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.
diff --git a/Lib/numbers.py b/Lib/numbers.py
index 3c13290..8e02203 100644
--- a/Lib/numbers.py
+++ b/Lib/numbers.py
@@ -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 @@
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 @@
"""-self"""
raise NotImplementedError
+ @abstractmethod
def __pos__(self):
"""+self"""
raise NotImplementedError
@@ -122,12 +125,28 @@
@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
diff --git a/Lib/rational.py b/Lib/rational.py
new file mode 100755
index 0000000..d455dc6
--- /dev/null
+++ b/Lib/rational.py
@@ -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
diff --git a/Lib/test/test_rational.py b/Lib/test/test_rational.py
new file mode 100644
index 0000000..654c46d
--- /dev/null
+++ b/Lib/test/test_rational.py
@@ -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()