Add rational.Rational as an implementation of numbers.Rational with infinite
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
parent
ca9c6e433c
commit
d7b00334f3
|
|
@ -1,310 +0,0 @@
|
|||
'''\
|
||||
This module implements rational numbers.
|
||||
|
||||
The entry point of this module is the function
|
||||
rat(numerator, denominator)
|
||||
If either numerator or denominator is of an integral or rational type,
|
||||
the result is a rational number, else, the result is the simplest of
|
||||
the types float and complex which can hold numerator/denominator.
|
||||
If denominator is omitted, it defaults to 1.
|
||||
Rational numbers can be used in calculations with any other numeric
|
||||
type. The result of the calculation will be rational if possible.
|
||||
|
||||
There is also a test function with calling sequence
|
||||
test()
|
||||
The documentation string of the test function contains the expected
|
||||
output.
|
||||
'''
|
||||
|
||||
# Contributed by Sjoerd Mullender
|
||||
|
||||
from types import *
|
||||
|
||||
def gcd(a, b):
|
||||
'''Calculate the Greatest Common Divisor.'''
|
||||
while b:
|
||||
a, b = b, a%b
|
||||
return a
|
||||
|
||||
def rat(num, den = 1):
|
||||
# must check complex before float
|
||||
if isinstance(num, complex) or isinstance(den, complex):
|
||||
# numerator or denominator is complex: return a complex
|
||||
return complex(num) / complex(den)
|
||||
if isinstance(num, float) or isinstance(den, float):
|
||||
# numerator or denominator is float: return a float
|
||||
return float(num) / float(den)
|
||||
# otherwise return a rational
|
||||
return Rat(num, den)
|
||||
|
||||
class Rat:
|
||||
'''This class implements rational numbers.'''
|
||||
|
||||
def __init__(self, num, den = 1):
|
||||
if den == 0:
|
||||
raise ZeroDivisionError, 'rat(x, 0)'
|
||||
|
||||
# normalize
|
||||
|
||||
# must check complex before float
|
||||
if (isinstance(num, complex) or
|
||||
isinstance(den, complex)):
|
||||
# numerator or denominator is complex:
|
||||
# normalized form has denominator == 1+0j
|
||||
self.__num = complex(num) / complex(den)
|
||||
self.__den = complex(1)
|
||||
return
|
||||
if isinstance(num, float) or isinstance(den, float):
|
||||
# numerator or denominator is float:
|
||||
# normalized form has denominator == 1.0
|
||||
self.__num = float(num) / float(den)
|
||||
self.__den = 1.0
|
||||
return
|
||||
if (isinstance(num, self.__class__) or
|
||||
isinstance(den, self.__class__)):
|
||||
# numerator or denominator is rational
|
||||
new = num / den
|
||||
if not isinstance(new, self.__class__):
|
||||
self.__num = new
|
||||
if isinstance(new, complex):
|
||||
self.__den = complex(1)
|
||||
else:
|
||||
self.__den = 1.0
|
||||
else:
|
||||
self.__num = new.__num
|
||||
self.__den = new.__den
|
||||
else:
|
||||
# make sure numerator and denominator don't
|
||||
# have common factors
|
||||
# this also makes sure that denominator > 0
|
||||
g = gcd(num, den)
|
||||
self.__num = num / g
|
||||
self.__den = den / g
|
||||
# try making numerator and denominator of IntType if they fit
|
||||
try:
|
||||
numi = int(self.__num)
|
||||
deni = int(self.__den)
|
||||
except (OverflowError, TypeError):
|
||||
pass
|
||||
else:
|
||||
if self.__num == numi and self.__den == deni:
|
||||
self.__num = numi
|
||||
self.__den = deni
|
||||
|
||||
def __repr__(self):
|
||||
