XEmacs 21.4.15

[chise/xemacs-chise.git.1] / man / lispref / numbers.texi

@c -*-texinfo-*-
@c This is part of the XEmacs Lisp Reference Manual.
@c Copyright (C) 1990, 1991, 1992, 1993, 1994 Free Software Foundation, Inc.
@c See the file lispref.texi for copying conditions.
@setfilename ../../info/numbers.info
@node Numbers, Strings and Characters, Lisp Data Types, 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, #b0111, #xFEED, #o744.  Their values are exact.  The
number prefixes `#b', `#o', and `#x' are supported to represent numbers
in binary, octal, and hexadecimal notation (or radix).  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 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
@section Integer Basics

  The range of values for an integer depends on the machine.  The
minimum range is @minus{}134217728 to 134217727 (28 bits; i.e.,
@ifinfo
-2**27
@end ifinfo
@tex
$-2^{27}$
@end tex
to
@ifinfo
2**27 - 1),
@end ifinfo
@tex
$2^{27}-1$),
@end tex
but some machines may provide a wider range.  Many examples in this
chapter assume an integer has 28 bits.
@cindex overflow

  The Lisp reader reads an integer as a sequence of 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.}
 268435457       ; @r{Also the integer 1, due to overflow.}
 0               ; @r{The integer 0.}
-0               ; @r{The integer 0.}
@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 28-bit binary, the decimal integer 5 looks like this:

@example
0000  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 1111
@end example

@noindent
@cindex two's complement
@minus{}1 is represented as 28 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  1111 1011
@end example

  In this implementation, the largest 28-bit binary integer is the
decimal integer 134,217,727.  In binary, it looks like this:

@example
0111  1111 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 134,217,727, the value is the
negative integer @minus{}134,217,728:

@example
(+ 1 134217727)
     @result{} -134217728
     @result{} 1000  0000 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 arguments to
such functions may be either integers or markers, which is why we often
give these arguments the name @var{int-or-marker}.  When the argument
value is a marker, its position value is used and its buffer 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}.

@cindex IEEE floating point
@cindex positive infinity
@cindex negative infinity
@cindex infinity
@cindex NaN
   Most modern computers support the IEEE floating point standard, which
provides for 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{(sqrt -1.0)} returns a
NaN.  For practical purposes, there's no significant difference between
different NaN values in XEmacs Lisp, and there's no rule for precisely
which NaN value should be used in a particular case, so this manual
doesn't try to distinguish them.  XEmacs Lisp has no read syntax for NaNs
or infinities; perhaps we should create a syntax in the future.

  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}, @code{integer-char-or-marker-p},
@code{number-or-marker-p} and @code{number-char-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.
@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{} @code{(= x 0)}.
@end defun

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

  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 XEmacs 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 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
(defconst 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.  XEmacs Lisp
can have just one integer object for any given value because it has a
limited range of integer values.
@end quotation

In addition to numbers, all of the following functions also accept
characters and markers as arguments, and treat them as their number
equivalents.

@defun =  number &rest more-numbers
This function returns @code{t} if all of its arguments are numerically
equal, @code{nil} otherwise.

@example
(= 5)
     @result{} t
(= 5 6)
     @result{} nil
(= 5 5.0)
     @result{} t
(= 5 5 6)
     @result{} nil
@end example
@end defun

@defun /=  number &rest more-numbers
This function returns @code{t} if no two arguments are numerically
equal, @code{nil} otherwise.

@example
(/= 5 6)
     @result{} t
(/= 5 5 6)
     @result{} nil
(/= 5 6 1)
     @result{} t
@end example
@end defun

@defun <  number &rest more-numbers
This function returns @code{t} if the sequence of its arguments is
monotonically increasing, @code{nil} otherwise.

@example
(< 5 6)
     @result{} t
(< 5 6 6)
     @result{} nil
(< 5 6 7)
     @result{} t
@end example
@end defun

@defun <=  number &rest more-numbers
This function returns @code{t} if the sequence of its arguments is
monotonically nondecreasing, @code{nil} otherwise.

