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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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257
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13 @minus{}3, 0, #b0111, #xFEED, #o744. Their values are exact. The
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14 number prefixes `#b', `#o', and `#x' are supported to represent numbers
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15 in binary, octal, and hexadecimal notation (or radix). Floating point
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16 numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or
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17 2.71828. They can also be expressed in exponential notation: 1.5e2
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18 equals 150; in this example, @samp{e2} stands for ten to the second
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19 power, and is multiplied by 1.5. Floating point values are not exact;
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20 they have a fixed, limited amount of precision.
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21
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22 @menu
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23 * Integer Basics:: Representation and range of integers.
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24 * Float Basics:: Representation and range of floating point.
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25 * Predicates on Numbers:: Testing for numbers.
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26 * Comparison of Numbers:: Equality and inequality predicates.
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27 * Numeric Conversions:: Converting float to integer and vice versa.
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28 * Arithmetic Operations:: How to add, subtract, multiply and divide.
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29 * Rounding Operations:: Explicitly rounding floating point numbers.
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30 * Bitwise Operations:: Logical and, or, not, shifting.
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31 * Math Functions:: Trig, exponential and logarithmic functions.
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32 * Random Numbers:: Obtaining random integers, predictable or not.
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33 @end menu
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34
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35 @node Integer Basics
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36 @section Integer Basics
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37
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38 The range of values for an integer depends on the machine. The
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39 minimum range is @minus{}134217728 to 134217727 (28 bits; i.e.,
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40 @ifinfo
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41 -2**27
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42 @end ifinfo
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43 @tex
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44 $-2^{27}$
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45 @end tex
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46 to
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47 @ifinfo
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48 2**27 - 1),
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49 @end ifinfo
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50 @tex
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51 $2^{27}-1$),
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52 @end tex
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53 but some machines may provide a wider range. Many examples in this
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54 chapter assume an integer has 28 bits.
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55 @cindex overflow
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56
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57 The Lisp reader reads an integer as a sequence of digits with optional
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58 initial sign and optional final period.
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59
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60 @example
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61 1 ; @r{The integer 1.}
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62 1. ; @r{The integer 1.}
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63 +1 ; @r{Also the integer 1.}
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64 -1 ; @r{The integer @minus{}1.}
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65 268435457 ; @r{Also the integer 1, due to overflow.}
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66 0 ; @r{The integer 0.}
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67 -0 ; @r{The integer 0.}
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68 @end example
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69
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70 To understand how various functions work on integers, especially the
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71 bitwise operators (@pxref{Bitwise Operations}), it is often helpful to
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72 view the numbers in their binary form.
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73
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74 In 28-bit binary, the decimal integer 5 looks like this:
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75
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76 @example
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77 0000 0000 0000 0000 0000 0000 0101
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78 @end example
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79
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80 @noindent
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81 (We have inserted spaces between groups of 4 bits, and two spaces
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82 between groups of 8 bits, to make the binary integer easier to read.)
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83
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84 The integer @minus{}1 looks like this:
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85
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86 @example
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87 1111 1111 1111 1111 1111 1111 1111
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88 @end example
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89
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90 @noindent
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91 @cindex two's complement
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92 @minus{}1 is represented as 28 ones. (This is called @dfn{two's
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93 complement} notation.)
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94
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95 The negative integer, @minus{}5, is creating by subtracting 4 from
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96 @minus{}1. In binary, the decimal integer 4 is 100. Consequently,
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97 @minus{}5 looks like this:
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98
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99 @example
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100 1111 1111 1111 1111 1111 1111 1011
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101 @end example
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102
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103 In this implementation, the largest 28-bit binary integer is the
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104 decimal integer 134,217,727. In binary, it looks like this:
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105
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106 @example
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107 0111 1111 1111 1111 1111 1111 1111
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108 @end example
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109
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110 Since the arithmetic functions do not check whether integers go
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111 outside their range, when you add 1 to 134,217,727, the value is the
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112 negative integer @minus{}134,217,728:
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113
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114 @example
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115 (+ 1 134217727)
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116 @result{} -134217728
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117 @result{} 1000 0000 0000 0000 0000 0000 0000
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118 @end example
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119
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120 Many of the following functions accept markers for arguments as well
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121 as integers. (@xref{Markers}.) More precisely, the actual arguments to
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122 such functions may be either integers or markers, which is why we often
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123 give these arguments the name @var{int-or-marker}. When the argument
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124 value is a marker, its position value is used and its buffer is ignored.
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125
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126 @ignore
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127 In version 19, except where @emph{integer} is specified as an
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128 argument, all of the functions for markers and integers also work for
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129 floating point numbers.
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130 @end ignore
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131
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132 @node Float Basics
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133 @section Floating Point Basics
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134
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135 XEmacs supports floating point numbers. The precise range of floating
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136 point numbers is machine-specific; it is the same as the range of the C
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137 data type @code{double} on the machine in question.
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138
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139 The printed representation for floating point numbers requires either
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140 a decimal point (with at least one digit following), an exponent, or
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141 both. For example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2},
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142 @samp{1.5e3}, and @samp{.15e4} are five ways of writing a floating point
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143 number whose value is 1500. They are all equivalent. You can also use
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144 a minus sign to write negative floating point numbers, as in
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145 @samp{-1.0}.
