@c -*-texinfo-*-
@c This is part of the GNU Emacs Lisp Reference Manual.
@c Copyright (C) 1990-1995, 1998-1999, 2001-2018 Free Software
@c Foundation, Inc.
@c See the file elisp.texi for copying conditions.
@node Numbers
@chapter Numbers
@cindex integers
@cindex numbers

  GNU Emacs supports two numeric data types: @dfn{integers} and
@dfn{floating-point numbers}.  Integers are whole numbers such as
@minus{}3, 0, 7, 13, and 511.  Floating-point numbers are numbers with
fractional parts, such as @minus{}4.5, 0.0, and 2.71828.  They can
also be expressed in exponential notation: @samp{1.5e2} is the same as
@samp{150.0}; here, @samp{e2} stands for ten to the second power, and
that is multiplied by 1.5.  Integer computations are exact, though
they may overflow.  Floating-point computations often involve rounding
errors, as the numbers have a fixed amount of precision.

@menu
* Integer Basics::            Representation and range of integers.
* Float Basics::              Representation and range of floating point.
* Predicates on Numbers::     Testing for numbers.
* Comparison of Numbers::     Equality and inequality predicates.
* Numeric Conversions::       Converting float to integer and vice versa.
* Arithmetic Operations::     How to add, subtract, multiply and divide.
* Rounding Operations::       Explicitly rounding floating-point numbers.
* Bitwise Operations::        Logical and, or, not, shifting.
* Math Functions::            Trig, exponential and logarithmic functions.
* Random Numbers::            Obtaining random integers, predictable or not.
@end menu

@node Integer Basics
@section Integer Basics

  The range of values for an integer depends on the machine.  The
minimum range is @minus{}536,870,912 to 536,870,911 (30 bits; i.e.,
@ifnottex
@minus{}2**29
@end ifnottex
@tex
@math{-2^{29}}
@end tex
to
@ifnottex
2**29 @minus{} 1),
@end ifnottex
@tex
@math{2^{29}-1}),
@end tex
but many machines provide a wider range.  Many examples in this
chapter assume the minimum integer width of 30 bits.
@cindex overflow

  The Lisp reader reads an integer as a nonempty sequence
of decimal digits with optional initial sign and optional
final period.

@example
 1               ; @r{The integer 1.}
 1.              ; @r{The integer 1.}
+1               ; @r{Also the integer 1.}
-1               ; @r{The integer @minus{}1.}
 0               ; @r{The integer 0.}
-0               ; @r{The integer 0.}
@end example

@cindex integers in specific radix
@cindex radix for reading an integer
@cindex base for reading an integer
@cindex hex numbers
@cindex octal numbers
@cindex reading numbers in hex, octal, and binary
  The syntax for integers in bases other than 10 consists of @samp{#}
followed by a radix indication followed by one or more digits.  The
radix indications are @samp{b} for binary, @samp{o} for octal,
@samp{x} for hex, and @samp{@var{radix}r} for radix @var{radix}.
Thus, @samp{#b@var{integer}} reads
@var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads
@var{integer} in radix @var{radix}.  Allowed values of @var{radix} run
from 2 to 36, and allowed digits are the first @var{radix} characters
taken from @samp{0}--@samp{9}, @samp{A}--@samp{Z}.
Letter case is ignored and there is no initial sign or final period.
For example:

@example
#b101100 @result{} 44
#o54 @result{} 44
#x2c @result{} 44
#24r1k @result{} 44
@end example

  If an integer is outside the Emacs range, the Lisp reader ordinarily
signals an overflow.  However, if a too-large plain integer ends in a
period, the Lisp reader treats it as a floating-point number instead.
This lets an Emacs Lisp program specify a large integer that is
quietly approximated by a floating-point number on machines with
limited word width.  For example, @samp{536870912.} is a
floating-point number if Emacs integers are only 30 bits wide and is
an integer otherwise.

  To understand how various functions work on integers, especially the
bitwise operators (@pxref{Bitwise Operations}), it is often helpful to
view the numbers in their binary form.

  In 30-bit binary, the decimal integer 5 looks like this:

@example
0000...000101 (30 bits total)
@end example

@noindent
(The @samp{...} stands for enough bits to fill out a 30-bit word; in
this case, @samp{...} stands for twenty 0 bits.  Later examples also
use the @samp{...} notation to make binary integers easier to read.)

  The integer @minus{}1 looks like this:

@example
1111...111111 (30 bits total)
@end example

@noindent
@cindex two's complement
@minus{}1 is represented as 30 ones.  (This is called @dfn{two's
complement} notation.)

