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1 # Copyright (c) 2004 Python Software Foundation. |
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2 # All rights reserved. |
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3 |
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4 # Written by Eric Price <eprice at tjhsst.edu> |
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5 # and Facundo Batista <facundo at taniquetil.com.ar> |
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6 # and Raymond Hettinger <python at rcn.com> |
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7 # and Aahz <aahz at pobox.com> |
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8 # and Tim Peters |
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9 |
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10 # This module is currently Py2.3 compatible and should be kept that way |
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11 # unless a major compelling advantage arises. IOW, 2.3 compatibility is |
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12 # strongly preferred, but not guaranteed. |
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13 |
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14 # Also, this module should be kept in sync with the latest updates of |
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15 # the IBM specification as it evolves. Those updates will be treated |
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16 # as bug fixes (deviation from the spec is a compatibility, usability |
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17 # bug) and will be backported. At this point the spec is stabilizing |
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18 # and the updates are becoming fewer, smaller, and less significant. |
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19 |
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20 """ |
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21 This is a Py2.3 implementation of decimal floating point arithmetic based on |
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22 the General Decimal Arithmetic Specification: |
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23 |
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24 www2.hursley.ibm.com/decimal/decarith.html |
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25 |
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26 and IEEE standard 854-1987: |
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27 |
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28 www.cs.berkeley.edu/~ejr/projects/754/private/drafts/854-1987/dir.html |
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29 |
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30 Decimal floating point has finite precision with arbitrarily large bounds. |
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31 |
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32 The purpose of this module is to support arithmetic using familiar |
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33 "schoolhouse" rules and to avoid some of the tricky representation |
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34 issues associated with binary floating point. The package is especially |
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35 useful for financial applications or for contexts where users have |
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36 expectations that are at odds with binary floating point (for instance, |
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37 in binary floating point, 1.00 % 0.1 gives 0.09999999999999995 instead |
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38 of the expected Decimal('0.00') returned by decimal floating point). |
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39 |
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40 Here are some examples of using the decimal module: |
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41 |
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42 >>> from decimal import * |
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43 >>> setcontext(ExtendedContext) |
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44 >>> Decimal(0) |
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45 Decimal('0') |
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46 >>> Decimal('1') |
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47 Decimal('1') |
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48 >>> Decimal('-.0123') |
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49 Decimal('-0.0123') |
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50 >>> Decimal(123456) |
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51 Decimal('123456') |
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52 >>> Decimal('123.45e12345678901234567890') |
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53 Decimal('1.2345E+12345678901234567892') |
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54 >>> Decimal('1.33') + Decimal('1.27') |
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55 Decimal('2.60') |
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56 >>> Decimal('12.34') + Decimal('3.87') - Decimal('18.41') |
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57 Decimal('-2.20') |
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58 >>> dig = Decimal(1) |
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59 >>> print dig / Decimal(3) |
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60 0.333333333 |
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61 >>> getcontext().prec = 18 |
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62 >>> print dig / Decimal(3) |
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63 0.333333333333333333 |
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64 >>> print dig.sqrt() |
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65 1 |
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66 >>> print Decimal(3).sqrt() |
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67 1.73205080756887729 |
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68 >>> print Decimal(3) ** 123 |
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69 4.85192780976896427E+58 |
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70 >>> inf = Decimal(1) / Decimal(0) |
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71 >>> print inf |
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72 Infinity |
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73 >>> neginf = Decimal(-1) / Decimal(0) |
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74 >>> print neginf |
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75 -Infinity |
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76 >>> print neginf + inf |
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77 NaN |
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78 >>> print neginf * inf |
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79 -Infinity |
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80 >>> print dig / 0 |
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81 Infinity |
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82 >>> getcontext().traps[DivisionByZero] = 1 |
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83 >>> print dig / 0 |
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84 Traceback (most recent call last): |
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85 ... |
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86 ... |
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87 ... |
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88 DivisionByZero: x / 0 |
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89 >>> c = Context() |
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90 >>> c.traps[InvalidOperation] = 0 |
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91 >>> print c.flags[InvalidOperation] |
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92 0 |
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93 >>> c.divide(Decimal(0), Decimal(0)) |
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94 Decimal('NaN') |
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95 >>> c.traps[InvalidOperation] = 1 |
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96 >>> print c.flags[InvalidOperation] |
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97 1 |
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98 >>> c.flags[InvalidOperation] = 0 |
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99 >>> print c.flags[InvalidOperation] |
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100 0 |
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101 >>> print c.divide(Decimal(0), Decimal(0)) |
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102 Traceback (most recent call last): |
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103 ... |
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104 ... |
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105 ... |
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106 InvalidOperation: 0 / 0 |
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107 >>> print c.flags[InvalidOperation] |
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108 1 |
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109 >>> c.flags[InvalidOperation] = 0 |
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110 >>> c.traps[InvalidOperation] = 0 |
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111 >>> print c.divide(Decimal(0), Decimal(0)) |
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112 NaN |
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113 >>> print c.flags[InvalidOperation] |
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114 1 |
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115 >>> |
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116 """ |
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117 |
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118 __all__ = [ |
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119 # Two major classes |
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120 'Decimal', 'Context', |
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121 |
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122 # Contexts |
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123 'DefaultContext', 'BasicContext', 'ExtendedContext', |
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124 |
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125 # Exceptions |
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126 'DecimalException', 'Clamped', 'InvalidOperation', 'DivisionByZero', |
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127 'Inexact', 'Rounded', 'Subnormal', 'Overflow', 'Underflow', |
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128 |
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129 # Constants for use in setting up contexts |
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130 'ROUND_DOWN', 'ROUND_HALF_UP', 'ROUND_HALF_EVEN', 'ROUND_CEILING', |
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131 'ROUND_FLOOR', 'ROUND_UP', 'ROUND_HALF_DOWN', 'ROUND_05UP', |
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132 |
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133 # Functions for manipulating contexts |
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134 'setcontext', 'getcontext', 'localcontext' |
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135 ] |
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136 |
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137 import copy as _copy |
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138 |
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139 try: |
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140 from collections import namedtuple as _namedtuple |
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141 DecimalTuple = _namedtuple('DecimalTuple', 'sign digits exponent') |
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142 except ImportError: |
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143 DecimalTuple = lambda *args: args |
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144 |
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145 # Rounding |
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146 ROUND_DOWN = 'ROUND_DOWN' |
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147 ROUND_HALF_UP = 'ROUND_HALF_UP' |
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148 ROUND_HALF_EVEN = 'ROUND_HALF_EVEN' |
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149 ROUND_CEILING = 'ROUND_CEILING' |
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150 ROUND_FLOOR = 'ROUND_FLOOR' |
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151 ROUND_UP = 'ROUND_UP' |
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152 ROUND_HALF_DOWN = 'ROUND_HALF_DOWN' |
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153 ROUND_05UP = 'ROUND_05UP' |
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154 |
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155 # Errors |
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156 |
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157 class DecimalException(ArithmeticError): |
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158 """Base exception class. |
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159 |
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160 Used exceptions derive from this. |
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161 If an exception derives from another exception besides this (such as |
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162 Underflow (Inexact, Rounded, Subnormal) that indicates that it is only |
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163 called if the others are present. This isn't actually used for |
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164 anything, though. |
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165 |
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166 handle -- Called when context._raise_error is called and the |
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167 trap_enabler is set. First argument is self, second is the |
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168 context. More arguments can be given, those being after |
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169 the explanation in _raise_error (For example, |
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170 context._raise_error(NewError, '(-x)!', self._sign) would |
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171 call NewError().handle(context, self._sign).) |
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172 |
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173 To define a new exception, it should be sufficient to have it derive |
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174 from DecimalException. |
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175 """ |
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176 def handle(self, context, *args): |
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177 pass |
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178 |
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179 |
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180 class Clamped(DecimalException): |
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181 """Exponent of a 0 changed to fit bounds. |
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182 |
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183 This occurs and signals clamped if the exponent of a result has been |
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184 altered in order to fit the constraints of a specific concrete |
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185 representation. This may occur when the exponent of a zero result would |
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186 be outside the bounds of a representation, or when a large normal |
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187 number would have an encoded exponent that cannot be represented. In |
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188 this latter case, the exponent is reduced to fit and the corresponding |
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189 number of zero digits are appended to the coefficient ("fold-down"). |
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190 """ |
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191 |
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192 class InvalidOperation(DecimalException): |
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193 """An invalid operation was performed. |
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194 |
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195 Various bad things cause this: |
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196 |
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197 Something creates a signaling NaN |
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198 -INF + INF |
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199 0 * (+-)INF |
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200 (+-)INF / (+-)INF |
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201 x % 0 |
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202 (+-)INF % x |
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203 x._rescale( non-integer ) |
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204 sqrt(-x) , x > 0 |
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205 0 ** 0 |
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206 x ** (non-integer) |
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207 x ** (+-)INF |
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208 An operand is invalid |
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209 |
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210 The result of the operation after these is a quiet positive NaN, |
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211 except when the cause is a signaling NaN, in which case the result is |
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212 also a quiet NaN, but with the original sign, and an optional |
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213 diagnostic information. |
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214 """ |
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215 def handle(self, context, *args): |
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216 if args: |
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217 ans = _dec_from_triple(args[0]._sign, args[0]._int, 'n', True) |
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218 return ans._fix_nan(context) |
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219 return NaN |
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220 |
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221 class ConversionSyntax(InvalidOperation): |
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222 """Trying to convert badly formed string. |
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223 |
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224 This occurs and signals invalid-operation if an string is being |
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225 converted to a number and it does not conform to the numeric string |
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226 syntax. The result is [0,qNaN]. |
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227 """ |
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228 def handle(self, context, *args): |
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229 return NaN |
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230 |
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231 class DivisionByZero(DecimalException, ZeroDivisionError): |
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232 """Division by 0. |
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233 |
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234 This occurs and signals division-by-zero if division of a finite number |
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235 by zero was attempted (during a divide-integer or divide operation, or a |
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236 power operation with negative right-hand operand), and the dividend was |
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237 not zero. |
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238 |
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239 The result of the operation is [sign,inf], where sign is the exclusive |
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240 or of the signs of the operands for divide, or is 1 for an odd power of |
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241 -0, for power. |
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242 """ |
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243 |
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244 def handle(self, context, sign, *args): |
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245 return Infsign[sign] |
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246 |
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247 class DivisionImpossible(InvalidOperation): |
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248 """Cannot perform the division adequately. |
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249 |
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250 This occurs and signals invalid-operation if the integer result of a |
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251 divide-integer or remainder operation had too many digits (would be |
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252 longer than precision). The result is [0,qNaN]. |
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253 """ |
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254 |
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255 def handle(self, context, *args): |
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256 return NaN |
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257 |
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258 class DivisionUndefined(InvalidOperation, ZeroDivisionError): |
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259 """Undefined result of division. |
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260 |
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261 This occurs and signals invalid-operation if division by zero was |
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262 attempted (during a divide-integer, divide, or remainder operation), and |
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263 the dividend is also zero. The result is [0,qNaN]. |
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264 """ |
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265 |
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266 def handle(self, context, *args): |
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267 return NaN |
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268 |
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269 class Inexact(DecimalException): |
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270 """Had to round, losing information. |
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271 |
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272 This occurs and signals inexact whenever the result of an operation is |
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273 not exact (that is, it needed to be rounded and any discarded digits |
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274 were non-zero), or if an overflow or underflow condition occurs. The |
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275 result in all cases is unchanged. |
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276 |
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277 The inexact signal may be tested (or trapped) to determine if a given |
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278 operation (or sequence of operations) was inexact. |
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279 """ |
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280 |
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281 class InvalidContext(InvalidOperation): |
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282 """Invalid context. Unknown rounding, for example. |
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283 |
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284 This occurs and signals invalid-operation if an invalid context was |
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285 detected during an operation. This can occur if contexts are not checked |
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286 on creation and either the precision exceeds the capability of the |
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287 underlying concrete representation or an unknown or unsupported rounding |
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288 was specified. These aspects of the context need only be checked when |
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289 the values are required to be used. The result is [0,qNaN]. |
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290 """ |
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291 |
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292 def handle(self, context, *args): |
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293 return NaN |
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294 |
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295 class Rounded(DecimalException): |
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296 """Number got rounded (not necessarily changed during rounding). |
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297 |
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298 This occurs and signals rounded whenever the result of an operation is |
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299 rounded (that is, some zero or non-zero digits were discarded from the |
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300 coefficient), or if an overflow or underflow condition occurs. The |
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301 result in all cases is unchanged. |
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302 |
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303 The rounded signal may be tested (or trapped) to determine if a given |
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304 operation (or sequence of operations) caused a loss of precision. |
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305 """ |
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306 |
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307 class Subnormal(DecimalException): |
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308 """Exponent < Emin before rounding. |
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309 |
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310 This occurs and signals subnormal whenever the result of a conversion or |
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311 operation is subnormal (that is, its adjusted exponent is less than |
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312 Emin, before any rounding). The result in all cases is unchanged. |
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313 |
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314 The subnormal signal may be tested (or trapped) to determine if a given |
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315 or operation (or sequence of operations) yielded a subnormal result. |
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316 """ |
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317 |
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318 class Overflow(Inexact, Rounded): |
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319 """Numerical overflow. |
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320 |
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321 This occurs and signals overflow if the adjusted exponent of a result |
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322 (from a conversion or from an operation that is not an attempt to divide |
