# Originally contributed by Sjoerd Mullender.
# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.
"""Rational, infinite-precision, real numbers."""
import math
import numbers
import operator
import re
__all__ = ["Rational"]
RationalAbc = numbers.Rational
def gcd(a, b):
"""Calculate the Greatest Common Divisor of a and b.
Unless b==0, the result will have the same sign as b (so that when
b is divided by it, the result comes out positive).
"""
while b:
a, b = b, a%b
return a
_RATIONAL_FORMAT = re.compile(r"""
\A\s* # optional whitespace at the start, then
(?P<sign>[-+]?) # an optional sign, then
(?=\d|\.\d) # lookahead for digit or .digit
(?P<num>\d*) # numerator (possibly empty)
(?: # followed by an optional
/(?P<denom>\d+) # / and denominator
| # or
\.(?P<decimal>\d*) # decimal point and fractional part
)?
\s*\Z # and optional whitespace to finish
""", re.VERBOSE)
class Rational(RationalAbc):
"""This class implements rational numbers.
Rational(8, 6) will produce a rational number equivalent to
4/3. Both arguments must be Integral. The numerator defaults to 0
and the denominator defaults to 1 so that Rational(3) == 3 and
Rational() == 0.
Rationals can also be constructed from strings of the form
'[-+]?[0-9]+((/|.)[0-9]+)?', optionally surrounded by spaces.
"""
__slots__ = ('_numerator', '_denominator')
# We're immutable, so use __new__ not __init__
def __new__(cls, numerator=0, denominator=1):
"""Constructs a Rational.
Takes a string like '3/2' or '1.5', another Rational, or a
numerator/denominator pair.
"""
self = super(Rational, cls).__new__(cls)
if denominator == 1:
if isinstance(numerator, str):
# Handle construction from strings.
input = numerator
m = _RATIONAL_FORMAT.match(input)
if m is None:
raise ValueError('Invalid literal for Rational: ' + input)
numerator = m.group('num')
decimal = m.group('decimal')
if decimal:
# The literal is a decimal number.
numerator = int(numerator + decimal)
denominator = 10**len(decimal)
else:
# The literal is an integer or fraction.
numerator = int(numerator)
# Default denominator to 1.
denominator = int(m.group('denom') or 1)
if m.group('sign') == '-':
numerator = -numerator
elif (not isinstance(numerator, numbers.Integral) and
isinstance(numerator, RationalAbc)):
# Handle copies from other rationals.
other_rational = numerator
numerator = other_rational.numerator
denominator = other_rational.denominator
if (not isinstance(numerator, numbers.Integral) or
not isinstance(denominator, numbers.Integral)):
raise TypeError("Rational(%(numerator)s, %(denominator)s):"
" Both arguments must be integral." % locals())
if denominator == 0:
raise ZeroDivisionError('Rational(%s, 0)' % numerator)
g = gcd(numerator, denominator)
self._numerator = int(numerator // g)
self._denominator = int(denominator // g)
return self
@classmethod
def from_float(cls, f):
"""Converts a finite float to a rational number, exactly.
Beware that Rational.from_float(0.3) != Rational(3, 10).
"""
if not isinstance(f, float):
raise TypeError("%s.from_float() only takes floats, not %r (%s)" %
(cls.__name__, f, type(f).__name__))
if math.isnan(f) or math.isinf(f):
raise TypeError("Cannot convert %r to %s." % (f, cls.__name__))
return cls(*f.as_integer_ratio())
@classmethod
def from_decimal(cls, dec):
"""Converts a finite Decimal instance to a rational number, exactly."""
from decimal import Decimal
if not isinstance(dec, Decimal):
raise TypeError(
"%s.from_decimal() only takes Decimals, not %r (%s)" %
(cls.__name__, dec, type(dec).__name__))
if not dec.is_finite():