  Subtracting 4 from @minus{}1 returns the negative integer @minus{}5.
In binary, the decimal integer 4 is 100.  Consequently,
@minus{}5 looks like this:

@example
1111...111011 (30 bits total)
@end example

  In this implementation, the largest 30-bit binary integer is
536,870,911 in decimal.  In binary, it looks like this:

@example
0111...111111 (30 bits total)
@end example

  Since the arithmetic functions do not check whether integers go
outside their range, when you add 1 to 536,870,911, the value is the
negative integer @minus{}536,870,912:

@example
(+ 1 536870911)
     @result{} -536870912
     @result{} 1000...000000 (30 bits total)
@end example

  Many of the functions described in this chapter accept markers for
arguments in place of numbers.  (@xref{Markers}.)  Since the actual
arguments to such functions may be either numbers or markers, we often
give these arguments the name @var{number-or-marker}.  When the argument
value is a marker, its position value is used and its buffer is ignored.

@cindex largest Lisp integer
@cindex maximum Lisp integer
@defvar most-positive-fixnum
The value of this variable is the largest integer that Emacs Lisp can
handle.  Typical values are
@ifnottex
2**29 @minus{} 1
@end ifnottex
@tex
@math{2^{29}-1}
@end tex
on 32-bit and
@ifnottex
2**61 @minus{} 1
@end ifnottex
@tex
@math{2^{61}-1}
@end tex
on 64-bit platforms.
@end defvar

@cindex smallest Lisp integer
@cindex minimum Lisp integer
@defvar most-negative-fixnum
The value of this variable is the smallest integer that Emacs Lisp can
handle.  It is negative.  Typical values are
@ifnottex
@minus{}2**29
@end ifnottex
@tex
@math{-2^{29}}
@end tex
on 32-bit and
@ifnottex
@minus{}2**61
@end ifnottex
@tex
@math{-2^{61}}
@end tex
on 64-bit platforms.
@end defvar

  In Emacs Lisp, text characters are represented by integers.  Any
integer between zero and the value of @code{(max-char)}, inclusive, is
considered to be valid as a character.  @xref{Character Codes}.

@node Float Basics
@section Floating-Point Basics

@cindex @acronym{IEEE} floating point
  Floating-point numbers are useful for representing numbers that are
not integral.  The range of floating-point numbers is
the same as the range of the C data type @code{double} on the machine
you are using.  On all computers currently supported by Emacs, this is
double-precision @acronym{IEEE} floating point.

  The read syntax for floating-point numbers requires either a decimal
point, an exponent, or both.  Optional signs (@samp{+} or @samp{-})
precede the number and its exponent.  For example, @samp{1500.0},
@samp{+15e2}, @samp{15.0e+2}, @samp{+1500000e-3}, and @samp{.15e4} are
five ways of writing a floating-point number whose value is 1500.
They are all equivalent.  Like Common Lisp, Emacs Lisp requires at
least one digit after any decimal point in a floating-point number;
@samp{1500.} is an integer, not a floating-point number.

  Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero
with respect to @code{equal} and @code{=}.  This follows the
@acronym{IEEE} floating-point standard, which says @code{-0.0} and
@code{0.0} are numerically equal even though other operations can
distinguish them.

@cindex positive infinity
@cindex negative infinity
@cindex infinity
@cindex NaN
  The @acronym{IEEE} floating-point standard supports positive
infinity and negative infinity as floating-point values.  It also
provides for a class of values called NaN, or ``not a number'';
numerical functions return such values in cases where there is no
correct answer.  For example, @code{(/ 0.0 0.0)} returns a NaN@.
Although NaN values carry a sign, for practical purposes there is no other
significant difference between different NaN values in Emacs Lisp.

Here are read syntaxes for these special floating-point values:

@table @asis
@item infinity
@samp{1.0e+INF} and @samp{-1.0e+INF}
@item not-a-number
@samp{0.0e+NaN} and @samp{-0.0e+NaN}
@end table

  The following functions are specialized for handling floating-point
numbers:

@defun isnan x
This predicate returns @code{t} if its floating-point argument is a NaN,
@code{nil} otherwise.
@end defun

@defun frexp x
This function returns a cons cell @code{(@var{s} . @var{e})},
where @var{s} and @var{e} are respectively the significand and
exponent of the floating-point number @var{x}.

