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1 @c -*-texinfo-*-
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2 @c This is part of the XEmacs Lisp Reference Manual.
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3 @c Copyright (C) 1990, 1991, 1992, 1993, 1994 Free Software Foundation, Inc.
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4 @c See the file lispref.texi for copying conditions.
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5 @setfilename ../../info/numbers.info
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6 @node Numbers, Strings and Characters, Lisp Data Types, Top
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7 @chapter Numbers
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8 @cindex integers
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9 @cindex numbers
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10
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11 XEmacs supports two numeric data types: @dfn{integers} and
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12 @dfn{floating point numbers}. Integers are whole numbers such as
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13 @minus{}3, 0, 7, 13, and 511. Their values are exact. Floating point
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14 numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or
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15 2.71828. They can also be expressed in exponential notation:
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16 1.5e2 equals 150; in this example, @samp{e2} stands for ten to the
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17 second power, and is multiplied by 1.5. Floating point values are not
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18 exact; they have a fixed, limited amount of precision.
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19
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20 @menu
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21 * Integer Basics:: Representation and range of integers.
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22 * Float Basics:: Representation and range of floating point.
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23 * Predicates on Numbers:: Testing for numbers.
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24 * Comparison of Numbers:: Equality and inequality predicates.
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25 * Numeric Conversions:: Converting float to integer and vice versa.
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26 * Arithmetic Operations:: How to add, subtract, multiply and divide.
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27 * Rounding Operations:: Explicitly rounding floating point numbers.
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28 * Bitwise Operations:: Logical and, or, not, shifting.
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29 * Math Functions:: Trig, exponential and logarithmic functions.
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30 * Random Numbers:: Obtaining random integers, predictable or not.
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31 @end menu
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32
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33 @node Integer Basics
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34 @section Integer Basics
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35
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36 The range of values for an integer depends on the machine. The
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37 minimum range is @minus{}134217728 to 134217727 (28 bits; i.e.,
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38 @ifinfo
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39 -2**27
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40 @end ifinfo
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41 @tex
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42 $-2^{27}$
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43 @end tex
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44 to
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45 @ifinfo
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46 2**27 - 1),
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47 @end ifinfo
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48 @tex
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49 $2^{27}-1$),
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50 @end tex
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51 but some machines may provide a wider range. Many examples in this
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52 chapter assume an integer has 28 bits.
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53 @cindex overflow
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54
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55 The Lisp reader reads an integer as a sequence of digits with optional
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56 initial sign and optional final period.
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57
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58 @example
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59 1 ; @r{The integer 1.}
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60 1. ; @r{The integer 1.}
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61 +1 ; @r{Also the integer 1.}
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62 -1 ; @r{The integer @minus{}1.}
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63 268435457 ; @r{Also the integer 1, due to overflow.}
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64 0 ; @r{The integer 0.}
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65 -0 ; @r{The integer 0.}
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66 @end example
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67
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68 To understand how various functions work on integers, especially the
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69 bitwise operators (@pxref{Bitwise Operations}), it is often helpful to
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70 view the numbers in their binary form.
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71
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72 In 28-bit binary, the decimal integer 5 looks like this:
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73
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74 @example
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75 0000 0000 0000 0000 0000 0000 0101
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76 @end example
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77
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78 @noindent
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79 (We have inserted spaces between groups of 4 bits, and two spaces
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80 between groups of 8 bits, to make the binary integer easier to read.)
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81
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82 The integer @minus{}1 looks like this:
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83
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84 @example
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85 1111 1111 1111 1111 1111 1111 1111
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86 @end example
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87
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88 @noindent
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89 @cindex two's complement
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90 @minus{}1 is represented as 28 ones. (This is called @dfn{two's
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91 complement} notation.)
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92
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93 The negative integer, @minus{}5, is creating by subtracting 4 from
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94 @minus{}1. In binary, the decimal integer 4 is 100. Consequently,
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95 @minus{}5 looks like this:
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96
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97 @example
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98 1111 1111 1111 1111 1111 1111 1011
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99 @end example
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100
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101 In this implementation, the largest 28-bit binary integer is the
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102 decimal integer 134,217,727. In binary, it looks like this:
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103
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104 @example
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105 0111 1111 1111 1111 1111 1111 1111
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106 @end example
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107
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108 Since the arithmetic functions do not check whether integers go
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109 outside their range, when you add 1 to 134,217,727, the value is the
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110 negative integer @minus{}134,217,728:
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111
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112 @example
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113 (+ 1 134217727)
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114 @result{} -134217728
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115 @result{} 1000 0000 0000 0000 0000 0000 0000
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116 @end example
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117
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118 Many of the following functions accept markers for arguments as well
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119 as integers. (@xref{Markers}.) More precisely, the actual arguments to
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120 such functions may be either integers or markers, which is why we often
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121 give these arguments the name @var{int-or-marker}. When the argument
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122 value is a marker, its position value is used and its buffer is ignored.
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123
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124 @ignore
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125 In version 19, except where @emph{integer} is specified as an
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126 argument, all of the functions for markers and integers also work for
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127 floating point numbers.
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128 @end ignore
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129
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130 @node Float Basics
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131 @section Floating Point Basics
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132
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133 XEmacs supports floating point numbers. The precise range of floating
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134 point numbers is machine-specific; it is the same as the range of the C
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135 data type @code{double} on the machine in question.
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136
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137 The printed representation for floating point numbers requires either
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138 a decimal point (with at least one digit following), an exponent, or
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139 both. For example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2},
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140 @samp{1.5e3}, and @samp{.15e4} are five ways of writing a floating point
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141 number whose value is 1500. They are all equivalent. You can also use
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142 a minus sign to write negative floating point numbers, as in
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143 @samp{-1.0}.
