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c8d2630995
Any objects that have the as_integer_ratio() method (e.g. numpy.float128) can now be converted to a fraction.
1042 lines
39 KiB
Python
1042 lines
39 KiB
Python
# Originally contributed by Sjoerd Mullender.
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# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.
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"""Fraction, infinite-precision, rational numbers."""
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import functools
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import math
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import numbers
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import operator
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import re
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import sys
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__all__ = ['Fraction']
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# Constants related to the hash implementation; hash(x) is based
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# on the reduction of x modulo the prime _PyHASH_MODULUS.
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_PyHASH_MODULUS = sys.hash_info.modulus
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# Value to be used for rationals that reduce to infinity modulo
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# _PyHASH_MODULUS.
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_PyHASH_INF = sys.hash_info.inf
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@functools.lru_cache(maxsize = 1 << 14)
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def _hash_algorithm(numerator, denominator):
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# To make sure that the hash of a Fraction agrees with the hash
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# of a numerically equal integer, float or Decimal instance, we
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# follow the rules for numeric hashes outlined in the
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# documentation. (See library docs, 'Built-in Types').
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try:
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dinv = pow(denominator, -1, _PyHASH_MODULUS)
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except ValueError:
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# ValueError means there is no modular inverse.
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hash_ = _PyHASH_INF
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else:
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# The general algorithm now specifies that the absolute value of
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# the hash is
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# (|N| * dinv) % P
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# where N is self._numerator and P is _PyHASH_MODULUS. That's
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# optimized here in two ways: first, for a non-negative int i,
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# hash(i) == i % P, but the int hash implementation doesn't need
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# to divide, and is faster than doing % P explicitly. So we do
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# hash(|N| * dinv)
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# instead. Second, N is unbounded, so its product with dinv may
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# be arbitrarily expensive to compute. The final answer is the
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# same if we use the bounded |N| % P instead, which can again
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# be done with an int hash() call. If 0 <= i < P, hash(i) == i,
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# so this nested hash() call wastes a bit of time making a
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# redundant copy when |N| < P, but can save an arbitrarily large
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# amount of computation for large |N|.
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hash_ = hash(hash(abs(numerator)) * dinv)
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result = hash_ if numerator >= 0 else -hash_
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return -2 if result == -1 else result
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_RATIONAL_FORMAT = re.compile(r"""
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\A\s* # optional whitespace at the start,
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(?P<sign>[-+]?) # an optional sign, then
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(?=\d|\.\d) # lookahead for digit or .digit
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(?P<num>\d*|\d+(_\d+)*) # numerator (possibly empty)
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(?: # followed by
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(?:\s*/\s*(?P<denom>\d+(_\d+)*))? # an optional denominator
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| # or
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(?:\.(?P<decimal>\d*|\d+(_\d+)*))? # an optional fractional part
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(?:E(?P<exp>[-+]?\d+(_\d+)*))? # and optional exponent
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)
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\s*\Z # and optional whitespace to finish
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""", re.VERBOSE | re.IGNORECASE)
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# Helpers for formatting
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def _round_to_exponent(n, d, exponent, no_neg_zero=False):
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"""Round a rational number to the nearest multiple of a given power of 10.
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Rounds the rational number n/d to the nearest integer multiple of
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10**exponent, rounding to the nearest even integer multiple in the case of
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a tie. Returns a pair (sign: bool, significand: int) representing the
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rounded value (-1)**sign * significand * 10**exponent.
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If no_neg_zero is true, then the returned sign will always be False when
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the significand is zero. Otherwise, the sign reflects the sign of the
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input.
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d must be positive, but n and d need not be relatively prime.
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"""
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if exponent >= 0:
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d *= 10**exponent
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else:
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n *= 10**-exponent
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# The divmod quotient is correct for round-ties-towards-positive-infinity;
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# In the case of a tie, we zero out the least significant bit of q.
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q, r = divmod(n + (d >> 1), d)
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if r == 0 and d & 1 == 0:
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q &= -2
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sign = q < 0 if no_neg_zero else n < 0
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return sign, abs(q)
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def _round_to_figures(n, d, figures):
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"""Round a rational number to a given number of significant figures.
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Rounds the rational number n/d to the given number of significant figures
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using the round-ties-to-even rule, and returns a triple
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(sign: bool, significand: int, exponent: int) representing the rounded
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value (-1)**sign * significand * 10**exponent.
