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error. (Float Basics): Use `(/ 0.0 0.0)' instead of `(sqrt -1.0)'.
1195 lines
36 KiB
Plaintext
1195 lines
36 KiB
Plaintext
@c -*-texinfo-*-
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@c This is part of the GNU Emacs Lisp Reference Manual.
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@c Copyright (C) 1990, 1991, 1992, 1993, 1994, 1995, 1998, 1999, 2003
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@c Free Software Foundation, Inc.
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@c See the file elisp.texi for copying conditions.
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@setfilename ../info/numbers
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@node Numbers, Strings and Characters, Lisp Data Types, Top
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@chapter Numbers
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@cindex integers
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@cindex numbers
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GNU Emacs supports two numeric data types: @dfn{integers} and
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@dfn{floating point numbers}. Integers are whole numbers such as
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@minus{}3, 0, 7, 13, and 511. Their values are exact. Floating point
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numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or
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2.71828. They can also be expressed in exponential notation: 1.5e2
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equals 150; in this example, @samp{e2} stands for ten to the second
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power, and that is multiplied by 1.5. Floating point values are not
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exact; they have a fixed, limited amount of precision.
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@menu
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* Integer Basics:: Representation and range of integers.
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* Float Basics:: Representation and range of floating point.
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* Predicates on Numbers:: Testing for numbers.
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* Comparison of Numbers:: Equality and inequality predicates.
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* Numeric Conversions:: Converting float to integer and vice versa.
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* Arithmetic Operations:: How to add, subtract, multiply and divide.
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* Rounding Operations:: Explicitly rounding floating point numbers.
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* Bitwise Operations:: Logical and, or, not, shifting.
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* Math Functions:: Trig, exponential and logarithmic functions.
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* Random Numbers:: Obtaining random integers, predictable or not.
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@end menu
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@node Integer Basics
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@comment node-name, next, previous, up
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@section Integer Basics
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The range of values for an integer depends on the machine. The
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minimum range is @minus{}268435456 to 268435455 (29 bits; i.e.,
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@ifnottex
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-2**28
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@end ifnottex
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@tex
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@math{-2^{28}}
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@end tex
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to
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@ifnottex
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2**28 - 1),
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@end ifnottex
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@tex
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@math{2^{28}-1}),
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@end tex
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but some machines may provide a wider range. Many examples in this
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chapter assume an integer has 29 bits.
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@cindex overflow
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The Lisp reader reads an integer as a sequence of digits with optional
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initial sign and optional final period.
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@example
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1 ; @r{The integer 1.}
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1. ; @r{The integer 1.}
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+1 ; @r{Also the integer 1.}
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-1 ; @r{The integer @minus{}1.}
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536870913 ; @r{Also the integer 1, due to overflow.}
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0 ; @r{The integer 0.}
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-0 ; @r{The integer 0.}
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@end example
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@cindex integers in specific radix
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@cindex radix for reading an integer
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@cindex base for reading an integer
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@cindex hex numbers
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@cindex octal numbers
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@cindex reading numbers in hex, octal, and binary
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In addition, the Lisp reader recognizes a syntax for integers in
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bases other than 10: @samp{#B@var{integer}} reads @var{integer} in
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binary (radix 2), @samp{#O@var{integer}} reads @var{integer} in octal
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(radix 8), @samp{#X@var{integer}} reads @var{integer} in hexadecimal
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(radix 16), and @samp{#@var{radix}r@var{integer}} reads @var{integer}
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in radix @var{radix} (where @var{radix} is between 2 and 36,
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inclusively). Case is not significant for the letter after @samp{#}
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(@samp{B}, @samp{O}, etc.) that denotes the radix.
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To understand how various functions work on integers, especially the
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bitwise operators (@pxref{Bitwise Operations}), it is often helpful to
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view the numbers in their binary form.
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In 29-bit binary, the decimal integer 5 looks like this:
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@example
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0 0000 0000 0000 0000 0000 0000 0101
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@end example
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@noindent
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(We have inserted spaces between groups of 4 bits, and two spaces
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between groups of 8 bits, to make the binary integer easier to read.)
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The integer @minus{}1 looks like this:
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@example
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1 1111 1111 1111 1111 1111 1111 1111
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@end example
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@noindent
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@cindex two's complement
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@minus{}1 is represented as 29 ones. (This is called @dfn{two's
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complement} notation.)
