"""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

\A\s* # optional whitespace at the start, then

(?P<sign>[-+]?) # an optional sign, then

(?=\d|\.\d) # lookahead for digit or .digit

(?P<num>\d*) # numerator (possibly empty)

(?: # followed by

(?:/(?P<denom>\d+))? # an optional denominator

| # or

(?:\.(?P<decimal>\d*))? # an optional fractional part

(?:E(?P<exp>[-+]?\d+))? # and optional exponent

)

\s*\Z # and optional whitespace to finish

"""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')

# We're immutable, so use __new__ not __init__

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

elif isinstance(numerator, float):

# Exact conversion from float

value = Fraction.from_float(numerator)

self._numerator = value._numerator

self._denominator = value._denominator

return self

elif isinstance(numerator, Decimal):

value = Fraction.from_decimal(numerator)

self._numerator = value._numerator

self._denominator = value._denominator

return self

elif isinstance(numerator, basestring):

# Handle construction from strings.

m = _RATIONAL_FORMAT.match(numerator)

if m is None:

raise ValueError('Invalid literal for Fraction: %r' %

numerator)

numerator = int(m.group('num') or '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")

elif (isinstance(numerator, Rational) and

isinstance(denominator, Rational)):

numerator, denominator = (

numerator.numerator * denominator.denominator,

denominator.numerator * numerator.denominator

)

else:

raise TypeError("both arguments should be "

"Rational instances")

if denominator == 0:

raise ZeroDivisionError('Fraction(%s, 0)' % numerator)

g = gcd(numerator, denominator)

self._numerator = numerator // g

self._denominator = denominator // g

return self

@classmethod

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)

elif not isinstance(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())

@classmethod

def from_decimal(cls, dec):

"""Converts a finite Decimal instance to a rational number, exactly."""

from 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():

# Catches infinities and nans.

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)

else:

return cls(digits, 10 ** -exp)

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)

"""

# Algorithm notes: For any real number x, define a *best upper

# approximation* to x to be a rational number p/q such that:

#

# (1) p/q >= x, and

# (2) if p/q > r/s >= x then s > q, for any rational r/s.

#

# Define *best lower approximation* similarly. Then it can be

# proved that a rational number is a best upper or lower

# approximation to x if, and only if, it is a convergent or

# semiconvergent of the (unique shortest) continued fraction

# associated to x.

#

# To find a best rational approximation with denominator <= M,

# we find the best upper and lower approximations with

# denominator <= M and take whichever of these is closer to x.

# In the event of a tie, the bound with smaller denominator is

# chosen. If both denominators are equal (which can happen

# only when max_denominator == 1 and self is midway between

# two integers) the lower bound---i.e., the floor of self, is

# taken.

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 = 0, 1, 1, 0

n, d = self._numerator, 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

else:

return bound1

@property

def numerator(a):

return a._numerator

@property

def denominator(a):

return a._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)

else:

return '%s/%s' % (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)

elif isinstance(b, float):

return fallback_operator(float(a), b)

elif isinstance(b, complex):

return fallback_operator(complex(a), b)

else:

return NotImplemented

forward.__name__ = '__' + fallback_operator.__name__ + '__'

forward.__doc__ = monomorphic_operator.__doc__

def reverse(b, a):

if isinstance(a, Rational):

# Includes ints.

return monomorphic_operator(a, b)

elif isinstance(a, numbers.Real):

return fallback_operator(float(a), float(b))

elif isinstance(a, numbers.Complex):

return fallback_operator(complex(a), complex(b))

else:

return NotImplemented

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"""

# Will be math.floor(a / b) in 3.0.

div = a / b

if isinstance(div, Rational):

# trunc(math.floor(div)) doesn't work if the rational is

# more precise than a float because the intermediate

# rounding may cross an integer boundary.

return div.numerator // div.denominator

else:

return math.floor(div)

def __rfloordiv__(b, a):

"""a // b"""

# Will be math.floor(a / b) in 3.0.

div = a / b

if isinstance(div, Rational):

# trunc(math.floor(div)) doesn't work if the rational is

# more precise than a float because the intermediate

# rounding may cross an integer boundary.

return div.numerator // div.denominator

else:

return math.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)

else:

return Fraction(a._denominator ** -power,

a._numerator ** -power)

else:

# A fractional power will generally produce an

# irrational number.

return float(a) ** float(b)

else:

return float(a) ** b

def __rpow__(b, a):

"""a ** b"""

if b._denominator == 1 and b._numerator >= 0:

# If a is an int, keep it that way if possible.

return a ** b._numerator

if isinstance(a, Rational):

return Fraction(a.numerator, a.denominator) ** b

if b._denominator == 1:

return a ** b._numerator

return a ** 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)

else:

return a._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.

"""

# XXX since this method is expensive, consider caching the result

if self._denominator == 1:

# Get integers right.

return hash(self._numerator)

# Expensive check, but definitely correct.

if self == float(self):

return hash(float(self))

else:

# Use tuple's hash to avoid a high collision rate on

# simple fractions.

return hash((self._numerator, self._denominator))

def __eq__(a, b):

"""a == b"""

if isinstance(b, Rational):

return (a._numerator == b.numerator and

a._denominator == b.denominator)

if isinstance(b, numbers.Complex) and b.imag == 0:

b = b.real

if isinstance(b, float):

if math.isnan(b) or math.isinf(b):

# comparisons with an infinity or nan should behave in

# the same way for any finite a, so treat a as zero.

return 0.0 == b

else:

return a == a.from_float(b)

else:

# Since a doesn't know how to compare with b, let's give b

# a chance to compare itself with a.

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.

"""

# convert other to a Rational instance where reasonable.

if isinstance(other, Rational):

return op(self._numerator * other.denominator,

self._denominator * other.numerator)

# comparisons with complex should raise a TypeError, for consistency

# with int<->complex, float<->complex, and complex<->complex comparisons.

if isinstance(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(0.0, other)

else:

return op(self, self.from_float(other))

else:

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

# support for pickling, copy, and deepcopy

def __reduce__(self):

return (self.__class__, (str(self),))

def __copy__(self):

if type(self) == Fraction:

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) == Fraction:

return self # My components are also immutable