@c -*-texinfo-*-
@c This is part of the GNU Emacs Lisp Reference Manual.
-@c Copyright (C) 1990-1995, 1998-1999, 2001-2012
-@c Free Software Foundation, Inc.
+@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
@chapter 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
+@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
* 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.
The range of values for an integer depends on the machine. The
minimum range is @minus{}536870912 to 536870911 (30 bits; i.e.,
@ifnottex
--2**29
+@minus{}2**29
@end ifnottex
@tex
@math{-2^{29}}
@end tex
to
@ifnottex
-2**29 - 1),
+2**29 @minus{} 1),
@end ifnottex
@tex
@math{2^{29}-1}),
@end tex
-but some machines provide a wider range. Many examples in this
-chapter assume that an integer has 30 bits and that floating point
-numbers are IEEE double precision.
+but many machines provide a wider range. Many examples in this
+chapter assume the minimum integer width of 30 bits.
@cindex overflow
The Lisp reader reads an integer as a sequence of digits with optional
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.
@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.
@end defvar
- @xref{Character Codes, max-char}, for the maximum value of a valid
-character codepoint.
+ 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}.
@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
+ 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 where possible (the standard is
-supported by most modern computers).
+@acronym{IEEE} floating-point standard, which is supported by all
+modern computers.
- The read syntax for floating point numbers requires either a decimal
+ 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
+@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}.
+sign to write negative floating-point numbers, as in @samp{-1.0}.
- Emacs Lisp treats @code{-0.0} as equal to ordinary zero (with
+ Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero (with
respect to @code{equal} and @code{=}), even though the two are
-distinguishable in the @acronym{IEEE} floating point standard.
+distinguishable in the @acronym{IEEE} floating-point standard.
@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
+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.)
-Here are the read syntaxes for these special floating point values:
+
+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:
@table @asis
@item positive infinity
@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
+precisely, the value is the logarithm of |@var{number}| base 2, 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
@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.
- At present, each integer value has a unique Lisp object in Emacs Lisp.
+ 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
-error if the unknown value is not a number---it accepts arguments of any
-type. By contrast, @code{=} signals an error if the arguments are not
-numbers or markers. However, it is a good idea to use @code{=} if you
-can, even for comparing integers, just in case we change the
-representation of integers in a future Emacs version.
-
- Sometimes it is useful to compare numbers with @code{equal}; it
+concerned. It is sometimes convenient to use @code{eq} for comparing
+an unknown value with an integer, because @code{eq} does not report an
+error if the unknown value is not a number---it accepts arguments of
+any type. By contrast, @code{=} signals an error if the arguments are
+not numbers or markers. However, it is better programming practice to
+use @code{=} if you can, even for comparing integers.
+
+ 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
@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 every argument is strictly less than the
+respective next 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}
+@defun <= number-or-marker &rest number-or-markers
+This function tests whether every argument is less than or equal to
+the respective next 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}
+@defun > number-or-marker &rest number-or-markers
+This function tests whether every argument is strictly greater than
+the respective next 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}
+@defun >= number-or-marker &rest number-or-markers
+This function tests whether every argument is greater than or equal to
+the respective next 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 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
+ 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
@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. An @code{arith-error} results if @var{divisor} is 0.
+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
@section Arithmetic Operations
@cindex arithmetic operations
- Emacs Lisp provides the traditional four arithmetic operations:
-addition, subtraction, multiplication, and division. Remainder and modulus
-functions supplement the division functions. The functions to
-add or subtract 1 are provided because they are traditional in Lisp and
-commonly used.
-
- All of these functions except @code{%} return a floating point value
-if any argument is floating.
+ Emacs Lisp provides the traditional four arithmetic operations
+(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 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
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.