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146
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147 @cindex IEEE floating point
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148 @cindex positive infinity
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149 @cindex negative infinity
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150 @cindex infinity
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151 @cindex NaN
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152 Most modern computers support the IEEE floating point standard, which
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153 provides for positive infinity and negative infinity as floating point
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154 values. It also provides for a class of values called NaN or
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155 ``not-a-number''; numerical functions return such values in cases where
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156 there is no correct answer. For example, @code{(sqrt -1.0)} returns a
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157 NaN. For practical purposes, there's no significant difference between
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158 different NaN values in XEmacs Lisp, and there's no rule for precisely
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159 which NaN value should be used in a particular case, so this manual
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160 doesn't try to distinguish them. XEmacs Lisp has no read syntax for NaNs
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161 or infinities; perhaps we should create a syntax in the future.
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162
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163 You can use @code{logb} to extract the binary exponent of a floating
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164 point number (or estimate the logarithm of an integer):
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165
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166 @defun logb number
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167 This function returns the binary exponent of @var{number}. More
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168 precisely, the value is the logarithm of @var{number} base 2, rounded
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169 down to an integer.
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170 @end defun
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171
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172 @node Predicates on Numbers
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173 @section Type Predicates for Numbers
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174
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175 The functions in this section test whether the argument is a number or
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176 whether it is a certain sort of number. The functions @code{integerp}
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177 and @code{floatp} can take any type of Lisp object as argument (the
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178 predicates would not be of much use otherwise); but the @code{zerop}
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179 predicate requires a number as its argument. See also
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180 @code{integer-or-marker-p}, @code{integer-char-or-marker-p},
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181 @code{number-or-marker-p} and @code{number-char-or-marker-p}, in
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182 @ref{Predicates on Markers}.
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183
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184 @defun floatp object
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185 This predicate tests whether its argument is a floating point
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186 number and returns @code{t} if so, @code{nil} otherwise.
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187
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188 @code{floatp} does not exist in Emacs versions 18 and earlier.
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189 @end defun
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190
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191 @defun integerp object
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192 This predicate tests whether its argument is an integer, and returns
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193 @code{t} if so, @code{nil} otherwise.
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194 @end defun
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195
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196 @defun numberp object
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197 This predicate tests whether its argument is a number (either integer or
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198 floating point), and returns @code{t} if so, @code{nil} otherwise.
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199 @end defun
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200
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201 @defun natnump object
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202 @cindex natural numbers
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203 The @code{natnump} predicate (whose name comes from the phrase
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204 ``natural-number-p'') tests to see whether its argument is a nonnegative
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205 integer, and returns @code{t} if so, @code{nil} otherwise. 0 is
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206 considered non-negative.
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207 @end defun
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208
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209 @defun zerop number
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210 This predicate tests whether its argument is zero, and returns @code{t}
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211 if so, @code{nil} otherwise. The argument must be a number.
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212
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213 These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}.
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214 @end defun
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215
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216 @node Comparison of Numbers
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217 @section Comparison of Numbers
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218 @cindex number equality
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219
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220 To test numbers for numerical equality, you should normally use
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221 @code{=}, not @code{eq}. There can be many distinct floating point
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222 number objects with the same numeric value. If you use @code{eq} to
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223 compare them, then you test whether two values are the same
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224 @emph{object}. By contrast, @code{=} compares only the numeric values
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225 of the objects.
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226
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227 At present, each integer value has a unique Lisp object in XEmacs Lisp.
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228 Therefore, @code{eq} is equivalent to @code{=} where integers are
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229 concerned. It is sometimes convenient to use @code{eq} for comparing an
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230 unknown value with an integer, because @code{eq} does not report an
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231 error if the unknown value is not a number---it accepts arguments of any
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232 type. By contrast, @code{=} signals an error if the arguments are not
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233 numbers or markers. However, it is a good idea to use @code{=} if you
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234 can, even for comparing integers, just in case we change the
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235 representation of integers in a future XEmacs version.
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236
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237 There is another wrinkle: because floating point arithmetic is not
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238 exact, it is often a bad idea to check for equality of two floating
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239 point values. Usually it is better to test for approximate equality.
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240 Here's a function to do this:
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241
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242 @example
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243 (defvar fuzz-factor 1.0e-6)
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244 (defun approx-equal (x y)
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245 (or (and (= x 0) (= y 0))
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246 (< (/ (abs (- x y))
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247 (max (abs x) (abs y)))
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248 fuzz-factor)))
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249 @end example
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250
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251 @cindex CL note---integers vrs @code{eq}
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252 @quotation
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253 @b{Common Lisp note:} Comparing numbers in Common Lisp always requires
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254 @code{=} because Common Lisp implements multi-word integers, and two
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255 distinct integer objects can have the same numeric value. XEmacs Lisp
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256 can have just one integer object for any given value because it has a
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257 limited range of integer values.
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258 @end quotation
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259
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260 @defun = number-or-marker1 number-or-marker2
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261 This function tests whether its arguments are numerically equal, and
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262 returns @code{t} if so, @code{nil} otherwise.
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263 @end defun
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264
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265 @defun /= number-or-marker1 number-or-marker2
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266 This function tests whether its arguments are numerically not equal. It
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267 returns @code{t} if so, and @code{nil} otherwise.
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268 @end defun
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269
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270 @defun < number-or-marker1 number-or-marker2
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271 This function tests whether its first argument is strictly less than
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272 its second argument. It returns @code{t} if so, @code{nil} otherwise.
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273 @end defun
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274
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275 @defun <= number-or-marker1 number-or-marker2
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276 This function tests whether its first argument is less than or equal
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277 to its second argument. It returns @code{t} if so, @code{nil}
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278 otherwise.
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279 @end defun
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280
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281 @defun > number-or-marker1 number-or-marker2
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282 This function tests whether its first argument is strictly greater
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283 than its second argument. It returns @code{t} if so, @code{nil}
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284 otherwise.
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285 @end defun
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286
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287 @defun >= number-or-marker1 number-or-marker2
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288 This function tests whether its first argument is greater than or
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289 equal to its second argument. It returns @code{t} if so, @code{nil}
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290 otherwise.