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The negative integer, @minus{}5, is creating by subtracting 4 from
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@minus{}1. In binary, the decimal integer 4 is 100. Consequently,
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@minus{}5 looks like this:
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@example
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1 1111 1111 1111 1111 1111 1111 1011
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@end example
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In this implementation, the largest 29-bit binary integer value is
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268,435,455 in decimal. In binary, it looks like this:
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@example
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0 1111 1111 1111 1111 1111 1111 1111
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@end example
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Since the arithmetic functions do not check whether integers go
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outside their range, when you add 1 to 268,435,455, the value is the
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negative integer @minus{}268,435,456:
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@example
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(+ 1 268435455)
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@result{} -268435456
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@result{} 1 0000 0000 0000 0000 0000 0000 0000
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@end example
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Many of the functions described in this chapter accept markers for
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arguments in place of numbers. (@xref{Markers}.) Since the actual
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arguments to such functions may be either numbers or markers, we often
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give these arguments the name @var{number-or-marker}. When the argument
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value is a marker, its position value is used and its buffer is ignored.
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@defvar most-positive-fixnum
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The value of this variable is the largest integer that Emacs Lisp
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can handle.
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@end defvar
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@defvar most-negative-fixnum
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The value of this variable is the smallest integer that Emacs Lisp can
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handle. It is negative.
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@end defvar
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@node Float Basics
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@section Floating Point Basics
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Floating point numbers are useful for representing numbers that are
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not integral. The precise range of floating point numbers is
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machine-specific; it is the same as the range of the C data type
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@code{double} on the machine you are using.
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The read-syntax for floating point numbers requires either a decimal
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point (with at least one digit following), an exponent, or both. For
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example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and
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@samp{.15e4} are five ways of writing a floating point number whose
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value is 1500. They are all equivalent. You can also use a minus sign
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to write negative floating point numbers, as in @samp{-1.0}.
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@cindex @acronym{IEEE} floating point
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@cindex positive infinity
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@cindex negative infinity
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@cindex infinity
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@cindex NaN
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Most modern computers support the @acronym{IEEE} floating point standard,
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which provides for positive infinity and negative infinity as floating point
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values. It also provides for a class of values called NaN or
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``not-a-number''; numerical functions return such values in cases where
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there is no correct answer. For example, @code{(/ 0.0 0.0)} returns a
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NaN. For practical purposes, there's no significant difference between
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different NaN values in Emacs Lisp, and there's no rule for precisely
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which NaN value should be used in a particular case, so Emacs Lisp
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doesn't try to distinguish them. Here are the read syntaxes for
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these special floating point values:
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@table @asis
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@item positive infinity
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@samp{1.0e+INF}
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@item negative infinity
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@samp{-1.0e+INF}
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@item Not-a-number
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@samp{0.0e+NaN}.
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@end table
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In addition, the value @code{-0.0} is distinguishable from ordinary
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zero in @acronym{IEEE} floating point (although @code{equal} and
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@code{=} consider them equal values).
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You can use @code{logb} to extract the binary exponent of a floating
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point number (or estimate the logarithm of an integer):
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@defun logb number
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This function returns the binary exponent of @var{number}. More
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precisely, the value is the logarithm of @var{number} base 2, rounded
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down to an integer.
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@example
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(logb 10)
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@result{} 3
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(logb 10.0e20)
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@result{} 69
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@end example
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@end defun
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@node Predicates on Numbers
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@section Type Predicates for Numbers
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The functions in this section test whether the argument is a number or
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whether it is a certain sort of number. The functions @code{integerp}
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and @code{floatp} can take any type of Lisp object as argument (the
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predicates would not be of much use otherwise); but the @code{zerop}
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predicate requires a number as its argument. See also
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@code{integer-or-marker-p} and @code{number-or-marker-p}, in
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@ref{Predicates on Markers}.
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@defun floatp object
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This predicate tests whether its argument is a floating point
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number and returns @code{t} if so, @code{nil} otherwise.
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@code{floatp} does not exist in Emacs versions 18 and earlier.
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@end defun
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@defun integerp object
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This predicate tests whether its argument is an integer, and returns
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@code{t} if so, @code{nil} otherwise.
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@end defun
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@defun numberp object
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This predicate tests whether its argument is a number (either integer or
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floating point), and returns @code{t} if so, @code{nil} otherwise.
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@end defun
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@defun wholenump object
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@cindex natural numbers
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The @code{wholenump} predicate (whose name comes from the phrase
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``whole-number-p'') tests to see whether its argument is a nonnegative
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integer, and returns @code{t} if so, @code{nil} otherwise. 0 is
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considered non-negative.
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@findex natnump
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@code{natnump} is an obsolete synonym for @code{wholenump}.
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@end defun
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@defun zerop number
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This predicate tests whether its argument is zero, and returns @code{t}
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if so, @code{nil} otherwise. The argument must be a number.
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These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}.
