Mercurial > cpython-lru_cache-weakref
annotate Lib/fractions.py @ 78069:a8aa77458e12 3.2
Add c_size_t and c_ssize_t to the ctypes summary table.
| author | Antoine Pitrou <solipsis@pitrou.net> |
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| date | Thu, 12 Jul 2012 20:31:50 +0200 |
| parents | 4c4fe7139c68 |
| children | a2b54b6d9759 |
| rev | line source |
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1 # Originally contributed by Sjoerd Mullender. |
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2 # Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>. |
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3 |
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4 """Fraction, infinite-precision, real numbers.""" |
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5 |
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6 from decimal import Decimal |
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7 import math |
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8 import numbers |
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9 import operator |
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10 import re |
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11 import sys |
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12 |
| 46743 | 13 __all__ = ['Fraction', 'gcd'] |
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15 |
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17 def gcd(a, b): |
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18 """Calculate the Greatest Common Divisor of a and b. |
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19 |
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20 Unless b==0, the result will have the same sign as b (so that when |
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21 b is divided by it, the result comes out positive). |
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22 """ |
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23 while b: |
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24 a, b = b, a%b |
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25 return a |
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26 |
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27 # Constants related to the hash implementation; hash(x) is based |
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28 # on the reduction of x modulo the prime _PyHASH_MODULUS. |
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29 _PyHASH_MODULUS = sys.hash_info.modulus |
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30 # Value to be used for rationals that reduce to infinity modulo |
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31 # _PyHASH_MODULUS. |
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32 _PyHASH_INF = sys.hash_info.inf |
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33 |
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34 _RATIONAL_FORMAT = re.compile(r""" |
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35 \A\s* # optional whitespace at the start, then |
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36 (?P<sign>[-+]?) # an optional sign, then |
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37 (?=\d|\.\d) # lookahead for digit or .digit |
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38 (?P<num>\d*) # numerator (possibly empty) |
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39 (?: # followed by |
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40 (?:/(?P<denom>\d+))? # an optional denominator |
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41 | # or |
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42 (?:\.(?P<decimal>\d*))? # an optional fractional part |
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43 (?:E(?P<exp>[-+]?\d+))? # and optional exponent |
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44 ) |
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45 \s*\Z # and optional whitespace to finish |
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46 """, re.VERBOSE | re.IGNORECASE) |
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47 |
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48 |
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49 class Fraction(numbers.Rational): |
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50 """This class implements rational numbers. |
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51 |
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52 In the two-argument form of the constructor, Fraction(8, 6) will |
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53 produce a rational number equivalent to 4/3. Both arguments must |
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54 be Rational. The numerator defaults to 0 and the denominator |
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55 defaults to 1 so that Fraction(3) == 3 and Fraction() == 0. |
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56 |
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57 Fractions can also be constructed from: |
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58 |
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59 - numeric strings similar to those accepted by the |
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60 float constructor (for example, '-2.3' or '1e10') |
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61 |
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62 - strings of the form '123/456' |
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63 |
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64 - float and Decimal instances |
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65 |
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66 - other Rational instances (including integers) |
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67 |
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68 """ |
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69 |
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70 __slots__ = ('_numerator', '_denominator') |
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71 |
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72 # We're immutable, so use __new__ not __init__ |
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73 def __new__(cls, numerator=0, denominator=None): |
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74 """Constructs a Rational. |
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75 |
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76 Takes a string like '3/2' or '1.5', another Rational instance, a |
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77 numerator/denominator pair, or a float. |
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78 |
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79 Examples |
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80 -------- |
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81 |
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82 >>> Fraction(10, -8) |
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83 Fraction(-5, 4) |
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84 >>> Fraction(Fraction(1, 7), 5) |
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85 Fraction(1, 35) |
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86 >>> Fraction(Fraction(1, 7), Fraction(2, 3)) |
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87 Fraction(3, 14) |
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88 >>> Fraction('314') |
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89 Fraction(314, 1) |
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90 >>> Fraction('-35/4') |
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91 Fraction(-35, 4) |
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92 >>> Fraction('3.1415') # conversion from numeric string |
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93 Fraction(6283, 2000) |
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94 >>> Fraction('-47e-2') # string may include a decimal exponent |
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95 Fraction(-47, 100) |
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96 >>> Fraction(1.47) # direct construction from float (exact conversion) |
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97 Fraction(6620291452234629, 4503599627370496) |
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98 >>> Fraction(2.25) |
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99 Fraction(9, 4) |
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100 >>> Fraction(Decimal('1.47')) |
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101 Fraction(147, 100) |
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102 |
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103 """ |
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104 self = super(Fraction, cls).__new__(cls) |
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105 |