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144
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145 @cindex IEEE floating point
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146 @cindex positive infinity
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147 @cindex negative infinity
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148 @cindex infinity
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149 @cindex NaN
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150 Most modern computers support the IEEE floating point standard, which
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151 provides for positive infinity and negative infinity as floating point
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152 values. It also provides for a class of values called NaN or
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153 ``not-a-number''; numerical functions return such values in cases where
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154 there is no correct answer. For example, @code{(sqrt -1.0)} returns a
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155 NaN. For practical purposes, there's no significant difference between
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156 different NaN values in XEmacs Lisp, and there's no rule for precisely
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157 which NaN value should be used in a particular case, so this manual
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158 doesn't try to distinguish them. XEmacs Lisp has no read syntax for NaNs
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159 or infinities; perhaps we should create a syntax in the future.
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160
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161 You can use @code{logb} to extract the binary exponent of a floating
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162 point number (or estimate the logarithm of an integer):
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163
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164 @defun logb number
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165 This function returns the binary exponent of @var{number}. More
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166 precisely, the value is the logarithm of @var{number} base 2, rounded
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167 down to an integer.
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168 @end defun
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169
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170 @node Predicates on Numbers
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171 @section Type Predicates for Numbers
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172
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173 The functions in this section test whether the argument is a number or
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174 whether it is a certain sort of number. The functions @code{integerp}
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175 and @code{floatp} can take any type of Lisp object as argument (the
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176 predicates would not be of much use otherwise); but the @code{zerop}
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177 predicate requires a number as its argument. See also
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178 @code{integer-or-marker-p}, @code{integer-char-or-marker-p},
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179 @code{number-or-marker-p} and @code{number-char-or-marker-p}, in
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180 @ref{Predicates on Markers}.
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181
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182 @defun floatp object
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183 This predicate tests whether its argument is a floating point
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184 number and returns @code{t} if so, @code{nil} otherwise.
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185
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186 @code{floatp} does not exist in Emacs versions 18 and earlier.
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187 @end defun
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188
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189 @defun integerp object
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190 This predicate tests whether its argument is an integer, and returns
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191 @code{t} if so, @code{nil} otherwise.
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192 @end defun
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193
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194 @defun numberp object
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195 This predicate tests whether its argument is a number (either integer or
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196 floating point), and returns @code{t} if so, @code{nil} otherwise.
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197 @end defun
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198
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199 @defun natnump object
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200 @cindex natural numbers
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201 The @code{wholenump} predicate (whose name comes from the phrase
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202 ``natural-number-p'') tests to see whether its argument is a nonnegative
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203 integer, and returns @code{t} if so, @code{nil} otherwise. 0 is
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204 considered non-negative.
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205 @end defun
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206
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207 @defun zerop number
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208 This predicate tests whether its argument is zero, and returns @code{t}
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209 if so, @code{nil} otherwise. The argument must be a number.
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210
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211 These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}.
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212 @end defun
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213
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214 @node Comparison of Numbers
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215 @section Comparison of Numbers
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216 @cindex number equality
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217
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218 To test numbers for numerical equality, you should normally use
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219 @code{=}, not @code{eq}. There can be many distinct floating point
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220 number objects with the same numeric value. If you use @code{eq} to
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221 compare them, then you test whether two values are the same
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222 @emph{object}. By contrast, @code{=} compares only the numeric values
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223 of the objects.
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224
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225 At present, each integer value has a unique Lisp object in XEmacs Lisp.
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226 Therefore, @code{eq} is equivalent @code{=} where integers are
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227 concerned. It is sometimes convenient to use @code{eq} for comparing an
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228 unknown value with an integer, because @code{eq} does not report an
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229 error if the unknown value is not a number---it accepts arguments of any
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230 type. By contrast, @code{=} signals an error if the arguments are not
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231 numbers or markers. However, it is a good idea to use @code{=} if you
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232 can, even for comparing integers, just in case we change the
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233 representation of integers in a future XEmacs version.
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234
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235 There is another wrinkle: because floating point arithmetic is not
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236 exact, it is often a bad idea to check for equality of two floating
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237 point values. Usually it is better to test for approximate equality.
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238 Here's a function to do this:
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239
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240 @example
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241 (defvar fuzz-factor 1.0e-6)
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242 (defun approx-equal (x y)
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243 (or (and (= x 0) (= y 0))
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244 (< (/ (abs (- x y))
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245 (max (abs x) (abs y)))
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246 fuzz-factor)))
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247 @end example
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248
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249 @cindex CL note---integers vrs @code{eq}
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250 @quotation
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251 @b{Common Lisp note:} Comparing numbers in Common Lisp always requires
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252 @code{=} because Common Lisp implements multi-word integers, and two
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253 distinct integer objects can have the same numeric value. XEmacs Lisp
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254 can have just one integer object for any given value because it has a
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255 limited range of integer values.
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256 @end quotation
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257
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258 @defun = number-or-marker1 number-or-marker2
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259 This function tests whether its arguments are numerically equal, and
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260 returns @code{t} if so, @code{nil} otherwise.
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261 @end defun
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262
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263 @defun /= number-or-marker1 number-or-marker2
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264 This function tests whether its arguments are numerically not equal. It
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265 returns @code{t} if so, and @code{nil} otherwise.
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266 @end defun
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267
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268 @defun < number-or-marker1 number-or-marker2
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269 This function tests whether its first argument is strictly less than
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270 its second argument. It returns @code{t} if so, @code{nil} otherwise.
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271 @end defun
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272
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273 @defun <= number-or-marker1 number-or-marker2
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274 This function tests whether its first argument is less than or equal
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275 to its second argument. It returns @code{t} if so, @code{nil}
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276 otherwise.