return 'Rat(%s,%s)' % (self.__num, self.__den)
|
||||
|
||||
def __str__(self):
|
||||
if self.__den == 1:
|
||||
return str(self.__num)
|
||||
else:
|
||||
return '(%s/%s)' % (str(self.__num), str(self.__den))
|
||||
|
||||
# a + b
|
||||
def __add__(a, b):
|
||||
try:
|
||||
return rat(a.__num * b.__den + b.__num * a.__den,
|
||||
a.__den * b.__den)
|
||||
except OverflowError:
|
||||
return rat(long(a.__num) * long(b.__den) +
|
||||
long(b.__num) * long(a.__den),
|
||||
long(a.__den) * long(b.__den))
|
||||
|
||||
def __radd__(b, a):
|
||||
return Rat(a) + b
|
||||
|
||||
# a - b
|
||||
def __sub__(a, b):
|
||||
try:
|
||||
return rat(a.__num * b.__den - b.__num * a.__den,
|
||||
a.__den * b.__den)
|
||||
except OverflowError:
|
||||
return rat(long(a.__num) * long(b.__den) -
|
||||
long(b.__num) * long(a.__den),
|
||||
long(a.__den) * long(b.__den))
|
||||
|
||||
def __rsub__(b, a):
|
||||
return Rat(a) - b
|
||||
|
||||
# a * b
|
||||
def __mul__(a, b):
|
||||
try:
|
||||
return rat(a.__num * b.__num, a.__den * b.__den)
|
||||
except OverflowError:
|
||||
return rat(long(a.__num) * long(b.__num),
|
||||
long(a.__den) * long(b.__den))
|
||||
|
||||
def __rmul__(b, a):
|
||||
return Rat(a) * b
|
||||
|
||||
# a / b
|
||||
def __div__(a, b):
|
||||
try:
|
||||
return rat(a.__num * b.__den, a.__den * b.__num)
|
||||
except OverflowError:
|
||||
return rat(long(a.__num) * long(b.__den),
|
||||
long(a.__den) * long(b.__num))
|
||||
|
||||
def __rdiv__(b, a):
|
||||
return Rat(a) / b
|
||||
|
||||
# a % b
|
||||
def __mod__(a, b):
|
||||
div = a / b
|
||||
try:
|
||||
div = int(div)
|
||||
except OverflowError:
|
||||
div = long(div)
|
||||
return a - b * div
|
||||
|
||||
def __rmod__(b, a):
|
||||
return Rat(a) % b
|
||||
|
||||
# a ** b
|
||||
def __pow__(a, b):
|
||||
if b.__den != 1:
|
||||
if isinstance(a.__num, complex):
|
||||
a = complex(a)
|
||||
else:
|
||||
a = float(a)
|
||||
if isinstance(b.__num, complex):
|
||||
b = complex(b)
|
||||
else:
|
||||
b = float(b)
|
||||
return a ** b
|
||||
try:
|
||||
return rat(a.__num ** b.__num, a.__den ** b.__num)
|
||||
except OverflowError:
|
||||
return rat(long(a.__num) ** b.__num,
|
||||
long(a.__den) ** b.__num)
|
||||
|
||||
def __rpow__(b, a):
|
||||
return Rat(a) ** b
|
||||
|
||||
# -a
|
||||
def __neg__(a):
|
||||
try:
|
||||
return rat(-a.__num, a.__den)
|
||||
except OverflowError:
|
||||
# a.__num == sys.maxint
|
||||
return rat(-long(a.__num), a.__den)
|
||||
|
||||
# abs(a)
|
||||
def __abs__(a):
|
||||
return rat(abs(a.__num), a.__den)
|
||||
|
||||
# int(a)
|
||||
def __int__(a):
|
||||
return int(a.__num / a.__den)
|
||||
|
||||
# long(a)
|
||||
def __long__(a):
|
||||
return long(a.__num) / long(a.__den)
|
||||
|
||||
# float(a)
|
||||
def __float__(a):
|
||||
return float(a.__num) / float(a.__den)
|
||||
|
||||
# complex(a)
|
||||
def __complex__(a):
|
||||
return complex(a.__num) / complex(a.__den)
|
||||
|
||||
# cmp(a,b)
|
||||
def __cmp__(a, b):
|
||||
diff = Rat(a - b)
|
||||
if diff.__num < 0:
|
||||
return -1
|
||||
elif diff.__num > 0:
|
||||
return 1
|
||||
else:
|
||||
return 0
|
||||
|
||||
def __rcmp__(b, a):
|
||||
return cmp(Rat(a), b)
|
||||
|
||||
# a != 0
|
||||
def __nonzero__(a):
|
||||
return a.__num != 0
|
||||
|
||||
# coercion
|
||||
def __coerce__(a, b):
|
||||
return a, Rat(b)
|
||||
|
||||
def test():
|
||||
'''\
|
||||
Test function for rat module.
