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## [Emacs-diffs] Changes to numbers.texi

 From: Glenn Morris Subject: [Emacs-diffs] Changes to numbers.texi Date: Thu, 06 Sep 2007 04:13:31 +0000

```CVSROOT:        /sources/emacs
Module name:    emacs
Changes by:     Glenn Morris <gm>       07/09/06 04:13:30

Index: numbers.texi
===================================================================
RCS file: numbers.texi
diff -N numbers.texi
--- numbers.texi        7 Apr 2007 01:54:45 -0000       1.45
+++ /dev/null   1 Jan 1970 00:00:00 -0000
@@ -1,1211 +0,0 @@
address@hidden -*-texinfo-*-
address@hidden This is part of the GNU Emacs Lisp Reference Manual.
address@hidden Copyright (C) 1990, 1991, 1992, 1993, 1994, 1995, 1998, 1999,
2001,
address@hidden   2002, 2003, 2004, 2005, 2006, 2007  Free Software Foundation,
Inc.
address@hidden See the file elisp.texi for copying conditions.
address@hidden ../info/numbers
address@hidden Numbers, Strings and Characters, Lisp Data Types, Top
address@hidden Numbers
address@hidden integers
address@hidden numbers
-
-  GNU Emacs supports two numeric data types: @dfn{integers} and
address@hidden point numbers}.  Integers are whole numbers such as
address@hidden, 0, 7, 13, and 511.  Their values are exact.  Floating point
-numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or
-2.71828.  They can also be expressed in exponential notation: 1.5e2
-equals 150; in this example, @samp{e2} stands for ten to the second
-power, and that is multiplied by 1.5.  Floating point values are not
-exact; they have a fixed, limited amount of precision.
-
address@hidden
-* 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.
address@hidden menu
-
address@hidden Integer Basics
address@hidden  node-name,  next,  previous,  up
address@hidden Integer Basics
-
-  The range of values for an integer depends on the machine.  The
-minimum range is @minus{}268435456 to 268435455 (29 bits; i.e.,
address@hidden
--2**28
address@hidden ifnottex
address@hidden
address@hidden
address@hidden tex
-to
address@hidden
-2**28 - 1),
address@hidden ifnottex
address@hidden
address@hidden),
address@hidden tex
-but some machines may provide a wider range.  Many examples in this
-chapter assume an integer has 29 bits.
address@hidden overflow
-
-  The Lisp reader reads an integer as a sequence of digits with optional
-initial sign and optional final period.
-
address@hidden
- 1               ; @r{The integer 1.}
- 1.              ; @r{The integer 1.}
-+1               ; @r{Also the integer 1.}
--1               ; @r{The integer @minus{}1.}
- 536870913       ; @r{Also the integer 1, due to overflow.}
- 0               ; @r{The integer 0.}
--0               ; @r{The integer 0.}
address@hidden example
-
address@hidden integers in specific radix
address@hidden radix for reading an integer
address@hidden base for reading an integer
address@hidden hex numbers
address@hidden octal numbers
address@hidden reading numbers in hex, octal, and binary
-  The syntax for integers in bases other than 10 uses @samp{#}
-followed by a letter that specifies the radix: @samp{b} for binary,
address@hidden for octal, @samp{x} for hex, or @address@hidden to
-specify radix @var{radix}.  Case is not significant for the letter
-that specifies the radix.  Thus, @address@hidden reads
address@hidden in binary, and @address@hidden@var{integer}} reads
address@hidden in radix @var{radix}.  Allowed values of @var{radix} run
-from 2 to 36.  For example:
-
address@hidden
-#b101100 @result{} 44
-#o54 @result{} 44
-#x2c @result{} 44
-#24r1k @result{} 44
address@hidden 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 29-bit binary, the decimal integer 5 looks like this:
-
address@hidden
-0 0000  0000 0000  0000 0000  0000 0101
address@hidden example
-
address@hidden
-(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:
-
address@hidden
-1 1111  1111 1111  1111 1111  1111 1111
address@hidden example
-
address@hidden
address@hidden two's complement
address@hidden is represented as 29 ones.  (This is called @dfn{two's
-complement} notation.)