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323 by zero), after rounding, would be greater than the largest value that |
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324 can be handled by the implementation (the value Emax). |
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325 |
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326 The result depends on the rounding mode: |
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327 |
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328 For round-half-up and round-half-even (and for round-half-down and |
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329 round-up, if implemented), the result of the operation is [sign,inf], |
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330 where sign is the sign of the intermediate result. For round-down, the |
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331 result is the largest finite number that can be represented in the |
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332 current precision, with the sign of the intermediate result. For |
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333 round-ceiling, the result is the same as for round-down if the sign of |
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334 the intermediate result is 1, or is [0,inf] otherwise. For round-floor, |
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335 the result is the same as for round-down if the sign of the intermediate |
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336 result is 0, or is [1,inf] otherwise. In all cases, Inexact and Rounded |
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337 will also be raised. |
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338 """ |
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339 |
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340 def handle(self, context, sign, *args): |
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341 if context.rounding in (ROUND_HALF_UP, ROUND_HALF_EVEN, |
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342 ROUND_HALF_DOWN, ROUND_UP): |
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343 return Infsign[sign] |
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344 if sign == 0: |
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345 if context.rounding == ROUND_CEILING: |
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346 return Infsign[sign] |
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347 return _dec_from_triple(sign, '9'*context.prec, |
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348 context.Emax-context.prec+1) |
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349 if sign == 1: |
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350 if context.rounding == ROUND_FLOOR: |
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351 return Infsign[sign] |
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352 return _dec_from_triple(sign, '9'*context.prec, |
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353 context.Emax-context.prec+1) |
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354 |
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355 |
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356 class Underflow(Inexact, Rounded, Subnormal): |
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357 """Numerical underflow with result rounded to 0. |
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358 |
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359 This occurs and signals underflow if a result is inexact and the |
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360 adjusted exponent of the result would be smaller (more negative) than |
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361 the smallest value that can be handled by the implementation (the value |
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362 Emin). That is, the result is both inexact and subnormal. |
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363 |
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364 The result after an underflow will be a subnormal number rounded, if |
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365 necessary, so that its exponent is not less than Etiny. This may result |
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366 in 0 with the sign of the intermediate result and an exponent of Etiny. |
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367 |
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368 In all cases, Inexact, Rounded, and Subnormal will also be raised. |
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369 """ |
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370 |
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371 # List of public traps and flags |
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372 _signals = [Clamped, DivisionByZero, Inexact, Overflow, Rounded, |
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373 Underflow, InvalidOperation, Subnormal] |
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374 |
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375 # Map conditions (per the spec) to signals |
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376 _condition_map = {ConversionSyntax:InvalidOperation, |
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377 DivisionImpossible:InvalidOperation, |
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378 DivisionUndefined:InvalidOperation, |
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379 InvalidContext:InvalidOperation} |
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380 |
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381 ##### Context Functions ################################################## |
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382 |
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383 # The getcontext() and setcontext() function manage access to a thread-local |
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384 # current context. Py2.4 offers direct support for thread locals. If that |
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385 # is not available, use threading.currentThread() which is slower but will |
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386 # work for older Pythons. If threads are not part of the build, create a |
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387 # mock threading object with threading.local() returning the module namespace. |
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388 |
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389 try: |
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390 import threading |
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391 except ImportError: |
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392 # Python was compiled without threads; create a mock object instead |
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393 import sys |
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394 class MockThreading(object): |
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395 def local(self, sys=sys): |
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396 return sys.modules[__name__] |
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397 threading = MockThreading() |
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398 del sys, MockThreading |
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399 |
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400 try: |
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401 threading.local |
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402 |
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403 except AttributeError: |
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404 |
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405 # To fix reloading, force it to create a new context |
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406 # Old contexts have different exceptions in their dicts, making problems. |
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407 if hasattr(threading.currentThread(), '__decimal_context__'): |
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408 del threading.currentThread().__decimal_context__ |
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409 |
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410 def setcontext(context): |
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411 """Set this thread's context to context.""" |
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412 if context in (DefaultContext, BasicContext, ExtendedContext): |
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413 context = context.copy() |
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414 context.clear_flags() |
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415 threading.currentThread().__decimal_context__ = context |
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416 |
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417 def getcontext(): |
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418 """Returns this thread's context. |
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419 |
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420 If this thread does not yet have a context, returns |
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421 a new context and sets this thread's context. |
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422 New contexts are copies of DefaultContext. |
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423 """ |
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424 try: |
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425 return threading.currentThread().__decimal_context__ |
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426 except AttributeError: |
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427 context = Context() |
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428 threading.currentThread().__decimal_context__ = context |
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429 return context |
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430 |
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431 else: |
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432 |
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433 local = threading.local() |
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434 if hasattr(local, '__decimal_context__'): |
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435 del local.__decimal_context__ |
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436 |
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437 def getcontext(_local=local): |
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438 """Returns this thread's context. |
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439 |
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440 If this thread does not yet have a context, returns |
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441 a new context and sets this thread's context. |
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442 New contexts are copies of DefaultContext. |
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443 """ |
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444 try: |
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445 return _local.__decimal_context__ |
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446 except AttributeError: |
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447 context = Context() |
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448 _local.__decimal_context__ = context |
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449 return context |
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450 |
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451 def setcontext(context, _local=local): |
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452 """Set this thread's context to context.""" |
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453 if context in (DefaultContext, BasicContext, ExtendedContext): |
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454 context = context.copy() |
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455 context.clear_flags() |
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456 _local.__decimal_context__ = context |
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457 |
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458 del threading, local # Don't contaminate the namespace |
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459 |
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460 def localcontext(ctx=None): |
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461 """Return a context manager for a copy of the supplied context |
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462 |
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463 Uses a copy of the current context if no context is specified |
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464 The returned context manager creates a local decimal context |
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465 in a with statement: |
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466 def sin(x): |
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467 with localcontext() as ctx: |
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468 ctx.prec += 2 |
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469 # Rest of sin calculation algorithm |
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470 # uses a precision 2 greater than normal |
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471 return +s # Convert result to normal precision |
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472 |
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473 def sin(x): |
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474 with localcontext(ExtendedContext): |
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475 # Rest of sin calculation algorithm |
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476 # uses the Extended Context from the |
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477 # General Decimal Arithmetic Specification |
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478 return +s # Convert result to normal context |
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479 |
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480 >>> setcontext(DefaultContext) |
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481 >>> print getcontext().prec |
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482 28 |
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483 >>> with localcontext(): |
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484 ... ctx = getcontext() |
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485 ... ctx.prec += 2 |
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486 ... print ctx.prec |
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487 ... |
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488 30 |
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489 >>> with localcontext(ExtendedContext): |
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490 ... print getcontext().prec |
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491 ... |
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492 9 |
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493 >>> print getcontext().prec |
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494 28 |
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495 """ |
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496 if ctx is None: ctx = getcontext() |
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497 return _ContextManager(ctx) |
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498 |
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499 |
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500 ##### Decimal class ####################################################### |
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501 |
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502 class Decimal(object): |
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503 """Floating point class for decimal arithmetic.""" |
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504 |
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505 __slots__ = ('_exp','_int','_sign', '_is_special') |
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506 # Generally, the value of the Decimal instance is given by |
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507 # (-1)**_sign * _int * 10**_exp |
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508 # Special values are signified by _is_special == True |
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509 |
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510 # We're immutable, so use __new__ not __init__ |
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511 def __new__(cls, value="0", context=None): |
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512 """Create a decimal point instance. |
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513 |
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514 >>> Decimal('3.14') # string input |
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515 Decimal('3.14') |
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516 >>> Decimal((0, (3, 1, 4), -2)) # tuple (sign, digit_tuple, exponent) |
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517 Decimal('3.14') |
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518 >>> Decimal(314) # int or long |
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519 Decimal('314') |
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520 >>> Decimal(Decimal(314)) # another decimal instance |
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521 Decimal('314') |
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522 >>> Decimal(' 3.14 \\n') # leading and trailing whitespace okay |
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523 Decimal('3.14') |
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524 """ |
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525 |
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526 # Note that the coefficient, self._int, is actually stored as |
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527 # a string rather than as a tuple of digits. This speeds up |
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528 # the "digits to integer" and "integer to digits" conversions |
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529 # that are used in almost every arithmetic operation on |
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530 # Decimals. This is an internal detail: the as_tuple function |
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531 # and the Decimal constructor still deal with tuples of |
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532 # digits. |
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533 |
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534 self = object.__new__(cls) |
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535 |
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536 # From a string |
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537 # REs insist on real strings, so we can too. |
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538 if isinstance(value, basestring): |
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539 m = _parser(value.strip()) |
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540 if m is None: |
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541 if context is None: |
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542 context = getcontext() |
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543 return context._raise_error(ConversionSyntax, |
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544 "Invalid literal for Decimal: %r" % value) |
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545 |
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546 if m.group('sign') == "-": |
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547 self._sign = 1 |
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548 else: |
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549 self._sign = 0 |
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550 intpart = m.group('int') |
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551 if intpart is not None: |
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552 # finite number |
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553 fracpart = m.group('frac') |
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554 exp = int(m.group('exp') or '0') |
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555 if fracpart is not None: |
|
556 self._int = str((intpart+fracpart).lstrip('0') or '0') |
|
557 self._exp = exp - len(fracpart) |
|
558 else: |
|
559 self._int = str(intpart.lstrip('0') or '0') |
|
560 self._exp = exp |
|
561 self._is_special = False |
|
562 else: |
|
563 diag = m.group('diag') |
|
564 if diag is not None: |
|
565 # NaN |
|
566 self._int = str(diag.lstrip('0')) |
|
567 if m.group('signal'): |
|
568 self._exp = 'N' |
|
569 else: |
|
570 self._exp = 'n' |
|
571 else: |
|
572 # infinity |
|
573 self._int = '0' |
|
574 self._exp = 'F' |
|
575 self._is_special = True |
|
576 return self |
|
577 |
|
578 # From an integer |
|
579 if isinstance(value, (int,long)): |
|
580 if value >= 0: |
|
581 self._sign = 0 |
|
582 else: |
|
583 self._sign = 1 |
|
584 self._exp = 0 |
|
585 self._int = str(abs(value)) |
|
586 self._is_special = False |
|
587 return self |
|
588 |
|
589 # From another decimal |
|
590 if isinstance(value, Decimal): |
|
591 self._exp = value._exp |
|
592 self._sign = value._sign |
|
593 self._int = value._int |
|
594 self._is_special = value._is_special |
|
595 return self |
|
596 |
|
597 # From an internal working value |
|
598 if isinstance(value, _WorkRep): |
|
599 self._sign = value.sign |
|
600 self._int = str(value.int) |
|
601 self._exp = int(value.exp) |
|
602 self._is_special = False |
|
603 return self |
|
604 |
|
605 # tuple/list conversion (possibly from as_tuple()) |
|
606 if isinstance(value, (list,tuple)): |
|
607 if len(value) != 3: |
|
608 raise ValueError('Invalid tuple size in creation of Decimal ' |
|
609 'from list or tuple. The list or tuple ' |
|
610 'should have exactly three elements.') |
|
611 # process sign. The isinstance test rejects floats |
|
612 if not (isinstance(value[0], (int, long)) and value[0] in (0,1)): |
|
613 raise ValueError("Invalid sign. The first value in the tuple " |
|
614 "should be an integer; either 0 for a " |
|
615 "positive number or 1 for a negative number.") |
|
616 self._sign = value[0] |
|
617 if value[2] == 'F': |
|
618 # infinity: value[1] is ignored |
|
619 self._int = '0' |
|
620 self._exp = value[2] |
|
621 self._is_special = True |
|
622 else: |
|
623 # process and validate the digits in value[1] |
|
624 digits = [] |
|
625 for digit in value[1]: |
|
626 if isinstance(digit, (int, long)) and 0 <= digit <= 9: |
|
627 # skip leading zeros |
|
628 if digits or digit != 0: |
|
629 digits.append(digit) |
|
630 else: |
|
631 raise ValueError("The second value in the tuple must " |
|
632 "be composed of integers in the range " |
|
633 "0 through 9.") |
|
634 if value[2] in ('n', 'N'): |
|
635 # NaN: digits form the diagnostic |
|
636 self._int = ''.join(map(str, digits)) |
|
637 self._exp = value[2] |
|
638 self._is_special = True |
|
639 elif isinstance(value[2], (int, long)): |
|
640 # finite number: digits give the coefficient |
|
641 self._int = ''.join(map(str, digits or [0])) |
|
642 self._exp = value[2] |
|
643 self._is_special = False |
|
644 else: |
|
645 raise ValueError("The third value in the tuple must " |
|
646 "be an integer, or one of the " |
|
647 "strings 'F', 'n', 'N'.") |
|
648 return self |
|
649 |
|
650 if isinstance(value, float): |
|
651 raise TypeError("Cannot convert float to Decimal. " + |
|
652 "First convert the float to a string") |
|
653 |
|
654 raise TypeError("Cannot convert %r to Decimal" % value) |
|
655 |
|
656 def _isnan(self): |
|
657 """Returns whether the number is not actually one. |
|
658 |
|
659 0 if a number |
|
660 1 if NaN |
|
661 2 if sNaN |
|
662 """ |
|
663 if self._is_special: |
|
664 exp = self._exp |
|
665 if exp == 'n': |
|
666 return 1 |
|
667 elif exp == 'N': |
|
668 return 2 |
|
669 return 0 |
|
670 |
|
671 def _isinfinity(self): |
|
672 """Returns whether the number is infinite |
|
673 |
|
674 0 if finite or not a number |
|
675 1 if +INF |
|
676 -1 if -INF |
|
677 """ |
|
678 if self._exp == 'F': |
|
679 if self._sign: |
|
680 return -1 |
|
681 return 1 |
|
682 return 0 |
|
683 |
|
684 def _check_nans(self, other=None, context=None): |
|
685 """Returns whether the number is not actually one. |
|
686 |
|
687 if self, other are sNaN, signal |
|
688 if self, other are NaN return nan |
|
689 return 0 |
|
690 |
|
691 Done before operations. |
|
692 """ |
|
693 |
|
694 self_is_nan = self._isnan() |
|
695 if other is None: |
|
696 other_is_nan = False |
|
697 else: |
|
698 other_is_nan = other._isnan() |
|
699 |
|
700 if self_is_nan or other_is_nan: |
|
701 if context is None: |
|
702 context = getcontext() |
|
703 |
|
704 if self_is_nan == 2: |
|
705 return context._raise_error(InvalidOperation, 'sNaN', |
|
706 self) |
|
707 if other_is_nan == 2: |
|
708 return context._raise_error(InvalidOperation, 'sNaN', |
|
709 other) |
|
710 if self_is_nan: |
|
711 return self._fix_nan(context) |
|
712 |
|
713 return other._fix_nan(context) |
|
714 return 0 |
|
715 |
|
716 def _compare_check_nans(self, other, context): |
|
717 """Version of _check_nans used for the signaling comparisons |
|
718 compare_signal, __le__, __lt__, __ge__, __gt__. |
|
719 |
|
720 Signal InvalidOperation if either self or other is a (quiet |
|
721 or signaling) NaN. Signaling NaNs take precedence over quiet |
|
722 NaNs. |
|
723 |
|
724 Return 0 if neither operand is a NaN. |
|
725 |
|
726 """ |
|
727 if context is None: |
|
728 context = getcontext() |
|
729 |
|
730 if self._is_special or other._is_special: |
|
731 if self.is_snan(): |
|
732 return context._raise_error(InvalidOperation, |
|
733 'comparison involving sNaN', |
|
734 self) |
|
735 elif other.is_snan(): |
|
736 return context._raise_error(InvalidOperation, |
|
737 'comparison involving sNaN', |
|
738 other) |
|
739 elif self.is_qnan(): |
|
740 return context._raise_error(InvalidOperation, |
|
741 'comparison involving NaN', |
|
742 self) |
|
743 elif other.is_qnan(): |
|
744 return context._raise_error(InvalidOperation, |
|
745 'comparison involving NaN', |
|
746 other) |
|
747 return 0 |
|
748 |
|
749 def __nonzero__(self): |
|
750 """Return True if self is nonzero; otherwise return False. |
|
751 |
|
752 NaNs and infinities are considered nonzero. |
|
753 """ |
|
754 return self._is_special or self._int != '0' |
|
755 |
|
756 def _cmp(self, other): |
|
757 """Compare the two non-NaN decimal instances self and other. |
|
758 |
|
759 Returns -1 if self < other, 0 if self == other and 1 |
|
760 if self > other. This routine is for internal use only.""" |
|
761 |
|
762 if self._is_special or other._is_special: |
|
763 return cmp(self._isinfinity(), other._isinfinity()) |
|
764 |
|
765 # check for zeros; note that cmp(0, -0) should return 0 |
|
766 if not self: |
|
767 if not other: |
|
768 return 0 |
|
769 else: |
|
770 return -((-1)**other._sign) |
|
771 if not other: |
|
772 return (-1)**self._sign |
|
773 |
|
774 # If different signs, neg one is less |
|
775 if other._sign < self._sign: |
|
776 return -1 |
|
777 if self._sign < other._sign: |
|
778 return 1 |
|
779 |
|
780 self_adjusted = self.adjusted() |
|
781 other_adjusted = other.adjusted() |
|
782 if self_adjusted == other_adjusted: |
|
783 self_padded = self._int + '0'*(self._exp - other._exp) |
|
784 other_padded = other._int + '0'*(other._exp - self._exp) |
|
785 return cmp(self_padded, other_padded) * (-1)**self._sign |
|
786 elif self_adjusted > other_adjusted: |
|
787 return (-1)**self._sign |
|
788 else: # self_adjusted < other_adjusted |
|
789 return -((-1)**self._sign) |
|
790 |
|
791 # Note: The Decimal standard doesn't cover rich comparisons for |
|
792 # Decimals. In particular, the specification is silent on the |
|
793 # subject of what should happen for a comparison involving a NaN. |
|
