@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 Foundation, Inc. @c See the file elisp.texi for copying conditions. @node Numbers @chapter Numbers @cindex integers @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. @menu * Integer Basics:: Representation and range of integers. * Float Basics:: Representation and range of floating point. * Predicates on Numbers:: Testing for numbers. * Comparison of Numbers:: Equality and inequality predicates. * Numeric Conversions:: Converting float to integer and vice versa. * Arithmetic Operations:: How to add, subtract, multiply and divide. * 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. @end menu @node Integer Basics @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., @ifnottex -2**29 @end ifnottex @tex @math{-2^{29}} @end tex to @ifnottex 2**29 @minus{} 1), @end ifnottex @tex @math{2^{29}-1}), @end tex 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 initial sign and optional final period. An integer that is out of the Emacs range is treated as a floating-point number. @example 1 ; @r{The integer 1.} 1. ; @r{The integer 1.} +1 ; @r{Also the integer 1.} -1 ; @r{The integer @minus{}1.} 1073741825 ; @r{The floating point number 1073741825.0.} 0 ; @r{The integer 0.} -0 ; @r{The integer 0.} @end example @cindex integers in specific radix @cindex radix for reading an integer @cindex base for reading an integer @cindex hex numbers @cindex octal numbers @cindex reading numbers in hex, octal, and binary The syntax for integers in bases other than 10 uses @samp{#} followed by a letter that specifies the radix: @samp{b} for binary, @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to specify radix @var{radix}. Case is not significant for the letter that specifies the radix. Thus, @samp{#b@var{integer}} reads @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads @var{integer} in radix @var{radix}. Allowed values of @var{radix} run from 2 to 36. For example: @example #b101100 @result{} 44 #o54 @result{} 44 #x2c @result{} 44 #24r1k @result{} 44 @end example To understand how various functions work on integers, especially the bitwise operators (@pxref{Bitwise Operations}), it is often helpful to view the numbers in their binary form. In 30-bit binary, the decimal integer 5 looks like this: @example 0000...000101 (30 bits total) @end example @noindent (The @samp{...} stands for enough bits to fill out a 30-bit word; in this case, @samp{...} stands for twenty 0 bits. Later examples also use the @samp{...} notation to make binary integers easier to read.) The integer @minus{}1 looks like this: @example 1111...111111 (30 bits total) @end example @noindent @cindex two's complement @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, @minus{}5 looks like this: @example 1111...111011 (30 bits total) @end example In this implementation, the largest 30-bit binary integer value is 536,870,911 in decimal. In binary, it looks like this: @example 0111...111111 (30 bits total) @end example Since the arithmetic functions do not check whether integers go outside their range, when you add 1 to 536,870,911, the value is the negative integer @minus{}536,870,912: @example (+ 1 536870911) @result{} -536870912 @result{} 1000...000000 (30 bits total) @end example Many of the functions described in this chapter accept markers for arguments in place of numbers. (@xref{Markers}.) Since the actual arguments to such functions may be either numbers or markers, we often give these arguments the name @var{number-or-marker}. When the argument value is a marker, its position value is used and its buffer is ignored. @cindex largest Lisp integer number @cindex maximum Lisp integer number @defvar most-positive-fixnum The value of this variable is the largest integer that Emacs Lisp can handle. @end defvar @cindex smallest Lisp integer number @cindex minimum Lisp integer number @defvar most-negative-fixnum The value of this variable is the smallest integer that Emacs Lisp can handle. It is negative. @end defvar 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 @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. @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 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.) 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 @samp{1.0e+INF} @item negative infinity @samp{-1.0e+INF} @item Not-a-number @samp{0.0e+NaN} or @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 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}: @smallexample @var{x} = @var{sig} * 2^@var{exp} @end smallexample @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)}. @end defun @defun ldexp sig &optional exp 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. @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 down to an integer. @example (logb 10) @result{} 3 (logb 10.0e20) @result{} 69 @end example @end defun @node Predicates on Numbers @section Type Predicates for Numbers @cindex predicates for numbers The functions in this section test for numbers, or for a specific type of number. The functions @code{integerp} and @code{floatp} can take any type of Lisp object as argument (they would not be of much use otherwise), but the @code{zerop} predicate requires a number as its argument. See also @code{integer-or-marker-p} and @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. @end defun @defun integerp object This predicate tests whether its argument is an integer, and returns @code{t} if so, @code{nil} otherwise. @end defun @defun numberp object This predicate tests whether its argument is a number (either integer or floating point), and returns @code{t} if so, @code{nil} otherwise. @end defun @defun natnump object @cindex natural numbers This predicate (whose name comes from the phrase natural number'') tests to see whether its argument is a nonnegative integer, and returns @code{t} if so, @code{nil} otherwise. 