|
||||
|
||||
The expected output is (module some differences in floating
|
||||
precission):
|
||||
-1
|
||||
-1
|
||||
0 0L 0.1 (0.1+0j)
|
||||
[Rat(1,2), Rat(-3,10), Rat(1,25), Rat(1,4)]
|
||||
[Rat(-3,10), Rat(1,25), Rat(1,4), Rat(1,2)]
|
||||
0
|
||||
(11/10)
|
||||
(11/10)
|
||||
1.1
|
||||
OK
|
||||
2 1.5 (3/2) (1.5+1.5j) (15707963/5000000)
|
||||
2 2 2.0 (2+0j)
|
||||
|
||||
4 0 4 1 4 0
|
||||
3.5 0.5 3.0 1.33333333333 2.82842712475 1
|
||||
(7/2) (1/2) 3 (4/3) 2.82842712475 1
|
||||
(3.5+1.5j) (0.5-1.5j) (3+3j) (0.666666666667-0.666666666667j) (1.43248815986+2.43884761145j) 1
|
||||
1.5 1 1.5 (1.5+0j)
|
||||
|
||||
3.5 -0.5 3.0 0.75 2.25 -1
|
||||
3.0 0.0 2.25 1.0 1.83711730709 0
|
||||
3.0 0.0 2.25 1.0 1.83711730709 1
|
||||
(3+1.5j) -1.5j (2.25+2.25j) (0.5-0.5j) (1.50768393746+1.04970907623j) -1
|
||||
(3/2) 1 1.5 (1.5+0j)
|
||||
|
||||
(7/2) (-1/2) 3 (3/4) (9/4) -1
|
||||
3.0 0.0 2.25 1.0 1.83711730709 -1
|
||||
3 0 (9/4) 1 1.83711730709 0
|
||||
(3+1.5j) -1.5j (2.25+2.25j) (0.5-0.5j) (1.50768393746+1.04970907623j) -1
|
||||
(1.5+1.5j) (1.5+1.5j)
|
||||
|
||||
(3.5+1.5j) (-0.5+1.5j) (3+3j) (0.75+0.75j) 4.5j -1
|
||||
(3+1.5j) 1.5j (2.25+2.25j) (1+1j) (1.18235814075+2.85446505899j) 1
|
||||
(3+1.5j) 1.5j (2.25+2.25j) (1+1j) (1.18235814075+2.85446505899j) 1
|
||||
(3+3j) 0j 4.5j (1+0j) (-0.638110484918+0.705394566962j) 0
|
||||
'''
|
||||
print rat(-1L, 1)
|
||||
print rat(1, -1)
|
||||
a = rat(1, 10)
|
||||
print int(a), long(a), float(a), complex(a)
|
||||
b = rat(2, 5)
|
||||
l = [a+b, a-b, a*b, a/b]
|
||||
print l
|
||||
l.sort()
|
||||
print l
|
||||
print rat(0, 1)
|
||||
print a+1
|
||||
print a+1L
|
||||
print a+1.0
|
||||
try:
|
||||
print rat(1, 0)
|
||||
raise SystemError, 'should have been ZeroDivisionError'
|
||||
except ZeroDivisionError:
|
||||
print 'OK'
|
||||
print rat(2), rat(1.5), rat(3, 2), rat(1.5+1.5j), rat(31415926,10000000)
|
||||
list = [2, 1.5, rat(3,2), 1.5+1.5j]
|
||||
for i in list:
|
||||
print i,
|
||||
if not isinstance(i, complex):
|
||||
print int(i), float(i),
|
||||
print complex(i)
|
||||
print
|
||||
for j in list:
|
||||
print i + j, i - j, i * j, i / j, i ** j,
|
||||
if not (isinstance(i, complex) or
|
||||
isinstance(j, complex)):
|
||||
print cmp(i, j)
|
||||
print
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
test()
|
||||
|
|
@ -21,6 +21,7 @@ The following modules are documented in this chapter:
|
|||
math.rst
|
||||
cmath.rst
|
||||
decimal.rst
|
||||
rational.rst
|
||||
random.rst
|
||||
itertools.rst
|
||||
functools.rst
|
||||
|
|
|
|||
|
|
@ -0,0 +1,65 @@
|
|||
|
||||
:mod:`rational` --- Rational numbers
|
||||
====================================
|
||||
|
||||
.. module:: rational
|
||||
:synopsis: Rational numbers.