@example
(<= 5 6)
     @result{} t
(<= 5 6 6)
     @result{} t
(<= 5 6 5)
     @result{} nil
@end example
@end defun

@defun >  number &rest more-numbers
This function returns @code{t} if the sequence of its arguments is
monotonically decreasing, @code{nil} otherwise.
@end defun

@defun >=  number &rest more-numbers
This function returns @code{t} if the sequence of its arguments is
monotonically nonincreasing, @code{nil} otherwise.
@end defun

@defun max number &rest more-numbers
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 &rest more-numbers
This function returns the smallest of its arguments.

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

@node Numeric Conversions
@section Numeric Conversions
@cindex rounding in 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.  These functions accept integer arguments
also, and return such arguments unchanged.

@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.  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.
@end defun

@node Arithmetic Operations
@section Arithmetic Operations

  XEmacs 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 XEmacs Lisp, arithmetic functions
do not check for overflow.  Thus @code{(1+ 134217727)} may evaluate to
@minus{}134217728, depending on your hardware.

@defun 1+ number
This function returns @var{number} plus one.  @var{number} may be a
number, character or marker.  Markers and characters are converted to
integers.

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

Now that the @code{cl} package is always available from lisp code, a
more convenient and natural way to increment a variable is
@w{@code{(incf foo)}}.
@end defun

@defun 1- number
This function returns @var{number} minus one.  @var{number} may be a
number, character or marker.  Markers and characters are converted to
integers.
@end defun

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

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

If any of the arguments are characters or markers, they are first
converted to integers.

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

@defun - &optional number &rest other-numbers
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{other-numbers} from @var{number}, cumulatively.  If there are
no arguments, an error is signaled.

If any of the arguments are characters or markers, they are first
converted to integers.

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

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

If any of the arguments are characters or markers, they are first
converted to integers.

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

@defun / dividend &rest divisors
The @code{/} function serves two purposes: inversion and division.  When
@code{/} has a single argument, the value is the inverse of the
argument.  When there are multiple arguments, @code{/} divides
@var{dividend} by each of the @var{divisors}, cumulatively, returning
the quotient.  If there are no arguments, an error is signaled.

If none of the arguments are floats, then the result is an integer.
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.

If any of the arguments are characters or markers, they are first
converted to integers.

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

@example
@group
(/ 6 2)
     @result{} 3
@end group
(/ 5 2)
     @result{} 2
(/ 25 3 2)
     @result{} 4
(/ 3.0)
     @result{} 0.3333333333333333
(/ -17 6)
     @result{} -2
@end example

The result of @code{(/ -17 6)} could in principle be -3 on some
machines.
@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 number
This function rounds @var{number} to the next lower integral value, and
returns that value as a floating point number.
@end defun

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

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

@defun fround number
This function rounds @var{number} to the nearest integral value,
and returns that value as a floating point number.
@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 XEmacs 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 XEmacs Lisp arithmetic functions, does
not check for overflow, so shifting left can discard significant bits
and change the sign of the number.  For example, left shifting
134,217,727 produces @minus{}2 on a 28-bit machine:

@example
(lsh 134217727 1)          ; @r{left shift}
     @result{} -2
@end example

In binary, in the 28-bit implementation, the argument looks like this:

@example
@group
;; @r{Decimal 134,217,727}
0111  1111 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  1111 1110
@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  1111 1111  1111 1111  1111 1010
     @result{}
1111  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{} 134217725
;; @r{Decimal @minus{}6 becomes decimal 134,217,725.}
1111  1111 1111  1111 1111  1111 1010
     @result{}
0111  1111 1111  1111 1111  1111 1101
@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{             28-bit binary values}

