@c -*-texinfo-*-
@c This is part of the GNU Emacs Lisp Reference Manual.
@c Copyright (C) 1990-1995, 1998-1999, 2001-2012
@c   Free Software Foundation, Inc.
@c See the file elisp.texi for copying conditions.
@setfilename ../../info/numbers
@node Numbers, Strings and Characters, Lisp Data Types, Top
@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.  Their values are exact.  Floating point
numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or
2.71828.  They can also be expressed in exponential notation: 1.5e2
equals 150; in this example, @samp{e2} stands for ten to the second
power, and that is multiplied by 1.5.  Floating point values are not
exact; they have a fixed, limited 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
@comment  node-name,  next,  previous,  up
@section Integer Basics

  The range of values for an integer depends on the machine.  The
minimum range is @minus{}536870912 to 536870911 (30 bits; i.e.,
@ifnottex
-2**29
@end ifnottex
@tex
@math{-2^{29}}
@end tex
to
@ifnottex
2**29 - 1),
@end ifnottex
@tex
@math{2^{29}-1}),
@end tex
but some machines provide a wider range.  Many examples in this
chapter assume that an integer has 30 bits and that floating point
numbers are IEEE double precision.
@cindex overflow

  The Lisp reader reads an integer as a sequence of digits with optional
initial sign and optional final period.  An integer that is out of the
Emacs range is treated as a floating-point number.

@example
 1               ; @r{The integer 1.}
 1.              ; @r{The integer 1.}
+1               ; @r{Also the integer 1.}
-1               ; @r{The integer @minus{}1.}
 1073741825      ; @r{The floating point number 1073741825.0.}
 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 uses @samp{#}
followed by a letter that specifies the radix: @samp{b} for binary,
@samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to
specify radix @var{radix}.  Case is not significant for the letter
that specifies the 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.  For example:

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

  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.)

  The negative integer, @minus{}5, is creating by subtracting 4 from
@minus{}1.  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 value 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 number
@cindex maximum Lisp integer number
@defvar most-positive-fixnum
The value of this variable is the largest integer that Emacs Lisp
can handle.
@end defvar

@cindex smallest Lisp integer number
@cindex minimum Lisp integer number
@defvar most-negative-fixnum
The value of this variable is the smallest integer that Emacs Lisp can
handle.  It is negative.
@end defvar

  @xref{Character Codes, max-char}, for the maximum value of a valid
character codepoint.

@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 precise range of floating point numbers is
machine-specific; it is the same as the range of the C data type
@code{double} on the machine you are using.  Emacs uses the
@acronym{IEEE} floating point standard where possible (the standard is
supported by most modern computers).

  The read syntax for floating point numbers requires either a decimal
point (with at least one digit following), an exponent, or both.  For
example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and
@samp{.15e4} are five ways of writing a floating point number whose
value is 1500.  They are all equivalent.  You can also use a minus
sign to write negative floating point numbers, as in @samp{-1.0}.

  Emacs Lisp treats @code{-0.0} as equal to ordinary zero (with
respect to @code{equal} and @code{=}), even though the two are
distinguishable in the @acronym{IEEE} floating point standard.

@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.  (NaN
values can also carry a sign, but for practical purposes there's no
significant difference between different NaN values in Emacs Lisp.)
Here are the read syntaxes for these special floating point values:

@table @asis
@item positive infinity
@samp{1.0e+INF}
@item negative infinity
@samp{-1.0e+INF}
@item Not-a-number
@samp{0.0e+NaN} or @samp{-0.0e+NaN}.
@end table

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

  The following functions are specialized for handling floating point
numbers:

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

@smallexample
@var{x} = @var{sig} * 2^@var{exp}
@end smallexample

@var{sig} is a floating point number between 0.5 (inclusive) and 1.0
(exclusive).  If @var{x} is zero, the return value is @code{(0 . 0)}.
@end defun

@defun ldexp sig &optional exp
This function returns a floating point number corresponding to the
significand @var{sig} and exponent @var{exp}.
@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
numbers.
@end defun

@defun logb number
This function returns the binary exponent of @var{number}.  More
precisely, the value is the logarithm of @var{number} base 2, 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 a floating point
number and returns @code{t} if so, @code{nil} otherwise.

@code{floatp} does not exist in Emacs versions 18 and earlier.
@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 number
This 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
number 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.

  At present, each integer value has a unique Lisp object in Emacs Lisp.
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 a good idea to use @code{=} if you
can, even for comparing integers, just in case we change the
representation of integers in a future Emacs version.

  Sometimes it is useful to compare numbers with @code{equal}; it
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 two 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 (and (= x 0) (= y 0))
      (< (/ (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 integer values.
@end quotation

@defun = number-or-marker1 number-or-marker2
This function tests whether 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-marker1 number-or-marker2
This function tests whether its first argument is strictly less than
its second argument.  It returns @code{t} if so, @code{nil} otherwise.
@end defun

@defun <=  number-or-marker1 number-or-marker2
This function tests whether its first argument is less than or equal
to its second argument.  It returns @code{t} if so, @code{nil}
otherwise.
@end defun

@defun >  number-or-marker1 number-or-marker2
This function tests whether its first argument is strictly greater
than its second argument.  It returns @code{t} if so, @code{nil}
otherwise.
@end defun

@defun >=  number-or-marker1 number-or-marker2
This function tests whether its first argument is greater than or
equal to its second 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.
If any of the arguments is floating-point, the value is returned
as floating point, even if it was given as an integer.