794 # We take the following approach: |
|
795 # |
|
796 # == comparisons involving a NaN always return False |
|
797 # != comparisons involving a NaN always return True |
|
798 # <, >, <= and >= comparisons involving a (quiet or signaling) |
|
799 # NaN signal InvalidOperation, and return False if the |
|
800 # InvalidOperation is not trapped. |
|
801 # |
|
802 # This behavior is designed to conform as closely as possible to |
|
803 # that specified by IEEE 754. |
|
804 |
|
805 def __eq__(self, other): |
|
806 other = _convert_other(other) |
|
807 if other is NotImplemented: |
|
808 return other |
|
809 if self.is_nan() or other.is_nan(): |
|
810 return False |
|
811 return self._cmp(other) == 0 |
|
812 |
|
813 def __ne__(self, other): |
|
814 other = _convert_other(other) |
|
815 if other is NotImplemented: |
|
816 return other |
|
817 if self.is_nan() or other.is_nan(): |
|
818 return True |
|
819 return self._cmp(other) != 0 |
|
820 |
|
821 def __lt__(self, other, context=None): |
|
822 other = _convert_other(other) |
|
823 if other is NotImplemented: |
|
824 return other |
|
825 ans = self._compare_check_nans(other, context) |
|
826 if ans: |
|
827 return False |
|
828 return self._cmp(other) < 0 |
|
829 |
|
830 def __le__(self, other, context=None): |
|
831 other = _convert_other(other) |
|
832 if other is NotImplemented: |
|
833 return other |
|
834 ans = self._compare_check_nans(other, context) |
|
835 if ans: |
|
836 return False |
|
837 return self._cmp(other) <= 0 |
|
838 |
|
839 def __gt__(self, other, context=None): |
|
840 other = _convert_other(other) |
|
841 if other is NotImplemented: |
|
842 return other |
|
843 ans = self._compare_check_nans(other, context) |
|
844 if ans: |
|
845 return False |
|
846 return self._cmp(other) > 0 |
|
847 |
|
848 def __ge__(self, other, context=None): |
|
849 other = _convert_other(other) |
|
850 if other is NotImplemented: |
|
851 return other |
|
852 ans = self._compare_check_nans(other, context) |
|
853 if ans: |
|
854 return False |
|
855 return self._cmp(other) >= 0 |
|
856 |
|
857 def compare(self, other, context=None): |
|
858 """Compares one to another. |
|
859 |
|
860 -1 => a < b |
|
861 0 => a = b |
|
862 1 => a > b |
|
863 NaN => one is NaN |
|
864 Like __cmp__, but returns Decimal instances. |
|
865 """ |
|
866 other = _convert_other(other, raiseit=True) |
|
867 |
|
868 # Compare(NaN, NaN) = NaN |
|
869 if (self._is_special or other and other._is_special): |
|
870 ans = self._check_nans(other, context) |
|
871 if ans: |
|
872 return ans |
|
873 |
|
874 return Decimal(self._cmp(other)) |
|
875 |
|
876 def __hash__(self): |
|
877 """x.__hash__() <==> hash(x)""" |
|
878 # Decimal integers must hash the same as the ints |
|
879 # |
|
880 # The hash of a nonspecial noninteger Decimal must depend only |
|
881 # on the value of that Decimal, and not on its representation. |
|
882 # For example: hash(Decimal('100E-1')) == hash(Decimal('10')). |
|
883 if self._is_special: |
|
884 if self._isnan(): |
|
885 raise TypeError('Cannot hash a NaN value.') |
|
886 return hash(str(self)) |
|
887 if not self: |
|
888 return 0 |
|
889 if self._isinteger(): |
|
890 op = _WorkRep(self.to_integral_value()) |
|
891 # to make computation feasible for Decimals with large |
|
892 # exponent, we use the fact that hash(n) == hash(m) for |
|
893 # any two nonzero integers n and m such that (i) n and m |
|
894 # have the same sign, and (ii) n is congruent to m modulo |
|
895 # 2**64-1. So we can replace hash((-1)**s*c*10**e) with |
|
896 # hash((-1)**s*c*pow(10, e, 2**64-1). |
|
897 return hash((-1)**op.sign*op.int*pow(10, op.exp, 2**64-1)) |
|
898 # The value of a nonzero nonspecial Decimal instance is |
|
899 # faithfully represented by the triple consisting of its sign, |
|
900 # its adjusted exponent, and its coefficient with trailing |
|
901 # zeros removed. |
|
902 return hash((self._sign, |
|
903 self._exp+len(self._int), |
|
904 self._int.rstrip('0'))) |
|
905 |
|
906 def as_tuple(self): |
|
907 """Represents the number as a triple tuple. |
|
908 |
|
909 To show the internals exactly as they are. |
|
910 """ |
|
911 return DecimalTuple(self._sign, tuple(map(int, self._int)), self._exp) |
|
912 |
|
913 def __repr__(self): |
|
914 """Represents the number as an instance of Decimal.""" |
|
915 # Invariant: eval(repr(d)) == d |
|
916 return "Decimal('%s')" % str(self) |
|
917 |
|
918 def __str__(self, eng=False, context=None): |
|
919 """Return string representation of the number in scientific notation. |
|
920 |
|
921 Captures all of the information in the underlying representation. |
|
922 """ |
|
923 |
|
924 sign = ['', '-'][self._sign] |
|
925 if self._is_special: |
|
926 if self._exp == 'F': |
|
927 return sign + 'Infinity' |
|
928 elif self._exp == 'n': |
|
929 return sign + 'NaN' + self._int |
|
930 else: # self._exp == 'N' |
|
931 return sign + 'sNaN' + self._int |
|
932 |
|
933 # number of digits of self._int to left of decimal point |
|
934 leftdigits = self._exp + len(self._int) |
|
935 |
|
936 # dotplace is number of digits of self._int to the left of the |
|
937 # decimal point in the mantissa of the output string (that is, |
|
938 # after adjusting the exponent) |
|
939 if self._exp <= 0 and leftdigits > -6: |
|
940 # no exponent required |
|
941 dotplace = leftdigits |
|
942 elif not eng: |
|
943 # usual scientific notation: 1 digit on left of the point |
|
944 dotplace = 1 |
|
945 elif self._int == '0': |
|
946 # engineering notation, zero |
|
947 dotplace = (leftdigits + 1) % 3 - 1 |
|
948 else: |
|
949 # engineering notation, nonzero |
|
950 dotplace = (leftdigits - 1) % 3 + 1 |
|
951 |
|
952 if dotplace <= 0: |
|
953 intpart = '0' |
|
954 fracpart = '.' + '0'*(-dotplace) + self._int |
|
955 elif dotplace >= len(self._int): |
|
956 intpart = self._int+'0'*(dotplace-len(self._int)) |
|
957 fracpart = '' |
|
958 else: |
|
959 intpart = self._int[:dotplace] |
|
960 fracpart = '.' + self._int[dotplace:] |
|
961 if leftdigits == dotplace: |
|
962 exp = '' |
|
963 else: |
|
964 if context is None: |
|
965 context = getcontext() |
|
966 exp = ['e', 'E'][context.capitals] + "%+d" % (leftdigits-dotplace) |
|
967 |
|
968 return sign + intpart + fracpart + exp |
|
969 |
|
970 def to_eng_string(self, context=None): |
|
971 """Convert to engineering-type string. |
|
972 |
|
973 Engineering notation has an exponent which is a multiple of 3, so there |
|
974 are up to 3 digits left of the decimal place. |
|
975 |
|
976 Same rules for when in exponential and when as a value as in __str__. |
|
977 """ |
|
978 return self.__str__(eng=True, context=context) |
|
979 |
|
980 def __neg__(self, context=None): |
|
981 """Returns a copy with the sign switched. |
|
982 |
|
983 Rounds, if it has reason. |
|
984 """ |
|
985 if self._is_special: |
|
986 ans = self._check_nans(context=context) |
|
987 if ans: |
|
988 return ans |
|
989 |
|
990 if not self: |
|
991 # -Decimal('0') is Decimal('0'), not Decimal('-0') |
|
992 ans = self.copy_abs() |
|
993 else: |
|
994 ans = self.copy_negate() |
|
995 |
|
996 if context is None: |
|
997 context = getcontext() |
|
998 return ans._fix(context) |
|
999 |
|
1000 def __pos__(self, context=None): |
|
1001 """Returns a copy, unless it is a sNaN. |
|
1002 |
|
1003 Rounds the number (if more then precision digits) |
|
1004 """ |
|
1005 if self._is_special: |
|
1006 ans = self._check_nans(context=context) |
|
1007 if ans: |
|
1008 return ans |
|
1009 |
|
1010 if not self: |
|
1011 # + (-0) = 0 |
|
1012 ans = self.copy_abs() |
|
1013 else: |
|
1014 ans = Decimal(self) |
|
1015 |
|
1016 if context is None: |
|
1017 context = getcontext() |
|
1018 return ans._fix(context) |
|
1019 |
|
1020 def __abs__(self, round=True, context=None): |
|
1021 """Returns the absolute value of self. |
|
1022 |
|
1023 If the keyword argument 'round' is false, do not round. The |
|
1024 expression self.__abs__(round=False) is equivalent to |
|
1025 self.copy_abs(). |
|
1026 """ |
|
1027 if not round: |
|
1028 return self.copy_abs() |
|
1029 |
|
1030 if self._is_special: |
|
1031 ans = self._check_nans(context=context) |
|
1032 if ans: |
|
1033 return ans |
|
1034 |
|
1035 if self._sign: |
|
1036 ans = self.__neg__(context=context) |
|
1037 else: |
|
1038 ans = self.__pos__(context=context) |
|
1039 |
|
1040 return ans |
|
1041 |
|
1042 def __add__(self, other, context=None): |
|
1043 """Returns self + other. |
|
1044 |
|
1045 -INF + INF (or the reverse) cause InvalidOperation errors. |
|
1046 """ |
|
1047 other = _convert_other(other) |
|
1048 if other is NotImplemented: |
|
1049 return other |
|
1050 |
|
1051 if context is None: |
|
1052 context = getcontext() |
|
1053 |
|
1054 if self._is_special or other._is_special: |
|
1055 ans = self._check_nans(other, context) |
|
1056 if ans: |
|
1057 return ans |
|
1058 |
|
1059 if self._isinfinity(): |
|
1060 # If both INF, same sign => same as both, opposite => error. |
|
1061 if self._sign != other._sign and other._isinfinity(): |
|
1062 return context._raise_error(InvalidOperation, '-INF + INF') |
|
1063 return Decimal(self) |
|
1064 if other._isinfinity(): |
|
1065 return Decimal(other) # Can't both be infinity here |
|
1066 |
|
1067 exp = min(self._exp, other._exp) |
|
1068 negativezero = 0 |
|
1069 if context.rounding == ROUND_FLOOR and self._sign != other._sign: |
|
1070 # If the answer is 0, the sign should be negative, in this case. |
|
1071 negativezero = 1 |
|
1072 |
|
1073 if not self and not other: |
|
1074 sign = min(self._sign, other._sign) |
|
1075 if negativezero: |
|
1076 sign = 1 |
|
1077 ans = _dec_from_triple(sign, '0', exp) |
|
1078 ans = ans._fix(context) |
|
1079 return ans |
|
1080 if not self: |
|
1081 exp = max(exp, other._exp - context.prec-1) |
|
1082 ans = other._rescale(exp, context.rounding) |
|
1083 ans = ans._fix(context) |
|
1084 return ans |
|
1085 if not other: |
|
1086 exp = max(exp, self._exp - context.prec-1) |
|
1087 ans = self._rescale(exp, context.rounding) |
|
1088 ans = ans._fix(context) |
|
1089 return ans |
|
1090 |
|
1091 op1 = _WorkRep(self) |
|
1092 op2 = _WorkRep(other) |
|
1093 op1, op2 = _normalize(op1, op2, context.prec) |
|
1094 |
|
1095 result = _WorkRep() |
|
1096 if op1.sign != op2.sign: |
|
1097 # Equal and opposite |
|
1098 if op1.int == op2.int: |
|
1099 ans = _dec_from_triple(negativezero, '0', exp) |
|
1100 ans = ans._fix(context) |
|
1101 return ans |
|
1102 if op1.int < op2.int: |
|
1103 op1, op2 = op2, op1 |
|
1104 # OK, now abs(op1) > abs(op2) |
|
1105 if op1.sign == 1: |
|
1106 result.sign = 1 |
|
1107 op1.sign, op2.sign = op2.sign, op1.sign |
|
1108 else: |
|
1109 result.sign = 0 |
|
1110 # So we know the sign, and op1 > 0. |
|
1111 elif op1.sign == 1: |
|
1112 result.sign = 1 |
|
1113 op1.sign, op2.sign = (0, 0) |
|
1114 else: |
|
1115 result.sign = 0 |
|
1116 # Now, op1 > abs(op2) > 0 |
|
1117 |
|
1118 if op2.sign == 0: |
|
1119 result.int = op1.int + op2.int |
|
1120 else: |
|
1121 result.int = op1.int - op2.int |
|
1122 |
|
1123 result.exp = op1.exp |
|
1124 ans = Decimal(result) |
|
1125 ans = ans._fix(context) |
|
1126 return ans |
|
1127 |
|
1128 __radd__ = __add__ |
|
1129 |
|
1130 def __sub__(self, other, context=None): |
|
1131 """Return self - other""" |
|
1132 other = _convert_other(other) |
|
1133 if other is NotImplemented: |
|
1134 return other |
|
1135 |
|
1136 if self._is_special or other._is_special: |
|
1137 ans = self._check_nans(other, context=context) |
|
1138 if ans: |
|
1139 return ans |
|
1140 |
|
1141 # self - other is computed as self + other.copy_negate() |
|
1142 return self.__add__(other.copy_negate(), context=context) |
|
1143 |
|
1144 def __rsub__(self, other, context=None): |
|
1145 """Return other - self""" |
|
1146 other = _convert_other(other) |
|
1147 if other is NotImplemented: |
|
1148 return other |
|
1149 |
|
1150 return other.__sub__(self, context=context) |
|
1151 |
|
1152 def __mul__(self, other, context=None): |
|
1153 """Return self * other. |
|
1154 |
|
1155 (+-) INF * 0 (or its reverse) raise InvalidOperation. |
|
1156 """ |
|
1157 other = _convert_other(other) |
|
1158 if other is NotImplemented: |
|
1159 return other |
|
1160 |
|
1161 if context is None: |
|
1162 context = getcontext() |
|
1163 |
|
1164 resultsign = self._sign ^ other._sign |
|
1165 |
|
1166 if self._is_special or other._is_special: |
|
1167 ans = self._check_nans(other, context) |
|
1168 if ans: |
|
1169 return ans |
|
1170 |
|
1171 if self._isinfinity(): |
|
1172 if not other: |
|
1173 return context._raise_error(InvalidOperation, '(+-)INF * 0') |
|
1174 return Infsign[resultsign] |
|
1175 |
|
1176 if other._isinfinity(): |
|
1177 if not self: |
|
1178 return context._raise_error(InvalidOperation, '0 * (+-)INF') |
|
1179 return Infsign[resultsign] |
|
1180 |
|
1181 resultexp = self._exp + other._exp |
|
1182 |
|
1183 # Special case for multiplying by zero |
|
1184 if not self or not other: |
|
1185 ans = _dec_from_triple(resultsign, '0', resultexp) |
|
1186 # Fixing in case the exponent is out of bounds |
|
1187 ans = ans._fix(context) |
|
1188 return ans |
|
1189 |
|
1190 # Special case for multiplying by power of 10 |
|
1191 if self._int == '1': |
|
1192 ans = _dec_from_triple(resultsign, other._int, resultexp) |
|
1193 ans = ans._fix(context) |
|
1194 return ans |
|
1195 if other._int == '1': |
|
1196 ans = _dec_from_triple(resultsign, self._int, resultexp) |
|
1197 ans = ans._fix(context) |
|
1198 return ans |
|
1199 |
|
1200 op1 = _WorkRep(self) |
|
1201 op2 = _WorkRep(other) |
|
1202 |
|
1203 ans = _dec_from_triple(resultsign, str(op1.int * op2.int), resultexp) |
|
1204 ans = ans._fix(context) |
|
1205 |
|
1206 return ans |
|
1207 __rmul__ = __mul__ |
|
1208 |
|
1209 def __truediv__(self, other, context=None): |
|
1210 """Return self / other.""" |
|
1211 other = _convert_other(other) |
|
1212 if other is NotImplemented: |
|
1213 return NotImplemented |
|
1214 |
|
1215 if context is None: |
|
1216 context = getcontext() |
|
1217 |
|
1218 sign = self._sign ^ other._sign |
|
1219 |
|
1220 if self._is_special or other._is_special: |
|
1221 ans = self._check_nans(other, context) |
|
1222 if ans: |
|
1223 return ans |
|
1224 |
|
1225 if self._isinfinity() and other._isinfinity(): |
|
1226 return context._raise_error(InvalidOperation, '(+-)INF/(+-)INF') |
|
1227 |
|
1228 if self._isinfinity(): |
|
1229 return Infsign[sign] |
|
1230 |
|
1231 if other._isinfinity(): |
|
1232 context._raise_error(Clamped, 'Division by infinity') |
|
1233 return _dec_from_triple(sign, '0', context.Etiny()) |
|
1234 |
|
1235 # Special cases for zeroes |
|
1236 if not other: |
|
1237 if not self: |
|
1238 return context._raise_error(DivisionUndefined, '0 / 0') |
|
1239 return context._raise_error(DivisionByZero, 'x / 0', sign) |
|
1240 |
|
1241 if not self: |
|
1242 exp = self._exp - other._exp |
|
1243 coeff = 0 |
|
1244 else: |
|
1245 # OK, so neither = 0, INF or NaN |
|
1246 shift = len(other._int) - len(self._int) + context.prec + 1 |
|
1247 exp = self._exp - other._exp - shift |
|
1248 op1 = _WorkRep(self) |
|
1249 op2 = _WorkRep(other) |
|
1250 if shift >= 0: |
|
1251 coeff, remainder = divmod(op1.int * 10**shift, op2.int) |
|
1252 else: |
|
1253 coeff, remainder = divmod(op1.int, op2.int * 10**-shift) |
|
1254 if remainder: |
|
1255 # result is not exact; adjust to ensure correct rounding |
|
1256 if coeff % 5 == 0: |
|
1257 coeff += 1 |
|
1258 else: |
|
1259 # result is exact; get as close to ideal exponent as possible |
|
1260 ideal_exp = self._exp - other._exp |
|
1261 while exp < ideal_exp and coeff % 10 == 0: |
|
1262 coeff //= 10 |
|
1263 exp += 1 |
|
1264 |
|
1265 ans = _dec_from_triple(sign, str(coeff), exp) |
|
1266 return ans._fix(context) |
|
1267 |
|
1268 def _divide(self, other, context): |
|
1269 """Return (self // other, self % other), to context.prec precision. |
|
1270 |
|
1271 Assumes that neither self nor other is a NaN, that self is not |
|
1272 infinite and that other is nonzero. |
|
1273 """ |
|
1274 sign = self._sign ^ other._sign |
|
1275 if other._isinfinity(): |
|
1276 ideal_exp = self._exp |
|
1277 else: |
|
1278 ideal_exp = min(self._exp, other._exp) |
|
1279 |
|
1280 expdiff = self.adjusted() - other.adjusted() |
|
1281 if not self or other._isinfinity() or expdiff <= -2: |
|
1282 return (_dec_from_triple(sign, '0', 0), |
|
1283 self._rescale(ideal_exp, context.rounding)) |
|
1284 if expdiff <= context.prec: |
|
1285 op1 = _WorkRep(self) |
|
1286 op2 = _WorkRep(other) |
|
1287 if op1.exp >= op2.exp: |
|
1288 op1.int *= 10**(op1.exp - op2.exp) |
|
1289 else: |
|
1290 op2.int *= 10**(op2.exp - op1.exp) |
|
1291 q, r = divmod(op1.int, op2.int) |
|
1292 if q < 10**context.prec: |
|
1293 return (_dec_from_triple(sign, str(q), 0), |
|
1294 _dec_from_triple(self._sign, str(r), ideal_exp)) |
|
1295 |
|
1296 # Here the quotient is too large to be representable |
|
1297 ans = context._raise_error(DivisionImpossible, |
|
1298 'quotient too large in //, % or divmod') |
|
1299 return ans, ans |
|
1300 |
|
1301 def __rtruediv__(self, other, context=None): |
|
1302 """Swaps self/other and returns __truediv__.""" |
|
1303 other = _convert_other(other) |
|
1304 if other is NotImplemented: |
|
1305 return other |
|
1306 return other.__truediv__(self, context=context) |
|
1307 |
|
1308 __div__ = __truediv__ |
|
1309 __rdiv__ = __rtruediv__ |
|
1310 |
|
1311 def __divmod__(self, other, context=None): |
|
1312 """ |
|
1313 Return (self // other, self % other) |
|
1314 """ |
|
1315 other = _convert_other(other) |
|
1316 if other is NotImplemented: |
|
1317 return other |
|
1318 |
|
1319 if context is None: |
|
1320 context = getcontext() |
|
1321 |
|
1322 ans = self._check_nans(other, context) |
|
1323 if ans: |
|
1324 return (ans, ans) |
|
1325 |
|
1326 sign = self._sign ^ other._sign |
|
1327 if self._isinfinity(): |
|
1328 if other._isinfinity(): |
|
1329 ans = context._raise_error(InvalidOperation, 'divmod(INF, INF)') |
|
1330 return ans, ans |
|
1331 else: |
|
1332 return (Infsign[sign], |
|
1333 context._raise_error(InvalidOperation, 'INF % x')) |
|
1334 |
|
1335 if not other: |
|
1336 if not self: |
|
1337 ans = context._raise_error(DivisionUndefined, 'divmod(0, 0)') |
|
1338 return ans, ans |
|
1339 else: |
|
1340 return (context._raise_error(DivisionByZero, 'x // 0', sign), |
|
1341 context._raise_error(InvalidOperation, 'x % 0')) |
|
1342 |
|
1343 quotient, remainder = self._divide(other, context) |
|
1344 remainder = remainder._fix(context) |
|
1345 return quotient, remainder |
|
1346 |
|
1347 def __rdivmod__(self, other, context=None): |
|
1348 """Swaps self/other and returns __divmod__.""" |
|
1349 other = _convert_other(other) |
|
1350 if other is NotImplemented: |
|
1351 return other |
|
1352 return other.__divmod__(self, context=context) |
|
1353 |
|
1354 def __mod__(self, other, context=None): |
|
1355 """ |
|
1356 self % other |
|
1357 """ |
|
1358 other = _convert_other(other) |
|
1359 if other is NotImplemented: |
|
1360 return other |
|
1361 |
|
1362 if context is None: |
|
1363 context = getcontext() |
|
1364 |
|
1365 ans = self._check_nans(other, context) |
|
1366 if ans: |
|
1367 return ans |
|
1368 |
|
1369 if self._isinfinity(): |
|
1370 return context._raise_error(InvalidOperation, 'INF % x') |
|
1371 elif not other: |
|
1372 if self: |
|
1373 return context._raise_error(InvalidOperation, 'x % 0') |
|
1374 else: |
|
1375 return context._raise_error(DivisionUndefined, '0 % 0') |
|
1376 |
|
1377 remainder = self._divide(other, context)[1] |
|
1378 remainder = remainder._fix(context) |
|
1379 return remainder |
|
1380 |
|
1381 def __rmod__(self, other, context=None): |
|
1382 """Swaps self/other and returns __mod__.""" |
|
1383 other = _convert_other(other) |
|
1384 if other is NotImplemented: |
|
1385 return other |
|
1386 return other.__mod__(self, context=context) |
|
1387 |
|
1388 def remainder_near(self, other, context=None): |
|
1389 """ |
|
1390 Remainder nearest to 0- abs(remainder-near) <= other/2 |
|
1391 """ |
|
1392 if context is None: |
|
1393 context = getcontext() |
|
1394 |
|
1395 other = _convert_other(other, raiseit=True) |
|
1396 |
|
1397 ans = self._check_nans(other, context) |
|
1398 if ans: |
|
1399 return ans |
|
1400 |
|
1401 # self == +/-infinity -> InvalidOperation |
|
1402 if self._isinfinity(): |
|
1403 return context._raise_error(InvalidOperation, |
|
1404 'remainder_near(infinity, x)') |
|
1405 |
|
1406 # other == 0 -> either InvalidOperation or DivisionUndefined |
|
1407 if not other: |
|
1408 if self: |
|
1409 return context._raise_error(InvalidOperation, |
|
1410 'remainder_near(x, 0)') |
|
1411 else: |
|
1412 return context._raise_error(DivisionUndefined, |
|
1413 'remainder_near(0, 0)') |
|
1414 |
|
1415 # other = +/-infinity -> remainder = self |
|
1416 if other._isinfinity(): |
|
1417 ans = Decimal(self) |
|
1418 return ans._fix(context) |
|
1419 |
|
1420 # self = 0 -> remainder = self, with ideal exponent |
|
1421 ideal_exponent = min(self._exp, other._exp) |
|
1422 if not self: |
|
1423 ans = _dec_from_triple(self._sign, '0', ideal_exponent) |
|
1424 return ans._fix(context) |
|
1425 |
|
1426 # catch most cases of large or small quotient |
|
1427 expdiff = self.adjusted() - other.adjusted() |
|
1428 if expdiff >= context.prec + 1: |
|
1429 # expdiff >= prec+1 => abs(self/other) > 10**prec |
|
1430 return context._raise_error(DivisionImpossible) |
|
1431 if expdiff <= -2: |
|
1432 # expdiff <= -2 => abs(self/other) < 0.1 |
|
1433 ans = self._rescale(ideal_exponent, context.rounding) |
|
1434 return ans._fix(context) |
|
1435 |
|
1436 # adjust both arguments to have the same exponent, then divide |
|
1437 op1 = _WorkRep(self) |
|
1438 op2 = _WorkRep(other) |
|
1439 if op1.exp >= op2.exp: |
|
1440 op1.int *= 10**(op1.exp - op2.exp) |
|
1441 else: |
|
1442 op2.int *= 10**(op2.exp - op1.exp) |
|
1443 q, r = divmod(op1.int, op2.int) |
|
1444 # remainder is r*10**ideal_exponent; other is +/-op2.int * |
|
1445 # 10**ideal_exponent. Apply correction to ensure that |
|
1446 # abs(remainder) <= abs(other)/2 |
|
1447 if 2*r + (q&1) > op2.int: |
|
1448 r -= op2.int |
|
1449 q += 1 |
|
1450 |
|
1451 if q >= 10**context.prec: |
|
1452 return context._raise_error(DivisionImpossible) |
|
1453 |
|
1454 # result has same sign as self unless r is negative |
|
1455 sign = self._sign |
|
1456 if r < 0: |
|
1457 sign = 1-sign |
|
1458 r = -r |
|
1459 |
|
1460 ans = _dec_from_triple(sign, str(r), ideal_exponent) |
|
1461 return ans._fix(context) |
|
1462 |
|
1463 def __floordiv__(self, other, context=None): |
|
1464 """self // other""" |
|
1465 other = _convert_other(other) |
|
1466 if other is NotImplemented: |
|
1467 return other |
|
1468 |
|
1469 if context is None: |
|
1470 context = getcontext() |
|
1471 |
|
1472 ans = self._check_nans(other, context) |
|
1473 if ans: |
|
1474 return ans |
|
1475 |
|
1476 if self._isinfinity(): |
|
1477 if other._isinfinity(): |
|
1478 return context._raise_error(InvalidOperation, 'INF // INF') |
|
1479 else: |
|
1480 return Infsign[self._sign ^ other._sign] |
|
1481 |
|
1482 if not other: |
|
1483 if self: |
|
1484 return context._raise_error(DivisionByZero, 'x // 0', |
|
1485 self._sign ^ other._sign) |
|
1486 else: |
|
1487 return context._raise_error(DivisionUndefined, '0 // 0') |
|
1488 |
|
1489 return self._divide(other, context)[0] |
|
1490 |
|
1491 def __rfloordiv__(self, other, context=None): |
|
1492 """Swaps self/other and returns __floordiv__.""" |
|
1493 other = _convert_other(other) |
|
1494 if other is NotImplemented: |
|
1495 return other |
|
1496 return other.__floordiv__(self, context=context) |
|
1497 |
|
1498 def __float__(self): |
|
1499 """Float representation.""" |
|
1500 return float(str(self)) |
|
1501 |
|
1502 def __int__(self): |
|
1503 """Converts self to an int, truncating if necessary.""" |
|
1504 if self._is_special: |
|
1505 if self._isnan(): |
|
1506 context = getcontext() |
|
1507 return context._raise_error(InvalidContext) |
|
1508 elif self._isinfinity(): |
|
1509 raise OverflowError("Cannot convert infinity to int") |
|
1510 s = (-1)**self._sign |
|
1511 if self._exp >= 0: |
|
1512 return s*int(self._int)*10**self._exp |
|
1513 else: |
|
1514 return s*int(self._int[:self._exp] or '0') |
|
1515 |
|
1516 __trunc__ = __int__ |
|
1517 |
|
1518 @property |
|
1519 def real(self): |
|
1520 return self |
|
1521 |
|
1522 @property |
|
1523 def imag(self): |
|
1524 return Decimal(0) |
|
1525 |
|
1526 def conjugate(self): |
|
1527 return self |
|
1528 |
|
1529 def __complex__(self): |
|
1530 return complex(float(self)) |
|
1531 |
|
1532 def __long__(self): |
|
1533 """Converts to a long. |
|
1534 |
|
1535 Equivalent to long(int(self)) |
|
1536 """ |
|
1537 return long(self.__int__()) |
|
1538 |
|
1539 def _fix_nan(self, context): |
|
1540 """Decapitate the payload of a NaN to fit the context""" |
|
1541 payload = self._int |
|
1542 |
|
1543 # maximum length of payload is precision if _clamp=0, |
|
1544 # precision-1 if _clamp=1. |
|
1545 max_payload_len = context.prec - context._clamp |
|
1546 if len(payload) > max_payload_len: |
|
1547 payload = payload[len(payload)-max_payload_len:].lstrip('0') |
|
1548 return _dec_from_triple(self._sign, payload, self._exp, True) |
|
1549 return Decimal(self) |
|
1550 |
|
1551 def _fix(self, context): |
|
1552 """Round if it is necessary to keep self within prec precision. |
|
1553 |
|
1554 Rounds and fixes the exponent. Does not raise on a sNaN. |
|
1555 |
|
1556 Arguments: |
|
1557 self - Decimal instance |
|
1558 context - context used. |
|
1559 """ |
|
1560 |
|
1561 if self._is_special: |
|
1562 if self._isnan(): |
|
1563 # decapitate payload if necessary |
|
1564 return self._fix_nan(context) |
|
1565 else: |
|
1566 # self is +/-Infinity; return unaltered |
|
1567 return Decimal(self) |
|
1568 |
|
1569 # if self is zero then exponent should be between Etiny and |
|
1570 # Emax if _clamp==0, and between Etiny and Etop if _clamp==1. |
|
1571 Etiny = context.Etiny() |
|
1572 Etop = context.Etop() |
|
1573 if not self: |
|
1574 exp_max = [context.Emax, Etop][context._clamp] |
|
1575 new_exp = min(max(self._exp, Etiny), exp_max) |
|
1576 if new_exp != self._exp: |
|
1577 context._raise_error(Clamped) |
|
1578 return _dec_from_triple(self._sign, '0', new_exp) |
|
1579 else: |
|
1580 return Decimal(self) |
|
1581 |
|
1582 # exp_min is the smallest allowable exponent of the result, |
|
1583 # equal to max(self.adjusted()-context.prec+1, Etiny) |
|
1584 exp_min = len(self._int) + self._exp - context.prec |
|
1585 if exp_min > Etop: |
|
1586 # overflow: exp_min > Etop iff self.adjusted() > Emax |
|
1587 context._raise_error(Inexact) |
|
1588 context._raise_error(Rounded) |
|
1589 return context._raise_error(Overflow, 'above Emax', self._sign) |
|
1590 self_is_subnormal = exp_min < Etiny |
|
1591 if self_is_subnormal: |
|
1592 context._raise_error(Subnormal) |
|
1593 exp_min = Etiny |
|
1594 |
|
1595 # round if self has too many digits |
|
1596 if self._exp < exp_min: |
|
1597 context._raise_error(Rounded) |
|
1598 digits = len(self._int) + self._exp - exp_min |
|
1599 if digits < 0: |
|
1600 self = _dec_from_triple(self._sign, '1', exp_min-1) |
|
1601 digits = 0 |
|
1602 this_function = getattr(self, self._pick_rounding_function[context.rounding]) |
|
1603 changed = this_function(digits) |
|
1604 coeff = self._int[:digits] or '0' |
|
1605 if changed == 1: |
|
1606 coeff = str(int(coeff)+1) |
|
1607 ans = _dec_from_triple(self._sign, coeff, exp_min) |
|
1608 |
|
1609 if changed: |
|
1610 context._raise_error(Inexact) |
|
1611 if self_is_subnormal: |
|
1612 context._raise_error(Underflow) |
|
1613 if not ans: |
|
1614 # raise Clamped on underflow to 0 |
|
1615 context._raise_error(Clamped) |
|
1616 elif len(ans._int) == context.prec+1: |
|
1617 # we get here only if rescaling rounds the |
|
1618 # cofficient up to exactly 10**context.prec |
|
1619 if ans._exp < Etop: |
|
1620 ans = _dec_from_triple(ans._sign, |
|
1621 ans._int[:-1], ans._exp+1) |
|
1622 else: |
|
1623 # Inexact and Rounded have already been raised |
|
1624 ans = context._raise_error(Overflow, 'above Emax', |
|
1625 self._sign) |
|
1626 return ans |
|
1627 |
|
1628 # fold down if _clamp == 1 and self has too few digits |
|
1629 if context._clamp == 1 and self._exp > Etop: |
|
1630 context._raise_error(Clamped) |
|
1631 self_padded = self._int + '0'*(self._exp - Etop) |
|
1632 return _dec_from_triple(self._sign, self_padded, Etop) |
|
1633 |
|
1634 # here self was representable to begin with; return unchanged |
|
1635 return Decimal(self) |
|
1636 |
|
1637 _pick_rounding_function = {} |
|
1638 |
|
1639 # for each of the rounding functions below: |
|
1640 # self is a finite, nonzero Decimal |