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277 @end defun
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278
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279 @defun > number-or-marker1 number-or-marker2
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280 This function tests whether its first argument is strictly greater
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281 than its second argument. It returns @code{t} if so, @code{nil}
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282 otherwise.
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283 @end defun
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284
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285 @defun >= number-or-marker1 number-or-marker2
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286 This function tests whether its first argument is greater than or
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287 equal to its second argument. It returns @code{t} if so, @code{nil}
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288 otherwise.
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289 @end defun
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290
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291 @defun max number-or-marker &rest numbers-or-markers
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292 This function returns the largest of its arguments.
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293
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294 @example
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295 (max 20)
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296 @result{} 20
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297 (max 1 2.5)
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298 @result{} 2.5
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299 (max 1 3 2.5)
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300 @result{} 3
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301 @end example
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302 @end defun
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303
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304 @defun min number-or-marker &rest numbers-or-markers
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305 This function returns the smallest of its arguments.
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306
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307 @example
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308 (min -4 1)
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309 @result{} -4
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310 @end example
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311 @end defun
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312
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313 @node Numeric Conversions
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314 @section Numeric Conversions
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315 @cindex rounding in conversions
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316
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317 To convert an integer to floating point, use the function @code{float}.
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318
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319 @defun float number
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320 This returns @var{number} converted to floating point.
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321 If @var{number} is already a floating point number, @code{float} returns
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322 it unchanged.
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323 @end defun
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324
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325 There are four functions to convert floating point numbers to integers;
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326 they differ in how they round. These functions accept integer arguments
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327 also, and return such arguments unchanged.
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328
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329 @defun truncate number
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330 This returns @var{number}, converted to an integer by rounding towards
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331 zero.
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332 @end defun
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333
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334 @defun floor number &optional divisor
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335 This returns @var{number}, converted to an integer by rounding downward
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336 (towards negative infinity).
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337
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338 If @var{divisor} is specified, @var{number} is divided by @var{divisor}
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339 before the floor is taken; this is the division operation that
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340 corresponds to @code{mod}. An @code{arith-error} results if
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341 @var{divisor} is 0.
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342 @end defun
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343
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344 @defun ceiling number
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345 This returns @var{number}, converted to an integer by rounding upward
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346 (towards positive infinity).
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347 @end defun
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348
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349 @defun round number
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350 This returns @var{number}, converted to an integer by rounding towards the
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351 nearest integer. Rounding a value equidistant between two integers
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352 may choose the integer closer to zero, or it may prefer an even integer,
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353 depending on your machine.
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354 @end defun
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355
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356 @node Arithmetic Operations
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357 @section Arithmetic Operations
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358
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359 XEmacs Lisp provides the traditional four arithmetic operations:
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360 addition, subtraction, multiplication, and division. Remainder and modulus
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361 functions supplement the division functions. The functions to
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362 add or subtract 1 are provided because they are traditional in Lisp and
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363 commonly used.
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364
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365 All of these functions except @code{%} return a floating point value
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366 if any argument is floating.
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367
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368 It is important to note that in XEmacs Lisp, arithmetic functions
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369 do not check for overflow. Thus @code{(1+ 134217727)} may evaluate to
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370 @minus{}134217728, depending on your hardware.
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371
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372 @defun 1+ number-or-marker
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373 This function returns @var{number-or-marker} plus 1.
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374 For example,
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375
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376 @example
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377 (setq foo 4)
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378 @result{} 4
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379 (1+ foo)
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380 @result{} 5
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381 @end example
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382
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383 This function is not analogous to the C operator @code{++}---it does not
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384 increment a variable. It just computes a sum. Thus, if we continue,
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385
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386 @example
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387 foo
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388 @result{} 4
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389 @end example
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390
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391 If you want to increment the variable, you must use @code{setq},
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392 like this:
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393
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394 @example
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395 (setq foo (1+ foo))
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396 @result{} 5
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397 @end example
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398 @end defun
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399
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400 @defun 1- number-or-marker
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401 This function returns @var{number-or-marker} minus 1.
|
|
|
402 @end defun
|
|
|
403
|
|
|
404 @defun abs number
|
|
|
405 This returns the absolute value of @var{number}.
|
|
|
406 @end defun
|
|
|
407
|
|
|
408 @defun + &rest numbers-or-markers
|
|
|
409 This function adds its arguments together. When given no arguments,
|
|
|
410 @code{+} returns 0.
|
|
|
411
|
|
|
412 @example
|
|
|
413 (+)
|
|
|
414 @result{} 0
|
|
|
415 (+ 1)
|
|
|
416 @result{} 1
|
|
|
417 (+ 1 2 3 4)
|
|
|
418 @result{} 10
|
|
|
419 @end example
|
|
|
420 @end defun
|
|
|
421
|
|
|
422 @defun - &optional number-or-marker &rest other-numbers-or-markers
|
|
|
423 The @code{-} function serves two purposes: negation and subtraction.
|
|
|
424 When @code{-} has a single argument, the value is the negative of the
|
|
|
425 argument. When there are multiple arguments, @code{-} subtracts each of
|
|
|
426 the @var{other-numbers-or-markers} from @var{number-or-marker},
|
|
|
427 cumulatively. If there are no arguments, the result is 0.
|
|
|
428
|
|
|
429 @example
|
|
|
430 (- 10 1 2 3 4)
|
|
|
431 @result{} 0
|
|
|
432 (- 10)
|
|
|
433 @result{} -10
|
|
|
434 (-)
|
|
|
435 @result{} 0
|
|
|
436 @end example
|
|
|
437 @end defun
|
|
|
438
|
|
|
439 @defun * &rest numbers-or-markers
|
|
|
440 This function multiplies its arguments together, and returns the
|
|
|
441 product. When given no arguments, @code{*} returns 1.