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In the special case where n = 0, returns a significand of zero and
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an exponent of 1 - figures, for compatibility with formatting.
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Otherwise, the returned significand satisfies
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10**(figures - 1) <= significand < 10**figures.
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d must be positive, but n and d need not be relatively prime.
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figures must be positive.
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"""
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# Special case for n == 0.
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if n == 0:
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return False, 0, 1 - figures
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# Find integer m satisfying 10**(m - 1) <= abs(n)/d <= 10**m. (If abs(n)/d
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# is a power of 10, either of the two possible values for m is fine.)
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str_n, str_d = str(abs(n)), str(d)
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m = len(str_n) - len(str_d) + (str_d <= str_n)
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# Round to a multiple of 10**(m - figures). The significand we get
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# satisfies 10**(figures - 1) <= significand <= 10**figures.
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exponent = m - figures
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sign, significand = _round_to_exponent(n, d, exponent)
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# Adjust in the case where significand == 10**figures, to ensure that
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# 10**(figures - 1) <= significand < 10**figures.
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if len(str(significand)) == figures + 1:
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significand //= 10
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exponent += 1
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return sign, significand, exponent
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# Pattern for matching non-float-style format specifications.
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_GENERAL_FORMAT_SPECIFICATION_MATCHER = re.compile(r"""
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(?:
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(?P<fill>.)?
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(?P<align>[<>=^])
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)?
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(?P<sign>[-+ ]?)
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# Alt flag forces a slash and denominator in the output, even for
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# integer-valued Fraction objects.
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(?P<alt>\#)?
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# We don't implement the zeropad flag since there's no single obvious way
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# to interpret it.
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(?P<minimumwidth>0|[1-9][0-9]*)?
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(?P<thousands_sep>[,_])?
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""", re.DOTALL | re.VERBOSE).fullmatch
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# Pattern for matching float-style format specifications;
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# supports 'e', 'E', 'f', 'F', 'g', 'G' and '%' presentation types.
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_FLOAT_FORMAT_SPECIFICATION_MATCHER = re.compile(r"""
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(?:
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(?P<fill>.)?
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(?P<align>[<>=^])
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)?
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(?P<sign>[-+ ]?)
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(?P<no_neg_zero>z)?
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(?P<alt>\#)?
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# A '0' that's *not* followed by another digit is parsed as a minimum width
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# rather than a zeropad flag.
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(?P<zeropad>0(?=[0-9]))?
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(?P<minimumwidth>0|[1-9][0-9]*)?
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(?P<thousands_sep>[,_])?
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(?:\.(?P<precision>0|[1-9][0-9]*))?
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(?P<presentation_type>[eEfFgG%])
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""", re.DOTALL | re.VERBOSE).fullmatch
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class Fraction(numbers.Rational):
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"""This class implements rational numbers.
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In the two-argument form of the constructor, Fraction(8, 6) will
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produce a rational number equivalent to 4/3. Both arguments must
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be Rational. The numerator defaults to 0 and the denominator
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defaults to 1 so that Fraction(3) == 3 and Fraction() == 0.
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Fractions can also be constructed from:
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- numeric strings similar to those accepted by the
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float constructor (for example, '-2.3' or '1e10')
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- strings of the form '123/456'
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- float and Decimal instances
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- other Rational instances (including integers)
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"""
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__slots__ = ('_numerator', '_denominator')
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# We're immutable, so use __new__ not __init__
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def __new__(cls, numerator=0, denominator=None):
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"""Constructs a Rational.
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Takes a string like '3/2' or '1.5', another Rational instance, a
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numerator/denominator pair, or a float.
