1 | =head1 NAME |
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2 | |
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3 | perlnumber - semantics of numbers and numeric operations in Perl |
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4 | |
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5 | =head1 SYNOPSIS |
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6 | |
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7 | $n = 1234; # decimal integer |
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8 | $n = 0b1110011; # binary integer |
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9 | $n = 01234; # octal integer |
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10 | $n = 0x1234; # hexadecimal integer |
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11 | $n = 12.34e-56; # exponential notation |
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12 | $n = "-12.34e56"; # number specified as a string |
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13 | $n = "1234"; # number specified as a string |
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14 | $n = v49.50.51.52; # number specified as a string, which in |
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15 | # turn is specified in terms of numbers :-) |
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16 | |
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17 | =head1 DESCRIPTION |
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18 | |
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19 | This document describes how Perl internally handles numeric values. |
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20 | |
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21 | Perl's operator overloading facility is completely ignored here. Operator |
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22 | overloading allows user-defined behaviors for numbers, such as operations |
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23 | over arbitrarily large integers, floating points numbers with arbitrary |
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24 | precision, operations over "exotic" numbers such as modular arithmetic or |
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25 | p-adic arithmetic, and so on. See L<overload> for details. |
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26 | |
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27 | =head1 Storing numbers |
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28 | |
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29 | Perl can internally represent numbers in 3 different ways: as native |
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30 | integers, as native floating point numbers, and as decimal strings. |
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31 | Decimal strings may have an exponential notation part, as in C<"12.34e-56">. |
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32 | I<Native> here means "a format supported by the C compiler which was used |
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33 | to build perl". |
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34 | |
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35 | The term "native" does not mean quite as much when we talk about native |
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36 | integers, as it does when native floating point numbers are involved. |
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37 | The only implication of the term "native" on integers is that the limits for |
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38 | the maximal and the minimal supported true integral quantities are close to |
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39 | powers of 2. However, "native" floats have a most fundamental |
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40 | restriction: they may represent only those numbers which have a relatively |
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41 | "short" representation when converted to a binary fraction. For example, |
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42 | 0.9 cannot be respresented by a native float, since the binary fraction |
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43 | for 0.9 is infinite: |
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44 | |
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45 | binary0.1110011001100... |
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46 | |
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47 | with the sequence C<1100> repeating again and again. In addition to this |
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48 | limitation, the exponent of the binary number is also restricted when it |
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49 | is represented as a floating point number. On typical hardware, floating |
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50 | point values can store numbers with up to 53 binary digits, and with binary |
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51 | exponents between -1024 and 1024. In decimal representation this is close |
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52 | to 16 decimal digits and decimal exponents in the range of -304..304. |
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53 | The upshot of all this is that Perl cannot store a number like |
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54 | 12345678901234567 as a floating point number on such architectures without |
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55 | loss of information. |
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56 | |
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57 | Similarly, decimal strings can represent only those numbers which have a |
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58 | finite decimal expansion. Being strings, and thus of arbitrary length, there |
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59 | is no practical limit for the exponent or number of decimal digits for these |
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60 | numbers. (But realize that what we are discussing the rules for just the |
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61 | I<storage> of these numbers. The fact that you can store such "large" numbers |
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62 | does not mean that that the I<operations> over these numbers will use all |
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63 | of the significant digits. |
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64 | See L<"Numeric operators and numeric conversions"> for details.) |
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65 | |
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66 | In fact numbers stored in the native integer format may be stored either |
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67 | in the signed native form, or in the unsigned native form. Thus the limits |
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68 | for Perl numbers stored as native integers would typically be -2**31..2**32-1, |
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69 | with appropriate modifications in the case of 64-bit integers. Again, this |
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70 | does not mean that Perl can do operations only over integers in this range: |
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71 | it is possible to store many more integers in floating point format. |
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72 | |
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73 | Summing up, Perl numeric values can store only those numbers which have |
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74 | a finite decimal expansion or a "short" binary expansion. |
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75 | |
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76 | =head1 Numeric operators and numeric conversions |
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77 | |
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78 | As mentioned earlier, Perl can store a number in any one of three formats, |
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79 | but most operators typically understand only one of those formats. When |
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80 | a numeric value is passed as an argument to such an operator, it will be |
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81 | converted to the format understood by the operator. |
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82 | |
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83 | Six such conversions are possible: |
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84 | |
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85 | native integer --> native floating point (*) |
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86 | native integer --> decimal string |
