arithmetic operators obey the current floating-point rounding direction and report floating-point arithmetic errors as specified in math_errhandling unless part of a static initializer (in which case floating-point exceptions are not raised and the rounding mode is to nearest).
Floating-point contraction
Unless
#pragma STDC FP_CONTRACT is supported and set to OFF, all floating-point arithmetic may be performed as if the intermediate results have infinite range and precision, that is, optimizations that omit rounding errors and floating-point exceptions are allowed. For example, C++ allows the implementation of (x * y) + z with a single fused multiply-add CPU instruction or optimization of a = x * x * x * x; as tmp = x * x; a = tmp * tmp.
Unrelated to contracting, intermediate results of floating-point arithmetic may have range and precision that is different from the one indicated by its type, see FLT_EVAL_METHOD.
Formally, the C++ standard makes no guarantee on the accuracy of floating-point operations.
Unary arithmetic operators
The unary arithmetic operator expressions have the form
+ expression
|
(1) | ||||||||
- expression
|
(2) | ||||||||
Unary + and - operators have higher precedence than all binary arithmetic operators, so expression cannot contain top-level binary arithmetic operators. These operators associate from right to left:
+a - b; // equivalent to (+a) - b, NOT +(a - b) -c + d; // equivalent to (-c) + d, NOT -(c + d) +-e; // equivalent to +(-e), the unary + is a no-op if “e” is a built-in type // because any possible promotion is performed during negation already
Built-in unary arithmetic operators
-a, where N is the number of bits after promotion.
- In other words, the result is the two’s complement of the operand (where operand and result are considered as unsigned).
Overloads
In overload resolution against user-defined operators, for every cv-unqualified promoted arithmetic type A and for every type T, the following function signatures participate in overload resolution:
| A operator+(A) |
||
| T* operator+(T*) |
||
| A operator-(A) |
||
#include <iostream> int main() { char c = 0x6a; int n1 = 1; unsigned char n2 = 1; unsigned int n3 = 1; std::cout << "char: " << c << " int: " << +c << "\n" "-1, where 1 is signed: " << -n1 << "\n" "-1, where 1 is unsigned char: " << -n2 << "\n" "-1, where 1 is unsigned int: " << -n3 << '\n'; char a[3]; std::cout << "size of array: " << sizeof a << "\n" "size of pointer: " << sizeof +a << '\n'; }
Possible output:
char: j int: 106 -1, where 1 is signed: -1 -1, where 1 is unsigned char: -1 -1, where 1 is unsigned int: 4294967295 size of array: 3 size of pointer: 8
Additive operators
The additive operator expressions have the form
lhs + rhs
|
(1) | ||||||||
lhs - rhs
|
(2) | ||||||||
Binary + and - operators have higher precedence than all other binary arithmetic operators except *, / and %. These operators associate from left to right:
a + b * c; // equivalent to a + (b * c), NOT (a + b) * c d / e - f; // equivalent to (d / e) - f, NOT d / (e - f) g + h >> i; // equivalent to (g + h) >> i, NOT g + (h >> i) j - k + l - m; // equivalent to ((j - k) + l) - m
Built-in additive operators
For built-in binary plus and binary minus operators, both of lhs and rhs must be prvalues, and one of the following conditions must be satisfied:
- Both operands have arithmetic or unscoped enumeration type. In this case, usual arithmetic conversions are performed on both operands.
- Exactly one operand has integral or unscoped enumeration type. In this case, integral promotion is applied to that operand.
In the remaining description in this section, "operand(s)", lhs and rhs refer to the converted or promoted operand(s).
- Both operands have arithmetic type. In this case, the result is the sum of the operands.
- One operand is a pointer to a completely-defined object type, and the other operand has integral type. In this case, the integral value is added to the pointer (see pointer arithmetic).
- Both operands have arithmetic type. In this case, the result is the difference resulting from the subtraction of rhs from lhs.
- lhs is a pointer to a completely-defined object type, and rhs has integral type. In this case, the integral value is subtracted from the pointer (see pointer arithmetic).
- Both operands are pointers to cv-qualified or cv-unqualified versions of the same completely-defined object type. In this case rhs is subtracted from lhs (see pointer arithmetic).
If both operands have a floating-point type, and the type supports IEEE floating-point arithmetic (see std::numeric_limits::is_iec559):
- If one operand is NaN, the result is NaN.
- Infinity minus infinity is NaN, and FE_INVALID is raised.
- Infinity plus the negative infinity is NaN, and FE_INVALID is raised.