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@end defun
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@node Comparison of Numbers
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@section Comparison of Numbers
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@cindex number equality
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To test numbers for numerical equality, you should normally use
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@code{=}, not @code{eq}. There can be many distinct floating point
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number objects with the same numeric value. If you use @code{eq} to
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compare them, then you test whether two values are the same
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@emph{object}. By contrast, @code{=} compares only the numeric values
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of the objects.
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At present, each integer value has a unique Lisp object in Emacs Lisp.
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Therefore, @code{eq} is equivalent to @code{=} where integers are
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concerned. It is sometimes convenient to use @code{eq} for comparing an
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unknown value with an integer, because @code{eq} does not report an
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error if the unknown value is not a number---it accepts arguments of any
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type. By contrast, @code{=} signals an error if the arguments are not
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numbers or markers. However, it is a good idea to use @code{=} if you
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can, even for comparing integers, just in case we change the
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representation of integers in a future Emacs version.
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Sometimes it is useful to compare numbers with @code{equal}; it treats
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two numbers as equal if they have the same data type (both integers, or
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both floating point) and the same value. By contrast, @code{=} can
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treat an integer and a floating point number as equal.
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There is another wrinkle: because floating point arithmetic is not
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exact, it is often a bad idea to check for equality of two floating
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point values. Usually it is better to test for approximate equality.
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Here's a function to do this:
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@example
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(defvar fuzz-factor 1.0e-6)
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(defun approx-equal (x y)
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(or (and (= x 0) (= y 0))
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(< (/ (abs (- x y))
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(max (abs x) (abs y)))
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fuzz-factor)))
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@end example
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@cindex CL note---integers vrs @code{eq}
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@quotation
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@b{Common Lisp note:} Comparing numbers in Common Lisp always requires
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@code{=} because Common Lisp implements multi-word integers, and two
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distinct integer objects can have the same numeric value. Emacs Lisp
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can have just one integer object for any given value because it has a
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limited range of integer values.
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@end quotation
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@defun = number-or-marker1 number-or-marker2
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This function tests whether its arguments are numerically equal, and
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returns @code{t} if so, @code{nil} otherwise.
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@end defun
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@defun /= number-or-marker1 number-or-marker2
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This function tests whether its arguments are numerically equal, and
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returns @code{t} if they are not, and @code{nil} if they are.
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@end defun
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@defun < number-or-marker1 number-or-marker2
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This function tests whether its first argument is strictly less than
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its second argument. It returns @code{t} if so, @code{nil} otherwise.
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@end defun
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@defun <= number-or-marker1 number-or-marker2
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This function tests whether its first argument is less than or equal
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to its second argument. It returns @code{t} if so, @code{nil}
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otherwise.
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@end defun
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@defun > number-or-marker1 number-or-marker2
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This function tests whether its first argument is strictly greater
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than its second argument. It returns @code{t} if so, @code{nil}
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otherwise.
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@end defun
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@defun >= number-or-marker1 number-or-marker2
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This function tests whether its first argument is greater than or
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equal to its second argument. It returns @code{t} if so, @code{nil}
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otherwise.
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@end defun
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@defun max number-or-marker &rest numbers-or-markers
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This function returns the largest of its arguments.
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If any of the argument is floating-point, the value is returned
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as floating point, even if it was given as an integer.
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@example
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(max 20)
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@result{} 20
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(max 1 2.5)
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@result{} 2.5
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(max 1 3 2.5)
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@result{} 3.0
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@end example
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@end defun
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@defun min number-or-marker &rest numbers-or-markers
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This function returns the smallest of its arguments.
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If any of the argument is floating-point, the value is returned
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as floating point, even if it was given as an integer.
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@example
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(min -4 1)
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@result{} -4
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@end example
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@end defun
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@defun abs number
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This function returns the absolute value of @var{number}.
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@end defun
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@node Numeric Conversions
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@section Numeric Conversions
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@cindex rounding in conversions
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To convert an integer to floating point, use the function @code{float}.
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@defun float number
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This returns @var{number} converted to floating point.
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If @var{number} is already a floating point number, @code{float} returns
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it unchanged.
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@end defun
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There are four functions to convert floating point numbers to integers;
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they differ in how they round. All accept an argument @var{number}
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and an optional argument @var{divisor}. Both arguments may be
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integers or floating point numbers. @var{divisor} may also be
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@code{nil}. If @var{divisor} is @code{nil} or omitted, these
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functions convert @var{number} to an integer, or return it unchanged
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if it already is an integer. If @var{divisor} is non-@code{nil}, they
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divide @var{number} by @var{divisor} and convert the result to an
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integer. An @code{arith-error} results if @var{divisor} is 0.
