Difference between revisions of "cpp/language/operator arithmetic"

Latest revision as of 23:30, 24 June 2024

Returns the result of specific arithmetic operation.

[edit] General explanation

All built-in arithmetic operators compute the result of specific arithmetic operation and returns its result. The arguments are not modified.

[edit] 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.

[edit] 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. -ftrapv in GCC and Clang),
• it saturates to minimal or maximal value (on many DSPs),
• it is completely optimized out by the compiler.

[edit] 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).

[edit] 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.

[edit] Unary arithmetic operators

The unary arithmetic operator expressions have the form

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

[edit] Built-in unary arithmetic operators

[edit] 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:

#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';
}

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

[edit] Additive operators

The additive operator expressions have the form

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

[edit] 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).

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.

[edit] 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.

• 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.

• 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​.

• 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​.

• 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 and jth 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

[edit] 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:

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';
}

 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

[edit] Multiplicative operators

The multiplicative operator expressions have the form

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

[edit] 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).

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.

[edit] 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:

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';
}

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

[edit] Bitwise logic operators

The bitwise logic operator expressions have the form

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)

[edit] 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).

[edit] 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:

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';
}

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

[edit] Bitwise shift operators

The bitwise shift operator expressions have the form

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)

[edit] 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.

The type of the result is that of lhs.

[edit] 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:

#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';
}

0x123 << 1 = 0x246
0x123 << 63 = 0x8000000000000000
0x10 << 10 = 0x4000
-1000 >> 1 = -500

[edit] Standard library

Arithmetic operators are overloaded for many standard library types.

[edit] Unary arithmetic operators

[edit] Additive operators

[edit] Multiplicative operators