Pointer arithmetic
When an expression J that has integral type is added to or subtracted from an expression P of pointer type, the result has the type of P.
- If P evaluates to a null pointer value and J evaluates to 0, the result is a null pointer value.
- Otherwise, if P points to the
ith element of an array object x with n elements, given the value of J as j, P is added or subtracted as follows:
- The expressions P + J and J + P
- point to the
i+jth element of x if i + j is in[0,n), and - are pointers past the end of the last element of x if i + j is n.
- point to the
- The expression P - J
- points to the
i-jth element of x if i - j is in[0,n), and - is a pointer past the end of the last element of x if i - j is n.
- points to the
- Other j values result in undefined behavior.
- Otherwise, if P points to a complete object, a base class subobject or a member subobject y, given the value of J as j, P is added or subtracted as follows:
- The expressions P + J and J + P
- point to y if j is 0, and
- are pointers past the end of y if j is 1.
- The expression P - J
- points to y if j is 0, and
- is a pointer past the end of y if j is -1.
- Other j values result in undefined behavior.
- Otherwise, if P is a pointer past the end of an object z, given the value of J as j:
- If z is an array object with n elements, P is added or subtracted as follows:
- The expressions P + J and J + P
- point to the
n+jth element of z if n + j is in[0,n), and - are pointers past the end of the last element of z if j is 0.
- point to the
- The expression P - J
- points to the
n-jth element of z if n - j is in[0,n), and - is a pointer past the end of the last element of z if j is 0.
- points to the
- Other j values result in undefined behavior.
- Otherwise, P is added or subtracted as follows:
- The expressions P + J and J + P
- point to z if j is -1, and
- are pointers past the end of z if j is 0.
- The expression P - J
- points to z if j is 1, and
- is a pointer past the end of z if j is 0.
- Other j values result in undefined behavior.
- Otherwise, the behavior is undefined.
When two pointer expressions P and Q are subtracted, the type of the result is std::ptrdiff_t.
- If P and Q both evaluate to null pointer values, the result is 0.
- Otherwise, if P and Q point to, respectively, the
ith andjth array elements of the same array object x, the expression P - Q has the value i − j.
- If i − j is not representable by std::ptrdiff_t, the behavior is undefined.
- Otherwise, if P and Q point to the same complete object, base class subobject or member subobject, the result is 0.
- Otherwise, the behavior is undefined.
These pointer arithmetic operators allow pointers to satisfy the LegacyRandomAccessIterator requirements.
For addition and subtraction, if P or Q have type “pointer to (possibly cv-qualified) T”, where T and the array element type are not similar, the behavior is undefined:
int arr[5] = {1, 2, 3, 4, 5}; unsigned int *p = reinterpret_cast<unsigned int*>(arr + 1); unsigned int k = *p; // OK, the value of “k” is 2 unsigned int *q = p + 1; // undefined behavior: “p” points to int, not unsigned int
Overloads
In overload resolution against user-defined operators, for every pair of promoted arithmetic types L and R and for every object type T, the following function signatures participate in overload resolution:
| LR operator+(L, R) |
||
| LR operator-(L, R) |
||
| T* operator+(T*, std::ptrdiff_t) |
||
| T* operator+(std::ptrdiff_t, T*) |
||
| T* operator-(T*, std::ptrdiff_t) |
||
| std::ptrdiff_t operator-(T*, T*) |
||
where LR is the result of usual arithmetic conversions on L and R.
#include <iostream> int main() { char c = 2; unsigned int un = 2; int n = -10; std::cout << " 2 + (-10), where 2 is a char = " << c + n << "\n" " 2 + (-10), where 2 is unsigned = " << un + n << "\n" " -10 - 2.12 = " << n - 2.12 << '\n'; char a[4] = {'a', 'b', 'c', 'd'}; char* p = &a[1]; std::cout << "Pointer addition examples: " << *p << *(p + 2) << *(2 + p) << *(p - 1) << '\n'; char* p2 = &a[4]; std::cout << "Pointer difference: " << p2 - p << '\n'; }
Output:
2 + (-10), where 2 is a char = -8 2 + (-10), where 2 is unsigned = 4294967288 -10 - 2.12 = -12.12 Pointer addition examples: bdda Pointer difference: 3
Multiplicative operators
The multiplicative operator expressions have the form
lhs * rhs
|
(1) | ||||||||
lhs / rhs
|
(2) | ||||||||
lhs % rhs
|
(3) | ||||||||
Multiplicative operators have higher precedence than all other binary arithmetic operators. These operators associate from left to right:
a + b * c; // equivalent to a + (b * c), NOT (a + b) * c d / e - f; // equivalent to (d / e) - f, NOT d / (e - f) g % h >> i; // equivalent to (g % h) >> i, NOT g % (h >> i) j * k / l % m; // equivalent to ((j * k) / l) % m
Built-in multiplicative operators
For built-in multiplication and division operators, both operands must have arithmetic or unscoped enumeration type. For the built-in remainder operator, both operands must have integral or unscoped enumeration type. Usual arithmetic conversions are performed on both operands.
