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492 lines
16 KiB
Python
Executable file
492 lines
16 KiB
Python
Executable file
# Originally contributed by Sjoerd Mullender.
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# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.
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"""Rational, infinite-precision, real numbers."""
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from __future__ import division
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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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__all__ = ["Rational"]
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RationalAbc = numbers.Rational
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def _gcd(a, b):
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"""Calculate the Greatest Common Divisor.
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Unless b==0, the result will have the same sign as b (so that when
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b is divided by it, the result comes out positive).
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"""
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while b:
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a, b = b, a%b
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return a
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def _binary_float_to_ratio(x):
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"""x -> (top, bot), a pair of ints s.t. x = top/bot.
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The conversion is done exactly, without rounding.
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bot > 0 guaranteed.
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Some form of binary fp is assumed.
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Pass NaNs or infinities at your own risk.
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>>> _binary_float_to_ratio(10.0)
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(10, 1)
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>>> _binary_float_to_ratio(0.0)
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(0, 1)
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>>> _binary_float_to_ratio(-.25)
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(-1, 4)
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"""
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if x == 0:
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return 0, 1
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f, e = math.frexp(x)
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signbit = 1
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if f < 0:
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f = -f
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signbit = -1
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assert 0.5 <= f < 1.0
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# x = signbit * f * 2**e exactly
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# Suck up CHUNK bits at a time; 28 is enough so that we suck
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# up all bits in 2 iterations for all known binary double-
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# precision formats, and small enough to fit in an int.
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CHUNK = 28
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top = 0
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# invariant: x = signbit * (top + f) * 2**e exactly
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while f:
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f = math.ldexp(f, CHUNK)
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digit = trunc(f)
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assert digit >> CHUNK == 0
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top = (top << CHUNK) | digit
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f = f - digit
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assert 0.0 <= f < 1.0
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e = e - CHUNK
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assert top
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# Add in the sign bit.
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top = signbit * top
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# now x = top * 2**e exactly; fold in 2**e
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if e>0:
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return (top * 2**e, 1)
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else:
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return (top, 2 ** -e)
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_RATIONAL_FORMAT = re.compile(
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r'^\s*(?P<sign>[-+]?)(?P<num>\d+)(?:/(?P<denom>\d+))?\s*$')
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class Rational(RationalAbc):
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"""This class implements rational numbers.
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Rational(8, 6) will produce a rational number equivalent to
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4/3. Both arguments must be Integral. The numerator defaults to 0
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and the denominator defaults to 1 so that Rational(3) == 3 and
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Rational() == 0.
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Rationals can also be constructed from strings of the form
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'[-+]?[0-9]+(/[0-9]+)?', optionally surrounded by spaces.
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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=1):
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"""Constructs a Rational.
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Takes a string, another Rational, or a numerator/denominator pair.
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"""
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self = super(Rational, cls).__new__(cls)
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if denominator == 1:
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if isinstance(numerator, basestring):
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# Handle construction from strings.
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input = numerator
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m = _RATIONAL_FORMAT.match(input)
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if m is None:
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raise ValueError('Invalid literal for Rational: ' + input)
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numerator = int(m.group('num'))
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# Default denominator to 1. That's the only optional group.
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denominator = int(m.group('denom') or 1)
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if m.group('sign') == '-':
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numerator = -numerator
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elif (not isinstance(numerator, numbers.Integral) and
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isinstance(numerator, RationalAbc)):
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# Handle copies from other rationals.
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other_rational = numerator
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numerator = other_rational.numerator
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denominator = other_rational.denominator
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if (not isinstance(numerator, numbers.Integral) or
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not isinstance(denominator, numbers.Integral)):
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raise TypeError("Rational(%(numerator)s, %(denominator)s):"
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" Both arguments must be integral." % locals())
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if denominator == 0:
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raise ZeroDivisionError('Rational(%s, 0)' % numerator)
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g = _gcd(numerator, denominator)
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self.numerator = int(numerator // g)
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self.denominator = int(denominator // g)
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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 Rational.from_float(0.3) != Rational(3, 10).
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"""
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if 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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if math.isnan(f) or math.isinf(f):
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raise TypeError("Cannot convert %r to %s." % (f, cls.__name__))
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return cls(*_binary_float_to_ratio(f))
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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 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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if not dec.is_finite():
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# Catches infinities and nans.