|
||||
.. moduleauthor:: Jeffrey Yasskin <jyasskin at gmail.com>
|
||||
.. sectionauthor:: Jeffrey Yasskin <jyasskin at gmail.com>
|
||||
.. versionadded:: 2.6
|
||||
|
||||
|
||||
The :mod:`rational` module defines an immutable, infinite-precision
|
||||
Rational number class.
|
||||
|
||||
|
||||
.. class:: Rational(numerator=0, denominator=1)
|
||||
Rational(other_rational)
|
||||
|
||||
The first version requires that *numerator* and *denominator* are
|
||||
instances of :class:`numbers.Integral` and returns a new
|
||||
``Rational`` representing ``numerator/denominator``. If
|
||||
*denominator* is :const:`0`, raises a :exc:`ZeroDivisionError`. The
|
||||
second version requires that *other_rational* is an instance of
|
||||
:class:`numbers.Rational` and returns an instance of
|
||||
:class:`Rational` with the same value.
|
||||
|
||||
Implements all of the methods and operations from
|
||||
:class:`numbers.Rational` and is hashable.
|
||||
|
||||
|
||||
.. method:: Rational.from_float(flt)
|
||||
|
||||
This classmethod constructs a :class:`Rational` representing the
|
||||
exact value of *flt*, which must be a :class:`float`. Beware that
|
||||
``Rational.from_float(0.3)`` is not the same value as ``Rational(3,
|
||||
10)``
|
||||
|
||||
|
||||
.. method:: Rational.__floor__()
|
||||
|
||||
Returns the greatest :class:`int` ``<= self``. Will be accessible
|
||||
through :func:`math.floor` in Py3k.
|
||||
|
||||
|
||||
.. method:: Rational.__ceil__()
|
||||
|
||||
Returns the least :class:`int` ``>= self``. Will be accessible
|
||||
through :func:`math.ceil` in Py3k.
|
||||
|
||||
|
||||
.. method:: Rational.__round__()
|
||||
Rational.__round__(ndigits)
|
||||
|
||||
The first version returns the nearest :class:`int` to ``self``,
|
||||
rounding half to even. The second version rounds ``self`` to the
|
||||
nearest multiple of ``Rational(1, 10**ndigits)`` (logically, if
|
||||
``ndigits`` is negative), again rounding half toward even. Will be
|
||||
accessible through :func:`round` in Py3k.
|
||||
|
||||
|
||||
.. seealso::
|
||||
|
||||
Module :mod:`numbers`
|
||||
The abstract base classes making up the numeric tower.
|
||||
|
||||
|
|
@ -5,6 +5,7 @@
|
|||
|
||||
TODO: Fill out more detailed documentation on the operators."""
|
||||
|
||||
from __future__ import division
|
||||
from abc import ABCMeta, abstractmethod, abstractproperty
|
||||
|
||||
__all__ = ["Number", "Exact", "Inexact",
|
||||
|
|
@ -63,7 +64,8 @@ class Complex(Number):
|
|||
def __complex__(self):
|
||||
"""Return a builtin complex instance. Called for complex(self)."""
|
||||
|
||||
def __bool__(self):
|
||||
# Will be __bool__ in 3.0.
|
||||
def __nonzero__(self):
|
||||
"""True if self != 0. Called for bool(self)."""
|
||||
return self != 0
|
||||
|
||||
|
|
@ -98,6 +100,7 @@ class Complex(Number):
|
|||
"""-self"""
|
||||
raise NotImplementedError
|
||||
|
||||
@abstractmethod
|
||||
def __pos__(self):
|
||||
"""+self"""
|
||||
raise NotImplementedError
|
||||
|
|
@ -122,12 +125,28 @@ class Complex(Number):
|
|||
|
||||
@abstractmethod
|
||||
def __div__(self, other):
|
||||
"""self / other; should promote to float or complex when necessary."""
|
||||
"""self / other without __future__ division
|
||||
|
||||
May promote to float.