In the remaining description in this section, "operand(s)", lhs and rhs refer to the converted operand(s).
- Multiplication of a NaN by any number gives NaN.
- Multiplication of infinity by zero gives NaN and FE_INVALID is raised.
- If one operand is NaN, the result is NaN.
- Dividing a non-zero number by ±0.0 gives the correctly-signed infinity and FE_DIVBYZERO is raised.
- Dividing 0.0 by 0.0 gives NaN and FE_INVALID is raised.
Note: Until CWG issue 614 was resolved (N2757), if one or both operands to binary operator % were negative, the sign of the remainder was implementation-defined, as it depends on the rounding direction of integer division. The function std::div provided well-defined behavior in that case.
Note: for floating-point remainder, see std::remainder and std::fmod.
Overloads
In overload resolution against user-defined operators, for every pair of promoted arithmetic types LA and RA and for every pair of promoted integral types LI and RI the following function signatures participate in overload resolution:
| LRA operator*(LA, RA) |
||
| LRA operator/(LA, RA) |
||
| LRI operator%(LI, RI) |
||
where LRx is the result of usual arithmetic conversions on Lx and Rx.
#include <iostream> int main() { char c = 2; unsigned int un = 2; int n = -10; std::cout << "2 * (-10), where 2 is a char = " << c * n << "\n" "2 * (-10), where 2 is unsigned = " << un * n << "\n" "-10 / 2.12 = " << n / 2.12 << "\n" "-10 / 21 = " << n / 21 << "\n" "-10 % 21 = " << n % 21 << '\n'; }
Output:
2 * (-10), where 2 is a char = -20 2 * (-10), where 2 is unsigned = 4294967276 -10 / 2.12 = -4.71698 -10 / 21 = 0 -10 % 21 = -10
Bitwise logic operators
The bitwise logic operator expressions have the form
~ rhs
|
(1) | ||||||||
lhs & rhs
|
(2) | ||||||||
lhs | rhs
|
(3) | ||||||||
lhs ^ rhs
|
(4) | ||||||||
The bitwise NOT operator has higher precedence than all binary arithmetic operators. It associates from right to left:
~a - b; // equivalent to (~a) - b, NOT ~(a - b) ~c * d; // equivalent to (~c) * d, NOT ~(c * d) ~-e; // equivalent to ~(-e)
There is an ambiguity in the grammar when ~ is followed by a type name or decltype specifier(since C++11): it can either be operator~ or start a destructor identifier). The ambiguity is resolved by treating ~ as operator~. ~ can start a destructor identifier only in places where forming an operator~ is syntactically invalid.
All other bitwise logic operators have lower precedence than all other binary arithmetic operators. Bitwise AND has higher precedence than bitwise XOR, which has higher precedence than bitwise OR. They associate from left to right:
a & b * c; // equivalent to a & (b * c), NOT (a & b) * c d / e ^ f; // equivalent to (d / e) ^ f, NOT d / (e ^ f) g << h | i; // equivalent to (g << h) | i, NOT g << (h | i) j & k & l; // equivalent to (j & k) & l m | n ^ o // equivalent to m | (n ^ o)
Built-in bitwise logic operators
For the built-in bitwise NOT operator, rhs must be a prvalue of integral or unscoped enumeration type, and integral promotion is performed on rhs. For other built-in bitwise logic operators, both operands must have integral or unscoped enumeration type, and usual arithmetic conversions are performed on both operands.
In the remaining description in this section, "operand(s)", lhs and rhs refer to the converted or promoted operand(s).
- In other words, the result is the one’s complement of the operand (where operand and result are considered as unsigned).
Overloads
In overload resolution against user-defined operators, for every pair of promoted integral types L and R the following function signatures participate in overload resolution:
| R operator~(R) |
||
| LR operator&(L, R) |
||
| LR operator^(L, R) |
||
| LR operator|(L, R) |
||
where LR is the result of usual arithmetic conversions on L and R.
