# Originally contributed by Sjoerd Mullender.

# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.

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

from decimal import Decimal

import functools

import math

import numbers

import operator

import re

import sys

__all__ = ['Fraction']

# 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

@functools.lru_cache(maxsize = 1 << 14)

def _hash_algorithm(numerator, denominator):

# 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').

try:

dinv = pow(denominator, -1, _PyHASH_MODULUS)

except ValueError:

# ValueError means there is no modular inverse.

hash_ = _PyHASH_INF

else:

# The general algorithm now specifies that the absolute value of

# the hash is

# (|N| * dinv) % P

# where N is self._numerator and P is _PyHASH_MODULUS. That's

# optimized here in two ways: first, for a non-negative int i,

# hash(i) == i % P, but the int hash implementation doesn't need

# to divide, and is faster than doing % P explicitly. So we do

# hash(|N| * dinv)

# instead. Second, N is unbounded, so its product with dinv may

# be arbitrarily expensive to compute. The final answer is the

# same if we use the bounded |N| % P instead, which can again

# be done with an int hash() call. If 0 <= i < P, hash(i) == i,

# so this nested hash() call wastes a bit of time making a

# redundant copy when |N| < P, but can save an arbitrarily large

# amount of computation for large |N|.

hash_ = hash(hash(abs(numerator)) * dinv)

result = hash_ if numerator >= 0 else -hash_

return -2 if result == -1 else result

_RATIONAL_FORMAT = re.compile(r"""

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

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

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

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

(?: # followed by

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

| # or

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

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

)

\s*\Z # and optional whitespace to finish

""", re.VERBOSE | re.IGNORECASE)

# Helpers for formatting

def _round_to_exponent(n, d, exponent, no_neg_zero=False):

"""Round a rational number to the nearest multiple of a given power of 10.

Rounds the rational number n/d to the nearest integer multiple of

10**exponent, rounding to the nearest even integer multiple in the case of

a tie. Returns a pair (sign: bool, significand: int) representing the

rounded value (-1)**sign * significand * 10**exponent.

If no_neg_zero is true, then the returned sign will always be False when

the significand is zero. Otherwise, the sign reflects the sign of the

input.

d must be positive, but n and d need not be relatively prime.

"""

if exponent >= 0:

d *= 10**exponent

else:

n *= 10**-exponent

# The divmod quotient is correct for round-ties-towards-positive-infinity;

# In the case of a tie, we zero out the least significant bit of q.

q, r = divmod(n + (d >> 1), d)

if r == 0 and d & 1 == 0:

q &= -2

sign = q < 0 if no_neg_zero else n < 0

return sign, abs(q)

def _round_to_figures(n, d, figures):

"""Round a rational number to a given number of significant figures.

Rounds the rational number n/d to the given number of significant figures

using the round-ties-to-even rule, and returns a triple

(sign: bool, significand: int, exponent: int) representing the rounded

value (-1)**sign * significand * 10**exponent.

In the special case where n = 0, returns a significand of zero and

an exponent of 1 - figures, for compatibility with formatting.

Otherwise, the returned significand satisfies

10**(figures - 1) <= significand < 10**figures.

d must be positive, but n and d need not be relatively prime.

figures must be positive.

"""

# Special case for n == 0.

if n == 0:

return False, 0, 1 - figures

# Find integer m satisfying 10**(m - 1) <= abs(n)/d <= 10**m. (If abs(n)/d

# is a power of 10, either of the two possible values for m is fine.)

str_n, str_d = str(abs(n)), str(d)

m = len(str_n) - len(str_d) + (str_d <= str_n)

# Round to a multiple of 10**(m - figures). The significand we get

# satisfies 10**(figures - 1) <= significand <= 10**figures.

exponent = m - figures

sign, significand = _round_to_exponent(n, d, exponent)

# Adjust in the case where significand == 10**figures, to ensure that

# 10**(figures - 1) <= significand < 10**figures.

if len(str(significand)) == figures + 1:

significand //= 10

exponent += 1

return sign, significand, exponent

# Pattern for matching non-float-style format specifications.

_GENERAL_FORMAT_SPECIFICATION_MATCHER = re.compile(r"""

(?:

(?P<fill>.)?

(?P<align>[<>=^])

)?

(?P<sign>[-+ ]?)

# Alt flag forces a slash and denominator in the output, even for

# integer-valued Fraction objects.

(?P<alt>\#)?

# We don't implement the zeropad flag since there's no single obvious way

# to interpret it.

(?P<minimumwidth>0|[1-9][0-9]*)?

(?P<thousands_sep>[,_])?