|
||||
"""
|
||||
raise NotImplementedError
|
||||
|
||||
@abstractmethod
|
||||
def __rdiv__(self, other):
|
||||
"""other / self"""
|
||||
"""other / self without __future__ division"""
|
||||
raise NotImplementedError
|
||||
|
||||
@abstractmethod
|
||||
def __truediv__(self, other):
|
||||
"""self / other with __future__ division.
|
||||
|
||||
Should promote to float when necessary.
|
||||
"""
|
||||
raise NotImplementedError
|
||||
|
||||
@abstractmethod
|
||||
def __rtruediv__(self, other):
|
||||
"""other / self with __future__ division"""
|
||||
raise NotImplementedError
|
||||
|
||||
@abstractmethod
|
||||
|
|
|
|||
|
|
@ -0,0 +1,410 @@
|
|||
# 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
|
||||
|
|
@ -0,0 +1,279 @@
|
|||
"""Tests for Lib/rational.py."""
|
||||
|
||||
from decimal import Decimal
|
||||
from test.test_support import run_unittest, verbose
|
||||
import math
|
||||
import operator
|
||||
import rational
|
||||
import unittest
|
||||
R = rational.Rational
|
||||
|
||||
def _components(r):
|
||||
return (r.numerator, r.denominator)
|
||||
|
||||
class RationalTest(unittest.TestCase):
|
||||
|
||||
def assertTypedEquals(self, expected, actual):
|
||||
"""Asserts that both the types and values are the same."""
|
||||
self.assertEquals(type(expected), type(actual))
|
||||
self.assertEquals(expected, actual)
|
||||
|
||||
def assertRaisesMessage(self, exc_type, message,
|
||||
callable, *args, **kwargs):
|
||||
"""Asserts that callable(*args, **kwargs) raises exc_type(message)."""
|
||||
try:
|
||||
callable(*args, **kwargs)
|
||||
except exc_type, e:
|
||||
self.assertEquals(message, str(e))
|
||||
else:
|
||||
self.fail("%s not raised" % exc_type.__name__)
|
||||
|
||||
def testInit(self):
|
||||
self.assertEquals((0, 1), _components(R()))
|
||||
self.assertEquals((7, 1), _components(R(7)))
|
||||
self.assertEquals((7, 3), _components(R(R(7, 3))))
|
||||
|
||||
self.assertEquals((-1, 1), _components(R(-1, 1)))
|
||||
self.assertEquals((-1, 1), _components(R(1, -1)))
|
||||
self.assertEquals((1, 1), _components(R(-2, -2)))
|
||||
self.assertEquals((1, 2), _components(R(5, 10)))
|
||||
self.assertEquals((7, 15), _components(R(7, 15)))
|
||||
self.assertEquals((10**23, 1), _components(R(10**23)))
|
||||
|
||||
self.assertRaisesMessage(ZeroDivisionError, "Rational(12, 0)",
|
||||
R, 12, 0)
|
||||
self.assertRaises(TypeError, R, 1.5)
|
||||
self.assertRaises(TypeError, R, 1.5 + 3j)
|
||||
|
||||
def testFromFloat(self):
|
||||
self.assertRaisesMessage(
|
||||
TypeError, "Rational.from_float() only takes floats, not 3 (int)",
|
||||
R.from_float, 3)
|
||||
|
||||
self.assertEquals((0, 1), _components(R.from_float(-0.0)))
|
||||
self.assertEquals((10, 1), _components(R.from_float(10.0)))
|
||||
self.assertEquals((-5, 2), _components(R.from_float(-2.5)))
|
||||
self.assertEquals((99999999999999991611392, 1),
|
||||
_components(R.from_float(1e23)))
|
||||
self.assertEquals(float(10**23), float(R.from_float(1e23)))
|
||||
self.assertEquals((3602879701896397, 1125899906842624),
|
||||
_components(R.from_float(3.2)))
|
||||
self.assertEquals(3.2, float(R.from_float(3.2)))
|
||||
|
||||
inf = 1e1000
|
||||
nan = inf - inf
|
||||
self.assertRaisesMessage(
|
||||
TypeError, "Cannot convert inf to Rational.",
|
||||
R.from_float, inf)
|
||||
self.assertRaisesMessage(
|
||||
TypeError, "Cannot convert -inf to Rational.",
|
||||
R.from_float, -inf)
|
||||
self.assertRaisesMessage(
|
||||
TypeError, "Cannot convert nan to Rational.",
|
||||
R.from_float, nan)
|
||||
|
||||
def testConversions(self):
|
||||
self.assertTypedEquals(-1, trunc(R(-11, 10)))
|
||||
self.assertTypedEquals(-2, R(-11, 10).__floor__())
|
||||
self.assertTypedEquals(-1, R(-11, 10).__ceil__())
|
||||
self.assertTypedEquals(-1, R(-10, 10).__ceil__())
|
||||
|
||||
self.assertTypedEquals(0, R(-1, 10).__round__())
|
||||
self.assertTypedEquals(0, R(-5, 10).__round__())
|
||||
self.assertTypedEquals(-2, R(-15, 10).__round__())
|
||||
self.assertTypedEquals(-1, R(-7, 10).__round__())
|
||||
|
||||
self.assertEquals(False, bool(R(0, 1)))
|
||||
self.assertEquals(True, bool(R(3, 2)))
|
||||
self.assertTypedEquals(0.1, float(R(1, 10)))
