Merged revisions 60481,60485,60489-60492,60494-60496,60498-60499,60501-60503,60505-60506,60508-60509,60523-60524,60532,60543,60545,60547-60548,60552,60554,60556-60559,60561-60562,60569,60571-60572,60574,60576-60583,60585-60586,60589,60591,60594-60595,60597-60598,60600-60601,60606-60612,60615,60617,60619-60621,60623-60625,60627-60629,60631,60633,60635,60647,60650,60652,60654,60656,60658-60659,60664-60666,60668-60670,60672,60676,60678,60680-60683,60685-60686,60688,60690,60692-60694,60697-60700,60705-60706,60708,60711,60714,60720,60724-60730,60732,60736,60742,60744,60746,60748,60750-60766,60769-60786 via svnmerge from

svn+ssh://pythondev@svn.python.org/python/trunk

........
  r60752 | mark.dickinson | 2008-02-12 22:31:59 +0100 (Tue, 12 Feb 2008) | 5 lines

  Implementation of Fraction.limit_denominator.

  Remove Fraction.to_continued_fraction and
  Fraction.from_continued_fraction
........
  r60754 | mark.dickinson | 2008-02-12 22:40:53 +0100 (Tue, 12 Feb 2008) | 3 lines

  Revert change in r60712:  turn alternate constructors back into
  classmethods instead of staticmethods.
........
  r60755 | mark.dickinson | 2008-02-12 22:46:54 +0100 (Tue, 12 Feb 2008) | 4 lines

  Replace R=fractions.Fraction with F=fractions.Fraction in
  test_fractions.py.  This should have been part of the name
  change from Rational to Fraction.
........
  r60758 | georg.brandl | 2008-02-13 08:20:22 +0100 (Wed, 13 Feb 2008) | 3 lines

  #2063: correct order of utime and stime in os.times()
  result on Windows.
........
  r60762 | jeffrey.yasskin | 2008-02-13 18:58:04 +0100 (Wed, 13 Feb 2008) | 7 lines

  Working on issue #1762: Brought
    ./python.exe -m timeit -s 'from fractions import Fraction; f = Fraction(3, 2)' 'isinstance(3, Fraction); isinstance(f, Fraction)'
  from 12.3 usec/loop to 3.44 usec/loop and
    ./python.exe -m timeit -s 'from fractions import Fraction' 'Fraction(3, 2)'
  from 48.8 usec to 23.6 usec by avoiding genexps and sets in __instancecheck__
  and inlining the common case from __subclasscheck__.
........
  r60765 | brett.cannon | 2008-02-13 20:15:44 +0100 (Wed, 13 Feb 2008) | 5 lines

  Fix --enable-universalsdk and its comment line so that zsh's flag completion
  works.

  Thanks to Jeroen Ruigrok van der Werven for the fix.
........
  r60771 | kurt.kaiser | 2008-02-14 01:08:55 +0100 (Thu, 14 Feb 2008) | 2 lines

  Bring NEWS.txt up to date from check-in msgs.
........
  r60772 | raymond.hettinger | 2008-02-14 02:08:02 +0100 (Thu, 14 Feb 2008) | 3 lines

  Update notes on Decimal.
........
  r60773 | raymond.hettinger | 2008-02-14 03:41:22 +0100 (Thu, 14 Feb 2008) | 1 line

  Fix decimal repr which should have used single quotes like other reprs.
........
  r60785 | jeffrey.yasskin | 2008-02-14 07:12:24 +0100 (Thu, 14 Feb 2008) | 11 lines

  Performance optimizations on Fraction's constructor.

    ./python.exe -m timeit -s 'from fractions import Fraction' 'Fraction(3)`
  31.7 usec/loop -> 9.2 usec/loop

    ./python.exe -m timeit -s 'from fractions import Fraction' 'Fraction(3, 2)'`
  27.7 usec/loop -> 9.32 usec/loop

    ./python.exe -m timeit -s 'from fractions import Fraction; f = Fraction(3, 2)' 'Fraction(f)'
  31.9 usec/loop -> 14.3 usec/loop
........
  r60786 | jeffrey.yasskin | 2008-02-14 08:49:25 +0100 (Thu, 14 Feb 2008) | 5 lines

  Change simple instances (in Fraction) of self.numerator and self.denominator to
  self._numerator and self._denominator. This speeds abs() up from 12.2us to
  10.8us and trunc() from 2.07us to 1.11us. This doesn't change _add and friends
  because they're more complicated.
........

