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+@c -*-texinfo-*-
+@c This is part of the GNU Emacs Lisp Reference Manual.
+@c Copyright (C) 1990, 1991, 1992, 1993, 1994 Free Software Foundation, Inc. 
+@c See the file elisp.texi for copying conditions.
+@setfilename ../info/numbers
+@node Numbers, Strings and Characters, Types of Lisp Object, 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 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.
+
+  Support for floating point numbers is a new feature in Emacs 19, and it
+is controlled by a separate compilation option, so you may encounter a site
+where Emacs does not support them.
+
+@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.
+* Transcendental 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
+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).  Many examples in this
+chapter assume an integer has 24 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.}
+ 16777217        ; @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 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, it 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 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
+
+@cindex @code{LISP_FLOAT_TYPE} configuration macro
+  Emacs version 19 supports floating point numbers, if compiled with the
+macro @code{LISP_FLOAT_TYPE} defined.  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 value called NaN or ``not-a-number''
+which is the result you get from numerical functions in cases where
+there is no correct answer.  For example, @code{(sqrt -1.0)} returns
+NaN.  There is no read syntax for NaN 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} 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{} @code{(= 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 test for
+numerical equality, 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 Emacs 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:} 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 /= 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.
+
+@example
+(max 20)
+     @result{} 20
+(max 1 2.5)
+     @result{} 2.5
+(max 1 3 2.5)
+     @result{} 3
+@end example
+@end defun
+
+@defun min number-or-marker &rest numbers-or-markers
+This function returns the smallest of its arguments.
+
+@example
+(min -4 1)
+     @result{} -4
+@end example
+@end defun
+
+@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.
+@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 GNU Emacs Lisp, arithmetic functions
+do not check for overflow.  Thus @code{(1+ 8388607)} may evaluate to
+@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, @code{-} subtracts each of
+the @var{other-numbers-or-markers} 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 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 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
+
+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
+(mod 9 4)
+     @result{} 1
+(mod -9 4)
+     @result{} 3
+(mod 9 -4)
+     @result{} -3
+(mod -9 -4)
+     @result{} -1
+(mod 5.5 2.5)
+     @result{} .5
+@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 Rounding Operations
+@section Rounding Operations
+@cindex rounding without conversion
+
+The functions @code{ffloor}, @code{fceil}, @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{fceil}, the nearest integer above;
+@code{ftrucate}, 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 fceil float
+This function rounds @var{float} to the next higher integral value, and
+returns that value as a floating point number.
+@end defun
+
+@defun ftrunc 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
+
+  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       
+@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 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 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 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
+(lognot 5)             
+     @result{} -6
+;;  5  =  @r{0000 0000  0000 0000  0000 0101}
+;; @r{becomes}
+;; -6  =  @r{1111 1111  1111 1111  1111 1010}
+@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}.  @i{e} is a fundamental mathematical constant also called the
+base of natural logarithms.
+@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}.  If @var{arg} is negative,
+the value is a NaN.
+@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 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
+-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
+Emacs'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 @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 Emacs's process @sc{id} number.
+@c "Emacs'" 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