|
|
|
442
|
|
|
443 @example
|
|
|
444 (*)
|
|
|
445 @result{} 1
|
|
|
446 (* 1)
|
|
|
447 @result{} 1
|
|
|
448 (* 1 2 3 4)
|
|
|
449 @result{} 24
|
|
|
450 @end example
|
|
|
451 @end defun
|
|
|
452
|
|
|
453 @defun / dividend divisor &rest divisors
|
|
|
454 This function divides @var{dividend} by @var{divisor} and returns the
|
|
|
455 quotient. If there are additional arguments @var{divisors}, then it
|
|
|
456 divides @var{dividend} by each divisor in turn. Each argument may be a
|
|
|
457 number or a marker.
|
|
|
458
|
|
|
459 If all the arguments are integers, then the result is an integer too.
|
|
|
460 This means the result has to be rounded. On most machines, the result
|
|
|
461 is rounded towards zero after each division, but some machines may round
|
|
|
462 differently with negative arguments. This is because the Lisp function
|
|
|
463 @code{/} is implemented using the C division operator, which also
|
|
|
464 permits machine-dependent rounding. As a practical matter, all known
|
|
|
465 machines round in the standard fashion.
|
|
|
466
|
|
|
467 @cindex @code{arith-error} in division
|
|
|
468 If you divide by 0, an @code{arith-error} error is signaled.
|
|
|
469 (@xref{Errors}.)
|
|
|
470
|
|
|
471 @example
|
|
|
472 @group
|
|
|
473 (/ 6 2)
|
|
|
474 @result{} 3
|
|
|
475 @end group
|
|
|
476 (/ 5 2)
|
|
|
477 @result{} 2
|
|
|
478 (/ 25 3 2)
|
|
|
479 @result{} 4
|
|
|
480 (/ -17 6)
|
|
|
481 @result{} -2
|
|
|
482 @end example
|
|
|
483
|
|
|
484 The result of @code{(/ -17 6)} could in principle be -3 on some
|
|
|
485 machines.
|
|
|
486 @end defun
|
|
|
487
|
|
|
488 @defun % dividend divisor
|
|
|
489 @cindex remainder
|
|
|
490 This function returns the integer remainder after division of @var{dividend}
|
|
|
491 by @var{divisor}. The arguments must be integers or markers.
|
|
|
492
|
|
|
493 For negative arguments, the remainder is in principle machine-dependent
|
|
|
494 since the quotient is; but in practice, all known machines behave alike.
|
|
|
495
|
|
|
496 An @code{arith-error} results if @var{divisor} is 0.
|
|
|
497
|
|
|
498 @example
|
|
|
499 (% 9 4)
|
|
|
500 @result{} 1
|
|
|
501 (% -9 4)
|
|
|
502 @result{} -1
|
|
|
503 (% 9 -4)
|
|
|
504 @result{} 1
|
|
|
505 (% -9 -4)
|
|
|
506 @result{} -1
|
|
|
507 @end example
|
|
|
508
|
|
|
509 For any two integers @var{dividend} and @var{divisor},
|
|
|
510
|
|
|
511 @example
|
|
|
512 @group
|
|
|
513 (+ (% @var{dividend} @var{divisor})
|
|
|
514 (* (/ @var{dividend} @var{divisor}) @var{divisor}))
|
|
|
515 @end group
|
|
|
516 @end example
|
|
|
517
|
|
|
518 @noindent
|
|
|
519 always equals @var{dividend}.
|
|
|
520 @end defun
|
|
|
521
|
|
|
522 @defun mod dividend divisor
|
|
|
523 @cindex modulus
|
|
|
524 This function returns the value of @var{dividend} modulo @var{divisor};
|
|
|
525 in other words, the remainder after division of @var{dividend}
|
|
|
526 by @var{divisor}, but with the same sign as @var{divisor}.
|
|
|
527 The arguments must be numbers or markers.
|
|
|
528
|
|
|
529 Unlike @code{%}, @code{mod} returns a well-defined result for negative
|
|
|
530 arguments. It also permits floating point arguments; it rounds the
|
|
|
531 quotient downward (towards minus infinity) to an integer, and uses that
|
|
|
532 quotient to compute the remainder.
|
|
|
533
|
|
|
534 An @code{arith-error} results if @var{divisor} is 0.
|
|
|
535
|
|
|
536 @example
|
|
|
537 @group
|
|
|
538 (mod 9 4)
|
|
|
539 @result{} 1
|
|
|
540 @end group
|
|
|
541 @group
|
|
|
542 (mod -9 4)
|
|
|
543 @result{} 3
|
|
|
544 @end group
|
|
|
545 @group
|
|
|
546 (mod 9 -4)
|
|
|
547 @result{} -3
|
|
|
548 @end group
|
|
|
549 @group
|
|
|
550 (mod -9 -4)
|
|
|
551 @result{} -1
|
|
|
552 @end group
|
|
|
553 @group
|
|
|
554 (mod 5.5 2.5)
|
|
|
555 @result{} .5
|
|
|
556 @end group
|
|
|
557 @end example
|
|
|
558
|
|
|
559 For any two numbers @var{dividend} and @var{divisor},
|
|
|
560
|
|
|
561 @example
|
|
|
562 @group
|
|
|
563 (+ (mod @var{dividend} @var{divisor})
|
|
|
564 (* (floor @var{dividend} @var{divisor}) @var{divisor}))
|
|
|
565 @end group
|
|
|
566 @end example
|
|
|
567
|
|
|
568 @noindent
|
|
|
569 always equals @var{dividend}, subject to rounding error if either
|
|
|
570 argument is floating point. For @code{floor}, see @ref{Numeric
|
|
|
571 Conversions}.
|
|
|
572 @end defun
|
|
|
573
|
|
|
574 @node Rounding Operations
|
|
|
575 @section Rounding Operations
|
|
|
576 @cindex rounding without conversion
|
|
|
577
|
|
|
578 The functions @code{ffloor}, @code{fceiling}, @code{fround} and
|
|
|
579 @code{ftruncate} take a floating point argument and return a floating
|
|
|
580 point result whose value is a nearby integer. @code{ffloor} returns the
|
|
|
581 nearest integer below; @code{fceiling}, the nearest integer above;
|
|
|
582 @code{ftruncate}, the nearest integer in the direction towards zero;
|
|
|
583 @code{fround}, the nearest integer.