|
1641 # prec is an integer satisfying 0 <= prec < len(self._int) |
|
1642 # |
|
1643 # each function returns either -1, 0, or 1, as follows: |
|
1644 # 1 indicates that self should be rounded up (away from zero) |
|
1645 # 0 indicates that self should be truncated, and that all the |
|
1646 # digits to be truncated are zeros (so the value is unchanged) |
|
1647 # -1 indicates that there are nonzero digits to be truncated |
|
1648 |
|
1649 def _round_down(self, prec): |
|
1650 """Also known as round-towards-0, truncate.""" |
|
1651 if _all_zeros(self._int, prec): |
|
1652 return 0 |
|
1653 else: |
|
1654 return -1 |
|
1655 |
|
1656 def _round_up(self, prec): |
|
1657 """Rounds away from 0.""" |
|
1658 return -self._round_down(prec) |
|
1659 |
|
1660 def _round_half_up(self, prec): |
|
1661 """Rounds 5 up (away from 0)""" |
|
1662 if self._int[prec] in '56789': |
|
1663 return 1 |
|
1664 elif _all_zeros(self._int, prec): |
|
1665 return 0 |
|
1666 else: |
|
1667 return -1 |
|
1668 |
|
1669 def _round_half_down(self, prec): |
|
1670 """Round 5 down""" |
|
1671 if _exact_half(self._int, prec): |
|
1672 return -1 |
|
1673 else: |
|
1674 return self._round_half_up(prec) |
|
1675 |
|
1676 def _round_half_even(self, prec): |
|
1677 """Round 5 to even, rest to nearest.""" |
|
1678 if _exact_half(self._int, prec) and \ |
|
1679 (prec == 0 or self._int[prec-1] in '02468'): |
|
1680 return -1 |
|
1681 else: |
|
1682 return self._round_half_up(prec) |
|
1683 |
|
1684 def _round_ceiling(self, prec): |
|
1685 """Rounds up (not away from 0 if negative.)""" |
|
1686 if self._sign: |
|
1687 return self._round_down(prec) |
|
1688 else: |
|
1689 return -self._round_down(prec) |
|
1690 |
|
1691 def _round_floor(self, prec): |
|
1692 """Rounds down (not towards 0 if negative)""" |
|
1693 if not self._sign: |
|
1694 return self._round_down(prec) |
|
1695 else: |
|
1696 return -self._round_down(prec) |
|
1697 |
|
1698 def _round_05up(self, prec): |
|
1699 """Round down unless digit prec-1 is 0 or 5.""" |
|
1700 if prec and self._int[prec-1] not in '05': |
|
1701 return self._round_down(prec) |
|
1702 else: |
|
1703 return -self._round_down(prec) |
|
1704 |
|
1705 def fma(self, other, third, context=None): |
|
1706 """Fused multiply-add. |
|
1707 |
|
1708 Returns self*other+third with no rounding of the intermediate |
|
1709 product self*other. |
|
1710 |
|
1711 self and other are multiplied together, with no rounding of |
|
1712 the result. The third operand is then added to the result, |
|
1713 and a single final rounding is performed. |
|
1714 """ |
|
1715 |
|
1716 other = _convert_other(other, raiseit=True) |
|
1717 |
|
1718 # compute product; raise InvalidOperation if either operand is |
|
1719 # a signaling NaN or if the product is zero times infinity. |
|
1720 if self._is_special or other._is_special: |
|
1721 if context is None: |
|
1722 context = getcontext() |
|
1723 if self._exp == 'N': |
|
1724 return context._raise_error(InvalidOperation, 'sNaN', self) |
|
1725 if other._exp == 'N': |
|
1726 return context._raise_error(InvalidOperation, 'sNaN', other) |
|
1727 if self._exp == 'n': |
|
1728 product = self |
|
1729 elif other._exp == 'n': |
|
1730 product = other |
|
1731 elif self._exp == 'F': |
|
1732 if not other: |
|
1733 return context._raise_error(InvalidOperation, |
|
1734 'INF * 0 in fma') |
|
1735 product = Infsign[self._sign ^ other._sign] |
|
1736 elif other._exp == 'F': |
|
1737 if not self: |
|
1738 return context._raise_error(InvalidOperation, |
|
1739 '0 * INF in fma') |
|
1740 product = Infsign[self._sign ^ other._sign] |
|
1741 else: |
|
1742 product = _dec_from_triple(self._sign ^ other._sign, |
|
1743 str(int(self._int) * int(other._int)), |
|
1744 self._exp + other._exp) |
|
1745 |
|
1746 third = _convert_other(third, raiseit=True) |
|
1747 return product.__add__(third, context) |
|
1748 |
|
1749 def _power_modulo(self, other, modulo, context=None): |
|
1750 """Three argument version of __pow__""" |
|
1751 |
|
1752 # if can't convert other and modulo to Decimal, raise |
|
1753 # TypeError; there's no point returning NotImplemented (no |
|
1754 # equivalent of __rpow__ for three argument pow) |
|
1755 other = _convert_other(other, raiseit=True) |
|
1756 modulo = _convert_other(modulo, raiseit=True) |
|
1757 |
|
1758 if context is None: |
|
1759 context = getcontext() |
|
1760 |
|
1761 # deal with NaNs: if there are any sNaNs then first one wins, |
|
1762 # (i.e. behaviour for NaNs is identical to that of fma) |
|
1763 self_is_nan = self._isnan() |
|
1764 other_is_nan = other._isnan() |
|
1765 modulo_is_nan = modulo._isnan() |
|
1766 if self_is_nan or other_is_nan or modulo_is_nan: |
|
1767 if self_is_nan == 2: |
|
1768 return context._raise_error(InvalidOperation, 'sNaN', |
|
1769 self) |
|
1770 if other_is_nan == 2: |
|
1771 return context._raise_error(InvalidOperation, 'sNaN', |
|
1772 other) |
|
1773 if modulo_is_nan == 2: |
|
1774 return context._raise_error(InvalidOperation, 'sNaN', |
|
1775 modulo) |
|
1776 if self_is_nan: |
|
1777 return self._fix_nan(context) |
|
1778 if other_is_nan: |
|
1779 return other._fix_nan(context) |
|
1780 return modulo._fix_nan(context) |
|
1781 |
|
1782 # check inputs: we apply same restrictions as Python's pow() |
|
1783 if not (self._isinteger() and |
|
1784 other._isinteger() and |
|
1785 modulo._isinteger()): |
|
1786 return context._raise_error(InvalidOperation, |
|
1787 'pow() 3rd argument not allowed ' |
|
1788 'unless all arguments are integers') |
|
1789 if other < 0: |
|
1790 return context._raise_error(InvalidOperation, |
|
1791 'pow() 2nd argument cannot be ' |
|
1792 'negative when 3rd argument specified') |
|
1793 if not modulo: |
|
1794 return context._raise_error(InvalidOperation, |
|
1795 'pow() 3rd argument cannot be 0') |
|
1796 |
|
1797 # additional restriction for decimal: the modulus must be less |
|
1798 # than 10**prec in absolute value |
|
1799 if modulo.adjusted() >= context.prec: |
|
1800 return context._raise_error(InvalidOperation, |
|
1801 'insufficient precision: pow() 3rd ' |
|
1802 'argument must not have more than ' |
|
1803 'precision digits') |
|
1804 |
|
1805 # define 0**0 == NaN, for consistency with two-argument pow |
|
1806 # (even though it hurts!) |
|
1807 if not other and not self: |
|
1808 return context._raise_error(InvalidOperation, |
|
1809 'at least one of pow() 1st argument ' |
|
1810 'and 2nd argument must be nonzero ;' |
|
1811 '0**0 is not defined') |
|
1812 |
|
1813 # compute sign of result |
|
1814 if other._iseven(): |
|
1815 sign = 0 |
|
1816 else: |
|
1817 sign = self._sign |
|
1818 |
|
1819 # convert modulo to a Python integer, and self and other to |
|
1820 # Decimal integers (i.e. force their exponents to be >= 0) |
|
1821 modulo = abs(int(modulo)) |
|
1822 base = _WorkRep(self.to_integral_value()) |
|
1823 exponent = _WorkRep(other.to_integral_value()) |
|
1824 |
|
1825 # compute result using integer pow() |
|
1826 base = (base.int % modulo * pow(10, base.exp, modulo)) % modulo |
|
1827 for i in xrange(exponent.exp): |
|
1828 base = pow(base, 10, modulo) |
|
1829 base = pow(base, exponent.int, modulo) |
|
1830 |
|
1831 return _dec_from_triple(sign, str(base), 0) |
|
1832 |
|
1833 def _power_exact(self, other, p): |
|
1834 """Attempt to compute self**other exactly. |
|
1835 |
|
1836 Given Decimals self and other and an integer p, attempt to |
|
1837 compute an exact result for the power self**other, with p |
|
1838 digits of precision. Return None if self**other is not |
|
1839 exactly representable in p digits. |
|
1840 |
|
1841 Assumes that elimination of special cases has already been |
|
1842 performed: self and other must both be nonspecial; self must |
|
1843 be positive and not numerically equal to 1; other must be |
|
1844 nonzero. For efficiency, other._exp should not be too large, |
|
1845 so that 10**abs(other._exp) is a feasible calculation.""" |
|
1846 |
|
1847 # In the comments below, we write x for the value of self and |
|
1848 # y for the value of other. Write x = xc*10**xe and y = |
|
1849 # yc*10**ye. |
|
1850 |
|
1851 # The main purpose of this method is to identify the *failure* |
|
1852 # of x**y to be exactly representable with as little effort as |
|
1853 # possible. So we look for cheap and easy tests that |
|
1854 # eliminate the possibility of x**y being exact. Only if all |
|
1855 # these tests are passed do we go on to actually compute x**y. |
|
1856 |
|
1857 # Here's the main idea. First normalize both x and y. We |
|
1858 # express y as a rational m/n, with m and n relatively prime |
|
1859 # and n>0. Then for x**y to be exactly representable (at |
|
1860 # *any* precision), xc must be the nth power of a positive |
|
1861 # integer and xe must be divisible by n. If m is negative |
|
1862 # then additionally xc must be a power of either 2 or 5, hence |
|
1863 # a power of 2**n or 5**n. |
|
1864 # |
|
1865 # There's a limit to how small |y| can be: if y=m/n as above |
|
1866 # then: |
|
1867 # |
|
1868 # (1) if xc != 1 then for the result to be representable we |
|
1869 # need xc**(1/n) >= 2, and hence also xc**|y| >= 2. So |
|
1870 # if |y| <= 1/nbits(xc) then xc < 2**nbits(xc) <= |
|
1871 # 2**(1/|y|), hence xc**|y| < 2 and the result is not |
|
1872 # representable. |
|
1873 # |
|
1874 # (2) if xe != 0, |xe|*(1/n) >= 1, so |xe|*|y| >= 1. Hence if |
|
1875 # |y| < 1/|xe| then the result is not representable. |
|
1876 # |
|
1877 # Note that since x is not equal to 1, at least one of (1) and |
|
1878 # (2) must apply. Now |y| < 1/nbits(xc) iff |yc|*nbits(xc) < |
|
1879 # 10**-ye iff len(str(|yc|*nbits(xc)) <= -ye. |
|
1880 # |
|
1881 # There's also a limit to how large y can be, at least if it's |
|
1882 # positive: the normalized result will have coefficient xc**y, |
|
1883 # so if it's representable then xc**y < 10**p, and y < |
|
1884 # p/log10(xc). Hence if y*log10(xc) >= p then the result is |
|
1885 # not exactly representable. |
|
1886 |
|
1887 # if len(str(abs(yc*xe)) <= -ye then abs(yc*xe) < 10**-ye, |
|
1888 # so |y| < 1/xe and the result is not representable. |
|
1889 # Similarly, len(str(abs(yc)*xc_bits)) <= -ye implies |y| |
|
1890 # < 1/nbits(xc). |
|
1891 |
|
1892 x = _WorkRep(self) |
|
1893 xc, xe = x.int, x.exp |
|
1894 while xc % 10 == 0: |
|
1895 xc //= 10 |
|
1896 xe += 1 |
|
1897 |
|
1898 y = _WorkRep(other) |
|
1899 yc, ye = y.int, y.exp |
|
1900 while yc % 10 == 0: |
|
1901 yc //= 10 |
|
1902 ye += 1 |
|
1903 |
|
1904 # case where xc == 1: result is 10**(xe*y), with xe*y |
|
1905 # required to be an integer |
|
1906 if xc == 1: |
|
1907 if ye >= 0: |
|
1908 exponent = xe*yc*10**ye |
|
1909 else: |
|
1910 exponent, remainder = divmod(xe*yc, 10**-ye) |
|
1911 if remainder: |
|
1912 return None |
|
1913 if y.sign == 1: |
|
1914 exponent = -exponent |
|
1915 # if other is a nonnegative integer, use ideal exponent |
|
1916 if other._isinteger() and other._sign == 0: |
|
1917 ideal_exponent = self._exp*int(other) |
|
1918 zeros = min(exponent-ideal_exponent, p-1) |
|
1919 else: |
|
1920 zeros = 0 |
|
1921 return _dec_from_triple(0, '1' + '0'*zeros, exponent-zeros) |
|
1922 |
|
1923 # case where y is negative: xc must be either a power |
|
1924 # of 2 or a power of 5. |
|
1925 if y.sign == 1: |
|
1926 last_digit = xc % 10 |
|
1927 if last_digit in (2,4,6,8): |
|
1928 # quick test for power of 2 |
|
1929 if xc & -xc != xc: |
|
1930 return None |
|
1931 # now xc is a power of 2; e is its exponent |
|
1932 e = _nbits(xc)-1 |
|
1933 # find e*y and xe*y; both must be integers |
|
1934 if ye >= 0: |
|
1935 y_as_int = yc*10**ye |
|
1936 e = e*y_as_int |
|
1937 xe = xe*y_as_int |
|
1938 else: |
|
1939 ten_pow = 10**-ye |
|
1940 e, remainder = divmod(e*yc, ten_pow) |
|
1941 if remainder: |
|
1942 return None |
|
1943 xe, remainder = divmod(xe*yc, ten_pow) |
|
1944 if remainder: |
|
1945 return None |
|
1946 |
|
1947 if e*65 >= p*93: # 93/65 > log(10)/log(5) |
|
1948 return None |
|
1949 xc = 5**e |
|
1950 |
|
1951 elif last_digit == 5: |
|
1952 # e >= log_5(xc) if xc is a power of 5; we have |
|
1953 # equality all the way up to xc=5**2658 |
|
1954 e = _nbits(xc)*28//65 |
|
1955 xc, remainder = divmod(5**e, xc) |
|
1956 if remainder: |
|
1957 return None |
|
1958 while xc % 5 == 0: |
|
1959 xc //= 5 |
|
1960 e -= 1 |
|
1961 if ye >= 0: |
|
1962 y_as_integer = yc*10**ye |
|
1963 e = e*y_as_integer |
|
1964 xe = xe*y_as_integer |
|
1965 else: |
|
1966 ten_pow = 10**-ye |
|
1967 e, remainder = divmod(e*yc, ten_pow) |
|
1968 if remainder: |
|
1969 return None |
|
1970 xe, remainder = divmod(xe*yc, ten_pow) |
|
1971 if remainder: |
|
1972 return None |
|
1973 if e*3 >= p*10: # 10/3 > log(10)/log(2) |
|
1974 return None |
|
1975 xc = 2**e |
|
1976 else: |
|
1977 return None |
|
1978 |
|
1979 if xc >= 10**p: |
|
1980 return None |
|
1981 xe = -e-xe |
|
1982 return _dec_from_triple(0, str(xc), xe) |
|
1983 |
|
1984 # now y is positive; find m and n such that y = m/n |
|
1985 if ye >= 0: |
|
1986 m, n = yc*10**ye, 1 |
|
1987 else: |
|
1988 if xe != 0 and len(str(abs(yc*xe))) <= -ye: |
|
1989 return None |
|
1990 xc_bits = _nbits(xc) |
|
1991 if xc != 1 and len(str(abs(yc)*xc_bits)) <= -ye: |
|
1992 return None |
|
1993 m, n = yc, 10**(-ye) |
|
1994 while m % 2 == n % 2 == 0: |
|
1995 m //= 2 |
|
1996 n //= 2 |
|
1997 while m % 5 == n % 5 == 0: |
|
1998 m //= 5 |
|
1999 n //= 5 |
|
2000 |
|
2001 # compute nth root of xc*10**xe |
|
2002 if n > 1: |
|
2003 # if 1 < xc < 2**n then xc isn't an nth power |
|
2004 if xc != 1 and xc_bits <= n: |
|
2005 return None |
|
2006 |
|
2007 xe, rem = divmod(xe, n) |
|
2008 if rem != 0: |
|
2009 return None |
|
2010 |
|
2011 # compute nth root of xc using Newton's method |
|
2012 a = 1L << -(-_nbits(xc)//n) # initial estimate |
|
2013 while True: |
|
2014 q, r = divmod(xc, a**(n-1)) |
|
2015 if a <= q: |
|
2016 break |
|
2017 else: |
|
2018 a = (a*(n-1) + q)//n |
|
2019 if not (a == q and r == 0): |
|
2020 return None |
|
2021 xc = a |
|
2022 |
|
2023 # now xc*10**xe is the nth root of the original xc*10**xe |
|
2024 # compute mth power of xc*10**xe |
|
2025 |
|
2026 # if m > p*100//_log10_lb(xc) then m > p/log10(xc), hence xc**m > |
|
2027 # 10**p and the result is not representable. |
|
2028 if xc > 1 and m > p*100//_log10_lb(xc): |
|
2029 return None |
|
2030 xc = xc**m |
|
2031 xe *= m |
|
2032 if xc > 10**p: |
|
2033 return None |
|
2034 |
|
2035 # by this point the result *is* exactly representable |
|
2036 # adjust the exponent to get as close as possible to the ideal |
|
2037 # exponent, if necessary |
|
2038 str_xc = str(xc) |
|
2039 if other._isinteger() and other._sign == 0: |
|
2040 ideal_exponent = self._exp*int(other) |
|
2041 zeros = min(xe-ideal_exponent, p-len(str_xc)) |
|
2042 else: |
|
2043 zeros = 0 |
|
2044 return _dec_from_triple(0, str_xc+'0'*zeros, xe-zeros) |
|
2045 |
|
2046 def __pow__(self, other, modulo=None, context=None): |
|
2047 """Return self ** other [ % modulo]. |
|
2048 |
|
2049 With two arguments, compute self**other. |
|
2050 |
|
2051 With three arguments, compute (self**other) % modulo. For the |
|
2052 three argument form, the following restrictions on the |
|
2053 arguments hold: |
|
2054 |
|
2055 - all three arguments must be integral |
|
2056 - other must be nonnegative |
|
2057 - either self or other (or both) must be nonzero |
|
2058 - modulo must be nonzero and must have at most p digits, |
|
2059 where p is the context precision. |
|
2060 |
|
2061 If any of these restrictions is violated the InvalidOperation |
|
2062 flag is raised. |
|
2063 |
|
2064 The result of pow(self, other, modulo) is identical to the |
|
2065 result that would be obtained by computing (self**other) % |
|
2066 modulo with unbounded precision, but is computed more |
|
2067 efficiently. It is always exact. |
|
2068 """ |
|
2069 |
|
2070 if modulo is not None: |
|
2071 return self._power_modulo(other, modulo, context) |
|
2072 |
|
2073 other = _convert_other(other) |
|
2074 if other is NotImplemented: |
|
2075 return other |
|
2076 |
|
2077 if context is None: |
|
2078 context = getcontext() |
|
2079 |
|
2080 # either argument is a NaN => result is NaN |
|
2081 ans = self._check_nans(other, context) |
|
2082 if ans: |
|
2083 return ans |
|
2084 |
|
2085 # 0**0 = NaN (!), x**0 = 1 for nonzero x (including +/-Infinity) |
|
2086 if not other: |
|
2087 if not self: |
|
2088 return context._raise_error(InvalidOperation, '0 ** 0') |
|
2089 else: |
|
2090 return Dec_p1 |
|
2091 |
|
2092 # result has sign 1 iff self._sign is 1 and other is an odd integer |
|
2093 result_sign = 0 |
|
2094 if self._sign == 1: |
|
2095 if other._isinteger(): |
|
2096 if not other._iseven(): |
|
2097 result_sign = 1 |
|
2098 else: |
|
2099 # -ve**noninteger = NaN |
|
2100 # (-0)**noninteger = 0**noninteger |
|
2101 if self: |
|
2102 return context._raise_error(InvalidOperation, |
|
2103 'x ** y with x negative and y not an integer') |
|
2104 # negate self, without doing any unwanted rounding |
|
2105 self = self.copy_negate() |
|
2106 |
|
2107 # 0**(+ve or Inf)= 0; 0**(-ve or -Inf) = Infinity |
|
2108 if not self: |
|
2109 if other._sign == 0: |
|
2110 return _dec_from_triple(result_sign, '0', 0) |
|
2111 else: |
|
2112 return Infsign[result_sign] |
|
2113 |
|
2114 # Inf**(+ve or Inf) = Inf; Inf**(-ve or -Inf) = 0 |
|
2115 if self._isinfinity(): |
|
2116 if other._sign == 0: |
|
2117 return Infsign[result_sign] |
|
2118 else: |
|
2119 return _dec_from_triple(result_sign, '0', 0) |
|
2120 |
|
2121 # 1**other = 1, but the choice of exponent and the flags |
|
2122 # depend on the exponent of self, and on whether other is a |
|
2123 # positive integer, a negative integer, or neither |
|
2124 if self == Dec_p1: |
|
2125 if other._isinteger(): |
|
2126 # exp = max(self._exp*max(int(other), 0), |
|
2127 # 1-context.prec) but evaluating int(other) directly |
|
2128 # is dangerous until we know other is small (other |
|
2129 # could be 1e999999999) |
|
2130 if other._sign == 1: |
|
2131 multiplier = 0 |
|
2132 elif other > context.prec: |
|
2133 multiplier = context.prec |
|
2134 else: |
|
2135 multiplier = int(other) |
|
2136 |
|
2137 exp = self._exp * multiplier |
|
2138 if exp < 1-context.prec: |
|
2139 exp = 1-context.prec |
|
2140 context._raise_error(Rounded) |
|
2141 else: |
|
2142 context._raise_error(Inexact) |
|
2143 context._raise_error(Rounded) |
|
2144 exp = 1-context.prec |
|
2145 |
|
2146 return _dec_from_triple(result_sign, '1'+'0'*-exp, exp) |
|
2147 |
|
2148 # compute adjusted exponent of self |
|
2149 self_adj = self.adjusted() |
|
2150 |
|
2151 # self ** infinity is infinity if self > 1, 0 if self < 1 |
|
2152 # self ** -infinity is infinity if self < 1, 0 if self > 1 |
|
2153 if other._isinfinity(): |
|
2154 if (other._sign == 0) == (self_adj < 0): |
|
2155 return _dec_from_triple(result_sign, '0', 0) |
|
2156 else: |
|
2157 return Infsign[result_sign] |
|
2158 |
|
2159 # from here on, the result always goes through the call |
|
2160 # to _fix at the end of this function. |
|
2161 ans = None |
|
2162 |
|
2163 # crude test to catch cases of extreme overflow/underflow. If |
|
2164 # log10(self)*other >= 10**bound and bound >= len(str(Emax)) |
|
2165 # then 10**bound >= 10**len(str(Emax)) >= Emax+1 and hence |
|
2166 # self**other >= 10**(Emax+1), so overflow occurs. The test |
|
2167 # for underflow is similar. |
|
2168 bound = self._log10_exp_bound() + other.adjusted() |
|
2169 if (self_adj >= 0) == (other._sign == 0): |
|
2170 # self > 1 and other +ve, or self < 1 and other -ve |
|
2171 # possibility of overflow |
|
2172 if bound >= len(str(context.Emax)): |
|
2173 ans = _dec_from_triple(result_sign, '1', context.Emax+1) |
|
2174 else: |
|
2175 # self > 1 and other -ve, or self < 1 and other +ve |
|
2176 # possibility of underflow to 0 |
|
2177 Etiny = context.Etiny() |
|
2178 if bound >= len(str(-Etiny)): |
|
2179 ans = _dec_from_triple(result_sign, '1', Etiny-1) |
|
2180 |
|
2181 # try for an exact result with precision +1 |
|
2182 if ans is None: |
|
2183 ans = self._power_exact(other, context.prec + 1) |
|
2184 if ans is not None and result_sign == 1: |
|
2185 ans = _dec_from_triple(1, ans._int, ans._exp) |
|
2186 |
|
2187 # usual case: inexact result, x**y computed directly as exp(y*log(x)) |
|
2188 if ans is None: |
|
2189 p = context.prec |
|
2190 x = _WorkRep(self) |
|
2191 xc, xe = x.int, x.exp |
|
2192 y = _WorkRep(other) |
|
2193 yc, ye = y.int, y.exp |
|
2194 if y.sign == 1: |
|
2195 yc = -yc |
|
2196 |
|
2197 # compute correctly rounded result: start with precision +3, |
|
2198 # then increase precision until result is unambiguously roundable |
|
2199 extra = 3 |
|
2200 while True: |
|
2201 coeff, exp = _dpower(xc, xe, yc, ye, p+extra) |
|
2202 if coeff % (5*10**(len(str(coeff))-p-1)): |
|
2203 break |
|
2204 extra += 3 |
|
2205 |
|
2206 ans = _dec_from_triple(result_sign, str(coeff), exp) |
|
2207 |
|
2208 # the specification says that for non-integer other we need to |
|
2209 # raise Inexact, even when the result is actually exact. In |
|
2210 # the same way, we need to raise Underflow here if the result |
|
2211 # is subnormal. (The call to _fix will take care of raising |
|
2212 # Rounded and Subnormal, as usual.) |
|
2213 if not other._isinteger(): |
|
2214 context._raise_error(Inexact) |
|
2215 # pad with zeros up to length context.prec+1 if necessary |
|
2216 if len(ans._int) <= context.prec: |
|
2217 expdiff = context.prec+1 - len(ans._int) |
|
2218 ans = _dec_from_triple(ans._sign, ans._int+'0'*expdiff, |
|
2219 ans._exp-expdiff) |
|
2220 if ans.adjusted() < context.Emin: |
|
2221 context._raise_error(Underflow) |
|
2222 |
|
2223 # unlike exp, ln and log10, the power function respects the |
|
2224 # rounding mode; no need to use ROUND_HALF_EVEN here |
|
2225 ans = ans._fix(context) |
|
2226 return ans |
|
2227 |
|
2228 def __rpow__(self, other, context=None): |
|
2229 """Swaps self/other and returns __pow__.""" |
|
2230 other = _convert_other(other) |
|
2231 if other is NotImplemented: |
|
2232 return other |
|
2233 return other.__pow__(self, context=context) |
|
2234 |
|
2235 def normalize(self, context=None): |
|
2236 """Normalize- strip trailing 0s, change anything equal to 0 to 0e0""" |
|
2237 |
|
2238 if context is None: |
|
2239 context = getcontext() |
|
2240 |
|
2241 if self._is_special: |
|
2242 ans = self._check_nans(context=context) |
|
2243 if ans: |
|
2244 return ans |
|
2245 |
|
2246 dup = self._fix(context) |
|
2247 if dup._isinfinity(): |
|
2248 return dup |
|
2249 |
|
2250 if not dup: |
|
2251 return _dec_from_triple(dup._sign, '0', 0) |
|
2252 exp_max = [context.Emax, context.Etop()][context._clamp] |
|
2253 end = len(dup._int) |
|
2254 exp = dup._exp |
|
2255 while dup._int[end-1] == '0' and exp < exp_max: |
|
2256 exp += 1 |
|
2257 end -= 1 |
|
2258 return _dec_from_triple(dup._sign, dup._int[:end], exp) |
|
2259 |
|
2260 def quantize(self, exp, rounding=None, context=None, watchexp=True): |
|
2261 """Quantize self so its exponent is the same as that of exp. |
|
2262 |
|
2263 Similar to self._rescale(exp._exp) but with error checking. |
|
2264 """ |
|
2265 exp = _convert_other(exp, raiseit=True) |
|
2266 |
|
2267 if context is None: |
|
2268 context = getcontext() |
|
2269 if rounding is None: |
|
2270 rounding = context.rounding |
|
2271 |
|
2272 if self._is_special or exp._is_special: |
|
2273 ans = self._check_nans(exp, context) |
|
2274 if ans: |
|
2275 return ans |
|
2276 |
|
2277 if exp._isinfinity() or self._isinfinity(): |
|
2278 if exp._isinfinity() and self._isinfinity(): |
|
2279 return Decimal(self) # if both are inf, it is OK |
|
2280 return context._raise_error(InvalidOperation, |
|
2281 'quantize with one INF') |
|
2282 |
|
2283 # if we're not watching exponents, do a simple rescale |
|
2284 if not watchexp: |
|
2285 ans = self._rescale(exp._exp, rounding) |
|
2286 # raise Inexact and Rounded where appropriate |
|
2287 if ans._exp > self._exp: |
|
2288 context._raise_error(Rounded) |
|
2289 if ans != self: |
|
2290 context._raise_error(Inexact) |
|
2291 return ans |
|
2292 |
|
2293 # exp._exp should be between Etiny and Emax |
|
2294 if not (context.Etiny() <= exp._exp <= context.Emax): |
|
2295 return context._raise_error(InvalidOperation, |
|
2296 'target exponent out of bounds in quantize') |
|
2297 |
|
2298 if not self: |
|
2299 ans = _dec_from_triple(self._sign, '0', exp._exp) |
|
2300 return ans._fix(context) |
|
2301 |
|
2302 self_adjusted = self.adjusted() |
|
2303 if self_adjusted > context.Emax: |
|
2304 return context._raise_error(InvalidOperation, |
|
2305 'exponent of quantize result too large for current context') |
|
2306 if self_adjusted - exp._exp + 1 > context.prec: |
|
2307 return context._raise_error(InvalidOperation, |
|
2308 'quantize result has too many digits for current context') |
|
2309 |
|
2310 ans = self._rescale(exp._exp, rounding) |
|
2311 if ans.adjusted() > context.Emax: |
|
2312 return context._raise_error(InvalidOperation, |
|
2313 'exponent of quantize result too large for current context') |
|
2314 if len(ans._int) > context.prec: |
|
2315 return context._raise_error(InvalidOperation, |
|
2316 'quantize result has too many digits for current context') |
|
2317 |
|
2318 # raise appropriate flags |
|
2319 if ans._exp > self._exp: |
|
2320 context._raise_error(Rounded) |
|
2321 if ans != self: |
|
2322 context._raise_error(Inexact) |
|
2323 if ans and ans.adjusted() < context.Emin: |
|
2324 context._raise_error(Subnormal) |
|
2325 |
|
2326 # call to fix takes care of any necessary folddown |
|
2327 ans = ans._fix(context) |
|
2328 return ans |
|
2329 |
|
2330 def same_quantum(self, other): |
|
2331 """Return True if self and other have the same exponent; otherwise |
|
2332 return False. |
|
2333 |
|
2334 If either operand is a special value, the following rules are used: |
|
2335 * return True if both operands are infinities |
|
2336 * return True if both operands are NaNs |
|
2337 * otherwise, return False. |
|
2338 """ |
|
2339 other = _convert_other(other, raiseit=True) |
|
2340 if self._is_special or other._is_special: |
|
2341 return (self.is_nan() and other.is_nan() or |
|
2342 self.is_infinite() and other.is_infinite()) |
|
2343 return self._exp == other._exp |
|
2344 |
|
2345 def _rescale(self, exp, rounding): |
|
2346 """Rescale self so that the exponent is exp, either by padding with zeros |
|
2347 or by truncating digits, using the given rounding mode. |
|
2348 |
|
2349 Specials are returned without change. This operation is |
|
2350 quiet: it raises no flags, and uses no information from the |
|
2351 context. |
|
2352 |
|
2353 exp = exp to scale to (an integer) |
|
2354 rounding = rounding mode |
|
2355 """ |
|
2356 if self._is_special: |
|
2357 return Decimal(self) |
|
2358 if not self: |
|
2359 return _dec_from_triple(self._sign, '0', exp) |
|
2360 |
|
2361 if self._exp >= exp: |
|
2362 # pad answer with zeros if necessary |
|
2363 return _dec_from_triple(self._sign, |
|
2364 self._int + '0'*(self._exp - exp), exp) |
|
2365 |
|
2366 # too many digits; round and lose data. If self.adjusted() < |
|
2367 # exp-1, replace self by 10**(exp-1) before rounding |
|
2368 digits = len(self._int) + self._exp - exp |
|
2369 if digits < 0: |
|
2370 self = _dec_from_triple(self._sign, '1', exp-1) |
|
2371 digits = 0 |
|
2372 this_function = getattr(self, self._pick_rounding_function[rounding]) |
|
2373 changed = this_function(digits) |
|
2374 coeff = self._int[:digits] or '0' |
|
2375 if changed == 1: |
|
2376 coeff = str(int(coeff)+1) |
|
2377 return _dec_from_triple(self._sign, coeff, exp) |
|
2378 |
|
2379 def _round(self, places, rounding): |
|
2380 """Round a nonzero, nonspecial Decimal to a fixed number of |
|
2381 significant figures, using the given rounding mode. |
|
2382 |
|
2383 Infinities, NaNs and zeros are returned unaltered. |
|
2384 |
|
2385 This operation is quiet: it raises no flags, and uses no |
|
2386 information from the context. |
|
2387 |
|
2388 """ |
|
2389 if places <= 0: |
|
2390 raise ValueError("argument should be at least 1 in _round") |
|
2391 if self._is_special or not self: |
|
2392 return Decimal(self) |
|
2393 ans = self._rescale(self.adjusted()+1-places, rounding) |
|
2394 # it can happen that the rescale alters the adjusted exponent; |
|
2395 # for example when rounding 99.97 to 3 significant figures. |
|
2396 # When this happens we end up with an extra 0 at the end of |
|
2397 # the number; a second rescale fixes this. |
|
2398 if ans.adjusted() != self.adjusted(): |
|
2399 ans = ans._rescale(ans.adjusted()+1-places, rounding) |
|
2400 return ans |
|
2401 |
|
2402 def to_integral_exact(self, rounding=None, context=None): |
