O.2 — Bitwise operators

The bitwise operators

C++ provides 6 bit manipulation operators, often called bitwise operators:

The bitwise operators are defined for integral types and std::bitset. We’ll use std::bitset in our examples because it’s easier to print the output in binary.

Avoid using the bitwise operators with signed operands, as many operators will return implementation-defined results prior to C++20 or have other potential gotchas that are easily avoided by using unsigned operands (or std::bitset).

Bitwise left shift (<<) and bitwise right shift (>>) operators

The bitwise left shift (<<) operator shifts bits to the left. The left operand is the expression to shift the bits of, and the right operand is an integer number of bits to shift left by.

So when we say x << 1, we are saying “shift the bits in the variable x left by 1 place”. New bits shifted in from the right side receive the value 0.

0011 << 1 is 0110
0011 << 2 is 1100
0011 << 3 is 1000

Note that in the third case, we shifted a bit off the end of the number! Bits that are shifted off the end of the binary number are lost forever.

The bitwise right shift (>>) operator shifts bits to the right.

1100 >> 1 is 0110
1100 >> 2 is 0011
1100 >> 3 is 0001

Note that in the third case we shifted a bit off the right end of the number, so it is lost.

Here’s an example of doing some bit shifting:

#include <bitset>
#include <iostream>

int main()
{
    std::bitset<4> x { 0b1100 };

    std::cout << x << '\n';
    std::cout << (x >> 1) << '\n'; // shift right by 1, yielding 0110
    std::cout << (x << 1) << '\n'; // shift left by 1, yielding 1000

    return 0;
}

This prints:

1100
0110
1000

What!? Aren’t operator<< and operator>> used for input and output?

They sure are.

Programs today typically do not make much use of the bitwise left and right shift operators to shift bits. Rather, you tend to see the bitwise left shift operator used with std::cout (or other stream objects) to output text. Consider the following program:

#include <bitset>
#include <iostream>

int main()
{
    unsigned int x { 0b0100 };
    x = x << 1; // use operator<< for left shift
    std::cout << std::bitset<4>{ x } << '\n'; // use operator<< for output

    return 0;
}

This program prints:

1000

In the above program, how does operator<< know to shift bits in one case and output x in another case? The answer is that std::cout has overloaded (provided an alternate definition for) operator<< that does console output rather than bit shifting.

When the compiler sees that the left operand of operator<< is std::cout, it knows that it should call the version of operator<< that std::cout overloaded to do output. If the left operand is an integral type, then operator<< knows it should do its usual bit-shifting behavior.

The same applies for operator>>.

Note that if you’re using operator << for both output and left shift, parenthesization is required:

#include <bitset>
#include <iostream>

int main()
{
	std::bitset<4> x{ 0b0110 };

	std::cout << x << 1 << '\n'; // print value of x (0110), then 1
	std::cout << (x << 1) << '\n'; // print x left shifted by 1 (1100)

	return 0;
}

This prints:

01101
1100

The first line prints the value of x (0110), and then the literal 1. The second line prints the value of x left-shifted by 1 (1100).

We will talk more about operator overloading in a future chapter, including discussion of how to overload operators for your own purposes.

Bitwise NOT

The bitwise NOT operator (~) is perhaps the easiest to understand of all the bitwise operators. It simply flips each bit from a 0 to a 1, or vice versa. Note that the result of a bitwise NOT is dependent on what size your data type is.

Flipping 4 bits:
~0100 is 1011

Flipping 8 bits:
~0000 0100 is 1111 1011

In both the 4-bit and 8-bit cases, we start with the same number (binary 0100 is the same as 0000 0100 in the same way that decimal 7 is the same as 07), but we end up with a different result.

We can see this in action in the following program:

#include <bitset>
#include <iostream>

int main()
{
	std::cout << ~std::bitset<4>{ 0b0100 } << ' ' << ~std::bitset<8>{ 0b0100 } << '\n';

	return 0;
}

This prints:
1011 11111011

Bitwise OR

Bitwise OR (|) works much like its logical OR counterpart. However, instead of applying the OR to the operands to produce a single result, bitwise OR applies to each bit! For example, consider the expression 0b0101 | 0b0110.