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291 @end defun
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292
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293 @defun max number-or-marker &rest numbers-or-markers
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294 This function returns the largest of its arguments.
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295
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296 @example
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297 (max 20)
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298 @result{} 20
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299 (max 1 2.5)
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300 @result{} 2.5
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301 (max 1 3 2.5)
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302 @result{} 3
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303 @end example
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304 @end defun
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305
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306 @defun min number-or-marker &rest numbers-or-markers
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307 This function returns the smallest of its arguments.
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308
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309 @example
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310 (min -4 1)
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311 @result{} -4
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312 @end example
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313 @end defun
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314
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315 @node Numeric Conversions
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316 @section Numeric Conversions
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317 @cindex rounding in conversions
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318
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319 To convert an integer to floating point, use the function @code{float}.
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320
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321 @defun float number
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322 This returns @var{number} converted to floating point.
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323 If @var{number} is already a floating point number, @code{float} returns
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324 it unchanged.
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325 @end defun
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326
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327 There are four functions to convert floating point numbers to integers;
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328 they differ in how they round. These functions accept integer arguments
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329 also, and return such arguments unchanged.
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330
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331 @defun truncate number
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332 This returns @var{number}, converted to an integer by rounding towards
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333 zero.
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334 @end defun
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335
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336 @defun floor number &optional divisor
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337 This returns @var{number}, converted to an integer by rounding downward
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338 (towards negative infinity).
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339
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340 If @var{divisor} is specified, @var{number} is divided by @var{divisor}
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341 before the floor is taken; this is the division operation that
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342 corresponds to @code{mod}. An @code{arith-error} results if
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343 @var{divisor} is 0.
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344 @end defun
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345
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346 @defun ceiling number
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347 This returns @var{number}, converted to an integer by rounding upward
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348 (towards positive infinity).
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349 @end defun
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350
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351 @defun round number
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352 This returns @var{number}, converted to an integer by rounding towards the
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353 nearest integer. Rounding a value equidistant between two integers
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354 may choose the integer closer to zero, or it may prefer an even integer,
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355 depending on your machine.
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356 @end defun
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357
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358 @node Arithmetic Operations
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359 @section Arithmetic Operations
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360
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361 XEmacs Lisp provides the traditional four arithmetic operations:
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362 addition, subtraction, multiplication, and division. Remainder and modulus
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363 functions supplement the division functions. The functions to
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364 add or subtract 1 are provided because they are traditional in Lisp and
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365 commonly used.
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366
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367 All of these functions except @code{%} return a floating point value
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368 if any argument is floating.
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369
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370 It is important to note that in XEmacs Lisp, arithmetic functions
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371 do not check for overflow. Thus @code{(1+ 134217727)} may evaluate to
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372 @minus{}134217728, depending on your hardware.
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373
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374 @defun 1+ number-or-marker
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375 This function returns @var{number-or-marker} plus 1.
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376 For example,
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377
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378 @example
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379 (setq foo 4)
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380 @result{} 4
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381 (1+ foo)
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382 @result{} 5
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383 @end example
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384
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385 This function is not analogous to the C operator @code{++}---it does not
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386 increment a variable. It just computes a sum. Thus, if we continue,
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387
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388 @example
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389 foo
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390 @result{} 4
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391 @end example
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392
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393 If you want to increment the variable, you must use @code{setq},
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394 like this:
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395
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396 @example
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397 (setq foo (1+ foo))
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398 @result{} 5
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399 @end example
|
|
74
|
400
|
|
|
401 Now that the @code{cl} package is always available from lisp code, a
|
|
|
402 more convenient and natural way to increment a variable is
|
|
|
403 @w{@code{(incf foo)}}.
|
|
0
|
404 @end defun
|
|
|
405
|
|
|
406 @defun 1- number-or-marker
|
|
|
407 This function returns @var{number-or-marker} minus 1.
|
|
|
408 @end defun
|
|
|
409
|
|
|
410 @defun abs number
|
|
|
411 This returns the absolute value of @var{number}.
|
|
|
412 @end defun
|
|
|
413
|
|
|
414 @defun + &rest numbers-or-markers
|
|
|
415 This function adds its arguments together. When given no arguments,
|
|
|
416 @code{+} returns 0.
|
|
|
417
|
|
|
418 @example
|
|
|
419 (+)
|
|
|
420 @result{} 0
|
|
|
421 (+ 1)
|
|
|
422 @result{} 1
|
|
|
423 (+ 1 2 3 4)
|
|
|
424 @result{} 10
|
|
|
425 @end example
|
|
|
426 @end defun
|
|
|
427
|
|
|
428 @defun - &optional number-or-marker &rest other-numbers-or-markers
|
|
|
429 The @code{-} function serves two purposes: negation and subtraction.
|
|
|
430 When @code{-} has a single argument, the value is the negative of the
|
|
|
431 argument. When there are multiple arguments, @code{-} subtracts each of
|
|
|
432 the @var{other-numbers-or-markers} from @var{number-or-marker},
|
|
|
433 cumulatively. If there are no arguments, the result is 0.
|
|
|
434
|
|
|
435 @example
|
|
|
436 (- 10 1 2 3 4)
|
|
|
437 @result{} 0
|
|
|
438 (- 10)
|
|
|
439 @result{} -10
|
|
|
440 (-)
|
|
|
441 @result{} 0
|
|
|
442 @end example
|
|
|
443 @end defun
|
|
|
444
|
|
|
445 @defun * &rest numbers-or-markers
|
|
|
446 This function multiplies its arguments together, and returns the
|
|
|
447 product. When given no arguments, @code{*} returns 1.