|
2403 """Rounds to a nearby integer. |
|
2404 |
|
2405 If no rounding mode is specified, take the rounding mode from |
|
2406 the context. This method raises the Rounded and Inexact flags |
|
2407 when appropriate. |
|
2408 |
|
2409 See also: to_integral_value, which does exactly the same as |
|
2410 this method except that it doesn't raise Inexact or Rounded. |
|
2411 """ |
|
2412 if self._is_special: |
|
2413 ans = self._check_nans(context=context) |
|
2414 if ans: |
|
2415 return ans |
|
2416 return Decimal(self) |
|
2417 if self._exp >= 0: |
|
2418 return Decimal(self) |
|
2419 if not self: |
|
2420 return _dec_from_triple(self._sign, '0', 0) |
|
2421 if context is None: |
|
2422 context = getcontext() |
|
2423 if rounding is None: |
|
2424 rounding = context.rounding |
|
2425 context._raise_error(Rounded) |
|
2426 ans = self._rescale(0, rounding) |
|
2427 if ans != self: |
|
2428 context._raise_error(Inexact) |
|
2429 return ans |
|
2430 |
|
2431 def to_integral_value(self, rounding=None, context=None): |
|
2432 """Rounds to the nearest integer, without raising inexact, rounded.""" |
|
2433 if context is None: |
|
2434 context = getcontext() |
|
2435 if rounding is None: |
|
2436 rounding = context.rounding |
|
2437 if self._is_special: |
|
2438 ans = self._check_nans(context=context) |
|
2439 if ans: |
|
2440 return ans |
|
2441 return Decimal(self) |
|
2442 if self._exp >= 0: |
|
2443 return Decimal(self) |
|
2444 else: |
|
2445 return self._rescale(0, rounding) |
|
2446 |
|
2447 # the method name changed, but we provide also the old one, for compatibility |
|
2448 to_integral = to_integral_value |
|
2449 |
|
2450 def sqrt(self, context=None): |
|
2451 """Return the square root of self.""" |
|
2452 if context is None: |
|
2453 context = getcontext() |
|
2454 |
|
2455 if self._is_special: |
|
2456 ans = self._check_nans(context=context) |
|
2457 if ans: |
|
2458 return ans |
|
2459 |
|
2460 if self._isinfinity() and self._sign == 0: |
|
2461 return Decimal(self) |
|
2462 |
|
2463 if not self: |
|
2464 # exponent = self._exp // 2. sqrt(-0) = -0 |
|
2465 ans = _dec_from_triple(self._sign, '0', self._exp // 2) |
|
2466 return ans._fix(context) |
|
2467 |
|
2468 if self._sign == 1: |
|
2469 return context._raise_error(InvalidOperation, 'sqrt(-x), x > 0') |
|
2470 |
|
2471 # At this point self represents a positive number. Let p be |
|
2472 # the desired precision and express self in the form c*100**e |
|
2473 # with c a positive real number and e an integer, c and e |
|
2474 # being chosen so that 100**(p-1) <= c < 100**p. Then the |
|
2475 # (exact) square root of self is sqrt(c)*10**e, and 10**(p-1) |
|
2476 # <= sqrt(c) < 10**p, so the closest representable Decimal at |
|
2477 # precision p is n*10**e where n = round_half_even(sqrt(c)), |
|
2478 # the closest integer to sqrt(c) with the even integer chosen |
|
2479 # in the case of a tie. |
|
2480 # |
|
2481 # To ensure correct rounding in all cases, we use the |
|
2482 # following trick: we compute the square root to an extra |
|
2483 # place (precision p+1 instead of precision p), rounding down. |
|
2484 # Then, if the result is inexact and its last digit is 0 or 5, |
|
2485 # we increase the last digit to 1 or 6 respectively; if it's |
|
2486 # exact we leave the last digit alone. Now the final round to |
|
2487 # p places (or fewer in the case of underflow) will round |
|
2488 # correctly and raise the appropriate flags. |
|
2489 |
|
2490 # use an extra digit of precision |
|
2491 prec = context.prec+1 |
|
2492 |
|
2493 # write argument in the form c*100**e where e = self._exp//2 |
|
2494 # is the 'ideal' exponent, to be used if the square root is |
|
2495 # exactly representable. l is the number of 'digits' of c in |
|
2496 # base 100, so that 100**(l-1) <= c < 100**l. |
|
2497 op = _WorkRep(self) |
|
2498 e = op.exp >> 1 |
|
2499 if op.exp & 1: |
|
2500 c = op.int * 10 |
|
2501 l = (len(self._int) >> 1) + 1 |
|
2502 else: |
|
2503 c = op.int |
|
2504 l = len(self._int)+1 >> 1 |
|
2505 |
|
2506 # rescale so that c has exactly prec base 100 'digits' |
|
2507 shift = prec-l |
|
2508 if shift >= 0: |
|
2509 c *= 100**shift |
|
2510 exact = True |
|
2511 else: |
|
2512 c, remainder = divmod(c, 100**-shift) |
|
2513 exact = not remainder |
|
2514 e -= shift |
|
2515 |
|
2516 # find n = floor(sqrt(c)) using Newton's method |
|
2517 n = 10**prec |
|
2518 while True: |
|
2519 q = c//n |
|
2520 if n <= q: |
|
2521 break |
|
2522 else: |
|
2523 n = n + q >> 1 |
|
2524 exact = exact and n*n == c |
|
2525 |
|
2526 if exact: |
|
2527 # result is exact; rescale to use ideal exponent e |
|
2528 if shift >= 0: |
|
2529 # assert n % 10**shift == 0 |
|
2530 n //= 10**shift |
|
2531 else: |
|
2532 n *= 10**-shift |
|
2533 e += shift |
|
2534 else: |
|
2535 # result is not exact; fix last digit as described above |
|
2536 if n % 5 == 0: |
|
2537 n += 1 |
|
2538 |
|
2539 ans = _dec_from_triple(0, str(n), e) |
|
2540 |
|
2541 # round, and fit to current context |
|
2542 context = context._shallow_copy() |
|
2543 rounding = context._set_rounding(ROUND_HALF_EVEN) |
|
2544 ans = ans._fix(context) |
|
2545 context.rounding = rounding |
|
2546 |
|
2547 return ans |
|
2548 |
|
2549 def max(self, other, context=None): |
|
2550 """Returns the larger value. |
|
2551 |
|
2552 Like max(self, other) except if one is not a number, returns |
|
2553 NaN (and signals if one is sNaN). Also rounds. |
|
2554 """ |
|
2555 other = _convert_other(other, raiseit=True) |
|
2556 |
|
2557 if context is None: |
|
2558 context = getcontext() |
|
2559 |
|
2560 if self._is_special or other._is_special: |
|
2561 # If one operand is a quiet NaN and the other is number, then the |
|
2562 # number is always returned |
|
2563 sn = self._isnan() |
|
2564 on = other._isnan() |
|
2565 if sn or on: |
|
2566 if on == 1 and sn != 2: |
|
2567 return self._fix_nan(context) |
|
2568 if sn == 1 and on != 2: |
|
2569 return other._fix_nan(context) |
|
2570 return self._check_nans(other, context) |
|
2571 |
|
2572 c = self._cmp(other) |
|
2573 if c == 0: |
|
2574 # If both operands are finite and equal in numerical value |
|
2575 # then an ordering is applied: |
|
2576 # |
|
2577 # If the signs differ then max returns the operand with the |
|
2578 # positive sign and min returns the operand with the negative sign |
|
2579 # |
|
2580 # If the signs are the same then the exponent is used to select |
|
2581 # the result. This is exactly the ordering used in compare_total. |
|
2582 c = self.compare_total(other) |
|
2583 |
|
2584 if c == -1: |
|
2585 ans = other |
|
2586 else: |
|
2587 ans = self |
|
2588 |
|
2589 return ans._fix(context) |
|
2590 |
|
2591 def min(self, other, context=None): |
|
2592 """Returns the smaller value. |
|
2593 |
|
2594 Like min(self, other) except if one is not a number, returns |
|
2595 NaN (and signals if one is sNaN). Also rounds. |
|
2596 """ |
|
2597 other = _convert_other(other, raiseit=True) |
|
2598 |
|
2599 if context is None: |
|
2600 context = getcontext() |
|
2601 |
|
2602 if self._is_special or other._is_special: |
|
2603 # If one operand is a quiet NaN and the other is number, then the |
|
2604 # number is always returned |
|
2605 sn = self._isnan() |
|
2606 on = other._isnan() |
|
2607 if sn or on: |
|
2608 if on == 1 and sn != 2: |
|
2609 return self._fix_nan(context) |
|
2610 if sn == 1 and on != 2: |
|
2611 return other._fix_nan(context) |
|
2612 return self._check_nans(other, context) |
|
2613 |
|
2614 c = self._cmp(other) |
|
2615 if c == 0: |
|
2616 c = self.compare_total(other) |
|
2617 |
|
2618 if c == -1: |
|
2619 ans = self |
|
2620 else: |
|
2621 ans = other |
|
2622 |
|
2623 return ans._fix(context) |
|
2624 |
|
2625 def _isinteger(self): |
|
2626 """Returns whether self is an integer""" |
|
2627 if self._is_special: |
|
2628 return False |
|
2629 if self._exp >= 0: |
|
2630 return True |
|
2631 rest = self._int[self._exp:] |
|
2632 return rest == '0'*len(rest) |
|
2633 |
|
2634 def _iseven(self): |
|
2635 """Returns True if self is even. Assumes self is an integer.""" |
|
2636 if not self or self._exp > 0: |
|
2637 return True |
|
2638 return self._int[-1+self._exp] in '02468' |
|
2639 |
|
2640 def adjusted(self): |
|
2641 """Return the adjusted exponent of self""" |
|
2642 try: |
|
2643 return self._exp + len(self._int) - 1 |
|
2644 # If NaN or Infinity, self._exp is string |
|
2645 except TypeError: |
|
2646 return 0 |
|
2647 |
|
2648 def canonical(self, context=None): |
|
2649 """Returns the same Decimal object. |
|
2650 |
|
2651 As we do not have different encodings for the same number, the |
|
2652 received object already is in its canonical form. |
|
2653 """ |
|
2654 return self |
|
2655 |
|
2656 def compare_signal(self, other, context=None): |
|
2657 """Compares self to the other operand numerically. |
|
2658 |
|
2659 It's pretty much like compare(), but all NaNs signal, with signaling |
|
2660 NaNs taking precedence over quiet NaNs. |
|
2661 """ |
|
2662 other = _convert_other(other, raiseit = True) |
|
2663 ans = self._compare_check_nans(other, context) |
|
2664 if ans: |
|
2665 return ans |
|
2666 return self.compare(other, context=context) |
|
2667 |
|
2668 def compare_total(self, other): |
|
2669 """Compares self to other using the abstract representations. |
|
2670 |
|
2671 This is not like the standard compare, which use their numerical |
|
2672 value. Note that a total ordering is defined for all possible abstract |
|
2673 representations. |
|
2674 """ |
|
2675 # if one is negative and the other is positive, it's easy |
|
2676 if self._sign and not other._sign: |
|
2677 return Dec_n1 |
|
2678 if not self._sign and other._sign: |
|
2679 return Dec_p1 |
|
2680 sign = self._sign |
|
2681 |
|
2682 # let's handle both NaN types |
|
2683 self_nan = self._isnan() |
|
2684 other_nan = other._isnan() |
|
2685 if self_nan or other_nan: |
|
2686 if self_nan == other_nan: |
|
2687 if self._int < other._int: |
|
2688 if sign: |
|
2689 return Dec_p1 |
|
2690 else: |
|
2691 return Dec_n1 |
|
2692 if self._int > other._int: |
|
2693 if sign: |
|
2694 return Dec_n1 |
|
2695 else: |
|
2696 return Dec_p1 |
|
2697 return Dec_0 |
|
2698 |
|
2699 if sign: |
|
2700 if self_nan == 1: |
|
2701 return Dec_n1 |
|
2702 if other_nan == 1: |
|
2703 return Dec_p1 |
|
2704 if self_nan == 2: |
|
2705 return Dec_n1 |
|
2706 if other_nan == 2: |
|
2707 return Dec_p1 |
|
2708 else: |
|
2709 if self_nan == 1: |
|
2710 return Dec_p1 |
|
2711 if other_nan == 1: |
|
2712 return Dec_n1 |
|
2713 if self_nan == 2: |
|
2714 return Dec_p1 |
|
2715 if other_nan == 2: |
|
2716 return Dec_n1 |
|
2717 |
|
2718 if self < other: |
|
2719 return Dec_n1 |
|
2720 if self > other: |
|
2721 return Dec_p1 |
|
2722 |
|
2723 if self._exp < other._exp: |
|
2724 if sign: |
|
2725 return Dec_p1 |
|
2726 else: |
|
2727 return Dec_n1 |
|
2728 if self._exp > other._exp: |
|
2729 if sign: |
|
2730 return Dec_n1 |
|
2731 else: |
|
2732 return Dec_p1 |
|
2733 return Dec_0 |
|
2734 |
|
2735 |
|
2736 def compare_total_mag(self, other): |
|
2737 """Compares self to other using abstract repr., ignoring sign. |
|
2738 |
|
2739 Like compare_total, but with operand's sign ignored and assumed to be 0. |
|
2740 """ |
|
2741 s = self.copy_abs() |
|
2742 o = other.copy_abs() |
|
2743 return s.compare_total(o) |
|
2744 |
|
2745 def copy_abs(self): |
|
2746 """Returns a copy with the sign set to 0. """ |
|
2747 return _dec_from_triple(0, self._int, self._exp, self._is_special) |
|
2748 |
|
2749 def copy_negate(self): |
|
2750 """Returns a copy with the sign inverted.""" |
|
2751 if self._sign: |
|
2752 return _dec_from_triple(0, self._int, self._exp, self._is_special) |
|
2753 else: |
|
2754 return _dec_from_triple(1, self._int, self._exp, self._is_special) |
|
2755 |
|
2756 def copy_sign(self, other): |
|
2757 """Returns self with the sign of other.""" |
|
2758 return _dec_from_triple(other._sign, self._int, |
|
2759 self._exp, self._is_special) |
|
2760 |
|
2761 def exp(self, context=None): |
|
2762 """Returns e ** self.""" |
|
2763 |
|
2764 if context is None: |
|
2765 context = getcontext() |
|
2766 |
|
2767 # exp(NaN) = NaN |
|
2768 ans = self._check_nans(context=context) |
|
2769 if ans: |
|
2770 return ans |
|
2771 |
|
2772 # exp(-Infinity) = 0 |
|
2773 if self._isinfinity() == -1: |
|
2774 return Dec_0 |
|
2775 |
|
2776 # exp(0) = 1 |
|
2777 if not self: |
|
2778 return Dec_p1 |
|
2779 |
|
2780 # exp(Infinity) = Infinity |
|
2781 if self._isinfinity() == 1: |
|
2782 return Decimal(self) |
|
2783 |
|
2784 # the result is now guaranteed to be inexact (the true |
|
2785 # mathematical result is transcendental). There's no need to |
|
2786 # raise Rounded and Inexact here---they'll always be raised as |
|
2787 # a result of the call to _fix. |
|
2788 p = context.prec |
|
2789 adj = self.adjusted() |
|
2790 |
|
2791 # we only need to do any computation for quite a small range |
|
2792 # of adjusted exponents---for example, -29 <= adj <= 10 for |
|
2793 # the default context. For smaller exponent the result is |
|
2794 # indistinguishable from 1 at the given precision, while for |
|
2795 # larger exponent the result either overflows or underflows. |
|
2796 if self._sign == 0 and adj > len(str((context.Emax+1)*3)): |
|
2797 # overflow |
|
2798 ans = _dec_from_triple(0, '1', context.Emax+1) |
|
2799 elif self._sign == 1 and adj > len(str((-context.Etiny()+1)*3)): |
|
2800 # underflow to 0 |
|
2801 ans = _dec_from_triple(0, '1', context.Etiny()-1) |
|
2802 elif self._sign == 0 and adj < -p: |
|
2803 # p+1 digits; final round will raise correct flags |
|
2804 ans = _dec_from_triple(0, '1' + '0'*(p-1) + '1', -p) |
|
2805 elif self._sign == 1 and adj < -p-1: |
|
2806 # p+1 digits; final round will raise correct flags |
|
2807 ans = _dec_from_triple(0, '9'*(p+1), -p-1) |
|
2808 # general case |
|
2809 else: |
|
2810 op = _WorkRep(self) |
|
2811 c, e = op.int, op.exp |
|
2812 if op.sign == 1: |
|
2813 c = -c |
|
2814 |
|
2815 # compute correctly rounded result: increase precision by |
|
2816 # 3 digits at a time until we get an unambiguously |
|
2817 # roundable result |
|
2818 extra = 3 |
|
2819 while True: |
|
2820 coeff, exp = _dexp(c, e, p+extra) |
|
2821 if coeff % (5*10**(len(str(coeff))-p-1)): |
|
2822 break |
|
2823 extra += 3 |
|
2824 |
|
2825 ans = _dec_from_triple(0, str(coeff), exp) |
|
2826 |
|
2827 # at this stage, ans should round correctly with *any* |
|
2828 # rounding mode, not just with ROUND_HALF_EVEN |
|
2829 context = context._shallow_copy() |
|
2830 rounding = context._set_rounding(ROUND_HALF_EVEN) |
|
2831 ans = ans._fix(context) |
|
2832 context.rounding = rounding |
|
2833 |
|
2834 return ans |
|
2835 |
|
2836 def is_canonical(self): |
|
2837 """Return True if self is canonical; otherwise return False. |
|
2838 |
|
2839 Currently, the encoding of a Decimal instance is always |
|
2840 canonical, so this method returns True for any Decimal. |
|
2841 """ |
|
2842 return True |
|
2843 |
|
2844 def is_finite(self): |
|
2845 """Return True if self is finite; otherwise return False. |
|
2846 |
|
2847 A Decimal instance is considered finite if it is neither |
|
2848 infinite nor a NaN. |
|
2849 """ |
|
2850 return not self._is_special |
|
2851 |
|
2852 def is_infinite(self): |
|
2853 """Return True if self is infinite; otherwise return False.""" |
|
2854 return self._exp == 'F' |
|
2855 |
|
2856 def is_nan(self): |
|
2857 """Return True if self is a qNaN or sNaN; otherwise return False.""" |
|
2858 return self._exp in ('n', 'N') |
|
2859 |
|
2860 def is_normal(self, context=None): |
|
2861 """Return True if self is a normal number; otherwise return False.""" |
|
2862 if self._is_special or not self: |
|
2863 return False |
|
2864 if context is None: |
|
2865 context = getcontext() |
|
2866 return context.Emin <= self.adjusted() <= context.Emax |
|
2867 |
|
2868 def is_qnan(self): |
|
2869 """Return True if self is a quiet NaN; otherwise return False.""" |
|
2870 return self._exp == 'n' |
|
2871 |
|
2872 def is_signed(self): |
|
2873 """Return True if self is negative; otherwise return False.""" |
|
2874 return self._sign == 1 |
|
2875 |
|
2876 def is_snan(self): |
|
2877 """Return True if self is a signaling NaN; otherwise return False.""" |
|
2878 return self._exp == 'N' |
|
2879 |
|
2880 def is_subnormal(self, context=None): |
|
2881 """Return True if self is subnormal; otherwise return False.""" |
|
2882 if self._is_special or not self: |
|
2883 return False |
|
2884 if context is None: |
|
2885 context = getcontext() |
|
2886 return self.adjusted() < context.Emin |
|
2887 |
|
2888 def is_zero(self): |
|
2889 """Return True if self is a zero; otherwise return False.""" |
|
2890 return not self._is_special and self._int == '0' |
|
2891 |
|
2892 def _ln_exp_bound(self): |
|
2893 """Compute a lower bound for the adjusted exponent of self.ln(). |
|
2894 In other words, compute r such that self.ln() >= 10**r. Assumes |
|
2895 that self is finite and positive and that self != 1. |
|
2896 """ |
|
2897 |
|
2898 # for 0.1 <= x <= 10 we use the inequalities 1-1/x <= ln(x) <= x-1 |
|
2899 adj = self._exp + len(self._int) - 1 |
|
2900 if adj >= 1: |
|
2901 # argument >= 10; we use 23/10 = 2.3 as a lower bound for ln(10) |
|
2902 return len(str(adj*23//10)) - 1 |
|
2903 if adj <= -2: |
|
2904 # argument <= 0.1 |
|
2905 return len(str((-1-adj)*23//10)) - 1 |
|
2906 op = _WorkRep(self) |
|
2907 c, e = op.int, op.exp |
|
2908 if adj == 0: |
|
2909 # 1 < self < 10 |
|
2910 num = str(c-10**-e) |
|
2911 den = str(c) |
|
2912 return len(num) - len(den) - (num < den) |
|
2913 # adj == -1, 0.1 <= self < 1 |
|
2914 return e + len(str(10**-e - c)) - 1 |
|
2915 |
|
2916 |
|
2917 def ln(self, context=None): |
|
2918 """Returns the natural (base e) logarithm of self.""" |
|
2919 |
|
2920 if context is None: |
|
2921 context = getcontext() |
|
2922 |
|
2923 # ln(NaN) = NaN |
|
2924 ans = self._check_nans(context=context) |
|
2925 if ans: |
|
2926 return ans |
|
2927 |
|
2928 # ln(0.0) == -Infinity |
|
2929 if not self: |
|
2930 return negInf |
|
2931 |
|
2932 # ln(Infinity) = Infinity |
|
2933 if self._isinfinity() == 1: |
|
2934 return Inf |
|
2935 |
|
2936 # ln(1.0) == 0.0 |
|
2937 if self == Dec_p1: |
|
2938 return Dec_0 |
|
2939 |
|
2940 # ln(negative) raises InvalidOperation |
|
2941 if self._sign == 1: |
|
2942 return context._raise_error(InvalidOperation, |
|
2943 'ln of a negative value') |
|
2944 |
|
2945 # result is irrational, so necessarily inexact |
|
2946 op = _WorkRep(self) |
|
2947 c, e = op.int, op.exp |
|
2948 p = context.prec |
|
2949 |
|
2950 # correctly rounded result: repeatedly increase precision by 3 |
|
2951 # until we get an unambiguously roundable result |
|
2952 places = p - self._ln_exp_bound() + 2 # at least p+3 places |
|
2953 while True: |
|
2954 coeff = _dlog(c, e, places) |
|
2955 # assert len(str(abs(coeff)))-p >= 1 |
|
2956 if coeff % (5*10**(len(str(abs(coeff)))-p-1)): |
|
2957 break |
|
2958 places += 3 |
|
2959 ans = _dec_from_triple(int(coeff<0), str(abs(coeff)), -places) |
|
2960 |
|
2961 context = context._shallow_copy() |
|
2962 rounding = context._set_rounding(ROUND_HALF_EVEN) |
|
2963 ans = ans._fix(context) |
|
2964 context.rounding = rounding |
|
2965 return ans |
|
2966 |
|
2967 def _log10_exp_bound(self): |
|
2968 """Compute a lower bound for the adjusted exponent of self.log10(). |
|
2969 In other words, find r such that self.log10() >= 10**r. |
|
2970 Assumes that self is finite and positive and that self != 1. |
|
2971 """ |
|
2972 |
|
2973 # For x >= 10 or x < 0.1 we only need a bound on the integer |
|
2974 # part of log10(self), and this comes directly from the |
|
2975 # exponent of x. For 0.1 <= x <= 10 we use the inequalities |
|
2976 # 1-1/x <= log(x) <= x-1. If x > 1 we have |log10(x)| > |
|
2977 # (1-1/x)/2.31 > 0. If x < 1 then |log10(x)| > (1-x)/2.31 > 0 |
|
2978 |
|
2979 adj = self._exp + len(self._int) - 1 |
|
2980 if adj >= 1: |
|
2981 # self >= 10 |
|
2982 return len(str(adj))-1 |
|
2983 if adj <= -2: |
|
2984 # self < 0.1 |
|
2985 return len(str(-1-adj))-1 |
|
2986 op = _WorkRep(self) |
|
2987 c, e = op.int, op.exp |
|
2988 if adj == 0: |
|
2989 # 1 < self < 10 |
|
2990 num = str(c-10**-e) |
|
2991 den = str(231*c) |
|
2992 return len(num) - len(den) - (num < den) + 2 |
|
2993 # adj == -1, 0.1 <= self < 1 |
|
2994 num = str(10**-e-c) |
|
2995 return len(num) + e - (num < "231") - 1 |
|
2996 |
|
2997 def log10(self, context=None): |
|
2998 """Returns the base 10 logarithm of self.""" |
|
2999 |
|
3000 if context is None: |
|
3001 context = getcontext() |
|
3002 |
|
3003 # log10(NaN) = NaN |
|
3004 ans = self._check_nans(context=context) |
|
3005 if ans: |
|
3006 return ans |
|
3007 |
|
3008 # log10(0.0) == -Infinity |
|
3009 if not self: |
|
3010 return negInf |
|
3011 |
|
3012 # log10(Infinity) = Infinity |
|
3013 if self._isinfinity() == 1: |
|
3014 return Inf |
|
3015 |
|
3016 # log10(negative or -Infinity) raises InvalidOperation |
|
3017 if self._sign == 1: |
|
3018 return context._raise_error(InvalidOperation, |
|
3019 'log10 of a negative value') |
|
3020 |
|
3021 # log10(10**n) = n |
|
3022 if self._int[0] == '1' and self._int[1:] == '0'*(len(self._int) - 1): |
|
3023 # answer may need rounding |
|
3024 ans = Decimal(self._exp + len(self._int) - 1) |
|
3025 else: |
|
3026 # result is irrational, so necessarily inexact |
|
3027 op = _WorkRep(self) |
|
3028 c, e = op.int, op.exp |
|
3029 p = context.prec |
|
3030 |
|
3031 # correctly rounded result: repeatedly increase precision |
|
3032 # until result is unambiguously roundable |
|
3033 places = p-self._log10_exp_bound()+2 |
|
3034 while True: |
|
3035 coeff = _dlog10(c, e, places) |
|
3036 # assert len(str(abs(coeff)))-p >= 1 |
|
3037 if coeff % (5*10**(len(str(abs(coeff)))-p-1)): |
|
3038 break |
|
3039 places += 3 |
|
3040 ans = _dec_from_triple(int(coeff<0), str(abs(coeff)), -places) |
|
3041 |
|
3042 context = context._shallow_copy() |
|
3043 rounding = context._set_rounding(ROUND_HALF_EVEN) |
|
3044 ans = ans._fix(context) |
|
3045 context.rounding = rounding |
|
3046 return ans |
|
3047 |
|
3048 def logb(self, context=None): |
|
3049 """ Returns the exponent of the magnitude of self's MSD. |
|
3050 |
|
3051 The result is the integer which is the exponent of the magnitude |
|
3052 of the most significant digit of self (as though it were truncated |
|
3053 to a single digit while maintaining the value of that digit and |
|
3054 without limiting the resulting exponent). |
|
3055 """ |
|
3056 # logb(NaN) = NaN |
|
3057 ans = self._check_nans(context=context) |
|
3058 if ans: |
|
3059 return ans |
|
3060 |
|
3061 if context is None: |
|
3062 context = getcontext() |
|
3063 |
|
3064 # logb(+/-Inf) = +Inf |
|
3065 if self._isinfinity(): |
|
3066 return Inf |
|
3067 |
|
3068 # logb(0) = -Inf, DivisionByZero |
|
3069 if not self: |
|
3070 return context._raise_error(DivisionByZero, 'logb(0)', 1) |
|
3071 |
|
3072 # otherwise, simply return the adjusted exponent of self, as a |
|
3073 # Decimal. Note that no attempt is made to fit the result |
|
3074 # into the current context. |
|
3075 return Decimal(self.adjusted()) |
|
3076 |
|
3077 def _islogical(self): |
|
3078 """Return True if self is a logical operand. |
|
3079 |
|
3080 For being logical, it must be a finite number with a sign of 0, |
|
3081 an exponent of 0, and a coefficient whose digits must all be |
|
3082 either 0 or 1. |
|
3083 """ |
|
3084 if self._sign != 0 or self._exp != 0: |
|
3085 return False |
|
3086 for dig in self._int: |
|
3087 if dig not in '01': |
|
3088 return False |
|
3089 return True |
|
3090 |
|
3091 def _fill_logical(self, context, opa, opb): |
|
3092 dif = context.prec - len(opa) |
|
3093 if dif > 0: |
|
3094 opa = '0'*dif + opa |
|
3095 elif dif < 0: |
|
3096 opa = opa[-context.prec:] |
|
3097 dif = context.prec - len(opb) |
|
3098 if dif > 0: |
|
3099 opb = '0'*dif + opb |
|
3100 elif dif < 0: |
|
3101 opb = opb[-context.prec:] |
|
3102 return opa, opb |
|
3103 |
|
3104 def logical_and(self, other, context=None): |
|
3105 """Applies an 'and' operation between self and other's digits.""" |
|
3106 if context is None: |
|
3107 context = getcontext() |
|
3108 if not self._islogical() or not other._islogical(): |
|
3109 return context._raise_error(InvalidOperation) |
|
3110 |
|
3111 # fill to context.prec |
|
3112 (opa, opb) = self._fill_logical(context, self._int, other._int) |
|
3113 |
|
3114 # make the operation, and clean starting zeroes |
|
3115 result = "".join([str(int(a)&int(b)) for a,b in zip(opa,opb)]) |
|
3116 return _dec_from_triple(0, result.lstrip('0') or '0', 0) |
|
3117 |
|
3118 def logical_invert(self, context=None): |
|
3119 """Invert all its digits.""" |
|
3120 if context is None: |
|
3121 context = getcontext() |
|
3122 return self.logical_xor(_dec_from_triple(0,'1'*context.prec,0), |
|
3123 context) |
|
3124 |
|
3125 def logical_or(self, other, context=None): |
|
3126 """Applies an 'or' operation between self and other's digits.""" |
|
3127 if context is None: |
|
3128 context = getcontext() |
|
3129 if not self._islogical() or not other._islogical(): |
|
3130 return context._raise_error(InvalidOperation) |
|
3131 |
|
3132 # fill to context.prec |
|
3133 (opa, opb) = self._fill_logical(context, self._int, other._int) |
|
3134 |
|
3135 # make the operation, and clean starting zeroes |
|
3136 result = "".join(str(int(a)|int(b)) for a,b in zip(opa,opb)) |
|
3137 return _dec_from_triple(0, result.lstrip('0') or '0', 0) |
|
3138 |
|
3139 def logical_xor(self, other, context=None): |
|
3140 """Applies an 'xor' operation between self and other's digits.""" |
|
3141 if context is None: |
|
3142 context = getcontext() |
|
3143 if not self._islogical() or not other._islogical(): |
|
3144 return context._raise_error(InvalidOperation) |
|
3145 |
|
3146 # fill to context.prec |
|
3147 (opa, opb) = self._fill_logical(context, self._int, other._int) |
|
3148 |
|
3149 # make the operation, and clean starting zeroes |
|
3150 result = "".join(str(int(a)^int(b)) for a,b in zip(opa,opb)) |
|
3151 return _dec_from_triple(0, result.lstrip('0') or '0', 0) |
|
3152 |
|
3153 def max_mag(self, other, context=None): |
|
3154 """Compares the values numerically with their sign ignored.""" |
|
3155 other = _convert_other(other, raiseit=True) |
|
3156 |
|
3157 if context is None: |
|
3158 context = getcontext() |
|
3159 |
|
3160 if self._is_special or other._is_special: |
|
3161 # If one operand is a quiet NaN and the other is number, then the |
|
3162 # number is always returned |
|
3163 sn = self._isnan() |
|
3164 on = other._isnan() |
|
3165 if sn or on: |
|
3166 if on == 1 and sn != 2: |
|
3167 return self._fix_nan(context) |
|
3168 if sn == 1 and on != 2: |
|
3169 return other._fix_nan(context) |
|
3170 return self._check_nans(other, context) |
|
3171 |
|
3172 c = self.copy_abs()._cmp(other.copy_abs()) |
|
3173 if c == 0: |
|
3174 c = self.compare_total(other) |
|
3175 |
|
3176 if c == -1: |
|
3177 ans = other |
|
3178 else: |
|
3179 ans = self |
|
3180 |
|
3181 return ans._fix(context) |
|
3182 |
|
3183 def min_mag(self, other, context=None): |
|
3184 """Compares the values numerically with their sign ignored.""" |
|
3185 other = _convert_other(other, raiseit=True) |
|
3186 |
|
3187 if context is None: |
|
3188 context = getcontext() |
|
3189 |
|
3190 if self._is_special or other._is_special: |
|
3191 # If one operand is a quiet NaN and the other is number, then the |
|
3192 # number is always returned |
|
3193 sn = self._isnan() |
|
3194 on = other._isnan() |
|
3195 if sn or on: |
|
3196 if on == 1 and sn != 2: |
|
3197 return self._fix_nan(context) |
|
3198 if sn == 1 and on != 2: |
|
3199 return other._fix_nan(context) |
|
3200 return self._check_nans(other, context) |
|
3201 |
|
3202 c = self.copy_abs()._cmp(other.copy_abs()) |
|
3203 if c == 0: |
|
3204 c = self.compare_total(other) |
|
3205 |
|
3206 if c == -1: |
|
3207 ans = self |
|
3208 else: |
|
3209 ans = other |
|
3210 |
|
3211 return ans._fix(context) |
|
3212 |
|
3213 def next_minus(self, context=None): |
|
3214 """Returns the largest representable number smaller than itself.""" |
|
3215 if context is None: |
|
3216 context = getcontext() |
|
3217 |
|
3218 ans = self._check_nans(context=context) |
|
3219 if ans: |
|
3220 return ans |
|
3221 |
|
3222 if self._isinfinity() == -1: |
|
3223 return negInf |
|
3224 if self._isinfinity() == 1: |
|
3225 return _dec_from_triple(0, '9'*context.prec, context.Etop()) |
|
3226 |
|
3227 context = context.copy() |
|
3228 context._set_rounding(ROUND_FLOOR) |
|
3229 context._ignore_all_flags() |
|
3230 new_self = self._fix(context) |
