@c -*-texinfo-*-
@c This is part of the GNU Emacs Lisp Reference Manual.
-@c Copyright (C) 1990-1995, 1998-1999, 2001-2013 Free Software
+@c Copyright (C) 1990-1995, 1998-1999, 2001-2014 Free Software
@c Foundation, Inc.
@c See the file elisp.texi for copying conditions.
@node Numbers
@cindex numbers
GNU Emacs supports two numeric data types: @dfn{integers} and
-@dfn{floating point numbers}. Integers are whole numbers such as
-@minus{}3, 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.
+@dfn{floating-point numbers}. Integers are whole numbers such as
+@minus{}3, 0, 7, 13, and 511. Floating-point numbers are numbers with
+fractional parts, such as @minus{}4.5, 0.0, and 2.71828. They can
+also be expressed in exponential notation: @samp{1.5e2} is the same as
+@samp{150.0}; here, @samp{e2} stands for ten to the second power, and
+that is multiplied by 1.5. Integer computations are exact, though
+they may overflow. Floating-point computations often involve rounding
+errors, as the numbers have a fixed amount of precision.
@menu
* Integer Basics:: Representation and range of integers.
* 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.
+* 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.
@section Integer Basics
The range of values for an integer depends on the machine. The
-minimum range is @minus{}536870912 to 536870911 (30 bits; i.e.,
+minimum range is @minus{}536,870,912 to 536,870,911 (30 bits; i.e.,
@ifnottex
--2**29
+@minus{}2**29
@end ifnottex
@tex
@math{-2^{29}}
1. ; @r{The integer 1.}
+1 ; @r{Also the integer 1.}
-1 ; @r{The integer @minus{}1.}
- 1073741825 ; @r{The floating point number 1073741825.0.}
+ 9000000000000000000
+ ; @r{The floating-point number 9e18.}
0 ; @r{The integer 0.}
-0 ; @r{The integer 0.}
@end example
@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{}1. In binary, the decimal integer 4 is 100. Consequently,
+ Subtracting 4 from @minus{}1 returns the negative integer @minus{}5.
+In binary, the decimal integer 4 is 100. Consequently,
@minus{}5 looks like this:
@example
1111...111011 (30 bits total)
@end example
- In this implementation, the largest 30-bit binary integer value is
+ In this implementation, the largest 30-bit binary integer is
536,870,911 in decimal. In binary, it looks like this:
@example
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
+@cindex largest Lisp integer
+@cindex maximum Lisp integer
@defvar most-positive-fixnum
-The value of this variable is the largest integer that Emacs Lisp
-can handle.
+The value of this variable is the largest integer that Emacs Lisp can
+handle. Typical values are
+@ifnottex
+2**29 @minus{} 1
+@end ifnottex
+@tex
+@math{2^{29}-1}
+@end tex
+on 32-bit and
+@ifnottex
+2**61 @minus{} 1
+@end ifnottex
+@tex
+@math{2^{61}-1}
+@end tex
+on 64-bit platforms.
@end defvar
-@cindex smallest Lisp integer number
-@cindex minimum Lisp integer number
+@cindex smallest Lisp integer
+@cindex minimum Lisp integer
@defvar most-negative-fixnum
The value of this variable is the smallest integer that Emacs Lisp can
-handle. It is negative.
+handle. It is negative. Typical values are
+@ifnottex
+@minus{}2**29
+@end ifnottex
+@tex
+@math{-2^{29}}
+@end tex
+on 32-bit and
+@ifnottex
+@minus{}2**61
+@end ifnottex
+@tex
+@math{-2^{61}}
+@end tex
+on 64-bit platforms.
@end defvar
In Emacs Lisp, text characters are represented by integers. Any
-integer between zero and the value of @code{max-char}, inclusive, is
-considered to be valid as a character. @xref{String Basics}.
+integer between zero and the value of @code{(max-char)}, inclusive, is
+considered to be valid as a character. @xref{Character Codes}.
@node Float Basics
-@section Floating Point Basics
+@section Floating-Point Basics
@cindex @acronym{IEEE} floating point
- 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
-@code{double} on the machine you are using. Emacs uses the
-@acronym{IEEE} floating point standard, which is supported by all
-modern computers.
