@c -*-texinfo-*-
@c This is part of the XEmacs Lisp Reference Manual.
@c Copyright (C) 1990, 1991, 1992, 1993 Free Software Foundation, Inc. 
@c See the file lispref.texi for copying conditions.
@setfilename ../../info/numbers.info
@node Numbers, Strings and Characters, Types of Lisp Object, Top
@chapter Numbers
@cindex integers
@cindex numbers

  XEmacs 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 an exponential notation as well:
thus, 1.5e2 equals 150; in this example, @samp{e2} stands for ten to the
second power, and 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.
* Bitwise Operations::        Logical and, or, not, shifting.
* Transcendental 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
range is @minus{}8388608 to 8388607 (24 bits; i.e.,
@ifinfo 
-2**23
@end ifinfo
@tex 
$-2^{23}$ 
@end tex
to 
@ifinfo 
2**23 - 1
@end ifinfo
@tex 
$2^{23}-1$
@end tex
) on most
machines, but on others it is @minus{}16777216 to 16777215
(25 bits), or @minus{}33554432 to 33554431 (26 bits).  All
of the examples shown below assume an integer has 24 bits.@refill
@cindex overflow

  The Lisp reader reads numbers as a sequence of digits with an optional
sign.

@example
 1               ; @r{The integer 1.}
+1               ; @r{Also the integer 1.}
-1               ; @r{The integer @minus{}1.}
 16777217        ; @r{Also the integer 1, due to overflow.}
 0               ; @r{The number 0.}
-0               ; @r{The number 0.}
 1.              ; @r{The integer 1.}
@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 24 bit binary, the decimal integer 5 looks like this:

@example
0000 0000  0000 0000  0000 0101
@end example

@noindent
(We have inserted spaces between groups of 4 bits, and two spaces
between groups of 8 bits, to make the binary integer easier to read.)

  The integer @minus{}1 looks like this:

@example
1111 1111  1111 1111  1111 1111
@end example

@noindent
@cindex two's complement
@minus{}1 is represented as 24 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 1111  1111 1111  1111 1011
@end example

  In this implementation, the largest 24 bit binary integer is the decimal
integer 8,388,607.  In binary, this number looks like this:

@example
0111 1111  1111 1111  1111 1111
@end example

  Since the arithmetic functions do not check whether integers go
outside their range, when you add 1 to 8,388,607, the value is negative
integer @minus{}8,388,608:

@example
(+ 1 8388607)
     @result{} -8388608
     @result{} 1000 0000  0000 0000  0000 0000
@end example

  Many of the following functions accept markers for arguments as well
as integers.  (@xref{Markers}.)  More precisely, the actual parameters
to such functions may be either integers or markers, which is why we
often give these parameters the name @var{int-or-marker}.  When the
actual parameter is a marker, the position value of the marker is used
and the buffer of the marker is ignored.

@ignore
  In version 19, except where @emph{integer} is specified as an
argument, all of the functions for markers and integers also work for
floating point numbers.
@end ignore

@node Float Basics
@section Floating Point Basics

  XEmacs supports floating point numbers.  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 in question.

  The printed representation 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}.

  You can use @code{logb} to extract the binary exponent of a floating
point number (or estimate the logarithm of an integer):

@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.
@end defun

@node Predicates on Numbers
@section Type Predicates for Numbers

  The functions in this section test whether the argument is a number or
whether it is a certain sort of number.  The functions @code{integerp}
and @code{floatp} can take any type of Lisp object as argument (the
predicates 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
The @code{natnump} predicate (whose name comes from the phrase
``natural-number-p'') tests to see whether its argument is a nonnegative
integer, and returns @code{t} if so, @code{nil} otherwise.  0 is
considered non-negative.

Markers are not converted to integers, hence @code{natnump} of a marker
is always @code{nil}.

People have pointed out that this function is misnamed, because the term
``natural number'' is usually understood as excluding zero.  We are open
to suggestions for a better name to use in a future version.
@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.

These two forms are equivalent: @code{(zerop x) @equiv{} (= x 0)}.
@end defun

@node Comparison of Numbers
@section Comparison of Numbers
@cindex number equality

  Floating point numbers in Emacs Lisp actually take up storage, and
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}.  If you want to compare
just the numeric values, use @code{=}.

  If you use @code{eq} to compare two integers, it always returns
@code{t} if they have the same value.  This is sometimes useful, because
@code{eq} accepts arguments of any type and never causes an error, 
whereas @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 XEmacs version.

  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)
  (< (/ (abs (- x y))
        (max (abs x) (abs y)))
     fuzz-factor))
@end example

@cindex CL note---integers vrs @code{eq}
@quotation
@b{Common Lisp note:} because of the way numbers are implemented in
Common Lisp, you generally need to use @samp{@code{=}} to test for
equality between numbers of any kind.
@end quotation

@defun = number-or-marker1 number-or-marker2
This function tests whether its arguments are the same number, and
returns @code{t} if so, @code{nil} otherwise.
@end defun

@defun /= number-or-marker1 number-or-marker2
This function tests whether its arguments are not the same number, and
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 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.