-
-  The negative integer, @minus{}5, is creating by subtracting 4 from
address@hidden  In binary, the decimal integer 4 is 100.  Consequently,
address@hidden looks like this:
-
address@hidden
-1 1111  1111 1111  1111 1111  1111 1011
address@hidden example
-
-  In this implementation, the largest 29-bit binary integer value is
-268,435,455 in decimal.  In binary, it looks like this:
-
address@hidden
-0 1111  1111 1111  1111 1111  1111 1111
address@hidden example
-
-  Since the arithmetic functions do not check whether integers go
-outside their range, when you add 1 to 268,435,455, the value is the
-negative integer @minus{}268,435,456:
-
address@hidden
-(+ 1 268435455)
-     @result{} -268435456
-     @result{} 1 0000  0000 0000  0000 0000  0000 0000
address@hidden example
-
-  Many of the functions described in this chapter accept markers for
-arguments in place of numbers.  (@xref{Markers}.)  Since the actual
-arguments to such functions may be either numbers or markers, we often
-give these arguments the name @var{number-or-marker}.  When the argument
-value is a marker, its position value is used and its buffer is ignored.
-
address@hidden most-positive-fixnum
-The value of this variable is the largest integer that Emacs Lisp
-can handle.
address@hidden defvar
-
address@hidden most-negative-fixnum
-The value of this variable is the smallest integer that Emacs Lisp can
-handle.  It is negative.
address@hidden defvar
-
address@hidden Float Basics
address@hidden Floating Point Basics
-
-  Floating point numbers are useful for representing numbers that are
-not integral.  The precise range of floating point numbers is
-machine-specific; it is the same as the range of the C data type
address@hidden on the machine you are using.
-
-  The read-syntax for floating point numbers requires either a decimal
-point (with at least one digit following), an exponent, or both.  For
-example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and
address@hidden 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}.
-
address@hidden @acronym{IEEE} floating point
address@hidden positive infinity
address@hidden negative infinity
address@hidden infinity
address@hidden NaN
-  Most modern computers support the @acronym{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{(/ 0.0 0.0)} returns a
-NaN.  For practical purposes, there's no significant difference between
-different NaN values in Emacs Lisp, and there's no rule for precisely
-which NaN value should be used in a particular case, so Emacs Lisp
-doesn't try to distinguish them (but it does report the sign, if you
-print it).  Here are the read syntaxes for these special floating
-point values:
-
address@hidden @asis
address@hidden positive infinity
address@hidden
address@hidden negative infinity
address@hidden
address@hidden Not-a-number
address@hidden or @samp{-0.0e+NaN}.
address@hidden table
-
-  To test whether a floating point value is a NaN, compare it with
-itself using @code{=}.  That returns @code{nil} for a NaN, and
address@hidden for any other floating point value.
-
-  The value @code{-0.0} is distinguishable from ordinary zero in
address@hidden floating point, but Emacs Lisp @code{equal} and
address@hidden consider them equal values.
-
-  You can use @code{logb} to extract the binary exponent of a floating
-point number (or estimate the logarithm of an integer):
-
address@hidden 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.
-
address@hidden
-(logb 10)
-     @result{} 3
-(logb 10.0e20)
-     @result{} 69
address@hidden example
address@hidden defun
-
address@hidden Predicates on Numbers
address@hidden Type Predicates for Numbers
address@hidden predicates for numbers
-
-  The functions in this section test for numbers, or for a specific
-type of number.  The functions @code{integerp} and @code{floatp} can
-take any type of Lisp object as argument (they would not be of much
-use otherwise), but the @code{zerop} predicate requires a number as
-its argument.  See also @code{integer-or-marker-p} and
address@hidden, in @ref{Predicates on Markers}.
-
address@hidden floatp object
-This predicate tests whether its argument is a floating point
-number and returns @code{t} if so, @code{nil} otherwise.
-
address@hidden does not exist in Emacs versions 18 and earlier.
address@hidden defun
-
address@hidden integerp object
-This predicate tests whether its argument is an integer, and returns
address@hidden if so, @code{nil} otherwise.
address@hidden defun
-
address@hidden 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.
address@hidden defun
-
address@hidden wholenump object
address@hidden natural numbers
-The @code{wholenump} predicate (whose name comes from the phrase
-``whole-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.
-
address@hidden natnump
address@hidden is an obsolete synonym for @code{wholenump}.
address@hidden defun
-
address@hidden 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.