-
- 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
-@samp{.15e4} 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}.
-
- Emacs Lisp treats @code{-0.0} as equal to ordinary zero (with
-respect to @code{equal} and @code{=}), even though the two are
-distinguishable in the @acronym{IEEE} floating point standard.
+ Floating-point numbers are useful for representing numbers that are
+not integral. The range of floating-point numbers is
+the same as the range of the C data type @code{double} on the machine
+you are using. On all computers currently supported by Emacs, this is
+double-precision @acronym{IEEE} floating point.
+
+ The read syntax for floating-point numbers requires either a decimal
+point, an exponent, or both. Optional signs (@samp{+} or @samp{-})
+precede the number and its exponent. For example, @samp{1500.0},
+@samp{+15e2}, @samp{15.0e+2}, @samp{+1500000e-3}, and @samp{.15e4} are
+five ways of writing a floating-point number whose value is 1500.
+They are all equivalent. Like Common Lisp, Emacs Lisp requires at
+least one digit after any decimal point in a floating-point number;
+@samp{1500.} is an integer, not a floating-point number.
+
+ Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero
+with respect to @code{equal} and @code{=}. This follows the
+@acronym{IEEE} floating-point standard, which says @code{-0.0} and
+@code{0.0} are numerically equal even though other operations can
+distinguish them.
@cindex positive infinity
@cindex negative infinity
@cindex infinity
@cindex NaN
- The @acronym{IEEE} floating point standard supports positive
-infinity and negative infinity as floating point values. It also
+ The @acronym{IEEE} floating-point standard supports 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@. (NaN
-values can also carry a sign, but for practical purposes there's no
-significant difference between different NaN values in Emacs Lisp.)
+correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN@.
+Although NaN values carry a sign, for practical purposes there is no other
+significant difference between different NaN values in Emacs Lisp.
-When a function is documented to return a NaN, it returns an
-implementation-defined value when Emacs is running on one of the
-now-rare platforms that do not use @acronym{IEEE} floating point. For
-example, @code{(log -1.0)} typically returns a NaN, but on
-non-@acronym{IEEE} platforms it returns an implementation-defined
-value.
-
-Here are the read syntaxes for these special floating point values:
+Here are read syntaxes for these special floating-point values:
@table @asis
-@item positive infinity
-@samp{1.0e+INF}
-@item negative infinity
-@samp{-1.0e+INF}
-@item Not-a-number
-@samp{0.0e+NaN} or @samp{-0.0e+NaN}.
+@item infinity
+@samp{1.0e+INF} and @samp{-1.0e+INF}
+@item not-a-number
+@samp{0.0e+NaN} and @samp{-0.0e+NaN}
@end table
-@defun isnan number
-This predicate tests whether its argument is NaN, and returns @code{t}
-if so, @code{nil} otherwise. The argument must be a number.
-@end defun
-
- The following functions are specialized for handling floating point
+ The following functions are specialized for handling floating-point
numbers:
-@defun frexp x
-This function returns a cons cell @code{(@var{sig} . @var{exp})},
-where @var{sig} and @var{exp} are respectively the significand and
-exponent of the floating point number @var{x}:
+@defun isnan x
+This predicate returns @code{t} if its floating-point argument is a NaN,
+@code{nil} otherwise.
+@end defun
-@smallexample
-@var{x} = @var{sig} * 2^@var{exp}
-@end smallexample
+@defun frexp x
+This function returns a cons cell @code{(@var{s} . @var{e})},
+where @var{s} and @var{e} are respectively the significand and
+exponent of the floating-point number @var{x}.
-@var{sig} is a floating point number between 0.5 (inclusive) and 1.0
-(exclusive). If @var{x} is zero, the return value is @code{(0 . 0)}.
+If @var{x} is finite, then @var{s} is a floating-point number between 0.5
+(inclusive) and 1.0 (exclusive), @var{e} is an integer, and
+@ifnottex
+@var{x} = @var{s} * 2**@var{e}.
+@end ifnottex
+@tex
+@math{x = s 2^e}.
+@end tex
+If @var{x} is zero or infinity, then @var{s} is the same as @var{x}.
+If @var{x} is a NaN, then @var{s} is also a NaN.