@example
(max 20)
     @result{} 20
(max 1 2.5)
     @result{} 2.5
(max 1 3 2.5)
     @result{} 3.0
@end example
@end defun

@defun min number-or-marker &rest numbers-or-markers
This function returns the smallest of its arguments.
If any of the arguments is floating-point, the value is returned
as floating point, even if it was given as an integer.

@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 a floating point number, @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.  An @code{arith-error} results if @var{divisor} is 0.

@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
may choose the integer closer to zero, or it may prefer an even integer,
depending on your machine.

@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.  Remainder and modulus
functions supplement the division functions.  The functions to
add or subtract 1 are provided because they are traditional in Lisp and
commonly used.

  All of these functions except @code{%} return a floating point value
if any argument is floating.

  It is important to note that in Emacs Lisp, arithmetic functions
do not check for 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 / dividend divisor &rest divisors
This function divides @var{dividend} by @var{divisor} and returns the
quotient.  If there are additional arguments @var{divisors}, then it
divides @var{dividend} by each divisor in turn.  Each argument may be a
number or a marker.

If all the arguments are integers, then the result is an integer too.
This means the result has to be rounded.  On most machines, the result
is rounded towards zero after each division, but some machines may round
differently with negative arguments.  This is because the Lisp function
@code{/} is implemented using the C division operator, which also
permits machine-dependent rounding.  As a practical matter, all known
machines round in the standard fashion.

@cindex @code{arith-error} in division
If you divide an integer by 0, an @code{arith-error} error is signaled.
(@xref{Errors}.)  Floating point division by zero returns either
infinity or a NaN if your machine supports @acronym{IEEE} floating point;
otherwise, it signals an @code{arith-error} error.

@example
@group
(/ 6 2)
     @result{} 3
@end group
(/ 5 2)
     @result{} 2
(/ 5.0 2)
     @result{} 2.5
(/ 5 2.0)
     @result{} 2.5
(/ 5.0 2.0)
     @result{} 2.5
(/ 25 3 2)
     @result{} 4
@group
(/ -17 6)
     @result{} -2   @r{(could in theory be @minus{}3 on some machines)}
@end group
@end example
@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 negative arguments, the remainder is in principle machine-dependent
since the quotient is; but in practice, all known machines behave alike.

An @code{arith-error} results if @var{divisor} is 0.

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

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}.
@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} returns a well-defined result for negative
arguments.  It also permits floating point arguments; it rounds the
quotient downward (towards minus infinity) to an integer, and uses that
quotient to compute the remainder.

An @code{arith-error} results if @var{divisor} is 0.

@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.  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.
@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 ``logical and'' of the arguments: the
@var{n}th bit is set in the result if, and only if, the @var{n}th 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.

@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 ``inclusive or'' of its arguments: the @var{n}th bit
is set in the result if, and only if, the @var{n}th 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 @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 ``exclusive or'' of its arguments: the
@var{n}th bit is set in the result if, and only if, the @var{n}th 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
@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 logical 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

@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 ordinary trigonometric functions, with argument 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, however, @var{arg} is out of
range (outside [@minus{}1, 1]), it signals a @code{domain-error} error.
@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, however, @var{arg} is out
of range (outside [@minus{}1, 1]), it signals a @code{domain-error} error.
@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} is negative, it signals a
@code{domain-error} error.
@end defun

@ignore
@defun expm1 arg
This function returns @code{(1- (exp @var{arg}))}, but it is more
accurate than that when @var{arg} is negative and @code{(exp @var{arg})}
is close to 1.
@end defun

@defun log1p arg
This function returns @code{(log (1+ @var{arg}))}, but it is more
accurate than that when @var{arg} is so small that adding 1 to it would
lose accuracy.
@end defun
@end ignore

@defun log10 arg
This function returns the logarithm of @var{arg}, with base 10.  If
@var{arg} is negative, it signals a @code{domain-error} error.
@code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}, at least
approximately.
@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.
@end defun

@defun sqrt arg
This returns the square root of @var{arg}.  If @var{arg} is negative,
it signals a @code{domain-error} error.
@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.

In Emacs, pseudo-random numbers are generated from a ``seed'' number.
Starting from any given seed, the @code{random} function always
generates the same sequence of numbers.  Emacs always starts with the
same seed value, so the sequence of values of @code{random} is actually
the same in each Emacs run!  For example, in one operating system, the
first call to @code{(random)} after you start Emacs always returns
@minus{}1457731, and the second one always returns @minus{}7692030.  This
repeatability is helpful for debugging.

If you want random numbers that don't always come out the same, execute
@code{(random t)}.  This chooses a new seed based on the current time of
day and on Emacs's process @acronym{ID} number.

@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}.

If @var{limit} is @code{t}, it means to choose a new seed based on the
current time of day and on Emacs's process @acronym{ID} number.

On some machines, any integer representable in Lisp may be the result
of @code{random}.  On other machines, the result can never be larger
than a certain maximum or less than a certain (negative) minimum.
@end defun