""", re.DOTALL | re.VERBOSE).fullmatch

# Pattern for matching float-style format specifications;

# supports 'e', 'E', 'f', 'F', 'g', 'G' and '%' presentation types.

_FLOAT_FORMAT_SPECIFICATION_MATCHER = re.compile(r"""

(?:

(?P<fill>.)?

(?P<align>[<>=^])

)?

(?P<sign>[-+ ]?)

(?P<no_neg_zero>z)?

(?P<alt>\#)?

# A '0' that's *not* followed by another digit is parsed as a minimum width

# rather than a zeropad flag.

(?P<zeropad>0(?=[0-9]))?

(?P<minimumwidth>0|[1-9][0-9]*)?

(?P<thousands_sep>[,_])?

(?:\.(?P<precision>0|[1-9][0-9]*))?

(?P<presentation_type>[eEfFgG%])

""", re.DOTALL | re.VERBOSE).fullmatch

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):

"""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:

decimal = decimal.replace('_', '')

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)

g = math.gcd(numerator, denominator)

if denominator < 0:

g = -g

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._from_coprime_ints(*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._from_coprime_ints(*dec.as_integer_ratio())

@classmethod

def _from_coprime_ints(cls, numerator, denominator, /):

"""Convert a pair of ints to a rational number, for internal use.

The ratio of integers should be in lowest terms and the denominator

should be positive.

"""

obj = super(Fraction, cls).__new__(cls)

obj._numerator = numerator

obj._denominator = denominator

return obj

def is_integer(self):

"""Return True if the Fraction is an integer."""

return self._denominator == 1

def as_integer_ratio(self):

"""Return a pair of integers, whose ratio is equal to the original Fraction.

The ratio is in lowest terms and has 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

# Determine which of the candidates (p0+k*p1)/(q0+k*q1) and p1/q1 is

# closer to self. The distance between them is 1/(q1*(q0+k*q1)), while

# the distance from p1/q1 to self is d/(q1*self._denominator). So we

# need to compare 2*(q0+k*q1) with self._denominator/d.

if 2*d*(q0+k*q1) <= self._denominator:

return Fraction._from_coprime_ints(p1, q1)

else:

return Fraction._from_coprime_ints(p0+k*p1, q0+k*q1)

@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 _format_general(self, match):

"""Helper method for __format__.

Handles fill, alignment, signs, and thousands separators in the

case of no presentation type.

"""

# Validate and parse the format specifier.

fill = match["fill"] or " "

align = match["align"] or ">"

pos_sign = "" if match["sign"] == "-" else match["sign"]

alternate_form = bool(match["alt"])

minimumwidth = int(match["minimumwidth"] or "0")

thousands_sep = match["thousands_sep"] or ''

# Determine the body and sign representation.

n, d = self._numerator, self._denominator

if d > 1 or alternate_form:

body = f"{abs(n):{thousands_sep}}/{d:{thousands_sep}}"

else:

body = f"{abs(n):{thousands_sep}}"

sign = '-' if n < 0 else pos_sign

# Pad with fill character if necessary and return.

padding = fill * (minimumwidth - len(sign) - len(body))

if align == ">":

return padding + sign + body

elif align == "<":

return sign + body + padding

elif align == "^":

half = len(padding) // 2

return padding[:half] + sign + body + padding[half:]

else: # align == "="

return sign + padding + body

def _format_float_style(self, match):

"""Helper method for __format__; handles float presentation types."""

fill = match["fill"] or " "

align = match["align"] or ">"

pos_sign = "" if match["sign"] == "-" else match["sign"]

no_neg_zero = bool(match["no_neg_zero"])

alternate_form = bool(match["alt"])

zeropad = bool(match["zeropad"])

minimumwidth = int(match["minimumwidth"] or "0")

thousands_sep = match["thousands_sep"]

precision = int(match["precision"] or "6")

presentation_type = match["presentation_type"]

trim_zeros = presentation_type in "gG" and not alternate_form

trim_point = not alternate_form

exponent_indicator = "E" if presentation_type in "EFG" else "e"

# Round to get the digits we need, figure out where to place the point,

# and decide whether to use scientific notation. 'point_pos' is the

# relative to the _end_ of the digit string: that is, it's the number

# of digits that should follow the point.

if presentation_type in "fF%":

exponent = -precision

if presentation_type == "%":

exponent -= 2

negative, significand = _round_to_exponent(

self._numerator, self._denominator, exponent, no_neg_zero)

scientific = False

point_pos = precision

else: # presentation_type in "eEgG"

figures = (

max(precision, 1)

if presentation_type in "gG"

else precision + 1

)

negative, significand, exponent = _round_to_figures(

self._numerator, self._denominator, figures)

scientific = (

presentation_type in "eE"

or exponent > 0

or exponent + figures <= -4

)

point_pos = figures - 1 if scientific else -exponent

# Get the suffix - the part following the digits, if any.

if presentation_type == "%":

suffix = "%"

elif scientific:

suffix = f"{exponent_indicator}{exponent + point_pos:+03d}"

else:

suffix = ""