1 files changed, 203 insertions, 218 deletions

diff --git a/Lib/test/test_fractions.py b/Lib/test/test_fractions.py
index 00fd549d72..dacb58796c 100644
--- a/Lib/test/test_fractions.py
+++ b/Lib/test/test_fractions.py
@@ -8,7 +8,7 @@ import fractions
 import unittest
 from copy import copy, deepcopy
 from pickle import dumps, loads
-R = fractions.Fraction
+F = fractions.Fraction
 gcd = fractions.gcd
 
 
@@ -49,77 +49,76 @@ class FractionTest(unittest.TestCase):
             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((0, 1), _components(F()))
+        self.assertEquals((7, 1), _components(F(7)))
+        self.assertEquals((7, 3), _components(F(F(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.assertEquals((-1, 1), _components(F(-1, 1)))
+        self.assertEquals((-1, 1), _components(F(1, -1)))
+        self.assertEquals((1, 1), _components(F(-2, -2)))
+        self.assertEquals((1, 2), _components(F(5, 10)))
+        self.assertEquals((7, 15), _components(F(7, 15)))
+        self.assertEquals((10**23, 1), _components(F(10**23)))
 
         self.assertRaisesMessage(ZeroDivisionError, "Fraction(12, 0)",
-                                 R, 12, 0)
-        self.assertRaises(TypeError, R, 1.5)
-        self.assertRaises(TypeError, R, 1.5 + 3j)
+                                 F, 12, 0)
+        self.assertRaises(AttributeError, F, 1.5)
+        self.assertRaises(AttributeError, F, 1.5 + 3j)
 
-        self.assertRaises(TypeError, R, R(1, 2), 3)
-        self.assertRaises(TypeError, R, "3/2", 3)
+        self.assertRaises(AttributeError, F, F(1, 2), 3)
+        self.assertRaises(AttributeError, F, "3/2", 3)
 
     def testFromString(self):
-        self.assertEquals((5, 1), _components(R("5")))
-        self.assertEquals((3, 2), _components(R("3/2")))
-        self.assertEquals((3, 2), _components(R(" \n  +3/2")))
-        self.assertEquals((-3, 2), _components(R("-3/2  ")))
-        self.assertEquals((3, 2), _components(R("    03/02 \n  ")))
-        self.assertEquals((3, 2), _components(R("    03/02 \n  ")))
-        self.assertEquals((16, 5), _components(R(" 3.2 ")))
-        self.assertEquals((-16, 5), _components(R(" -3.2 ")))
-        self.assertEquals((-3, 1), _components(R(" -3. ")))
-        self.assertEquals((3, 5), _components(R(" .6 ")))
+        self.assertEquals((5, 1), _components(F("5")))
+        self.assertEquals((3, 2), _components(F("3/2")))
+        self.assertEquals((3, 2), _components(F(" \n  +3/2")))
+        self.assertEquals((-3, 2), _components(F("-3/2  ")))
+        self.assertEquals((13, 2), _components(F("    013/02 \n  ")))
+        self.assertEquals((16, 5), _components(F(" 3.2 ")))
+        self.assertEquals((-16, 5), _components(F(" -3.2 ")))
+        self.assertEquals((-3, 1), _components(F(" -3. ")))
+        self.assertEquals((3, 5), _components(F(" .6 ")))
 
