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Edit File : fractions.py \| Size : 23.76 KB
# 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+aq1 if q2 > max_denominator: break p0, q0, p1, q1 = p1, q1, p0+ap1, q2 n, d = d, n-ad k = (max_denominator-q0)//q1 bound1 = Fraction(p0+kp1, q0+kq1) 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) Back

Edit File : fractions.py | Size : 23.76 KB

# 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)

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