|
||||
|
||||
# Check that __float__ isn't implemented by converting the
|
||||
# numerator and denominator to float before dividing.
|
||||
self.assertRaises(OverflowError, float, long('2'*400+'7'))
|
||||
self.assertAlmostEquals(2.0/3,
|
||||
float(R(long('2'*400+'7'), long('3'*400+'1'))))
|
||||
|
||||
self.assertTypedEquals(0.1+0j, complex(R(1,10)))
|
||||
|
||||
def testRound(self):
|
||||
self.assertTypedEquals(R(-200), R(-150).__round__(-2))
|
||||
self.assertTypedEquals(R(-200), R(-250).__round__(-2))
|
||||
self.assertTypedEquals(R(30), R(26).__round__(-1))
|
||||
self.assertTypedEquals(R(-2, 10), R(-15, 100).__round__(1))
|
||||
self.assertTypedEquals(R(-2, 10), R(-25, 100).__round__(1))
|
||||
|
||||
|
||||
def testArithmetic(self):
|
||||
self.assertEquals(R(1, 2), R(1, 10) + R(2, 5))
|
||||
self.assertEquals(R(-3, 10), R(1, 10) - R(2, 5))
|
||||
self.assertEquals(R(1, 25), R(1, 10) * R(2, 5))
|
||||
self.assertEquals(R(1, 4), R(1, 10) / R(2, 5))
|
||||
self.assertTypedEquals(2, R(9, 10) // R(2, 5))
|
||||
self.assertTypedEquals(10**23, R(10**23, 1) // R(1))
|
||||
self.assertEquals(R(2, 3), R(-7, 3) % R(3, 2))
|
||||
self.assertEquals(R(8, 27), R(2, 3) ** R(3))
|
||||
self.assertEquals(R(27, 8), R(2, 3) ** R(-3))
|
||||
self.assertTypedEquals(2.0, R(4) ** R(1, 2))
|
||||
# Will return 1j in 3.0:
|
||||
self.assertRaises(ValueError, pow, R(-1), R(1, 2))
|
||||
|
||||
def testMixedArithmetic(self):
|
||||
self.assertTypedEquals(R(11, 10), R(1, 10) + 1)
|
||||
self.assertTypedEquals(1.1, R(1, 10) + 1.0)
|
||||
self.assertTypedEquals(1.1 + 0j, R(1, 10) + (1.0 + 0j))
|
||||
self.assertTypedEquals(R(11, 10), 1 + R(1, 10))
|
||||
self.assertTypedEquals(1.1, 1.0 + R(1, 10))
|
||||
self.assertTypedEquals(1.1 + 0j, (1.0 + 0j) + R(1, 10))
|
||||
|
||||
self.assertTypedEquals(R(-9, 10), R(1, 10) - 1)
|
||||
self.assertTypedEquals(-0.9, R(1, 10) - 1.0)
|
||||
self.assertTypedEquals(-0.9 + 0j, R(1, 10) - (1.0 + 0j))
|
||||
self.assertTypedEquals(R(9, 10), 1 - R(1, 10))
|
||||
self.assertTypedEquals(0.9, 1.0 - R(1, 10))
|
||||
self.assertTypedEquals(0.9 + 0j, (1.0 + 0j) - R(1, 10))
|
||||
|
||||
self.assertTypedEquals(R(1, 10), R(1, 10) * 1)
|
||||
self.assertTypedEquals(0.1, R(1, 10) * 1.0)
|
||||
self.assertTypedEquals(0.1 + 0j, R(1, 10) * (1.0 + 0j))
|
||||
self.assertTypedEquals(R(1, 10), 1 * R(1, 10))
|
||||
self.assertTypedEquals(0.1, 1.0 * R(1, 10))
|
||||
self.assertTypedEquals(0.1 + 0j, (1.0 + 0j) * R(1, 10))
|
||||
|
||||
self.assertTypedEquals(R(1, 10), R(1, 10) / 1)
|
||||
self.assertTypedEquals(0.1, R(1, 10) / 1.0)
|
||||
self.assertTypedEquals(0.1 + 0j, R(1, 10) / (1.0 + 0j))
|
||||
self.assertTypedEquals(R(10, 1), 1 / R(1, 10))
|
||||
self.assertTypedEquals(10.0, 1.0 / R(1, 10))
|
||||