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@defun truncate number &optional divisor
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This returns @var{number}, converted to an integer by rounding towards
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zero.
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@example
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(truncate 1.2)
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@result{} 1
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(truncate 1.7)
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@result{} 1
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(truncate -1.2)
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@result{} -1
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(truncate -1.7)
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@result{} -1
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@end example
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@end defun
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@defun floor number &optional divisor
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This returns @var{number}, converted to an integer by rounding downward
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(towards negative infinity).
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If @var{divisor} is specified, this uses the kind of division
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operation that corresponds to @code{mod}, rounding downward.
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@example
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(floor 1.2)
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@result{} 1
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(floor 1.7)
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@result{} 1
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(floor -1.2)
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@result{} -2
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(floor -1.7)
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@result{} -2
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(floor 5.99 3)
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@result{} 1
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@end example
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@end defun
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@defun ceiling number &optional divisor
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This returns @var{number}, converted to an integer by rounding upward
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(towards positive infinity).
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@example
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(ceiling 1.2)
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@result{} 2
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(ceiling 1.7)
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@result{} 2
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(ceiling -1.2)
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@result{} -1
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(ceiling -1.7)
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@result{} -1
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@end example
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@end defun
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@defun round number &optional divisor
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This returns @var{number}, converted to an integer by rounding towards the
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nearest integer. Rounding a value equidistant between two integers
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may choose the integer closer to zero, or it may prefer an even integer,
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depending on your machine.
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@example
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(round 1.2)
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@result{} 1
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(round 1.7)
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@result{} 2
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(round -1.2)
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@result{} -1
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(round -1.7)
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@result{} -2
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@end example
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@end defun
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@node Arithmetic Operations
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@section Arithmetic Operations
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Emacs Lisp provides the traditional four arithmetic operations:
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addition, subtraction, multiplication, and division. Remainder and modulus
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functions supplement the division functions. The functions to
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add or subtract 1 are provided because they are traditional in Lisp and
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commonly used.
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All of these functions except @code{%} return a floating point value
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if any argument is floating.
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It is important to note that in Emacs Lisp, arithmetic functions
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do not check for overflow. Thus @code{(1+ 268435455)} may evaluate to
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@minus{}268435456, depending on your hardware.
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@defun 1+ number-or-marker
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This function returns @var{number-or-marker} plus 1.
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For example,
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@example
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(setq foo 4)
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@result{} 4
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(1+ foo)
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@result{} 5
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@end example
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This function is not analogous to the C operator @code{++}---it does not
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increment a variable. It just computes a sum. Thus, if we continue,
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@example
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foo
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@result{} 4
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@end example
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If you want to increment the variable, you must use @code{setq},
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like this:
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@example
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(setq foo (1+ foo))
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@result{} 5
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@end example
|
|
@end defun
|
|
|
|
@defun 1- number-or-marker
|
|
This function returns @var{number-or-marker} minus 1.
|
|
@end defun
|
|
|
|
@defun + &rest numbers-or-markers
|
|
This function adds its arguments together. When given no arguments,
|
|
@code{+} returns 0.
|
|
|
|
@example
|
|
(+)
|
|
@result{} 0
|
|
(+ 1)
|
|
@result{} 1
|
|
(+ 1 2 3 4)
|
|
@result{} 10
|
|
@end example
|
|
@end defun
|
|
|
|
@defun - &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.
|
|
|
|
@example
|
|
(- 10 1 2 3 4)
|
|
@result{} 0
|
|
(- 10)
|
|
@result{} -10
|
|
(-)
|
|
@result{} 0
|
|
@end example
|
|
@end defun
|
|
|
|
@defun * &rest numbers-or-markers
|
|
This function multiplies its arguments together, and returns the
|
|
product. When given no arguments, @code{*} returns 1.
|
|
|
|
@example
|
|
(*)
|
|
@result{} 1
|
|
(* 1)
|
|
@result{} 1
|
|
(* 1 2 3 4)
|
|
@result{} 24
|
|
@end example
|
|
@end defun
|
|
|
|
@defun / dividend divisor &rest divisors
|
|
This function divides @var{dividend} by @var{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
|
|
@code{/} is implemented using the C division operator, which also
|
|
permits machine-dependent rounding. As a practical matter, all known
|
|
machines round in the standard fashion.
|
|
|
|
@cindex @code{arith-error} in division
|
|
If you divide 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.
|
|
|
|
@example
|
|
@group
|
|
(/ 6 2)
|
|
@result{} 3
|
|
@end 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
|
|
(/ -17 6)
|
|
@result{} -2
|
|
@end example
|
|
|
|
The result of @code{(/ -17 6)} could in principle be -3 on some
|
|
machines.