-
address@hidden(zerop x)} is equivalent to @code{(= x 0)}.
address@hidden defun
-
address@hidden Comparison of Numbers
address@hidden Comparison of Numbers
address@hidden number comparison
address@hidden comparing numbers
-
-  To test numbers for numerical equality, you should normally use
address@hidden, 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
address@hidden  By contrast, @code{=} compares only the numeric values
-of the objects.
-
-  At present, each integer value has a unique Lisp object in Emacs Lisp.
-Therefore, @code{eq} is equivalent to @code{=} where integers are
-concerned.  It is sometimes convenient to use @code{eq} for comparing an
-unknown value with an integer, because @code{eq} does not report an
-error if the unknown value is not a number---it accepts arguments of any
-type.  By contrast, @code{=} signals an error if the arguments are not
-numbers or markers.  However, it is a good idea to use @code{=} if you
-can, even for comparing integers, just in case we change the
-representation of integers in a future Emacs version.
-
-  Sometimes it is useful to compare numbers with @code{equal}; it
-treats two numbers as equal if they have the same data type (both
-integers, or both floating point) and the same value.  By contrast,
address@hidden can treat an integer and a floating point number as equal.
address@hidden Predicates}.
-
-  There is another wrinkle: because floating point arithmetic is not
-exact, it is often a bad idea to check for equality of two floating
-point values.  Usually it is better to test for approximate equality.
-Here's a function to do this:
-
address@hidden
-(defvar fuzz-factor 1.0e-6)
-(defun approx-equal (x y)
-  (or (and (= x 0) (= y 0))
-      (< (/ (abs (- x y))
-            (max (abs x) (abs y)))
-         fuzz-factor)))
address@hidden example
-
address@hidden CL note---integers vrs @code{eq}
address@hidden
address@hidden Lisp note:} Comparing numbers in Common Lisp always requires
address@hidden 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.
address@hidden quotation
-
address@hidden = number-or-marker1 number-or-marker2
-This function tests whether its arguments are numerically equal, and
-returns @code{t} if so, @code{nil} otherwise.
address@hidden defun
-
address@hidden eql value1 value2
-This function acts like @code{eq} except when both arguments are
-numbers.  It compares numbers by type and numeric value, so that
address@hidden(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and
address@hidden(eql 1 1)} both return @code{t}.
address@hidden defun
-
address@hidden /= 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.
address@hidden defun
-
address@hidden <  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.
address@hidden defun
-
address@hidden <=  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.
address@hidden defun
-
address@hidden >  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.
address@hidden defun
-
address@hidden >=  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.
address@hidden defun
-
address@hidden max number-or-marker &rest numbers-or-markers
-This function returns the largest of its arguments.
-If any of the arguments is floating-point, the value is returned
-as floating point, even if it was given as an integer.
-
address@hidden
-(max 20)
-     @result{} 20
-(max 1 2.5)
-     @result{} 2.5
-(max 1 3 2.5)
-     @result{} 3.0
address@hidden example
address@hidden defun
-
address@hidden min number-or-marker &rest numbers-or-markers
-This function returns the smallest of its arguments.
-If any of the arguments is floating-point, the value is returned
-as floating point, even if it was given as an integer.
-
address@hidden
-(min -4 1)
-     @result{} -4
address@hidden example
address@hidden defun
-
address@hidden abs number
-This function returns the absolute value of @var{number}.
address@hidden defun
-
address@hidden Numeric Conversions
address@hidden Numeric Conversions
address@hidden rounding in conversions
address@hidden number conversions
address@hidden converting numbers
-
-To convert an integer to floating point, use the function @code{float}.
-
address@hidden float number
-This returns @var{number} converted to floating point.
-If @var{number} is already a floating point number, @code{float} returns
-it unchanged.
address@hidden defun
-
-There are four functions to convert floating point numbers to integers;
-they differ in how they round.  All accept an argument @var{number}
-and an optional argument @var{divisor}.  Both arguments may be
-integers or floating point numbers.  @var{divisor} may also be
address@hidden  If @var{divisor} is @code{nil} or omitted, these
-functions convert @var{number} to an integer, or return it unchanged
-if it already is an integer.  If @var{divisor} is address@hidden, they
-divide @var{number} by @var{divisor} and convert the result to an
-integer.  An @code{arith-error} results if @var{divisor} is 0.