If @var{x} is finite, then @var{s} is a floating-point number between 0.5
(inclusive) and 1.0 (exclusive), @var{e} is an integer, and
@ifnottex
@var{x} = @var{s} * 2**@var{e}.
@end ifnottex
@tex
@math{x = s 2^e}.
@end tex
If @var{x} is zero or infinity, then @var{s} is the same as @var{x}.
If @var{x} is a NaN, then @var{s} is also a NaN@.
If @var{x} is zero, then @var{e} is 0.
@end defun

@defun ldexp s e
Given a numeric significand @var{s} and an integer exponent @var{e},
this function returns the floating point number
@ifnottex
@var{s} * 2**@var{e}.
@end ifnottex
@tex
@math{s 2^e}.
@end tex
@end defun

@defun copysign x1 x2
This function copies the sign of @var{x2} to the value of @var{x1},
and returns the result.  @var{x1} and @var{x2} must be floating point.
@end defun

@defun logb x
This function returns the binary exponent of @var{x}.  More
precisely, the value is the logarithm base 2 of @math{|x|}, rounded
down to an integer.

@example
(logb 10)
     @result{} 3
(logb 10.0e20)
     @result{} 69
@end example
@end defun

@node Predicates on Numbers
@section Type Predicates for Numbers
@cindex predicates for numbers

  The functions in this section test for numbers, or for a specific
type of number.  The functions @code{integerp} and @code{floatp} can
take any type of Lisp object as argument (they would not be of much
use otherwise), but the @code{zerop} predicate requires a number as
its argument.  See also @code{integer-or-marker-p} and
@code{number-or-marker-p}, in @ref{Predicates on Markers}.

@defun floatp object
This predicate tests whether its argument is floating point
and returns @code{t} if so, @code{nil} otherwise.
@end defun

@defun integerp object
This predicate tests whether its argument is an integer, and returns
@code{t} if so, @code{nil} otherwise.
@end defun

@defun numberp object
This predicate tests whether its argument is a number (either integer or
floating point), and returns @code{t} if so, @code{nil} otherwise.
@end defun

@defun natnump object
@cindex natural numbers
This predicate (whose name comes from the phrase ``natural number'')
tests to see whether its argument is a nonnegative integer, and
returns @code{t} if so, @code{nil} otherwise.  0 is considered
non-negative.

@findex wholenump
@code{wholenump} is a synonym for @code{natnump}.
@end defun

@defun zerop number
This predicate tests whether its argument is zero, and returns @code{t}
if so, @code{nil} otherwise.  The argument must be a number.

@code{(zerop x)} is equivalent to @code{(= x 0)}.
@end defun

@node Comparison of Numbers
@section Comparison of Numbers
@cindex number comparison
@cindex comparing numbers

  To test numbers for numerical equality, you should normally use
@code{=}, not @code{eq}.  There can be many distinct floating-point
objects with the same numeric value.  If you use @code{eq} to
compare them, then you test whether two values are the same
@emph{object}.  By contrast, @code{=} compares only the numeric values
of the objects.

  In Emacs Lisp, each integer is a unique Lisp object.
Therefore, @code{eq} is equivalent to @code{=} where integers are
concerned.  It is sometimes convenient to use @code{eq} for comparing
an unknown value with an integer, because @code{eq} does not report an
error if the unknown value is not a number---it accepts arguments of
any type.  By contrast, @code{=} signals an error if the arguments are
not numbers or markers.  However, it is better programming practice to
use @code{=} if you can, even for comparing integers.

  Sometimes it is useful to compare numbers with @code{equal}, which
treats two numbers as equal if they have the same data type (both
integers, or both floating point) and the same value.  By contrast,
@code{=} can treat an integer and a floating-point number as equal.
@xref{Equality Predicates}.

  There is another wrinkle: because floating-point arithmetic is not
exact, it is often a bad idea to check for equality of floating-point
values.  Usually it is better to test for approximate equality.
Here's a function to do this:

@example
(defvar fuzz-factor 1.0e-6)
(defun approx-equal (x y)
  (or (= x y)
      (< (/ (abs (- x y))
            (max (abs x) (abs y)))
         fuzz-factor)))
@end example

@cindex CL note---integers vrs @code{eq}
@quotation
@b{Common Lisp note:} Comparing numbers in Common Lisp always requires
@code{=} because Common Lisp implements multi-word integers, and two
distinct integer objects can have the same numeric value.  Emacs Lisp
can have just one integer object for any given value because it has a
limited range of integers.
@end quotation

@defun = number-or-marker &rest number-or-markers
This function tests whether all its arguments are numerically equal,
and returns @code{t} if so, @code{nil} otherwise.
@end defun

@defun eql value1 value2
This function acts like @code{eq} except when both arguments are
numbers.  It compares numbers by type and numeric value, so that
@code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and
@code{(eql 1 1)} both return @code{t}.
@end defun

