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Learning C (Part 4): Bitwise Operators

(This post is part of the learning-c series.)

This article will discuss bitwise operators and common procedures done with them.

What do these bits look like?

Consider an unsigned short x with the value 15. Note that $1111_{2} = 15_{10}$. Consequently, if the size of unsigned short was 16 bits on our computer, we could expect x to appear like so:

If we were to "align" another unsigned short with our variable x, we could perform various logical operations between the two as if they were part of a truth table. For example, suppose we have an unsigned short y with value 9, which is equal to $1001_{2}$. We could perform a logical AND between x and y. The result is as follows:

Basic operations

NOT

The one's complement operator, ~, informally known as the unary NOT operator, takes the bits of an integer and flips them: 0's become 1's, 1's become 0's.

For example, consider an unsigned char c with 8 bits, with value 0. This means every bit composing c is set to 0. Thus ~c would transform all bits from 0 to 1. In this case, c actually goes from being the smallest possible unsigned char value to the largest possible unsigned char value!

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#include <stdio.h>
#include <limits.h>
int main(){
    unsigned char c = 0;
    printf("c: %u\n", c);
    printf("~c: %u, UCHAR_MAX: %u\n", (unsigned char) ~c, UCHAR_MAX);
    return 0;
}

raw source | show output ⇓

c: 0
~c: 255, UCHAR_MAX: 255

As a more interesting example, consider $57_{10} = 111001_{2}$ stored in an unsigned char variable c.

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#include <stdio.h>
int main(){
    unsigned char c = 57;
    printf("c: %u\n", c);
    printf("~c: %u\n", (unsigned char) ~c);
    return 0;
}

raw source | show output ⇓

c: 57
~c: 198

It's important to note that the type of integer the result of ~c will be stored in affects the outcome greatly. If we were to store the result of ~c into an unsigned int, for example, the leading 0's of the unsigned int would be converted to 1s, leading to a much larger result.

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#include <stdio.h>
int main(){
    unsigned char c = 57;
    unsigned int i;
    printf("c: %u\n", c);
    printf("~c: %u\n", (unsigned char) ~c);
    printf("~c: %u\n", (unsigned int) ~c);
    return 0;
}

raw source | show output ⇓

c: 57
~c: 198
~c: 4294967238

AND

And is used as a means to turn specific bits off. ANDing a bit at a particular location with 0 makes it 0, and ANDing with 1 preserves the value.

Suppose we have an unsigned char c with value $31_{10} = 11111_{2}$. Now suppose we want to turn the last 3 bits of c off (AKA set them to 0). The expected result would be $11000_{2} = 24_{10}$.

The way we do this is by ANDing with a sequence of bits that are all 1 except for the last 3, which will be 0. We can craft such a sequence in two steps:

  1. Create a sequence of bits that are all 0 except the last 3. We must have exactly enough to "fill" up the integer.
  2. NOT the sequence, causing all values to be 1 except the last 3

When we assign a value to an integral type (char, short, int, long), C will pad the value the necessary amount of 0s to make it fit the type correctly. Thus, if an unsigned char is 8 bits on our computer, the statement

unsigned char c = 0;

will completely fill c up with 8 bits of 0s. Consequently, we can take advantage of this to construct a sequence of 1s that fills up all 8 bits just by using the unary NOT operator

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#include <stdio.h>
int main(){
    unsigned char c;
    c = 0;
    printf("c: %u\n", c);
    c = 1;
    printf("~c: %u\n", c);
    c = ~0;
    printf("~c: %u\n", c);
    return 0;
}

raw source | show output ⇓

c: 0
~c: 1
~c: 255

We can now construct the sequence of bits that are all 1s except the last 3. To demonstrate, such a sequence for an 8 bit unsigned char would be $11111000_{2} = 248_{10}$. So we should start off with the sequence $00000111_{2}$ and then complement it.

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#include <stdio.h>
int main(){
    unsigned char c = ~0x7;
    printf("c: %u\n", c);
    return 0;
}

raw source | show output ⇓

c: 248

Now to AND this bitmask with our original $31_{10} = 11111_{2}$, with the expected result being $11000_{2} = 24_{10}$

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#include <stdio.h>
#define BITMASK ~0x7
int main(){
    unsigned char c = 31;
    printf("c: %u\n", c);
    c = c & BITMASK;
    printf("c: %u\n", c);
    return 0;
}

raw source | show output ⇓

c: 31
c: 24

Generating the bitmask is much simpler using shifts, which I'm unfortunately too bored to discuss in too much detail.

SHIFTS

There is a left shift, x << n, which shifts each bit of an integral to the left n times. There is also a right shift. Here are the only details we'll cover, since the shifts themselves are pretty straight forward:

  1. Left shifting always causes vacated bits to be filled with 0s
  2. Right shifting an unsigned quantity always fills the vacated bits with 0s
    • Right shifting a signed quantity may fill vacated bits with different symbols.
  3. x << n is equivalent to $x \times 2^n$
  4. x >> n is equivalent to $x / 2^n$ (integer division)

getbits() function

Suppose we have an integral value, and we would like to extract a certain number of bits at a particular position. Specifically, suppose we want to extract n bits from an integral starting at position p, where the rightmost bit is position 0.

For example:

So how do we actually achieve the desired result? It takes two simple steps:

  1. Shift our integral to the right until the last bit we are interested in sits at position 0.
  2. Set all bits outside of our range of interest to 0

In the example above, we can see that getbits(x,5,3) took 3 shifts to the right to get the last bit we were interested in (the 0 at position 3) to position 0. It took getbits(x,3,2) 2 shifts, and getbits(x,6,4) 3 shifts. Thus the number of shifts needed can be generalized to p+1-n, and you can use the examples above to verify that claim.

Why we do need to set bits outside of our range of interest to 0? Because there may be bits set to 1 to the left of our p. We only want the n bits starting from p to be returned, so we must set all bits to the left of our range after the shift to 0.

The way to generate this bitmask is to:

  1. Start off with a sequence of all 1s
  2. Shift to the left n times. This will place 0s in the vacant slots
  3. Take the complement of the sequence, which will leave us with all 0s followed by 1s in the last n bits.

The getbits() function can thus be defined as so:

unsigned getbits(unsigned x, int p, int n) {
    return (x >> (p+1-n)) & ~(~0 << n);
}

Tagged as c

(This post is part of the learning-c series.)

Date published - December 03, 2013