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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 04:22:23 +0000 |
CVSROOT: /sources/emacs
Module name: emacs
Changes by: Glenn Morris <gm> 07/09/06 04:22:23
Index: numbers.texi
===================================================================
RCS file: numbers.texi
diff -N numbers.texi
--- /dev/null 1 Jan 1970 00:00:00 -0000
+++ numbers.texi 6 Sep 2007 04:22:23 -0000 1.1
@@ -0,0 +1,1211 @@
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