#include <bitset> #include <cstdint> #include <iomanip> #include <iostream> int main() { std::uint16_t mask = 0x00f0; std::uint32_t x0 = 0x12345678; std::uint32_t x1 = x0 | mask; std::uint32_t x2 = x0 & ~mask; std::uint32_t x3 = x0 & mask; std::uint32_t x4 = x0 ^ mask; std::uint32_t x5 = ~x0; using bin16 = std::bitset<16>; using bin32 = std::bitset<32>; std::cout << std::hex << std::showbase << "Mask: " << mask << std::setw(49) << bin16(mask) << "\n" "Value: " << x0 << std::setw(42) << bin32(x0) << "\n" "Setting bits: " << x1 << std::setw(35) << bin32(x1) << "\n" "Clearing bits: " << x2 << std::setw(34) << bin32(x2) << "\n" "Selecting bits: " << x3 << std::setw(39) << bin32(x3) << "\n" "XOR-ing bits: " << x4 << std::setw(35) << bin32(x4) << "\n" "Inverting bits: " << x5 << std::setw(33) << bin32(x5) << '\n'; }
Output:
Mask: 0xf0 0000000011110000 Value: 0x12345678 00010010001101000101011001111000 Setting bits: 0x123456f8 00010010001101000101011011111000 Clearing bits: 0x12345608 00010010001101000101011000001000 Selecting bits: 0x70 00000000000000000000000001110000 XOR-ing bits: 0x12345688 00010010001101000101011010001000 Inverting bits: 0xedcba987 11101101110010111010100110000111
Bitwise shift operators
The bitwise shift operator expressions have the form
lhs << rhs
|
(1) | ||||||||
lhs >> rhs
|
(2) | ||||||||
Bitwise shift operators have higher precedence than bitwise logic operators, but have lower precedence than additive and multiplicative operators. These operators associate from left to right:
a >> b * c; // equivalent to a >> (b * c), NOT (a >> b) * c d << e & f; // equivalent to (d << e) & f, NOT d << (e & f) g << h >> i; // equivalent to (g << h) >> i, NOT g << (h >> i)
Built-in bitwise shift operators
For the built-in bitwise shift operators, both operands must be prvalues of integral or unscoped enumeration type. Integral promotions are performed on both operands.
In the remaining description in this section, "operand(s)", a, b, lhs and rhs refer to the converted or promoted operand(s).
If the value of rhs is negative or is not less than the number of bits in lhs, the behavior is undefined.
|
For unsigned a, the value of a << b is the value of a * 2b For signed and non-negative a, if a * 2b For negative a, the behavior of a << b is undefined. For unsigned a and for signed and non-negative a, the value of a >> b is the integer part of a/2b For negative a, the value of a >> b is implementation-defined (in most implementations, this performs arithmetic right shift, so that the result remains negative). |
(until C++20) |
|
The value of a << b is the unique value congruent to a * 2b The value of a >> b is a/2b |
(since C++20) |
The type of the result is that of lhs.
Overloads
In overload resolution against user-defined operators, for every pair of promoted integral types L and R, the following function signatures participate in overload resolution:
| L operator<<(L, R) |
||
| L operator>>(L, R) |
||
#include <iostream> enum { ONE = 1, TWO = 2 }; int main() { std::cout << std::hex << std::showbase; char c = 0x10; unsigned long long ull = 0x123; std::cout << "0x123 << 1 = " << (ull << 1) << "\n" "0x123 << 63 = " << (ull << 63) << "\n" // overflow in unsigned "0x10 << 10 = " << (c << 10) << '\n'; // char is promoted to int long long ll = -1000; std::cout << std::dec << "-1000 >> 1 = " << (ll >> ONE) << '\n'; }
Output:
0x123 << 1 = 0x246 0x123 << 63 = 0x8000000000000000 0x10 << 10 = 0x4000 -1000 >> 1 = -500
Standard library
Arithmetic operators are overloaded for many standard library types.
