@c -*-texinfo-*-
@c This is part of the GNU Emacs Lisp Reference Manual.
-@c Copyright (C) 1990, 1991, 1992, 1993, 1994, 1995, 1998, 1999, 2001,
-@c 2002, 2003, 2004, 2005, 2006, 2007, 2008 Free Software Foundation, Inc.
+@c Copyright (C) 1990-1995, 1998-1999, 2001-2012
+@c Free Software Foundation, Inc.
@c See the file elisp.texi for copying conditions.
@setfilename ../../info/numbers
@node Numbers, Strings and Characters, Lisp Data Types, Top
@menu
* Integer Basics:: Representation and range of integers.
-* Float Basics:: Representation and range of floating point.
+* 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.
+* 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.
@section 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.,
+minimum range is @minus{}536870912 to 536870911 (30 bits; i.e.,
@ifnottex
--2**28
+-2**29
@end ifnottex
@tex
-@math{-2^{28}}
+@math{-2^{29}}
@end tex
to
@ifnottex
-2**28 - 1),
+2**29 - 1),
@end ifnottex
@tex
-@math{2^{28}-1}),
+@math{2^{29}-1}),
@end tex
-but some machines may provide a wider range. Many examples in this
-chapter assume an integer has 29 bits.
+but some machines provide a wider range. Many examples in this
+chapter assume that an integer has 30 bits and that floating point
+numbers are IEEE double precision.
@cindex overflow
The Lisp reader reads an integer as a sequence of digits with optional
-initial sign and optional final period.
+initial sign and optional final period. An integer that is out of the
+Emacs range is treated as a floating-point number.
@example
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.}
+ 1073741825 ; @r{The floating point number 1073741825.0.}
0 ; @r{The integer 0.}
-0 ; @r{The integer 0.}
@end example
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:
+ In 30-bit binary, the decimal integer 5 looks like this:
@example
-0 0000 0000 0000 0000 0000 0000 0101
+0000...000101 (30 bits total)
@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 @samp{...} stands for enough bits to fill out a 30-bit word; in
+this case, @samp{...} stands for twenty 0 bits. Later examples also
+use the @samp{...} notation to make binary integers easier to read.)
The integer @minus{}1 looks like this:
@example
-1 1111 1111 1111 1111 1111 1111 1111
+1111...111111 (30 bits total)
@end example
@noindent
@cindex two's complement
-@minus{}1 is represented as 29 ones. (This is called @dfn{two's
+@minus{}1 is represented as 30 ones. (This is called @dfn{two's
complement} notation.)
The negative integer, @minus{}5, is creating by subtracting 4 from
@minus{}5 looks like this:
@example
-1 1111 1111 1111 1111 1111 1111 1011
+1111...111011 (30 bits total)
@end example
- In this implementation, the largest 29-bit binary integer value is
-268,435,455 in decimal. In binary, it looks like this:
+ In this implementation, the largest 30-bit binary integer value is
+536,870,911 in decimal. In binary, it looks like this:
@example
-0 1111 1111 1111 1111 1111 1111 1111
+0111...111111 (30 bits total)
@end 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:
+outside their range, when you add 1 to 536,870,911, the value is the
+negative integer @minus{}536,870,912:
@example
-(+ 1 268435455)
- @result{} -268435456
- @result{} 1 0000 0000 0000 0000 0000 0000 0000
+(+ 1 536870911)
+ @result{} -536870912
+ @result{} 1000...000000 (30 bits total)
@end example
Many of the functions described in this chapter accept markers for
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.
+@cindex largest Lisp integer number
+@cindex maximum Lisp integer number
@defvar most-positive-fixnum
The value of this variable is the largest integer that Emacs Lisp
can handle.
@end defvar
+@cindex smallest Lisp integer number
+@cindex minimum Lisp integer number
@defvar most-negative-fixnum
The value of this variable is the smallest integer that Emacs Lisp can
handle. It is negative.
@end defvar
+ @xref{Character Codes, max-char}, for the maximum value of a valid
+character codepoint.
+
@node Float Basics
@section Floating Point Basics
@samp{1.0e+INF}
@item negative infinity
@samp{-1.0e+INF}
-@item Not-a-number
+@item Not-a-number
@samp{0.0e+NaN} or @samp{-0.0e+NaN}.
@end table
@end example
@end defun
+@defvar float-e
+The mathematical constant @math{e} (2.71828@dots{}).
+@end defvar
+
+@defvar float-pi
+The mathematical constant @math{pi} (3.14159@dots{}).
+@end defvar
+
@node Predicates on Numbers
@section Type Predicates for Numbers
@cindex predicates for numbers
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
-@minus{}268435456, depending on your hardware.
+do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to
+@minus{}536870912, depending on your hardware.
@defun 1+ number-or-marker
This function returns @var{number-or-marker} plus 1.