@example
(max 20)
     @result{} 20
(max 1 2)
     @result{} 2
(max 1 3 2)
     @result{} 3
@end example
@end defun

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

@node Numeric Conversions
@section Numeric Conversions

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.  You can call these functions with an
integer argument also; if you do, they return it without change.

@defun truncate number
This returns @var{number}, converted to an integer by rounding towards
zero.
@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, @var{number} is divided by @var{divisor}
before the floor is taken; this is the division operation that
corresponds to @code{mod}.  An @code{arith-error} results if
@var{divisor} is 0.
@end defun

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

@defun round number
This returns @var{number}, converted to an integer by rounding towards the
nearest integer.
@end defun

@node Arithmetic Operations
@section 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+ 8388607)} may equal
@minus{}8388608, 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,

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

@defun + &rest numbers-or-markers
This function adds its arguments together.  When given no arguments,
@code{+} returns 0.  It does not check for overflow.

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

@defun - &optional number-or-marker &rest other-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, each of the
@var{other-numbers-or-markers} is subtracted from @var{number-or-marker},
cumulatively.  If there are no arguments, the result is 0.  This
function does not check for overflow.

@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.  It does
not check for overflow.

@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{divisors} and returns the
quotient.  If there are additional arguments @var{divisors}, then
@var{dividend} is divided 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 has the
same possibility for machine-dependent rounding.  As a practical matter,
all known machines round in the standard fashion.

@cindex @code{arith-error} in division
If you divide by 0, an @code{arith-error} error is signaled.
(@xref{Errors}.)

@example
(/ 6 2)
     @result{} 3
(/ 5 2)
     @result{} 2
(/ 25 3 2)
     @result{} 4
(/ -17 6)
     @result{} -2
@end example

Since the division operator in Emacs Lisp is implemented using the
division operator in C, the result of dividing negative numbers may in
principle vary from machine to machine, depending on how they round the
result.  Thus, the result of @code{(/ -17 6)} could be -3 on some
machines.  In practice, all known machines round the quotient towards
0.
@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 value 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{%}, the result is well-defined for negative arguments.
Also, floating point arguments are permitted.

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

@example
(mod 9 4)
     @result{} 1
(mod -9 4)
     @result{} 3
(mod 9 -4)
     @result{} -3
(mod -9 -4)
     @result{} -1
@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.
@end defun

@node Bitwise Operations
@section Bitwise Operations on Integers

  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.  If @var{count} is negative,
@code{lsh} shifts zeros into the most-significant bit, producing a
positive result even if @var{integer1} is negative.  Contrast this with
@code{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 @code{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.)

@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.

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 @minus{}2:

@example
(lsh 8388607 1)          ; @r{left shift}
     @result{} -2
@end example

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

@example
@group
;; @r{Decimal 8,388,607}
0111 1111  1111 1111  1111 1111         
@end group
@end example

@noindent
which becomes the following when left shifted:

@example
@group
;; @r{Decimal @minus{}2}
1111 1111  1111 1111  1111 1110         
@end group
@end example

Shifting the 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 the pattern of bits one place to the right
looks like this:

@example
@group
(lsh 6 -1)
     @result{} 3

;; @r{Decimal 6 becomes decimal 3.}
00000110 @result{} 00000011       

(lsh 5 -1)
     @result{} 2

;; @r{Decimal 5 becomes decimal 2.}
00000101 @result{} 00000010       
@end group
@end example

@noindent
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.
@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 a one in the leftmost position, while @code{lsh} puts
a zero in the leftmost position.

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}
;; @r{becomes decimal @minus{}3.}

1111 1111  1111 1111  1111 1010
     @result{} 
1111 1111  1111 1111  1111 1101
@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{} 8388605       

;; @r{Decimal @minus{}6}
;; @r{becomes decimal 8,388,605.}

1111 1111  1111 1111  1111 1010
     @result{} 
0111 1111  1111 1111  1111 1101
@end group
@end example

@noindent
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:

@c !!! Check if lined up in smallbook format!  XDVI shows problem
@c     with smallbook but not with regular book! --rjc 16mar92
@smallexample
@group
                   ;  @r{             24-bit binary values}

(lsh 5 2)          ;   5  =  @r{0000 0000  0000 0000  0000 0101}
     @result{} 20         ;  20  =  @r{0000 0000  0000 0000  0001 0100}
@end group
@group
(ash 5 2)
     @result{} 20
(lsh -5 2)         ;  -5  =  @r{1111 1111  1111 1111  1111 1011}
     @result{} -20        ; -20  =  @r{1111 1111  1111 1111  1110 1100}
(ash -5 2)
     @result{} -20
@end group
@group
(lsh 5 -2)         ;   5  =  @r{0000 0000  0000 0000  0000 0101}
     @result{} 1          ;   1  =  @r{0000 0000  0000 0000  0000 0001}
@end group
@group
(ash 5 -2)
     @result{} 1
@end group
@group
(lsh -5 -2)        ;  -5  =  @r{1111 1111  1111 1111  1111 1011}
     @result{} 4194302    ;         @r{0011 1111  1111 1111  1111 1110}
@end group
@group
(ash -5 -2)        ;  -5  =  @r{1111 1111  1111 1111  1111 1011}
     @result{} -2         ;  -2  =  @r{1111 1111  1111 1111  1111 1110}
@end group
@end smallexample
@end defun