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Examples
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--------
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>>> Fraction(10, -8)
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Fraction(-5, 4)
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>>> Fraction(Fraction(1, 7), 5)
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Fraction(1, 35)
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>>> Fraction(Fraction(1, 7), Fraction(2, 3))
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Fraction(3, 14)
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>>> Fraction('314')
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Fraction(314, 1)
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>>> Fraction('-35/4')
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Fraction(-35, 4)
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>>> Fraction('3.1415') # conversion from numeric string
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Fraction(6283, 2000)
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>>> Fraction('-47e-2') # string may include a decimal exponent
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Fraction(-47, 100)
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>>> Fraction(1.47) # direct construction from float (exact conversion)
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Fraction(6620291452234629, 4503599627370496)
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>>> Fraction(2.25)
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Fraction(9, 4)
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>>> Fraction(Decimal('1.47'))
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Fraction(147, 100)
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"""
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self = super(Fraction, cls).__new__(cls)
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if denominator is None:
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if type(numerator) is int:
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self._numerator = numerator
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self._denominator = 1
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return self
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elif isinstance(numerator, numbers.Rational):
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self._numerator = numerator.numerator
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self._denominator = numerator.denominator
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return self
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elif (isinstance(numerator, float) or
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(not isinstance(numerator, type) and
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hasattr(numerator, 'as_integer_ratio'))):
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# Exact conversion
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self._numerator, self._denominator = numerator.as_integer_ratio()
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return self
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elif isinstance(numerator, str):
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# Handle construction from strings.
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m = _RATIONAL_FORMAT.match(numerator)
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if m is None:
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raise ValueError('Invalid literal for Fraction: %r' %
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numerator)
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numerator = int(m.group('num') or '0')
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denom = m.group('denom')
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if denom:
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denominator = int(denom)
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else:
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denominator = 1
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decimal = m.group('decimal')
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if decimal:
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decimal = decimal.replace('_', '')
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scale = 10**len(decimal)
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numerator = numerator * scale + int(decimal)
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denominator *= scale
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exp = m.group('exp')
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if exp:
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exp = int(exp)
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if exp >= 0:
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numerator *= 10**exp
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else:
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denominator *= 10**-exp
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if m.group('sign') == '-':
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numerator = -numerator
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else:
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raise TypeError("argument should be a string or a number")
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elif type(numerator) is int is type(denominator):
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pass # *very* normal case
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elif (isinstance(numerator, numbers.Rational) and
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isinstance(denominator, numbers.Rational)):
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numerator, denominator = (
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numerator.numerator * denominator.denominator,
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denominator.numerator * numerator.denominator
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)
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else:
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raise TypeError("both arguments should be "
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"Rational instances")
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if denominator == 0:
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raise ZeroDivisionError('Fraction(%s, 0)' % numerator)
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g = math.gcd(numerator, denominator)
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if denominator < 0:
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g = -g
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numerator //= g
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denominator //= g
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self._numerator = numerator
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self._denominator = denominator
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return self
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@classmethod
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def from_float(cls, f):
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"""Converts a finite float to a rational number, exactly.
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Beware that Fraction.from_float(0.3) != Fraction(3, 10).
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"""
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if isinstance(f, numbers.Integral):
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return cls(f)
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elif not isinstance(f, float):
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raise TypeError("%s.from_float() only takes floats, not %r (%s)" %
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(cls.__name__, f, type(f).__name__))
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return cls._from_coprime_ints(*f.as_integer_ratio())
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@classmethod
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def from_decimal(cls, dec):
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"""Converts a finite Decimal instance to a rational number, exactly."""
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from decimal import Decimal
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if isinstance(dec, numbers.Integral):
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dec = Decimal(int(dec))
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elif not isinstance(dec, Decimal):
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raise TypeError(
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"%s.from_decimal() only takes Decimals, not %r (%s)" %
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(cls.__name__, dec, type(dec).__name__))
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return cls._from_coprime_ints(*dec.as_integer_ratio())
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@classmethod
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def _from_coprime_ints(cls, numerator, denominator, /):
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"""Convert a pair of ints to a rational number, for internal use.
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The ratio of integers should be in lowest terms and the denominator
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should be positive.
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"""
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obj = super(Fraction, cls).__new__(cls)
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obj._numerator = numerator
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obj._denominator = denominator
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return obj
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def is_integer(self):
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"""Return True if the Fraction is an integer."""
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return self._denominator == 1
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def as_integer_ratio(self):
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"""Return a pair of integers, whose ratio is equal to the original Fraction.
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The ratio is in lowest terms and has a positive denominator.
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"""
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return (self._numerator, self._denominator)
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def limit_denominator(self, max_denominator=1000000):
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"""Closest Fraction to self with denominator at most max_denominator.