@defun /= number-or-marker1 number-or-marker2
This function tests whether its arguments are numerically equal, and
returns @code{t} if they are not, and @code{nil} if they are.
@end defun

@defun <  number-or-marker &rest number-or-markers
This function tests whether each argument is strictly less than the
following argument.  It returns @code{t} if so, @code{nil} otherwise.
@end defun

@defun <= number-or-marker &rest number-or-markers
This function tests whether each argument is less than or equal to
the following argument.  It returns @code{t} if so, @code{nil} otherwise.
@end defun

@defun > number-or-marker &rest number-or-markers
This function tests whether each argument is strictly greater than
the following argument.  It returns @code{t} if so, @code{nil} otherwise.
@end defun

@defun >= number-or-marker &rest number-or-markers
This function tests whether each argument is greater than or equal to
the following argument.  It returns @code{t} if so, @code{nil} otherwise.
@end defun

@defun max number-or-marker &rest numbers-or-markers
This function returns the largest of its arguments.

@example
(max 20)
     @result{} 20
(max 1 2.5)
     @result{} 2.5
(max 1 3 2.5)
     @result{} 3
@end example
@end defun

@defun min number-or-marker &rest numbers-or-markers
This function returns the smallest of its arguments.

@example
(min -4 1)
     @result{} -4
@end example
@end defun

@defun abs number
This function returns the absolute value of @var{number}.
@end defun

@node Numeric Conversions
@section Numeric Conversions
@cindex rounding in conversions
@cindex number conversions
@cindex converting numbers

To convert an integer to floating point, use the function @code{float}.

@defun float number
This returns @var{number} converted to floating point.
If @var{number} is already floating point, @code{float} returns
it unchanged.
@end defun

  There are four functions to convert floating-point numbers to
integers; they differ in how they round.  All accept an argument
@var{number} and an optional argument @var{divisor}.  Both arguments
may be integers or floating-point numbers.  @var{divisor} may also be
@code{nil}.  If @var{divisor} is @code{nil} or omitted, these
functions convert @var{number} to an integer, or return it unchanged
if it already is an integer.  If @var{divisor} is non-@code{nil}, they
divide @var{number} by @var{divisor} and convert the result to an
integer.  If @var{divisor} is zero (whether integer or
floating point), Emacs signals an @code{arith-error} error.

@defun truncate number &optional divisor
This returns @var{number}, converted to an integer by rounding towards
zero.

@example
(truncate 1.2)
     @result{} 1
(truncate 1.7)
     @result{} 1
(truncate -1.2)
     @result{} -1
(truncate -1.7)
     @result{} -1
@end example
@end defun

@defun floor number &optional divisor
This returns @var{number}, converted to an integer by rounding downward
(towards negative infinity).

If @var{divisor} is specified, this uses the kind of division
operation that corresponds to @code{mod}, rounding downward.

@example
(floor 1.2)
     @result{} 1
(floor 1.7)
     @result{} 1
(floor -1.2)
     @result{} -2
(floor -1.7)
     @result{} -2
(floor 5.99 3)
     @result{} 1
@end example
@end defun

@defun ceiling number &optional divisor
This returns @var{number}, converted to an integer by rounding upward
(towards positive infinity).

@example
(ceiling 1.2)
     @result{} 2
(ceiling 1.7)
     @result{} 2
(ceiling -1.2)
     @result{} -1
(ceiling -1.7)
     @result{} -1
@end example
@end defun

@defun round number &optional divisor
This returns @var{number}, converted to an integer by rounding towards the
nearest integer.  Rounding a value equidistant between two integers
returns the even integer.

@example
(round 1.2)
     @result{} 1
(round 1.7)
     @result{} 2
(round -1.2)
     @result{} -1
(round -1.7)
     @result{} -2
@end example
@end defun

@node Arithmetic Operations
@section Arithmetic Operations
@cindex arithmetic operations

  Emacs Lisp provides the traditional four arithmetic operations
(addition, subtraction, multiplication, and division), as well as
remainder and modulus functions, and functions to add or subtract 1.
Except for @code{%}, each of these functions accepts both integer and
floating-point arguments, and returns a floating-point number if any
argument is floating point.

  Emacs Lisp arithmetic functions do not check for integer overflow.
Thus @code{(1+ 536870911)} may evaluate to
@minus{}536870912, depending on your hardware.