# Catches infinities and nans.
raise TypeError("Cannot convert %s to %s." % (dec, cls.__name__))
sign, digits, exp = dec.as_tuple()
digits = int(''.join(map(str, digits)))
if sign:
digits = -digits
if exp >= 0:
return cls(digits * 10 ** exp)
else:
return cls(digits, 10 ** -exp)
@classmethod
def from_continued_fraction(cls, seq):
'Build a Rational from a continued fraction expessed as a sequence'
n, d = 1, 0
for e in reversed(seq):
n, d = d, n
n += e * d
return cls(n, d) if seq else cls(0)
def as_continued_fraction(self):
'Return continued fraction expressed as a list'
n = self.numerator
d = self.denominator
cf = []
while d:
e = int(n // d)
cf.append(e)
n -= e * d
n, d = d, n
return cf
def approximate(self, max_denominator):
'Best rational approximation with a denominator <= max_denominator'
# XXX First cut at algorithm
# Still needs rounding rules as specified at
# http://en.wikipedia.org/wiki/Continued_fraction
if self.denominator <= max_denominator:
return self
cf = self.as_continued_fraction()
result = Rational(0)
for i in range(1, len(cf)):
new = self.from_continued_fraction(cf[:i])
if new.denominator > max_denominator:
break
result = new
return result
@property
def numerator(a):
return a._numerator
@property
def denominator(a):
return a._denominator
def __repr__(self):
"""repr(self)"""
return ('Rational(%r,%r)' % (self.numerator, self.denominator))
def __str__(self):
"""str(self)"""
if self.denominator == 1:
return str(self.numerator)
else:
return '%s/%s' % (self.numerator, self.denominator)
def _operator_fallbacks(monomorphic_operator, fallback_operator):
"""Generates forward and reverse operators given a purely-rational
operator and a function from the operator module.
Use this like:
__op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op)
In general, we want to implement the arithmetic operations so
that mixed-mode operations either call an implementation whose
author knew about the types of both arguments, or convert both
to the nearest built in type and do the operation there. In
Rational, 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, Rational)):
return Rational(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, RationalAbc):
return Rational(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
Rational. I'll refer to all of the above code that doesn't
refer to Rational, float, or complex as "boilerplate". 'r'
will be an instance of Rational, which is a subtype of
RationalAbc (r : Rational <: RationalAbc), and b : B <:
Complex. The first three involve 'r + b':
1. If B <: Rational, int, float, or complex, we handle
that specially, and all is well.
2. If Rational 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 RationalAbc
here, even though we could get an exact answer, in case
the other type wants to do something special.
3. If B <: Rational, Python tries B.__radd__ before
Rational.__add__. This is ok, because it was
implemented with knowledge of Rational, 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 Rational in its implementation, and that it
uses similar boilerplate code:
4. If B <: RationalAbc, then __radd_ converts both to the
builtin rational type (hey look, that's us) and
proceeds.
5. Otherwise, __radd__ tries to find the nearest common
base ABC, and fall back to its builtin type. Since this
class doesn't subclass a concrete type, there's no
implementation to fall back to, so we need to try as
hard as possible to return an actual value, or the user
will get a TypeError.
"""
def forward(a, b):
if isinstance(b, (int, Rational)):
return monomorphic_operator(a, b)
elif isinstance(b, float):
return fallback_operator(float(a), b)
elif isinstance(b, complex):
return fallback_operator(complex(a), b)
else:
return NotImplemented
forward.__name__ = '__' + fallback_operator.__name__ + '__'
forward.__doc__ = monomorphic_operator.__doc__
def reverse(b, a):
if isinstance(a, RationalAbc):