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106 if denominator is None: |
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107 if isinstance(numerator, numbers.Rational): |
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108 self._numerator = numerator.numerator |
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109 self._denominator = numerator.denominator |
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110 return self |
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111 |
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112 elif isinstance(numerator, float): |
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113 # Exact conversion from float |
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114 value = Fraction.from_float(numerator) |
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115 self._numerator = value._numerator |
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116 self._denominator = value._denominator |
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117 return self |
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118 |
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119 elif isinstance(numerator, Decimal): |
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120 value = Fraction.from_decimal(numerator) |
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121 self._numerator = value._numerator |
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122 self._denominator = value._denominator |
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123 return self |
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124 |
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125 elif isinstance(numerator, str): |
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126 # Handle construction from strings. |
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127 m = _RATIONAL_FORMAT.match(numerator) |
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128 if m is None: |
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129 raise ValueError('Invalid literal for Fraction: %r' % |
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130 numerator) |
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131 numerator = int(m.group('num') or '0') |
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132 denom = m.group('denom') |
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133 if denom: |
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134 denominator = int(denom) |
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135 else: |
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136 denominator = 1 |
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137 decimal = m.group('decimal') |
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138 if decimal: |
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139 scale = 10**len(decimal) |
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140 numerator = numerator * scale + int(decimal) |
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141 denominator *= scale |
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142 exp = m.group('exp') |
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143 if exp: |
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144 exp = int(exp) |
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145 if exp >= 0: |
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146 numerator *= 10**exp |
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147 else: |
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148 denominator *= 10**-exp |
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149 if m.group('sign') == '-': |
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150 numerator = -numerator |
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151 |
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152 else: |
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153 raise TypeError("argument should be a string " |
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154 "or a Rational instance") |
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155 |
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156 elif (isinstance(numerator, numbers.Rational) and |
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157 isinstance(denominator, numbers.Rational)): |
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158 numerator, denominator = ( |
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159 numerator.numerator * denominator.denominator, |
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160 denominator.numerator * numerator.denominator |
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161 ) |
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162 else: |
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163 raise TypeError("both arguments should be " |
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164 "Rational instances") |
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165 |
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166 if denominator == 0: |
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167 raise ZeroDivisionError('Fraction(%s, 0)' % numerator) |
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168 g = gcd(numerator, denominator) |
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169 self._numerator = numerator // g |
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170 self._denominator = denominator // g |
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171 return self |
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172 |
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173 @classmethod |
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174 def from_float(cls, f): |
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175 """Converts a finite float to a rational number, exactly. |
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176 |
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177 Beware that Fraction.from_float(0.3) != Fraction(3, 10). |
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178 |
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179 """ |
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180 if isinstance(f, numbers.Integral): |
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181 return cls(f) |
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182 elif not isinstance(f, float): |
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183 raise TypeError("%s.from_float() only takes floats, not %r (%s)" % |
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184 (cls.__name__, f, type(f).__name__)) |
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185 if math.isnan(f) or math.isinf(f): |
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186 raise TypeError("Cannot convert %r to %s." % (f, cls.__name__)) |
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187 return cls(*f.as_integer_ratio()) |
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188 |
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189 @classmethod |
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190 def from_decimal(cls, dec): |
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191 """Converts a finite Decimal instance to a rational number, exactly.""" |
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192 from decimal import Decimal |
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193 if isinstance(dec, numbers.Integral): |
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194 dec = Decimal(int(dec)) |
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195 elif not isinstance(dec, Decimal): |
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196 raise TypeError( |
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197 "%s.from_decimal() only takes Decimals, not %r (%s)" % |
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198 (cls.__name__, dec, type(dec).__name__)) |
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199 if not dec.is_finite(): |
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200 # Catches infinities and nans. |
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201 raise TypeError("Cannot convert %s to %s." % (dec, cls.__name__)) |
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202 sign, digits, exp = dec.as_tuple() |
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203 digits = int(''.join(map(str, digits))) |
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204 if sign: |
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205 digits = -digits |
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206 if exp >= 0: |
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207 return cls(digits * 10 ** exp) |
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208 else: |
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209 return cls(digits, 10 ** -exp) |
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210 |
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211 def limit_denominator(self, max_denominator=1000000): |
