# Originally contributed by Sjoerd Mullender.
# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.

"""Fraction, infinite-precision, real numbers."""

from decimal import Decimal
import math
import numbers
import operator
import re
import sys

__all__ = ['Fraction', 'gcd']



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).
 """
 import warnings
 warnings.warn('fractions.gcd() is deprecated. Use math.gcd() instead.',
 DeprecationWarning, 2)
 if type(a) is int is type(b):
 if (b or a) < 0:
 return -math.gcd(a, b)
 return math.gcd(a, b)
 return _gcd(a, b)

def _gcd(a, b):
 # Supports non-integers for backward compatibility.
 while b:
 a, b = b, a%b
 return a

# Constants related to the hash implementation; hash(x) is based
# on the reduction of x modulo the prime _PyHASH_MODULUS.
_PyHASH_MODULUS = sys.hash_info.modulus
# Value to be used for rationals that reduce to infinity modulo
# _PyHASH_MODULUS.
_PyHASH_INF = sys.hash_info.inf

_RATIONAL_FORMAT = re.compile(r"""
 \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
""", re.VERBOSE | re.IGNORECASE)


class Fraction(numbers.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')

 # We're immutable, so use __new__ not __init__
 def __new__(cls, numerator=0, denominator=None, *, _normalize=True):
 """Constructs a Rational.

 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 type(numerator) is int:
 self._numerator = numerator
 self._denominator = 1
 return self

 elif isinstance(numerator, numbers.Rational):
 self._numerator = numerator.numerator
 self._denominator = numerator.denominator
 return self

 elif isinstance(numerator, (float, Decimal)):
 # Exact conversion
 self._numerator, self._denominator = numerator.as_integer_ratio()
 return self

 elif isinstance(numerator, str):
 # 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 type(numerator) is int is type(denominator):
 pass # *very* normal case

 elif (isinstance(numerator, numbers.Rational) and
 isinstance(denominator, numbers.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)
 if _normalize:
 if type(numerator) is int is type(denominator):
 # *very* normal case
 g = math.gcd(numerator, denominator)
 if denominator < 0:
 g = -g
 else:
 g = _gcd(numerator, denominator)
 numerator //= g
 denominator //= g
 self._numerator = numerator
 self._denominator = denominator
 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__))
 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__))
 return cls(*dec.as_integer_ratio())

 def as_integer_ratio(self):
 """Return the integer ratio as a tuple.

 Return a tuple of two integers, whose ratio is equal to the
 Fraction and with a positive denominator.
 """
 return (self._numerator, self._denominator)

 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 '%s(%s, %s)' % (self.__class__.__name__,
 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, 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, numbers.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, 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, numbers.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"""
 da, db = a.denominator, b.denominator
 return Fraction(a.numerator * db + b.numerator * da,
 da * db)

 __add__, __radd__ = _operator_fallbacks(_add, operator.add)

 def _sub(a, b):
 """a - b"""
 da, db = a.denominator, b.denominator
 return Fraction(a.numerator * db - b.numerator * da,
 da * db)

 __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)

 def _floordiv(a, b):
 """a // b"""
 return (a.numerator * b.denominator) // (a.denominator * b.numerator)

 __floordiv__, __rfloordiv__ = _operator_fallbacks(_floordiv, operator.floordiv)

 def _divmod(a, b):
 """(a // b, a % b)"""
 da, db = a.denominator, b.denominator
 div, n_mod = divmod(a.numerator * db, da * b.numerator)
 return div, Fraction(n_mod, da * db)

 __divmod__, __rdivmod__ = _operator_fallbacks(_divmod, divmod)

 def _mod(a, b):
 """a % b"""
 da, db = a.denominator, b.denominator
 return Fraction((a.numerator * db) % (b.numerator * da), da * db)

 __mod__, __rmod__ = _operator_fallbacks(_mod, operator.mod)

 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, numbers.Rational):
 if b.denominator == 1:
 power = b.numerator
 if power >= 0:
 return Fraction(a._numerator ** power,
 a._denominator ** power,
 _normalize=False)
 elif a._numerator >= 0:
 return Fraction(a._denominator ** -power,
 a._numerator ** -power,
 _normalize=False)
 else:
 return Fraction((-a._denominator) ** -power,
 (-a._numerator) ** -power,
 _normalize=False)
 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, numbers.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, _normalize=False)

 def __neg__(a):
 """-a"""
 return Fraction(-a._numerator, a._denominator, _normalize=False)

 def __abs__(a):
 """abs(a)"""
 return Fraction(abs(a._numerator), a._denominator, _normalize=False)

 def __trunc__(a):
 """trunc(a)"""
 if a._numerator < 0:
 return -(-a._numerator // a._denominator)
 else:
 return a._numerator // a._denominator

 def __floor__(a):
 """math.floor(a)"""
 return a.numerator // a.denominator

 def __ceil__(a):
 """math.ceil(a)"""
 # The negations cleverly convince floordiv to return the ceiling.
 return -(-a.numerator // a.denominator)

 def __round__(self, ndigits=None):
 """round(self, ndigits)

 Rounds half toward even.
 """
 if ndigits is None:
 floor, remainder = divmod(self.numerator, self.denominator)
 if remainder * 2 < self.denominator:
 return floor
 elif remainder * 2 > self.denominator:
 return floor + 1
 # Deal with the half case:
 elif floor % 2 == 0:
 return floor
 else:
 return floor + 1
 shift = 10**abs(ndigits)
 # See _operator_fallbacks.forward to check that the results of
 # these operations will always be Fraction and therefore have
 # round().
 if ndigits > 0:
 return Fraction(round(self * shift), shift)
 else:
 return Fraction(round(self / shift) * shift)

 def __hash__(self):
 """hash(self)"""

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

 # In order to make sure that the hash of a Fraction agrees
 # with the hash of a numerically equal integer, float or
 # Decimal instance, we follow the rules for numeric hashes
 # outlined in the documentation. (See library docs, 'Built-in
 # Types').

 # dinv is the inverse of self._denominator modulo the prime
 # _PyHASH_MODULUS, or 0 if self._denominator is divisible by
 # _PyHASH_MODULUS.
 dinv = pow(self._denominator, _PyHASH_MODULUS - 2, _PyHASH_MODULUS)
 if not dinv:
 hash_ = _PyHASH_INF
 else:
 hash_ = abs(self._numerator) * dinv % _PyHASH_MODULUS
 result = hash_ if self >= 0 else -hash_
 return -2 if result == -1 else result

 def __eq__(a, b):
 """a == b"""
 if type(b) is int:
 return a._numerator == b and a._denominator == 1
 if isinstance(b, numbers.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, numbers.Rational):
 return op(self._numerator * other.denominator,
 self._denominator * other.numerator)
 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 __bool__(a):
 """a != 0"""
 # bpo-39274: Use bool() because (a._numerator != 0) can return an
 # object which is not a bool.
 return bool(a._numerator)

 # 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
 return self.__class__(self._numerator, self._denominator)