GNU Info
(elisp)Bitwise Operations
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, bringing zeros into the vacated bits. If COUNT
is negative, `lsh' shifts zeros into the leftmost
(most-significant) bit, producing a positive result even if
INTEGER1 is negative. Contrast this with `ash', below.
Here are two examples of `lsh', shifting a pattern of bits one
place to the left. We 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.
Shifting a 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 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 one place to the right
divides the value of a positive integer by two, rounding downward.
The function `lsh', like all Emacs Lisp arithmetic functions, does
not check for overflow, so shifting left can discard significant
bits and change the sign of the number. For example, left shifting
134,217,727 produces -2 on a 28-bit machine:
(lsh 134217727 1) ; left shift
=> -2
In binary, in the 28-bit implementation, the argument looks like
this:
;; Decimal 134,217,727
0111 1111 1111 1111 1111 1111 1111
which becomes the following when left shifted:
;; Decimal -2
1111 1111 1111 1111 1111 1111 1110
- 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 ones in the
empty bit positions on the left, while `lsh' puts zeros in those
bit positions.
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 1111 1010
=>
1111 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) => 134217725
;; Decimal -6 becomes decimal 134,217,725.
1111 1111 1111 1111 1111 1111 1010
=>
0111 1111 1111 1111 1111 1111 1101
Here are other examples:
; 28-bit binary values
(lsh 5 2) ; 5 = 0000 0000 0000 0000 0000 0000 0101
=> 20 ; = 0000 0000 0000 0000 0000 0001 0100
(ash 5 2)
=> 20
(lsh -5 2) ; -5 = 1111 1111 1111 1111 1111 1111 1011
=> -20 ; = 1111 1111 1111 1111 1111 1110 1100
(ash -5 2)
=> -20
(lsh 5 -2) ; 5 = 0000 0000 0000 0000 0000 0000 0101
=> 1 ; = 0000 0000 0000 0000 0000 0000 0001
(ash 5 -2)
=> 1
(lsh -5 -2) ; -5 = 1111 1111 1111 1111 1111 1111 1011
=> 4194302 ; = 0011 1111 1111 1111 1111 1111 1110
(ash -5 -2) ; -5 = 1111 1111 1111 1111 1111 1111 1011
=> -2 ; = 1111 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.
; 28-bit binary values
(logand 14 13) ; 14 = 0000 0000 0000 0000 0000 0000 1110
; 13 = 0000 0000 0000 0000 0000 0000 1101
=> 12 ; 12 = 0000 0000 0000 0000 0000 0000 1100
(logand 14 13 4) ; 14 = 0000 0000 0000 0000 0000 0000 1110
; 13 = 0000 0000 0000 0000 0000 0000 1101
; 4 = 0000 0000 0000 0000 0000 0000 0100
=> 4 ; 4 = 0000 0000 0000 0000 0000 0000 0100
(logand)
=> -1 ; -1 = 1111 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.
; 28-bit binary values
(logior 12 5) ; 12 = 0000 0000 0000 0000 0000 0000 1100
; 5 = 0000 0000 0000 0000 0000 0000 0101
=> 13 ; 13 = 0000 0000 0000 0000 0000 0000 1101
(logior 12 5 7) ; 12 = 0000 0000 0000 0000 0000 0000 1100
; 5 = 0000 0000 0000 0000 0000 0000 0101
; 7 = 0000 0000 0000 0000 0000 0000 0111
=> 15 ; 15 = 0000 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, which is an identity element for this operation. If
`logxor' is passed just one argument, it returns that argument.
; 28-bit binary values
(logxor 12 5) ; 12 = 0000 0000 0000 0000 0000 0000 1100
; 5 = 0000 0000 0000 0000 0000 0000 0101
=> 9 ; 9 = 0000 0000 0000 0000 0000 0000 1001
(logxor 12 5 7) ; 12 = 0000 0000 0000 0000 0000 0000 1100
; 5 = 0000 0000 0000 0000 0000 0000 0101
; 7 = 0000 0000 0000 0000 0000 0000 0111
=> 14 ; 14 = 0000 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.
(lognot 5)
=> -6
;; 5 = 0000 0000 0000 0000 0000 0000 0101
;; becomes
;; -6 = 1111 1111 1111 1111 1111 1111 1010
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