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212 """Closest Fraction to self with denominator at most max_denominator. |
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213 |
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214 >>> Fraction('3.141592653589793').limit_denominator(10) |
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215 Fraction(22, 7) |
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216 >>> Fraction('3.141592653589793').limit_denominator(100) |
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217 Fraction(311, 99) |
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218 >>> Fraction(4321, 8765).limit_denominator(10000) |
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219 Fraction(4321, 8765) |
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220 |
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221 """ |
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222 # Algorithm notes: For any real number x, define a *best upper |
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223 # approximation* to x to be a rational number p/q such that: |
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224 # |
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225 # (1) p/q >= x, and |
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226 # (2) if p/q > r/s >= x then s > q, for any rational r/s. |
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227 # |
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228 # Define *best lower approximation* similarly. Then it can be |
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229 # proved that a rational number is a best upper or lower |
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230 # approximation to x if, and only if, it is a convergent or |
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231 # semiconvergent of the (unique shortest) continued fraction |
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232 # associated to x. |
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233 # |
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234 # To find a best rational approximation with denominator <= M, |
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235 # we find the best upper and lower approximations with |
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236 # denominator <= M and take whichever of these is closer to x. |
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237 # In the event of a tie, the bound with smaller denominator is |
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238 # chosen. If both denominators are equal (which can happen |
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239 # only when max_denominator == 1 and self is midway between |
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240 # two integers) the lower bound---i.e., the floor of self, is |
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241 # taken. |
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242 |
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243 if max_denominator < 1: |
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244 raise ValueError("max_denominator should be at least 1") |
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245 if self._denominator <= max_denominator: |
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246 return Fraction(self) |
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247 |
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248 p0, q0, p1, q1 = 0, 1, 1, 0 |
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249 n, d = self._numerator, self._denominator |
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250 while True: |
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251 a = n//d |
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252 q2 = q0+a*q1 |
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253 if q2 > max_denominator: |
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254 break |
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255 p0, q0, p1, q1 = p1, q1, p0+a*p1, q2 |
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256 n, d = d, n-a*d |
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257 |
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258 k = (max_denominator-q0)//q1 |
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259 bound1 = Fraction(p0+k*p1, q0+k*q1) |
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260 bound2 = Fraction(p1, q1) |
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261 if abs(bound2 - self) <= abs(bound1-self): |
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262 return bound2 |
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263 else: |
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264 return bound1 |
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265 |
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266 @property |
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267 def numerator(a): |
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268 return a._numerator |
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269 |
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270 @property |
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271 def denominator(a): |
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272 return a._denominator |
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273 |
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274 def __repr__(self): |
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275 """repr(self)""" |
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276 return ('Fraction(%s, %s)' % (self._numerator, self._denominator)) |
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277 |
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278 def __str__(self): |
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279 """str(self)""" |
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280 if self._denominator == 1: |
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281 return str(self._numerator) |
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282 else: |
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283 return '%s/%s' % (self._numerator, self._denominator) |
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284 |
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285 def _operator_fallbacks(monomorphic_operator, fallback_operator): |
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286 """Generates forward and reverse operators given a purely-rational |
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287 operator and a function from the operator module. |
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288 |
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289 Use this like: |
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290 __op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op) |
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291 |
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292 In general, we want to implement the arithmetic operations so |
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293 that mixed-mode operations either call an implementation whose |
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294 author knew about the types of both arguments, or convert both |
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295 to the nearest built in type and do the operation there. In |
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296 Fraction, that means that we define __add__ and __radd__ as: |
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297 |
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298 def __add__(self, other): |
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299 # Both types have numerators/denominator attributes, |
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300 # so do the operation directly |
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301 if isinstance(other, (int, Fraction)): |
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302 return Fraction(self.numerator * other.denominator + |
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303 other.numerator * self.denominator, |
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304 self.denominator * other.denominator) |
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305 # float and complex don't have those operations, but we |
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306 # know about those types, so special case them. |
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307 elif isinstance(other, float): |
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308 return float(self) + other |
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309 elif isinstance(other, complex): |
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310 return complex(self) + other |
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311 # Let the other type take over. |
