Bitwise operations in C

In the C programming language, operations can be performed on a bit level using bitwise operators.

Bitwise operations are contrasted by byte-level operations which characterize the bitwise operators' logical counterparts, the AND, OR and NOT operators. Instead of performing on individual bits, these operators perform on strings of eight bits (known as bytes) at a time. The reason for this is that a byte is normally the smallest unit of addressable memory (i.e. data with a unique memory address.)

This applies to bitwise operators as well, which means that even though they operate on only one bit at a time they cannot accept anything smaller than a byte as their input.

Bitwise operators

C provides six operators for bit manipulation.[1]

Symbol Operator
& bitwise AND
| bitwise inclusive OR
^ bitwise XOR (eXclusive OR)
<< left shift
>> right shift
~ bitwise NOT (one's complement) (unary)

Bitwise AND "&"

bit a bit b a & b (a AND b)
0 0 0
0 1 0
1 0 0
1 1 1

The bitwise AND operator is a single ampersand: &. It is just a representation of AND which does its work on the bits of the operands rather than the truth value of the operands. Bitwise binary AND does the logical AND (as shown in the table above) of the bits in each position of a number in its binary form.

For instance, working with a byte (the char type): 0

The most significant bit of the first number is 1 and that of the second number is also 1 so the most significant bit of the result is 1; in the second most significant bit, the bit of second number is zero, so we have the result as 0. [2]

Bitwise OR "|"

bit a bit b a | b (a OR b)
0 0 0
0 1 1
1 0 1
1 1 1

Similar to bitwise AND, bitwise OR only operates at the bit level. Its result is a 1 if one of the either bits is 1 and zero only when both bits are 0. Its symbol is '|' which can be called a pipe.

```     11001110
| 10011000
= 11011110
```

[2]

Bitwise XOR "^"

bit a bit b a ^ b (a XOR b)
0 0 0
0 1 1
1 0 1
1 1 0

The Bitwise XOR (exclusive or) performs a logical XOR function, which is equivalent to adding two bits and discarding the carry. The result is zero only when we have two zeroes or two ones.[3] XOR can be used to toggle the bits between 1 and 0. Thus i = i ^ 1 when used in a loop toggles its values between 1 and 0.[4]

Bitwise NOT "~" / one's complement (unary)

bit a ~a (complement of a)
0 1
1 0

The one's complement (~) or the bitwise complement gets us the complement of a given number. Thus we get the bits inverted, for every bit 1 the result is bit 0 and conversely for every bit 0 we have a bit 1. This operation should not be confused with logical negation "!".

Summary of bitwise operators

bit a bit b a & b (a AND b) a | b (a OR b) a ^ b (a XOR b)
0 0 0 0 0
0 1 0 1 1
1 0 0 1 1
1 1 1 1 0

Shift operators

There are two bitwise shift operators. They are

• Right shift (>>)
• Left shift (<<)

Right shift >>

The symbol of right shift operator is >>. For its operation, it requires two operands. It shifts each bit in its left operand to the right. The number following the operator decides the number of places the bits are shifted (i.e. the right operand). Thus by doing ch >> 3 all the bits will be shifted to the right by three places and so on.

Example:

If the variable ch contains the bit pattern 11100101, then ch >> 1 will produce the result 01110010, and ch >> 2 will produce 00111001.

Here blank spaces are generated simultaneously on the left when the bits are shifted to the right. When performed on an unsigned type, the operation performed is a logical shift, causing the blanks to be filled by 0s (zeros). When performed on a signed type, an arithmetic shift is performed, causing the blank to be filled with the sign bit of the left operand.

Right shift can be used to divide a bit pattern by 2 as shown:

``` i = 14; // Bit pattern 1110
j = i >> 1; // here we have the bit pattern shifted by 1 thus we get 111 = 7 which is 14/2
```

Right shift operator usage

Typical usage of a right shift operator in C can be seen from the following code.

Example:

```    #include <stdio.h>

void showbits(unsigned int x)
{
int i;
for(i=(sizeof(int)*8)-1; i>=0; i--)
(x&(1<<i))?putchar('1'):putchar('0');

printf("\n");
}

int main()
{
int j = 5225, m, n;
printf("The decimal %d is equal to binary - ", j);
/* assume we have a function that prints a binary string when given
a decimal integer
*/
showbits(j);

/* the loop for right shift operation */
for ( m = 0; m <= 5; m++ ) {
n = j >> m;
printf("%d right shift %d gives ", j, m);
showbits(n);
}
return 0;
}
```

The output of the above program will be

```  The decimal 5225 is equal to binary - 0001010001101001
5225 right shift 0 gives 0001010001101001
5225 right shift 1 gives 0000101000110100
5225 right shift 2 gives 0000010100011010
5225 right shift 3 gives 0000001010001101
5225 right shift 4 gives 0000000101000110
5225 right shift 5 gives 0000000010100011
```

Left shift <<

The symbol of left shift operator is <<. It shifts each bit in its left-hand operand to the left by the number of positions indicated by the right-hand operand. It works opposite to that of right shift operator. Thus by doing ch << 1 in the above example we have 11001010. Blank spaces generated are filled up by zeroes as above.