|
|
|
584
|
|
|
585 @defun ffloor float
|
|
|
586 This function rounds @var{float} to the next lower integral value, and
|
|
|
587 returns that value as a floating point number.
|
|
|
588 @end defun
|
|
|
589
|
|
|
590 @defun fceiling float
|
|
|
591 This function rounds @var{float} to the next higher integral value, and
|
|
|
592 returns that value as a floating point number.
|
|
|
593 @end defun
|
|
|
594
|
|
|
595 @defun ftruncate float
|
|
|
596 This function rounds @var{float} towards zero to an integral value, and
|
|
|
597 returns that value as a floating point number.
|
|
|
598 @end defun
|
|
|
599
|
|
|
600 @defun fround float
|
|
|
601 This function rounds @var{float} to the nearest integral value,
|
|
|
602 and returns that value as a floating point number.
|
|
|
603 @end defun
|
|
|
604
|
|
|
605 @node Bitwise Operations
|
|
|
606 @section Bitwise Operations on Integers
|
|
|
607
|
|
|
608 In a computer, an integer is represented as a binary number, a
|
|
|
609 sequence of @dfn{bits} (digits which are either zero or one). A bitwise
|
|
|
610 operation acts on the individual bits of such a sequence. For example,
|
|
|
611 @dfn{shifting} moves the whole sequence left or right one or more places,
|
|
|
612 reproducing the same pattern ``moved over''.
|
|
|
613
|
|
|
614 The bitwise operations in XEmacs Lisp apply only to integers.
|
|
|
615
|
|
|
616 @defun lsh integer1 count
|
|
|
617 @cindex logical shift
|
|
|
618 @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the
|
|
|
619 bits in @var{integer1} to the left @var{count} places, or to the right
|
|
|
620 if @var{count} is negative, bringing zeros into the vacated bits. If
|
|
|
621 @var{count} is negative, @code{lsh} shifts zeros into the leftmost
|
|
|
622 (most-significant) bit, producing a positive result even if
|
|
|
623 @var{integer1} is negative. Contrast this with @code{ash}, below.
|
|
|
624
|
|
|
625 Here are two examples of @code{lsh}, shifting a pattern of bits one
|
|
|
626 place to the left. We show only the low-order eight bits of the binary
|
|
|
627 pattern; the rest are all zero.
|
|
|
628
|
|
|
629 @example
|
|
|
630 @group
|
|
|
631 (lsh 5 1)
|
|
|
632 @result{} 10
|
|
|
633 ;; @r{Decimal 5 becomes decimal 10.}
|
|
|
634 00000101 @result{} 00001010
|
|
|
635
|
|
|
636 (lsh 7 1)
|
|
|
637 @result{} 14
|
|
|
638 ;; @r{Decimal 7 becomes decimal 14.}
|
|
|
639 00000111 @result{} 00001110
|
|
|
640 @end group
|
|
|
641 @end example
|
|
|
642
|
|
|
643 @noindent
|
|
|
644 As the examples illustrate, shifting the pattern of bits one place to
|
|
|
645 the left produces a number that is twice the value of the previous
|
|
|
646 number.
|
|
|
647
|
|
|
648 Shifting a pattern of bits two places to the left produces results
|
|
|
649 like this (with 8-bit binary numbers):
|
|
|
650
|
|
|
651 @example
|
|
|
652 @group
|
|
|
653 (lsh 3 2)
|
|
|
654 @result{} 12
|
|
|
655 ;; @r{Decimal 3 becomes decimal 12.}
|
|
|
656 00000011 @result{} 00001100
|
|
|
657 @end group
|
|
|
658 @end example
|
|
|
659
|
|
|
660 On the other hand, shifting one place to the right looks like this:
|
|
|
661
|
|
|
662 @example
|
|
|
663 @group
|
|
|
664 (lsh 6 -1)
|
|
|
665 @result{} 3
|
|
|
666 ;; @r{Decimal 6 becomes decimal 3.}
|
|
|
667 00000110 @result{} 00000011
|
|
|
668 @end group
|
|
|
669
|
|
|
670 @group
|
|
|
671 (lsh 5 -1)
|
|
|
672 @result{} 2
|
|
|
673 ;; @r{Decimal 5 becomes decimal 2.}
|
|
|
674 00000101 @result{} 00000010
|
|
|
675 @end group
|
|
|
676 @end example
|
|
|
677
|
|
|
678 @noindent
|
|
|
679 As the example illustrates, shifting one place to the right divides the
|
|
|
680 value of a positive integer by two, rounding downward.