@defun logand &rest ints-or-markers
@cindex logical and
@cindex bitwise and
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{               24-bit binary values}

(logand 14 13)     ; 14  =  @r{0000 0000  0000 0000  0000 1110}
                   ; 13  =  @r{0000 0000  0000 0000  0000 1101}
     @result{} 12         ; 12  =  @r{0000 0000  0000 0000  0000 1100}
@end group

@group
(logand 14 13 4)   ; 14  =  @r{0000 0000  0000 0000  0000 1110}
                   ; 13  =  @r{0000 0000  0000 0000  0000 1101}
                   ;  4  =  @r{0000 0000  0000 0000  0000 0100}
     @result{} 4          ;  4  =  @r{0000 0000  0000 0000  0000 0100}
@end group

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

@defun logior &rest ints-or-markers
@cindex logical inclusive or
@cindex bitwise or
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{              24-bit binary values}

(logior 12 5)      ; 12  =  @r{0000 0000  0000 0000  0000 1100}
                   ;  5  =  @r{0000 0000  0000 0000  0000 0101}
     @result{} 13         ; 13  =  @r{0000 0000  0000 0000  0000 1101}
@end group

@group
(logior 12 5 7)    ; 12  =  @r{0000 0000  0000 0000  0000 1100}
                   ;  5  =  @r{0000 0000  0000 0000  0000 0101}
                   ;  7  =  @r{0000 0000  0000 0000  0000 0111}
     @result{} 15         ; 15  =  @r{0000 0000  0000 0000  0000 1111}
@end group
@end smallexample
@end defun

@defun logxor &rest ints-or-markers
@cindex bitwise exclusive or
@cindex logical exclusive or
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.  If @code{logxor} is passed just one argument, it returns
that argument.

@smallexample
@group
                   ; @r{              24-bit binary values}

(logxor 12 5)      ; 12  =  @r{0000 0000  0000 0000  0000 1100}
                   ;  5  =  @r{0000 0000  0000 0000  0000 0101}
     @result{} 9          ;  9  =  @r{0000 0000  0000 0000  0000 1001}
@end group

@group
(logxor 12 5 7)    ; 12  =  @r{0000 0000  0000 0000  0000 1100}
                   ;  5  =  @r{0000 0000  0000 0000  0000 0101}
                   ;  7  =  @r{0000 0000  0000 0000  0000 0111}
     @result{} 14         ; 14  =  @r{0000 0000  0000 0000  0000 1110}
@end group
@end smallexample
@end defun

@defun lognot integer
@cindex logical not
@cindex bitwise not
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
;;  5  =  @r{0000 0000  0000 0000  0000 0101}
;; @r{becomes}
;; -6  =  @r{1111 1111  1111 1111  1111 1010}

(lognot 5)             
     @result{} -6
@end example
@end defun

@node Transcendental Functions
@section Transcendental Functions
@cindex transcendental functions
@cindex mathematical functions

These mathematical functions are available if floating point is
supported.  They 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 @minus{} pi / 2
and pi / 2 (inclusive) whose sine is @var{arg}; if, however, @var{arg}
is out of range (outside [-1, 1]), then the result is a NaN.
@end defun

@defun acos arg
The value of @code{(acos @var{arg})} is a number between 0 and pi
(inclusive) whose cosine is @var{arg}; if, however, @var{arg}
is out of range (outside [-1, 1]), then the result is a NaN.
@end defun

@defun atan arg
The value of @code{(atan @var{arg})} is a number between @minus{} pi / 2
and pi / 2 (exclusive) whose tangent is @var{arg}.
@end defun

@defun exp arg
This is the exponential function; it returns @i{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 base @var{e} is used.  If @var{arg}
is negative, the result is a NaN.
@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, the result is a NaN.
@end defun

@defun expt x y
This function returns @var{x} raised to power @var{y}.
@end defun

@defun sqrt arg
This returns the square root of @var{arg}.
@end defun

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

In a computer, 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 XEmacs, 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.  XEmacs always starts with the
same seed value, so the sequence of values of @code{random} is actually
the same in each XEmacs run!  For example, in one operating system, the
first call to @code{(random)} after you start XEmacs always returns
-1457731, and the second one always returns -7692030.  This is helpful
for debugging.

If you want truly unpredictable random numbers, execute @code{(random
t)}.  This chooses a new seed based on the current time of day and on
XEmacs' process @sc{id} number.

@defun random &optional limit
This function returns a pseudo-random integer.  When called more than
once, it returns a series of pseudo-random integers.

If @var{limit} is @code{nil}, then the value may in principle be any
integer.  If @var{limit} is a positive integer, the value is chosen to
be nonnegative and less than @var{limit} (only in Emacs 19).

If @var{limit} is @code{t}, it means to choose a new seed based on the
current time of day and on XEmacs's process @sc{id} number.
@c "XEmacs'" is incorrect usage!

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
