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svn+ssh://pythondev@svn.python.org/python/trunk ........ r60441 | christian.heimes | 2008-01-30 12:46:00 +0100 (Wed, 30 Jan 2008) | 1 line Removed unused var ........ r60448 | christian.heimes | 2008-01-30 18:21:22 +0100 (Wed, 30 Jan 2008) | 1 line Fixed some references leaks in sys. ........ r60450 | christian.heimes | 2008-01-30 19:58:29 +0100 (Wed, 30 Jan 2008) | 1 line The previous change was causing a segfault after multiple calls to Py_Initialize() and Py_Finalize(). ........ r60463 | raymond.hettinger | 2008-01-30 23:17:31 +0100 (Wed, 30 Jan 2008) | 1 line Update itertool recipes ........ r60464 | christian.heimes | 2008-01-30 23:54:18 +0100 (Wed, 30 Jan 2008) | 1 line Bug #1234: Fixed semaphore errors on AIX 5.2 ........ r60469 | raymond.hettinger | 2008-01-31 02:38:15 +0100 (Thu, 31 Jan 2008) | 6 lines Fix defect in __ixor__ which would get the wrong answer if the input iterable had a duplicate element (two calls to toggle() reverse each other). Borrow the correct code from sets.py. ........ r60470 | raymond.hettinger | 2008-01-31 02:42:11 +0100 (Thu, 31 Jan 2008) | 1 line Missing return ........ r60471 | jeffrey.yasskin | 2008-01-31 08:44:11 +0100 (Thu, 31 Jan 2008) | 4 lines Added more documentation on how mixed-mode arithmetic should be implemented. I also noticed and fixed a bug in Rational's forward operators (they were claiming all instances of numbers.Rational instead of just the concrete types). ........
522 lines
18 KiB
Python
Executable file
522 lines
18 KiB
Python
Executable file
# Originally contributed by Sjoerd Mullender.
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# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.
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"""Rational, infinite-precision, real numbers."""
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import math
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import numbers
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import operator
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import re
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__all__ = ["Rational"]
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RationalAbc = numbers.Rational
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def gcd(a, b):
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"""Calculate the Greatest Common Divisor of a and b.
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Unless b==0, the result will have the same sign as b (so that when
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b is divided by it, the result comes out positive).
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"""
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while b:
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a, b = b, a%b
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return a
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_RATIONAL_FORMAT = re.compile(
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r'^\s*(?P<sign>[-+]?)(?P<num>\d+)'
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r'(?:/(?P<denom>\d+)|\.(?P<decimal>\d+))?\s*$')
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class Rational(RationalAbc):
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"""This class implements rational numbers.
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Rational(8, 6) will produce a rational number equivalent to
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4/3. Both arguments must be Integral. The numerator defaults to 0
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and the denominator defaults to 1 so that Rational(3) == 3 and
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Rational() == 0.
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Rationals can also be constructed from strings of the form
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'[-+]?[0-9]+((/|.)[0-9]+)?', optionally surrounded by spaces.
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"""
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__slots__ = ('numerator', 'denominator')
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# We're immutable, so use __new__ not __init__
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def __new__(cls, numerator=0, denominator=1):
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"""Constructs a Rational.
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Takes a string like '3/2' or '1.5', another Rational, or a
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numerator/denominator pair.
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"""
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self = super(Rational, cls).__new__(cls)
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if denominator == 1:
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if isinstance(numerator, str):
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# Handle construction from strings.
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input = numerator
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m = _RATIONAL_FORMAT.match(input)
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if m is None:
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raise ValueError('Invalid literal for Rational: ' + input)
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numerator = m.group('num')
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decimal = m.group('decimal')
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if decimal:
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# The literal is a decimal number.
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numerator = int(numerator + decimal)
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denominator = 10**len(decimal)
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else:
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# The literal is an integer or fraction.
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numerator = int(numerator)
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# Default denominator to 1.
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denominator = int(m.group('denom') or 1)
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if m.group('sign') == '-':
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numerator = -numerator
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elif (not isinstance(numerator, numbers.Integral) and
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isinstance(numerator, RationalAbc)):
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# Handle copies from other rationals.