|
|
@end defun
|
|
|
|
@defun % dividend divisor
|
|
@cindex remainder
|
|
This function returns the integer remainder after division of @var{dividend}
|
|
by @var{divisor}. The arguments must be integers or markers.
|
|
|
|
For negative arguments, the remainder is in principle machine-dependent
|
|
since the quotient is; but in practice, all known machines behave alike.
|
|
|
|
An @code{arith-error} results if @var{divisor} is 0.
|
|
|
|
@example
|
|
(% 9 4)
|
|
@result{} 1
|
|
(% -9 4)
|
|
@result{} -1
|
|
(% 9 -4)
|
|
@result{} 1
|
|
(% -9 -4)
|
|
@result{} -1
|
|
@end example
|
|
|
|
For any two integers @var{dividend} and @var{divisor},
|
|
|
|
@example
|
|
@group
|
|
(+ (% @var{dividend} @var{divisor})
|
|
(* (/ @var{dividend} @var{divisor}) @var{divisor}))
|
|
@end group
|
|
@end example
|
|
|
|
@noindent
|
|
always equals @var{dividend}.
|
|
@end defun
|
|
|
|
@defun mod dividend divisor
|
|
@cindex modulus
|
|
This function returns the value of @var{dividend} modulo @var{divisor};
|
|
in other words, the remainder after division of @var{dividend}
|
|
by @var{divisor}, but with the same sign as @var{divisor}.
|
|
The arguments must be numbers or markers.
|
|
|
|
Unlike @code{%}, @code{mod} returns a well-defined result for negative
|
|
arguments. It also permits floating point arguments; it rounds the
|
|
quotient downward (towards minus infinity) to an integer, and uses that
|
|
quotient to compute the remainder.
|
|
|
|
An @code{arith-error} results if @var{divisor} is 0.
|
|
|
|
@example
|
|
@group
|
|
(mod 9 4)
|
|
@result{} 1
|
|
@end group
|
|
@group
|
|
(mod -9 4)
|
|
@result{} 3
|
|
@end group
|
|
@group
|
|
(mod 9 -4)
|
|
@result{} -3
|
|
@end group
|
|
@group
|
|
(mod -9 -4)
|
|
@result{} -1
|
|
@end group
|
|
@group
|
|
(mod 5.5 2.5)
|
|
@result{} .5
|
|
@end group
|
|
@end example
|
|
|
|
For any two numbers @var{dividend} and @var{divisor},
|
|
|
|
@example
|
|
@group
|
|
(+ (mod @var{dividend} @var{divisor})
|
|
(* (floor @var{dividend} @var{divisor}) @var{divisor}))
|
|
@end group
|
|
@end example
|
|
|
|
@noindent
|
|
always equals @var{dividend}, subject to rounding error if either
|
|
argument is floating point. For @code{floor}, see @ref{Numeric
|
|
Conversions}.
|
|
@end defun
|
|
|
|
@node Rounding Operations
|
|
@section Rounding Operations
|
|
@cindex rounding without conversion
|
|
|
|
The functions @code{ffloor}, @code{fceiling}, @code{fround}, and
|
|
@code{ftruncate} take a floating point argument and return a floating
|
|
point result whose value is a nearby integer. @code{ffloor} returns the
|
|
nearest integer below; @code{fceiling}, the nearest integer above;
|
|
@code{ftruncate}, the nearest integer in the direction towards zero;
|
|
@code{fround}, the nearest integer.
|
|
|
|
@defun ffloor float
|
|
This function rounds @var{float} to the next lower integral value, and
|
|
returns that value as a floating point number.
|
|
@end defun
|
|
|
|
@defun fceiling float
|
|
This function rounds @var{float} to the next higher integral value, and
|
|
returns that value as a floating point number.
|
|
@end defun
|
|
|
|
@defun ftruncate float
|
|
This function rounds @var{float} towards zero to an integral value, and
|
|
returns that value as a floating point number.
|
|
@end defun
|
|
|
|
@defun fround float
|
|
This function rounds @var{float} to the nearest integral value,
|
|
and returns that value as a floating point number.
|
|
@end defun
|
|
|
|
@node Bitwise Operations
|
|
@section Bitwise Operations on Integers
|
|
|
|
In a computer, an integer is represented as a binary number, a
|
|
sequence of @dfn{bits} (digits which are either zero or one). A bitwise
|
|
operation acts on the individual bits of such a sequence. For example,
|
|
@dfn{shifting} moves the whole sequence left or right one or more places,
|
|
reproducing the same pattern ``moved over''.
|
|
|
|
The bitwise operations in Emacs Lisp apply only to integers.