|
|
|
681
|
|
|
682 The function @code{lsh}, like all XEmacs Lisp arithmetic functions, does
|
|
|
683 not check for overflow, so shifting left can discard significant bits
|
|
|
684 and change the sign of the number. For example, left shifting
|
|
|
685 134,217,727 produces @minus{}2 on a 28-bit machine:
|
|
|
686
|
|
|
687 @example
|
|
|
688 (lsh 134217727 1) ; @r{left shift}
|
|
|
689 @result{} -2
|
|
|
690 @end example
|
|
|
691
|
|
|
692 In binary, in the 28-bit implementation, the argument looks like this:
|
|
|
693
|
|
|
694 @example
|
|
|
695 @group
|
|
|
696 ;; @r{Decimal 134,217,727}
|
|
|
697 0111 1111 1111 1111 1111 1111 1111
|
|
|
698 @end group
|
|
|
699 @end example
|
|
|
700
|
|
|
701 @noindent
|
|
|
702 which becomes the following when left shifted:
|
|
|
703
|
|
|
704 @example
|
|
|
705 @group
|
|
|
706 ;; @r{Decimal @minus{}2}
|
|
|
707 1111 1111 1111 1111 1111 1111 1110
|
|
|
708 @end group
|
|
|
709 @end example
|
|
|
710 @end defun
|
|
|
711
|
|
|
712 @defun ash integer1 count
|
|
|
713 @cindex arithmetic shift
|
|
|
714 @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1}
|
|
|
715 to the left @var{count} places, or to the right if @var{count}
|
|
|
716 is negative.
|
|
|
717
|
|
|
718 @code{ash} gives the same results as @code{lsh} except when
|
|
|
719 @var{integer1} and @var{count} are both negative. In that case,
|
|
|
720 @code{ash} puts ones in the empty bit positions on the left, while
|
|
|
721 @code{lsh} puts zeros in those bit positions.
|
|
|
722
|
|
|
723 Thus, with @code{ash}, shifting the pattern of bits one place to the right
|
|
|
724 looks like this:
|
|
|
725
|
|
|
726 @example
|
|
|
727 @group
|
|
|
728 (ash -6 -1) @result{} -3
|
|
|
729 ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
|
|
|
730 1111 1111 1111 1111 1111 1111 1010
|
|
|
731 @result{}
|
|
|
732 1111 1111 1111 1111 1111 1111 1101
|
|
|
733 @end group
|
|
|
734 @end example
|
|
|
735
|
|
|
736 In contrast, shifting the pattern of bits one place to the right with
|
|
|
737 @code{lsh} looks like this:
|
|
|
738
|
|
|
739 @example
|
|
|
740 @group
|
|
|
741 (lsh -6 -1) @result{} 134217725
|
|
|
742 ;; @r{Decimal @minus{}6 becomes decimal 134,217,725.}
|
|
|
743 1111 1111 1111 1111 1111 1111 1010
|
|
|
744 @result{}
|
|
|
745 0111 1111 1111 1111 1111 1111 1101
|
|
|
746 @end group
|
|
|
747 @end example
|
|
|
748
|
|
|
749 Here are other examples:
|
|
|
750
|
|
|
751 @c !!! Check if lined up in smallbook format! XDVI shows problem
|
|
|
752 @c with smallbook but not with regular book! --rjc 16mar92
|
|
|
753 @smallexample
|
|
|
754 @group
|
|
|
755 ; @r{ 28-bit binary values}
|
|
|
756
|
|
|
757 (lsh 5 2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
758 @result{} 20 ; = @r{0000 0000 0000 0000 0000 0001 0100}
|
|
|
759 @end group
|
|
|
760 @group
|
|
|
761 (ash 5 2)
|
|
|
762 @result{} 20
|
|
|
763 (lsh -5 2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011}
|
|
|
764 @result{} -20 ; = @r{1111 1111 1111 1111 1111 1110 1100}
|
|
|
765 (ash -5 2)
|
|
|
766 @result{} -20
|
|
|
767 @end group
|
|
|
768 @group
|
|
|
769 (lsh 5 -2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
770 @result{} 1 ; = @r{0000 0000 0000 0000 0000 0000 0001}
|
|
|
771 @end group
|
|
|
772 @group
|
|
|
773 (ash 5 -2)
|
|
|
774 @result{} 1
|
|
|
775 @end group
|
|
|
776 @group
|
|
|
777 (lsh -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011}
|
|
|
778 @result{} 4194302 ; = @r{0011 1111 1111 1111 1111 1111 1110}
|
|
|
779 @end group
|
|
|
780 @group
|
|
|
781 (ash -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011}
|
|
|
782 @result{} -2 ; = @r{1111 1111 1111 1111 1111 1111 1110}
|
|
|
783 @end group
|
|
|
784 @end smallexample
|
|
|
785 @end defun
|
|
|
786
|
|
|
787 @defun logand &rest ints-or-markers
|
|
|
788 @cindex logical and
|
|
|
789 @cindex bitwise and
|
|
|
790 This function returns the ``logical and'' of the arguments: the
|
|
|
791 @var{n}th bit is set in the result if, and only if, the @var{n}th bit is
|
|
|
792 set in all the arguments. (``Set'' means that the value of the bit is 1
|
|
|
793 rather than 0.)
|
|
|
794
|
|
|
795 For example, using 4-bit binary numbers, the ``logical and'' of 13 and
|
|
|
796 12 is 12: 1101 combined with 1100 produces 1100.
|
|
|
797 In both the binary numbers, the leftmost two bits are set (i.e., they
|
|
|
798 are 1's), so the leftmost two bits of the returned value are set.
|
|
|
799 However, for the rightmost two bits, each is zero in at least one of
|
|
|
800 the arguments, so the rightmost two bits of the returned value are 0's.
|
|
|
801
|
|
|
802 @noindent
|
|
|
803 Therefore,
|
|
|
804
|
|
|
805 @example
|
|
|
806 @group
|
|
|
807 (logand 13 12)
|
|
|
808 @result{} 12
|
|
|
809 @end group
|
|
|
810 @end example
|
|
|
811
|
|
|
812 If @code{logand} is not passed any argument, it returns a value of
|
|
|
813 @minus{}1. This number is an identity element for @code{logand}
|
|
|
814 because its binary representation consists entirely of ones. If
|
|
|
815 @code{logand} is passed just one argument, it returns that argument.