|
|
|
448
|
|
|
449 @example
|
|
|
450 (*)
|
|
|
451 @result{} 1
|
|
|
452 (* 1)
|
|
|
453 @result{} 1
|
|
|
454 (* 1 2 3 4)
|
|
|
455 @result{} 24
|
|
|
456 @end example
|
|
|
457 @end defun
|
|
|
458
|
|
|
459 @defun / dividend divisor &rest divisors
|
|
|
460 This function divides @var{dividend} by @var{divisor} and returns the
|
|
|
461 quotient. If there are additional arguments @var{divisors}, then it
|
|
|
462 divides @var{dividend} by each divisor in turn. Each argument may be a
|
|
|
463 number or a marker.
|
|
|
464
|
|
|
465 If all the arguments are integers, then the result is an integer too.
|
|
|
466 This means the result has to be rounded. On most machines, the result
|
|
|
467 is rounded towards zero after each division, but some machines may round
|
|
|
468 differently with negative arguments. This is because the Lisp function
|
|
|
469 @code{/} is implemented using the C division operator, which also
|
|
|
470 permits machine-dependent rounding. As a practical matter, all known
|
|
|
471 machines round in the standard fashion.
|
|
|
472
|
|
|
473 @cindex @code{arith-error} in division
|
|
|
474 If you divide by 0, an @code{arith-error} error is signaled.
|
|
|
475 (@xref{Errors}.)
|
|
|
476
|
|
|
477 @example
|
|
|
478 @group
|
|
|
479 (/ 6 2)
|
|
|
480 @result{} 3
|
|
|
481 @end group
|
|
|
482 (/ 5 2)
|
|
|
483 @result{} 2
|
|
|
484 (/ 25 3 2)
|
|
|
485 @result{} 4
|
|
|
486 (/ -17 6)
|
|
|
487 @result{} -2
|
|
|
488 @end example
|
|
|
489
|
|
|
490 The result of @code{(/ -17 6)} could in principle be -3 on some
|
|
|
491 machines.
|
|
|
492 @end defun
|
|
|
493
|
|
|
494 @defun % dividend divisor
|
|
|
495 @cindex remainder
|
|
|
496 This function returns the integer remainder after division of @var{dividend}
|
|
|
497 by @var{divisor}. The arguments must be integers or markers.
|
|
|
498
|
|
|
499 For negative arguments, the remainder is in principle machine-dependent
|
|
|
500 since the quotient is; but in practice, all known machines behave alike.
|
|
|
501
|
|
|
502 An @code{arith-error} results if @var{divisor} is 0.
|
|
|
503
|
|
|
504 @example
|
|
|
505 (% 9 4)
|
|
|
506 @result{} 1
|
|
|
507 (% -9 4)
|
|
|
508 @result{} -1
|
|
|
509 (% 9 -4)
|
|
|
510 @result{} 1
|
|
|
511 (% -9 -4)
|
|
|
512 @result{} -1
|
|
|
513 @end example
|
|
|
514
|
|
|
515 For any two integers @var{dividend} and @var{divisor},
|
|
|
516
|
|
|
517 @example
|
|
|
518 @group
|
|
|
519 (+ (% @var{dividend} @var{divisor})
|
|
|
520 (* (/ @var{dividend} @var{divisor}) @var{divisor}))
|
|
|
521 @end group
|
|
|
522 @end example
|
|
|
523
|
|
|
524 @noindent
|
|
|
525 always equals @var{dividend}.
|
|
|
526 @end defun
|
|
|
527
|
|
|
528 @defun mod dividend divisor
|
|
|
529 @cindex modulus
|
|
|
530 This function returns the value of @var{dividend} modulo @var{divisor};
|
|
|
531 in other words, the remainder after division of @var{dividend}
|
|
|
532 by @var{divisor}, but with the same sign as @var{divisor}.
|
|
|
533 The arguments must be numbers or markers.
|
|
|
534
|
|
|
535 Unlike @code{%}, @code{mod} returns a well-defined result for negative
|
|
|
536 arguments. It also permits floating point arguments; it rounds the
|
|
|
537 quotient downward (towards minus infinity) to an integer, and uses that
|
|
|
538 quotient to compute the remainder.
|
|
|
539
|
|
|
540 An @code{arith-error} results if @var{divisor} is 0.
|
|
|
541
|
|
|
542 @example
|
|
|
543 @group
|
|
|
544 (mod 9 4)
|
|
|
545 @result{} 1
|
|
|
546 @end group
|
|
|
547 @group
|
|
|
548 (mod -9 4)
|
|
|
549 @result{} 3
|
|
|
550 @end group
|
|
|
551 @group
|
|
|
552 (mod 9 -4)
|
|
|
553 @result{} -3
|
|
|
554 @end group
|
|
|
555 @group
|
|
|
556 (mod -9 -4)
|
|
|
557 @result{} -1
|
|
|
558 @end group
|
|
|
559 @group
|
|
|
560 (mod 5.5 2.5)
|
|
|
561 @result{} .5
|
|
|
562 @end group
|
|
|
563 @end example
|
|
|
564
|
|
|
565 For any two numbers @var{dividend} and @var{divisor},
|
|
|
566
|
|
|
567 @example
|
|
|
568 @group
|
|
|
569 (+ (mod @var{dividend} @var{divisor})
|
|
|
570 (* (floor @var{dividend} @var{divisor}) @var{divisor}))
|
|
|
571 @end group
|
|
|
572 @end example
|
|
|
573
|
|
|
574 @noindent
|
|
|
575 always equals @var{dividend}, subject to rounding error if either
|
|
|
576 argument is floating point. For @code{floor}, see @ref{Numeric
|
|
|
577 Conversions}.
|
|
|
578 @end defun
|
|
|
579
|
|
|
580 @node Rounding Operations
|
|
|
581 @section Rounding Operations
|
|
|
582 @cindex rounding without conversion
|
|
|
583
|
|
|
584 The functions @code{ffloor}, @code{fceiling}, @code{fround} and
|
|
|
585 @code{ftruncate} take a floating point argument and return a floating
|
|
|
586 point result whose value is a nearby integer. @code{ffloor} returns the
|
|
|
587 nearest integer below; @code{fceiling}, the nearest integer above;
|
|
|
588 @code{ftruncate}, the nearest integer in the direction towards zero;
|
|
|
589 @code{fround}, the nearest integer.