|
|
|
|
@defun lsh integer1 count
|
|
@cindex logical shift
|
|
@code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the
|
|
bits in @var{integer1} to the left @var{count} places, or to the right
|
|
if @var{count} is negative, bringing zeros into the vacated bits. If
|
|
@var{count} is negative, @code{lsh} shifts zeros into the leftmost
|
|
(most-significant) bit, producing a positive result even if
|
|
@var{integer1} 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.
|
|
|
|
@example
|
|
@group
|
|
(lsh 5 1)
|
|
@result{} 10
|
|
;; @r{Decimal 5 becomes decimal 10.}
|
|
00000101 @result{} 00001010
|
|
|
|
(lsh 7 1)
|
|
@result{} 14
|
|
;; @r{Decimal 7 becomes decimal 14.}
|
|
00000111 @result{} 00001110
|
|
@end group
|
|
@end example
|
|
|
|
@noindent
|
|
As the examples illustrate, shifting the pattern of bits one place to
|
|
the left produces a number that is twice the value of the previous
|
|
number.
|
|
|
|
Shifting a pattern of bits two places to the left produces results
|
|
like this (with 8-bit binary numbers):
|
|
|
|
@example
|
|
@group
|
|
(lsh 3 2)
|
|
@result{} 12
|
|
;; @r{Decimal 3 becomes decimal 12.}
|
|
00000011 @result{} 00001100
|
|
@end group
|
|
@end example
|
|
|
|
On the other hand, shifting one place to the right looks like this:
|
|
|
|
@example
|
|
@group
|
|
(lsh 6 -1)
|
|
@result{} 3
|
|
;; @r{Decimal 6 becomes decimal 3.}
|
|
00000110 @result{} 00000011
|
|
@end group
|
|
|
|
@group
|
|
(lsh 5 -1)
|
|
@result{} 2
|
|
;; @r{Decimal 5 becomes decimal 2.}
|
|
00000101 @result{} 00000010
|
|
@end group
|
|
@end example
|
|
|
|
@noindent
|
|
As the example illustrates, shifting 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:
|
|
|
|
@example
|
|
(lsh 268435455 1) ; @r{left shift}
|
|
@result{} -2
|
|
@end example
|
|
|
|
In binary, in the 29-bit implementation, the argument looks like this:
|
|
|
|
@example
|
|
@group
|
|
;; @r{Decimal 268,435,455}
|
|
0 1111 1111 1111 1111 1111 1111 1111
|
|
@end group
|
|
@end example
|
|
|
|
@noindent
|
|
which becomes the following when left shifted:
|
|
|
|
@example
|
|
@group
|
|
;; @r{Decimal @minus{}2}
|
|
1 1111 1111 1111 1111 1111 1111 1110
|
|
@end group
|
|
@end example
|
|
@end defun
|
|
|
|
@defun ash integer1 count
|
|
@cindex arithmetic shift
|
|
@code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1}
|
|
to the left @var{count} places, or to the right if @var{count}
|
|
is negative.
|
|
|
|
@code{ash} gives the same results as @code{lsh} except when
|
|
@var{integer1} and @var{count} are both negative. In that case,
|
|
@code{ash} puts ones in the empty bit positions on the left, while
|
|
@code{lsh} puts zeros in those bit positions.
|
|
|
|
Thus, with @code{ash}, shifting the pattern of bits one place to the right
|
|
looks like this:
|
|
|
|
@example
|
|
@group
|
|
(ash -6 -1) @result{} -3
|
|
;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
|
|
1 1111 1111 1111 1111 1111 1111 1010
|
|
@result{}
|
|
1 1111 1111 1111 1111 1111 1111 1101
|
|
@end group
|
|
@end example
|
|
|
|
In contrast, shifting the pattern of bits one place to the right with
|
|
@code{lsh} looks like this:
|
|
|
|
@example
|
|
@group
|
|
(lsh -6 -1) @result{} 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
|
|
@end group
|
|
@end example
|
|
|
|
Here are other examples:
|
|
|
|
@c !!! Check if lined up in smallbook format! XDVI shows problem
|
|
@c with smallbook but not with regular book! --rjc 16mar92
|
|
@smallexample
|
|
@group
|
|
; @r{ 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}
|
|
@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}
|
|
(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}
|
|
@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}
|
|
@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}
|
|
@end group
|
|
@end smallexample
|
|
@end defun
|
|
|
|
@defun logand &rest ints-or-markers
|
|
@cindex logical and
|
|
@cindex bitwise and
|
|
This function returns the ``logical and'' of the arguments: the
|
|
@var{n}th bit is set in the result if, and only if, the @var{n}th bit is
|
|
set in all the arguments. (``Set'' means that the value of the bit is 1
|
|
rather than 0.)