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raise TypeError("Cannot convert %s to %s." % (dec, cls.__name__))
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sign, digits, exp = dec.as_tuple()
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digits = int(''.join(map(str, digits)))
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if sign:
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digits = -digits
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if exp >= 0:
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return cls(digits * 10 ** exp)
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else:
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return cls(digits, 10 ** -exp)
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@classmethod
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def from_continued_fraction(cls, seq):
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'Build a Rational from a continued fraction expessed as a sequence'
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n, d = 1, 0
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for e in reversed(seq):
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n, d = d, n
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n += e * d
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return cls(n, d) if seq else cls(0)
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def as_continued_fraction(self):
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'Return continued fraction expressed as a list'
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n = self.numerator
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d = self.denominator
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cf = []
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while d:
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e = int(n // d)
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cf.append(e)
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n -= e * d
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n, d = d, n
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return cf
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@classmethod
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def approximate_from_float(cls, f, max_denominator):
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'Best rational approximation to f with a denominator <= max_denominator'
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# XXX First cut at algorithm
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# Still needs rounding rules as specified at
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# http://en.wikipedia.org/wiki/Continued_fraction
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cf = cls.from_float(f).as_continued_fraction()
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result = Rational(0)
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for i in range(1, len(cf)):
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new = cls.from_continued_fraction(cf[:i])
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if new.denominator > max_denominator:
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break
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result = new
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return result
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def __repr__(self):
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"""repr(self)"""
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return ('Rational(%r,%r)' % (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 _operator_fallbacks(monomorphic_operator, fallback_operator):
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"""Generates forward and reverse operators given a purely-rational
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operator and a function from the operator module.
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Use this like:
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__op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op)
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"""
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def forward(a, b):
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if isinstance(b, RationalAbc):
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# Includes ints.
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return monomorphic_operator(a, b)
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elif isinstance(b, float):
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return fallback_operator(float(a), b)
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elif isinstance(b, complex):
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return fallback_operator(complex(a), b)
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else:
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return NotImplemented
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forward.__name__ = '__' + fallback_operator.__name__ + '__'
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forward.__doc__ = monomorphic_operator.__doc__
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def reverse(b, a):
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if isinstance(a, RationalAbc):
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# Includes ints.
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return monomorphic_operator(a, b)
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elif isinstance(a, numbers.Real):
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return fallback_operator(float(a), float(b))
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elif isinstance(a, numbers.Complex):
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return fallback_operator(complex(a), complex(b))
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else:
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return NotImplemented
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reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
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reverse.__doc__ = monomorphic_operator.__doc__
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return forward, reverse
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def _add(a, b):
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"""a + b"""
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return Rational(a.numerator * b.denominator +
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b.numerator * a.denominator,
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a.denominator * b.denominator)
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__add__, __radd__ = _operator_fallbacks(_add, operator.add)
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def _sub(a, b):
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"""a - b"""
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return Rational(a.numerator * b.denominator -
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b.numerator * a.denominator,
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a.denominator * b.denominator)
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__sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub)
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def _mul(a, b):
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"""a * b"""
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return Rational(a.numerator * b.numerator, a.denominator * b.denominator)
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__mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul)
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def _div(a, b):
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"""a / b"""
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return Rational(a.numerator * b.denominator,
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a.denominator * b.numerator)
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__truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv)
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__div__, __rdiv__ = _operator_fallbacks(_div, operator.div)
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@classmethod
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def _floordiv(cls, a, b):
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div = a / b
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if isinstance(div, RationalAbc):
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# trunc(math.floor(div)) doesn't work if the rational is
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# more precise than a float because the intermediate
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# rounding may cross an integer boundary.
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return div.numerator // div.denominator
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else:
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return math.floor(div)
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def __floordiv__(a, b):
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"""a // b"""
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# Will be math.floor(a / b) in 3.0.
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return a._floordiv(a, b)
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def __rfloordiv__(b, a):
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"""a // b"""
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# Will be math.floor(a / b) in 3.0.
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return b._floordiv(a, b)
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@classmethod
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def _mod(cls, a, b):
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div = a // b
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return a - b * div
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def __mod__(a, b):
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"""a % b"""
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return a._mod(a, b)
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def __rmod__(b, a):
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"""a % b"""
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return b._mod(a, b)
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def __pow__(a, b):
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"""a ** b
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If b is not an integer, the result will be a float or complex
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since roots are generally irrational. If b is an integer, the
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result will be rational.
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"""
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if isinstance(b, RationalAbc):
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if b.denominator == 1:
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power = b.numerator
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if power >= 0:
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return Rational(a.numerator ** power,
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a.denominator ** power)
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else:
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return Rational(a.denominator ** -power,
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a.numerator ** -power)
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else:
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# A fractional power will generally produce an
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# irrational number.