|
|
|
590
|
|
|
591 @defun ffloor float
|
|
|
592 This function rounds @var{float} to the next lower integral value, and
|
|
|
593 returns that value as a floating point number.
|
|
|
594 @end defun
|
|
|
595
|
|
|
596 @defun fceiling float
|
|
|
597 This function rounds @var{float} to the next higher integral value, and
|
|
|
598 returns that value as a floating point number.
|
|
|
599 @end defun
|
|
|
600
|
|
|
601 @defun ftruncate float
|
|
|
602 This function rounds @var{float} towards zero to an integral value, and
|
|
|
603 returns that value as a floating point number.
|
|
|
604 @end defun
|
|
|
605
|
|
|
606 @defun fround float
|
|
|
607 This function rounds @var{float} to the nearest integral value,
|
|
|
608 and returns that value as a floating point number.
|
|
|
609 @end defun
|
|
|
610
|
|
|
611 @node Bitwise Operations
|
|
|
612 @section Bitwise Operations on Integers
|
|
|
613
|
|
|
614 In a computer, an integer is represented as a binary number, a
|
|
|
615 sequence of @dfn{bits} (digits which are either zero or one). A bitwise
|
|
|
616 operation acts on the individual bits of such a sequence. For example,
|
|
|
617 @dfn{shifting} moves the whole sequence left or right one or more places,
|
|
|
618 reproducing the same pattern ``moved over''.
|
|
|
619
|
|
|
620 The bitwise operations in XEmacs Lisp apply only to integers.
|
|
|
621
|
|
|
622 @defun lsh integer1 count
|
|
|
623 @cindex logical shift
|
|
|
624 @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the
|
|
|
625 bits in @var{integer1} to the left @var{count} places, or to the right
|
|
|
626 if @var{count} is negative, bringing zeros into the vacated bits. If
|
|
|
627 @var{count} is negative, @code{lsh} shifts zeros into the leftmost
|
|
|
628 (most-significant) bit, producing a positive result even if
|
|
|
629 @var{integer1} is negative. Contrast this with @code{ash}, below.
|
|
|
630
|
|
|
631 Here are two examples of @code{lsh}, shifting a pattern of bits one
|
|
|
632 place to the left. We show only the low-order eight bits of the binary
|
|
|
633 pattern; the rest are all zero.
|
|
|
634
|
|
|
635 @example
|
|
|
636 @group
|
|
|
637 (lsh 5 1)
|
|
|
638 @result{} 10
|
|
|
639 ;; @r{Decimal 5 becomes decimal 10.}
|
|
|
640 00000101 @result{} 00001010
|
|
|
641
|
|
|
642 (lsh 7 1)
|
|
|
643 @result{} 14
|
|
|
644 ;; @r{Decimal 7 becomes decimal 14.}
|
|
|
645 00000111 @result{} 00001110
|
|
|
646 @end group
|
|
|
647 @end example
|
|
|
648
|
|
|
649 @noindent
|
|
|
650 As the examples illustrate, shifting the pattern of bits one place to
|
|
|
651 the left produces a number that is twice the value of the previous
|
|
|
652 number.
|
|
|
653
|
|
|
654 Shifting a pattern of bits two places to the left produces results
|
|
|
655 like this (with 8-bit binary numbers):
|
|
|
656
|
|
|
657 @example
|
|
|
658 @group
|
|
|
659 (lsh 3 2)
|
|
|
660 @result{} 12
|
|
|
661 ;; @r{Decimal 3 becomes decimal 12.}
|
|
|
662 00000011 @result{} 00001100
|
|
|
663 @end group
|
|
|
664 @end example
|
|
|
665
|
|
|
666 On the other hand, shifting one place to the right looks like this:
|
|
|
667
|
|
|
668 @example
|
|
|
669 @group
|
|
|
670 (lsh 6 -1)
|
|
|
671 @result{} 3
|
|
|
672 ;; @r{Decimal 6 becomes decimal 3.}
|
|
|
673 00000110 @result{} 00000011
|
|
|
674 @end group
|
|
|
675
|
|
|
676 @group
|
|
|
677 (lsh 5 -1)
|
|
|
678 @result{} 2
|
|
|
679 ;; @r{Decimal 5 becomes decimal 2.}
|
|
|
680 00000101 @result{} 00000010
|
|
|
681 @end group
|
|
|
682 @end example
|
|
|
683
|
|
|
684 @noindent
|
|
|
685 As the example illustrates, shifting one place to the right divides the
|
|
|
686 value of a positive integer by two, rounding downward.