-
address@hidden truncate number &optional divisor
-This returns @var{number}, converted to an integer by rounding towards
-zero.
-
address@hidden
-(truncate 1.2)
-     @result{} 1
-(truncate 1.7)
-     @result{} 1
-(truncate -1.2)
-     @result{} -1
-(truncate -1.7)
-     @result{} -1
address@hidden example
address@hidden defun
-
address@hidden floor number &optional divisor
-This returns @var{number}, converted to an integer by rounding downward
-(towards negative infinity).
-
-If @var{divisor} is specified, this uses the kind of division
-operation that corresponds to @code{mod}, rounding downward.
-
address@hidden
-(floor 1.2)
-     @result{} 1
-(floor 1.7)
-     @result{} 1
-(floor -1.2)
-     @result{} -2
-(floor -1.7)
-     @result{} -2
-(floor 5.99 3)
-     @result{} 1
address@hidden example
address@hidden defun
-
address@hidden ceiling number &optional divisor
-This returns @var{number}, converted to an integer by rounding upward
-(towards positive infinity).
-
address@hidden
-(ceiling 1.2)
-     @result{} 2
-(ceiling 1.7)
-     @result{} 2
-(ceiling -1.2)
-     @result{} -1
-(ceiling -1.7)
-     @result{} -1
address@hidden example
address@hidden defun
-
address@hidden round number &optional divisor
-This returns @var{number}, converted to an integer by rounding towards the
-nearest integer.  Rounding a value equidistant between two integers
-may choose the integer closer to zero, or it may prefer an even integer,
-depending on your machine.
-
address@hidden
-(round 1.2)
-     @result{} 1
-(round 1.7)
-     @result{} 2
-(round -1.2)
-     @result{} -1
-(round -1.7)
-     @result{} -2
address@hidden example
address@hidden defun
-
address@hidden Arithmetic Operations
address@hidden Arithmetic Operations
address@hidden 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+ 268435455)} may evaluate to
address@hidden, depending on your hardware.
-
address@hidden 1+ number-or-marker
-This function returns @var{number-or-marker} plus 1.
-For example,
-
address@hidden
-(setq foo 4)
-     @result{} 4
-(1+ foo)
-     @result{} 5
address@hidden 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,
-
address@hidden
-foo
-     @result{} 4
address@hidden example
-
-If you want to increment the variable, you must use @code{setq},
-like this:
-
address@hidden
-(setq foo (1+ foo))
-     @result{} 5
address@hidden example
address@hidden defun
-
address@hidden 1- number-or-marker
-This function returns @var{number-or-marker} minus 1.
address@hidden defun
-
address@hidden + &rest numbers-or-markers
-This function adds its arguments together.  When given no arguments,
address@hidden returns 0.
-
address@hidden
-(+)
-     @result{} 0
-(+ 1)
-     @result{} 1
-(+ 1 2 3 4)
-     @result{} 10
address@hidden example
address@hidden defun
-
address@hidden - &optional number-or-marker &rest more-numbers-or-markers
-The @code{-} function serves two purposes: negation and subtraction.
-When @code{-} has a single argument, the value is the negative of the
-argument.  When there are multiple arguments, @code{-} subtracts each of
-the @var{more-numbers-or-markers} from @var{number-or-marker},
-cumulatively.  If there are no arguments, the result is 0.
-
address@hidden
-(- 10 1 2 3 4)
-     @result{} 0
-(- 10)
-     @result{} -10
-(-)
-     @result{} 0
address@hidden example
address@hidden defun
-
address@hidden * &rest numbers-or-markers
-This function multiplies its arguments together, and returns the
-product.  When given no arguments, @code{*} returns 1.
-
address@hidden
-(*)
-     @result{} 1
-(* 1)
-     @result{} 1
-(* 1 2 3 4)
-     @result{} 24
address@hidden example
address@hidden defun
-
address@hidden / dividend divisor &rest divisors
-This function divides @var{dividend} by @var{divisor} and returns the
-quotient.  If there are additional arguments @var{divisors}, then it
-divides @var{dividend} by each divisor in turn.  Each argument may be a
-number or a marker.
-
-If all the arguments are integers, then the result is an integer too.
-This means the result has to be rounded.  On most machines, the result
-is rounded towards zero after each division, but some machines may round
-differently with negative arguments.  This is because the Lisp function
address@hidden/} 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.