Unary arithmetic operators
| implements unary + and unary - (public member function of std::chrono::duration<Rep,Period>) | |
| applies unary operators to complex numbers (function template) | |
| applies a unary arithmetic operator to each element of the valarray (public member function of std::valarray<T>) |
Additive operators
| (C++11) |
performs add and subtract operations involving a time point (function template) |
| implements arithmetic operations with durations as arguments (function template) | |
| (C++20) |
adds or subtracts a year_month_day and some number of years or months (function) |
| concatenates two strings, a string and a char, or a string and string_view (function template) | |
| advances or decrements the iterator (public member function of std::reverse_iterator<Iter>)
| |
| advances or decrements the iterator (public member function of std::move_iterator<Iter>)
| |
| performs complex number arithmetic on two complex values or a complex and a scalar (function template) | |
| applies binary operators to each element of two valarrays, or a valarray and a value (function template) |
Multiplicative operators
| implements arithmetic operations with durations as arguments (function template) | |
| performs complex number arithmetic on two complex values or a complex and a scalar (function template) | |
| applies binary operators to each element of two valarrays, or a valarray and a value (function template) |
Bitwise logic operators
| performs binary AND, OR, XOR and NOT (public member function of std::bitset<N>) | |
| performs binary logic operations on bitsets (function template) | |
| applies a unary arithmetic operator to each element of the valarray (public member function of std::valarray<T>)
| |
| applies binary operators to each element of two valarrays, or a valarray and a value (function template) |
Bitwise shift operators
| applies binary operators to each element of two valarrays, or a valarray and a value (function template) | |
| performs binary shift left and shift right (public member function of std::bitset<N>)
|
Stream insertion/extraction operators
Throughout the standard library, bitwise shift operators are commonly overloaded with I/O stream (std::ios_base& or one of the classes derived from it) as both the left operand and return type. Such operators are known as stream insertion and stream extraction operators:
| extracts formatted data (public member function of std::basic_istream<CharT,Traits>) | |
| extracts characters and character arrays (function template) | |
| inserts formatted data (public member function of std::basic_ostream<CharT,Traits>) | |
| inserts character data or insert into rvalue stream (function template) | |
| serializes and deserializes a complex number (function template) | |
| performs stream input and output of bitsets (function template) | |
| performs stream input and output on strings (function template) | |
| (C++11) |
performs stream input and output on pseudo-random number engine (function template) |
| (C++11) |
performs stream input and output on pseudo-random number distribution (function template) |
Defect reports
The following behavior-changing defect reports were applied retroactively to previously published C++ standards.
| DR | Applied to | Behavior as published | Correct behavior |
|---|---|---|---|
| CWG 614 | C++98 | the algebraic quotient of integer division was rounded in implementation-defined direction |
the algebraic quotient of integer division is truncated towards zero (fractional part is discarded) |
| CWG 1450 | C++98 | the result of a / b was unspecified if it is not representable in the result type |
the behavior of both a / b and a % b is undefined in this case |
| CWG 1457 | C++98 | the behavior of shifting the leftmost 1 bit of apositive signed value into the sign bit was undefined |
made well-defined |
| CWG 1504 | C++98 | a pointer to a base class subobject of an array element could be used in pointer arithmetic |
the behavior is undefined in this case |
| CWG 1515 | C++98 | only unsigned integers which declared unsigned should obey the laws of arithmetic modulo 2n |
applies to all unsigned integers |
| CWG 1642 | C++98 | arithmetic operators allow their operands to be lvalues | some operands must be rvalues |
| CWG 1865 | C++98 | the resolution of CWG issue 1504 made the behaviors of pointer arithmetic involving pointers to array element undefined if the pointed-to type and the array element type have different cv-qualifications in non-top levels |
made well-defined |
| CWG 1971 | C++98 | it was unclear whether the rule resolving the ambiguity of ~ applies to cases such as ~X(0) |
the rule applies to such cases |
| CWG 2419 | C++98 | a pointer to non-array object was only treated as a pointer to the first element of an array with size 1 in pointer arithmetic if the pointer is obtained by & |
applies to all pointers to non-array objects |
| CWG 2626 | C++98 | the result of built-in operator~ was simply'one's complement' without proper definition |
the result is phrased in terms of the base-2 representation |
| CWG 2724 | C++20 | the rounding direction of arithmetic right shift was unclear | made clear |
| CWG 2853 | C++98 | a pointer past the end of an object could not be added or subtracted with an integer |
it can |
See also
| Common operators | ||||||
|---|---|---|---|---|---|---|
| assignment | increment decrement |
arithmetic | logical | comparison | member access |
other |
|
a = b |
++a |
+a |
!a |
a == b |
a[...] |
function call |
| a(...) | ||||||
| comma | ||||||
| a, b | ||||||
| conditional | ||||||
| a ? b : c | ||||||
| Special operators | ||||||
|
static_cast converts one type to another related type | ||||||
| C documentation for Arithmetic operators
|