         self.assertRaisesMessage(
             ZeroDivisionError, "Fraction(3, 0)",
-            R, "3/0")
+            F, "3/0")
         self.assertRaisesMessage(
             ValueError, "Invalid literal for Fraction: 3/",
-            R, "3/")
+            F, "3/")
         self.assertRaisesMessage(
             ValueError, "Invalid literal for Fraction: 3 /2",
-            R, "3 /2")
+            F, "3 /2")
         self.assertRaisesMessage(
             # Denominators don't need a sign.
             ValueError, "Invalid literal for Fraction: 3/+2",
-            R, "3/+2")
+            F, "3/+2")
         self.assertRaisesMessage(
             # Imitate float's parsing.
             ValueError, "Invalid literal for Fraction: + 3/2",
-            R, "+ 3/2")
+            F, "+ 3/2")
         self.assertRaisesMessage(
             # Avoid treating '.' as a regex special character.
             ValueError, "Invalid literal for Fraction: 3a2",
-            R, "3a2")
+            F, "3a2")
         self.assertRaisesMessage(
             # Only parse ordinary decimals, not scientific form.
             ValueError, "Invalid literal for Fraction: 3.2e4",
-            R, "3.2e4")
+            F, "3.2e4")
         self.assertRaisesMessage(
             # Don't accept combinations of decimals and rationals.
             ValueError, "Invalid literal for Fraction: 3/7.2",
-            R, "3/7.2")
+            F, "3/7.2")
         self.assertRaisesMessage(
             # Don't accept combinations of decimals and rationals.
             ValueError, "Invalid literal for Fraction: 3.2/7",
-            R, "3.2/7")
+            F, "3.2/7")
         self.assertRaisesMessage(
             # Allow 3. and .3, but not .
             ValueError, "Invalid literal for Fraction: .",
-            R, ".")
+            F, ".")
 
     def testImmutable(self):
-        r = R(7, 3)
+        r = F(7, 3)
         r.__init__(2, 15)
         self.assertEquals((7, 3), _components(r))
 
@@ -132,267 +131,253 @@ class FractionTest(unittest.TestCase):
         r._denominator = 2
         self.assertEquals((4, 2), _components(r))
         # Which breaks some important operations:
-        self.assertNotEquals(R(4, 2), r)
+        self.assertNotEquals(F(4, 2), r)
 
     def testFromFloat(self):
         self.assertRaisesMessage(
             TypeError, "Fraction.from_float() only takes floats, not 3 (int)",
-            R.from_float, 3)
+            F.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((0, 1), _components(F.from_float(-0.0)))
+        self.assertEquals((10, 1), _components(F.from_float(10.0)))
+        self.assertEquals((-5, 2), _components(F.from_float(-2.5)))
         self.assertEquals((99999999999999991611392, 1),
-                          _components(R.from_float(1e23)))
-        self.assertEquals(float(10**23), float(R.from_float(1e23)))
+                          _components(F.from_float(1e23)))
+        self.assertEquals(float(10**23), float(F.from_float(1e23)))
         self.assertEquals((3602879701896397, 1125899906842624),
-                          _components(R.from_float(3.2)))
-        self.assertEquals(3.2, float(R.from_float(3.2)))
+                          _components(F.from_float(3.2)))
+        self.assertEquals(3.2, float(F.from_float(3.2)))
 
         inf = 1e1000
         nan = inf - inf
         self.assertRaisesMessage(
             TypeError, "Cannot convert inf to Fraction.",
-            R.from_float, inf)
+            F.from_float, inf)
         self.assertRaisesMessage(
             TypeError, "Cannot convert -inf to Fraction.",
-            R.from_float, -inf)
+            F.from_float, -inf)
         self.assertRaisesMessage(
             TypeError, "Cannot convert nan to Fraction.",
-            R.from_float, nan)
+            F.from_float, nan)
 
     def testFromDecimal(self):
         self.assertRaisesMessage(
             TypeError,
             "Fraction.from_decimal() only takes Decimals, not 3 (int)",
-            R.from_decimal, 3)
-        self.assertEquals(R(0), R.from_decimal(Decimal("-0")))
-        self.assertEquals(R(5, 10), R.from_decimal(Decimal("0.5")))
-        self.assertEquals(R(5, 1000), R.from_decimal(Decimal("5e-3")))
-        self.assertEquals(R(5000), R.from_decimal(Decimal("5e3")))
-        self.assertEquals(1 - R(1, 10**30),
-                          R.from_decimal(Decimal("0." + "9" * 30)))
+            F.from_decimal, 3)
+        self.assertEquals(F(0), F.from_decimal(Decimal("-0")))
+        self.assertEquals(F(5, 10), F.from_decimal(Decimal("0.5")))
+        self.assertEquals(F(5, 1000), F.from_decimal(Decimal("5e-3")))
+        self.assertEquals(F(5000), F.from_decimal(Decimal("5e3")))
+        self.assertEquals(1 - F(1, 10**30),
+                          F.from_decimal(Decimal("0." + "9" * 30)))
 