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other_rational = numerator
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numerator = other_rational.numerator
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denominator = other_rational.denominator
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if (not isinstance(numerator, numbers.Integral) or
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not isinstance(denominator, numbers.Integral)):
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raise TypeError("Rational(%(numerator)s, %(denominator)s):"
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" Both arguments must be integral." % locals())
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if denominator == 0:
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raise ZeroDivisionError('Rational(%s, 0)' % numerator)
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g = gcd(numerator, denominator)
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self.numerator = int(numerator // g)
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self.denominator = int(denominator // g)
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return self
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@classmethod
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def from_float(cls, f):
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"""Converts a finite float to a rational number, exactly.
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Beware that Rational.from_float(0.3) != Rational(3, 10).
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"""
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if not isinstance(f, float):
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raise TypeError("%s.from_float() only takes floats, not %r (%s)" %
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(cls.__name__, f, type(f).__name__))
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if math.isnan(f) or math.isinf(f):
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raise TypeError("Cannot convert %r to %s." % (f, cls.__name__))
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return cls(*f.as_integer_ratio())
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@classmethod
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def from_decimal(cls, dec):
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"""Converts a finite Decimal instance to a rational number, exactly."""
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from decimal import Decimal
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if not isinstance(dec, Decimal):
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raise TypeError(
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"%s.from_decimal() only takes Decimals, not %r (%s)" %
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(cls.__name__, dec, type(dec).__name__))
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if not dec.is_finite():
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# Catches infinities and nans.
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raise TypeError("Cannot convert %s to %s." % (dec, cls.__name__))
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sign, digits, exp = dec.as_tuple()
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digits = int(''.join(map(str, digits)))
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if sign:
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digits = -digits
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if exp >= 0:
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return cls(digits * 10 ** exp)
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else:
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return cls(digits, 10 ** -exp)
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@classmethod
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def from_continued_fraction(cls, seq):
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'Build a Rational from a continued fraction expessed as a sequence'
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n, d = 1, 0
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for e in reversed(seq):
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n, d = d, n
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n += e * d
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return cls(n, d) if seq else cls(0)
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def as_continued_fraction(self):
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'Return continued fraction expressed as a list'
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n = self.numerator
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d = self.denominator
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cf = []
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while d:
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e = int(n // d)
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cf.append(e)
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n -= e * d
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n, d = d, n
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return cf
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def approximate(self, max_denominator):
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'Best rational approximation with a denominator <= max_denominator'
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# XXX First cut at algorithm
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# Still needs rounding rules as specified at
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# http://en.wikipedia.org/wiki/Continued_fraction
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if self.denominator <= max_denominator:
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return self
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cf = self.as_continued_fraction()
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result = Rational(0)
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for i in range(1, len(cf)):
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new = self.from_continued_fraction(cf[:i])
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if new.denominator > max_denominator:
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break
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result = new
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return result
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def __repr__(self):
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"""repr(self)"""
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return ('Rational(%r,%r)' % (self.numerator, self.denominator))
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def __str__(self):
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"""str(self)"""
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if self.denominator == 1:
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return str(self.numerator)
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else:
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return '%s/%s' % (self.numerator, self.denominator)
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def _operator_fallbacks(monomorphic_operator, fallback_operator):
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"""Generates forward and reverse operators given a purely-rational
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operator and a function from the operator module.
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Use this like:
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__op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op)
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In general, we want to implement the arithmetic operations so
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that mixed-mode operations either call an implementation whose
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author knew about the types of both arguments, or convert both
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to the nearest built in type and do the operation there. In
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Rational, that means that we define __add__ and __radd__ as:
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def __add__(self, other):
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if isinstance(other, (int, Rational)):
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# Do the real operation.
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return Rational(self.numerator * other.denominator +
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other.numerator * self.denominator,
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self.denominator * other.denominator)
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# float and complex don't follow this protocol, and
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# Rational knows about them, so special case them.
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elif isinstance(other, float):
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return float(self) + other
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elif isinstance(other, complex):
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return complex(self) + other
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else:
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# Let the other type take over.
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return NotImplemented
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def __radd__(self, other):
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# radd handles more types than add because there's
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# nothing left to fall back to.
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if isinstance(other, RationalAbc):
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return Rational(self.numerator * other.denominator +
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other.numerator * self.denominator,
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self.denominator * other.denominator)
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elif isinstance(other, Real):
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return float(other) + float(self)
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elif isinstance(other, Complex):
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return complex(other) + complex(self)
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else:
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return NotImplemented
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There are 5 different cases for a mixed-type addition on
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Rational. I'll refer to all of the above code that doesn't
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refer to Rational, float, or complex as "boilerplate". 'r'
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will be an instance of Rational, which is a subtype of
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RationalAbc (r : Rational <: RationalAbc), and b : B <:
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Complex. The first three involve 'r + b':
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1. If B <: Rational, int, float, or complex, we handle
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that specially, and all is well.