@defun 1+ number-or-marker
This function returns @var{number-or-marker} plus 1.
For example,

@example
(setq foo 4)
     @result{} 4
(1+ foo)
     @result{} 5
@end example

This function is not analogous to the C operator @code{++}---it does not
increment a variable.  It just computes a sum.  Thus, if we continue,

@example
foo
     @result{} 4
@end example

If you want to increment the variable, you must use @code{setq},
like this:

@example
(setq foo (1+ foo))
     @result{} 5
@end example
@end defun

@defun 1- number-or-marker
This function returns @var{number-or-marker} minus 1.
@end defun

@defun + &rest numbers-or-markers
This function adds its arguments together.  When given no arguments,
@code{+} returns 0.

@example
(+)
     @result{} 0
(+ 1)
     @result{} 1
(+ 1 2 3 4)
     @result{} 10
@end example
@end defun

@defun - &optional number-or-marker &rest more-numbers-or-markers
The @code{-} function serves two purposes: negation and subtraction.
When @code{-} has a single argument, the value is the negative of the
argument.  When there are multiple arguments, @code{-} subtracts each of
the @var{more-numbers-or-markers} from @var{number-or-marker},
cumulatively.  If there are no arguments, the result is 0.

@example
(- 10 1 2 3 4)
     @result{} 0
(- 10)
     @result{} -10
(-)
     @result{} 0
@end example
@end defun

@defun * &rest numbers-or-markers
This function multiplies its arguments together, and returns the
product.  When given no arguments, @code{*} returns 1.

@example
(*)
     @result{} 1
(* 1)
     @result{} 1
(* 1 2 3 4)
     @result{} 24
@end example
@end defun

@defun / number &rest divisors
With one or more @var{divisors}, this function divides @var{number}
by each divisor in @var{divisors} in turn, and returns the quotient.
With no @var{divisors}, this function returns 1/@var{number}, i.e.,
the multiplicative inverse of @var{number}.  Each argument may be a
number or a marker.

If all the arguments are integers, the result is an integer, obtained
by rounding the quotient towards zero after each division.

@example
@group
(/ 6 2)
     @result{} 3
@end group
@group
(/ 5 2)
     @result{} 2
@end group
@group
(/ 5.0 2)
     @result{} 2.5
@end group
@group
(/ 5 2.0)
     @result{} 2.5
@end group
@group
(/ 5.0 2.0)
     @result{} 2.5
@end group
@group
(/ 4.0)
     @result{} 0.25
@end group
@group
(/ 4)
     @result{} 0
@end group
@group
(/ 25 3 2)
     @result{} 4
@end group
@group
(/ -17 6)
     @result{} -2
@end group
@end example

@cindex @code{arith-error} in division
If you divide an integer by the integer 0, Emacs signals an
@code{arith-error} error (@pxref{Errors}).  Floating-point division of
a nonzero number by zero yields either positive or negative infinity
(@pxref{Float Basics}).
@end defun

@defun % dividend divisor
@cindex remainder
This function returns the integer remainder after division of @var{dividend}
by @var{divisor}.  The arguments must be integers or markers.

For any two integers @var{dividend} and @var{divisor},

@example
@group
(+ (% @var{dividend} @var{divisor})
   (* (/ @var{dividend} @var{divisor}) @var{divisor}))
@end group
@end example

@noindent
always equals @var{dividend} if @var{divisor} is nonzero.

@example
(% 9 4)
     @result{} 1
(% -9 4)
     @result{} -1
(% 9 -4)
     @result{} 1
(% -9 -4)
     @result{} -1
@end example
@end defun

@defun mod dividend divisor
@cindex modulus
This function returns the value of @var{dividend} modulo @var{divisor};
in other words, the remainder after division of @var{dividend}
by @var{divisor}, but with the same sign as @var{divisor}.
The arguments must be numbers or markers.

Unlike @code{%}, @code{mod} permits floating-point arguments; it
rounds the quotient downward (towards minus infinity) to an integer,
and uses that quotient to compute the remainder.

If @var{divisor} is zero, @code{mod} signals an @code{arith-error}
error if both arguments are integers, and returns a NaN otherwise.

@example
@group
(mod 9 4)
     @result{} 1
@end group
@group
(mod -9 4)
     @result{} 3
@end group
@group
(mod 9 -4)
     @result{} -3
@end group
@group
(mod -9 -4)
     @result{} -1
@end group
@group
(mod 5.5 2.5)
     @result{} .5
@end group
@end example

For any two numbers @var{dividend} and @var{divisor},

@example
@group
(+ (mod @var{dividend} @var{divisor})
   (* (floor @var{dividend} @var{divisor}) @var{divisor}))
@end group
@end example

@noindent
always equals @var{dividend}, subject to rounding error if either
argument is floating point and to an @code{arith-error} if @var{dividend} is an
integer and @var{divisor} is 0.  For @code{floor}, see @ref{Numeric
Conversions}.
@end defun

@node Rounding Operations
@section Rounding Operations
@cindex rounding without conversion

The functions @code{ffloor}, @code{fceiling}, @code{fround}, and
@code{ftruncate} take a floating-point argument and return a floating-point
result whose value is a nearby integer.  @code{ffloor} returns the
nearest integer below; @code{fceiling}, the nearest integer above;
@code{ftruncate}, the nearest integer in the direction towards zero;
@code{fround}, the nearest integer.