|
3231 if new_self != self: |
|
3232 return new_self |
|
3233 return self.__sub__(_dec_from_triple(0, '1', context.Etiny()-1), |
|
3234 context) |
|
3235 |
|
3236 def next_plus(self, context=None): |
|
3237 """Returns the smallest representable number larger than itself.""" |
|
3238 if context is None: |
|
3239 context = getcontext() |
|
3240 |
|
3241 ans = self._check_nans(context=context) |
|
3242 if ans: |
|
3243 return ans |
|
3244 |
|
3245 if self._isinfinity() == 1: |
|
3246 return Inf |
|
3247 if self._isinfinity() == -1: |
|
3248 return _dec_from_triple(1, '9'*context.prec, context.Etop()) |
|
3249 |
|
3250 context = context.copy() |
|
3251 context._set_rounding(ROUND_CEILING) |
|
3252 context._ignore_all_flags() |
|
3253 new_self = self._fix(context) |
|
3254 if new_self != self: |
|
3255 return new_self |
|
3256 return self.__add__(_dec_from_triple(0, '1', context.Etiny()-1), |
|
3257 context) |
|
3258 |
|
3259 def next_toward(self, other, context=None): |
|
3260 """Returns the number closest to self, in the direction towards other. |
|
3261 |
|
3262 The result is the closest representable number to self |
|
3263 (excluding self) that is in the direction towards other, |
|
3264 unless both have the same value. If the two operands are |
|
3265 numerically equal, then the result is a copy of self with the |
|
3266 sign set to be the same as the sign of other. |
|
3267 """ |
|
3268 other = _convert_other(other, raiseit=True) |
|
3269 |
|
3270 if context is None: |
|
3271 context = getcontext() |
|
3272 |
|
3273 ans = self._check_nans(other, context) |
|
3274 if ans: |
|
3275 return ans |
|
3276 |
|
3277 comparison = self._cmp(other) |
|
3278 if comparison == 0: |
|
3279 return self.copy_sign(other) |
|
3280 |
|
3281 if comparison == -1: |
|
3282 ans = self.next_plus(context) |
|
3283 else: # comparison == 1 |
|
3284 ans = self.next_minus(context) |
|
3285 |
|
3286 # decide which flags to raise using value of ans |
|
3287 if ans._isinfinity(): |
|
3288 context._raise_error(Overflow, |
|
3289 'Infinite result from next_toward', |
|
3290 ans._sign) |
|
3291 context._raise_error(Rounded) |
|
3292 context._raise_error(Inexact) |
|
3293 elif ans.adjusted() < context.Emin: |
|
3294 context._raise_error(Underflow) |
|
3295 context._raise_error(Subnormal) |
|
3296 context._raise_error(Rounded) |
|
3297 context._raise_error(Inexact) |
|
3298 # if precision == 1 then we don't raise Clamped for a |
|
3299 # result 0E-Etiny. |
|
3300 if not ans: |
|
3301 context._raise_error(Clamped) |
|
3302 |
|
3303 return ans |
|
3304 |
|
3305 def number_class(self, context=None): |
|
3306 """Returns an indication of the class of self. |
|
3307 |
|
3308 The class is one of the following strings: |
|
3309 sNaN |
|
3310 NaN |
|
3311 -Infinity |
|
3312 -Normal |
|
3313 -Subnormal |
|
3314 -Zero |
|
3315 +Zero |
|
3316 +Subnormal |
|
3317 +Normal |
|
3318 +Infinity |
|
3319 """ |
|
3320 if self.is_snan(): |
|
3321 return "sNaN" |
|
3322 if self.is_qnan(): |
|
3323 return "NaN" |
|
3324 inf = self._isinfinity() |
|
3325 if inf == 1: |
|
3326 return "+Infinity" |
|
3327 if inf == -1: |
|
3328 return "-Infinity" |
|
3329 if self.is_zero(): |
|
3330 if self._sign: |
|
3331 return "-Zero" |
|
3332 else: |
|
3333 return "+Zero" |
|
3334 if context is None: |
|
3335 context = getcontext() |
|
3336 if self.is_subnormal(context=context): |
|
3337 if self._sign: |
|
3338 return "-Subnormal" |
|
3339 else: |
|
3340 return "+Subnormal" |
|
3341 # just a normal, regular, boring number, :) |
|
3342 if self._sign: |
|
3343 return "-Normal" |
|
3344 else: |
|
3345 return "+Normal" |
|
3346 |
|
3347 def radix(self): |
|
3348 """Just returns 10, as this is Decimal, :)""" |
|
3349 return Decimal(10) |
|
3350 |
|
3351 def rotate(self, other, context=None): |
|
3352 """Returns a rotated copy of self, value-of-other times.""" |
|
3353 if context is None: |
|
3354 context = getcontext() |
|
3355 |
|
3356 ans = self._check_nans(other, context) |
|
3357 if ans: |
|
3358 return ans |
|
3359 |
|
3360 if other._exp != 0: |
|
3361 return context._raise_error(InvalidOperation) |
|
3362 if not (-context.prec <= int(other) <= context.prec): |
|
3363 return context._raise_error(InvalidOperation) |
|
3364 |
|
3365 if self._isinfinity(): |
|
3366 return Decimal(self) |
|
3367 |
|
3368 # get values, pad if necessary |
|
3369 torot = int(other) |
|
3370 rotdig = self._int |
|
3371 topad = context.prec - len(rotdig) |
|
3372 if topad: |
|
3373 rotdig = '0'*topad + rotdig |
|
3374 |
|
3375 # let's rotate! |
|
3376 rotated = rotdig[torot:] + rotdig[:torot] |
|
3377 return _dec_from_triple(self._sign, |
|
3378 rotated.lstrip('0') or '0', self._exp) |
|
3379 |
|
3380 def scaleb (self, other, context=None): |
|
3381 """Returns self operand after adding the second value to its exp.""" |
|
3382 if context is None: |
|
3383 context = getcontext() |
|
3384 |
|
3385 ans = self._check_nans(other, context) |
|
3386 if ans: |
|
3387 return ans |
|
3388 |
|
3389 if other._exp != 0: |
|
3390 return context._raise_error(InvalidOperation) |
|
3391 liminf = -2 * (context.Emax + context.prec) |
|
3392 limsup = 2 * (context.Emax + context.prec) |
|
3393 if not (liminf <= int(other) <= limsup): |
|
3394 return context._raise_error(InvalidOperation) |
|
3395 |
|
3396 if self._isinfinity(): |
|
3397 return Decimal(self) |
|
3398 |
|
3399 d = _dec_from_triple(self._sign, self._int, self._exp + int(other)) |
|
3400 d = d._fix(context) |
|
3401 return d |
|
3402 |
|
3403 def shift(self, other, context=None): |
|
3404 """Returns a shifted copy of self, value-of-other times.""" |
|
3405 if context is None: |
|
3406 context = getcontext() |
|
3407 |
|
3408 ans = self._check_nans(other, context) |
|
3409 if ans: |
|
3410 return ans |
|
3411 |
|
3412 if other._exp != 0: |
|
3413 return context._raise_error(InvalidOperation) |
|
3414 if not (-context.prec <= int(other) <= context.prec): |
|
3415 return context._raise_error(InvalidOperation) |
|
3416 |
|
3417 if self._isinfinity(): |
|
3418 return Decimal(self) |
|
3419 |
|
3420 # get values, pad if necessary |
|
3421 torot = int(other) |
|
3422 if not torot: |
|
3423 return Decimal(self) |
|
3424 rotdig = self._int |
|
3425 topad = context.prec - len(rotdig) |
|
3426 if topad: |
|
3427 rotdig = '0'*topad + rotdig |
|
3428 |
|
3429 # let's shift! |
|
3430 if torot < 0: |
|
3431 rotated = rotdig[:torot] |
|
3432 else: |
|
3433 rotated = rotdig + '0'*torot |
|
3434 rotated = rotated[-context.prec:] |
|
3435 |
|
3436 return _dec_from_triple(self._sign, |
|
3437 rotated.lstrip('0') or '0', self._exp) |
|
3438 |
|
3439 # Support for pickling, copy, and deepcopy |
|
3440 def __reduce__(self): |
|
3441 return (self.__class__, (str(self),)) |
|
3442 |
|
3443 def __copy__(self): |
|
3444 if type(self) == Decimal: |
|
3445 return self # I'm immutable; therefore I am my own clone |
|
3446 return self.__class__(str(self)) |
|
3447 |
|
3448 def __deepcopy__(self, memo): |
|
3449 if type(self) == Decimal: |
|
3450 return self # My components are also immutable |
|
3451 return self.__class__(str(self)) |
|
3452 |
|
3453 # PEP 3101 support. See also _parse_format_specifier and _format_align |
|
3454 def __format__(self, specifier, context=None): |
|
3455 """Format a Decimal instance according to the given specifier. |
|
3456 |
|
3457 The specifier should be a standard format specifier, with the |
|
3458 form described in PEP 3101. Formatting types 'e', 'E', 'f', |
|
3459 'F', 'g', 'G', and '%' are supported. If the formatting type |
|
3460 is omitted it defaults to 'g' or 'G', depending on the value |
|
3461 of context.capitals. |
|
3462 |
|
3463 At this time the 'n' format specifier type (which is supposed |
|
3464 to use the current locale) is not supported. |
|
3465 """ |
|
3466 |
|
3467 # Note: PEP 3101 says that if the type is not present then |
|
3468 # there should be at least one digit after the decimal point. |
|
3469 # We take the liberty of ignoring this requirement for |
|
3470 # Decimal---it's presumably there to make sure that |
|
3471 # format(float, '') behaves similarly to str(float). |
|
3472 if context is None: |
|
3473 context = getcontext() |
|
3474 |
|
3475 spec = _parse_format_specifier(specifier) |
|
3476 |
|
3477 # special values don't care about the type or precision... |
|
3478 if self._is_special: |
|
3479 return _format_align(str(self), spec) |
|
3480 |
|
3481 # a type of None defaults to 'g' or 'G', depending on context |
|
3482 # if type is '%', adjust exponent of self accordingly |
|
3483 if spec['type'] is None: |
|
3484 spec['type'] = ['g', 'G'][context.capitals] |
|
3485 elif spec['type'] == '%': |
|
3486 self = _dec_from_triple(self._sign, self._int, self._exp+2) |
|
3487 |
|
3488 # round if necessary, taking rounding mode from the context |
|
3489 rounding = context.rounding |
|
3490 precision = spec['precision'] |
|
3491 if precision is not None: |
|
3492 if spec['type'] in 'eE': |
|
3493 self = self._round(precision+1, rounding) |
|
3494 elif spec['type'] in 'gG': |
|
3495 if len(self._int) > precision: |
|
3496 self = self._round(precision, rounding) |
|
3497 elif spec['type'] in 'fF%': |
|
3498 self = self._rescale(-precision, rounding) |
|
3499 # special case: zeros with a positive exponent can't be |
|
3500 # represented in fixed point; rescale them to 0e0. |
|
3501 elif not self and self._exp > 0 and spec['type'] in 'fF%': |
|
3502 self = self._rescale(0, rounding) |
|
3503 |
|
3504 # figure out placement of the decimal point |
|
3505 leftdigits = self._exp + len(self._int) |
|
3506 if spec['type'] in 'fF%': |
|
3507 dotplace = leftdigits |
|
3508 elif spec['type'] in 'eE': |
|
3509 if not self and precision is not None: |
|
3510 dotplace = 1 - precision |
|
3511 else: |
|
3512 dotplace = 1 |
|
3513 elif spec['type'] in 'gG': |
|
3514 if self._exp <= 0 and leftdigits > -6: |
|
3515 dotplace = leftdigits |
|
3516 else: |
|
3517 dotplace = 1 |
|
3518 |
|
3519 # figure out main part of numeric string... |
|
3520 if dotplace <= 0: |
|
3521 num = '0.' + '0'*(-dotplace) + self._int |
|
3522 elif dotplace >= len(self._int): |
|
3523 # make sure we're not padding a '0' with extra zeros on the right |
|
3524 assert dotplace==len(self._int) or self._int != '0' |
|
3525 num = self._int + '0'*(dotplace-len(self._int)) |
|
3526 else: |
|
3527 num = self._int[:dotplace] + '.' + self._int[dotplace:] |
|
3528 |
|
3529 # ...then the trailing exponent, or trailing '%' |
|
3530 if leftdigits != dotplace or spec['type'] in 'eE': |
|
3531 echar = {'E': 'E', 'e': 'e', 'G': 'E', 'g': 'e'}[spec['type']] |
|
3532 num = num + "{0}{1:+}".format(echar, leftdigits-dotplace) |
|
3533 elif spec['type'] == '%': |
|
3534 num = num + '%' |
|
3535 |
|
3536 # add sign |
|
3537 if self._sign == 1: |
|
3538 num = '-' + num |
|
3539 return _format_align(num, spec) |
|
3540 |
|
3541 |
|
3542 def _dec_from_triple(sign, coefficient, exponent, special=False): |
|
3543 """Create a decimal instance directly, without any validation, |
|
3544 normalization (e.g. removal of leading zeros) or argument |
|
3545 conversion. |
|
3546 |
|
3547 This function is for *internal use only*. |
|
3548 """ |
|
3549 |
|
3550 self = object.__new__(Decimal) |
|
3551 self._sign = sign |
|
3552 self._int = coefficient |
|
3553 self._exp = exponent |
|
3554 self._is_special = special |
|
3555 |
|
3556 return self |
|
3557 |
|
3558 ##### Context class ####################################################### |
|
3559 |
|
3560 |
|
3561 # get rounding method function: |
|
3562 rounding_functions = [name for name in Decimal.__dict__.keys() |
|
3563 if name.startswith('_round_')] |
|
3564 for name in rounding_functions: |
|
3565 # name is like _round_half_even, goes to the global ROUND_HALF_EVEN value. |
|
3566 globalname = name[1:].upper() |
|
3567 val = globals()[globalname] |
|
3568 Decimal._pick_rounding_function[val] = name |
|
3569 |
|
3570 del name, val, globalname, rounding_functions |
|
3571 |
|
3572 class _ContextManager(object): |
|
3573 """Context manager class to support localcontext(). |
|
3574 |
|
3575 Sets a copy of the supplied context in __enter__() and restores |
|
3576 the previous decimal context in __exit__() |
|
3577 """ |
|
3578 def __init__(self, new_context): |
|
3579 self.new_context = new_context.copy() |
|
3580 def __enter__(self): |
|
3581 self.saved_context = getcontext() |
|
3582 setcontext(self.new_context) |
|
3583 return self.new_context |
|
3584 def __exit__(self, t, v, tb): |
|
3585 setcontext(self.saved_context) |
|
3586 |
|
3587 class Context(object): |
|
3588 """Contains the context for a Decimal instance. |
|
3589 |
|
3590 Contains: |
|
3591 prec - precision (for use in rounding, division, square roots..) |
|
3592 rounding - rounding type (how you round) |
|
3593 traps - If traps[exception] = 1, then the exception is |
|
3594 raised when it is caused. Otherwise, a value is |
|
3595 substituted in. |
|
3596 flags - When an exception is caused, flags[exception] is set. |
|
3597 (Whether or not the trap_enabler is set) |
|
3598 Should be reset by user of Decimal instance. |
|
3599 Emin - Minimum exponent |
|
3600 Emax - Maximum exponent |
|
3601 capitals - If 1, 1*10^1 is printed as 1E+1. |
|
3602 If 0, printed as 1e1 |
|
3603 _clamp - If 1, change exponents if too high (Default 0) |
|
3604 """ |
|
3605 |
|
3606 def __init__(self, prec=None, rounding=None, |
|
3607 traps=None, flags=None, |
|
3608 Emin=None, Emax=None, |
|
3609 capitals=None, _clamp=0, |
|
3610 _ignored_flags=None): |
|
3611 if flags is None: |
|
3612 flags = [] |
|
3613 if _ignored_flags is None: |
|
3614 _ignored_flags = [] |
|
3615 if not isinstance(flags, dict): |
|
3616 flags = dict([(s, int(s in flags)) for s in _signals]) |
|
3617 del s |
|
3618 if traps is not None and not isinstance(traps, dict): |
|
3619 traps = dict([(s, int(s in traps)) for s in _signals]) |
|
3620 del s |
|
3621 for name, val in locals().items(): |
|
3622 if val is None: |
|
3623 setattr(self, name, _copy.copy(getattr(DefaultContext, name))) |
|
3624 else: |
|
3625 setattr(self, name, val) |
|
3626 del self.self |
|
3627 |
|
3628 def __repr__(self): |
|
3629 """Show the current context.""" |
|
3630 s = [] |
|
3631 s.append('Context(prec=%(prec)d, rounding=%(rounding)s, ' |
|
3632 'Emin=%(Emin)d, Emax=%(Emax)d, capitals=%(capitals)d' |
|
3633 % vars(self)) |
|
3634 names = [f.__name__ for f, v in self.flags.items() if v] |
|
3635 s.append('flags=[' + ', '.join(names) + ']') |
|
3636 names = [t.__name__ for t, v in self.traps.items() if v] |
|
3637 s.append('traps=[' + ', '.join(names) + ']') |
|
3638 return ', '.join(s) + ')' |
|
3639 |
|
3640 def clear_flags(self): |
|
3641 """Reset all flags to zero""" |
|
3642 for flag in self.flags: |
|
3643 self.flags[flag] = 0 |
|
3644 |
|
3645 def _shallow_copy(self): |
|
3646 """Returns a shallow copy from self.""" |
|
3647 nc = Context(self.prec, self.rounding, self.traps, |
|
3648 self.flags, self.Emin, self.Emax, |
|
3649 self.capitals, self._clamp, self._ignored_flags) |
|
3650 return nc |
|
3651 |
|
3652 def copy(self): |
|
3653 """Returns a deep copy from self.""" |
|
3654 nc = Context(self.prec, self.rounding, self.traps.copy(), |
|
3655 self.flags.copy(), self.Emin, self.Emax, |
|
3656 self.capitals, self._clamp, self._ignored_flags) |
|
3657 return nc |
|
3658 __copy__ = copy |
|
3659 |
|
3660 def _raise_error(self, condition, explanation = None, *args): |
|
3661 """Handles an error |
|
3662 |
|
3663 If the flag is in _ignored_flags, returns the default response. |
|
3664 Otherwise, it sets the flag, then, if the corresponding |
|
3665 trap_enabler is set, it reaises the exception. Otherwise, it returns |
|
3666 the default value after setting the flag. |
|
3667 """ |
|
3668 error = _condition_map.get(condition, condition) |
|
3669 if error in self._ignored_flags: |
|
3670 # Don't touch the flag |
|
3671 return error().handle(self, *args) |
|
3672 |
|
3673 self.flags[error] = 1 |
|
3674 if not self.traps[error]: |
|
3675 # The errors define how to handle themselves. |
|
3676 return condition().handle(self, *args) |
|
3677 |
|
3678 # Errors should only be risked on copies of the context |
|
3679 # self._ignored_flags = [] |
|
3680 raise error(explanation) |
|
3681 |
|
3682 def _ignore_all_flags(self): |
|
3683 """Ignore all flags, if they are raised""" |
|
3684 return self._ignore_flags(*_signals) |
|
3685 |
|
3686 def _ignore_flags(self, *flags): |
|
3687 """Ignore the flags, if they are raised""" |
|
3688 # Do not mutate-- This way, copies of a context leave the original |
|
3689 # alone. |
|
3690 self._ignored_flags = (self._ignored_flags + list(flags)) |
|
3691 return list(flags) |
|
3692 |
|
3693 def _regard_flags(self, *flags): |
|
3694 """Stop ignoring the flags, if they are raised""" |
|
3695 if flags and isinstance(flags[0], (tuple,list)): |
|
3696 flags = flags[0] |
|
3697 for flag in flags: |
|
3698 self._ignored_flags.remove(flag) |
|
3699 |
|
3700 # We inherit object.__hash__, so we must deny this explicitly |
|
3701 __hash__ = None |
|
3702 |
|
3703 def Etiny(self): |
|
3704 """Returns Etiny (= Emin - prec + 1)""" |
|
3705 return int(self.Emin - self.prec + 1) |
|
3706 |
|
3707 def Etop(self): |
|
3708 """Returns maximum exponent (= Emax - prec + 1)""" |
|
3709 return int(self.Emax - self.prec + 1) |
|
3710 |
|
3711 def _set_rounding(self, type): |
|
3712 """Sets the rounding type. |
|
3713 |
|
3714 Sets the rounding type, and returns the current (previous) |
|
3715 rounding type. Often used like: |
|
3716 |
|
3717 context = context.copy() |
|
3718 # so you don't change the calling context |
|
3719 # if an error occurs in the middle. |
|
3720 rounding = context._set_rounding(ROUND_UP) |
|
3721 val = self.__sub__(other, context=context) |
|
3722 context._set_rounding(rounding) |
|
3723 |
|
3724 This will make it round up for that operation. |
|
3725 """ |
|
3726 rounding = self.rounding |
|
3727 self.rounding= type |
|
3728 return rounding |
|
3729 |
|
3730 def create_decimal(self, num='0'): |
|
3731 """Creates a new Decimal instance but using self as context. |
|
3732 |
|
3733 This method implements the to-number operation of the |
|
3734 IBM Decimal specification.""" |
|
3735 |
|
3736 if isinstance(num, basestring) and num != num.strip(): |
|
3737 return self._raise_error(ConversionSyntax, |
|
3738 "no trailing or leading whitespace is " |
|
3739 "permitted.") |
|
3740 |
|
3741 d = Decimal(num, context=self) |
|
3742 if d._isnan() and len(d._int) > self.prec - self._clamp: |
|
3743 return self._raise_error(ConversionSyntax, |
|
3744 "diagnostic info too long in NaN") |
|
3745 return d._fix(self) |
|
3746 |
|
3747 # Methods |
|
3748 def abs(self, a): |
|
3749 """Returns the absolute value of the operand. |
|
3750 |
|
3751 If the operand is negative, the result is the same as using the minus |
|
3752 operation on the operand. Otherwise, the result is the same as using |
|
3753 the plus operation on the operand. |
|
3754 |
|
3755 >>> ExtendedContext.abs(Decimal('2.1')) |
|
3756 Decimal('2.1') |
|
3757 >>> ExtendedContext.abs(Decimal('-100')) |
|
3758 Decimal('100') |
|
3759 >>> ExtendedContext.abs(Decimal('101.5')) |
|
3760 Decimal('101.5') |
|
3761 >>> ExtendedContext.abs(Decimal('-101.5')) |
|
3762 Decimal('101.5') |
|
3763 """ |
|
3764 return a.__abs__(context=self) |
|
3765 |
|
3766 def add(self, a, b): |
|
3767 """Return the sum of the two operands. |
|
3768 |
|
3769 >>> ExtendedContext.add(Decimal('12'), Decimal('7.00')) |
|
3770 Decimal('19.00') |
|
3771 >>> ExtendedContext.add(Decimal('1E+2'), Decimal('1.01E+4')) |
|
3772 Decimal('1.02E+4') |
|
3773 """ |
|
3774 return a.__add__(b, context=self) |
|
3775 |
|
3776 def _apply(self, a): |
|
3777 return str(a._fix(self)) |
|
3778 |
|
3779 def canonical(self, a): |
|
3780 """Returns the same Decimal object. |
|
3781 |
|
3782 As we do not have different encodings for the same number, the |
|
3783 received object already is in its canonical form. |
|
3784 |
|
3785 >>> ExtendedContext.canonical(Decimal('2.50')) |
|
3786 Decimal('2.50') |
|
3787 """ |
|
3788 return a.canonical(context=self) |
|
3789 |
|
3790 def compare(self, a, b): |
|
3791 """Compares values numerically. |
|
3792 |
|
3793 If the signs of the operands differ, a value representing each operand |
|
3794 ('-1' if the operand is less than zero, '0' if the operand is zero or |
|
3795 negative zero, or '1' if the operand is greater than zero) is used in |
|
3796 place of that operand for the comparison instead of the actual |
|
3797 operand. |
|
3798 |
|
3799 The comparison is then effected by subtracting the second operand from |
|
3800 the first and then returning a value according to the result of the |
|
3801 subtraction: '-1' if the result is less than zero, '0' if the result is |
|
3802 zero or negative zero, or '1' if the result is greater than zero. |
|
3803 |
|
3804 >>> ExtendedContext.compare(Decimal('2.1'), Decimal('3')) |
|
3805 Decimal('-1') |
|
3806 >>> ExtendedContext.compare(Decimal('2.1'), Decimal('2.1')) |
|
3807 Decimal('0') |
|
3808 >>> ExtendedContext.compare(Decimal('2.1'), Decimal('2.10')) |
|
3809 Decimal('0') |
|
3810 >>> ExtendedContext.compare(Decimal('3'), Decimal('2.1')) |
|
3811 Decimal('1') |
|
3812 >>> ExtendedContext.compare(Decimal('2.1'), Decimal('-3')) |
|
3813 Decimal('1') |
|
3814 >>> ExtendedContext.compare(Decimal('-3'), Decimal('2.1')) |
|
3815 Decimal('-1') |
|
3816 """ |
|
3817 return a.compare(b, context=self) |
|
3818 |
|
3819 def compare_signal(self, a, b): |
|
3820 """Compares the values of the two operands numerically. |
|
3821 |
|
3822 It's pretty much like compare(), but all NaNs signal, with signaling |
|
3823 NaNs taking precedence over quiet NaNs. |
|
3824 |
|
3825 >>> c = ExtendedContext |
|
3826 >>> c.compare_signal(Decimal('2.1'), Decimal('3')) |
|
3827 Decimal('-1') |
|
3828 >>> c.compare_signal(Decimal('2.1'), Decimal('2.1')) |
|
3829 Decimal('0') |
|
3830 >>> c.flags[InvalidOperation] = 0 |
|
3831 >>> print c.flags[InvalidOperation] |
|
3832 0 |
|
3833 >>> c.compare_signal(Decimal('NaN'), Decimal('2.1')) |
|
3834 Decimal('NaN') |
|
3835 >>> print c.flags[InvalidOperation] |
|
3836 1 |
|
3837 >>> c.flags[InvalidOperation] = 0 |
|
3838 >>> print c.flags[InvalidOperation] |
|
3839 0 |
|
3840 >>> c.compare_signal(Decimal('sNaN'), Decimal('2.1')) |
|
3841 Decimal('NaN') |
|
3842 >>> print c.flags[InvalidOperation] |
|
3843 1 |
|
3844 """ |
|
3845 return a.compare_signal(b, context=self) |
|
3846 |
|
3847 def compare_total(self, a, b): |
|
3848 """Compares two operands using their abstract representation. |
|
3849 |
|
3850 This is not like the standard compare, which use their numerical |
|
3851 value. Note that a total ordering is defined for all possible abstract |
|
3852 representations. |
|
3853 |
|
3854 >>> ExtendedContext.compare_total(Decimal('12.73'), Decimal('127.9')) |
|
3855 Decimal('-1') |
|
3856 >>> ExtendedContext.compare_total(Decimal('-127'), Decimal('12')) |
|
3857 Decimal('-1') |
|
3858 >>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.3')) |
|
3859 Decimal('-1') |
|
3860 >>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.30')) |
|
3861 Decimal('0') |
|
3862 >>> ExtendedContext.compare_total(Decimal('12.3'), Decimal('12.300')) |
|
3863 Decimal('1') |
|
3864 >>> ExtendedContext.compare_total(Decimal('12.3'), Decimal('NaN')) |
|
3865 Decimal('-1') |
|
3866 """ |
|
3867 return a.compare_total(b) |
|
3868 |
|
3869 def compare_total_mag(self, a, b): |
|
3870 """Compares two operands using their abstract representation ignoring sign. |
|
3871 |
|
3872 Like compare_total, but with operand's sign ignored and assumed to be 0. |
|
3873 """ |
|
3874 return a.compare_total_mag(b) |
|
3875 |
|
3876 def copy_abs(self, a): |
|
3877 """Returns a copy of the operand with the sign set to 0. |
|
3878 |
|
3879 >>> ExtendedContext.copy_abs(Decimal('2.1')) |
|
3880 Decimal('2.1') |
|
3881 >>> ExtendedContext.copy_abs(Decimal('-100')) |
|
3882 Decimal('100') |
|
3883 """ |
|
3884 return a.copy_abs() |
|
3885 |
|
3886 def copy_decimal(self, a): |
|
3887 """Returns a copy of the decimal objet. |
|
3888 |
|
3889 >>> ExtendedContext.copy_decimal(Decimal('2.1')) |
|
3890 Decimal('2.1') |
|
3891 >>> ExtendedContext.copy_decimal(Decimal('-1.00')) |
|
3892 Decimal('-1.00') |
|
3893 """ |
|
3894 return Decimal(a) |
|
3895 |
|
3896 def copy_negate(self, a): |
|
3897 """Returns a copy of the operand with the sign inverted. |
|
3898 |
|
3899 >>> ExtendedContext.copy_negate(Decimal('101.5')) |
|
3900 Decimal('-101.5') |
|
3901 >>> ExtendedContext.copy_negate(Decimal('-101.5')) |
|
3902 Decimal('101.5') |
|
3903 """ |
|
3904 return a.copy_negate() |
|
3905 |
|
3906 def copy_sign(self, a, b): |
|
3907 """Copies the second operand's sign to the first one. |
|
3908 |
|
3909 In detail, it returns a copy of the first operand with the sign |
|
3910 equal to the sign of the second operand. |
|
3911 |
|
3912 >>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('7.33')) |
|
3913 Decimal('1.50') |
|
3914 >>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('7.33')) |
|
3915 Decimal('1.50') |
|
3916 >>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('-7.33')) |
|
3917 Decimal('-1.50') |
|
3918 >>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('-7.33')) |
|
3919 Decimal('-1.50') |
|
3920 """ |
|
3921 return a.copy_sign(b) |
|
3922 |
|
3923 def divide(self, a, b): |
|
3924 """Decimal division in a specified context. |
|
3925 |
|
3926 >>> ExtendedContext.divide(Decimal('1'), Decimal('3')) |
|
3927 Decimal('0.333333333') |
|
3928 >>> ExtendedContext.divide(Decimal('2'), Decimal('3')) |
|
3929 Decimal('0.666666667') |
|
3930 >>> ExtendedContext.divide(Decimal('5'), Decimal('2')) |
|
3931 Decimal('2.5') |
|
3932 >>> ExtendedContext.divide(Decimal('1'), Decimal('10')) |
|
3933 Decimal('0.1') |
|
3934 >>> ExtendedContext.divide(Decimal('12'), Decimal('12')) |
|
3935 Decimal('1') |
|
3936 >>> ExtendedContext.divide(Decimal('8.00'), Decimal('2')) |
|
3937 Decimal('4.00') |
|
3938 >>> ExtendedContext.divide(Decimal('2.400'), Decimal('2.0')) |
|
3939 Decimal('1.20') |
|
3940 >>> ExtendedContext.divide(Decimal('1000'), Decimal('100')) |
|
3941 Decimal('10') |
|
3942 >>> ExtendedContext.divide(Decimal('1000'), Decimal('1')) |
|
3943 Decimal('1000') |
|
3944 >>> ExtendedContext.divide(Decimal('2.40E+6'), Decimal('2')) |
|
3945 Decimal('1.20E+6') |
|
3946 """ |
|
3947 return a.__div__(b, context=self) |
|
3948 |
|
3949 def divide_int(self, a, b): |
|
3950 """Divides two numbers and returns the integer part of the result. |
|
3951 |
|
3952 >>> ExtendedContext.divide_int(Decimal('2'), Decimal('3')) |
|
3953 Decimal('0') |
|
3954 >>> ExtendedContext.divide_int(Decimal('10'), Decimal('3')) |
|
3955 Decimal('3') |
|
3956 >>> ExtendedContext.divide_int(Decimal('1'), Decimal('0.3')) |
|
3957 Decimal('3') |
|
3958 """ |
|
3959 return a.__floordiv__(b, context=self) |
|
3960 |
|
3961 def divmod(self, a, b): |
|
3962 return a.__divmod__(b, context=self) |
|
3963 |
|
3964 def exp(self, a): |
|
3965 """Returns e ** a. |
|
3966 |
|
3967 >>> c = ExtendedContext.copy() |
|
3968 >>> c.Emin = -999 |
|
3969 >>> c.Emax = 999 |
|
3970 >>> c.exp(Decimal('-Infinity')) |
|
3971 Decimal('0') |
|
3972 >>> c.exp(Decimal('-1')) |
|
3973 Decimal('0.367879441') |
|
3974 >>> c.exp(Decimal('0')) |
|
3975 Decimal('1') |
|
3976 >>> c.exp(Decimal('1')) |
|
3977 Decimal('2.71828183') |
|
3978 >>> c.exp(Decimal('0.693147181')) |
|
3979 Decimal('2.00000000') |
|
3980 >>> c.exp(Decimal('+Infinity')) |
|
3981 Decimal('Infinity') |
|
3982 """ |
|
3983 return a.exp(context=self) |
|
3984 |
|
3985 def fma(self, a, b, c): |
|
3986 """Returns a multiplied by b, plus c. |
|
3987 |
|
3988 The first two operands are multiplied together, using multiply, |
|
3989 the third operand is then added to the result of that |
|
3990 multiplication, using add, all with only one final rounding. |
|
3991 |
|
3992 >>> ExtendedContext.fma(Decimal('3'), Decimal('5'), Decimal('7')) |
|
3993 Decimal('22') |
|
3994 >>> ExtendedContext.fma(Decimal('3'), Decimal('-5'), Decimal('7')) |
|
3995 Decimal('-8') |
|
3996 >>> ExtendedContext.fma(Decimal('888565290'), Decimal('1557.96930'), Decimal('-86087.7578')) |
|
3997 Decimal('1.38435736E+12') |
|
3998 """ |
|
3999 return a.fma(b, c, context=self) |
|
4000 |
|
4001 def is_canonical(self, a): |
|
4002 """Return True if the operand is canonical; otherwise return False. |
|
4003 |
|
4004 Currently, the encoding of a Decimal instance is always |
|
4005 canonical, so this method returns True for any Decimal. |
|
4006 |
|
4007 >>> ExtendedContext.is_canonical(Decimal('2.50')) |
|
4008 True |
|
4009 """ |
|
4010 return a.is_canonical() |