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87 | native floating_point --> native integer (*) |
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88 | native floating_point --> decimal string (*) |
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89 | decimal string --> native integer |
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90 | decimal string --> native floating point (*) |
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91 | |
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92 | These conversions are governed by the following general rules: |
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93 | |
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94 | =over |
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95 | |
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96 | =item * |
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97 | |
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98 | If the source number can be represented in the target form, that |
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99 | representation is used. |
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100 | |
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101 | =item * |
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102 | |
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103 | If the source number is outside of the limits representable in the target form, |
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104 | a representation of the closest limit is used. (I<Loss of information>) |
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105 | |
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106 | =item * |
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107 | |
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108 | If the source number is between two numbers representable in the target form, |
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109 | a representation of one of these numbers is used. (I<Loss of information>) |
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110 | |
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111 | =item * |
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112 | |
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113 | In C<< native floating point --> native integer >> conversions the magnitude |
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114 | of the result is less than or equal to the magnitude of the source. |
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115 | (I<"Rounding to zero".>) |
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116 | |
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117 | =item * |
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118 | |
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119 | If the C<< decimal string --> native integer >> conversion cannot be done |
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120 | without loss of information, the result is compatible with the conversion |
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121 | sequence C<< decimal_string --> native_floating_point --> native_integer >>. |
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122 | In particular, rounding is strongly biased to 0, though a number like |
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123 | C<"0.99999999999999999999"> has a chance of being rounded to 1. |
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124 | |
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125 | =back |
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126 | |
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127 | B<RESTRICTION>: The conversions marked with C<(*)> above involve steps |
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128 | performed by the C compiler. In particular, bugs/features of the compiler |
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129 | used may lead to breakage of some of the above rules. |
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130 | |
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131 | =head1 Flavors of Perl numeric operations |
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132 | |
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133 | Perl operations which take a numeric argument treat that argument in one |
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134 | of four different ways: they may force it to one of the integer/floating/ |
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135 | string formats, or they may behave differently depending on the format of |
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136 | the operand. Forcing a numeric value to a particular format does not |
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137 | change the number stored in the value. |
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138 | |
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139 | All the operators which need an argument in the integer format treat the |
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140 | argument as in modular arithmetic, e.g., C<mod 2**32> on a 32-bit |
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141 | architecture. C<sprintf "%u", -1> therefore provides the same result as |
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142 | C<sprintf "%u", ~0>. |
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143 | |
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144 | =over |
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145 | |
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146 | =item Arithmetic operators except, C<no integer> |
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147 | |
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148 | force the argument into the floating point format. |
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149 | |
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150 | =item Arithmetic operators except, C<use integer> |
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151 | |
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152 | =item Bitwise operators, C<no integer> |
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153 | |
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154 | force the argument into the integer format if it is not a string. |
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155 | |
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156 | =item Bitwise operators, C<use integer> |
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157 | |
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158 | force the argument into the integer format |
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159 | |
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160 | =item Operators which expect an integer |
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161 | |
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162 | force the argument into the integer format. This is applicable |
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163 | to the third and fourth arguments of C<sysread>, for example. |
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164 | |
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165 | =item Operators which expect a string |
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166 | |
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167 | force the argument into the string format. For example, this is |
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168 | applicable to C<printf "%s", $value>. |
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169 | |
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170 | =back |
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171 | |
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172 | Though forcing an argument into a particular form does not change the |
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173 | stored number, Perl remembers the result of such conversions. In |
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174 | particular, though the first such conversion may be time-consuming, |
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175 | repeated operations will not need to redo the conversion. |
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176 | |
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177 | =head1 AUTHOR |
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178 | |
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179 | Ilya Zakharevich C<ilya@math.ohio-state.edu> |
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180 | |
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181 | Editorial adjustments by Gurusamy Sarathy <gsar@ActiveState.com> |
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182 | |
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183 | =head1 SEE ALSO |
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184 | |
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185 | L<overload> |
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