@defun ffloor float
This function rounds @var{float} to the next lower integral value, and
returns that value as a floating-point number.
@end defun

@defun fceiling float
This function rounds @var{float} to the next higher integral value, and
returns that value as a floating-point number.
@end defun

@defun ftruncate float
This function rounds @var{float} towards zero to an integral value, and
returns that value as a floating-point number.
@end defun

@defun fround float
This function rounds @var{float} to the nearest integral value,
and returns that value as a floating-point number.
Rounding a value equidistant between two integers returns the even integer.
@end defun

@node Bitwise Operations
@section Bitwise Operations on Integers
@cindex bitwise arithmetic
@cindex logical arithmetic

  In a computer, an integer is represented as a binary number, a
sequence of @dfn{bits} (digits which are either zero or one).  A bitwise
operation acts on the individual bits of such a sequence.  For example,
@dfn{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.

@defun lsh integer1 count
@cindex logical shift
@code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the
bits in @var{integer1} to the left @var{count} places, or to the right
if @var{count} is negative, bringing zeros into the vacated bits.  If
@var{count} is negative, @code{lsh} shifts zeros into the leftmost
(most-significant) bit, producing a positive result even if
@var{integer1} is negative.  Contrast this with @code{ash}, below.

Here are two examples of @code{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.

@example
@group
(lsh 5 1)
     @result{} 10
;; @r{Decimal 5 becomes decimal 10.}
00000101 @result{} 00001010

(lsh 7 1)
     @result{} 14
;; @r{Decimal 7 becomes decimal 14.}
00000111 @result{} 00001110
@end group
@end example

@noindent
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):

@example
@group
(lsh 3 2)
     @result{} 12
;; @r{Decimal 3 becomes decimal 12.}
00000011 @result{} 00001100
@end group
@end example

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

@example
@group
(lsh 6 -1)
     @result{} 3
;; @r{Decimal 6 becomes decimal 3.}
00000110 @result{} 00000011
@end group

@group
(lsh 5 -1)
     @result{} 2
;; @r{Decimal 5 becomes decimal 2.}
00000101 @result{} 00000010
@end group
@end example

@noindent
As the example illustrates, shifting one place to the right divides the
value of a positive integer by two, rounding downward.

The function @code{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
536,870,911 produces @minus{}2 in the 30-bit implementation:

@example
(lsh 536870911 1)          ; @r{left shift}
     @result{} -2
@end example

In binary, the argument looks like this:

@example
@group
;; @r{Decimal 536,870,911}
0111...111111 (30 bits total)
@end group
@end example

@noindent
which becomes the following when left shifted:

@example
@group
;; @r{Decimal @minus{}2}
1111...111110 (30 bits total)
@end group
@end example
@end defun

@defun ash integer1 count
@cindex arithmetic shift
@code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1}
to the left @var{count} places, or to the right if @var{count}
is negative.

@code{ash} gives the same results as @code{lsh} except when
@var{integer1} and @var{count} are both negative.  In that case,
@code{ash} puts ones in the empty bit positions on the left, while
@code{lsh} puts zeros in those bit positions.

Thus, with @code{ash}, shifting the pattern of bits one place to the right
looks like this:

@example
@group
(ash -6 -1) @result{} -3
;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
1111...111010 (30 bits total)
     @result{}
1111...111101 (30 bits total)
@end group
@end example

In contrast, shifting the pattern of bits one place to the right with
@code{lsh} looks like this:

@example
@group
(lsh -6 -1) @result{} 536870909
;; @r{Decimal @minus{}6 becomes decimal 536,870,909.}
1111...111010 (30 bits total)
     @result{}
0111...111101 (30 bits total)
@end group
@end example

Here are other examples:

@c !!! Check if lined up in smallbook format!  XDVI shows problem
@c     with smallbook but not with regular book! --rjc 16mar92
@smallexample
@group
                   ;  @r{       30-bit binary values}