-
address@hidden @code{arith-error} in division
-If you divide an integer by 0, an @code{arith-error} error is signaled.
-(@xref{Errors}.)  Floating point division by zero returns either
-infinity or a NaN if your machine supports @acronym{IEEE} floating point;
-otherwise, it signals an @code{arith-error} error.
-
address@hidden
address@hidden
-(/ 6 2)
-     @result{} 3
address@hidden group
-(/ 5 2)
-     @result{} 2
-(/ 5.0 2)
-     @result{} 2.5
-(/ 5 2.0)
-     @result{} 2.5
-(/ 5.0 2.0)
-     @result{} 2.5
-(/ 25 3 2)
-     @result{} 4
address@hidden
-(/ -17 6)
-     @result{} -2   @r{(could in theory be @minus{}3 on some machines)}
address@hidden group
address@hidden example
address@hidden defun
-
address@hidden % dividend divisor
address@hidden 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.
-
address@hidden
-(% 9 4)
-     @result{} 1
-(% -9 4)
-     @result{} -1
-(% 9 -4)
-     @result{} 1
-(% -9 -4)
-     @result{} -1
address@hidden example
-
-For any two integers @var{dividend} and @var{divisor},
-
address@hidden
address@hidden
-(+ (% @var{dividend} @var{divisor})
-   (* (/ @var{dividend} @var{divisor}) @var{divisor}))
address@hidden group
address@hidden example
-
address@hidden
-always equals @var{dividend}.
address@hidden defun
-
address@hidden mod dividend divisor
address@hidden 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.
-
address@hidden
address@hidden
-(mod 9 4)
-     @result{} 1
address@hidden group
address@hidden
-(mod -9 4)
-     @result{} 3
address@hidden group
address@hidden
-(mod 9 -4)
-     @result{} -3
address@hidden group
address@hidden
-(mod -9 -4)
-     @result{} -1
address@hidden group
address@hidden
-(mod 5.5 2.5)
-     @result{} .5
address@hidden group
address@hidden example
-
-For any two numbers @var{dividend} and @var{divisor},
-
address@hidden
address@hidden
-(+ (mod @var{dividend} @var{divisor})
-   (* (floor @var{dividend} @var{divisor}) @var{divisor}))
address@hidden group
address@hidden example
-
address@hidden
-always equals @var{dividend}, subject to rounding error if either
-argument is floating point.  For @code{floor}, see @ref{Numeric
-Conversions}.
address@hidden defun
-
address@hidden Rounding Operations
address@hidden Rounding Operations
address@hidden rounding without conversion
-
-The functions @code{ffloor}, @code{fceiling}, @code{fround}, and
address@hidden 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;
address@hidden, the nearest integer in the direction towards zero;
address@hidden, the nearest integer.
-
address@hidden ffloor float
-This function rounds @var{float} to the next lower integral value, and
-returns that value as a floating point number.
address@hidden defun
-
address@hidden fceiling float
-This function rounds @var{float} to the next higher integral value, and
-returns that value as a floating point number.
address@hidden defun
-
address@hidden ftruncate float
-This function rounds @var{float} towards zero to an integral value, and
-returns that value as a floating point number.
address@hidden defun
-
address@hidden fround float
-This function rounds @var{float} to the nearest integral value,
-and returns that value as a floating point number.
address@hidden defun
-
address@hidden Bitwise Operations
address@hidden Bitwise Operations on Integers
address@hidden bitwise arithmetic
address@hidden logical arithmetic
-
-  In a computer, an integer is represented as a binary number, a
-sequence of @dfn{bits} (digits which are either zero or one).  A bitwise
-operation acts on the individual bits of such a sequence.  For example,
address@hidden 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.
-
address@hidden lsh integer1 count
address@hidden logical shift
address@hidden, 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
address@hidden is negative, @code{lsh} shifts zeros into the leftmost
-(most-significant) bit, producing a positive result even if
address@hidden 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.