|
|
|
|
For example, using 4-bit binary numbers, the ``logical and'' of 13 and
|
|
12 is 12: 1101 combined with 1100 produces 1100.
|
|
In both the binary numbers, the leftmost two bits are set (i.e., they
|
|
are 1's), so the leftmost two bits of the returned value are set.
|
|
However, for the rightmost two bits, each is zero in at least one of
|
|
the arguments, so the rightmost two bits of the returned value are 0's.
|
|
|
|
@noindent
|
|
Therefore,
|
|
|
|
@example
|
|
@group
|
|
(logand 13 12)
|
|
@result{} 12
|
|
@end group
|
|
@end example
|
|
|
|
If @code{logand} is not passed any argument, it returns a value of
|
|
@minus{}1. This number is an identity element for @code{logand}
|
|
because its binary representation consists entirely of ones. If
|
|
@code{logand} is passed just one argument, it returns that argument.
|
|
|
|
@smallexample
|
|
@group
|
|
; @r{ 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}
|
|
@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}
|
|
@end group
|
|
|
|
@group
|
|
(logand)
|
|
@result{} -1 ; -1 = @r{1 1111 1111 1111 1111 1111 1111 1111}
|
|
@end group
|
|
@end smallexample
|
|
@end defun
|
|
|
|
@defun logior &rest ints-or-markers
|
|
@cindex logical inclusive or
|
|
@cindex bitwise or
|
|
This function returns the ``inclusive or'' of its arguments: the @var{n}th bit
|
|
is set in the result if, and only if, the @var{n}th bit is set in at least
|
|
one of the arguments. If there are no arguments, the result is zero,
|
|
which is an identity element for this operation. If @code{logior} is
|
|
passed just one argument, it returns that argument.
|
|
|
|
@smallexample
|
|
@group
|
|
; @r{ 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}
|
|
@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}
|
|
@end group
|
|
@end smallexample
|
|
@end defun
|
|
|
|
@defun logxor &rest ints-or-markers
|
|
@cindex bitwise exclusive or
|
|
@cindex logical exclusive or
|
|
This function returns the ``exclusive or'' of its arguments: the
|
|
@var{n}th bit is set in the result if, and only if, the @var{n}th bit is
|
|
set in an odd number of the arguments. If there are no arguments, the
|
|
result is 0, which is an identity element for this operation. If
|
|
@code{logxor} is passed just one argument, it returns that argument.
|
|
|
|
@smallexample
|
|
@group
|
|
; @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}
|
|
@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}
|
|
@end group
|
|
@end smallexample
|
|
@end defun
|
|
|
|
@defun lognot integer
|
|
@cindex logical not
|
|
@cindex bitwise not
|
|
This function returns the logical complement of its argument: the @var{n}th
|
|
bit is one in the result if, and only if, the @var{n}th bit is zero in
|
|
@var{integer}, and vice-versa.
|
|
|
|
@example
|
|
(lognot 5)
|
|
@result{} -6
|
|
;; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101}
|
|
;; @r{becomes}
|
|
;; -6 = @r{1 1111 1111 1111 1111 1111 1111 1010}
|
|
@end example
|
|
@end defun
|
|
|
|
@node Math Functions
|
|
@section Standard Mathematical Functions
|
|
@cindex transcendental functions
|
|
@cindex mathematical functions
|
|
|
|
These mathematical functions allow integers as well as floating point
|
|
numbers as arguments.
|
|
|
|
@defun sin arg
|
|
@defunx cos arg
|
|
@defunx tan arg
|
|
These are the ordinary trigonometric functions, with argument measured
|
|
in radians.
|
|
@end defun
|
|
|
|
@defun asin arg
|
|
The value of @code{(asin @var{arg})} is a number between
|
|
@ifnottex
|
|
@minus{}pi/2
|
|
@end ifnottex
|
|
@tex
|
|
@math{-\pi/2}
|
|
@end tex
|
|
and
|
|
@ifnottex
|
|
pi/2
|
|
@end ifnottex
|
|
@tex
|
|
@math{\pi/2}
|
|
@end tex
|
|
(inclusive) whose sine is @var{arg}; if, however, @var{arg} is out of
|
|
range (outside [-1, 1]), it signals a @code{domain-error} error.
|
|
@end defun
|
|
|
|
@defun acos arg
|
|
The value of @code{(acos @var{arg})} is a number between 0 and
|
|
@ifnottex
|
|
pi
|
|
@end ifnottex
|
|
@tex
|
|
@math{\pi}
|
|
@end tex
|
|
(inclusive) whose cosine is @var{arg}; if, however, @var{arg} is out
|
|
of range (outside [-1, 1]), it signals a @code{domain-error} error.