|
|
|
687
|
|
|
688 The function @code{lsh}, like all XEmacs Lisp arithmetic functions, does
|
|
|
689 not check for overflow, so shifting left can discard significant bits
|
|
|
690 and change the sign of the number. For example, left shifting
|
|
|
691 134,217,727 produces @minus{}2 on a 28-bit machine:
|
|
|
692
|
|
|
693 @example
|
|
|
694 (lsh 134217727 1) ; @r{left shift}
|
|
|
695 @result{} -2
|
|
|
696 @end example
|
|
|
697
|
|
|
698 In binary, in the 28-bit implementation, the argument looks like this:
|
|
|
699
|
|
|
700 @example
|
|
|
701 @group
|
|
|
702 ;; @r{Decimal 134,217,727}
|
|
|
703 0111 1111 1111 1111 1111 1111 1111
|
|
|
704 @end group
|
|
|
705 @end example
|
|
|
706
|
|
|
707 @noindent
|
|
|
708 which becomes the following when left shifted:
|
|
|
709
|
|
|
710 @example
|
|
|
711 @group
|
|
|
712 ;; @r{Decimal @minus{}2}
|
|
|
713 1111 1111 1111 1111 1111 1111 1110
|
|
|
714 @end group
|
|
|
715 @end example
|
|
|
716 @end defun
|
|
|
717
|
|
|
718 @defun ash integer1 count
|
|
|
719 @cindex arithmetic shift
|
|
|
720 @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1}
|
|
|
721 to the left @var{count} places, or to the right if @var{count}
|
|
|
722 is negative.
|
|
|
723
|
|
|
724 @code{ash} gives the same results as @code{lsh} except when
|
|
|
725 @var{integer1} and @var{count} are both negative. In that case,
|
|
|
726 @code{ash} puts ones in the empty bit positions on the left, while
|
|
|
727 @code{lsh} puts zeros in those bit positions.
|
|
|
728
|
|
|
729 Thus, with @code{ash}, shifting the pattern of bits one place to the right
|
|
|
730 looks like this:
|
|
|
731
|
|
|
732 @example
|
|
|
733 @group
|
|
|
734 (ash -6 -1) @result{} -3
|
|
|
735 ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
|
|
|
736 1111 1111 1111 1111 1111 1111 1010
|
|
|
737 @result{}
|
|
|
738 1111 1111 1111 1111 1111 1111 1101
|
|
|
739 @end group
|
|
|
740 @end example
|
|
|
741
|
|
|
742 In contrast, shifting the pattern of bits one place to the right with
|
|
|
743 @code{lsh} looks like this:
|
|
|
744
|
|
|
745 @example
|
|
|
746 @group
|
|
|
747 (lsh -6 -1) @result{} 134217725
|
|
|
748 ;; @r{Decimal @minus{}6 becomes decimal 134,217,725.}
|
|
|
749 1111 1111 1111 1111 1111 1111 1010
|
|
|
750 @result{}
|
|
|
751 0111 1111 1111 1111 1111 1111 1101
|
|
|
752 @end group
|
|
|
753 @end example
|
|
|
754
|
|
|
755 Here are other examples:
|
|
|
756
|
|
|
757 @c !!! Check if lined up in smallbook format! XDVI shows problem
|
|
|
758 @c with smallbook but not with regular book! --rjc 16mar92
|
|
|
759 @smallexample
|
|
|
760 @group
|
|
|
761 ; @r{ 28-bit binary values}
|
|
|
762
|
|
|
763 (lsh 5 2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
764 @result{} 20 ; = @r{0000 0000 0000 0000 0000 0001 0100}
|
|
|
765 @end group
|
|
|
766 @group
|
|
|
767 (ash 5 2)
|
|
|
768 @result{} 20
|
|
|
769 (lsh -5 2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011}
|
|
|
770 @result{} -20 ; = @r{1111 1111 1111 1111 1111 1110 1100}
|
|
|
771 (ash -5 2)
|
|
|
772 @result{} -20
|
|
|
773 @end group
|
|
|
774 @group
|
|
|
775 (lsh 5 -2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
776 @result{} 1 ; = @r{0000 0000 0000 0000 0000 0000 0001}
|
|
|
777 @end group
|
|
|
778 @group
|
|
|
779 (ash 5 -2)
|
|
|
780 @result{} 1
|
|
|
781 @end group
|
|
|
782 @group
|
|
|
783 (lsh -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011}
|
|
|
784 @result{} 4194302 ; = @r{0011 1111 1111 1111 1111 1111 1110}
|
|
|
785 @end group
|
|
|
786 @group
|
|
|
787 (ash -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011}
|
|
|
788 @result{} -2 ; = @r{1111 1111 1111 1111 1111 1111 1110}
|
|
|
789 @end group
|
|
|
790 @end smallexample
|
|
|
791 @end defun
|
|
|
792
|
|
|
793 @defun logand &rest ints-or-markers
|
|
|
794 @cindex logical and
|
|
|
795 @cindex bitwise and
|
|
|
796 This function returns the ``logical and'' of the arguments: the
|
|
|
797 @var{n}th bit is set in the result if, and only if, the @var{n}th bit is
|
|
|
798 set in all the arguments. (``Set'' means that the value of the bit is 1
|
|
|
799 rather than 0.)
|
|
|
800
|
|
|
801 For example, using 4-bit binary numbers, the ``logical and'' of 13 and
|
|
|
802 12 is 12: 1101 combined with 1100 produces 1100.
|
|
|
803 In both the binary numbers, the leftmost two bits are set (i.e., they
|
|
|
804 are 1's), so the leftmost two bits of the returned value are set.
|
|
|
805 However, for the rightmost two bits, each is zero in at least one of
|
|
|
806 the arguments, so the rightmost two bits of the returned value are 0's.