self.assertTypedEquals(10.0 + 0j, (1.0 + 0j) / R(1, 10))
|
||||
|
||||
self.assertTypedEquals(0, R(1, 10) // 1)
|
||||
self.assertTypedEquals(0.0, R(1, 10) // 1.0)
|
||||
self.assertTypedEquals(10, 1 // R(1, 10))
|
||||
self.assertTypedEquals(10**23, 10**22 // R(1, 10))
|
||||
self.assertTypedEquals(10.0, 1.0 // R(1, 10))
|
||||
|
||||
self.assertTypedEquals(R(1, 10), R(1, 10) % 1)
|
||||
self.assertTypedEquals(0.1, R(1, 10) % 1.0)
|
||||
self.assertTypedEquals(R(0, 1), 1 % R(1, 10))
|
||||
self.assertTypedEquals(0.0, 1.0 % R(1, 10))
|
||||
|
||||
# No need for divmod since we don't override it.
|
||||
|
||||
# ** has more interesting conversion rules.
|
||||
self.assertTypedEquals(R(100, 1), R(1, 10) ** -2)
|
||||
self.assertTypedEquals(R(100, 1), R(10, 1) ** 2)
|
||||
self.assertTypedEquals(0.1, R(1, 10) ** 1.0)
|
||||
self.assertTypedEquals(0.1 + 0j, R(1, 10) ** (1.0 + 0j))
|
||||
self.assertTypedEquals(4 , 2 ** R(2, 1))
|
||||
# Will return 1j in 3.0:
|
||||
self.assertRaises(ValueError, pow, (-1), R(1, 2))
|
||||
self.assertTypedEquals(R(1, 4) , 2 ** R(-2, 1))
|
||||
self.assertTypedEquals(2.0 , 4 ** R(1, 2))
|
||||
self.assertTypedEquals(0.25, 2.0 ** R(-2, 1))
|
||||
self.assertTypedEquals(1.0 + 0j, (1.0 + 0j) ** R(1, 10))
|
||||
|
||||
def testMixingWithDecimal(self):
|
||||
"""Decimal refuses mixed comparisons."""
|
||||
self.assertRaisesMessage(
|
||||
TypeError,
|
||||
"unsupported operand type(s) for +: 'Rational' and 'Decimal'",
|
||||
operator.add, R(3,11), Decimal('3.1415926'))
|
||||
self.assertNotEquals(R(5, 2), Decimal('2.5'))
|
||||
|
||||
def testComparisons(self):
|
||||
self.assertTrue(R(1, 2) < R(2, 3))
|
||||
self.assertFalse(R(1, 2) < R(1, 2))
|
||||
self.assertTrue(R(1, 2) <= R(2, 3))
|
||||
self.assertTrue(R(1, 2) <= R(1, 2))
|
||||
self.assertFalse(R(2, 3) <= R(1, 2))
|
||||
self.assertTrue(R(1, 2) == R(1, 2))
|
||||
self.assertFalse(R(1, 2) == R(1, 3))
|
||||
|
||||
def testMixedLess(self):
|
||||
self.assertTrue(2 < R(5, 2))
|
||||
self.assertFalse(2 < R(4, 2))
|
||||
self.assertTrue(R(5, 2) < 3)
|
||||
self.assertFalse(R(4, 2) < 2)
|
||||
|
||||
self.assertTrue(R(1, 2) < 0.6)
|
||||
self.assertFalse(R(1, 2) < 0.4)
|
||||
self.assertTrue(0.4 < R(1, 2))
|
||||
self.assertFalse(0.5 < R(1, 2))
|
||||
|
||||
def testMixedLessEqual(self):
|
||||
self.assertTrue(0.5 <= R(1, 2))
|
||||
self.assertFalse(0.6 <= R(1, 2))
|
||||
self.assertTrue(R(1, 2) <= 0.5)
|
||||
self.assertFalse(R(1, 2) <= 0.4)
|
||||
self.assertTrue(2 <= R(4, 2))
|
||||
self.assertFalse(2 <= R(3, 2))
|
||||
self.assertTrue(R(4, 2) <= 2)
|
||||
self.assertFalse(R(5, 2) <= 2)
|
||||
|
||||
def testBigFloatComparisons(self):
|
||||
# Because 10**23 can't be represented exactly as a float:
|
||||
self.assertFalse(R(10**23) == float(10**23))