(lsh 5 2)          ;   5  =  @r{0000  0000 0000  0000 0000  0000 0101}
     @result{} 20         ;      =  @r{0000  0000 0000  0000 0000  0001 0100}
@end group
@group
(ash 5 2)
     @result{} 20
(lsh -5 2)         ;  -5  =  @r{1111  1111 1111  1111 1111  1111 1011}
     @result{} -20        ;      =  @r{1111  1111 1111  1111 1111  1110 1100}
(ash -5 2)
     @result{} -20
@end group
@group
(lsh 5 -2)         ;   5  =  @r{0000  0000 0000  0000 0000  0000 0101}
     @result{} 1          ;      =  @r{0000  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  1111 1011}
     @result{} 4194302    ;      =  @r{0011  1111 1111  1111 1111  1111 1110}
@end group
@group
(ash -5 -2)        ;  -5  =  @r{1111  1111 1111  1111 1111  1111 1011}
     @result{} -2         ;      =  @r{1111  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{               28-bit binary values}

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

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

@group
(logand)
     @result{} -1         ; -1  =  @r{1111  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{              28-bit binary values}

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

@group
(logior 12 5 7)    ; 12  =  @r{0000  0000 0000  0000 0000  0000 1100}
                   ;  5  =  @r{0000  0000 0000  0000 0000  0000 0101}
                   ;  7  =  @r{0000  0000 0000  0000 0000  0000 0111}
     @result{} 15         ; 15  =  @r{0000  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, which is an identity element for this operation.  If
@code{logxor} is passed just one argument, it returns that argument.

@smallexample
@group
                   ; @r{              28-bit binary values}

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

@group
(logxor 12 5 7)    ; 12  =  @r{0000  0000 0000  0000 0000  0000 1100}
                   ;  5  =  @r{0000  0000 0000  0000 0000  0000 0101}
                   ;  7  =  @r{0000  0000 0000  0000 0000  0000 0111}
     @result{} 14         ; 14  =  @r{0000  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
(lognot 5)
     @result{} -6
;;  5  =  @r{0000  0000 0000  0000 0000  0000 0101}
;; @r{becomes}
;; -6  =  @r{1111  1111 1111  1111 1111  1111 1010}
@end example
@end defun

@node Math Functions
@section Standard Mathematical Functions
@cindex transcendental functions
@cindex mathematical functions

These mathematical functions are available if floating point is
supported (which is the normal state of affairs).  They allow integers
as well as floating point numbers as arguments.

@defun sin number
@defunx cos number
@defunx tan number
These are the ordinary trigonometric functions, with argument measured
in radians.
@end defun

@defun asin number
The value of @code{(asin @var{number})} is a number between @minus{}pi/2
and pi/2 (inclusive) whose sine is @var{number}; if, however, @var{number}
is out of range (outside [-1, 1]), then the result is a NaN.
@end defun

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

@defun atan number &optional number2
The value of @code{(atan @var{number})} is a number between @minus{}pi/2
and pi/2 (exclusive) whose tangent is @var{number}.

If optional argument @var{number2} is supplied, the function returns
@code{atan2(@var{number},@var{number2})}.
@end defun

@defun sinh number
@defunx cosh number
@defunx tanh number
These are the ordinary hyperbolic trigonometric functions.
@end defun

@defun asinh number
@defunx acosh number
@defunx atanh number
These are the inverse hyperbolic trigonometric functions.
@end defun

@defun exp number
This is the exponential function; it returns @i{e} to the power
@var{number}.  @i{e} is a fundamental mathematical constant also called the
base of natural logarithms.
@end defun

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

@defun log10 number
This function returns the logarithm of @var{number}, with base 10.  If
@var{number} is negative, the result is a NaN.  @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, it is truncated to fit the range of possible
integer values.
@end defun

@defun sqrt number
This returns the square root of @var{number}.  If @var{number} is negative,
the value is a NaN.
@end defun

@defun cube-root number
This returns the cube root of @var{number}.
@end defun

@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 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
repeatability 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's process @sc{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 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