# emacs / lispref / numbers.texi

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1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 @c -*-texinfo-*- @c This is part of the GNU Emacs Lisp Reference Manual. @c Copyright (C) 1990, 1991, 1992, 1993, 1994, 1995, 1998, 1999 @c Free Software Foundation, Inc. @c See the file elisp.texi for copying conditions. @setfilename ../info/numbers @node Numbers, Strings and Characters, Lisp Data Types, Top @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 @comment node-name, next, previous, up @section Integer Basics The range of values for an integer depends on the machine. The minimum range is @minus{}134217728 to 134217727 (28 bits; i.e., @ifnottex -2**27 @end ifnottex @tex @math{-2^{27}} @end tex to @ifnottex 2**27 - 1), @end ifnottex @tex @math{2^{27}-1}), @end tex but some machines may provide a wider range. Many examples in this chapter assume an integer has 28 bits. @cindex overflow The Lisp reader reads an integer as a sequence of digits with optional initial sign and optional final period. @example 1 ; @r{The integer 1.} 1. ; @r{The integer 1.} +1 ; @r{Also the integer 1.} -1 ; @r{The integer @minus{}1.} 268435457 ; @r{Also the integer 1, due to overflow.} 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 In addition, the Lisp reader recognizes a syntax for integers in bases other than 10: @samp{#B@var{integer}} reads @var{integer} in binary (radix 2), @samp{#O@var{integer}} reads @var{integer} in octal (radix 8), @samp{#X@var{integer}} reads @var{integer} in hexadecimal (radix 16), and @samp{#@var{radix}r@var{integer}} reads @var{integer} in radix @var{radix} (where @var{radix} is between 2 and 36, inclusivley). Case is not significant for the letter after @samp{#} (@samp{B}, @samp{O}, etc.) that denotes the radix. 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 28-bit binary, the decimal integer 5 looks like this: @example 0000 0000 0000 0000 0000 0000 0101 @end example @noindent (We have inserted spaces between groups of 4 bits, and two spaces between groups of 8 bits, to make the binary integer easier to read.) The integer @minus{}1 looks like this: @example 1111 1111 1111 1111 1111 1111 1111 @end example @noindent @cindex two's complement @minus{}1 is represented as 28 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 1111 1111 1111 1111 1111 1011 @end example In this implementation, the largest 28-bit binary integer value is 134,217,727 in decimal. In binary, it looks like this: @example 0111 1111 1111 1111 1111 1111 1111 @end example Since the arithmetic functions do not check whether integers go outside their range, when you add 1 to 134,217,727, the value is the negative integer @minus{}134,217,728: @example (+ 1 134217727) @result{} -134217728 @result{} 1000 0000 0000 0000 0000 0000 0000 @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. @node Float Basics @section Floating Point Basics 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. 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}. @cindex IEEE floating point @cindex positive infinity @cindex negative infinity @cindex infinity @cindex NaN Most modern computers support the IEEE floating point standard, which provides for 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{(sqrt -1.0)} returns a NaN. For practical purposes, there's no significant difference between different NaN values in Emacs Lisp, and there's no rule for precisely which NaN value should be used in a particular case, so Emacs Lisp doesn't try to distinguish them. 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}. @end table In addition, the value @code{-0.0} is distinguishable from ordinary zero in IEEE floating point (although @code{equal} and @code{=} consider them equal values). You can use @code{logb} to extract the binary exponent of a floating point number (or estimate the logarithm of an integer): @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 The functions in this section test whether the argument is a number or whether it is a certain sort of number. The functions @code{integerp} and @code{floatp} can take any type of Lisp object as argument (the predicates 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. @code{floatp} does not exist in Emacs versions 18 and earlier. @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 wholenump object @cindex natural numbers The @code{wholenump} predicate (whose name comes from the phrase whole-number-p'') 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 natnump @code{natnump} is an obsolete synonym for @code{wholenump}. @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. These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}. @end defun @node Comparison of Numbers @section Comparison of Numbers @cindex number equality 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. At present, each integer value has a unique Lisp object in Emacs Lisp. 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 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. 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-marker1 number-or-marker2 This function tests whether its arguments are numerically equal, and returns @code{t} if so, @code{nil} otherwise. @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-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. @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. @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. @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. @end defun @defun max number-or-marker &rest numbers-or-markers This function returns the largest of its arguments. If any of the argument 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 argument 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 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. These functions accept integer arguments also, and return such arguments unchanged. @defun truncate number 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, @code{floor} divides @var{number} by @var{divisor} and then converts to an integer; this uses the kind of division operation that corresponds to @code{mod}, rounding downward. An @code{arith-error} results if @var{divisor} is 0. @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 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 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 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. It is important to note that in Emacs Lisp, arithmetic functions do not check for overflow. Thus @code{(1+ 134217727)} may evaluate to @minus{}134217728, 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, 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 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 134,217,727 produces @minus{}2 on a 28-bit machine: @example (lsh 134217727 1) ; @r{left shift} @result{} -2 @end example In binary, in the 28-bit implementation, the argument looks like this: @example @group ;; @r{Decimal 134,217,727} 0111 1111 1111 1111 1111 1111 1111 @end group @end example @noindent which becomes the following when left shifted: @example @group ;; @r{Decimal @minus{}2} 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.} 1111 1111 1111 1111 1111 1111 1010 @result{} 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{} 134217725 ;; @r{Decimal @minus{}6 becomes decimal 134,217,725.} 1111 1111 1111 1111 1111 1111 1010 @result{} 0111 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{ 28-bit binary values} (lsh 5 2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} @result{} 20 ; = @r{0000 0000 0000 0000 0000 0001 0100} @end group @group (ash 5 2) @result{} 20 (lsh -5 2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} @result{} -20 ; = @r{1111 1111 1111 1111 1111 1110 1100} (ash -5 2) @result{} -20 @end group @group (lsh 5 -2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} @result{} 1 ; = @r{0000 0000 0000 0000 0000 0000 0001} @end group @group (ash 5 -2) @result{} 1 @end group @group (lsh -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} @result{} 4194302 ; = @r{0011 1111 1111 1111 1111 1111 1110} @end group @group (ash -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} @result{} -2 ; = @r{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{ 28-bit binary values} (logand 14 13) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} @result{} 12 ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} @end group @group (logand 14 13 4) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} ; 4 = @r{0000 0000 0000 0000 0000 0000 0100} @result{} 4 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100} @end group @group (logand) @result{} -1 ; -1 = @r{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{ 28-bit binary values} (logior 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} @result{} 13 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} @end group @group (logior 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} ; 7 = @r{0000 0000 0000 0000 0000 0000 0111} @result{} 15 ; 15 = @r{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{ 28-bit binary values} (logxor 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} @result{} 9 ; 9 = @r{0000 0000 0000 0000 0000 0000 1001} @end group @group (logxor 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} ; 7 = @r{0000 0000 0000 0000 0000 0000 0111} @result{} 14 ; 14 = @r{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{0000 0000 0000 0000 0000 0000 0101} ;; @r{becomes} ;; -6 = @r{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]), then the result is 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, however, @var{arg} is out of range (outside [-1, 1]), then the result is a NaN. @end defun @defun atan arg The value of @code{(atan @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 (exclusive) whose tangent is @var{arg}. @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, the result is a NaN. @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, the result is a NaN. @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, the value is a NaN. @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 @sc{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 @sc{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