# String of output digits, padded sufficiently with zeros on the left

# so that we'll have at least one digit before the decimal point.

digits = f"{significand:0{point_pos + 1}d}"

# Before padding, the output has the form f"{sign}{leading}{trailing}",

# where `leading` includes thousands separators if necessary and

# `trailing` includes the decimal separator where appropriate.

sign = "-" if negative else pos_sign

leading = digits[: len(digits) - point_pos]

frac_part = digits[len(digits) - point_pos :]

if trim_zeros:

frac_part = frac_part.rstrip("0")

separator = "" if trim_point and not frac_part else "."

trailing = separator + frac_part + suffix

# Do zero padding if required.

if zeropad:

min_leading = minimumwidth - len(sign) - len(trailing)

# When adding thousands separators, they'll be added to the

# zero-padded portion too, so we need to compensate.

leading = leading.zfill(

3 * min_leading // 4 + 1 if thousands_sep else min_leading

)

# Insert thousands separators if required.

if thousands_sep:

first_pos = 1 + (len(leading) - 1) % 3

leading = leading[:first_pos] + "".join(

thousands_sep + leading[pos : pos + 3]

for pos in range(first_pos, len(leading), 3)

)

# We now have a sign and a body. Pad with fill character if necessary

# and return.

body = leading + trailing

padding = fill * (minimumwidth - len(sign) - len(body))

if align == ">":

return padding + sign + body

elif align == "<":

return sign + body + padding

elif align == "^":

half = len(padding) // 2

return padding[:half] + sign + body + padding[half:]

else: # align == "="

return sign + padding + body

def __format__(self, format_spec, /):

"""Format this fraction according to the given format specification."""

if match := _GENERAL_FORMAT_SPECIFICATION_MATCHER(format_spec):

return self._format_general(match)

if match := _FLOAT_FORMAT_SPECIFICATION_MATCHER(format_spec):

# Refuse the temptation to guess if both alignment _and_

# zero padding are specified.

if match["align"] is None or match["zeropad"] is None:

return self._format_float_style(match)

raise ValueError(

f"Invalid format specifier {format_spec!r} "

f"for object of type {type(self).__name__!r}"

)

def _operator_fallbacks(monomorphic_operator, fallback_operator,

handle_complex=True):

"""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, Fraction):

return monomorphic_operator(a, b)

elif isinstance(b, int):

return monomorphic_operator(a, Fraction(b))

elif isinstance(b, float):

return fallback_operator(float(a), b)

elif handle_complex and isinstance(b, complex):

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

elif isinstance(a, numbers.Real):

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

elif handle_complex and isinstance(a, numbers.Complex):

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

else:

return NotImplemented

reverse.__name__ = '__r' + fallback_operator.__name__ + '__'