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2. If Rational falls back to the boilerplate code, and it
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were to return a value from __add__, we'd miss the
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possibility that B defines a more intelligent __radd__,
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so the boilerplate should return NotImplemented from
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__add__. In particular, we don't handle RationalAbc
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here, even though we could get an exact answer, in case
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the other type wants to do something special.
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3. If B <: Rational, Python tries B.__radd__ before
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Rational.__add__. This is ok, because it was
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implemented with knowledge of Rational, so it can
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handle those instances before delegating to Real or
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Complex.
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The next two situations describe 'b + r'. We assume that b
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didn't know about Rational in its implementation, and that it
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uses similar boilerplate code:
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4. If B <: RationalAbc, then __radd_ converts both to the
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builtin rational type (hey look, that's us) and
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proceeds.
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5. Otherwise, __radd__ tries to find the nearest common
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base ABC, and fall back to its builtin type. Since this
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class doesn't subclass a concrete type, there's no
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implementation to fall back to, so we need to try as
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hard as possible to return an actual value, or the user
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will get a TypeError.
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"""
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def forward(a, b):
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if isinstance(b, (int, Rational)):
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return monomorphic_operator(a, b)
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elif isinstance(b, float):
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return fallback_operator(float(a), b)
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elif isinstance(b, complex):
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return fallback_operator(complex(a), b)
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else:
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return NotImplemented
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forward.__name__ = '__' + fallback_operator.__name__ + '__'
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forward.__doc__ = monomorphic_operator.__doc__
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def reverse(b, a):
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if isinstance(a, RationalAbc):
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# Includes ints.
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return monomorphic_operator(a, b)
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elif isinstance(a, numbers.Real):
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return fallback_operator(float(a), float(b))
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elif isinstance(a, numbers.Complex):
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return fallback_operator(complex(a), complex(b))
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else:
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return NotImplemented
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reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
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reverse.__doc__ = monomorphic_operator.__doc__
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return forward, reverse
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def _add(a, b):
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"""a + b"""
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return Rational(a.numerator * b.denominator +
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b.numerator * a.denominator,
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a.denominator * b.denominator)
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__add__, __radd__ = _operator_fallbacks(_add, operator.add)
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def _sub(a, b):
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"""a - b"""
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return Rational(a.numerator * b.denominator -
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b.numerator * a.denominator,
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a.denominator * b.denominator)
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__sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub)
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def _mul(a, b):
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"""a * b"""
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return Rational(a.numerator * b.numerator, a.denominator * b.denominator)
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__mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul)
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def _div(a, b):
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"""a / b"""
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return Rational(a.numerator * b.denominator,
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a.denominator * b.numerator)
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__truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv)
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def __floordiv__(a, b):
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"""a // b"""
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return math.floor(a / b)
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def __rfloordiv__(b, a):
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"""a // b"""
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return math.floor(a / b)
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def __mod__(a, b):
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"""a % b"""
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div = a // b
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return a - b * div
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def __rmod__(b, a):
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"""a % b"""
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div = a // b
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return a - b * div
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def __pow__(a, b):
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"""a ** b
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If b is not an integer, the result will be a float or complex
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since roots are generally irrational. If b is an integer, the
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result will be rational.
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"""
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if isinstance(b, RationalAbc):
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if b.denominator == 1:
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power = b.numerator
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if power >= 0:
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return Rational(a.numerator ** power,
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a.denominator ** power)
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else:
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return Rational(a.denominator ** -power,
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a.numerator ** -power)
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else:
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# A fractional power will generally produce an
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# irrational number.
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return float(a) ** float(b)
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else:
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return float(a) ** b
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def __rpow__(b, a):
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"""a ** b"""
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if b.denominator == 1 and b.numerator >= 0:
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# If a is an int, keep it that way if possible.
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return a ** b.numerator
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if isinstance(a, RationalAbc):
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return Rational(a.numerator, a.denominator) ** b
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if b.denominator == 1:
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return a ** b.numerator
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return a ** float(b)
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def __pos__(a):
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"""+a: Coerces a subclass instance to Rational"""
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return Rational(a.numerator, a.denominator)
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def __neg__(a):
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"""-a"""
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return Rational(-a.numerator, a.denominator)
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def __abs__(a):
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"""abs(a)"""
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return Rational(abs(a.numerator), a.denominator)
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def __trunc__(a):
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"""trunc(a)"""
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if a.numerator < 0:
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return -(-a.numerator // a.denominator)
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else:
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return a.numerator // a.denominator
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__int__ = __trunc__
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def __floor__(a):
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"""Will be math.floor(a) in 3.0."""
