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.. highlight:: cpp
.. _lang-bitwisemath:
Bitwise Operators (``&``, ``|``, ``^``, ``~``)
==============================================
The bitwise operators perform their calculations at the bit level of
variables. They help solve a wide range of common programming
problems.
Much of the material here is adapted for Maple from an (Arduino)
`tutorial on bitwise math
<http://www.arduino.cc/playground/Code/BitMath>`_\ . Another great
resource is the Wikipedia article on `bitwise operations
<http://en.wikipedia.org/wiki/Bitwise_operation>`_\ .
Below are descriptions and syntax for all of the operators.
.. contents:: Contents
:local:
.. _lang-bitwisemath-and:
Bitwise AND (``&``)
-------------------
The bitwise AND operator in C++ is a single ampersand, ``&``, used
between two other integer expressions. Bitwise AND operates on each
bit position of the surrounding expressions independently, according
to this rule: if both input bits are 1, the resulting output is 1,
otherwise the output is 0. Another way of expressing this is::
0 0 1 1 operand1
0 1 0 1 operand2
----------
0 0 0 1 (operand1 & operand2) = result
On the Maple, the type ``int`` is a 32-bit value, so using ``&``
between two ``int`` expressions causes 32 simultaneous AND operations
to occur. In a code fragment like::
int a = 92; // in binary: 00000000000000000000000001011100
int b = 101; // in binary: 00000000000000000000000001100101
int c = a & b; // result: 00000000000000000000000001000100,
// (or 68 in decimal).
Each of the 32 bits in ``a`` and ``b`` are processed using bitwise
AND, and all 32 resulting bits are stored in ``c``, resulting in the
value 1000100 in binary, which is 68 in decimal.
.. _lang-bitwisemath-or:
Bitwise OR (``|``)
------------------
The bitwise OR operator in C++ is the vertical bar symbol, ``|``. Like
the ``&`` operator, ``|`` operates independently on each bit in its
two surrounding integer expressions, but what it does is
different. The bitwise OR of two bits is 1 if either or both of the
input bits is 1, otherwise it is 0. For example::
0 0 1 1 operand1
0 1 0 1 operand2
----------
0 1 1 1 (operand1 | operand2) = result
Here is an example of bitwise OR used in a snippet of C++ code (using
``char``, which takes up 8 bits of memory, instead of ``int``, which
uses 32)::
char a = 92; // in binary: 01011100
char b = 101; // in binary: 01100101
char c = a | b; // result: 01111101, or 125 in decimal.
.. _lang-bitwisemath-xor:
Bitwise XOR (``^``)
-------------------
There is a somewhat unusual operator in C++ called bitwise EXCLUSIVE
OR, also known as bitwise XOR. (In English, this is usually pronounced
"zor" or "ex-or"). The bitwise XOR operator is written using the caret
symbol, ``^``. This operator is very similar to the bitwise OR
operator ``|``, except it evaluates to 0 for a given bit position when
both of the input bits for that position are 1::
0 0 1 1 operand1
0 1 0 1 operand2
----------
0 1 1 0 (operand1 ^ operand2) = result
Another way to look at bitwise XOR is that each bit in the result
is a 1 if the input bits are different, or 0 if they are the same.
Here is a simple example::
int x = 12; // binary (ignoring extra bits): 1100
int y = 10; // binary: 1010
int z = x ^ y; // binary: 0110, or decimal 6
The ^ operator is often used to toggle (i.e. change from 0 to 1, or 1
to 0) some of the bits in an integer expression. In a bitwise OR
operation if there is a 1 in the mask bit, that bit is inverted; if
there is a 0, the bit is not inverted and stays the same. Below is a
program to blink digital pin 13 (the LED pin on Maple)::
// Blink Maple LED pin
int led_pin = 13;
int toggle = 0;
// demo for Exclusive OR
void setup(){
pinMode(led_pin, OUTPUT);
}
void loop(){
toggle = toggle ^ 1;
digitalWrite(led_pin, toggle);
delay(100);
}
.. _lang-bitwisemath-not:
Bitwise NOT (``~``)
-------------------
The bitwise NOT operator in C++ is the tilde character ``~``. Unlike
``&`` and ``|``, the bitwise NOT operator is applied to a single
operand to its right. Bitwise NOT changes each bit to its opposite: 0
becomes 1, and 1 becomes 0. For example::
0 1 operand1
----
1 0 ~operand1 = result
Another example::
char a = 103; // binary: 01100111
char b = ~a; // binary: 10011000 = -104
You might be surprised to see a negative number like -104 as the
result of this operation. This is because the highest bit in an int
variable is the so-called "sign bit". If the highest bit is 1, the
number is interpreted as negative. This encoding of positive and
negative numbers is referred to as *two's complement*. For more
information, see the Wikipedia article on `two's
complement. <http://en.wikipedia.org/wiki/Twos_complement>`_
As an aside, it is interesting to note that (under two's complement
arithmetic) for any integer ``x``, ``~x`` is the same as ``-x-1``.
At times, the sign bit in a signed integer expression can cause
some unwanted surprises.
Uses
----
One of the most common uses of bitwise operations is to select or
manipulate a particular bit (or bits) from an integer value, often
called `bit masking
<http://en.wikipedia.org/wiki/Mask_%28computing%29>`_\ . See the
linked Wikipedia article for more information and examples.
If you really want to see bit-twiddling techniques in their full
glory, you could do much worse than to get yourself a copy of
`Hacker's Delight <http://www.hackersdelight.org/>`_\ .
See Also
--------
- :ref:`Boolean operations <lang-boolean>` (``&&``, ``||``)
- :ref:`Compound bitwise operations <lang-compoundbitwise>` (``&=``,
``|=``, ``^=``).
.. include:: cc-attribution.txt
|
