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Add rational.Rational as an implementation of numbers.Rational with infinite

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precision. This has been discussed at http://bugs.python.org/issue1682. It's
useful primarily for teaching, but it also demonstrates how to implement a
member of the numeric tower, including fallbacks for mixed-mode arithmetic.

I expect to write a couple more patches in this area:
 * Rational.from_decimal()
 * Rational.trim/approximate() (maybe with different names)
 * Maybe remove the parentheses from Rational.__str__()
 * Maybe rename one of the Rational classes
 * Maybe make Rational('3/2') work.
This commit is contained in:
Jeffrey Yasskin 2008-01-15 07:46:24 +00:00
parent ca9c6e433c
commit d7b00334f3
6 changed files with 777 additions and 313 deletions
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Lib/rational.py Executable file
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# Originally contributed by Sjoerd Mullender.
# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.
"""Rational, infinite-precision, real numbers."""
from __future__ import division
import math
import numbers
import operator
__all__ = ["Rational"]
RationalAbc = numbers.Rational
def _gcd(a, b):
"""Calculate the Greatest Common Divisor.
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
def _binary_float_to_ratio(x):
"""x -> (top, bot), a pair of ints s.t. x = top/bot.
The conversion is done exactly, without rounding.
bot > 0 guaranteed.
Some form of binary fp is assumed.
Pass NaNs or infinities at your own risk.
>>> _binary_float_to_ratio(10.0)
(10, 1)
>>> _binary_float_to_ratio(0.0)
(0, 1)
>>> _binary_float_to_ratio(-.25)
(-1, 4)
"""
if x == 0:
return 0, 1
f, e = math.frexp(x)
signbit = 1
if f < 0:
f = -f
signbit = -1
assert 0.5 <= f < 1.0
# x = signbit * f * 2**e exactly
# Suck up CHUNK bits at a time; 28 is enough so that we suck
# up all bits in 2 iterations for all known binary double-
# precision formats, and small enough to fit in an int.
CHUNK = 28
top = 0
# invariant: x = signbit * (top + f) * 2**e exactly
while f:
f = math.ldexp(f, CHUNK)
digit = trunc(f)
assert digit >> CHUNK == 0
top = (top << CHUNK) | digit
f = f - digit
assert 0.0 <= f < 1.0
e = e - CHUNK
assert top
# Add in the sign bit.
top = signbit * top
# now x = top * 2**e exactly; fold in 2**e
if e>0:
return (top * 2**e, 1)
else:
return (top, 2 ** -e)
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.
"""
__slots__ = ('_numerator', '_denominator')
def __init__(self, numerator=0, denominator=1):
if (not isinstance(numerator, numbers.Integral) and
isinstance(numerator, RationalAbc) and
denominator == 1):
# 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)
@classmethod
def from_float(cls, f):
"""Converts a float to a rational number, exactly."""
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(*_binary_float_to_ratio(f))
@property
def numerator(a):
return a._numerator
@property
def denominator(a):
return a._denominator
def __repr__(self):
"""repr(self)"""
return ('rational.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)
"""
def forward(a, b):
if isinstance(b, RationalAbc):
# Includes ints.
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)
__div__, __rdiv__ = _operator_fallbacks(_div, operator.div)
@classmethod
def _floordiv(cls, a, b):
div = a / b
if isinstance(div, RationalAbc):
# trunc(math.floor(div)) doesn't work if the rational is
# more precise than a float because the intermediate
# rounding may cross an integer boundary.
return div.numerator // div.denominator
else:
return math.floor(div)
def __floordiv__(a, b):
"""a // b"""
# Will be math.floor(a / b) in 3.0.
return a._floordiv(a, b)
def __rfloordiv__(b, a):
"""a // b"""
# Will be math.floor(a / b) in 3.0.
return b._floordiv(a, b)
@classmethod
def _mod(cls, a, b):
div = a // b
return a - b * div
def __mod__(a, b):
"""a % b"""
return a._mod(a, b)
def __rmod__(b, a):
"""a % b"""
return b._mod(a, b)
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((self * shift).__round__(), shift)
else:
return Rational((self / shift).__round__() * shift)
def __hash__(self):
"""hash(self)
Tricky because values that are exactly representable as a
float must have the same hash as that float.
"""
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 __nonzero__(a):
"""a != 0"""
return a.numerator != 0