To do (any) bitwise operations, it is easiest to line the two operands up like this:

0 1 0 1 OR
0 1 1 0

and then apply the operation to each column of bits.

If you remember, logical OR evaluates to true (1) if either the left, right, or both operands are true (1), and 0 otherwise. Bitwise OR evaluates to 1 if either the left, right, or both bits are 1, and 0 otherwise. Consequently, the expression evaluates like this:

0 1 0 1 OR
0 1 1 0
-------
0 1 1 1

Our result is 0111 binary.

#include <bitset>
#include <iostream>

int main()
{
	std::cout << (std::bitset<4>{ 0b0101 } | std::bitset<4>{ 0b0110 }) << '\n';

	return 0;
}

This prints:

0111

We can do the same thing to compound OR expressions, such as 0b0111 | 0b0011 | 0b0001. If any of the bits in a column are 1, the result of that column is 1.

0 1 1 1 OR
0 0 1 1 OR
0 0 0 1
--------
0 1 1 1

Here’s code for the above:

#include <bitset>
#include <iostream>

int main()
{
	std::cout << (std::bitset<4>{ 0b0111 } | std::bitset<4>{ 0b0011 } | std::bitset<4>{ 0b0001 }) << '\n';

	return 0;
}

This prints:

0111

Bitwise AND

Bitwise AND (&) works similarly to the above. Logical AND evaluates to true if both the left and right operand evaluate to true. Bitwise AND evaluates to true (1) if both bits in the column are 1. Consider the expression 0b0101 & 0b0110. Lining each of the bits up and applying an AND operation to each column of bits:

0 1 0 1 AND
0 1 1 0
--------
0 1 0 0

#include <bitset>
#include <iostream>

int main()
{
	std::cout << (std::bitset<4>{ 0b0101 } & std::bitset<4>{ 0b0110 }) << '\n';

	return 0;
}

This prints:

0100

Similarly, we can do the same thing to compound AND expressions, such as 0b0001 & 0b0011 & 0b0111. If all of the bits in a column are 1, the result of that column is 1.

0 0 0 1 AND
0 0 1 1 AND
0 1 1 1
--------
0 0 0 1

#include <bitset>
#include <iostream>

int main()
{
	std::cout << (std::bitset<4>{ 0b0001 } & std::bitset<4>{ 0b0011 } & std::bitset<4>{ 0b0111 }) << '\n';

	return 0;
}

This prints:

0001

Bitwise XOR

The last operator is the bitwise XOR (^), also known as exclusive or.

When evaluating two operands, XOR evaluates to true (1) if one and only one of its operands is true (1). If neither or both are true, it evaluates to 0. Consider the expression 0b0110 ^ 0b0011:

0 1 1 0 XOR
0 0 1 1
-------
0 1 0 1

It is also possible to evaluate compound XOR expression column style, such as 0b0001 ^ 0b0011 ^ 0b0111. If there are an even number of 1 bits in a column, the result is 0. If there are an odd number of 1 bits in a column, the result is 1.

0 0 0 1 XOR
0 0 1 1 XOR
0 1 1 1
--------
0 1 0 1

Bitwise assignment operators

Similar to the arithmetic assignment operators, C++ provides bitwise assignment operators in order to facilitate easy modification of variables.

For example, instead of writing x = x >> 1;, you can write x >>= 1;.

#include <bitset>
#include <iostream>

int main()
{
    std::bitset<4> bits { 0b0100 };
    bits >>= 1;
    std::cout << bits << '\n';

    return 0;
}

This program prints:

0010

Bitwise operators perform integral promotion on smaller integral types Advanced

If the operand(s) of a bitwise operator are an integral type smaller than an int, those operands will be promoted (converted) to int or unsigned int, and the result returned will also be an int or unsigned int. For example, if our operands are unsigned short, they will be promoted (converted) to unsigned int, and the result of the operation will be returned as an unsigned int.

In many cases, this won’t matter.

