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fractions.pyc
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2014-12-31
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# Source Generated with Decompyle++
# File: in.pyc (Python 2.7)
'''Rational, infinite-precision, real numbers.'''
from __future__ import division
from decimal import Decimal
import math
import numbers
import operator
import re
__all__ = [
'Fraction',
'gcd']
Rational = numbers.Rational
def gcd(a, b):
'''Calculate the Greatest Common Divisor of a and b.
Unless b==0, the result will have the same sign as b (so that when
b is divided by it, the result comes out positive).
'''
while b:
a = b
b = a % b
return a
_RATIONAL_FORMAT = re.compile('\n \\A\\s* # optional whitespace at the start, then\n (?P<sign>[-+]?) # an optional sign, then\n (?=\\d|\\.\\d) # lookahead for digit or .digit\n (?P<num>\\d*) # numerator (possibly empty)\n (?: # followed by\n (?:/(?P<denom>\\d+))? # an optional denominator\n | # or\n (?:\\.(?P<decimal>\\d*))? # an optional fractional part\n (?:E(?P<exp>[-+]?\\d+))? # and optional exponent\n )\n \\s*\\Z # and optional whitespace to finish\n', re.VERBOSE | re.IGNORECASE)
class Fraction(Rational):
"""This class implements rational numbers.
In the two-argument form of the constructor, Fraction(8, 6) will
produce a rational number equivalent to 4/3. Both arguments must
be Rational. The numerator defaults to 0 and the denominator
defaults to 1 so that Fraction(3) == 3 and Fraction() == 0.
Fractions can also be constructed from:
- numeric strings similar to those accepted by the
float constructor (for example, '-2.3' or '1e10')
- strings of the form '123/456'
- float and Decimal instances
- other Rational instances (including integers)
"""
__slots__ = ('_numerator', '_denominator')
def __new__(cls, numerator = 0, denominator = None):
"""Constructs a Fraction.
Takes a string like '3/2' or '1.5', another Rational instance, a
numerator/denominator pair, or a float.
Examples
--------
>>> Fraction(10, -8)
Fraction(-5, 4)
>>> Fraction(Fraction(1, 7), 5)
Fraction(1, 35)
>>> Fraction(Fraction(1, 7), Fraction(2, 3))
Fraction(3, 14)
>>> Fraction('314')
Fraction(314, 1)
>>> Fraction('-35/4')
Fraction(-35, 4)
>>> Fraction('3.1415') # conversion from numeric string
Fraction(6283, 2000)
>>> Fraction('-47e-2') # string may include a decimal exponent
Fraction(-47, 100)
>>> Fraction(1.47) # direct construction from float (exact conversion)
Fraction(6620291452234629, 4503599627370496)
>>> Fraction(2.25)
Fraction(9, 4)
>>> Fraction(Decimal('1.47'))
Fraction(147, 100)
"""
self = super(Fraction, cls).__new__(cls)
if denominator is None:
if isinstance(numerator, Rational):
self._numerator = numerator.numerator
self._denominator = numerator.denominator
return self
if None(numerator, float):
value = Fraction.from_float(numerator)
self._numerator = value._numerator
self._denominator = value._denominator
return self
if None(numerator, Decimal):
value = Fraction.from_decimal(numerator)
self._numerator = value._numerator
self._denominator = value._denominator
return self
if None(numerator, basestring):
m = _RATIONAL_FORMAT.match(numerator)
numerator = None(int if m is None else '0')
denom = m.group('denom')
if denom:
denominator = int(denom)
else:
denominator = 1
decimal = m.group('decimal')
if decimal:
scale = 10 ** len(decimal)
numerator = numerator * scale + int(decimal)
denominator *= scale
exp = m.group('exp')
if exp:
exp = int(exp)
if exp >= 0:
numerator *= 10 ** exp
else:
denominator *= 10 ** (-exp)
if m.group('sign') == '-':
numerator = -numerator
else:
raise TypeError('argument should be a string or a Rational instance')
if isinstance(numerator, Rational) and isinstance(denominator, Rational):
numerator = numerator.numerator * denominator.denominator
denominator = denominator.numerator * numerator.denominator
else:
raise TypeError('both arguments should be Rational instances')
if None == 0:
raise ZeroDivisionError('Fraction(%s, 0)' % numerator)
g = gcd(numerator, denominator)
self._numerator = numerator // g
self._denominator = denominator // g
return self
def from_float(cls, f):
'''Converts a finite float to a rational number, exactly.
Beware that Fraction.from_float(0.3) != Fraction(3, 10).
'''
if isinstance(f, numbers.Integral):
return cls(f)
if not None(f, float):
raise TypeError('%s.from_float() only takes floats, not %r (%s)' % (cls.__name__, f, type(f).__name__))
if math.isnan(f) or math.isinf(f):
raise TypeError('Cannot convert %r to %s.' % (f, cls.__name__))
return cls(*f.as_integer_ratio())
from_float = classmethod(from_float)
def from_decimal(cls, dec):
'''Converts a finite Decimal instance to a rational number, exactly.'''