         self.assertRaisesMessage(
             TypeError, "Cannot convert Infinity to Fraction.",
-            R.from_decimal, Decimal("inf"))
+            F.from_decimal, Decimal("inf"))
         self.assertRaisesMessage(
             TypeError, "Cannot convert -Infinity to Fraction.",
-            R.from_decimal, Decimal("-inf"))
+            F.from_decimal, Decimal("-inf"))
         self.assertRaisesMessage(
             TypeError, "Cannot convert NaN to Fraction.",
-            R.from_decimal, Decimal("nan"))
+            F.from_decimal, Decimal("nan"))
         self.assertRaisesMessage(
             TypeError, "Cannot convert sNaN to Fraction.",
-            R.from_decimal, Decimal("snan"))
-
-    def testFromContinuedFraction(self):
-        self.assertRaises(TypeError, R.from_continued_fraction, None)
-        phi = R.from_continued_fraction([1]*100)
-        self.assertEquals(round(phi - (1 + 5 ** 0.5) / 2, 10), 0.0)
-
-        minusphi = R.from_continued_fraction([-1]*100)
-        self.assertEquals(round(minusphi + (1 + 5 ** 0.5) / 2, 10), 0.0)
-
-        self.assertEquals(R.from_continued_fraction([0]), R(0))
-        self.assertEquals(R.from_continued_fraction([]), R(0))
-
-    def testAsContinuedFraction(self):
-        self.assertEqual(R.from_float(math.pi).as_continued_fraction()[:15],
-                         [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3])
-        self.assertEqual(R.from_float(-math.pi).as_continued_fraction()[:16],
-                         [-4, 1, 6, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3])
-        self.assertEqual(R(0).as_continued_fraction(), [0])
-
-    def testApproximateFrom(self):
-        self.assertEqual(R.from_float(math.pi).approximate(10000), R(355, 113))
-        self.assertEqual(R.from_float(-math.pi).approximate(10000), R(-355, 113))
-        self.assertEqual(R.from_float(0.0).approximate(10000), R(0))
+            F.from_decimal, Decimal("snan"))
+
+    def testLimitDenominator(self):
+        rpi = F('3.1415926535897932')
+        self.assertEqual(rpi.limit_denominator(10000), F(355, 113))
+        self.assertEqual(-rpi.limit_denominator(10000), F(-355, 113))
+        self.assertEqual(rpi.limit_denominator(113), F(355, 113))
+        self.assertEqual(rpi.limit_denominator(112), F(333, 106))
+        self.assertEqual(F(201, 200).limit_denominator(100), F(1))
+        self.assertEqual(F(201, 200).limit_denominator(101), F(102, 101))
+        self.assertEqual(F(0).limit_denominator(10000), F(0))
 
     def testConversions(self):
-        self.assertTypedEquals(-1, math.trunc(R(-11, 10)))
-        self.assertTypedEquals(-2, math.floor(R(-11, 10)))
-        self.assertTypedEquals(-1, math.ceil(R(-11, 10)))
-        self.assertTypedEquals(-1, math.ceil(R(-10, 10)))
-        self.assertTypedEquals(-1, int(R(-11, 10)))
-
-        self.assertTypedEquals(0, round(R(-1, 10)))
-        self.assertTypedEquals(0, round(R(-5, 10)))
-        self.assertTypedEquals(-2, round(R(-15, 10)))
-        self.assertTypedEquals(-1, round(R(-7, 10)))
-
-        self.assertEquals(False, bool(R(0, 1)))
-        self.assertEquals(True, bool(R(3, 2)))
-        self.assertTypedEquals(0.1, float(R(1, 10)))
+        self.assertTypedEquals(-1, math.trunc(F(-11, 10)))
+        self.assertTypedEquals(-2, math.floor(F(-11, 10)))
+        self.assertTypedEquals(-1, math.ceil(F(-11, 10)))
+        self.assertTypedEquals(-1, math.ceil(F(-10, 10)))
+        self.assertTypedEquals(-1, int(F(-11, 10)))
+        self.assertTypedEquals(0, round(F(-1, 10)))
+        self.assertTypedEquals(0, round(F(-5, 10)))
+        self.assertTypedEquals(-2, round(F(-15, 10)))
+        self.assertTypedEquals(-1, round(F(-7, 10)))
+
+        self.assertEquals(False, bool(F(0, 1)))
+        self.assertEquals(True, bool(F(3, 2)))
+        self.assertTypedEquals(0.1, float(F(1, 10)))
 
