X-Git-Url: https://code.delx.au/gnu-emacs/blobdiff_plain/114f9c96795aff3b51b9060d7c9c1b77debcc99a..ec1b9b17fa69a4ac051f64e21c85660d127c23a5:/doc/lispref/numbers.texi diff --git a/doc/lispref/numbers.texi b/doc/lispref/numbers.texi index a23721f1a8..e83da348e0 100644 --- a/doc/lispref/numbers.texi +++ b/doc/lispref/numbers.texi @@ -1,7 +1,8 @@ @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, 2009, 2010 Free Software Foundation, Inc. +@c 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 +@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 @@ -20,10 +21,10 @@ exact; they have a fixed, limited amount of precision. @menu * Integer Basics:: Representation and range of integers. -* Float Basics:: Representation and range of floating point. +* 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. @@ -36,22 +37,22 @@ exact; they have a fixed, limited amount of precision. @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. +chapter assume an integer has 30 bits. @cindex overflow The Lisp reader reads an integer as a sequence of digits with optional @@ -62,7 +63,7 @@ initial sign and optional final period. 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{Also the integer 1, due to overflow.} 0 ; @r{The integer 0.} -0 ; @r{The integer 0.} @end example @@ -93,10 +94,10 @@ from 2 to 36. For 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 +00 0000 0000 0000 0000 0000 0000 0101 @end example @noindent @@ -106,12 +107,12 @@ between groups of 8 bits, to make the binary integer easier to read.) The integer @minus{}1 looks like this: @example -1 1111 1111 1111 1111 1111 1111 1111 +11 1111 1111 1111 1111 1111 1111 1111 @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 @@ -119,24 +120,24 @@ complement} notation.) @minus{}5 looks like this: @example -1 1111 1111 1111 1111 1111 1111 1011 +11 1111 1111 1111 1111 1111 1111 1011 @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 +01 1111 1111 1111 1111 1111 1111 1111 @end example Since the arithmetic functions do not check whether integers go -outside their range, when you add 1 to 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{} 10 0000 0000 0000 0000 0000 0000 0000 @end example Many of the functions described in this chapter accept markers for @@ -223,6 +224,14 @@ down to an integer. @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 @@ -820,19 +829,19 @@ 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: +536,870,911 produces @minus{}2 on a 30-bit machine: @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, in the 30-bit implementation, 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} +01 1111 1111 1111 1111 1111 1111 1111 @end group @end example @@ -842,7 +851,7 @@ which becomes the following when left shifted: @example @group ;; @r{Decimal @minus{}2} -1 1111 1111 1111 1111 1111 1111 1110 +11 1111 1111 1111 1111 1111 1111 1110 @end group @end example @end defun @@ -865,9 +874,9 @@ looks like this: @group (ash -6 -1) @result{} -3 ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} -1 1111 1111 1111 1111 1111 1111 1010 +11 1111 1111 1111 1111 1111 1111 1010 @result{} -1 1111 1111 1111 1111 1111 1111 1101 +11 1111 1111 1111 1111 1111 1111 1101 @end group @end example @@ -876,11 +885,11 @@ In contrast, shifting the pattern of bits one place to the right with @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.} +11 1111 1111 1111 1111 1111 1111 1010 @result{} -0 1111 1111 1111 1111 1111 1111 1101 +01 1111 1111 1111 1111 1111 1111 1101 @end group @end example @@ -890,34 +899,34 @@ Here are other examples: @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{00 0000 0000 0000 0000 0000 0000 0101} + @result{} 20 ; = @r{00 0000 0000 0000 0000 0000 0001 0100} @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{11 1111 1111 1111 1111 1111 1111 1011} + @result{} -20 ; = @r{11 1111 1111 1111 1111 1111 1110 1100} (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{00 0000 0000 0000 0000 0000 0000 0101} + @result{} 1 ; = @r{00 0000 0000 0000 0000 0000 0000 0001} @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{11 1111 1111 1111 1111 1111 1111 1011} + @result{} 268435454 ; = @r{00 0111 1111 1111 1111 1111 1111 1110} @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{11 1111 1111 1111 1111 1111 1111 1011} + @result{} -2 ; = @r{11 1111 1111 1111 1111 1111 1111 1110} @end group @end smallexample @end defun @@ -952,23 +961,23 @@ because its binary representation consists entirely of ones. If @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{00 0000 0000 0000 0000 0000 0000 1110} + ; 13 = @r{00 0000 0000 0000 0000 0000 0000 1101} + @result{} 12 ; 12 = @r{00 0000 0000 0000 0000 0000 0000 1100} @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{00 0000 0000 0000 0000 0000 0000 1110} + ; 13 = @r{00 0000 0000 0000 0000 0000 0000 1101} + ; 4 = @r{00 0000 0000 0000 0000 0000 0000 0100} + @result{} 4 ; 4 = @r{00 0000 0000 0000 0000 0000 0000 0100} @end group @group (logand) - @result{} -1 ; -1 = @r{1 1111 1111 1111 1111 1111 1111 1111} + @result{} -1 ; -1 = @r{11 1111 1111 1111 1111 1111 1111 1111} @end group @end smallexample @end defun @@ -982,18 +991,18 @@ passed just one argument, it returns that argument. @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{00 0000 0000 0000 0000 0000 0000 1100} + ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} + @result{} 13 ; 13 = @r{00 0000 0000 0000 0000 0000 0000 1101} @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{00 0000 0000 0000 0000 0000 0000 1100} + ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} + ; 7 = @r{00 0000 0000 0000 0000 0000 0000 0111} + @result{} 15 ; 15 = @r{00 0000 0000 0000 0000 0000 0000 1111} @end group @end smallexample @end defun @@ -1007,18 +1016,18 @@ result is 0, which is an identity element for this operation. If @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{00 0000 0000 0000 0000 0000 0000 1100} + ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} + @result{} 9 ; 9 = @r{00 0000 0000 0000 0000 0000 0000 1001} @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{00 0000 0000 0000 0000 0000 0000 1100} + ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} + ; 7 = @r{00 0000 0000 0000 0000 0000 0000 0111} + @result{} 14 ; 14 = @r{00 0000 0000 0000 0000 0000 0000 1110} @end group @end smallexample @end defun @@ -1031,9 +1040,9 @@ bit is one in the result if, and only if, the @var{n}th bit is zero in @example (lognot 5) @result{} -6 -;; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} +;; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} ;; @r{becomes} -;; -6 = @r{1 1111 1111 1111 1111 1111 1111 1010} +;; -6 = @r{11 1111 1111 1111 1111 1111 1111 1010} @end example @end defun