+If @var{x} is zero, then @var{e} is 0.
@end defun
@defun ldexp sig &optional exp
-This function returns a floating point number corresponding to the
+This function returns a floating-point number corresponding to the
significand @var{sig} and exponent @var{exp}.
@end defun
@defun copysign x1 x2
This function copies the sign of @var{x2} to the value of @var{x1},
-and returns the result. @var{x1} and @var{x2} must be floating point
-numbers.
+and returns the result. @var{x1} and @var{x2} must be floating point.
@end defun
-@defun logb number
-This function returns the binary exponent of @var{number}. More
-precisely, the value is the logarithm of |@var{number}| base 2, rounded
+@defun logb x
+This function returns the binary exponent of @var{x}. More
+precisely, the value is the logarithm base 2 of @math{|x|}, rounded
down to an integer.
@example
@code{number-or-marker-p}, in @ref{Predicates on Markers}.
@defun floatp object
-This predicate tests whether its argument is a floating point
-number and returns @code{t} if so, @code{nil} otherwise.
+This predicate tests whether its argument is floating point
+and returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun integerp object
returns @code{t} if so, @code{nil} otherwise. 0 is considered
non-negative.
-@findex wholenump number
-This is a synonym for @code{natnump}.
+@findex wholenump
+@code{wholenump} is a synonym for @code{natnump}.
@end defun
@defun zerop number
@cindex comparing numbers
To test numbers for numerical equality, you should normally use
-@code{=}, not @code{eq}. There can be many distinct floating point
-number objects with the same numeric value. If you use @code{eq} to
+@code{=}, not @code{eq}. There can be many distinct floating-point
+objects with the same numeric value. If you use @code{eq} to
compare them, then you test whether two values are the same
@emph{object}. By contrast, @code{=} compares only the numeric values
of the objects.
- In Emacs Lisp, each integer value is a unique Lisp object.
+ In Emacs Lisp, each integer is a unique Lisp object.
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
Sometimes it is useful to compare numbers with @code{equal}, which
treats two numbers as equal if they have the same data type (both
integers, or both floating point) and the same value. By contrast,
-@code{=} can treat an integer and a floating point number as equal.
+@code{=} can treat an integer and a floating-point number as equal.
@xref{Equality 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.
+ There is another wrinkle: because floating-point arithmetic is not
+exact, it is often a bad idea to check for equality of floating-point
+values. Usually it is better to test for approximate equality.
Here's a function to do this:
@example
(defvar fuzz-factor 1.0e-6)
(defun approx-equal (x y)
- (or (and (= x 0) (= y 0))
+ (or (= x y)
(< (/ (abs (- x y))
(max (abs x) (abs y)))
fuzz-factor)))
@code{=} 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.
+limited range of integers.
@end quotation
-@defun = number-or-marker1 number-or-marker2
-This function tests whether its arguments are numerically equal, and
-returns @code{t} if so, @code{nil} otherwise.
+@defun = number-or-marker &rest number-or-markers
+This function tests whether all its arguments are numerically equal,
+and returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun eql value1 value2
returns @code{t} if they are not, and @code{nil} if they are.
@end defun
-@defun < 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.
+@defun < number-or-marker &rest number-or-markers
+This function tests whether each argument is strictly less than the
+following argument. It returns @code{t} if so, @code{nil} otherwise.
@end defun
-@defun <= 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.
+@defun <= number-or-marker &rest number-or-markers
+This function tests whether each argument is less than or equal to
+the following argument. It returns @code{t} if so, @code{nil} otherwise.
@end defun
-@defun > 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.
+@defun > number-or-marker &rest number-or-markers
+This function tests whether each argument is strictly greater than
+the following argument. It returns @code{t} if so, @code{nil} otherwise.
@end defun
-@defun >= 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.
+@defun >= number-or-marker &rest number-or-markers
+This function tests whether each argument is greater than or equal to
+the following argument. It returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun 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
+If any of the arguments is floating point, the value is returned
as floating point, even if it was given as an integer.
@example
@defun 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
+If any of the arguments is floating point, the value is returned
as floating point, even if it was given as an integer.
@example
@defun float number
This returns @var{number} converted to floating point.