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:
+536,870,911 produces @minus{}2 in the 30-bit implementation:
@example
-(lsh 268435455 1) ; @r{left shift}
+(lsh 536870911 1) ; @r{left shift}
@result{} -2
@end example
-In binary, in the 29-bit implementation, the argument looks like this:
+In binary, the argument looks like this:
@example
@group
-;; @r{Decimal 268,435,455}
-0 1111 1111 1111 1111 1111 1111 1111
+;; @r{Decimal 536,870,911}
+0111...111111 (30 bits total)
@end group
@end example
@example
@group
;; @r{Decimal @minus{}2}
-1 1111 1111 1111 1111 1111 1111 1110
+1111...111110 (30 bits total)
@end group
@end example
@end defun
@group
(ash -6 -1) @result{} -3
;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
-1 1111 1111 1111 1111 1111 1111 1010
+1111...111010 (30 bits total)
@result{}
-1 1111 1111 1111 1111 1111 1111 1101
+1111...111101 (30 bits total)
@end group
@end example
@example
@group
-(lsh -6 -1) @result{} 268435453
-;; @r{Decimal @minus{}6 becomes decimal 268,435,453.}
-1 1111 1111 1111 1111 1111 1111 1010
+(lsh -6 -1) @result{} 536870909
+;; @r{Decimal @minus{}6 becomes decimal 536,870,909.}
+1111...111010 (30 bits total)
@result{}
-0 1111 1111 1111 1111 1111 1111 1101
+0111...111101 (30 bits total)
@end group
@end example
@c with smallbook but not with regular book! --rjc 16mar92
@smallexample
@group
- ; @r{ 29-bit binary values}
+ ; @r{ 30-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}
+(lsh 5 2) ; 5 = @r{0000...000101}
+ @result{} 20 ; = @r{0000...010100}
@end group
@group
(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}
+(lsh -5 2) ; -5 = @r{1111...111011}
+ @result{} -20 ; = @r{1111...101100}
(ash -5 2)
@result{} -20
@end group
@group
-(lsh 5 -2) ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101}
- @result{} 1 ; = @r{0 0000 0000 0000 0000 0000 0000 0001}
+(lsh 5 -2) ; 5 = @r{0000...000101}
+ @result{} 1 ; = @r{0000...000001}
@end group
@group
(ash 5 -2)
@result{} 1
@end group
@group
-(lsh -5 -2) ; -5 = @r{1 1111 1111 1111 1111 1111 1111 1011}
- @result{} 134217726 ; = @r{0 0111 1111 1111 1111 1111 1111 1110}
+(lsh -5 -2) ; -5 = @r{1111...111011}
+ @result{} 268435454
+ ; = @r{0011...111110}
@end group
@group
-(ash -5 -2) ; -5 = @r{1 1111 1111 1111 1111 1111 1111 1011}
- @result{} -2 ; = @r{1 1111 1111 1111 1111 1111 1111 1110}
+(ash -5 -2) ; -5 = @r{1111...111011}
+ @result{} -2 ; = @r{1111...111110}
@end group
@end smallexample
@end defun
@smallexample
@group
- ; @r{ 29-bit binary values}
+ ; @r{ 30-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}
+(logand 14 13) ; 14 = @r{0000...001110}
+ ; 13 = @r{0000...001101}
+ @result{} 12 ; 12 = @r{0000...001100}
@end group
@group
-(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}
+(logand 14 13 4) ; 14 = @r{0000...001110}
+ ; 13 = @r{0000...001101}
+ ; 4 = @r{0000...000100}
+ @result{} 4 ; 4 = @r{0000...000100}
@end group
@group
(logand)
- @result{} -1 ; -1 = @r{1 1111 1111 1111 1111 1111 1111 1111}
+ @result{} -1 ; -1 = @r{1111...111111}
@end group
@end smallexample
@end defun
@smallexample
@group
- ; @r{ 29-bit binary values}
+ ; @r{ 30-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}
+(logior 12 5) ; 12 = @r{0000...001100}
+ ; 5 = @r{0000...000101}
+ @result{} 13 ; 13 = @r{0000...001101}
@end group
@group
-(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}
+(logior 12 5 7) ; 12 = @r{0000...001100}
+ ; 5 = @r{0000...000101}
+ ; 7 = @r{0000...000111}
+ @result{} 15 ; 15 = @r{0000...001111}
@end group
@end smallexample
@end defun
@smallexample
@group
- ; @r{ 29-bit binary values}
+ ; @r{ 30-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}
+(logxor 12 5) ; 12 = @r{0000...001100}
+ ; 5 = @r{0000...000101}
+ @result{} 9 ; 9 = @r{0000...001001}
@end group
@group
-(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}
+(logxor 12 5 7) ; 12 = @r{0000...001100}
+ ; 5 = @r{0000...000101}
+ ; 7 = @r{0000...000111}
+ @result{} 14 ; 14 = @r{0000...001110}
@end group
@end smallexample
@end defun
@example
(lognot 5)
@result{} -6
-;; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101}
+;; 5 = @r{0000...000101} (30 bits total)
;; @r{becomes}
-;; -6 = @r{1 1111 1111 1111 1111 1111 1111 1010}
+;; -6 = @r{1111...111010} (30 bits total)
@end example
@end defun
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
-
-@ignore
- arch-tag: 574e8dd2-d513-4616-9844-c9a27869782e
-@end ignore
HCoop / bpt / emacs.git / blobdiff