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>>> Fraction('3.141592653589793').limit_denominator(10)
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Fraction(22, 7)
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>>> Fraction('3.141592653589793').limit_denominator(100)
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Fraction(311, 99)
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>>> Fraction(4321, 8765).limit_denominator(10000)
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Fraction(4321, 8765)
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"""
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# Algorithm notes: For any real number x, define a *best upper
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# approximation* to x to be a rational number p/q such that:
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#
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# (1) p/q >= x, and
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# (2) if p/q > r/s >= x then s > q, for any rational r/s.
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#
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# Define *best lower approximation* similarly. Then it can be
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# proved that a rational number is a best upper or lower
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# approximation to x if, and only if, it is a convergent or
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# semiconvergent of the (unique shortest) continued fraction
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# associated to x.
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#
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# To find a best rational approximation with denominator <= M,
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# we find the best upper and lower approximations with
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# denominator <= M and take whichever of these is closer to x.
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# In the event of a tie, the bound with smaller denominator is
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# chosen. If both denominators are equal (which can happen
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# only when max_denominator == 1 and self is midway between
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# two integers) the lower bound---i.e., the floor of self, is
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# taken.
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if max_denominator < 1:
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raise ValueError("max_denominator should be at least 1")
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if self._denominator <= max_denominator:
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return Fraction(self)
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p0, q0, p1, q1 = 0, 1, 1, 0
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n, d = self._numerator, self._denominator
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while True:
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a = n//d
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q2 = q0+a*q1
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if q2 > max_denominator:
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break
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p0, q0, p1, q1 = p1, q1, p0+a*p1, q2
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n, d = d, n-a*d
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k = (max_denominator-q0)//q1
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# Determine which of the candidates (p0+k*p1)/(q0+k*q1) and p1/q1 is
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# closer to self. The distance between them is 1/(q1*(q0+k*q1)), while
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# the distance from p1/q1 to self is d/(q1*self._denominator). So we
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# need to compare 2*(q0+k*q1) with self._denominator/d.
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if 2*d*(q0+k*q1) <= self._denominator:
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return Fraction._from_coprime_ints(p1, q1)
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else:
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return Fraction._from_coprime_ints(p0+k*p1, q0+k*q1)
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@property
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def numerator(a):
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return a._numerator
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@property
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def denominator(a):
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return a._denominator
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def __repr__(self):
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"""repr(self)"""
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return '%s(%s, %s)' % (self.__class__.__name__,
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self._numerator, self._denominator)
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def __str__(self):
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"""str(self)"""
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if self._denominator == 1:
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return str(self._numerator)
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else:
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return '%s/%s' % (self._numerator, self._denominator)
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def _format_general(self, match):
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"""Helper method for __format__.
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Handles fill, alignment, signs, and thousands separators in the
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case of no presentation type.
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"""
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# Validate and parse the format specifier.
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fill = match["fill"] or " "
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align = match["align"] or ">"
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pos_sign = "" if match["sign"] == "-" else match["sign"]
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alternate_form = bool(match["alt"])
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minimumwidth = int(match["minimumwidth"] or "0")
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thousands_sep = match["thousands_sep"] or ''
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# Determine the body and sign representation.
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n, d = self._numerator, self._denominator
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if d > 1 or alternate_form:
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body = f"{abs(n):{thousands_sep}}/{d:{thousands_sep}}"
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else:
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body = f"{abs(n):{thousands_sep}}"
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sign = '-' if n < 0 else pos_sign
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# Pad with fill character if necessary and return.
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padding = fill * (minimumwidth - len(sign) - len(body))
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if align == ">":
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return padding + sign + body
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elif align == "<":
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return sign + body + padding
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elif align == "^":
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half = len(padding) // 2
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return padding[:half] + sign + body + padding[half:]
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else: # align == "="
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return sign + padding + body
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def _format_float_style(self, match):
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"""Helper method for __format__; handles float presentation types."""
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fill = match["fill"] or " "
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align = match["align"] or ">"
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pos_sign = "" if match["sign"] == "-" else match["sign"]
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no_neg_zero = bool(match["no_neg_zero"])
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alternate_form = bool(match["alt"])
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zeropad = bool(match["zeropad"])
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minimumwidth = int(match["minimumwidth"] or "0")
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thousands_sep = match["thousands_sep"]
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precision = int(match["precision"] or "6")
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presentation_type = match["presentation_type"]
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trim_zeros = presentation_type in "gG" and not alternate_form
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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):
|
|
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)
|