|
|
|
816
|
|
|
817 @smallexample
|
|
|
818 @group
|
|
|
819 ; @r{ 28-bit binary values}
|
|
|
820
|
|
|
821 (logand 14 13) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110}
|
|
|
822 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101}
|
|
|
823 @result{} 12 ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
824 @end group
|
|
|
825
|
|
|
826 @group
|
|
|
827 (logand 14 13 4) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110}
|
|
|
828 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101}
|
|
|
829 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100}
|
|
|
830 @result{} 4 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100}
|
|
|
831 @end group
|
|
|
832
|
|
|
833 @group
|
|
|
834 (logand)
|
|
|
835 @result{} -1 ; -1 = @r{1111 1111 1111 1111 1111 1111 1111}
|
|
|
836 @end group
|
|
|
837 @end smallexample
|
|
|
838 @end defun
|
|
|
839
|
|
|
840 @defun logior &rest ints-or-markers
|
|
|
841 @cindex logical inclusive or
|
|
|
842 @cindex bitwise or
|
|
|
843 This function returns the ``inclusive or'' of its arguments: the @var{n}th bit
|
|
|
844 is set in the result if, and only if, the @var{n}th bit is set in at least
|
|
|
845 one of the arguments. If there are no arguments, the result is zero,
|
|
|
846 which is an identity element for this operation. If @code{logior} is
|
|
|
847 passed just one argument, it returns that argument.
|
|
|
848
|
|
|
849 @smallexample
|
|
|
850 @group
|
|
|
851 ; @r{ 28-bit binary values}
|
|
|
852
|
|
|
853 (logior 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
854 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
855 @result{} 13 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101}
|
|
|
856 @end group
|
|
|
857
|
|
|
858 @group
|
|
|
859 (logior 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
860 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
861 ; 7 = @r{0000 0000 0000 0000 0000 0000 0111}
|
|
|
862 @result{} 15 ; 15 = @r{0000 0000 0000 0000 0000 0000 1111}
|
|
|
863 @end group
|
|
|
864 @end smallexample
|
|
|
865 @end defun
|
|
|
866
|
|
|
867 @defun logxor &rest ints-or-markers
|
|
|
868 @cindex bitwise exclusive or
|
|
|
869 @cindex logical exclusive or
|
|
|
870 This function returns the ``exclusive or'' of its arguments: the
|
|
|
871 @var{n}th bit is set in the result if, and only if, the @var{n}th bit is
|
|
|
872 set in an odd number of the arguments. If there are no arguments, the
|
|
|
873 result is 0, which is an identity element for this operation. If
|
|
|
874 @code{logxor} is passed just one argument, it returns that argument.
|
|
|
875
|
|
|
876 @smallexample
|
|
|
877 @group
|
|
|
878 ; @r{ 28-bit binary values}
|
|
|
879
|
|
|
880 (logxor 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
881 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
882 @result{} 9 ; 9 = @r{0000 0000 0000 0000 0000 0000 1001}
|
|
|
883 @end group
|
|
|
884
|
|
|
885 @group
|
|
|
886 (logxor 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
887 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
888 ; 7 = @r{0000 0000 0000 0000 0000 0000 0111}
|
|
|
889 @result{} 14 ; 14 = @r{0000 0000 0000 0000 0000 0000 1110}
|
|
|
890 @end group
|
|
|
891 @end smallexample
|
|
|
892 @end defun
|
|
|
893
|
|
|
894 @defun lognot integer
|
|
|
895 @cindex logical not
|
|
|
896 @cindex bitwise not
|
|
|
897 This function returns the logical complement of its argument: the @var{n}th
|
|
|
898 bit is one in the result if, and only if, the @var{n}th bit is zero in
|
|
|
899 @var{integer}, and vice-versa.
|
|
|
900
|
|
|
901 @example
|
|
|
902 (lognot 5)
|
|
|
903 @result{} -6
|
|
|
904 ;; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
905 ;; @r{becomes}
|
|
|
906 ;; -6 = @r{1111 1111 1111 1111 1111 1111 1010}
|
|
|
907 @end example
|
|
|
908 @end defun
|
|
|
909
|
|
|
910 @node Math Functions
|
|
|
911 @section Standard Mathematical Functions
|
|
|
912 @cindex transcendental functions
|
|
|
913 @cindex mathematical functions
|
|
|
914
|
|
|
915 These mathematical functions are available if floating point is
|
|
|
916 supported (which is the normal state of affairs). They allow integers
|
|
|
917 as well as floating point numbers as arguments.
|
|
|
918
|
|
|
919 @defun sin arg
|
|
|
920 @defunx cos arg
|
|
|
921 @defunx tan arg
|
|
|
922 These are the ordinary trigonometric functions, with argument measured
|
|
|
923 in radians.
|
|
|
924 @end defun
|
|
|
925
|
|
|
926 @defun asin arg
|
|
|
927 The value of @code{(asin @var{arg})} is a number between @minus{}pi/2
|
|
|
928 and pi/2 (inclusive) whose sine is @var{arg}; if, however, @var{arg}
|
|
|
929 is out of range (outside [-1, 1]), then the result is a NaN.
|
|
|
930 @end defun
|
|
|
931
|
|
|
932 @defun acos arg
|
|
|
933 The value of @code{(acos @var{arg})} is a number between 0 and pi
|
|
|
934 (inclusive) whose cosine is @var{arg}; if, however, @var{arg}
|
|
|
935 is out of range (outside [-1, 1]), then the result is a NaN.