-
address@hidden
address@hidden
-(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
address@hidden group
address@hidden example
-
address@hidden
-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):
-
address@hidden
address@hidden
-(lsh 3 2)
-     @result{} 12
-;; @r{Decimal 3 becomes decimal 12.}
-00000011 @result{} 00001100
address@hidden group
address@hidden example
-
-On the other hand, shifting one place to the right looks like this:
-
address@hidden
address@hidden
-(lsh 6 -1)
-     @result{} 3
-;; @r{Decimal 6 becomes decimal 3.}
-00000110 @result{} 00000011
address@hidden group
-
address@hidden
-(lsh 5 -1)
-     @result{} 2
-;; @r{Decimal 5 becomes decimal 2.}
-00000101 @result{} 00000010
address@hidden group
address@hidden example
-
address@hidden
-As the example illustrates, shifting one place to the right divides the
-value of a positive integer by two, rounding downward.
-
-The function @code{lsh}, like all Emacs Lisp arithmetic functions, does
-not check for overflow, so shifting left can discard significant bits
-and change the sign of the number.  For example, left shifting
-268,435,455 produces @minus{}2 on a 29-bit machine:
-
address@hidden
-(lsh 268435455 1)          ; @r{left shift}
-     @result{} -2
address@hidden example
-
-In binary, in the 29-bit implementation, the argument looks like this:
-
address@hidden
address@hidden
-;; @r{Decimal 268,435,455}
-0 1111  1111 1111  1111 1111  1111 1111
address@hidden group
address@hidden example
-
address@hidden
-which becomes the following when left shifted:
-
address@hidden
address@hidden
-;; @r{Decimal @minus{}2}
-1 1111  1111 1111  1111 1111  1111 1110
address@hidden group
address@hidden example
address@hidden defun
-
address@hidden ash integer1 count
address@hidden arithmetic shift
address@hidden (@dfn{arithmetic shift}) shifts the bits in @var{integer1}
-to the left @var{count} places, or to the right if @var{count}
-is negative.
-
address@hidden gives the same results as @code{lsh} except when
address@hidden and @var{count} are both negative.  In that case,
address@hidden puts ones in the empty bit positions on the left, while
address@hidden puts zeros in those bit positions.
-
-Thus, with @code{ash}, shifting the pattern of bits one place to the right
-looks like this:
-
address@hidden
address@hidden
-(ash -6 -1) @result{} -3
-;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
-1 1111  1111 1111  1111 1111  1111 1010
-     @result{}
-1 1111  1111 1111  1111 1111  1111 1101
address@hidden group
address@hidden example
-
-In contrast, shifting the pattern of bits one place to the right with
address@hidden looks like this:
-
address@hidden
address@hidden
-(lsh -6 -1) @result{} 268435453
-;; @r{Decimal @minus{}6 becomes decimal 268,435,453.}
-1 1111  1111 1111  1111 1111  1111 1010
-     @result{}
-0 1111  1111 1111  1111 1111  1111 1101
address@hidden group
address@hidden example
-
-Here are other examples:
-
address@hidden !!! Check if lined up in smallbook format!  XDVI shows problem
address@hidden     with smallbook but not with regular book! --rjc 16mar92
address@hidden
address@hidden
-                   ;  @r{             29-bit binary values}
-
-(lsh 5 2)          ;   5  =  @r{0 0000  0000 0000  0000 0000  0000 0101}
-     @result{} 20         ;      =  @r{0 0000  0000 0000  0000 0000  0001 0100}
address@hidden group
address@hidden
-(ash 5 2)
-     @result{} 20
-(lsh -5 2)         ;  -5  =  @r{1 1111  1111 1111  1111 1111  1111 1011}
-     @result{} -20        ;      =  @r{1 1111  1111 1111  1111 1111  1110 1100}
-(ash -5 2)
-     @result{} -20
address@hidden group
address@hidden
-(lsh 5 -2)         ;   5  =  @r{0 0000  0000 0000  0000 0000  0000 0101}
-     @result{} 1          ;      =  @r{0 0000  0000 0000  0000 0000  0000 0001}
address@hidden group
address@hidden
-(ash 5 -2)
-     @result{} 1
address@hidden group
address@hidden
-(lsh -5 -2)        ;  -5  =  @r{1 1111  1111 1111  1111 1111  1111 1011}
-     @result{} 134217726  ;      =  @r{0 0111  1111 1111  1111 1111  1111 1110}
address@hidden group
address@hidden
-(ash -5 -2)        ;  -5  =  @r{1 1111  1111 1111  1111 1111  1111 1011}
-     @result{} -2         ;      =  @r{1 1111  1111 1111  1111 1111  1111 1110}
address@hidden group
address@hidden smallexample
address@hidden defun
-
address@hidden logand &rest ints-or-markers
-This function returns the ``logical and'' of the arguments: the
address@hidden 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.
