Library import 16 (#2433)

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diff --git a/contrib/tools/python3/Lib/fractions.py b/contrib/tools/python3/Lib/fractions.py
new file mode 100644
index 0000000000..88b418fe38
--- /dev/null
+++ b/contrib/tools/python3/Lib/fractions.py
@@ -0,0 +1,988 @@
+# 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 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__(self, format_spec, /):
+        """Format this fraction according to the given format specification."""
+
+        # Backwards compatiblility with existing formatting.
+        if not format_spec:
+            return str(self)
+
+        # Validate and parse the format specifier.
+        match = _FLOAT_FORMAT_SPECIFICATION_MATCHER(format_spec)
+        if match is None:
+            raise ValueError(
+                f"Invalid format specifier {format_spec!r} "
+                f"for object of type {type(self).__name__!r}"
+            )
+        elif match["align"] is not None and match["zeropad"] is not None:
+            # Avoid the temptation to guess.
+            raise ValueError(
+                f"Invalid format specifier {format_spec!r} "
+                f"for object of type {type(self).__name__!r}; "
+                "can't use explicit alignment when zero-padding"
+            )
+        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 _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, 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 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(Fraction(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
+
+    # 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)
+
+    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._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)
+        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._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):
+                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__, (self._numerator, self._denominator))
+
+    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)