+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
(/ 6 2)
@result{} 3
@end group
+@group
(/ 5 2)
@result{} 2
+@end group
+@group
(/ 5.0 2)
@result{} 2.5
+@end group
+@group
(/ 5 2.0)
@result{} 2.5
+@end group
+@group
(/ 5.0 2.0)
@result{} 2.5
+@end group
+@group
(/ 25 3 2)
@result{} 4
+@end group
@group
(/ -17 6)
- @result{} -2 @r{(could in theory be @minus{}3 on some machines)}
+ @result{} -2
@end group
@end example
+
+@cindex @code{arith-error} in division
+If you divide an integer by the integer 0, Emacs signals an
+@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
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.
+For any two integers @var{dividend} and @var{divisor},
-An @code{arith-error} results if @var{divisor} is 0.
+@example
+@group
+(+ (% @var{dividend} @var{divisor})
+ (* (/ @var{dividend} @var{divisor}) @var{divisor}))
+@end group
+@end example
+
+@noindent
+always equals @var{dividend}. If @var{divisor} is zero, Emacs signals
+an @code{arith-error} error.
@example
(% 9 4)
(% -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
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.
+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.
-An @code{arith-error} results if @var{divisor} is 0.
+If @var{divisor} is zero, @code{mod} signals an @code{arith-error}
+error if both arguments are integers, and returns a NaN otherwise.
@example
@group
@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.
@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
@defunx cos arg
@defunx tan arg
-These are the ordinary trigonometric functions, with argument measured
-in radians.
+These are the basic trigonometric functions, with argument @var{arg}
+measured in radians.
@end defun
@defun asin arg
@tex
@math{\pi/2}
@end tex
-(inclusive) whose sine is @var{arg}; if, however, @var{arg} is out of
-range (outside [@minus{}1, 1]), it signals a @code{domain-error} error.
+(inclusive) whose sine is @var{arg}. If @var{arg} is out of range
+(outside [@minus{}1, 1]), @code{asin} returns a NaN.
@end defun
@defun acos arg
@tex
@math{\pi}
@end tex
-(inclusive) whose cosine is @var{arg}; if, however, @var{arg} is out
-of range (outside [@minus{}1, 1]), it signals a @code{domain-error} error.
+(inclusive) whose cosine is @var{arg}. If @var{arg} is out of range
+(outside [@minus{}1, 1]), @code{acos} returns a NaN.
@end defun
@defun atan y &optional x
@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 natural base
-@math{e} 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.
+@math{e} is used. If @var{arg} or @var{base} is negative, @code{log}
+returns a NaN.
@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, overflow causes truncation, so watch out.
+If @var{x} is a finite negative number and @var{y} is a finite
+non-integer, @code{expt} returns a NaN.
@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.
+@code{sqrt} returns a NaN.
@end defun
In addition, Emacs defines the following common mathematical
@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
-@minus{}1457731, and the second one always returns @minus{}7692030. This
-repeatability is helpful for debugging.
-
-If you want random numbers that don't always come out the same, execute
-@code{(random t)}. This chooses a new seed based on the current time of
-day and on Emacs's process @acronym{ID} number.
+ 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.
+
+ Pseudo-random numbers are generated from a ``seed''. Starting from
+any given seed, the @code{random} function always generates the same
+sequence of numbers. By default, Emacs initializes the random seed at
+startup, in such a way that the sequence of values of @code{random}
+(with overwhelming likelihood) differs in each Emacs run.
+
+ Sometimes you want the random number sequence to be repeatable. For
+example, when debugging a program whose behavior depends on the random
+number sequence, it is helpful to get the same behavior in each
+program run. To make the sequence repeat, execute @code{(random "")}.
+This sets the seed to a constant value for your particular Emacs
+executable (though it may differ for other Emacs builds). You can use
+other strings to choose various seed values.
@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}.
+nonnegative and less than @var{limit}. Otherwise, the value might be
+any integer representable in Lisp, i.e., an integer between
+@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.
-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.
+If @var{limit} is a string, it means to choose a new seed based on the
+string's contents.
+
@end defun
HCoop / bpt / emacs.git / blobdiff