-
address@hidden
-Therefore,
-
address@hidden
address@hidden
-(logand 13 12)
-     @result{} 12
address@hidden group
address@hidden example
-
-If @code{logand} is not passed any argument, it returns a value of
address@hidden  This number is an identity element for @code{logand}
-because its binary representation consists entirely of ones.  If
address@hidden is passed just one argument, it returns that argument.
-
address@hidden
address@hidden
-                   ; @r{               29-bit binary values}
-
-(logand 14 13)     ; 14  =  @r{0 0000  0000 0000  0000 0000  0000 1110}
-                   ; 13  =  @r{0 0000  0000 0000  0000 0000  0000 1101}
-     @result{} 12         ; 12  =  @r{0 0000  0000 0000  0000 0000  0000 1100}
address@hidden group
-
address@hidden
-(logand 14 13 4)   ; 14  =  @r{0 0000  0000 0000  0000 0000  0000 1110}
-                   ; 13  =  @r{0 0000  0000 0000  0000 0000  0000 1101}
-                   ;  4  =  @r{0 0000  0000 0000  0000 0000  0000 0100}
-     @result{} 4          ;  4  =  @r{0 0000  0000 0000  0000 0000  0000 0100}
address@hidden group
-
address@hidden
-(logand)
-     @result{} -1         ; -1  =  @r{1 1111  1111 1111  1111 1111  1111 1111}
address@hidden group
address@hidden smallexample
address@hidden defun
-
address@hidden logior &rest ints-or-markers
-This function returns the ``inclusive or'' of its arguments: the @var{n}th bit
-is set in the result if, and only if, the @var{n}th bit is set in at least
-one of the arguments.  If there are no arguments, the result is zero,
-which is an identity element for this operation.  If @code{logior} is
-passed just one argument, it returns that argument.
-
address@hidden
address@hidden
-                   ; @r{              29-bit binary values}
-
-(logior 12 5)      ; 12  =  @r{0 0000  0000 0000  0000 0000  0000 1100}
-                   ;  5  =  @r{0 0000  0000 0000  0000 0000  0000 0101}
-     @result{} 13         ; 13  =  @r{0 0000  0000 0000  0000 0000  0000 1101}
address@hidden group
-
address@hidden
-(logior 12 5 7)    ; 12  =  @r{0 0000  0000 0000  0000 0000  0000 1100}
-                   ;  5  =  @r{0 0000  0000 0000  0000 0000  0000 0101}
-                   ;  7  =  @r{0 0000  0000 0000  0000 0000  0000 0111}
-     @result{} 15         ; 15  =  @r{0 0000  0000 0000  0000 0000  0000 1111}
address@hidden group
address@hidden smallexample
address@hidden defun
-
address@hidden logxor &rest ints-or-markers
-This function returns the ``exclusive or'' of its arguments: the
address@hidden 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
address@hidden is passed just one argument, it returns that argument.
-
address@hidden
address@hidden
-                   ; @r{              29-bit binary values}
-
-(logxor 12 5)      ; 12  =  @r{0 0000  0000 0000  0000 0000  0000 1100}
-                   ;  5  =  @r{0 0000  0000 0000  0000 0000  0000 0101}
-     @result{} 9          ;  9  =  @r{0 0000  0000 0000  0000 0000  0000 1001}
address@hidden group
-
address@hidden
-(logxor 12 5 7)    ; 12  =  @r{0 0000  0000 0000  0000 0000  0000 1100}
-                   ;  5  =  @r{0 0000  0000 0000  0000 0000  0000 0101}
-                   ;  7  =  @r{0 0000  0000 0000  0000 0000  0000 0111}
-     @result{} 14         ; 14  =  @r{0 0000  0000 0000  0000 0000  0000 1110}
address@hidden group
address@hidden smallexample
address@hidden defun
-
address@hidden lognot integer
-This function returns the logical complement of its argument: the @var{n}th
-bit is one in the result if, and only if, the @var{n}th bit is zero in
address@hidden, and vice-versa.