|
4011 |
|
4012 def is_finite(self, a): |
|
4013 """Return True if the operand is finite; otherwise return False. |
|
4014 |
|
4015 A Decimal instance is considered finite if it is neither |
|
4016 infinite nor a NaN. |
|
4017 |
|
4018 >>> ExtendedContext.is_finite(Decimal('2.50')) |
|
4019 True |
|
4020 >>> ExtendedContext.is_finite(Decimal('-0.3')) |
|
4021 True |
|
4022 >>> ExtendedContext.is_finite(Decimal('0')) |
|
4023 True |
|
4024 >>> ExtendedContext.is_finite(Decimal('Inf')) |
|
4025 False |
|
4026 >>> ExtendedContext.is_finite(Decimal('NaN')) |
|
4027 False |
|
4028 """ |
|
4029 return a.is_finite() |
|
4030 |
|
4031 def is_infinite(self, a): |
|
4032 """Return True if the operand is infinite; otherwise return False. |
|
4033 |
|
4034 >>> ExtendedContext.is_infinite(Decimal('2.50')) |
|
4035 False |
|
4036 >>> ExtendedContext.is_infinite(Decimal('-Inf')) |
|
4037 True |
|
4038 >>> ExtendedContext.is_infinite(Decimal('NaN')) |
|
4039 False |
|
4040 """ |
|
4041 return a.is_infinite() |
|
4042 |
|
4043 def is_nan(self, a): |
|
4044 """Return True if the operand is a qNaN or sNaN; |
|
4045 otherwise return False. |
|
4046 |
|
4047 >>> ExtendedContext.is_nan(Decimal('2.50')) |
|
4048 False |
|
4049 >>> ExtendedContext.is_nan(Decimal('NaN')) |
|
4050 True |
|
4051 >>> ExtendedContext.is_nan(Decimal('-sNaN')) |
|
4052 True |
|
4053 """ |
|
4054 return a.is_nan() |
|
4055 |
|
4056 def is_normal(self, a): |
|
4057 """Return True if the operand is a normal number; |
|
4058 otherwise return False. |
|
4059 |
|
4060 >>> c = ExtendedContext.copy() |
|
4061 >>> c.Emin = -999 |
|
4062 >>> c.Emax = 999 |
|
4063 >>> c.is_normal(Decimal('2.50')) |
|
4064 True |
|
4065 >>> c.is_normal(Decimal('0.1E-999')) |
|
4066 False |
|
4067 >>> c.is_normal(Decimal('0.00')) |
|
4068 False |
|
4069 >>> c.is_normal(Decimal('-Inf')) |
|
4070 False |
|
4071 >>> c.is_normal(Decimal('NaN')) |
|
4072 False |
|
4073 """ |
|
4074 return a.is_normal(context=self) |
|
4075 |
|
4076 def is_qnan(self, a): |
|
4077 """Return True if the operand is a quiet NaN; otherwise return False. |
|
4078 |
|
4079 >>> ExtendedContext.is_qnan(Decimal('2.50')) |
|
4080 False |
|
4081 >>> ExtendedContext.is_qnan(Decimal('NaN')) |
|
4082 True |
|
4083 >>> ExtendedContext.is_qnan(Decimal('sNaN')) |
|
4084 False |
|
4085 """ |
|
4086 return a.is_qnan() |
|
4087 |
|
4088 def is_signed(self, a): |
|
4089 """Return True if the operand is negative; otherwise return False. |
|
4090 |
|
4091 >>> ExtendedContext.is_signed(Decimal('2.50')) |
|
4092 False |
|
4093 >>> ExtendedContext.is_signed(Decimal('-12')) |
|
4094 True |
|
4095 >>> ExtendedContext.is_signed(Decimal('-0')) |
|
4096 True |
|
4097 """ |
|
4098 return a.is_signed() |
|
4099 |
|
4100 def is_snan(self, a): |
|
4101 """Return True if the operand is a signaling NaN; |
|
4102 otherwise return False. |
|
4103 |
|
4104 >>> ExtendedContext.is_snan(Decimal('2.50')) |
|
4105 False |
|
4106 >>> ExtendedContext.is_snan(Decimal('NaN')) |
|
4107 False |
|
4108 >>> ExtendedContext.is_snan(Decimal('sNaN')) |
|
4109 True |
|
4110 """ |
|
4111 return a.is_snan() |
|
4112 |
|
4113 def is_subnormal(self, a): |
|
4114 """Return True if the operand is subnormal; otherwise return False. |
|
4115 |
|
4116 >>> c = ExtendedContext.copy() |
|
4117 >>> c.Emin = -999 |
|
4118 >>> c.Emax = 999 |
|
4119 >>> c.is_subnormal(Decimal('2.50')) |
|
4120 False |
|
4121 >>> c.is_subnormal(Decimal('0.1E-999')) |
|
4122 True |
|
4123 >>> c.is_subnormal(Decimal('0.00')) |
|
4124 False |
|
4125 >>> c.is_subnormal(Decimal('-Inf')) |
|
4126 False |
|
4127 >>> c.is_subnormal(Decimal('NaN')) |
|
4128 False |
|
4129 """ |
|
4130 return a.is_subnormal(context=self) |
|
4131 |
|
4132 def is_zero(self, a): |
|
4133 """Return True if the operand is a zero; otherwise return False. |
|
4134 |
|
4135 >>> ExtendedContext.is_zero(Decimal('0')) |
|
4136 True |
|
4137 >>> ExtendedContext.is_zero(Decimal('2.50')) |
|
4138 False |
|
4139 >>> ExtendedContext.is_zero(Decimal('-0E+2')) |
|
4140 True |
|
4141 """ |
|
4142 return a.is_zero() |
|
4143 |
|
4144 def ln(self, a): |
|
4145 """Returns the natural (base e) logarithm of the operand. |
|
4146 |
|
4147 >>> c = ExtendedContext.copy() |
|
4148 >>> c.Emin = -999 |
|
4149 >>> c.Emax = 999 |
|
4150 >>> c.ln(Decimal('0')) |
|
4151 Decimal('-Infinity') |
|
4152 >>> c.ln(Decimal('1.000')) |
|
4153 Decimal('0') |
|
4154 >>> c.ln(Decimal('2.71828183')) |
|
4155 Decimal('1.00000000') |
|
4156 >>> c.ln(Decimal('10')) |
|
4157 Decimal('2.30258509') |
|
4158 >>> c.ln(Decimal('+Infinity')) |
|
4159 Decimal('Infinity') |
|
4160 """ |
|
4161 return a.ln(context=self) |
|
4162 |
|
4163 def log10(self, a): |
|
4164 """Returns the base 10 logarithm of the operand. |
|
4165 |
|
4166 >>> c = ExtendedContext.copy() |
|
4167 >>> c.Emin = -999 |
|
4168 >>> c.Emax = 999 |
|
4169 >>> c.log10(Decimal('0')) |
|
4170 Decimal('-Infinity') |
|
4171 >>> c.log10(Decimal('0.001')) |
|
4172 Decimal('-3') |
|
4173 >>> c.log10(Decimal('1.000')) |
|
4174 Decimal('0') |
|
4175 >>> c.log10(Decimal('2')) |
|
4176 Decimal('0.301029996') |
|
4177 >>> c.log10(Decimal('10')) |
|
4178 Decimal('1') |
|
4179 >>> c.log10(Decimal('70')) |
|
4180 Decimal('1.84509804') |
|
4181 >>> c.log10(Decimal('+Infinity')) |
|
4182 Decimal('Infinity') |
|
4183 """ |
|
4184 return a.log10(context=self) |
|
4185 |
|
4186 def logb(self, a): |
|
4187 """ Returns the exponent of the magnitude of the operand's MSD. |
|
4188 |
|
4189 The result is the integer which is the exponent of the magnitude |
|
4190 of the most significant digit of the operand (as though the |
|
4191 operand were truncated to a single digit while maintaining the |
|
4192 value of that digit and without limiting the resulting exponent). |
|
4193 |
|
4194 >>> ExtendedContext.logb(Decimal('250')) |
|
4195 Decimal('2') |
|
4196 >>> ExtendedContext.logb(Decimal('2.50')) |
|
4197 Decimal('0') |
|
4198 >>> ExtendedContext.logb(Decimal('0.03')) |
|
4199 Decimal('-2') |
|
4200 >>> ExtendedContext.logb(Decimal('0')) |
|
4201 Decimal('-Infinity') |
|
4202 """ |
|
4203 return a.logb(context=self) |
|
4204 |
|
4205 def logical_and(self, a, b): |
|
4206 """Applies the logical operation 'and' between each operand's digits. |
|
4207 |
|
4208 The operands must be both logical numbers. |
|
4209 |
|
4210 >>> ExtendedContext.logical_and(Decimal('0'), Decimal('0')) |
|
4211 Decimal('0') |
|
4212 >>> ExtendedContext.logical_and(Decimal('0'), Decimal('1')) |
|
4213 Decimal('0') |
|
4214 >>> ExtendedContext.logical_and(Decimal('1'), Decimal('0')) |
|
4215 Decimal('0') |
|
4216 >>> ExtendedContext.logical_and(Decimal('1'), Decimal('1')) |
|
4217 Decimal('1') |
|
4218 >>> ExtendedContext.logical_and(Decimal('1100'), Decimal('1010')) |
|
4219 Decimal('1000') |
|
4220 >>> ExtendedContext.logical_and(Decimal('1111'), Decimal('10')) |
|
4221 Decimal('10') |
|
4222 """ |
|
4223 return a.logical_and(b, context=self) |
|
4224 |
|
4225 def logical_invert(self, a): |
|
4226 """Invert all the digits in the operand. |
|
4227 |
|
4228 The operand must be a logical number. |
|
4229 |
|
4230 >>> ExtendedContext.logical_invert(Decimal('0')) |
|
4231 Decimal('111111111') |
|
4232 >>> ExtendedContext.logical_invert(Decimal('1')) |
|
4233 Decimal('111111110') |
|
4234 >>> ExtendedContext.logical_invert(Decimal('111111111')) |
|
4235 Decimal('0') |
|
4236 >>> ExtendedContext.logical_invert(Decimal('101010101')) |
|
4237 Decimal('10101010') |
|
4238 """ |
|
4239 return a.logical_invert(context=self) |
|
4240 |
|
4241 def logical_or(self, a, b): |
|
4242 """Applies the logical operation 'or' between each operand's digits. |
|
4243 |
|
4244 The operands must be both logical numbers. |
|
4245 |
|
4246 >>> ExtendedContext.logical_or(Decimal('0'), Decimal('0')) |
|
4247 Decimal('0') |
|
4248 >>> ExtendedContext.logical_or(Decimal('0'), Decimal('1')) |
|
4249 Decimal('1') |
|
4250 >>> ExtendedContext.logical_or(Decimal('1'), Decimal('0')) |
|
4251 Decimal('1') |
|
4252 >>> ExtendedContext.logical_or(Decimal('1'), Decimal('1')) |
|
4253 Decimal('1') |
|
4254 >>> ExtendedContext.logical_or(Decimal('1100'), Decimal('1010')) |
|
4255 Decimal('1110') |
|
4256 >>> ExtendedContext.logical_or(Decimal('1110'), Decimal('10')) |
|
4257 Decimal('1110') |
|
4258 """ |
|
4259 return a.logical_or(b, context=self) |
|
4260 |
|
4261 def logical_xor(self, a, b): |
|
4262 """Applies the logical operation 'xor' between each operand's digits. |
|
4263 |
|
4264 The operands must be both logical numbers. |
|
4265 |
|
4266 >>> ExtendedContext.logical_xor(Decimal('0'), Decimal('0')) |
|
4267 Decimal('0') |
|
4268 >>> ExtendedContext.logical_xor(Decimal('0'), Decimal('1')) |
|
4269 Decimal('1') |
|
4270 >>> ExtendedContext.logical_xor(Decimal('1'), Decimal('0')) |
|
4271 Decimal('1') |
|
4272 >>> ExtendedContext.logical_xor(Decimal('1'), Decimal('1')) |
|
4273 Decimal('0') |
|
4274 >>> ExtendedContext.logical_xor(Decimal('1100'), Decimal('1010')) |
|
4275 Decimal('110') |
|
4276 >>> ExtendedContext.logical_xor(Decimal('1111'), Decimal('10')) |
|
4277 Decimal('1101') |
|
4278 """ |
|
4279 return a.logical_xor(b, context=self) |
|
4280 |
|
4281 def max(self, a,b): |
|
4282 """max compares two values numerically and returns the maximum. |
|
4283 |
|
4284 If either operand is a NaN then the general rules apply. |
|
4285 Otherwise, the operands are compared as though by the compare |
|
4286 operation. If they are numerically equal then the left-hand operand |
|
4287 is chosen as the result. Otherwise the maximum (closer to positive |
|
4288 infinity) of the two operands is chosen as the result. |
|
4289 |
|
4290 >>> ExtendedContext.max(Decimal('3'), Decimal('2')) |
|
4291 Decimal('3') |
|
4292 >>> ExtendedContext.max(Decimal('-10'), Decimal('3')) |
|
4293 Decimal('3') |
|
4294 >>> ExtendedContext.max(Decimal('1.0'), Decimal('1')) |
|
4295 Decimal('1') |
|
4296 >>> ExtendedContext.max(Decimal('7'), Decimal('NaN')) |
|
4297 Decimal('7') |
|
4298 """ |
|
4299 return a.max(b, context=self) |
|
4300 |
|
4301 def max_mag(self, a, b): |
|
4302 """Compares the values numerically with their sign ignored.""" |
|
4303 return a.max_mag(b, context=self) |
|
4304 |
|
4305 def min(self, a,b): |
|
4306 """min compares two values numerically and returns the minimum. |
|
4307 |
|
4308 If either operand is a NaN then the general rules apply. |
|
4309 Otherwise, the operands are compared as though by the compare |
|
4310 operation. If they are numerically equal then the left-hand operand |
|
4311 is chosen as the result. Otherwise the minimum (closer to negative |
|
4312 infinity) of the two operands is chosen as the result. |
|
4313 |
|
4314 >>> ExtendedContext.min(Decimal('3'), Decimal('2')) |
|
4315 Decimal('2') |
|
4316 >>> ExtendedContext.min(Decimal('-10'), Decimal('3')) |
|
4317 Decimal('-10') |
|
4318 >>> ExtendedContext.min(Decimal('1.0'), Decimal('1')) |
|
4319 Decimal('1.0') |
|
4320 >>> ExtendedContext.min(Decimal('7'), Decimal('NaN')) |
|
4321 Decimal('7') |
|
4322 """ |
|
4323 return a.min(b, context=self) |
|
4324 |
|
4325 def min_mag(self, a, b): |
|
4326 """Compares the values numerically with their sign ignored.""" |
|
4327 return a.min_mag(b, context=self) |
|
4328 |
|
4329 def minus(self, a): |
|
4330 """Minus corresponds to unary prefix minus in Python. |
|
4331 |
|
4332 The operation is evaluated using the same rules as subtract; the |
|
4333 operation minus(a) is calculated as subtract('0', a) where the '0' |
|
4334 has the same exponent as the operand. |
|
4335 |
|
4336 >>> ExtendedContext.minus(Decimal('1.3')) |
|
4337 Decimal('-1.3') |
|
4338 >>> ExtendedContext.minus(Decimal('-1.3')) |
|
4339 Decimal('1.3') |
|
4340 """ |
|
4341 return a.__neg__(context=self) |
|
4342 |
|
4343 def multiply(self, a, b): |
|
4344 """multiply multiplies two operands. |
|
4345 |
|
4346 If either operand is a special value then the general rules apply. |
|
4347 Otherwise, the operands are multiplied together ('long multiplication'), |
|
4348 resulting in a number which may be as long as the sum of the lengths |
|
4349 of the two operands. |
|
4350 |
|
4351 >>> ExtendedContext.multiply(Decimal('1.20'), Decimal('3')) |
|
4352 Decimal('3.60') |
|
4353 >>> ExtendedContext.multiply(Decimal('7'), Decimal('3')) |
|
4354 Decimal('21') |
|
4355 >>> ExtendedContext.multiply(Decimal('0.9'), Decimal('0.8')) |
|
4356 Decimal('0.72') |
|
4357 >>> ExtendedContext.multiply(Decimal('0.9'), Decimal('-0')) |
|
4358 Decimal('-0.0') |
|
4359 >>> ExtendedContext.multiply(Decimal('654321'), Decimal('654321')) |
|
4360 Decimal('4.28135971E+11') |
|
4361 """ |
|
4362 return a.__mul__(b, context=self) |
|
4363 |
|
4364 def next_minus(self, a): |
|
4365 """Returns the largest representable number smaller than a. |
|
4366 |
|
4367 >>> c = ExtendedContext.copy() |
|
4368 >>> c.Emin = -999 |
|
4369 >>> c.Emax = 999 |
|
4370 >>> ExtendedContext.next_minus(Decimal('1')) |
|
4371 Decimal('0.999999999') |
|
4372 >>> c.next_minus(Decimal('1E-1007')) |
|
4373 Decimal('0E-1007') |
|
4374 >>> ExtendedContext.next_minus(Decimal('-1.00000003')) |
|
4375 Decimal('-1.00000004') |
|
4376 >>> c.next_minus(Decimal('Infinity')) |
|
4377 Decimal('9.99999999E+999') |
|
4378 """ |
|
4379 return a.next_minus(context=self) |
|
4380 |
|
4381 def next_plus(self, a): |
|
4382 """Returns the smallest representable number larger than a. |
|
4383 |
|
4384 >>> c = ExtendedContext.copy() |
|
4385 >>> c.Emin = -999 |
|
4386 >>> c.Emax = 999 |
|
4387 >>> ExtendedContext.next_plus(Decimal('1')) |
|
4388 Decimal('1.00000001') |
|
4389 >>> c.next_plus(Decimal('-1E-1007')) |
|
4390 Decimal('-0E-1007') |
|
4391 >>> ExtendedContext.next_plus(Decimal('-1.00000003')) |
|
4392 Decimal('-1.00000002') |
|
4393 >>> c.next_plus(Decimal('-Infinity')) |
|
4394 Decimal('-9.99999999E+999') |
|
4395 """ |
|
4396 return a.next_plus(context=self) |
|
4397 |
|
4398 def next_toward(self, a, b): |
|
4399 """Returns the number closest to a, in direction towards b. |
|
4400 |
|
4401 The result is the closest representable number from the first |
|
4402 operand (but not the first operand) that is in the direction |
|
4403 towards the second operand, unless the operands have the same |
|
4404 value. |
|
4405 |
|
4406 >>> c = ExtendedContext.copy() |
|
4407 >>> c.Emin = -999 |
|
4408 >>> c.Emax = 999 |
|
4409 >>> c.next_toward(Decimal('1'), Decimal('2')) |
|
4410 Decimal('1.00000001') |
|
4411 >>> c.next_toward(Decimal('-1E-1007'), Decimal('1')) |
|
4412 Decimal('-0E-1007') |
|
4413 >>> c.next_toward(Decimal('-1.00000003'), Decimal('0')) |
|
4414 Decimal('-1.00000002') |
|
4415 >>> c.next_toward(Decimal('1'), Decimal('0')) |
|
4416 Decimal('0.999999999') |
|
4417 >>> c.next_toward(Decimal('1E-1007'), Decimal('-100')) |
|
4418 Decimal('0E-1007') |
|
4419 >>> c.next_toward(Decimal('-1.00000003'), Decimal('-10')) |
|
4420 Decimal('-1.00000004') |
|
4421 >>> c.next_toward(Decimal('0.00'), Decimal('-0.0000')) |
|
4422 Decimal('-0.00') |
|
4423 """ |
|
4424 return a.next_toward(b, context=self) |
|
4425 |
|
4426 def normalize(self, a): |
|
4427 """normalize reduces an operand to its simplest form. |
|
4428 |
|
4429 Essentially a plus operation with all trailing zeros removed from the |
|
4430 result. |
|
4431 |
|
4432 >>> ExtendedContext.normalize(Decimal('2.1')) |
|
4433 Decimal('2.1') |
|
4434 >>> ExtendedContext.normalize(Decimal('-2.0')) |
|
4435 Decimal('-2') |
|
4436 >>> ExtendedContext.normalize(Decimal('1.200')) |
|
4437 Decimal('1.2') |
|
4438 >>> ExtendedContext.normalize(Decimal('-120')) |
|
4439 Decimal('-1.2E+2') |
|
4440 >>> ExtendedContext.normalize(Decimal('120.00')) |
|
4441 Decimal('1.2E+2') |
|
4442 >>> ExtendedContext.normalize(Decimal('0.00')) |
|
4443 Decimal('0') |
|
4444 """ |
|
4445 return a.normalize(context=self) |
|
4446 |
|
4447 def number_class(self, a): |
|
4448 """Returns an indication of the class of the operand. |
|
4449 |
|
4450 The class is one of the following strings: |
|
4451 -sNaN |
|
4452 -NaN |
|
4453 -Infinity |
|
4454 -Normal |
|
4455 -Subnormal |
|
4456 -Zero |
|
4457 +Zero |
|
4458 +Subnormal |
|
4459 +Normal |
|
4460 +Infinity |
|
4461 |
|
4462 >>> c = Context(ExtendedContext) |
|
4463 >>> c.Emin = -999 |
|
4464 >>> c.Emax = 999 |
|
4465 >>> c.number_class(Decimal('Infinity')) |
|
4466 '+Infinity' |
|
4467 >>> c.number_class(Decimal('1E-10')) |
|
4468 '+Normal' |
|
4469 >>> c.number_class(Decimal('2.50')) |
|
4470 '+Normal' |
|
4471 >>> c.number_class(Decimal('0.1E-999')) |
|
4472 '+Subnormal' |
|
4473 >>> c.number_class(Decimal('0')) |
|
4474 '+Zero' |
|
4475 >>> c.number_class(Decimal('-0')) |
|
4476 '-Zero' |
|
4477 >>> c.number_class(Decimal('-0.1E-999')) |
|
4478 '-Subnormal' |
|
4479 >>> c.number_class(Decimal('-1E-10')) |
|
4480 '-Normal' |
|
4481 >>> c.number_class(Decimal('-2.50')) |
|
4482 '-Normal' |
|
4483 >>> c.number_class(Decimal('-Infinity')) |
|
4484 '-Infinity' |
|
4485 >>> c.number_class(Decimal('NaN')) |
|
4486 'NaN' |
|
4487 >>> c.number_class(Decimal('-NaN')) |
|
4488 'NaN' |
|
4489 >>> c.number_class(Decimal('sNaN')) |
|
4490 'sNaN' |
|
4491 """ |
|
4492 return a.number_class(context=self) |
|
4493 |
|
4494 def plus(self, a): |
|
4495 """Plus corresponds to unary prefix plus in Python. |
|
4496 |
|
4497 The operation is evaluated using the same rules as add; the |
|
4498 operation plus(a) is calculated as add('0', a) where the '0' |
|
4499 has the same exponent as the operand. |
|
4500 |
|
4501 >>> ExtendedContext.plus(Decimal('1.3')) |
|
4502 Decimal('1.3') |
|
4503 >>> ExtendedContext.plus(Decimal('-1.3')) |
|
4504 Decimal('-1.3') |
|
4505 """ |
|
4506 return a.__pos__(context=self) |
|
4507 |
|
4508 def power(self, a, b, modulo=None): |
|
4509 """Raises a to the power of b, to modulo if given. |
|
4510 |
|
4511 With two arguments, compute a**b. If a is negative then b |
|
4512 must be integral. The result will be inexact unless b is |
|
4513 integral and the result is finite and can be expressed exactly |
|
4514 in 'precision' digits. |
|
4515 |
|
4516 With three arguments, compute (a**b) % modulo. For the |
|
4517 three argument form, the following restrictions on the |
|
4518 arguments hold: |
|
4519 |
|
4520 - all three arguments must be integral |
|
4521 - b must be nonnegative |
|
4522 - at least one of a or b must be nonzero |
|
4523 - modulo must be nonzero and have at most 'precision' digits |
|
4524 |
|
4525 The result of pow(a, b, modulo) is identical to the result |
|
4526 that would be obtained by computing (a**b) % modulo with |
|
4527 unbounded precision, but is computed more efficiently. It is |
|
4528 always exact. |
|
4529 |
|
4530 >>> c = ExtendedContext.copy() |
|
4531 >>> c.Emin = -999 |
|
4532 >>> c.Emax = 999 |
|
4533 >>> c.power(Decimal('2'), Decimal('3')) |
|
4534 Decimal('8') |
|
4535 >>> c.power(Decimal('-2'), Decimal('3')) |
|
4536 Decimal('-8') |
|
4537 >>> c.power(Decimal('2'), Decimal('-3')) |
|
4538 Decimal('0.125') |
|
4539 >>> c.power(Decimal('1.7'), Decimal('8')) |
|
4540 Decimal('69.7575744') |
|
4541 >>> c.power(Decimal('10'), Decimal('0.301029996')) |
|
4542 Decimal('2.00000000') |
|
4543 >>> c.power(Decimal('Infinity'), Decimal('-1')) |
|
4544 Decimal('0') |
|
4545 >>> c.power(Decimal('Infinity'), Decimal('0')) |
|
4546 Decimal('1') |
|
4547 >>> c.power(Decimal('Infinity'), Decimal('1')) |
|
4548 Decimal('Infinity') |
|
4549 >>> c.power(Decimal('-Infinity'), Decimal('-1')) |
|
4550 Decimal('-0') |
|
4551 >>> c.power(Decimal('-Infinity'), Decimal('0')) |
|
4552 Decimal('1') |
|
4553 >>> c.power(Decimal('-Infinity'), Decimal('1')) |
|
4554 Decimal('-Infinity') |
|
4555 >>> c.power(Decimal('-Infinity'), Decimal('2')) |
|
4556 Decimal('Infinity') |
|
4557 >>> c.power(Decimal('0'), Decimal('0')) |
|
4558 Decimal('NaN') |
|
4559 |
|
4560 >>> c.power(Decimal('3'), Decimal('7'), Decimal('16')) |
|
4561 Decimal('11') |
|
4562 >>> c.power(Decimal('-3'), Decimal('7'), Decimal('16')) |
|
4563 Decimal('-11') |
|
4564 >>> c.power(Decimal('-3'), Decimal('8'), Decimal('16')) |
|
4565 Decimal('1') |
|
4566 >>> c.power(Decimal('3'), Decimal('7'), Decimal('-16')) |
|
4567 Decimal('11') |
|
4568 >>> c.power(Decimal('23E12345'), Decimal('67E189'), Decimal('123456789')) |
|
4569 Decimal('11729830') |
|
4570 >>> c.power(Decimal('-0'), Decimal('17'), Decimal('1729')) |
|
4571 Decimal('-0') |
|
4572 >>> c.power(Decimal('-23'), Decimal('0'), Decimal('65537')) |
|
4573 Decimal('1') |
|
4574 """ |
|
4575 return a.__pow__(b, modulo, context=self) |
|
4576 |
|
4577 def quantize(self, a, b): |
|
4578 """Returns a value equal to 'a' (rounded), having the exponent of 'b'. |
|
4579 |
|
4580 The coefficient of the result is derived from that of the left-hand |
|
4581 operand. It may be rounded using the current rounding setting (if the |
|
4582 exponent is being increased), multiplied by a positive power of ten (if |
|
4583 the exponent is being decreased), or is unchanged (if the exponent is |
|
4584 already equal to that of the right-hand operand). |
|
4585 |
|
4586 Unlike other operations, if the length of the coefficient after the |
|
4587 quantize operation would be greater than precision then an Invalid |
|
4588 operation condition is raised. This guarantees that, unless there is |
|
4589 an error condition, the exponent of the result of a quantize is always |
|
4590 equal to that of the right-hand operand. |
|
4591 |
|
4592 Also unlike other operations, quantize will never raise Underflow, even |
|
4593 if the result is subnormal and inexact. |
|
4594 |
|
4595 >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.001')) |
|
4596 Decimal('2.170') |
|
4597 >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.01')) |
|
4598 Decimal('2.17') |
|
4599 >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.1')) |
|
4600 Decimal('2.2') |
|
4601 >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('1e+0')) |
|
4602 Decimal('2') |
|
4603 >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('1e+1')) |
|
4604 Decimal('0E+1') |
|
4605 >>> ExtendedContext.quantize(Decimal('-Inf'), Decimal('Infinity')) |
|
4606 Decimal('-Infinity') |
|
4607 >>> ExtendedContext.quantize(Decimal('2'), Decimal('Infinity')) |
|
4608 Decimal('NaN') |
|
4609 >>> ExtendedContext.quantize(Decimal('-0.1'), Decimal('1')) |
|
4610 Decimal('-0') |
|
4611 >>> ExtendedContext.quantize(Decimal('-0'), Decimal('1e+5')) |
|
4612 Decimal('-0E+5') |
|
4613 >>> ExtendedContext.quantize(Decimal('+35236450.6'), Decimal('1e-2')) |
|
4614 Decimal('NaN') |
|
4615 >>> ExtendedContext.quantize(Decimal('-35236450.6'), Decimal('1e-2')) |
|
4616 Decimal('NaN') |
|
4617 >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e-1')) |
|
4618 Decimal('217.0') |
|
4619 >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e-0')) |
|
4620 Decimal('217') |
|
4621 >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e+1')) |
|
4622 Decimal('2.2E+2') |
|
4623 >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e+2')) |
|
4624 Decimal('2E+2') |
|
4625 """ |
|
4626 return a.quantize(b, context=self) |
|
4627 |
|
4628 def radix(self): |
|
4629 """Just returns 10, as this is Decimal, :) |
|
4630 |
|
4631 >>> ExtendedContext.radix() |
|
4632 Decimal('10') |
|
4633 """ |
|
4634 return Decimal(10) |
|
4635 |
|
4636 def remainder(self, a, b): |
|
4637 """Returns the remainder from integer division. |
|
4638 |
|
4639 The result is the residue of the dividend after the operation of |
|
4640 calculating integer division as described for divide-integer, rounded |
|
4641 to precision digits if necessary. The sign of the result, if |
|
4642 non-zero, is the same as that of the original dividend. |
|
4643 |
|
4644 This operation will fail under the same conditions as integer division |
|
4645 (that is, if integer division on the same two operands would fail, the |
|
4646 remainder cannot be calculated). |
|
4647 |
|
4648 >>> ExtendedContext.remainder(Decimal('2.1'), Decimal('3')) |
|
4649 Decimal('2.1') |
|
4650 >>> ExtendedContext.remainder(Decimal('10'), Decimal('3')) |
|
4651 Decimal('1') |
|
4652 >>> ExtendedContext.remainder(Decimal('-10'), Decimal('3')) |
|
4653 Decimal('-1') |
|
4654 >>> ExtendedContext.remainder(Decimal('10.2'), Decimal('1')) |
|
4655 Decimal('0.2') |
|
4656 >>> ExtendedContext.remainder(Decimal('10'), Decimal('0.3')) |
|
4657 Decimal('0.1') |
|
4658 >>> ExtendedContext.remainder(Decimal('3.6'), Decimal('1.3')) |
|
4659 Decimal('1.0') |
|
4660 """ |
|
4661 return a.__mod__(b, context=self) |
|
4662 |
|
4663 def remainder_near(self, a, b): |
|
4664 """Returns to be "a - b * n", where n is the integer nearest the exact |
|
4665 value of "x / b" (if two integers are equally near then the even one |
|
4666 is chosen). If the result is equal to 0 then its sign will be the |
|
4667 sign of a. |
|
4668 |
|
4669 This operation will fail under the same conditions as integer division |
|
4670 (that is, if integer division on the same two operands would fail, the |
|
4671 remainder cannot be calculated). |
|
4672 |
|
4673 >>> ExtendedContext.remainder_near(Decimal('2.1'), Decimal('3')) |
|
4674 Decimal('-0.9') |
|
4675 >>> ExtendedContext.remainder_near(Decimal('10'), Decimal('6')) |
|
4676 Decimal('-2') |
|
4677 >>> ExtendedContext.remainder_near(Decimal('10'), Decimal('3')) |
|
4678 Decimal('1') |
|
4679 >>> ExtendedContext.remainder_near(Decimal('-10'), Decimal('3')) |
|
4680 Decimal('-1') |
|
4681 >>> ExtendedContext.remainder_near(Decimal('10.2'), Decimal('1')) |
|
4682 Decimal('0.2') |
|
4683 >>> ExtendedContext.remainder_near(Decimal('10'), Decimal('0.3')) |
|
4684 Decimal('0.1') |
|
4685 >>> ExtendedContext.remainder_near(Decimal('3.6'), Decimal('1.3')) |
|
4686 Decimal('-0.3') |
|
4687 """ |
|
4688 return a.remainder_near(b, context=self) |
|
4689 |
|
4690 def rotate(self, a, b): |
|
4691 """Returns a rotated copy of a, b times. |
|
4692 |
|
4693 The coefficient of the result is a rotated copy of the digits in |
|
4694 the coefficient of the first operand. The number of places of |
|
4695 rotation is taken from the absolute value of the second operand, |
|
4696 with the rotation being to the left if the second operand is |
|
4697 positive or to the right otherwise. |
|
4698 |
|
4699 >>> ExtendedContext.rotate(Decimal('34'), Decimal('8')) |
|
4700 Decimal('400000003') |
|
4701 >>> ExtendedContext.rotate(Decimal('12'), Decimal('9')) |
|
4702 Decimal('12') |
|
4703 >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('-2')) |
|
4704 Decimal('891234567') |
|
4705 >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('0')) |
|
4706 Decimal('123456789') |
|
4707 >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('+2')) |
|
4708 Decimal('345678912') |
|
4709 """ |
|
4710 return a.rotate(b, context=self) |
|
4711 |
|
4712 def same_quantum(self, a, b): |
|
4713 """Returns True if the two operands have the same exponent. |
|
4714 |
|
4715 The result is never affected by either the sign or the coefficient of |
|
4716 either operand. |
|
4717 |
|
4718 >>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('0.001')) |
|
4719 False |
|
4720 >>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('0.01')) |
|
4721 True |
|
4722 >>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('1')) |
|
4723 False |
|
4724 >>> ExtendedContext.same_quantum(Decimal('Inf'), Decimal('-Inf')) |
|
4725 True |
|
4726 """ |
|
4727 return a.same_quantum(b) |
|
4728 |
|
4729 def scaleb (self, a, b): |
|
4730 """Returns the first operand after adding the second value its exp. |
|
4731 |
|
4732 >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('-2')) |
|
4733 Decimal('0.0750') |
|
4734 >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('0')) |
|
4735 Decimal('7.50') |
|
4736 >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('3')) |
|
4737 Decimal('7.50E+3') |
|
4738 """ |
|
4739 return a.scaleb (b, context=self) |
|
4740 |
|
4741 def shift(self, a, b): |
|
4742 """Returns a shifted copy of a, b times. |
|
4743 |
|
4744 The coefficient of the result is a shifted copy of the digits |
|
4745 in the coefficient of the first operand. The number of places |
|
4746 to shift is taken from the absolute value of the second operand, |
|
4747 with the shift being to the left if the second operand is |
|
4748 positive or to the right otherwise. Digits shifted into the |
|
4749 coefficient are zeros. |
|
4750 |
|
4751 >>> ExtendedContext.shift(Decimal('34'), Decimal('8')) |
|
4752 Decimal('400000000') |
|
4753 >>> ExtendedContext.shift(Decimal('12'), Decimal('9')) |
|
4754 Decimal('0') |
|
4755 >>> ExtendedContext.shift(Decimal('123456789'), Decimal('-2')) |
|
4756 Decimal('1234567') |
|
4757 >>> ExtendedContext.shift(Decimal('123456789'), Decimal('0')) |
|
4758 Decimal('123456789') |
|