|
|
@end defun
|
|
|
|
@defun atan y &optional x
|
|
The value of @code{(atan @var{y})} is a number between
|
|
@ifnottex
|
|
@minus{}pi/2
|
|
@end ifnottex
|
|
@tex
|
|
@math{-\pi/2}
|
|
@end tex
|
|
and
|
|
@ifnottex
|
|
pi/2
|
|
@end ifnottex
|
|
@tex
|
|
@math{\pi/2}
|
|
@end 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 @code{[@var{x}, @var{y}]} and the
|
|
@code{X} axis.
|
|
@end defun
|
|
|
|
@defun exp arg
|
|
This is the exponential function; it returns
|
|
@tex
|
|
@math{e}
|
|
@end tex
|
|
@ifnottex
|
|
@i{e}
|
|
@end ifnottex
|
|
to the power @var{arg}.
|
|
@tex
|
|
@math{e}
|
|
@end tex
|
|
@ifnottex
|
|
@i{e}
|
|
@end ifnottex
|
|
is a fundamental mathematical constant also called the base of natural
|
|
logarithms.
|
|
@end defun
|
|
|
|
@defun log arg &optional base
|
|
This function returns the logarithm of @var{arg}, with base @var{base}.
|
|
If you don't specify @var{base}, the base
|
|
@tex
|
|
@math{e}
|
|
@end tex
|
|
@ifnottex
|
|
@i{e}
|
|
@end ifnottex
|
|
is used. If @var{arg} is negative, it signals a @code{domain-error}
|
|
error.
|
|
@end defun
|
|
|
|
@ignore
|
|
@defun expm1 arg
|
|
This function returns @code{(1- (exp @var{arg}))}, but it is more
|
|
accurate than that when @var{arg} is negative and @code{(exp @var{arg})}
|
|
is close to 1.
|
|
@end defun
|
|
|
|
@defun log1p arg
|
|
This function returns @code{(log (1+ @var{arg}))}, but it is more
|
|
accurate than that when @var{arg} is so small that adding 1 to it would
|
|
lose accuracy.
|
|
@end defun
|
|
@end ignore
|
|
|
|
@defun log10 arg
|
|
This function returns the logarithm of @var{arg}, with base 10. If
|
|
@var{arg} is negative, it signals a @code{domain-error} error.
|
|
@code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}, at least
|
|
approximately.
|
|
@end defun
|
|
|
|
@defun 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, it is truncated to fit the range of possible
|
|
integer values.
|
|
@end defun
|
|
|
|
@defun sqrt arg
|
|
This returns the square root of @var{arg}. If @var{arg} is negative,
|
|
it signals a @code{domain-error} error.
|
|
@end defun
|
|
|
|
@node Random Numbers
|
|
@section Random Numbers
|
|
@cindex random numbers
|
|
|
|
A deterministic computer program cannot generate true random numbers.
|
|
For most purposes, @dfn{pseudo-random numbers} suffice. A series of
|
|
pseudo-random numbers is generated in a deterministic fashion. The
|
|
numbers are not truly random, but they have certain properties that
|
|
mimic a random series. For example, all possible values occur equally
|
|
often in a pseudo-random series.
|
|
|
|
In Emacs, pseudo-random numbers are generated from a ``seed'' number.
|
|
Starting from any given seed, the @code{random} function always
|
|
generates the same sequence of numbers. Emacs always starts with the
|
|
same seed value, so the sequence of values of @code{random} is actually
|
|
the same in each Emacs run! For example, in one operating system, the
|
|
first call to @code{(random)} after you start Emacs always returns
|
|
-1457731, and the second one always returns -7692030. This
|
|
repeatability is helpful for debugging.
|
|
|
|
If you want random numbers that don't always come out the same, execute
|
|
@code{(random t)}. This chooses a new seed based on the current time of
|
|
day and on Emacs's process @acronym{ID} number.
|
|
|
|
@defun random &optional limit
|
|
This function returns a pseudo-random integer. Repeated calls return a
|
|
series of pseudo-random integers.
|
|
|
|
If @var{limit} is 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.
|
|
@c "Emacs'" is incorrect usage!
|
|
|
|
On some machines, any integer representable in Lisp may be the result
|
|
of @code{random}. On other machines, the result can never be larger
|
|
than a certain maximum or less than a certain (negative) minimum.
|
|
@end defun
|
|
|
|
@ignore
|
|
arch-tag: 574e8dd2-d513-4616-9844-c9a27869782e
|
|
@end ignore
|