-
address@hidden
-(lognot 5)
-     @result{} -6
-;;  5  =  @r{0 0000  0000 0000  0000 0000  0000 0101}
-;; @r{becomes}
-;; -6  =  @r{1 1111  1111 1111  1111 1111  1111 1010}
address@hidden example
address@hidden defun
-
address@hidden Math Functions
address@hidden Standard Mathematical Functions
address@hidden transcendental functions
address@hidden mathematical functions
address@hidden floating-point functions
-
-  These mathematical functions allow integers as well as floating point
-numbers as arguments.
-
address@hidden sin arg
address@hidden cos arg
address@hidden tan arg
-These are the ordinary trigonometric functions, with argument measured
-in radians.
address@hidden defun
-
address@hidden asin arg
-The value of @code{(asin @var{arg})} is a number between
address@hidden
address@hidden/2
address@hidden ifnottex
address@hidden
address@hidden/2}
address@hidden tex
-and
address@hidden
-pi/2
address@hidden ifnottex
address@hidden
address@hidden/2}
address@hidden tex
-(inclusive) whose sine is @var{arg}; if, however, @var{arg} is out of
-range (outside address@hidden, 1]), it signals a @code{domain-error} error.
address@hidden defun
-
address@hidden acos arg
-The value of @code{(acos @var{arg})} is a number between 0 and
address@hidden
-pi
address@hidden ifnottex
address@hidden
address@hidden
address@hidden tex
-(inclusive) whose cosine is @var{arg}; if, however, @var{arg} is out
-of range (outside address@hidden, 1]), it signals a @code{domain-error} error.
address@hidden defun
-
address@hidden atan y &optional x
-The value of @code{(atan @var{y})} is a number between
address@hidden
address@hidden/2
address@hidden ifnottex
address@hidden
address@hidden/2}
address@hidden tex
-and
address@hidden
-pi/2
address@hidden ifnottex
address@hidden
address@hidden/2}
address@hidden tex
-(exclusive) whose tangent is @var{y}.  If the optional second
-argument @var{x} is given, the value of @code{(atan y x)} is the
-angle in radians between the vector @address@hidden, @var{y}]} and the
address@hidden axis.
address@hidden defun
-
address@hidden exp arg
-This is the exponential function; it returns
address@hidden
address@hidden
address@hidden tex
address@hidden
address@hidden
address@hidden ifnottex
-to the power @var{arg}.
address@hidden
address@hidden
address@hidden tex
address@hidden
address@hidden
address@hidden ifnottex
-is a fundamental mathematical constant also called the base of natural
-logarithms.
address@hidden defun
-
address@hidden 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
address@hidden
address@hidden
address@hidden tex
address@hidden
address@hidden
address@hidden ifnottex
-is used.  If @var{arg} is negative, it signals a @code{domain-error}
-error.
address@hidden defun
-
address@hidden
address@hidden 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.
address@hidden defun
-
address@hidden 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.
address@hidden defun
address@hidden ignore
-
address@hidden log10 arg
-This function returns the logarithm of @var{arg}, with base 10.  If
address@hidden is negative, it signals a @code{domain-error} error.
address@hidden(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}, at least
-approximately.
address@hidden defun
-
address@hidden expt x y
-This function returns @var{x} raised to power @var{y}.  If both
-arguments are integers and @var{y} is positive, the result is an
-integer; in this case, overflow causes truncation, so watch out.
address@hidden defun
-
address@hidden sqrt arg
-This returns the square root of @var{arg}.  If @var{arg} is negative,
-it signals a @code{domain-error} error.
address@hidden defun
-
address@hidden Random Numbers
address@hidden Random Numbers
address@hidden 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
address@hidden, and the second one always returns @minus{}7692030.  This
-repeatability is helpful for debugging.
-
-If you want random numbers that don't always come out the same, execute
address@hidden(random t)}.  This chooses a new seed based on the current time of
-day and on Emacs's process @acronym{ID} number.
-
address@hidden random &optional limit
-This function returns a pseudo-random integer.  Repeated calls return a
-series of pseudo-random integers.
-
-If @var{limit} is a positive integer, the value is chosen to be
-nonnegative and less than @var{limit}.
-
-If @var{limit} is @code{t}, it means to choose a new seed based on the
-current time of day and on Emacs's process @acronym{ID} number.
address@hidden "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.
address@hidden defun
-
address@hidden
-   arch-tag: 574e8dd2-d513-4616-9844-c9a27869782e
address@hidden ignore

```

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