(lsh 5 2)          ;   5  =  @r{0000...000101}
     @result{} 20         ;      =  @r{0000...010100}
@end group
@group
(ash 5 2)
     @result{} 20
(lsh -5 2)         ;  -5  =  @r{1111...111011}
     @result{} -20        ;      =  @r{1111...101100}
(ash -5 2)
     @result{} -20
@end group
@group
(lsh 5 -2)         ;   5  =  @r{0000...000101}
     @result{} 1          ;      =  @r{0000...000001}
@end group
@group
(ash 5 -2)
     @result{} 1
@end group
@group
(lsh -5 -2)        ;  -5  =  @r{1111...111011}
     @result{} 268435454
                   ;      =  @r{0011...111110}
@end group
@group
(ash -5 -2)        ;  -5  =  @r{1111...111011}
     @result{} -2         ;      =  @r{1111...111110}
@end group
@end smallexample
@end defun

@defun logand &rest ints-or-markers
This function returns the bitwise AND of the arguments: the @var{n}th
bit is 1 in the result if, and only if, the @var{n}th bit is 1 in all
the arguments.

For example, using 4-bit binary numbers, the bitwise AND of 13 and
12 is 12: 1101 combined with 1100 produces 1100.
In both the binary numbers, the leftmost two bits are both 1
so the leftmost two bits of the returned value are both 1.
However, for the rightmost two bits, each is 0 in at least one of
the arguments, so the rightmost two bits of the returned value are both 0.

@noindent
Therefore,

@example
@group
(logand 13 12)
     @result{} 12
@end group
@end example

If @code{logand} is not passed any argument, it returns a value of
@minus{}1.  This number is an identity element for @code{logand}
because its binary representation consists entirely of ones.  If
@code{logand} is passed just one argument, it returns that argument.

@smallexample
@group
                   ; @r{       30-bit binary values}

(logand 14 13)     ; 14  =  @r{0000...001110}
                   ; 13  =  @r{0000...001101}
     @result{} 12         ; 12  =  @r{0000...001100}
@end group

@group
(logand 14 13 4)   ; 14  =  @r{0000...001110}
                   ; 13  =  @r{0000...001101}
                   ;  4  =  @r{0000...000100}
     @result{} 4          ;  4  =  @r{0000...000100}
@end group

@group
(logand)
     @result{} -1         ; -1  =  @r{1111...111111}
@end group
@end smallexample
@end defun

@defun logior &rest ints-or-markers
This function returns the bitwise inclusive OR of its arguments: the @var{n}th
bit is 1 in the result if, and only if, the @var{n}th bit is 1 in at
least one of the arguments.  If there are no arguments, the result is 0,
which is an identity element for this operation.  If @code{logior} is
passed just one argument, it returns that argument.

@smallexample
@group
                   ; @r{       30-bit binary values}

(logior 12 5)      ; 12  =  @r{0000...001100}
                   ;  5  =  @r{0000...000101}
     @result{} 13         ; 13  =  @r{0000...001101}
@end group

@group
(logior 12 5 7)    ; 12  =  @r{0000...001100}
                   ;  5  =  @r{0000...000101}
                   ;  7  =  @r{0000...000111}
     @result{} 15         ; 15  =  @r{0000...001111}
@end group
@end smallexample
@end defun

@defun logxor &rest ints-or-markers
This function returns the bitwise exclusive OR of its arguments: the
@var{n}th bit is 1 in the result if, and only if, the @var{n}th bit is
1 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
@code{logxor} is passed just one argument, it returns that argument.

@smallexample
@group
                   ; @r{       30-bit binary values}

(logxor 12 5)      ; 12  =  @r{0000...001100}
                   ;  5  =  @r{0000...000101}
     @result{} 9          ;  9  =  @r{0000...001001}
@end group

@group
(logxor 12 5 7)    ; 12  =  @r{0000...001100}
                   ;  5  =  @r{0000...000101}
                   ;  7  =  @r{0000...000111}
     @result{} 14         ; 14  =  @r{0000...001110}
@end group
@end smallexample
@end defun

@defun lognot integer
This function returns the bitwise complement of its argument: the @var{n}th
bit is one in the result if, and only if, the @var{n}th bit is zero in
@var{integer}, and vice-versa.

@example
(lognot 5)
     @result{} -6
;;  5  =  @r{0000...000101} (30 bits total)
;; @r{becomes}
;; -6  =  @r{1111...111010} (30 bits total)
@end example
@end defun

@cindex popcount
@cindex Hamming weight
@cindex counting set bits
@defun logcount integer
This function returns the @dfn{Hamming weight} of @var{integer}: the
number of ones in the binary representation of @var{integer}.
If @var{integer} is negative, it returns the number of zero bits in
its two's complement binary representation.  The result is always
nonnegative.