Decimal = Decimal
import decimal
if isinstance(dec, numbers.Integral):
dec = Decimal(int(dec))
elif not isinstance(dec, Decimal):
raise TypeError('%s.from_decimal() only takes Decimals, not %r (%s)' % (cls.__name__, dec, type(dec).__name__))
if not dec.is_finite():
raise TypeError('Cannot convert %s to %s.' % (dec, cls.__name__))
(sign, digits, exp) = dec.as_tuple()
digits = int(''.join(map(str, digits)))
if sign:
digits = -digits
if exp >= 0:
return cls(digits * 10 ** exp)
return None(digits, 10 ** (-exp))
from_decimal = classmethod(from_decimal)
def limit_denominator(self, max_denominator = 1000000):
"""Closest Fraction to self with denominator at most max_denominator.
>>> Fraction('3.141592653589793').limit_denominator(10)
Fraction(22, 7)
>>> Fraction('3.141592653589793').limit_denominator(100)
Fraction(311, 99)
>>> Fraction(4321, 8765).limit_denominator(10000)
Fraction(4321, 8765)
"""
if max_denominator < 1:
raise ValueError('max_denominator should be at least 1')
if self._denominator <= max_denominator:
return Fraction(self)
(p0, q0, p1, q1) = None
n = self._numerator
d = self._denominator
while True:
a = n // d
q2 = q0 + a * q1
if q2 > max_denominator:
break
(p0, q0, p1, q1) = (p1, q1, p0 + a * p1, q2)
n = d
d = n - a * d
k = (max_denominator - q0) // q1
bound1 = Fraction(p0 + k * p1, q0 + k * q1)
bound2 = Fraction(p1, q1)
if abs(bound2 - self) <= abs(bound1 - self):
return bound2
return None
def numerator(a):
return a._numerator
numerator = property(numerator)
def denominator(a):
return a._denominator
denominator = property(denominator)
def __repr__(self):
'''repr(self)'''
return 'Fraction(%s, %s)' % (self._numerator, self._denominator)
def __str__(self):
'''str(self)'''
if self._denominator == 1:
return str(self._numerator)
return None % (self._numerator, self._denominator)
def _operator_fallbacks(monomorphic_operator, fallback_operator):
'''Generates forward and reverse operators given a purely-rational
operator and a function from the operator module.
Use this like:
__op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op)
In general, we want to implement the arithmetic operations so
that mixed-mode operations either call an implementation whose
author knew about the types of both arguments, or convert both
to the nearest built in type and do the operation there. In
Fraction, that means that we define __add__ and __radd__ as:
def __add__(self, other):
# Both types have numerators/denominator attributes,
# so do the operation directly
if isinstance(other, (int, long, Fraction)):
return Fraction(self.numerator * other.denominator +
other.numerator * self.denominator,
self.denominator * other.denominator)
# float and complex don\'t have those operations, but we
# know about those types, so special case them.
elif isinstance(other, float):
return float(self) + other
elif isinstance(other, complex):
return complex(self) + other
# Let the other type take over.
return NotImplemented
def __radd__(self, other):
# radd handles more types than add because there\'s
# nothing left to fall back to.
if isinstance(other, Rational):
return Fraction(self.numerator * other.denominator +
other.numerator * self.denominator,
self.denominator * other.denominator)
elif isinstance(other, Real):
return float(other) + float(self)
elif isinstance(other, Complex):
return complex(other) + complex(self)
return NotImplemented
There are 5 different cases for a mixed-type addition on
Fraction. I\'ll refer to all of the above code that doesn\'t
refer to Fraction, float, or complex as "boilerplate". \'r\'
will be an instance of Fraction, which is a subtype of
Rational (r : Fraction <: Rational), and b : B <:
Complex. The first three involve \'r + b\':
1. If B <: Fraction, int, float, or complex, we handle
that specially, and all is well.
2. If Fraction falls back to the boilerplate code, and it
were to return a value from __add__, we\'d miss the
possibility that B defines a more intelligent __radd__,
so the boilerplate should return NotImplemented from
__add__. In particular, we don\'t handle Rational
here, even though we could get an exact answer, in case
the other type wants to do something special.
3. If B <: Fraction, Python tries B.__radd__ before
Fraction.__add__. This is ok, because it was
implemented with knowledge of Fraction, so it can
handle those instances before delegating to Real or
Complex.
The next two situations describe \'b + r\'. We assume that b
didn\'t know about Fraction in its implementation, and that it
uses similar boilerplate code:
4. If B <: Rational, then __radd_ converts both to the
builtin rational type (hey look, that\'s us) and
proceeds.
5. Otherwise, __radd__ tries to find the nearest common
base ABC, and fall back to its builtin type. Since this
class doesn\'t subclass a concrete type, there\'s no
implementation to fall back to, so we need to try as
hard as possible to return an actual value, or the user
will get a TypeError.