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return a.numerator // a.denominator
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def __ceil__(a):
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"""Will be math.ceil(a) in 3.0."""
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# The negations cleverly convince floordiv to return the ceiling.
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return -(-a.numerator // a.denominator)
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def __round__(self, ndigits=None):
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"""Will be round(self, ndigits) in 3.0.
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Rounds half toward even.
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"""
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if ndigits is None:
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floor, remainder = divmod(self.numerator, self.denominator)
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if remainder * 2 < self.denominator:
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return floor
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elif remainder * 2 > self.denominator:
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return floor + 1
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# Deal with the half case:
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elif floor % 2 == 0:
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return floor
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else:
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return floor + 1
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shift = 10**abs(ndigits)
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# See _operator_fallbacks.forward to check that the results of
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# these operations will always be Rational and therefore have
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# round().
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if ndigits > 0:
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return Rational(round(self * shift), shift)
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else:
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return Rational(round(self / shift) * shift)
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def __hash__(self):
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"""hash(self)
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Tricky because values that are exactly representable as a
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float must have the same hash as that float.
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"""
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# XXX since this method is expensive, consider caching the result
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if self.denominator == 1:
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# Get integers right.
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return hash(self.numerator)
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# Expensive check, but definitely correct.
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if self == float(self):
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return hash(float(self))
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else:
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# Use tuple's hash to avoid a high collision rate on
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# simple fractions.
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return hash((self.numerator, self.denominator))
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def __eq__(a, b):
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"""a == b"""
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if isinstance(b, RationalAbc):
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return (a.numerator == b.numerator and
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a.denominator == b.denominator)
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if isinstance(b, numbers.Complex) and b.imag == 0:
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b = b.real
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if isinstance(b, float):
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return a == a.from_float(b)
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else:
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# XXX: If b.__eq__ is implemented like this method, it may
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# give the wrong answer after float(a) changes a's
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# value. Better ways of doing this are welcome.
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return float(a) == b
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def _subtractAndCompareToZero(a, b, op):
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"""Helper function for comparison operators.
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Subtracts b from a, exactly if possible, and compares the
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result with 0 using op, in such a way that the comparison
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won't recurse. If the difference raises a TypeError, returns
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NotImplemented instead.
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"""
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if isinstance(b, numbers.Complex) and b.imag == 0:
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b = b.real
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if isinstance(b, float):
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b = a.from_float(b)
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try:
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# XXX: If b <: Real but not <: RationalAbc, this is likely
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# to fall back to a float. If the actual values differ by
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# less than MIN_FLOAT, this could falsely call them equal,
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# which would make <= inconsistent with ==. Better ways of
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# doing this are welcome.
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diff = a - b
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except TypeError:
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return NotImplemented
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if isinstance(diff, RationalAbc):
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return op(diff.numerator, 0)
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return op(diff, 0)
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def __lt__(a, b):
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"""a < b"""
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|
return a._subtractAndCompareToZero(b, operator.lt)
|
|
|
|
def __gt__(a, b):
|
|
"""a > b"""
|
|
return a._subtractAndCompareToZero(b, operator.gt)
|
|
|
|
def __le__(a, b):
|
|
"""a <= b"""
|
|
return a._subtractAndCompareToZero(b, operator.le)
|
|
|
|
def __ge__(a, b):
|
|
"""a >= b"""
|
|
return a._subtractAndCompareToZero(b, operator.ge)
|
|
|
|
def __bool__(a):
|
|
"""a != 0"""
|
|
return a.numerator != 0
|
|
|
|
# support for pickling, copy, and deepcopy
|
|
|
|
def __reduce__(self):
|
|
return (self.__class__, (str(self),))
|
|
|
|
def __copy__(self):
|
|
if type(self) == Rational:
|
|
return self # I'm immutable; therefore I am my own clone
|
|
return self.__class__(self.numerator, self.denominator)
|
|
|
|
def __deepcopy__(self, memo):
|
|
if type(self) == Rational:
|
|
return self # My components are also immutable
|
|
return self.__class__(self.numerator, self.denominator)
|