-If @var{number} is already a floating point number, @code{float} returns
+If @var{number} is already floating point, @code{float} returns
it unchanged.
@end defun
- There are four functions to convert floating point numbers to
+ 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
+may be integers or floating-point numbers. @var{divisor} may also be
@code{nil}. 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 non-@code{nil}, they
divide @var{number} by @var{divisor} and convert the result to an
-integer. integer. If @var{divisor} is zero (whether integer or
-floating-point), Emacs signals an @code{arith-error} error.
+integer. If @var{divisor} is zero (whether integer or
+floating point), Emacs signals an @code{arith-error} error.
@defun truncate number &optional divisor
This returns @var{number}, converted to an integer by rounding towards
@defun 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.
+returns the even integer.
@example
(round 1.2)
(addition, subtraction, multiplication, and division), as well as
remainder and modulus functions, and functions to add or subtract 1.
Except for @code{%}, each of these functions accepts both integer and
-floating point arguments, and returns a floating point number if any
-argument is a floating point number.
+floating-point arguments, and returns a floating-point number if any
+argument is floating point.
- It is important to note that in Emacs Lisp, arithmetic functions
-do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to
+ Emacs Lisp arithmetic functions do not check for integer overflow.
+Thus @code{(1+ 536870911)} may evaluate to
@minus{}536870912, depending on your hardware.
@defun 1+ number-or-marker
If all the arguments are integers, the result is an integer, obtained
by rounding the quotient towards zero after each division.
-(Hypothetically, some machines may have different rounding behavior
-for negative arguments, because @code{/} is implemented using the C
-division operator, which permits machine-dependent rounding; but this
-does not happen in practice.)
@example
@group
@cindex @code{arith-error} in division
If you divide an integer by the integer 0, Emacs signals an
-@code{arith-error} error (@pxref{Errors}). If you divide a floating
-point number by 0, or divide by the floating point number 0.0, the
-result is either positive or negative infinity (@pxref{Float Basics}).
+@code{arith-error} error (@pxref{Errors}). Floating-point division of
+a nonzero number by zero yields either positive or negative infinity
+(@pxref{Float Basics}).
@end defun
@defun % dividend divisor
@end example
@noindent
-always equals @var{dividend}. If @var{divisor} is zero, Emacs signals
-an @code{arith-error} error.
+always equals @var{dividend} if @var{divisor} is nonzero.
@example
(% 9 4)
by @var{divisor}, but with the same sign as @var{divisor}.
The arguments must be numbers or markers.
-Unlike @code{%}, @code{mod} permits floating point arguments; it
+Unlike @code{%}, @code{mod} permits floating-point arguments; it
rounds the quotient downward (towards minus infinity) to an integer,
and uses that quotient to compute the remainder.
@noindent
always equals @var{dividend}, subject to rounding error if either
-argument is floating point. For @code{floor}, see @ref{Numeric
+argument is floating point and to an @code{arith-error} if @var{dividend} is an
+integer and @var{divisor} is 0. For @code{floor}, see @ref{Numeric
Conversions}.
@end defun
@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
+@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.
+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.
+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.
+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.
+and returns that value as a floating-point number.
+Rounding a value equidistant between two integers returns the even integer.
@end defun
@node Bitwise Operations
@cindex mathematical functions
@cindex floating-point functions
- These mathematical functions allow integers as well as floating point
+ These mathematical functions allow integers as well as floating-point
numbers as arguments.
@defun sin arg
returns a NaN.
@end defun
-@defun log10 arg
-This function returns the logarithm of @var{arg}, with base 10:
-@code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}.
-@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
@end defun
@defun sqrt arg
-This returns the square root of @var{arg}. If @var{arg} is negative,
-@code{sqrt} returns a NaN.
+This returns the square root of @var{arg}. If @var{arg} is finite
+and less than zero, @code{sqrt} returns a NaN.
@end defun
In addition, Emacs defines the following common mathematical
@code{most-negative-fixnum} and @code{most-positive-fixnum}
(@pxref{Integer Basics}).
-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.
+If @var{limit} is @code{t}, it means to choose a new seed as if Emacs
+were restarting.
If @var{limit} is a string, it means to choose a new seed based on the
string's contents.
HCoop / bpt / emacs.git / blobdiff