         # Check that __float__ isn't implemented by converting the
         # numerator and denominator to float before dividing.
         self.assertRaises(OverflowError, float, int('2'*400+'7'))
         self.assertAlmostEquals(2.0/3,
-                                float(R(int('2'*400+'7'), int('3'*400+'1'))))
+                                float(F(int('2'*400+'7'), int('3'*400+'1'))))
 
-        self.assertTypedEquals(0.1+0j, complex(R(1,10)))
+        self.assertTypedEquals(0.1+0j, complex(F(1,10)))
 
     def testRound(self):
-        self.assertTypedEquals(R(-200), round(R(-150), -2))
-        self.assertTypedEquals(R(-200), round(R(-250), -2))
-        self.assertTypedEquals(R(30), round(R(26), -1))
-        self.assertTypedEquals(R(-2, 10), round(R(-15, 100), 1))
-        self.assertTypedEquals(R(-2, 10), round(R(-25, 100), 1))
+        self.assertTypedEquals(F(-200), round(F(-150), -2))
+        self.assertTypedEquals(F(-200), round(F(-250), -2))
+        self.assertTypedEquals(F(30), round(F(26), -1))
+        self.assertTypedEquals(F(-2, 10), round(F(-15, 100), 1))
+        self.assertTypedEquals(F(-2, 10), round(F(-25, 100), 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))
-        z = pow(R(-1), R(1, 2))
+        self.assertEquals(F(1, 2), F(1, 10) + F(2, 5))
+        self.assertEquals(F(-3, 10), F(1, 10) - F(2, 5))
+        self.assertEquals(F(1, 25), F(1, 10) * F(2, 5))
+        self.assertEquals(F(1, 4), F(1, 10) / F(2, 5))
+        self.assertTypedEquals(2, F(9, 10) // F(2, 5))
+        self.assertTypedEquals(10**23, F(10**23, 1) // F(1))
+        self.assertEquals(F(2, 3), F(-7, 3) % F(3, 2))
+        self.assertEquals(F(8, 27), F(2, 3) ** F(3))
+        self.assertEquals(F(27, 8), F(2, 3) ** F(-3))
+        self.assertTypedEquals(2.0, F(4) ** F(1, 2))
+        z = pow(F(-1), F(1, 2))
         self.assertAlmostEquals(z.real, 0)
         self.assertEquals(z.imag, 1)
 