4759 >>> ExtendedContext.shift(Decimal('123456789'), Decimal('+2')) |
|
4760 Decimal('345678900') |
|
4761 """ |
|
4762 return a.shift(b, context=self) |
|
4763 |
|
4764 def sqrt(self, a): |
|
4765 """Square root of a non-negative number to context precision. |
|
4766 |
|
4767 If the result must be inexact, it is rounded using the round-half-even |
|
4768 algorithm. |
|
4769 |
|
4770 >>> ExtendedContext.sqrt(Decimal('0')) |
|
4771 Decimal('0') |
|
4772 >>> ExtendedContext.sqrt(Decimal('-0')) |
|
4773 Decimal('-0') |
|
4774 >>> ExtendedContext.sqrt(Decimal('0.39')) |
|
4775 Decimal('0.624499800') |
|
4776 >>> ExtendedContext.sqrt(Decimal('100')) |
|
4777 Decimal('10') |
|
4778 >>> ExtendedContext.sqrt(Decimal('1')) |
|
4779 Decimal('1') |
|
4780 >>> ExtendedContext.sqrt(Decimal('1.0')) |
|
4781 Decimal('1.0') |
|
4782 >>> ExtendedContext.sqrt(Decimal('1.00')) |
|
4783 Decimal('1.0') |
|
4784 >>> ExtendedContext.sqrt(Decimal('7')) |
|
4785 Decimal('2.64575131') |
|
4786 >>> ExtendedContext.sqrt(Decimal('10')) |
|
4787 Decimal('3.16227766') |
|
4788 >>> ExtendedContext.prec |
|
4789 9 |
|
4790 """ |
|
4791 return a.sqrt(context=self) |
|
4792 |
|
4793 def subtract(self, a, b): |
|
4794 """Return the difference between the two operands. |
|
4795 |
|
4796 >>> ExtendedContext.subtract(Decimal('1.3'), Decimal('1.07')) |
|
4797 Decimal('0.23') |
|
4798 >>> ExtendedContext.subtract(Decimal('1.3'), Decimal('1.30')) |
|
4799 Decimal('0.00') |
|
4800 >>> ExtendedContext.subtract(Decimal('1.3'), Decimal('2.07')) |
|
4801 Decimal('-0.77') |
|
4802 """ |
|
4803 return a.__sub__(b, context=self) |
|
4804 |
|
4805 def to_eng_string(self, a): |
|
4806 """Converts a number to a string, using scientific notation. |
|
4807 |
|
4808 The operation is not affected by the context. |
|
4809 """ |
|
4810 return a.to_eng_string(context=self) |
|
4811 |
|
4812 def to_sci_string(self, a): |
|
4813 """Converts a number to a string, using scientific notation. |
|
4814 |
|
4815 The operation is not affected by the context. |
|
4816 """ |
|
4817 return a.__str__(context=self) |
|
4818 |
|
4819 def to_integral_exact(self, a): |
|
4820 """Rounds to an integer. |
|
4821 |
|
4822 When the operand has a negative exponent, the result is the same |
|
4823 as using the quantize() operation using the given operand as the |
|
4824 left-hand-operand, 1E+0 as the right-hand-operand, and the precision |
|
4825 of the operand as the precision setting; Inexact and Rounded flags |
|
4826 are allowed in this operation. The rounding mode is taken from the |
|
4827 context. |
|
4828 |
|
4829 >>> ExtendedContext.to_integral_exact(Decimal('2.1')) |
|
4830 Decimal('2') |
|
4831 >>> ExtendedContext.to_integral_exact(Decimal('100')) |
|
4832 Decimal('100') |
|
4833 >>> ExtendedContext.to_integral_exact(Decimal('100.0')) |
|
4834 Decimal('100') |
|
4835 >>> ExtendedContext.to_integral_exact(Decimal('101.5')) |
|
4836 Decimal('102') |
|
4837 >>> ExtendedContext.to_integral_exact(Decimal('-101.5')) |
|
4838 Decimal('-102') |
|
4839 >>> ExtendedContext.to_integral_exact(Decimal('10E+5')) |
|
4840 Decimal('1.0E+6') |
|
4841 >>> ExtendedContext.to_integral_exact(Decimal('7.89E+77')) |
|
4842 Decimal('7.89E+77') |
|
4843 >>> ExtendedContext.to_integral_exact(Decimal('-Inf')) |
|
4844 Decimal('-Infinity') |
|
4845 """ |
|
4846 return a.to_integral_exact(context=self) |
|
4847 |
|
4848 def to_integral_value(self, a): |
|
4849 """Rounds to an integer. |
|
4850 |
|
4851 When the operand has a negative exponent, the result is the same |
|
4852 as using the quantize() operation using the given operand as the |
|
4853 left-hand-operand, 1E+0 as the right-hand-operand, and the precision |
|
4854 of the operand as the precision setting, except that no flags will |
|
4855 be set. The rounding mode is taken from the context. |
|
4856 |
|
4857 >>> ExtendedContext.to_integral_value(Decimal('2.1')) |
|
4858 Decimal('2') |
|
4859 >>> ExtendedContext.to_integral_value(Decimal('100')) |
|
4860 Decimal('100') |
|
4861 >>> ExtendedContext.to_integral_value(Decimal('100.0')) |
|
4862 Decimal('100') |
|
4863 >>> ExtendedContext.to_integral_value(Decimal('101.5')) |
|
4864 Decimal('102') |
|
4865 >>> ExtendedContext.to_integral_value(Decimal('-101.5')) |
|
4866 Decimal('-102') |
|
4867 >>> ExtendedContext.to_integral_value(Decimal('10E+5')) |
|
4868 Decimal('1.0E+6') |
|
4869 >>> ExtendedContext.to_integral_value(Decimal('7.89E+77')) |
|
4870 Decimal('7.89E+77') |
|
4871 >>> ExtendedContext.to_integral_value(Decimal('-Inf')) |
|
4872 Decimal('-Infinity') |
|
4873 """ |
|
4874 return a.to_integral_value(context=self) |
|
4875 |
|
4876 # the method name changed, but we provide also the old one, for compatibility |
|
4877 to_integral = to_integral_value |
|
4878 |
|
4879 class _WorkRep(object): |
|
4880 __slots__ = ('sign','int','exp') |
|
4881 # sign: 0 or 1 |
|
4882 # int: int or long |
|
4883 # exp: None, int, or string |
|
4884 |
|
4885 def __init__(self, value=None): |
|
4886 if value is None: |
|
4887 self.sign = None |
|
4888 self.int = 0 |
|
4889 self.exp = None |
|
4890 elif isinstance(value, Decimal): |
|
4891 self.sign = value._sign |
|
4892 self.int = int(value._int) |
|
4893 self.exp = value._exp |
|
4894 else: |
|
4895 # assert isinstance(value, tuple) |
|
4896 self.sign = value[0] |
|
4897 self.int = value[1] |
|
4898 self.exp = value[2] |
|
4899 |
|
4900 def __repr__(self): |
|
4901 return "(%r, %r, %r)" % (self.sign, self.int, self.exp) |
|
4902 |
|
4903 __str__ = __repr__ |
|
4904 |
|
4905 |
|
4906 |
|
4907 def _normalize(op1, op2, prec = 0): |
|
4908 """Normalizes op1, op2 to have the same exp and length of coefficient. |
|
4909 |
|
4910 Done during addition. |
|
4911 """ |
|
4912 if op1.exp < op2.exp: |
|
4913 tmp = op2 |
|
4914 other = op1 |
|
4915 else: |
|
4916 tmp = op1 |
|
4917 other = op2 |
|
4918 |
|
4919 # Let exp = min(tmp.exp - 1, tmp.adjusted() - precision - 1). |
|
4920 # Then adding 10**exp to tmp has the same effect (after rounding) |
|
4921 # as adding any positive quantity smaller than 10**exp; similarly |
|
4922 # for subtraction. So if other is smaller than 10**exp we replace |
|
4923 # it with 10**exp. This avoids tmp.exp - other.exp getting too large. |
|
4924 tmp_len = len(str(tmp.int)) |
|
4925 other_len = len(str(other.int)) |
|
4926 exp = tmp.exp + min(-1, tmp_len - prec - 2) |
|
4927 if other_len + other.exp - 1 < exp: |
|
4928 other.int = 1 |
|
4929 other.exp = exp |
|
4930 |
|
4931 tmp.int *= 10 ** (tmp.exp - other.exp) |
|
4932 tmp.exp = other.exp |
|
4933 return op1, op2 |
|
4934 |
|
4935 ##### Integer arithmetic functions used by ln, log10, exp and __pow__ ##### |
|
4936 |
|
4937 # This function from Tim Peters was taken from here: |
|
4938 # http://mail.python.org/pipermail/python-list/1999-July/007758.html |
|
4939 # The correction being in the function definition is for speed, and |
|
4940 # the whole function is not resolved with math.log because of avoiding |
|
4941 # the use of floats. |
|
4942 def _nbits(n, correction = { |
|
4943 '0': 4, '1': 3, '2': 2, '3': 2, |
|
4944 '4': 1, '5': 1, '6': 1, '7': 1, |
|
4945 '8': 0, '9': 0, 'a': 0, 'b': 0, |
|
4946 'c': 0, 'd': 0, 'e': 0, 'f': 0}): |
|
4947 """Number of bits in binary representation of the positive integer n, |
|
4948 or 0 if n == 0. |
|
4949 """ |
|
4950 if n < 0: |
|
4951 raise ValueError("The argument to _nbits should be nonnegative.") |
|
4952 hex_n = "%x" % n |
|
4953 return 4*len(hex_n) - correction[hex_n[0]] |
|
4954 |
|
4955 def _sqrt_nearest(n, a): |
|
4956 """Closest integer to the square root of the positive integer n. a is |
|
4957 an initial approximation to the square root. Any positive integer |
|
4958 will do for a, but the closer a is to the square root of n the |
|
4959 faster convergence will be. |
|
4960 |
|
4961 """ |
|
4962 if n <= 0 or a <= 0: |
|
4963 raise ValueError("Both arguments to _sqrt_nearest should be positive.") |
|
4964 |
|
4965 b=0 |
|
4966 while a != b: |
|
4967 b, a = a, a--n//a>>1 |
|
4968 return a |
|
4969 |
|
4970 def _rshift_nearest(x, shift): |
|
4971 """Given an integer x and a nonnegative integer shift, return closest |
|
4972 integer to x / 2**shift; use round-to-even in case of a tie. |
|
4973 |
|
4974 """ |
|
4975 b, q = 1L << shift, x >> shift |
|
4976 return q + (2*(x & (b-1)) + (q&1) > b) |
|
4977 |
|
4978 def _div_nearest(a, b): |
|
4979 """Closest integer to a/b, a and b positive integers; rounds to even |
|
4980 in the case of a tie. |
|
4981 |
|
4982 """ |
|
4983 q, r = divmod(a, b) |
|
4984 return q + (2*r + (q&1) > b) |
|
4985 |
|
4986 def _ilog(x, M, L = 8): |
|
4987 """Integer approximation to M*log(x/M), with absolute error boundable |
|
4988 in terms only of x/M. |
|
4989 |
|
4990 Given positive integers x and M, return an integer approximation to |
|
4991 M * log(x/M). For L = 8 and 0.1 <= x/M <= 10 the difference |
|
4992 between the approximation and the exact result is at most 22. For |
|
4993 L = 8 and 1.0 <= x/M <= 10.0 the difference is at most 15. In |
|
4994 both cases these are upper bounds on the error; it will usually be |
|
4995 much smaller.""" |
|
4996 |
|
4997 # The basic algorithm is the following: let log1p be the function |
|
4998 # log1p(x) = log(1+x). Then log(x/M) = log1p((x-M)/M). We use |
|
4999 # the reduction |
|
5000 # |
|
5001 # log1p(y) = 2*log1p(y/(1+sqrt(1+y))) |
|
5002 # |
|
5003 # repeatedly until the argument to log1p is small (< 2**-L in |
|
5004 # absolute value). For small y we can use the Taylor series |
|
5005 # expansion |
|
5006 # |
|
5007 # log1p(y) ~ y - y**2/2 + y**3/3 - ... - (-y)**T/T |
|
5008 # |
|
5009 # truncating at T such that y**T is small enough. The whole |
|
5010 # computation is carried out in a form of fixed-point arithmetic, |
|
5011 # with a real number z being represented by an integer |
|
5012 # approximation to z*M. To avoid loss of precision, the y below |
|
5013 # is actually an integer approximation to 2**R*y*M, where R is the |
|
5014 # number of reductions performed so far. |
|
5015 |
|
5016 y = x-M |
|
5017 # argument reduction; R = number of reductions performed |
|
5018 R = 0 |
|
5019 while (R <= L and long(abs(y)) << L-R >= M or |
|
5020 R > L and abs(y) >> R-L >= M): |
|
5021 y = _div_nearest(long(M*y) << 1, |
|
5022 M + _sqrt_nearest(M*(M+_rshift_nearest(y, R)), M)) |
|
5023 R += 1 |
|
5024 |
|
5025 # Taylor series with T terms |
|
5026 T = -int(-10*len(str(M))//(3*L)) |
|
5027 yshift = _rshift_nearest(y, R) |
|
5028 w = _div_nearest(M, T) |
|
5029 for k in xrange(T-1, 0, -1): |
|
5030 w = _div_nearest(M, k) - _div_nearest(yshift*w, M) |
|
5031 |
|
5032 return _div_nearest(w*y, M) |
|
5033 |
|
5034 def _dlog10(c, e, p): |
|
5035 """Given integers c, e and p with c > 0, p >= 0, compute an integer |
|
5036 approximation to 10**p * log10(c*10**e), with an absolute error of |
|
5037 at most 1. Assumes that c*10**e is not exactly 1.""" |
|
5038 |
|
5039 # increase precision by 2; compensate for this by dividing |
|
5040 # final result by 100 |
|
5041 p += 2 |
|
5042 |
|
5043 # write c*10**e as d*10**f with either: |
|
5044 # f >= 0 and 1 <= d <= 10, or |
|
5045 # f <= 0 and 0.1 <= d <= 1. |
|
5046 # Thus for c*10**e close to 1, f = 0 |
|
5047 l = len(str(c)) |
|
5048 f = e+l - (e+l >= 1) |
|
5049 |
|
5050 if p > 0: |
|
5051 M = 10**p |
|
5052 k = e+p-f |
|
5053 if k >= 0: |
|
5054 c *= 10**k |
|
5055 else: |
|
5056 c = _div_nearest(c, 10**-k) |
|
5057 |
|
5058 log_d = _ilog(c, M) # error < 5 + 22 = 27 |
|
5059 log_10 = _log10_digits(p) # error < 1 |
|
5060 log_d = _div_nearest(log_d*M, log_10) |
|
5061 log_tenpower = f*M # exact |
|
5062 else: |
|
5063 log_d = 0 # error < 2.31 |
|
5064 log_tenpower = _div_nearest(f, 10**-p) # error < 0.5 |
|
5065 |
|
5066 return _div_nearest(log_tenpower+log_d, 100) |
|
5067 |
|
5068 def _dlog(c, e, p): |
|
5069 """Given integers c, e and p with c > 0, compute an integer |
|
5070 approximation to 10**p * log(c*10**e), with an absolute error of |
|
5071 at most 1. Assumes that c*10**e is not exactly 1.""" |
|
5072 |
|
5073 # Increase precision by 2. The precision increase is compensated |
|
5074 # for at the end with a division by 100. |
|
5075 p += 2 |
|
5076 |
|
5077 # rewrite c*10**e as d*10**f with either f >= 0 and 1 <= d <= 10, |
|
5078 # or f <= 0 and 0.1 <= d <= 1. Then we can compute 10**p * log(c*10**e) |
|
5079 # as 10**p * log(d) + 10**p*f * log(10). |
|
5080 l = len(str(c)) |
|
5081 f = e+l - (e+l >= 1) |
|
5082 |
|
5083 # compute approximation to 10**p*log(d), with error < 27 |
|
5084 if p > 0: |
|
5085 k = e+p-f |
|
5086 if k >= 0: |
|
5087 c *= 10**k |
|
5088 else: |
|
5089 c = _div_nearest(c, 10**-k) # error of <= 0.5 in c |
|
5090 |
|
5091 # _ilog magnifies existing error in c by a factor of at most 10 |
|
5092 log_d = _ilog(c, 10**p) # error < 5 + 22 = 27 |
|
5093 else: |
|
5094 # p <= 0: just approximate the whole thing by 0; error < 2.31 |
|
5095 log_d = 0 |
|
5096 |
|
5097 # compute approximation to f*10**p*log(10), with error < 11. |
|
5098 if f: |
|
5099 extra = len(str(abs(f)))-1 |
|
5100 if p + extra >= 0: |
|
5101 # error in f * _log10_digits(p+extra) < |f| * 1 = |f| |
|
5102 # after division, error < |f|/10**extra + 0.5 < 10 + 0.5 < 11 |
|
5103 f_log_ten = _div_nearest(f*_log10_digits(p+extra), 10**extra) |
|
5104 else: |
|
5105 f_log_ten = 0 |
|
5106 else: |
|
5107 f_log_ten = 0 |
|
5108 |
|
5109 # error in sum < 11+27 = 38; error after division < 0.38 + 0.5 < 1 |
|
5110 return _div_nearest(f_log_ten + log_d, 100) |
|
5111 |
|
5112 class _Log10Memoize(object): |
|
5113 """Class to compute, store, and allow retrieval of, digits of the |
|
5114 constant log(10) = 2.302585.... This constant is needed by |
|
5115 Decimal.ln, Decimal.log10, Decimal.exp and Decimal.__pow__.""" |
|
5116 def __init__(self): |
|
5117 self.digits = "23025850929940456840179914546843642076011014886" |
|
5118 |
|
5119 def getdigits(self, p): |
|
5120 """Given an integer p >= 0, return floor(10**p)*log(10). |
|
5121 |
|
5122 For example, self.getdigits(3) returns 2302. |
|
5123 """ |
|
5124 # digits are stored as a string, for quick conversion to |
|
5125 # integer in the case that we've already computed enough |
|
5126 # digits; the stored digits should always be correct |
|
5127 # (truncated, not rounded to nearest). |
|
5128 if p < 0: |
|
5129 raise ValueError("p should be nonnegative") |
|
5130 |
|
5131 if p >= len(self.digits): |
|
5132 # compute p+3, p+6, p+9, ... digits; continue until at |
|
5133 # least one of the extra digits is nonzero |
|
5134 extra = 3 |
|
5135 while True: |
|
5136 # compute p+extra digits, correct to within 1ulp |
|
5137 M = 10**(p+extra+2) |
|
5138 digits = str(_div_nearest(_ilog(10*M, M), 100)) |
|
5139 if digits[-extra:] != '0'*extra: |
|
5140 break |
|
5141 extra += 3 |
|
5142 # keep all reliable digits so far; remove trailing zeros |
|
5143 # and next nonzero digit |
|
5144 self.digits = digits.rstrip('0')[:-1] |
|
5145 return int(self.digits[:p+1]) |
|
5146 |
|
5147 _log10_digits = _Log10Memoize().getdigits |
|
5148 |
|
5149 def _iexp(x, M, L=8): |
|
5150 """Given integers x and M, M > 0, such that x/M is small in absolute |
|
5151 value, compute an integer approximation to M*exp(x/M). For 0 <= |
|
5152 x/M <= 2.4, the absolute error in the result is bounded by 60 (and |
|
5153 is usually much smaller).""" |
|
5154 |
|
5155 # Algorithm: to compute exp(z) for a real number z, first divide z |
|
5156 # by a suitable power R of 2 so that |z/2**R| < 2**-L. Then |
|
5157 # compute expm1(z/2**R) = exp(z/2**R) - 1 using the usual Taylor |
|
5158 # series |
|
5159 # |
|
5160 # expm1(x) = x + x**2/2! + x**3/3! + ... |
|
5161 # |
|
5162 # Now use the identity |
|
5163 # |
|
5164 # expm1(2x) = expm1(x)*(expm1(x)+2) |
|
5165 # |
|
5166 # R times to compute the sequence expm1(z/2**R), |
|
5167 # expm1(z/2**(R-1)), ... , exp(z/2), exp(z). |
|
5168 |
|
5169 # Find R such that x/2**R/M <= 2**-L |
|
5170 R = _nbits((long(x)<<L)//M) |
|
5171 |
|
5172 # Taylor series. (2**L)**T > M |
|
5173 T = -int(-10*len(str(M))//(3*L)) |
|
5174 y = _div_nearest(x, T) |
|
5175 Mshift = long(M)<<R |
|
5176 for i in xrange(T-1, 0, -1): |
|
5177 y = _div_nearest(x*(Mshift + y), Mshift * i) |
|
5178 |
|
5179 # Expansion |
|
5180 for k in xrange(R-1, -1, -1): |
|
5181 Mshift = long(M)<<(k+2) |
|
5182 y = _div_nearest(y*(y+Mshift), Mshift) |
|
5183 |
|
5184 return M+y |
|
5185 |
|
5186 def _dexp(c, e, p): |
|
5187 """Compute an approximation to exp(c*10**e), with p decimal places of |
|
5188 precision. |
|
5189 |
|
5190 Returns integers d, f such that: |
|
5191 |
|
5192 10**(p-1) <= d <= 10**p, and |
|
5193 (d-1)*10**f < exp(c*10**e) < (d+1)*10**f |
|
5194 |
|
5195 In other words, d*10**f is an approximation to exp(c*10**e) with p |
|
5196 digits of precision, and with an error in d of at most 1. This is |
|
5197 almost, but not quite, the same as the error being < 1ulp: when d |
|
5198 = 10**(p-1) the error could be up to 10 ulp.""" |
|
5199 |
|
5200 # we'll call iexp with M = 10**(p+2), giving p+3 digits of precision |
|
5201 p += 2 |
|
5202 |
|
5203 # compute log(10) with extra precision = adjusted exponent of c*10**e |
|
5204 extra = max(0, e + len(str(c)) - 1) |
|
5205 q = p + extra |
|
5206 |
|
5207 # compute quotient c*10**e/(log(10)) = c*10**(e+q)/(log(10)*10**q), |
|
5208 # rounding down |
|
5209 shift = e+q |
|
5210 if shift >= 0: |
|
5211 cshift = c*10**shift |
|
5212 else: |
|
5213 cshift = c//10**-shift |
|
5214 quot, rem = divmod(cshift, _log10_digits(q)) |
|
5215 |
|
5216 # reduce remainder back to original precision |
|
5217 rem = _div_nearest(rem, 10**extra) |
|
5218 |
|
5219 # error in result of _iexp < 120; error after division < 0.62 |
|
5220 return _div_nearest(_iexp(rem, 10**p), 1000), quot - p + 3 |
|
5221 |
|
5222 def _dpower(xc, xe, yc, ye, p): |
|
5223 """Given integers xc, xe, yc and ye representing Decimals x = xc*10**xe and |
|
5224 y = yc*10**ye, compute x**y. Returns a pair of integers (c, e) such that: |
|
5225 |
|
5226 10**(p-1) <= c <= 10**p, and |
|
5227 (c-1)*10**e < x**y < (c+1)*10**e |
|
5228 |
|
5229 in other words, c*10**e is an approximation to x**y with p digits |
|
5230 of precision, and with an error in c of at most 1. (This is |
|
5231 almost, but not quite, the same as the error being < 1ulp: when c |
|
5232 == 10**(p-1) we can only guarantee error < 10ulp.) |
|
5233 |
|
5234 We assume that: x is positive and not equal to 1, and y is nonzero. |
|
5235 """ |
|
5236 |
|
5237 # Find b such that 10**(b-1) <= |y| <= 10**b |
|
5238 b = len(str(abs(yc))) + ye |
|
5239 |
|
5240 # log(x) = lxc*10**(-p-b-1), to p+b+1 places after the decimal point |
|
5241 lxc = _dlog(xc, xe, p+b+1) |
|
5242 |
|
5243 # compute product y*log(x) = yc*lxc*10**(-p-b-1+ye) = pc*10**(-p-1) |
|
5244 shift = ye-b |
|
5245 if shift >= 0: |
|
5246 pc = lxc*yc*10**shift |
|
5247 else: |
|
5248 pc = _div_nearest(lxc*yc, 10**-shift) |
|
5249 |
|
5250 if pc == 0: |
|
5251 # we prefer a result that isn't exactly 1; this makes it |
|
5252 # easier to compute a correctly rounded result in __pow__ |
|
5253 if ((len(str(xc)) + xe >= 1) == (yc > 0)): # if x**y > 1: |
|
5254 coeff, exp = 10**(p-1)+1, 1-p |
|
5255 else: |
|
5256 coeff, exp = 10**p-1, -p |
|
5257 else: |
|
5258 coeff, exp = _dexp(pc, -(p+1), p+1) |
|
5259 coeff = _div_nearest(coeff, 10) |
|
5260 exp += 1 |
|
5261 |
|
5262 return coeff, exp |
|
5263 |
|
5264 def _log10_lb(c, correction = { |
|
5265 '1': 100, '2': 70, '3': 53, '4': 40, '5': 31, |
|
5266 '6': 23, '7': 16, '8': 10, '9': 5}): |
|
5267 """Compute a lower bound for 100*log10(c) for a positive integer c.""" |
|
5268 if c <= 0: |
|
5269 raise ValueError("The argument to _log10_lb should be nonnegative.") |
|
5270 str_c = str(c) |
|
5271 return 100*len(str_c) - correction[str_c[0]] |
|
5272 |
|
5273 ##### Helper Functions #################################################### |
|
5274 |
|
5275 def _convert_other(other, raiseit=False): |
|
5276 """Convert other to Decimal. |
|
5277 |
|
5278 Verifies that it's ok to use in an implicit construction. |
|
5279 """ |
|
5280 if isinstance(other, Decimal): |
|
5281 return other |
|
5282 if isinstance(other, (int, long)): |
|
5283 return Decimal(other) |
|
5284 if raiseit: |
|
5285 raise TypeError("Unable to convert %s to Decimal" % other) |
|
5286 return NotImplemented |
|
5287 |
|
5288 ##### Setup Specific Contexts ############################################ |
|
5289 |
|
5290 # The default context prototype used by Context() |
|
5291 # Is mutable, so that new contexts can have different default values |
|
5292 |
|
5293 DefaultContext = Context( |
|
5294 prec=28, rounding=ROUND_HALF_EVEN, |
|
5295 traps=[DivisionByZero, Overflow, InvalidOperation], |
|
5296 flags=[], |
|
5297 Emax=999999999, |
|
5298 Emin=-999999999, |
|
5299 capitals=1 |
|
5300 ) |
|
5301 |
|
5302 # Pre-made alternate contexts offered by the specification |
|
5303 # Don't change these; the user should be able to select these |
|
5304 # contexts and be able to reproduce results from other implementations |
|
5305 # of the spec. |
|
5306 |
|
5307 BasicContext = Context( |
|
5308 prec=9, rounding=ROUND_HALF_UP, |
|
5309 traps=[DivisionByZero, Overflow, InvalidOperation, Clamped, Underflow], |
|
5310 flags=[], |
|
5311 ) |
|
5312 |
|
5313 ExtendedContext = Context( |
|
5314 prec=9, rounding=ROUND_HALF_EVEN, |
|
5315 traps=[], |
|
5316 flags=[], |
|
5317 ) |
|
5318 |
|
5319 |
|
5320 ##### crud for parsing strings ############################################# |
|
5321 # |
|
5322 # Regular expression used for parsing numeric strings. Additional |
|
5323 # comments: |
|
5324 # |
|
5325 # 1. Uncomment the two '\s*' lines to allow leading and/or trailing |
|
5326 # whitespace. But note that the specification disallows whitespace in |
|
5327 # a numeric string. |
|
5328 # |
|
5329 # 2. For finite numbers (not infinities and NaNs) the body of the |
|
5330 # number between the optional sign and the optional exponent must have |
|
5331 # at least one decimal digit, possibly after the decimal point. The |
|
5332 # lookahead expression '(?=\d|\.\d)' checks this. |
|
5333 # |
|
5334 # As the flag UNICODE is not enabled here, we're explicitly avoiding any |
|
5335 # other meaning for \d than the numbers [0-9]. |
|
5336 |
|
5337 import re |
|
5338 _parser = re.compile(r""" # A numeric string consists of: |
|
5339 # \s* |
|
5340 (?P<sign>[-+])? # an optional sign, followed by either... |
|
5341 ( |
|
5342 (?=[0-9]|\.[0-9]) # ...a number (with at least one digit) |
|
5343 (?P<int>[0-9]*) # having a (possibly empty) integer part |
|
5344 (\.(?P<frac>[0-9]*))? # followed by an optional fractional part |
|
5345 (E(?P<exp>[-+]?[0-9]+))? # followed by an optional exponent, or... |
|
5346 | |
|
5347 Inf(inity)? # ...an infinity, or... |
|
5348 | |
|
5349 (?P<signal>s)? # ...an (optionally signaling) |
|
5350 NaN # NaN |
|
5351 (?P<diag>[0-9]*) # with (possibly empty) diagnostic info. |
|
5352 ) |
|
5353 # \s* |
|
5354 \Z |
|
5355 """, re.VERBOSE | re.IGNORECASE).match |
|
5356 |
|
5357 _all_zeros = re.compile('0*$').match |
|
5358 _exact_half = re.compile('50*$').match |
|
5359 |
|
5360 ##### PEP3101 support functions ############################################## |
|
5361 # The functions parse_format_specifier and format_align have little to do |
|
5362 # with the Decimal class, and could potentially be reused for other pure |
|
5363 # Python numeric classes that want to implement __format__ |
|
5364 # |
|
5365 # A format specifier for Decimal looks like: |
|
5366 # |
|
5367 # [[fill]align][sign][0][minimumwidth][.precision][type] |
|
5368 # |
|
5369 |
|
5370 _parse_format_specifier_regex = re.compile(r"""\A |
|
5371 (?: |
|
5372 (?P<fill>.)? |
|
5373 (?P<align>[<>=^]) |
|
5374 )? |
|
5375 (?P<sign>[-+ ])? |
|
5376 (?P<zeropad>0)? |
|
5377 (?P<minimumwidth>(?!0)\d+)? |
|
5378 (?:\.(?P<precision>0|(?!0)\d+))? |
|
5379 (?P<type>[eEfFgG%])? |
|
5380 \Z |
|
5381 """, re.VERBOSE) |
|
5382 |
|
5383 del re |
|
5384 |
|
5385 def _parse_format_specifier(format_spec): |
|
5386 """Parse and validate a format specifier. |
|
5387 |
|
5388 Turns a standard numeric format specifier into a dict, with the |
|
5389 following entries: |
|
5390 |
|
5391 fill: fill character to pad field to minimum width |
|
5392 align: alignment type, either '<', '>', '=' or '^' |
|
5393 sign: either '+', '-' or ' ' |
|
5394 minimumwidth: nonnegative integer giving minimum width |
|
5395 precision: nonnegative integer giving precision, or None |
|
5396 type: one of the characters 'eEfFgG%', or None |
|
5397 unicode: either True or False (always True for Python 3.x) |
|
5398 |
|
5399 """ |
|
5400 m = _parse_format_specifier_regex.match(format_spec) |
|
5401 if m is None: |
|
5402 raise ValueError("Invalid format specifier: " + format_spec) |
|
5403 |
|
5404 # get the dictionary |
|
5405 format_dict = m.groupdict() |
|
5406 |
|
5407 # defaults for fill and alignment |
|
5408 fill = format_dict['fill'] |
|
5409 align = format_dict['align'] |
|
5410 if format_dict.pop('zeropad') is not None: |
|
5411 # in the face of conflict, refuse the temptation to guess |
|
5412 if fill is not None and fill != '0': |
|
5413 raise ValueError("Fill character conflicts with '0'" |
|
5414 " in format specifier: " + format_spec) |
|
5415 if align is not None and align != '=': |
|
5416 raise ValueError("Alignment conflicts with '0' in " |
|
5417 "format specifier: " + format_spec) |
|
5418 fill = '0' |
|
5419 align = '=' |
|
5420 format_dict['fill'] = fill or ' ' |
|
5421 format_dict['align'] = align or '<' |
|
5422 |
|
5423 if format_dict['sign'] is None: |
|
5424 format_dict['sign'] = '-' |
|
5425 |
|
5426 # turn minimumwidth and precision entries into integers. |
|
5427 # minimumwidth defaults to 0; precision remains None if not given |
|
5428 format_dict['minimumwidth'] = int(format_dict['minimumwidth'] or '0') |
|
5429 if format_dict['precision'] is not None: |
|
5430 format_dict['precision'] = int(format_dict['precision']) |
|
5431 |
|
5432 # if format type is 'g' or 'G' then a precision of 0 makes little |
|
5433 # sense; convert it to 1. Same if format type is unspecified. |
|
5434 if format_dict['precision'] == 0: |
|
5435 if format_dict['type'] in 'gG' or format_dict['type'] is None: |
|
5436 format_dict['precision'] = 1 |
|
5437 |
|
5438 # record whether return type should be str or unicode |
|
5439 format_dict['unicode'] = isinstance(format_spec, unicode) |
|
5440 |
|
5441 return format_dict |
|
5442 |
|
5443 def _format_align(body, spec_dict): |
|
5444 """Given an unpadded, non-aligned numeric string, add padding and |
|
5445 aligment to conform with the given format specifier dictionary (as |
|
5446 output from parse_format_specifier). |
|
5447 |
|
5448 It's assumed that if body is negative then it starts with '-'. |
|
5449 Any leading sign ('-' or '+') is stripped from the body before |
|
5450 applying the alignment and padding rules, and replaced in the |
|
5451 appropriate position. |
|
5452 |
|
5453 """ |
|
5454 # figure out the sign; we only examine the first character, so if |
|
5455 # body has leading whitespace the results may be surprising. |
|
5456 if len(body) > 0 and body[0] in '-+': |
|
5457 sign = body[0] |
|
5458 body = body[1:] |
|
5459 else: |
|
5460 sign = '' |
|
5461 |
|
5462 if sign != '-': |
|
5463 if spec_dict['sign'] in ' +': |
|
5464 sign = spec_dict['sign'] |
|
5465 else: |
|
5466 sign = '' |
|
5467 |
|
5468 # how much extra space do we have to play with? |
|
5469 minimumwidth = spec_dict['minimumwidth'] |
|
5470 fill = spec_dict['fill'] |
|
5471 padding = fill*(max(minimumwidth - (len(sign+body)), 0)) |
|
5472 |
|
5473 align = spec_dict['align'] |
|
5474 if align == '<': |
|
5475 result = padding + sign + body |
|
5476 elif align == '>': |
|
5477 result = sign + body + padding |
|
5478 elif align == '=': |
|
5479 result = sign + padding + body |
|
5480 else: #align == '^' |
|
5481 half = len(padding)//2 |
|
5482 result = padding[:half] + sign + body + padding[half:] |
|
5483 |
|
5484 # make sure that result is unicode if necessary |
|
5485 if spec_dict['unicode']: |
|
5486 result = unicode(result) |
|
5487 |
|
5488 return result |
|
5489 |
|
5490 ##### Useful Constants (internal use only) ################################ |
|
5491 |
|
5492 # Reusable defaults |
|
5493 Inf = Decimal('Inf') |
|
5494 negInf = Decimal('-Inf') |
|
5495 NaN = Decimal('NaN') |
|
5496 Dec_0 = Decimal(0) |
|
5497 Dec_p1 = Decimal(1) |
|
5498 Dec_n1 = Decimal(-1) |
|
5499 |
|
5500 # Infsign[sign] is infinity w/ that sign |
|
5501 Infsign = (Inf, negInf) |
|
5502 |
|
5503 |
|
5504 |
|
5505 if __name__ == '__main__': |
|
5506 import doctest, sys |
|
5507 doctest.testmod(sys.modules[__name__]) |