0 is considered non-negative. @findex wholenump number This is a synonym for @code{natnump}. @end defun @defun zerop number This predicate tests whether its argument is zero, and returns @code{t} if so, @code{nil} otherwise. The argument must be a number. @code{(zerop x)} is equivalent to @code{(= x 0)}. @end defun @node Comparison of Numbers @section Comparison of Numbers @cindex number comparison @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 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. 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 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. @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. 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)) (< (/ (abs (- x y)) (max (abs x) (abs y))) fuzz-factor))) @end example @cindex CL note---integers vrs @code{eq} @quotation @b{Common Lisp note:} Comparing numbers in Common Lisp always requires @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. @end quotation @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 This function acts like @code{eq} except when both arguments are numbers. It compares numbers by type and numeric value, so that @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and @code{(eql 1 1)} both return @code{t}. @end defun @defun /= number-or-marker1 number-or-marker2 This function tests whether its arguments are numerically equal, and returns @code{t} if they are not, and @code{nil} if they are. @end defun @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-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-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-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 as floating point, even if it was given as an integer. @example (max 20) @result{} 20 (max 1 2.5) @result{} 2.5 (max 1 3 2.5) @result{} 3.0 @end example @end defun @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 as floating point, even if it was given as an integer. @example (min -4 1) @result{} -4 @end example @end defun @defun abs number This function returns the absolute value of @var{number}. @end defun @node Numeric Conversions @section Numeric Conversions @cindex rounding in conversions @cindex number conversions @cindex converting numbers To convert an integer to floating point, use the function @code{float}. @defun float number This returns @var{number} converted to floating point. If @var{number} is already a floating point number, @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 @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. @defun truncate number &optional divisor This returns @var{number}, converted to an integer by rounding towards zero. @example (truncate 1.2) @result{} 1 (truncate 1.7) @result{} 1 (truncate -1.2) @result{} -1 (truncate -1.7) @result{} -1 @end example @end defun @defun floor number &optional divisor This returns @var{number}, converted to an integer by rounding downward (towards negative infinity). If @var{divisor} is specified, this uses the kind of division operation that corresponds to @code{mod}, rounding downward. @example (floor 1.2) @result{} 1 (floor 1.7) @result{} 1 (floor -1.2) @result{} -2 (floor -1.7) @result{} -2 (floor 5.99 3) @result{} 1 @end example @end defun @defun ceiling number &optional divisor This returns @var{number}, converted to an integer by rounding upward (towards positive infinity). @example (ceiling 1.2) @result{} 2 (ceiling 1.7) @result{} 2 (ceiling -1.2) @result{} -1 (ceiling -1.7) @result{} -1 @end example @end defun @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. @example (round 1.2) @result{} 1 (round 1.7) @result{} 2 (round -1.2) @result{} -1 (round -1.7) @result{} -2 @end example @end defun @node Arithmetic Operations @section Arithmetic Operations @cindex arithmetic operations 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 a floating point number. It is important to note that in Emacs Lisp, arithmetic functions do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to @minus{}536870912, depending on your hardware. @defun 1+ number-or-marker This function returns @var{number-or-marker} plus 1. For example, @example (setq foo 4) @result{} 4 (1+ foo) @result{} 5 @end example This function is not analogous to the C operator @code{++}---it does not increment a variable. It just computes a sum. Thus, if we continue, @example foo @result{} 4 @end example If you want to increment the variable, you must use @code{setq}, like this: @example (setq foo (1+ foo)) @result{} 5 @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, 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 @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}). 