'''
def forward(a, b):
if isinstance(b, (int, long, Fraction)):
return monomorphic_operator(a, b)
if None(b, float):
return fallback_operator(float(a), b)
if None(b, complex):
return fallback_operator(complex(a), b)
return None
forward.__name__ = '__' + fallback_operator.__name__ + '__'
forward.__doc__ = monomorphic_operator.__doc__
def reverse(b, a):
if isinstance(a, Rational):
return monomorphic_operator(a, b)
if None(a, numbers.Real):
return fallback_operator(float(a), float(b))
if None(a, numbers.Complex):
return fallback_operator(complex(a), complex(b))
return None
reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
reverse.__doc__ = monomorphic_operator.__doc__
return (forward, reverse)
def _add(a, b):
'''a + b'''
return Fraction(a.numerator * b.denominator + b.numerator * a.denominator, a.denominator * b.denominator)
(__add__, __radd__) = _operator_fallbacks(_add, operator.add)
def _sub(a, b):
'''a - b'''
return Fraction(a.numerator * b.denominator - b.numerator * a.denominator, a.denominator * b.denominator)
(__sub__, __rsub__) = _operator_fallbacks(_sub, operator.sub)
def _mul(a, b):
'''a * b'''
return Fraction(a.numerator * b.numerator, a.denominator * b.denominator)
(__mul__, __rmul__) = _operator_fallbacks(_mul, operator.mul)
def _div(a, b):
'''a / b'''
return Fraction(a.numerator * b.denominator, a.denominator * b.numerator)
(__truediv__, __rtruediv__) = _operator_fallbacks(_div, operator.truediv)
(__div__, __rdiv__) = _operator_fallbacks(_div, operator.div)
def __floordiv__(a, b):
'''a // b'''
div = a / b
if isinstance(div, Rational):
return div.numerator // div.denominator
return None.floor(div)
def __rfloordiv__(b, a):
'''a // b'''
div = a / b
if isinstance(div, Rational):
return div.numerator // div.denominator
return None.floor(div)
def __mod__(a, b):
'''a % b'''
div = a // b
return a - b * div
def __rmod__(b, a):
'''a % b'''
div = a // b
return a - b * div
def __pow__(a, b):
'''a ** b
If b is not an integer, the result will be a float or complex
since roots are generally irrational. If b is an integer, the
result will be rational.
'''
if isinstance(b, Rational):
if b.denominator == 1:
power = b.numerator
if power >= 0:
return Fraction(a._numerator ** power, a._denominator ** power)
return None(a._denominator ** (-power), a._numerator ** (-power))
return float(a) ** float(b)
return float(a) ** b
def __rpow__(b, a):
'''a ** b'''
if b._denominator == 1 and b._numerator >= 0:
return a ** b._numerator
if None(a, Rational):
return Fraction(a.numerator, a.denominator) ** b
if None._denominator == 1:
return a ** b._numerator
return None ** float(b)
def __pos__(a):
'''+a: Coerces a subclass instance to Fraction'''
return Fraction(a._numerator, a._denominator)
def __neg__(a):
'''-a'''
return Fraction(-(a._numerator), a._denominator)
def __abs__(a):
'''abs(a)'''
return Fraction(abs(a._numerator), a._denominator)
def __trunc__(a):
'''trunc(a)'''
if a._numerator < 0:
return -(-(a._numerator) // a._denominator)
return None._numerator // a._denominator
def __hash__(self):
'''hash(self)
Tricky because values that are exactly representable as a
float must have the same hash as that float.
'''
if self._denominator == 1:
return hash(self._numerator)
if None == float(self):
return hash(float(self))
return None((self._numerator, self._denominator))
def __eq__(a, b):
'''a == b'''
if isinstance(b, Rational):
if a._numerator == b.numerator:
pass
return a._denominator == b.denominator
if None(b, numbers.Complex) and b.imag == 0:
b = b.real
if isinstance(b, float):
if math.isnan(b) or math.isinf(b):
return 0 == b
return None == a.from_float(b)
return NotImplemented
def _richcmp(self, other, op):
'''Helper for comparison operators, for internal use only.
Implement comparison between a Rational instance `self`, and
either another Rational instance or a float `other`. If
`other` is not a Rational instance or a float, return
NotImplemented. `op` should be one of the six standard
comparison operators.
'''
if isinstance(other, Rational):
return op(self._numerator * other.denominator, self._denominator * other.numerator)
if None(other, complex):
raise TypeError('no ordering relation is defined for complex numbers')
if isinstance(other, float):
if math.isnan(other) or math.isinf(other):
return op(0, other)
return None(self, self.from_float(other))
return NotImplemented
def __lt__(a, b):
'''a < b'''
return a._richcmp(b, operator.lt)
def __gt__(a, b):
'''a > b'''
return a._richcmp(b, operator.gt)
def __le__(a, b):
'''a <= b'''
return a._richcmp(b, operator.le)
def __ge__(a, b):
'''a >= b'''
return a._richcmp(b, operator.ge)
def __nonzero__(a):
'''a != 0'''
return a._numerator != 0
def __reduce__(self):
return (self.__class__, (str(self),))
def __copy__(self):
if type(self) == Fraction:
return self
return None.__class__(self._numerator, self._denominator)
def __deepcopy__(self, memo):
if type(self) == Fraction:
return self
return None.__class__(self._numerator, self._denominator)