|
|
|
807
|
|
|
808 @noindent
|
|
|
809 Therefore,
|
|
|
810
|
|
|
811 @example
|
|
|
812 @group
|
|
|
813 (logand 13 12)
|
|
|
814 @result{} 12
|
|
|
815 @end group
|
|
|
816 @end example
|
|
|
817
|
|
|
818 If @code{logand} is not passed any argument, it returns a value of
|
|
|
819 @minus{}1. This number is an identity element for @code{logand}
|
|
|
820 because its binary representation consists entirely of ones. If
|
|
|
821 @code{logand} is passed just one argument, it returns that argument.
|
|
|
822
|
|
|
823 @smallexample
|
|
|
824 @group
|
|
|
825 ; @r{ 28-bit binary values}
|
|
|
826
|
|
|
827 (logand 14 13) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110}
|
|
|
828 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101}
|
|
|
829 @result{} 12 ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
830 @end group
|
|
|
831
|
|
|
832 @group
|
|
|
833 (logand 14 13 4) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110}
|
|
|
834 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101}
|
|
|
835 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100}
|
|
|
836 @result{} 4 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100}
|
|
|
837 @end group
|
|
|
838
|
|
|
839 @group
|
|
|
840 (logand)
|
|
|
841 @result{} -1 ; -1 = @r{1111 1111 1111 1111 1111 1111 1111}
|
|
|
842 @end group
|
|
|
843 @end smallexample
|
|
|
844 @end defun
|
|
|
845
|
|
|
846 @defun logior &rest ints-or-markers
|
|
|
847 @cindex logical inclusive or
|
|
|
848 @cindex bitwise or
|
|
|
849 This function returns the ``inclusive or'' of its arguments: the @var{n}th bit
|
|
|
850 is set in the result if, and only if, the @var{n}th bit is set in at least
|
|
|
851 one of the arguments. If there are no arguments, the result is zero,
|
|
|
852 which is an identity element for this operation. If @code{logior} is
|
|
|
853 passed just one argument, it returns that argument.
|
|
|
854
|
|
|
855 @smallexample
|
|
|
856 @group
|
|
|
857 ; @r{ 28-bit binary values}
|
|
|
858
|
|
|
859 (logior 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
860 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
861 @result{} 13 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101}
|
|
|
862 @end group
|
|
|
863
|
|
|
864 @group
|
|
|
865 (logior 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
866 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
867 ; 7 = @r{0000 0000 0000 0000 0000 0000 0111}
|
|
|
868 @result{} 15 ; 15 = @r{0000 0000 0000 0000 0000 0000 1111}
|
|
|
869 @end group
|
|
|
870 @end smallexample
|
|
|
871 @end defun
|
|
|
872
|
|
|
873 @defun logxor &rest ints-or-markers
|
|
|
874 @cindex bitwise exclusive or
|
|
|
875 @cindex logical exclusive or
|
|
|
876 This function returns the ``exclusive or'' of its arguments: the
|
|
|
877 @var{n}th bit is set in the result if, and only if, the @var{n}th bit is
|
|
|
878 set in an odd number of the arguments. If there are no arguments, the
|
|
|
879 result is 0, which is an identity element for this operation. If
|
|
|
880 @code{logxor} is passed just one argument, it returns that argument.
|
|
|
881
|
|
|
882 @smallexample
|
|
|
883 @group
|
|
|
884 ; @r{ 28-bit binary values}
|
|
|
885
|
|
|
886 (logxor 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
887 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
888 @result{} 9 ; 9 = @r{0000 0000 0000 0000 0000 0000 1001}
|
|
|
889 @end group
|
|
|
890
|
|
|
891 @group
|
|
|
892 (logxor 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
|
|
|
893 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
894 ; 7 = @r{0000 0000 0000 0000 0000 0000 0111}
|
|
|
895 @result{} 14 ; 14 = @r{0000 0000 0000 0000 0000 0000 1110}
|
|
|
896 @end group
|
|
|
897 @end smallexample
|
|
|
898 @end defun
|
|
|
899
|
|
|
900 @defun lognot integer
|
|
|
901 @cindex logical not
|
|
|
902 @cindex bitwise not
|
|
|
903 This function returns the logical complement of its argument: the @var{n}th
|
|
|
904 bit is one in the result if, and only if, the @var{n}th bit is zero in
|
|
|
905 @var{integer}, and vice-versa.
|
|
|
906
|
|
|
907 @example
|
|
|
908 (lognot 5)
|
|
|
909 @result{} -6
|
|
|
910 ;; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
|
|
|
911 ;; @r{becomes}
|
|
|
912 ;; -6 = @r{1111 1111 1111 1111 1111 1111 1010}
|
|
|
913 @end example
|
|
|
914 @end defun
|
|
|
915
|
|
|
916 @node Math Functions
|
|
|
917 @section Standard Mathematical Functions
|
|
|
918 @cindex transcendental functions
|
|
|
919 @cindex mathematical functions
|
|
|
920
|
|
|
921 These mathematical functions are available if floating point is
|
|
|
922 supported (which is the normal state of affairs). They allow integers
|
|
|
923 as well as floating point numbers as arguments.
|
|
|
924
|
|
|
925 @defun sin arg
|
|
|
926 @defunx cos arg
|
|
|
927 @defunx tan arg
|
|
|
928 These are the ordinary trigonometric functions, with argument measured
|
|
|
929 in radians.
|
|
|
930 @end defun
|
|
|
931
|
|
|
932 @defun asin arg
|
|
|
933 The value of @code{(asin @var{arg})} is a number between @minus{}pi/2
|
|
|
934 and pi/2 (inclusive) whose sine 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 acos arg
|
|
|
939 The value of @code{(acos @var{arg})} is a number between 0 and pi
|
|
|
940 (inclusive) whose cosine is @var{arg}; if, however, @var{arg}
|
|
|
941 is out of range (outside [-1, 1]), then the result is a NaN.