# Includes ints.
return monomorphic_operator(a, b)
elif isinstance(a, numbers.Real):
return fallback_operator(float(a), float(b))
elif isinstance(a, numbers.Complex):
return fallback_operator(complex(a), complex(b))
else:
return NotImplemented
reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
reverse.__doc__ = monomorphic_operator.__doc__
return forward, reverse
def _add(a, b):
"""a + b"""
return Rational(a.numerator * b.denominator +
b.numerator * a.denominator,
a.denominator * b.denominator)
__add__, __radd__ = _operator_fallbacks(_add, operator.add)
def _sub(a, b):
"""a - b"""
return Rational(a.numerator * b.denominator -
b.numerator * a.denominator,
a.denominator * b.denominator)
__sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub)
def _mul(a, b):
"""a * b"""
return Rational(a.numerator * b.numerator, a.denominator * b.denominator)
__mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul)
def _div(a, b):
"""a / b"""
return Rational(a.numerator * b.denominator,
a.denominator * b.numerator)
__truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv)
def __floordiv__(a, b):
"""a // b"""
return math.floor(a / b)
def __rfloordiv__(b, a):
"""a // b"""
return math.floor(a / b)
def __mod__(a, b):
"""a % b"""
div = a // b
return a - b * div
def __rmod__(b, a):
"""a % b"""
div = a // b
return a - b * div
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, RationalAbc):
if b.denominator == 1:
power = b.numerator
if power >= 0:
return Rational(a.numerator ** power,
a.denominator ** power)
else:
return Rational(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, RationalAbc):
return Rational(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 Rational"""
return Rational(a.numerator, a.denominator)
def __neg__(a):
"""-a"""
return Rational(-a.numerator, a.denominator)
def __abs__(a):
"""abs(a)"""
return Rational(abs(a.numerator), a.denominator)
def __trunc__(a):
"""trunc(a)"""
if a.numerator < 0:
return -(-a.numerator // a.denominator)
else:
return a.numerator // a.denominator
def __floor__(a):
"""Will be math.floor(a) in 3.0."""
return a.numerator // a.denominator
def __ceil__(a):
"""Will be math.ceil(a) in 3.0."""
# The negations cleverly convince floordiv to return the ceiling.
return -(-a.numerator // a.denominator)
def __round__(self, ndigits=None):
"""Will be round(self, ndigits) in 3.0.
Rounds half toward even.
"""
if ndigits is None:
floor, remainder = divmod(self.numerator, self.denominator)
if remainder * 2 < self.denominator:
return floor
elif remainder * 2 > self.denominator:
return floor + 1
# Deal with the half case:
elif floor % 2 == 0:
return floor
else:
return floor + 1
shift = 10**abs(ndigits)
# See _operator_fallbacks.forward to check that the results of
# these operations will always be Rational and therefore have
# round().
if ndigits > 0:
return Rational(round(self * shift), shift)
else:
return Rational(round(self / shift) * shift)
def __hash__(self):
"""hash(self)
Tricky because values that are exactly representable as a
float must have the same hash as that float.
"""
# XXX since this method is expensive, consider caching the result
if self.denominator == 1:
# Get integers right.
return hash(self.numerator)
# Expensive check, but definitely correct.
if self == float(self):
return hash(float(self))
else:
# Use tuple's hash to avoid a high collision rate on
# simple fractions.
return hash((self.numerator, self.denominator))
def __eq__(a, b):
"""a == b"""
if isinstance(b, RationalAbc):
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):
return a == a.from_float(b)
else:
# XXX: If b.__eq__ is implemented like this method, it may
# give the wrong answer after float(a) changes a's
# value. Better ways of doing this are welcome.
return float(a) == b
def _subtractAndCompareToZero(a, b, op):
"""Helper function for comparison operators.
Subtracts b from a, exactly if possible, and compares the
result with 0 using op, in such a way that the comparison
won't recurse. If the difference raises a TypeError, returns
NotImplemented instead.
"""
if isinstance(b, numbers.Complex) and b.imag == 0:
b = b.real
if isinstance(b, float):
b = a.from_float(b)
try:
# XXX: If b <: Real but not <: RationalAbc, this is likely
# to fall back to a float. If the actual values differ by
# less than MIN_FLOAT, this could falsely call them equal,
# which would make <= inconsistent with ==. Better ways of
# doing this are welcome.
diff = a - b
except TypeError:
return NotImplemented
if isinstance(diff, RationalAbc):
return op(diff.numerator, 0)
return op(diff, 0)
def __lt__(a, b):
"""a < b"""
return a._subtractAndCompareToZero(b, operator.lt)
def __gt__(a, b):
"""a > b"""
return a._subtractAndCompareToZero(b, operator.gt)
def __le__(a, b):
"""a <= b"""
return a._subtractAndCompareToZero(b, operator.le)
def __ge__(a, b):
"""a >= b"""
return a._subtractAndCompareToZero(b, operator.ge)
def __bool__(a):
"""a != 0"""
return a.numerator != 0
# support for pickling, copy, and deepcopy
def __reduce__(self):
return (self.__class__, (str(self),))
def __copy__(self):
if type(self) == Rational:
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) == Rational:
return self # My components are also immutable
return self.__class__(self.numerator, self.denominator)