|
|
|
936 @end defun
|
|
|
937
|
|
|
938 @defun atan arg
|
|
|
939 The value of @code{(atan @var{arg})} is a number between @minus{}pi/2
|
|
|
940 and pi/2 (exclusive) whose tangent is @var{arg}.
|
|
|
941 @end defun
|
|
|
942
|
|
|
943 @defun sinh arg
|
|
|
944 @defunx cosh arg
|
|
|
945 @defunx tanh arg
|
|
|
946 These are the ordinary hyperbolic trigonometric functions.
|
|
|
947 @end defun
|
|
|
948
|
|
|
949 @defun asinh arg
|
|
|
950 @defunx acosh arg
|
|
|
951 @defunx atanh arg
|
|
|
952 These are the inverse hyperbolic trigonometric functions.
|
|
|
953 @end defun
|
|
|
954
|
|
|
955 @defun exp arg
|
|
|
956 This is the exponential function; it returns @i{e} to the power
|
|
|
957 @var{arg}. @i{e} is a fundamental mathematical constant also called the
|
|
|
958 base of natural logarithms.
|
|
|
959 @end defun
|
|
|
960
|
|
|
961 @defun log arg &optional base
|
|
|
962 This function returns the logarithm of @var{arg}, with base @var{base}.
|
|
|
963 If you don't specify @var{base}, the base @var{e} is used. If @var{arg}
|
|
|
964 is negative, the result is a NaN.
|
|
|
965 @end defun
|
|
|
966
|
|
|
967 @ignore
|
|
|
968 @defun expm1 arg
|
|
|
969 This function returns @code{(1- (exp @var{arg}))}, but it is more
|
|
|
970 accurate than that when @var{arg} is negative and @code{(exp @var{arg})}
|
|
|
971 is close to 1.
|
|
|
972 @end defun
|
|
|
973
|
|
|
974 @defun log1p arg
|
|
|
975 This function returns @code{(log (1+ @var{arg}))}, but it is more
|
|
|
976 accurate than that when @var{arg} is so small that adding 1 to it would
|
|
|
977 lose accuracy.
|
|
|
978 @end defun
|
|
|
979 @end ignore
|
|
|
980
|
|
|
981 @defun log10 arg
|
|
|
982 This function returns the logarithm of @var{arg}, with base 10. If
|
|
|
983 @var{arg} is negative, the result is a NaN. @code{(log10 @var{x})}
|
|
|
984 @equiv{} @code{(log @var{x} 10)}, at least approximately.
|
|
|
985 @end defun
|
|
|
986
|
|
|
987 @defun expt x y
|
|
|
988 This function returns @var{x} raised to power @var{y}. If both
|
|
|
989 arguments are integers and @var{y} is positive, the result is an
|
|
|
990 integer; in this case, it is truncated to fit the range of possible
|
|
|
991 integer values.
|
|
|
992 @end defun
|
|
|
993
|
|
|
994 @defun sqrt arg
|
|
|
995 This returns the square root of @var{arg}. If @var{arg} is negative,
|
|
|
996 the value is a NaN.
|
|
|
997 @end defun
|
|
|
998
|
|
|
999 @defun cube-root arg
|
|
|
1000 This returns the cube root of @var{arg}.
|
|
|
1001 @end defun
|
|
|
1002
|
|
|
1003 @node Random Numbers
|
|
|
1004 @section Random Numbers
|
|
|
1005 @cindex random numbers
|
|
|
1006
|
|
|
1007 A deterministic computer program cannot generate true random numbers.
|
|
|
1008 For most purposes, @dfn{pseudo-random numbers} suffice. A series of
|
|
|
1009 pseudo-random numbers is generated in a deterministic fashion. The
|
|
|
1010 numbers are not truly random, but they have certain properties that
|
|
|
1011 mimic a random series. For example, all possible values occur equally
|
|
|
1012 often in a pseudo-random series.
|
|
|
1013
|
|
|
1014 In XEmacs, pseudo-random numbers are generated from a ``seed'' number.
|
|
|
1015 Starting from any given seed, the @code{random} function always
|
|
|
1016 generates the same sequence of numbers. XEmacs always starts with the
|
|
|
1017 same seed value, so the sequence of values of @code{random} is actually
|
|
|
1018 the same in each XEmacs run! For example, in one operating system, the
|
|
|
1019 first call to @code{(random)} after you start XEmacs always returns
|
|
|
1020 -1457731, and the second one always returns -7692030. This
|
|
|
1021 repeatability is helpful for debugging.
|
|
|
1022
|
|
|
1023 If you want truly unpredictable random numbers, execute @code{(random
|
|
|
1024 t)}. This chooses a new seed based on the current time of day and on
|
|
|
1025 XEmacs's process @sc{id} number.
|
|
|
1026
|
|
|
1027 @defun random &optional limit
|
|
|
1028 This function returns a pseudo-random integer. Repeated calls return a
|
|
|
1029 series of pseudo-random integers.
|
|
|
1030
|
|
|
1031 If @var{limit} is a positive integer, the value is chosen to be
|
|
|
1032 nonnegative and less than @var{limit}.
|
|
|
1033
|
|
|
1034 If @var{limit} is @code{t}, it means to choose a new seed based on the
|
|
|
1035 current time of day and on XEmacs's process @sc{id} number.
|
|
|
1036 @c "XEmacs'" is incorrect usage!
|
|
|
1037
|
|
|
1038 On some machines, any integer representable in Lisp may be the result
|
|
|
1039 of @code{random}. On other machines, the result can never be larger
|
|
|
1040 than a certain maximum or less than a certain (negative) minimum.
|
|
|
1041 @end defun
|