@example
(logcount 43)     ; 43 = #b101011
     @result{} 4
(logcount -43)    ; -43 = #b111...1010101
     @result{} 3
@end example
@end defun

@node Math Functions
@section Standard Mathematical Functions
@cindex transcendental functions
@cindex mathematical functions
@cindex floating-point functions

  These mathematical functions allow integers as well as floating-point
numbers as arguments.

@defun sin arg
@defunx cos arg
@defunx tan arg
These are the basic trigonometric functions, with argument @var{arg}
measured in radians.
@end defun

@defun asin arg
The value of @code{(asin @var{arg})} is a number between
@ifnottex
@minus{}pi/2
@end ifnottex
@tex
@math{-\pi/2}
@end tex
and
@ifnottex
pi/2
@end ifnottex
@tex
@math{\pi/2}
@end tex
(inclusive) whose sine is @var{arg}.  If @var{arg} is out of range
(outside [@minus{}1, 1]), @code{asin} returns a NaN.
@end defun

@defun acos arg
The value of @code{(acos @var{arg})} is a number between 0 and
@ifnottex
pi
@end ifnottex
@tex
@math{\pi}
@end tex
(inclusive) whose cosine is @var{arg}.  If @var{arg} is out of range
(outside [@minus{}1, 1]), @code{acos} returns a NaN.
@end defun

@defun atan y &optional x
The value of @code{(atan @var{y})} is a number between
@ifnottex
@minus{}pi/2
@end ifnottex
@tex
@math{-\pi/2}
@end tex
and
@ifnottex
pi/2
@end ifnottex
@tex
@math{\pi/2}
@end tex
(exclusive) whose tangent is @var{y}.  If the optional second
argument @var{x} is given, the value of @code{(atan y x)} is the
angle in radians between the vector @code{[@var{x}, @var{y}]} and the
@code{X} axis.
@end defun

@defun exp arg
This is the exponential function; it returns @math{e} to the power
@var{arg}.
@end defun

@defun log arg &optional base
This function returns the logarithm of @var{arg}, with base
@var{base}.  If you don't specify @var{base}, the natural base
@math{e} is used.  If @var{arg} or @var{base} is negative, @code{log}
returns a NaN.
@end defun

@defun expt x y
This function returns @var{x} raised to power @var{y}.  If both
arguments are integers and @var{y} is positive, the result is an
integer; in this case, overflow causes truncation, so watch out.
If @var{x} is a finite negative number and @var{y} is a finite
non-integer, @code{expt} returns a NaN.
@end defun

@defun sqrt arg
This returns the square root of @var{arg}.  If @var{arg} is finite
and less than zero, @code{sqrt} returns a NaN.
@end defun

In addition, Emacs defines the following common mathematical
constants:

@defvar float-e
The mathematical constant @math{e} (2.71828@dots{}).
@end defvar

@defvar float-pi
The mathematical constant @math{pi} (3.14159@dots{}).
@end defvar

@node Random Numbers
@section Random Numbers
@cindex random numbers

  A deterministic computer program cannot generate true random
numbers.  For most purposes, @dfn{pseudo-random numbers} suffice.  A
series of pseudo-random numbers is generated in a deterministic
fashion.  The numbers are not truly random, but they have certain
properties that mimic a random series.  For example, all possible
values occur equally often in a pseudo-random series.

@cindex seed, for random number generation
  Pseudo-random numbers are generated from a @dfn{seed value}.  Starting from
any given seed, the @code{random} function always generates the same
sequence of numbers.  By default, Emacs initializes the random seed at
startup, in such a way that the sequence of values of @code{random}
(with overwhelming likelihood) differs in each Emacs run.

  Sometimes you want the random number sequence to be repeatable.  For
example, when debugging a program whose behavior depends on the random
number sequence, it is helpful to get the same behavior in each
program run.  To make the sequence repeat, execute @code{(random "")}.
This sets the seed to a constant value for your particular Emacs
executable (though it may differ for other Emacs builds).  You can use
other strings to choose various seed values.

@defun random &optional limit
This function returns a pseudo-random integer.  Repeated calls return a
series of pseudo-random integers.

If @var{limit} is a positive integer, the value is chosen to be
nonnegative and less than @var{limit}.  Otherwise, the value might be
any integer representable in Lisp, i.e., an integer between
@code{most-negative-fixnum} and @code{most-positive-fixnum}
(@pxref{Integer Basics}).

If @var{limit} is @code{t}, it means to choose a new seed as if Emacs
were restarting, typically from the system entropy.  On systems
lacking entropy pools, choose the seed from less-random volatile data
such as the current time.

If @var{limit} is a string, it means to choose a new seed based on the
string's contents.

@end defun