However, when using bitwise operators on integral types narrower than int or unsigned int, there are two cases to watch out for:

• operator~ and operator<< are width-sensitive and may produce different results depending on the width of the operand.
• Initializing or assigning the result to a variable of the smaller integral type is a narrowing conversion (since converting an int or unsigned int to a smaller integral type may result in data loss). This is disallowed in list initialization, and your compiler may or may not complain about a narrowing assignment.

The following program exhibits these issues (assuming 32-bit ints):

#include <bitset>
#include <cstdint>
#include <iostream>

int main()
{
    std::uint8_t c { 0b00001111 };
    
    std::cout << std::bitset<32>(~c) << '\n';     // incorrect: prints 11111111111111111111111111110000
    std::cout << std::bitset<32>(c << 6) << '\n'; // incorrect: prints 0000000000000000001111000000
    std::uint8_t cneg { ~c };                     // error: narrowing conversion from unsigned int to std::uint8_t
    c = ~c;                                       // possible warning: narrowing conversion from unsigned int to std::uint8_t
 
    return 0;
}

These issues can be addressed by using static_cast to convert the result of your bitwise operation back to the narrower integral type. The following program produces the correct results:

#include <bitset>
#include <cstdint>
#include <iostream>

int main()
{
    std::uint8_t c { 0b00001111 };

    std::cout << std::bitset<32>(static_cast<std::uint8_t>(~c)) << '\n';     // correct: prints 00000000000000000000000011110000
    std::cout << std::bitset<32>(static_cast<std::uint8_t>(c << 6)) << '\n'; // correct: prints 0000000000000000000011000000
    std::uint8_t cneg { static_cast<std::uint8_t>(~c) };                     // compiles
    c = static_cast<std::uint8_t>(~c);                                       // no warning
 
    return 0;
}

Summary

Summarizing how to evaluate bitwise operations utilizing the column method:

When evaluating bitwise OR, if any bit in a column is 1, the result for that column is 1.
When evaluating bitwise AND, if all bits in a column are 1, the result for that column is 1.
When evaluating bitwise XOR, if there are an odd number of 1 bits in a column, the result for that column is 1.

In the next lesson, we’ll explore how these operators can be used in conjunction with bit masks to facilitate bit manipulation.

Quiz time

0 0 1 1 OR
0 1 0 1
--------
0 1 1 1

0 0 1 1 AND
0 1 0 1
--------
0 0 0 1

Inside the parenthesis:

0 0 1 1 OR
0 1 0 1
--------
0 1 1 1

Then:

0 1 1 1 AND
1 0 0 1
--------
0 0 0 1

#include <bitset>
#include <iostream>

// "rotl" stands for "rotate left"
std::bitset<4> rotl(std::bitset<4> bits)
{
// Your code here
}

int main()
{
	std::bitset<4> bits1{ 0b0001 };
	std::cout << rotl(bits1) << '\n';

	std::bitset<4> bits2{ 0b1001 };
	std::cout << rotl(bits2) << '\n';

	return 0;
}

0010
0011

#include <bitset>
#include <iostream>

std::bitset<4> rotl(std::bitset<4> bits)
{
	// keep track of whether the leftmost bit was a 1
	const bool leftbit{ bits.test(3) };

	bits <<= 1; // do left shift (which shifts the leftmost bit off the end)

	// if the left bit was a 1
	if (leftbit)
		bits.set(0); // set the rightmost bit to a 1

	return bits;
}

int main()
{
	std::bitset<4> bits1{ 0b0001 };
	std::cout << rotl(bits1) << '\n';

	std::bitset<4> bits2{ 0b1001 };
	std::cout << rotl(bits2) << '\n';

	return 0;
}

#include <bitset>
#include <iostream>

// h/t to reader Chris for this solution
std::bitset<4> rotl(std::bitset<4> bits)
{
	// bits << 1 does the left shift
	// bits >> 3 handle the rotation of the leftmost bit
	return (bits<<1) | (bits>>3);
}

int main()
{
	std::bitset<4> bits1{ 0b0001 };
	std::cout << rotl(bits1) << '\n';

	std::bitset<4> bits2{ 0b1001 };
	std::cout << rotl(bits2) << '\n';

	return 0;
}