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}). @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 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}. If @var{divisor} is zero, Emacs signals an @code{arith-error} error. @example (% 9 4) @result{} 1 (% -9 4) @result{} -1 (% 9 -4) @result{} 1 (% -9 -4) @result{} -1 @end example @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} permits floating point arguments; it rounds the quotient downward (towards minus infinity) to an integer, and uses that quotient to compute the remainder. 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 (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 @cindex bitwise arithmetic @cindex logical arithmetic 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 536,870,911 produces @minus{}2 in the 30-bit implementation: @example (lsh 536870911 1) ; @r{left shift} @result{} -2 @end example In binary, the argument looks like this: @example @group ;; @r{Decimal 536,870,911} 0111...111111 (30 bits total) @end group @end example @noindent which becomes the following when left shifted: @example @group ;; @r{Decimal @minus{}2} 1111...111110 (30 bits total) @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.} 1111...111010 (30 bits total) @result{} 1111...111101 (30 bits total) @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{} 536870909 ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.} 1111...111010 (30 bits total) @result{} 0111...111101 (30 bits total) @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{ 30-bit binary values} (lsh 5 2) ; 5 = @r{0000...000101} @result{} 20 ; = @r{0000...010100} @end group @group (ash 5 2) @result{} 20 (lsh -5 2) ; -5 = @r{1111...111011} @result{} -20 ; = @r{1111...101100} (ash -5 2) @result{} -20 @end group @group (lsh 5 -2) ; 5 = @r{0000...000101} @result{} 1 ; = @r{0000...000001} @end group @group (ash 5 -2) @result{} 1 @end group @group (lsh -5 -2) ; -5 = @r{1111...111011} @result{} 268435454 ; = @r{0011...111110} @end group @group (ash -5 -2) ; -5 = @r{1111...111011} @result{} -2 ; = @r{1111...111110} @end group @end smallexample @end defun @defun logand &rest ints-or-markers 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{ 30-bit binary values} (logand 14 13) ; 14 = @r{0000...001110} ; 13 = @r{0000...001101} @result{} 12 ; 12 = @r{0000...001100} @end group @group (logand 14 13 4) ; 14 = @r{0000...001110} ; 13 = @r{0000...001101} ; 4 = @r{0000...000100} @result{} 4 ; 4 = @r{0000...000100} @end group @group (logand) @result{} -1 ; -1 = @r{1111...111111} @end group @end smallexample @end defun @defun logior &rest ints-or-markers 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{ 30-bit binary values} (logior 12 5) ; 12 = @r{0000...001100} ; 5 = @r{0000...000101} @result{} 13 ; 13 = @r{0000...001101} @end group @group (logior 12 5 7) ; 12 = @r{0000...001100} ; 5 = @r{0000...000101} ; 7 = @r{0000...000111} @result{} 15 ; 15 = @r{0000...001111} @end group @end smallexample @end defun @defun logxor &rest ints-or-markers 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{ 30-bit binary values} (logxor 12 5) ; 12 = @r{0000...001100} ; 5 = @r{0000...000101} @result{} 9 ; 9 = @r{0000...001001} @end group @group (logxor 12 5 7) ; 12 = @r{0000...001100} ; 5 = @r{0000...000101} ; 7 = @r{0000...000111} @result{} 14 ; 14 = @r{0000...001110} @end group @end smallexample @end defun @defun lognot integer 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{0000...000101} (30 bits total) ;; @r{becomes} ;; -6 = @r{1111...111010} (30 bits total) @end example @end defun @node Math Functions @section Standard Mathematical Functions @cindex transcendental functions @cindex mathematical functions @cindex floating-point 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 basic trigonometric functions, with argument @var{arg} 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 @var{arg} is out of range (outside [@minus{}1, 1]), @code{asin} returns a NaN. @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 @var{arg} is out of range (outside [@minus{}1, 1]), @code{acos} returns a NaN. @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 @math{e} to the power @var{arg}. @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 natural base @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, @code{sqrt} returns a NaN. @end defun In addition, Emacs defines the following common mathematical constants: @defvar float-e The mathematical constant @math{e} (2.71828@dots{}). @end defvar @defvar float-pi The mathematical constant @math{pi} (3.14159@dots{}). @end defvar @node 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. 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}. 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. If @var{limit} is a string, it means to choose a new seed based on the string's contents. @end defun