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312 return NotImplemented |
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313 |
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314 def __radd__(self, other): |
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315 # radd handles more types than add because there's |
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316 # nothing left to fall back to. |
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317 if isinstance(other, numbers.Rational): |
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318 return Fraction(self.numerator * other.denominator + |
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319 other.numerator * self.denominator, |
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320 self.denominator * other.denominator) |
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321 elif isinstance(other, Real): |
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322 return float(other) + float(self) |
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323 elif isinstance(other, Complex): |
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324 return complex(other) + complex(self) |
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325 return NotImplemented |
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326 |
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327 |
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328 There are 5 different cases for a mixed-type addition on |
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329 Fraction. I'll refer to all of the above code that doesn't |
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330 refer to Fraction, float, or complex as "boilerplate". 'r' |
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331 will be an instance of Fraction, which is a subtype of |
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332 Rational (r : Fraction <: Rational), and b : B <: |
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333 Complex. The first three involve 'r + b': |
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334 |
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335 1. If B <: Fraction, int, float, or complex, we handle |
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336 that specially, and all is well. |
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337 2. If Fraction falls back to the boilerplate code, and it |
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338 were to return a value from __add__, we'd miss the |
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339 possibility that B defines a more intelligent __radd__, |
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340 so the boilerplate should return NotImplemented from |
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341 __add__. In particular, we don't handle Rational |
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342 here, even though we could get an exact answer, in case |
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343 the other type wants to do something special. |
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344 3. If B <: Fraction, Python tries B.__radd__ before |
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345 Fraction.__add__. This is ok, because it was |
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346 implemented with knowledge of Fraction, so it can |
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347 handle those instances before delegating to Real or |
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348 Complex. |
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349 |
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350 The next two situations describe 'b + r'. We assume that b |
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351 didn't know about Fraction in its implementation, and that it |
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352 uses similar boilerplate code: |
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353 |
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354 4. If B <: Rational, then __radd_ converts both to the |
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355 builtin rational type (hey look, that's us) and |
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356 proceeds. |
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357 5. Otherwise, __radd__ tries to find the nearest common |
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358 base ABC, and fall back to its builtin type. Since this |
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359 class doesn't subclass a concrete type, there's no |
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360 implementation to fall back to, so we need to try as |
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361 hard as possible to return an actual value, or the user |
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362 will get a TypeError. |
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363 |
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364 """ |
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365 def forward(a, b): |
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366 if isinstance(b, (int, Fraction)): |
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367 return monomorphic_operator(a, b) |
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368 elif isinstance(b, float): |
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369 return fallback_operator(float(a), b) |
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370 elif isinstance(b, complex): |
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371 return fallback_operator(complex(a), b) |
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372 else: |
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373 return NotImplemented |
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374 forward.__name__ = '__' + fallback_operator.__name__ + '__' |
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375 forward.__doc__ = monomorphic_operator.__doc__ |
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376 |
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377 def reverse(b, a): |
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378 if isinstance(a, numbers.Rational): |
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379 # Includes ints. |
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380 return monomorphic_operator(a, b) |
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381 elif isinstance(a, numbers.Real): |
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382 return fallback_operator(float(a), float(b)) |
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383 elif isinstance(a, numbers.Complex): |
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384 return fallback_operator(complex(a), complex(b)) |
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385 else: |
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386 return NotImplemented |
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387 reverse.__name__ = '__r' + fallback_operator.__name__ + '__' |
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388 reverse.__doc__ = monomorphic_operator.__doc__ |
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389 |
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390 return forward, reverse |
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391 |
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392 def _add(a, b): |
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393 """a + b""" |
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394 return Fraction(a.numerator * b.denominator + |
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395 b.numerator * a.denominator, |
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396 a.denominator * b.denominator) |
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397 |
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398 __add__, __radd__ = _operator_fallbacks(_add, operator.add) |
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399 |
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400 def _sub(a, b): |
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401 """a - b""" |
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402 return Fraction(a.numerator * b.denominator - |
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403 b.numerator * a.denominator, |
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404 a.denominator * b.denominator) |
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405 |
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406 __sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub) |
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407 |
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408 def _mul(a, b): |
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409 """a * b""" |
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410 return Fraction(a.numerator * b.numerator, a.denominator * b.denominator) |
