# (lispref.info)Bitwise Operations

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Bitwise Operations on Integers
==============================

In a computer, an integer is represented as a binary number, a
sequence of "bits" (digits which are either zero or one).  A bitwise
operation acts on the individual bits of such a sequence.  For example,
"shifting" moves the whole sequence left or right one or more places,
reproducing the same pattern "moved over".

The bitwise operations in Emacs Lisp apply only to integers.

- Function: lsh INTEGER1 COUNT
`lsh', which is an abbreviation for "logical shift", shifts the
bits in INTEGER1 to the left COUNT places, or to the right if
COUNT is negative.  If COUNT is negative, `lsh' shifts zeros into
the most-significant bit, producing a positive result even if
INTEGER1 is negative.  Contrast this with `ash', below.

Thus, the decimal number 5 is the binary number 00000101.  Shifted
once to the left, with a zero put in the one's place, the number
becomes 00001010, decimal 10.

Here are two examples of shifting the pattern of bits one place to
the left.  Since the contents of the rightmost place has been
moved one place to the left, a value has to be inserted into the
rightmost place.  With `lsh', a zero is placed into the rightmost
place.  (These examples show only the low-order eight bits of the
binary pattern; the rest are all zero.)

(lsh 5 1)
=> 10

;; Decimal 5 becomes decimal 10.
00000101 => 00001010

(lsh 7 1)
=> 14

;; Decimal 7 becomes decimal 14.
00000111 => 00001110

As the examples illustrate, shifting the pattern of bits one place
to the left produces a number that is twice the value of the
previous number.

Note, however that functions do not check for overflow, and a
returned value may be negative (and in any case, no more than a 24
bit value) when an integer is sufficiently left shifted.

For example, left shifting 8,388,607 produces -2:

(lsh 8388607 1)          ; left shift
=> -2

In binary, in the 24 bit implementation, the numbers looks like
this:

;; Decimal 8,388,607
0111 1111  1111 1111  1111 1111

which becomes the following when left shifted:

;; Decimal -2
1111 1111  1111 1111  1111 1110

Shifting the pattern of bits two places to the left produces
results like this (with 8-bit binary numbers):

(lsh 3 2)
=> 12

;; Decimal 3 becomes decimal 12.
00000011 => 00001100

On the other hand, shifting the pattern of bits one place to the
right looks like this:

(lsh 6 -1)
=> 3

;; Decimal 6 becomes decimal 3.
00000110 => 00000011

(lsh 5 -1)
=> 2

;; Decimal 5 becomes decimal 2.
00000101 => 00000010

As the example illustrates, shifting the pattern of bits one place
to the right divides the value of the binary number by two,
rounding downward.

- Function: ash INTEGER1 COUNT
`ash' ("arithmetic shift") shifts the bits in INTEGER1 to the left
COUNT places, or to the right if COUNT is negative.

`ash' gives the same results as `lsh' except when INTEGER1 and
COUNT are both negative.  In that case, `ash' puts a one in the
leftmost position, while `lsh' puts a zero in the leftmost
position.

Thus, with `ash', shifting the pattern of bits one place to the
right looks like this:

(ash -6 -1)
=> -3

;; Decimal -6
;; becomes decimal -3.

1111 1111  1111 1111  1111 1010
=>
1111 1111  1111 1111  1111 1101

In contrast, shifting the pattern of bits one place to the right
with `lsh' looks like this:

(lsh -6 -1)
=> 8388605

;; Decimal -6
;; becomes decimal 8,388,605.

1111 1111  1111 1111  1111 1010
=>
0111 1111  1111 1111  1111 1101

In this case, the 1 in the leftmost position is shifted one place
to the right, and a zero is shifted into the leftmost position.