Left shift can be used to multiply an integer in multiples of 2 as in

``` int i = 4; /* bit pattern equivalent is 100 */
int j = i << 2; /* makes it 10000, which multiplies the original number by 4 i.e. 16 */
```

The following program adds two operands using AND, XOR and left shift (<<).

```#include <stdio.h>

int main()
{
unsigned int x = 3, y = 1, sum, carry;
sum = x ^ y; // x XOR y
carry = x & y; // x AND y
while (carry != 0) {
carry = carry << 1; // left shift the carry
x = sum; // initialize x as sum
y = carry; // initialize y as carry
sum = x ^ y; // sum is calculated
carry = x & y; /* carry is calculated, the loop condition is
evaluated and the process is repeated until
carry is equal to 0.
*/
}
printf("%d\n", sum); // the program will print 4
return 0;
}
```

Bitwise assignment operators

C provides a compound assignment operator for each binary arithmetic and bitwise operation (i.e. each operation which accepts two operands). Each of the compound bitwise assignment operators perform the appropriate binary operation and store the result in the left operand.[5]

The bitwise assignment operators are as follows:

Symbol Operator
`&=` bitwise AND assignment
`|=` bitwise inclusive OR assignment
`^=` bitwise exclusive OR assignment
`<<=` left shift assignment
`>>=` right shift assignment

Logical equivalents

Four of the bitwise operators have equivalent logical operators. They are equivalent in that they have the same truth tables. However, logical operators treat each operand as having only one value, either true or false, rather than treating each bit of an operand as an independent value. Logical operators consider zero false and any nonzero value true. Another difference is that logical operators perform short-circuit evaluation.

The table below matches equivalent operators and shows a and b as operands of the operators.

Bitwise Logical
`a & b` `a && b`
`a | b` `a || b`
`a ^ b` `a != b`
`~a` `!a`

`!=` has the same truth table as `^` but unlike the true logical operators, by itself `!=` is not strictly speaking a logical operator. This is because a logical operator must treat any nonzero value the same. To be used as a logical operator `!=` requires that operands be normalized first. A logical not applied to both operands won’t change the truth table that results but will ensure all nonzero values are converted to the same value before comparison. This works because `!` on a zero always results in a one and `!` on any nonzero value always results in a zero.

The example below shows that the truth tables for these Bitwise and Logical operators are identical but also demonstrates how they act on their operands differently. The need to normalize operands for `!=` can be demonstrated by introducing a `char T2 = 0x02` and packing it in an array. Mixing `T2` with `T` will show them being treated the same unless the operand NOTs around `!=` are removed.

Example:

```/* Equivalent bitwise and logical operator tests */

void testOperator(char* name, unsigned char was, unsigned char expected);

main()
{
// -- Bitwise operators -- //

//Truth tables packed in bits
const unsigned char operand1    = 0x0A; //0000 1010
const unsigned char operand2    = 0x0C; //0000 1100
const unsigned char expectedAnd = 0x08; //0000 1000
const unsigned char expectedOr  = 0x0E; //0000 1110
const unsigned char expectedXor = 0x06; //0000 0110

const unsigned char operand3    = 0x01; //0000 0001
const unsigned char expectedNot = 0xFE; //1111 1110

testOperator("Bitwise AND", operand1 & operand2, expectedAnd);
testOperator("Bitwise  OR", operand1 | operand2, expectedOr);
testOperator("Bitwise XOR", operand1 ^ operand2, expectedXor);
testOperator("Bitwise NOT", ~operand3, expectedNot);
printf("\n");

// -- Logical operators -- //

const unsigned char F = 0x00; //Zero
const unsigned char T = 0x01; //Any nonzero value

//Truth tables packed in arrays
const unsigned char operandArray1[4]    = {T, F, T, F};
const unsigned char operandArray2[4]    = {T, T, F, F};
const unsigned char expectedArrayAnd[4] = {T, F, F, F};
const unsigned char expectedArrayOr[4]  = {T, T, T, F};
const unsigned char expectedArrayXor[4] = {F, T, T, F};

const unsigned char operandArray3[2]    = {F, T};
const unsigned char expectedArrayNot[2] = {T, F};

int i;
for (i = 0; i < 4; i++) {
testOperator("Logical AND", operandArray1[i] && operandArray2[i], expectedArrayAnd[i]);
}
printf("\n");

for (i = 0; i < 4; i++) {
testOperator("Logical  OR", operandArray1[i] || operandArray2[i], expectedArrayOr[i]);
}
printf("\n");

for (i = 0; i < 4; i++) {
//Needs ! on operand's in case nonzero values are different
testOperator("Logical XOR", !operandArray1[i] != !operandArray2[i], expectedArrayXor[i]);
}
printf("\n");

for (i = 0; i < 2; i++) {
testOperator("Logical NOT", !operandArray3[i], expectedArrayNot[i]);
}
printf("\n");
}

void testOperator(char* name, unsigned char was, unsigned char expected) {
char* result = (was == expected) ? "passed" : "failed";
printf("%s %s test, was: %X expected: %X \n", name, result, was, expected);
}
```

The output of the above program will be

```Bitwise AND passed, was: 8 expected: 8
Bitwise  OR passed, was: E expected: E
Bitwise XOR passed, was: 6 expected: 6
Bitwise NOT passed, was: FE expected: FE

Logical AND passed, was: 1 expected: 1
Logical AND passed, was: 0 expected: 0
Logical AND passed, was: 0 expected: 0
Logical AND passed, was: 0 expected: 0

Logical  OR passed, was: 1 expected: 1
Logical  OR passed, was: 1 expected: 1
Logical  OR passed, was: 1 expected: 1
Logical  OR passed, was: 0 expected: 0

Logical XOR passed, was: 0 expected: 0
Logical XOR passed, was: 1 expected: 1
Logical XOR passed, was: 1 expected: 1
Logical XOR passed, was: 0 expected: 0

Logical NOT passed, was: 1 expected: 1
Logical NOT passed, was: 0 expected: 0
```