reverse.__doc__ = monomorphic_operator.__doc__

return forward, reverse

# Rational arithmetic algorithms: Knuth, TAOCP, Volume 2, 4.5.1.

#

# Assume input fractions a and b are normalized.

#

# 1) Consider addition/subtraction.

#

# Let g = gcd(da, db). Then

#

# na nb na*db ± nb*da

# a ± b == -- ± -- == ------------- ==

# da db da*db

#

# na*(db//g) ± nb*(da//g) t

# == ----------------------- == -

# (da*db)//g d

#

# Now, if g > 1, we're working with smaller integers.

#

# Note, that t, (da//g) and (db//g) are pairwise coprime.

#

# Indeed, (da//g) and (db//g) share no common factors (they were

# removed) and da is coprime with na (since input fractions are

# normalized), hence (da//g) and na are coprime. By symmetry,

# (db//g) and nb are coprime too. Then,

#

# gcd(t, da//g) == gcd(na*(db//g), da//g) == 1

# gcd(t, db//g) == gcd(nb*(da//g), db//g) == 1

#

# Above allows us optimize reduction of the result to lowest

# terms. Indeed,

#

# g2 = gcd(t, d) == gcd(t, (da//g)*(db//g)*g) == gcd(t, g)

#

# t//g2 t//g2

# a ± b == ----------------------- == ----------------

# (da//g)*(db//g)*(g//g2) (da//g)*(db//g2)

#

# is a normalized fraction. This is useful because the unnormalized

# denominator d could be much larger than g.

#

# We should special-case g == 1 (and g2 == 1), since 60.8% of

# randomly-chosen integers are coprime:

# https://en.wikipedia.org/wiki/Coprime_integers#Probability_of_coprimality

# Note, that g2 == 1 always for fractions, obtained from floats: here

# g is a power of 2 and the unnormalized numerator t is an odd integer.

#

# 2) Consider multiplication

#

# Let g1 = gcd(na, db) and g2 = gcd(nb, da), then

#

# na*nb na*nb (na//g1)*(nb//g2)

# a*b == ----- == ----- == -----------------

# da*db db*da (db//g1)*(da//g2)

#

# Note, that after divisions we're multiplying smaller integers.

#

# Also, the resulting fraction is normalized, because each of

# two factors in the numerator is coprime to each of the two factors

# in the denominator.

#

# Indeed, pick (na//g1). It's coprime with (da//g2), because input

# fractions are normalized. It's also coprime with (db//g1), because

# common factors are removed by g1 == gcd(na, db).

#

# As for addition/subtraction, we should special-case g1 == 1

# and g2 == 1 for same reason. That happens also for multiplying

# rationals, obtained from floats.

def _add(a, b):

"""a + b"""

na, da = a._numerator, a._denominator

nb, db = b._numerator, b._denominator

g = math.gcd(da, db)

if g == 1:

return Fraction._from_coprime_ints(na * db + da * nb, da * db)

s = da // g

t = na * (db // g) + nb * s

g2 = math.gcd(t, g)

if g2 == 1:

return Fraction._from_coprime_ints(t, s * db)

return Fraction._from_coprime_ints(t // g2, s * (db // g2))

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

def _sub(a, b):

"""a - b"""

na, da = a._numerator, a._denominator

nb, db = b._numerator, b._denominator

g = math.gcd(da, db)

if g == 1:

return Fraction._from_coprime_ints(na * db - da * nb, da * db)

s = da // g

t = na * (db // g) - nb * s

g2 = math.gcd(t, g)

if g2 == 1:

return Fraction._from_coprime_ints(t, s * db)

return Fraction._from_coprime_ints(t // g2, s * (db // g2))

__sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub)

def _mul(a, b):

"""a * b"""

na, da = a._numerator, a._denominator

nb, db = b._numerator, b._denominator

g1 = math.gcd(na, db)

if g1 > 1:

na //= g1

db //= g1

g2 = math.gcd(nb, da)

if g2 > 1:

nb //= g2

da //= g2

return Fraction._from_coprime_ints(na * nb, db * da)

__mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul)

def _div(a, b):

"""a / b"""

# Same as _mul(), with inversed b.

nb, db = b._numerator, b._denominator

if nb == 0:

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

na, da = a._numerator, a._denominator

g1 = math.gcd(na, nb)

if g1 > 1:

na //= g1

nb //= g1

g2 = math.gcd(db, da)

if g2 > 1:

da //= g2

db //= g2

n, d = na * db, nb * da

if d < 0:

n, d = -n, -d

return Fraction._from_coprime_ints(n, d)

__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, False)

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, False)

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, False)

def __pow__(a, b, modulo=None):

"""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 modulo is not None:

return NotImplemented

if isinstance(b, numbers.Rational):

if b.denominator == 1:

power = b.numerator

if power >= 0:

return Fraction._from_coprime_ints(a._numerator ** power,

a._denominator ** power)

elif a._numerator > 0:

return Fraction._from_coprime_ints(a._denominator ** -power,

a._numerator ** -power)

elif a._numerator == 0:

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

a._denominator ** -power)

else:

return Fraction._from_coprime_ints((-a._denominator) ** -power,

(-a._numerator) ** -power)

else:

# A fractional power will generally produce an

# irrational number.

return float(a) ** float(b)

elif isinstance(b, (float, complex)):

return float(a) ** b

else:

return NotImplemented

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._from_coprime_ints(a._numerator, a._denominator)

def __neg__(a):

"""-a"""

return Fraction._from_coprime_ints(-a._numerator, a._denominator)

def __abs__(a):

"""abs(a)"""

return Fraction._from_coprime_ints(abs(a._numerator), a._denominator)

def __int__(a, _index=operator.index):

"""int(a)"""

if a._numerator < 0:

return _index(-(-a._numerator // a._denominator))

else:

return _index(a._numerator // a._denominator)

def __trunc__(a):

"""math.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:

d = self._denominator

floor, remainder = divmod(self._numerator, d)

if remainder * 2 < d:

return floor

elif remainder * 2 > d:

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)"""

return _hash_algorithm(self._numerator, self._denominator)

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):