Here are other examples:

;               24-bit binary values

(lsh 5 2)          ;   5  =  0000 0000  0000 0000  0000 0101
=> 20         ;  20  =  0000 0000  0000 0000  0001 0100

(ash 5 2)
=> 20
(lsh -5 2)         ;  -5  =  1111 1111  1111 1111  1111 1011
=> -20        ; -20  =  1111 1111  1111 1111  1110 1100
(ash -5 2)
=> -20

(lsh 5 -2)         ;   5  =  0000 0000  0000 0000  0000 0101
=> 1          ;   1  =  0000 0000  0000 0000  0000 0001

(ash 5 -2)
=> 1

(lsh -5 -2)        ;  -5  =  1111 1111  1111 1111  1111 1011
=> 4194302    ;         0011 1111  1111 1111  1111 1110

(ash -5 -2)        ;  -5  =  1111 1111  1111 1111  1111 1011
=> -2         ;  -2  =  1111 1111  1111 1111  1111 1110

- Function: logand &rest INTS-OR-MARKERS
This function returns the "logical and" of the arguments: the Nth
bit is set in the result if, and only if, the Nth bit is set in
all the arguments.  ("Set" means that the value of the bit is 1
rather than 0.)

For example, using 4-bit binary numbers, the "logical and" of 13
and 12 is 12: 1101 combined with 1100 produces 1100.

In both the binary numbers, the leftmost two bits are set (i.e.,
they are 1's), so the leftmost two bits of the returned value are
set.  However, for the rightmost two bits, each is zero in at
least one of the arguments, so the rightmost two bits of the
returned value are 0's.

Therefore,

(logand 13 12)
=> 12

If `logand' is not passed any argument, it returns a value of -1.
This number is an identity element for `logand' because its binary
representation consists entirely of ones.  If `logand' is passed
just one argument, it returns that argument.

;                24-bit binary values

(logand 14 13)     ; 14  =  0000 0000  0000 0000  0000 1110
; 13  =  0000 0000  0000 0000  0000 1101
=> 12         ; 12  =  0000 0000  0000 0000  0000 1100

(logand 14 13 4)   ; 14  =  0000 0000  0000 0000  0000 1110
; 13  =  0000 0000  0000 0000  0000 1101
;  4  =  0000 0000  0000 0000  0000 0100
=> 4          ;  4  =  0000 0000  0000 0000  0000 0100

(logand)
=> -1         ; -1  =  1111 1111  1111 1111  1111 1111

- Function: logior &rest INTS-OR-MARKERS
This function returns the "inclusive or" of its arguments: the Nth
bit is set in the result if, and only if, the Nth bit is set in at
least one of the arguments.  If there are no arguments, the result
is zero, which is an identity element for this operation.  If
`logior' is passed just one argument, it returns that argument.

;               24-bit binary values

(logior 12 5)      ; 12  =  0000 0000  0000 0000  0000 1100
;  5  =  0000 0000  0000 0000  0000 0101
=> 13         ; 13  =  0000 0000  0000 0000  0000 1101

(logior 12 5 7)    ; 12  =  0000 0000  0000 0000  0000 1100
;  5  =  0000 0000  0000 0000  0000 0101
;  7  =  0000 0000  0000 0000  0000 0111
=> 15         ; 15  =  0000 0000  0000 0000  0000 1111

- Function: logxor &rest INTS-OR-MARKERS
This function returns the "exclusive or" of its arguments: the Nth
bit is set in the result if, and only if, the Nth bit is set in an
odd number of the arguments.  If there are no arguments, the
result is 0.  If `logxor' is passed just one argument, it returns
that argument.

;               24-bit binary values

(logxor 12 5)      ; 12  =  0000 0000  0000 0000  0000 1100
;  5  =  0000 0000  0000 0000  0000 0101
=> 9          ;  9  =  0000 0000  0000 0000  0000 1001

(logxor 12 5 7)    ; 12  =  0000 0000  0000 0000  0000 1100
;  5  =  0000 0000  0000 0000  0000 0101
;  7  =  0000 0000  0000 0000  0000 0111
=> 14         ; 14  =  0000 0000  0000 0000  0000 1110

- Function: lognot INTEGER
This function returns the logical complement of its argument: the
Nth bit is one in the result if, and only if, the Nth bit is zero
in INTEGER, and vice-versa.

;;  5  =  0000 0000  0000 0000  0000 0101
;; becomes
;; -6  =  1111 1111  1111 1111  1111 1010

(lognot 5)
=> -6

```

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