|
|
|
942 @end defun
|
|
|
943
|
|
|
944 @defun atan arg
|
|
|
945 The value of @code{(atan @var{arg})} is a number between @minus{}pi/2
|
|
|
946 and pi/2 (exclusive) whose tangent is @var{arg}.
|
|
|
947 @end defun
|
|
|
948
|
|
|
949 @defun sinh arg
|
|
|
950 @defunx cosh arg
|
|
|
951 @defunx tanh arg
|
|
|
952 These are the ordinary hyperbolic trigonometric functions.
|
|
|
953 @end defun
|
|
|
954
|
|
|
955 @defun asinh arg
|
|
|
956 @defunx acosh arg
|
|
|
957 @defunx atanh arg
|
|
|
958 These are the inverse hyperbolic trigonometric functions.
|
|
|
959 @end defun
|
|
|
960
|
|
|
961 @defun exp arg
|
|
|
962 This is the exponential function; it returns @i{e} to the power
|
|
|
963 @var{arg}. @i{e} is a fundamental mathematical constant also called the
|
|
|
964 base of natural logarithms.
|
|
|
965 @end defun
|
|
|
966
|
|
|
967 @defun log arg &optional base
|
|
|
968 This function returns the logarithm of @var{arg}, with base @var{base}.
|
|
|
969 If you don't specify @var{base}, the base @var{e} is used. If @var{arg}
|
|
|
970 is negative, the result is a NaN.
|
|
|
971 @end defun
|
|
|
972
|
|
|
973 @ignore
|
|
|
974 @defun expm1 arg
|
|
|
975 This function returns @code{(1- (exp @var{arg}))}, but it is more
|
|
|
976 accurate than that when @var{arg} is negative and @code{(exp @var{arg})}
|
|
|
977 is close to 1.
|
|
|
978 @end defun
|
|
|
979
|
|
|
980 @defun log1p arg
|
|
|
981 This function returns @code{(log (1+ @var{arg}))}, but it is more
|
|
|
982 accurate than that when @var{arg} is so small that adding 1 to it would
|
|
|
983 lose accuracy.
|
|
|
984 @end defun
|
|
|
985 @end ignore
|
|
|
986
|
|
|
987 @defun log10 arg
|
|
|
988 This function returns the logarithm of @var{arg}, with base 10. If
|
|
|
989 @var{arg} is negative, the result is a NaN. @code{(log10 @var{x})}
|
|
|
990 @equiv{} @code{(log @var{x} 10)}, at least approximately.
|
|
|
991 @end defun
|
|
|
992
|
|
|
993 @defun expt x y
|
|
|
994 This function returns @var{x} raised to power @var{y}. If both
|
|
|
995 arguments are integers and @var{y} is positive, the result is an
|
|
|
996 integer; in this case, it is truncated to fit the range of possible
|
|
|
997 integer values.
|
|
|
998 @end defun
|
|
|
999
|
|
|
1000 @defun sqrt arg
|
|
|
1001 This returns the square root of @var{arg}. If @var{arg} is negative,
|
|
|
1002 the value is a NaN.
|
|
|
1003 @end defun
|
|
|
1004
|
|
|
1005 @defun cube-root arg
|
|
|
1006 This returns the cube root of @var{arg}.
|
|
|
1007 @end defun
|
|
|
1008
|
|
|
1009 @node Random Numbers
|
|
|
1010 @section Random Numbers
|
|
|
1011 @cindex random numbers
|
|
|
1012
|
|
|
1013 A deterministic computer program cannot generate true random numbers.
|
|
|
1014 For most purposes, @dfn{pseudo-random numbers} suffice. A series of
|
|
|
1015 pseudo-random numbers is generated in a deterministic fashion. The
|
|
|
1016 numbers are not truly random, but they have certain properties that
|
|
|
1017 mimic a random series. For example, all possible values occur equally
|
|
|
1018 often in a pseudo-random series.
|
|
|
1019
|
|
|
1020 In XEmacs, pseudo-random numbers are generated from a ``seed'' number.
|
|
|
1021 Starting from any given seed, the @code{random} function always
|
|
|
1022 generates the same sequence of numbers. XEmacs always starts with the
|
|
|
1023 same seed value, so the sequence of values of @code{random} is actually
|
|
|
1024 the same in each XEmacs run! For example, in one operating system, the
|
|
|
1025 first call to @code{(random)} after you start XEmacs always returns
|
|
|
1026 -1457731, and the second one always returns -7692030. This
|
|
|
1027 repeatability is helpful for debugging.
|
|
|
1028
|
|
|
1029 If you want truly unpredictable random numbers, execute @code{(random
|
|
|
1030 t)}. This chooses a new seed based on the current time of day and on
|
|
|
1031 XEmacs's process @sc{id} number.
|
|
|
1032
|
|
|
1033 @defun random &optional limit
|
|
|
1034 This function returns a pseudo-random integer. Repeated calls return a
|
|
|
1035 series of pseudo-random integers.
|
|
|
1036
|
|
|
1037 If @var{limit} is a positive integer, the value is chosen to be
|
|
|
1038 nonnegative and less than @var{limit}.
|
|
|
1039
|
|
|
1040 If @var{limit} is @code{t}, it means to choose a new seed based on the
|
|
|
1041 current time of day and on XEmacs's process @sc{id} number.
|
|
|
1042 @c "XEmacs'" is incorrect usage!
|
|
|
1043
|
|
|
1044 On some machines, any integer representable in Lisp may be the result
|
|
|
1045 of @code{random}. On other machines, the result can never be larger
|
|
|
1046 than a certain maximum or less than a certain (negative) minimum.
|
|
|
1047 @end defun
|