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411 |
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412 __mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul) |
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413 |
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414 def _div(a, b): |
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415 """a / b""" |
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416 return Fraction(a.numerator * b.denominator, |
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417 a.denominator * b.numerator) |
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418 |
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419 __truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv) |
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420 |
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421 def __floordiv__(a, b): |
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422 """a // b""" |
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423 return math.floor(a / b) |
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424 |
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425 def __rfloordiv__(b, a): |
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426 """a // b""" |
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427 return math.floor(a / b) |
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428 |
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429 def __mod__(a, b): |
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430 """a % b""" |
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431 div = a // b |
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432 return a - b * div |
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433 |
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434 def __rmod__(b, a): |
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435 """a % b""" |
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436 div = a // b |
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437 return a - b * div |
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438 |
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439 def __pow__(a, b): |
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440 """a ** b |
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441 |
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442 If b is not an integer, the result will be a float or complex |
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443 since roots are generally irrational. If b is an integer, the |
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444 result will be rational. |
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445 |
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446 """ |
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447 if isinstance(b, numbers.Rational): |
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448 if b.denominator == 1: |
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449 power = b.numerator |
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450 if power >= 0: |
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451 return Fraction(a._numerator ** power, |
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452 a._denominator ** power) |
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453 else: |
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454 return Fraction(a._denominator ** -power, |
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455 a._numerator ** -power) |
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456 else: |
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457 # A fractional power will generally produce an |
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458 # irrational number. |
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459 return float(a) ** float(b) |
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460 else: |
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461 return float(a) ** b |
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462 |
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463 def __rpow__(b, a): |
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464 """a ** b""" |
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465 if b._denominator == 1 and b._numerator >= 0: |
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466 # If a is an int, keep it that way if possible. |
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467 return a ** b._numerator |
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468 |
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469 if isinstance(a, numbers.Rational): |
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470 return Fraction(a.numerator, a.denominator) ** b |
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471 |
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472 if b._denominator == 1: |
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473 return a ** b._numerator |
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474 |
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475 return a ** float(b) |
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476 |
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477 def __pos__(a): |
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478 """+a: Coerces a subclass instance to Fraction""" |
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479 return Fraction(a._numerator, a._denominator) |
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480 |
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481 def __neg__(a): |
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482 """-a""" |
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483 return Fraction(-a._numerator, a._denominator) |
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484 |
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485 def __abs__(a): |
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486 """abs(a)""" |
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487 return Fraction(abs(a._numerator), a._denominator) |
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488 |
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489 def __trunc__(a): |
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490 """trunc(a)""" |
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491 if a._numerator < 0: |
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492 return -(-a._numerator // a._denominator) |
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493 else: |
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494 return a._numerator // a._denominator |
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495 |
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496 def __floor__(a): |
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497 """Will be math.floor(a) in 3.0.""" |
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498 return a.numerator // a.denominator |
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499 |
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500 def __ceil__(a): |
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501 """Will be math.ceil(a) in 3.0.""" |
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502 # The negations cleverly convince floordiv to return the ceiling. |
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503 return -(-a.numerator // a.denominator) |
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504 |
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505 def __round__(self, ndigits=None): |
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506 """Will be round(self, ndigits) in 3.0. |
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507 |
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508 Rounds half toward even. |
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509 """ |
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510 if ndigits is None: |
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511 floor, remainder = divmod(self.numerator, self.denominator) |
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512 if remainder * 2 < self.denominator: |
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513 return floor |
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514 elif remainder * 2 > self.denominator: |
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515 return floor + 1 |
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516 # Deal with the half case: |
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517 elif floor % 2 == 0: |
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518 return floor |
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519 else: |
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520 return floor + 1 |
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521 shift = 10**abs(ndigits) |
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522 # See _operator_fallbacks.forward to check that the results of |