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return float(a) ** float(b)
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else:
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return float(a) ** b
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def __rpow__(b, a):
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"""a ** b"""
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if b.denominator == 1 and b.numerator >= 0:
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# If a is an int, keep it that way if possible.
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return a ** b.numerator
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if isinstance(a, RationalAbc):
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return Rational(a.numerator, a.denominator) ** b
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if b.denominator == 1:
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return a ** b.numerator
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return a ** float(b)
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def __pos__(a):
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"""+a: Coerces a subclass instance to Rational"""
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return Rational(a.numerator, a.denominator)
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def __neg__(a):
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"""-a"""
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return Rational(-a.numerator, a.denominator)
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def __abs__(a):
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"""abs(a)"""
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return Rational(abs(a.numerator), a.denominator)
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def __trunc__(a):
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"""trunc(a)"""
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if a.numerator < 0:
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return -(-a.numerator // a.denominator)
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else:
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return a.numerator // a.denominator
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__int__ = __trunc__
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def __floor__(a):
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"""Will be math.floor(a) in 3.0."""
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return a.numerator // a.denominator
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def __ceil__(a):
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"""Will be math.ceil(a) in 3.0."""
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# The negations cleverly convince floordiv to return the ceiling.
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return -(-a.numerator // a.denominator)
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def __round__(self, ndigits=None):
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"""Will be round(self, ndigits) in 3.0.
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Rounds half toward even.
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"""
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if ndigits is None:
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floor, remainder = divmod(self.numerator, self.denominator)
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if remainder * 2 < self.denominator:
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return floor
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elif remainder * 2 > self.denominator:
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return floor + 1
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# Deal with the half case:
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elif floor % 2 == 0:
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return floor
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else:
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return floor + 1
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shift = 10**abs(ndigits)
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# See _operator_fallbacks.forward to check that the results of
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# these operations will always be Rational and therefore have
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# __round__().
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if ndigits > 0:
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return Rational((self * shift).__round__(), shift)
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else:
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return Rational((self / shift).__round__() * shift)
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def __hash__(self):
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"""hash(self)
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Tricky because values that are exactly representable as a
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float must have the same hash as that float.
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"""
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if self.denominator == 1:
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# Get integers right.
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return hash(self.numerator)
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# Expensive check, but definitely correct.
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if self == float(self):
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return hash(float(self))
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else:
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# Use tuple's hash to avoid a high collision rate on
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# simple fractions.
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return hash((self.numerator, self.denominator))
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def __eq__(a, b):
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"""a == b"""
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if isinstance(b, RationalAbc):
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return (a.numerator == b.numerator and
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a.denominator == b.denominator)
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if isinstance(b, numbers.Complex) and b.imag == 0:
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b = b.real
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if isinstance(b, float):
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return a == a.from_float(b)
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else:
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# XXX: If b.__eq__ is implemented like this method, it may
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# give the wrong answer after float(a) changes a's
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# value. Better ways of doing this are welcome.
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return float(a) == b
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def _subtractAndCompareToZero(a, b, op):
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"""Helper function for comparison operators.
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Subtracts b from a, exactly if possible, and compares the
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result with 0 using op, in such a way that the comparison
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won't recurse. If the difference raises a TypeError, returns
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NotImplemented instead.
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"""
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if isinstance(b, numbers.Complex) and b.imag == 0:
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b = b.real
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if isinstance(b, float):
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b = a.from_float(b)
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try:
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# XXX: If b <: Real but not <: RationalAbc, this is likely
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# to fall back to a float. If the actual values differ by
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# less than MIN_FLOAT, this could falsely call them equal,
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# which would make <= inconsistent with ==. Better ways of
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# doing this are welcome.
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diff = a - b
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except TypeError:
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return NotImplemented
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if isinstance(diff, RationalAbc):
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return op(diff.numerator, 0)
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return op(diff, 0)
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def __lt__(a, b):
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"""a < b"""
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return a._subtractAndCompareToZero(b, operator.lt)
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def __gt__(a, b):
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"""a > b"""
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return a._subtractAndCompareToZero(b, operator.gt)
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def __le__(a, b):
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"""a <= b"""
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return a._subtractAndCompareToZero(b, operator.le)
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def __ge__(a, b):
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"""a >= b"""
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return a._subtractAndCompareToZero(b, operator.ge)
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def __nonzero__(a):
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"""a != 0"""
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return a.numerator != 0
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