|
||||
# The first test demonstrates why these are important.
|
||||
self.assertFalse(1e23 < float(R(trunc(1e23) + 1)))
|
||||
self.assertTrue(1e23 < R(trunc(1e23) + 1))
|
||||
self.assertFalse(1e23 <= R(trunc(1e23) - 1))
|
||||
self.assertTrue(1e23 > R(trunc(1e23) - 1))
|
||||
self.assertFalse(1e23 >= R(trunc(1e23) + 1))
|
||||
|
||||
def testBigComplexComparisons(self):
|
||||
self.assertFalse(R(10**23) == complex(10**23))
|
||||
self.assertTrue(R(10**23) > complex(10**23))
|
||||
self.assertFalse(R(10**23) <= complex(10**23))
|
||||
|
||||
def testMixedEqual(self):
|
||||
self.assertTrue(0.5 == R(1, 2))
|
||||
self.assertFalse(0.6 == R(1, 2))
|
||||
self.assertTrue(R(1, 2) == 0.5)
|
||||
self.assertFalse(R(1, 2) == 0.4)
|
||||
self.assertTrue(2 == R(4, 2))
|
||||
self.assertFalse(2 == R(3, 2))
|
||||
self.assertTrue(R(4, 2) == 2)
|
||||
self.assertFalse(R(5, 2) == 2)
|
||||
|
||||
def testStringification(self):
|
||||
self.assertEquals("rational.Rational(7,3)", repr(R(7, 3)))
|
||||
self.assertEquals("(7/3)", str(R(7, 3)))
|
||||
self.assertEquals("7", str(R(7, 1)))
|
||||
|
||||
def testHash(self):
|
||||
self.assertEquals(hash(2.5), hash(R(5, 2)))
|
||||
self.assertEquals(hash(10**50), hash(R(10**50)))
|
||||
self.assertNotEquals(hash(float(10**23)), hash(R(10**23)))
|
||||
|
||||
def testApproximatePi(self):
|
||||
# Algorithm borrowed from
|
||||
# http://docs.python.org/lib/decimal-recipes.html
|
||||
three = R(3)
|
||||
lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
|
||||
while abs(s - lasts) > R(1, 10**9):
|
||||
lasts = s
|
||||
n, na = n+na, na+8
|
||||
d, da = d+da, da+32
|
||||
t = (t * n) / d
|
||||
s += t
|
||||
self.assertAlmostEquals(math.pi, s)
|
||||
|
||||
def testApproximateCos1(self):
|
||||
# Algorithm borrowed from
|
||||
# http://docs.python.org/lib/decimal-recipes.html
|
||||
x = R(1)
|
||||
i, lasts, s, fact, num, sign = 0, 0, R(1), 1, 1, 1
|
||||
while abs(s - lasts) > R(1, 10**9):
|
||||
lasts = s
|
||||
i += 2
|
||||
fact *= i * (i-1)
|
||||
num *= x * x
|
||||
sign *= -1
|
||||
s += num / fact * sign
|
||||
self.assertAlmostEquals(math.cos(1), s)
|
||||
|
||||
def test_main():
|
||||
run_unittest(RationalTest)
|
||||
|
||||
if __name__ == '__main__':
|
||||
test_main()
|
||||
Loading…
Reference in New Issue