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523 # these operations will always be Fraction and therefore have |
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524 # round(). |
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525 if ndigits > 0: |
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526 return Fraction(round(self * shift), shift) |
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527 else: |
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528 return Fraction(round(self / shift) * shift) |
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529 |
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530 def __hash__(self): |
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531 """hash(self)""" |
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532 |
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533 # XXX since this method is expensive, consider caching the result |
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534 |
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535 # In order to make sure that the hash of a Fraction agrees |
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536 # with the hash of a numerically equal integer, float or |
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537 # Decimal instance, we follow the rules for numeric hashes |
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538 # outlined in the documentation. (See library docs, 'Built-in |
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539 # Types'). |
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540 |
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541 # dinv is the inverse of self._denominator modulo the prime |
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542 # _PyHASH_MODULUS, or 0 if self._denominator is divisible by |
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543 # _PyHASH_MODULUS. |
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544 dinv = pow(self._denominator, _PyHASH_MODULUS - 2, _PyHASH_MODULUS) |
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545 if not dinv: |
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546 hash_ = _PyHASH_INF |
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547 else: |
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548 hash_ = abs(self._numerator) * dinv % _PyHASH_MODULUS |
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549 result = hash_ if self >= 0 else -hash_ |
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550 return -2 if result == -1 else result |
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551 |
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552 def __eq__(a, b): |
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553 """a == b""" |
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554 if isinstance(b, numbers.Rational): |
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555 return (a._numerator == b.numerator and |
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556 a._denominator == b.denominator) |
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557 if isinstance(b, numbers.Complex) and b.imag == 0: |
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558 b = b.real |
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559 if isinstance(b, float): |
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560 if math.isnan(b) or math.isinf(b): |
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561 # comparisons with an infinity or nan should behave in |
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562 # the same way for any finite a, so treat a as zero. |
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563 return 0.0 == b |
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564 else: |
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565 return a == a.from_float(b) |
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566 else: |
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567 # Since a doesn't know how to compare with b, let's give b |
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568 # a chance to compare itself with a. |
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569 return NotImplemented |
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570 |
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571 def _richcmp(self, other, op): |
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572 """Helper for comparison operators, for internal use only. |
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573 |
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574 Implement comparison between a Rational instance `self`, and |
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575 either another Rational instance or a float `other`. If |
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576 `other` is not a Rational instance or a float, return |
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577 NotImplemented. `op` should be one of the six standard |
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578 comparison operators. |
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579 |
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580 """ |
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581 # convert other to a Rational instance where reasonable. |
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582 if isinstance(other, numbers.Rational): |
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583 return op(self._numerator * other.denominator, |
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584 self._denominator * other.numerator) |
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585 if isinstance(other, float): |
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586 if math.isnan(other) or math.isinf(other): |
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587 return op(0.0, other) |
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588 else: |
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589 return op(self, self.from_float(other)) |
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590 else: |
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591 return NotImplemented |
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592 |
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593 def __lt__(a, b): |
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594 """a < b""" |
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595 return a._richcmp(b, operator.lt) |
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596 |
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597 def __gt__(a, b): |
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598 """a > b""" |
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599 return a._richcmp(b, operator.gt) |
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600 |
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601 def __le__(a, b): |
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602 """a <= b""" |
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603 return a._richcmp(b, operator.le) |
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604 |
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605 def __ge__(a, b): |
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606 """a >= b""" |
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607 return a._richcmp(b, operator.ge) |
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608 |
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609 def __bool__(a): |
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610 """a != 0""" |
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611 return a._numerator != 0 |
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612 |
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613 # support for pickling, copy, and deepcopy |
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614 |
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615 def __reduce__(self): |
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616 return (self.__class__, (str(self),)) |
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617 |
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618 def __copy__(self): |
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619 if type(self) == Fraction: |
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620 return self # I'm immutable; therefore I am my own clone |
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621 return self.__class__(self._numerator, self._denominator) |
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622 |
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623 def __deepcopy__(self, memo): |
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624 if type(self) == Fraction: |
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625 return self # My components are also immutable |
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626 return self.__class__(self._numerator, self._denominator) |