     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, R(1, 10) // 1.0)
-        self.assertTypedEquals(10, 1 // R(1, 10))
-        self.assertTypedEquals(10**23, 10**22 // R(1, 10))
-        self.assertTypedEquals(10, 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))
+        self.assertTypedEquals(F(11, 10), F(1, 10) + 1)
+        self.assertTypedEquals(1.1, F(1, 10) + 1.0)
+        self.assertTypedEquals(1.1 + 0j, F(1, 10) + (1.0 + 0j))
+        self.assertTypedEquals(F(11, 10), 1 + F(1, 10))
+        self.assertTypedEquals(1.1, 1.0 + F(1, 10))
+        self.assertTypedEquals(1.1 + 0j, (1.0 + 0j) + F(1, 10))
+
+        self.assertTypedEquals(F(-9, 10), F(1, 10) - 1)
+        self.assertTypedEquals(-0.9, F(1, 10) - 1.0)
+        self.assertTypedEquals(-0.9 + 0j, F(1, 10) - (1.0 + 0j))
+        self.assertTypedEquals(F(9, 10), 1 - F(1, 10))
+        self.assertTypedEquals(0.9, 1.0 - F(1, 10))
+        self.assertTypedEquals(0.9 + 0j, (1.0 + 0j) - F(1, 10))
+
+        self.assertTypedEquals(F(1, 10), F(1, 10) * 1)
+        self.assertTypedEquals(0.1, F(1, 10) * 1.0)
+        self.assertTypedEquals(0.1 + 0j, F(1, 10) * (1.0 + 0j))
+        self.assertTypedEquals(F(1, 10), 1 * F(1, 10))
+        self.assertTypedEquals(0.1, 1.0 * F(1, 10))
+        self.assertTypedEquals(0.1 + 0j, (1.0 + 0j) * F(1, 10))
+
+        self.assertTypedEquals(F(1, 10), F(1, 10) / 1)
+        self.assertTypedEquals(0.1, F(1, 10) / 1.0)
+        self.assertTypedEquals(0.1 + 0j, F(1, 10) / (1.0 + 0j))
+        self.assertTypedEquals(F(10, 1), 1 / F(1, 10))
+        self.assertTypedEquals(10.0, 1.0 / F(1, 10))
+        self.assertTypedEquals(10.0 + 0j, (1.0 + 0j) / F(1, 10))
+
+        self.assertTypedEquals(0, F(1, 10) // 1)
+        self.assertTypedEquals(0, F(1, 10) // 1.0)
+        self.assertTypedEquals(10, 1 // F(1, 10))
+        self.assertTypedEquals(10**23, 10**22 // F(1, 10))
+        self.assertTypedEquals(10, 1.0 // F(1, 10))
+
+        self.assertTypedEquals(F(1, 10), F(1, 10) % 1)
+        self.assertTypedEquals(0.1, F(1, 10) % 1.0)
+        self.assertTypedEquals(F(0, 1), 1 % F(1, 10))
+        self.assertTypedEquals(0.0, 1.0 % F(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))
-        z = pow(-1, R(1, 2))
+        self.assertTypedEquals(F(100, 1), F(1, 10) ** -2)
+        self.assertTypedEquals(F(100, 1), F(10, 1) ** 2)
+        self.assertTypedEquals(0.1, F(1, 10) ** 1.0)
+        self.assertTypedEquals(0.1 + 0j, F(1, 10) ** (1.0 + 0j))
+        self.assertTypedEquals(4 , 2 ** F(2, 1))
+        z = pow(-1, F(1, 2))
         self.assertAlmostEquals(0, z.real)
         self.assertEquals(1, z.imag)
-        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))
+        self.assertTypedEquals(F(1, 4) , 2 ** F(-2, 1))
+        self.assertTypedEquals(2.0 , 4 ** F(1, 2))
+        self.assertTypedEquals(0.25, 2.0 ** F(-2, 1))
+        self.assertTypedEquals(1.0 + 0j, (1.0 + 0j) ** F(1, 10))
 
     def testMixingWithDecimal(self):
         # Decimal refuses mixed comparisons.
         self.assertRaisesMessage(
             TypeError,
             "unsupported operand type(s) for +: 'Fraction' and 'Decimal'",
-            operator.add, R(3,11), Decimal('3.1415926'))
-        self.assertNotEquals(R(5, 2), Decimal('2.5'))
+            operator.add, F(3,11), Decimal('3.1415926'))
+        self.assertNotEquals(F(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))
-        self.assertFalse(R(1, 2) != R(1, 2))
-        self.assertTrue(R(1, 2) != R(1, 3))
+        self.assertTrue(F(1, 2) < F(2, 3))
+        self.assertFalse(F(1, 2) < F(1, 2))
+        self.assertTrue(F(1, 2) <= F(2, 3))
+        self.assertTrue(F(1, 2) <= F(1, 2))
+        self.assertFalse(F(2, 3) <= F(1, 2))
+        self.assertTrue(F(1, 2) == F(1, 2))
+        self.assertFalse(F(1, 2) == F(1, 3))
+        self.assertFalse(F(1, 2) != F(1, 2))
+        self.assertTrue(F(1, 2) != F(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(2 < F(5, 2))
+        self.assertFalse(2 < F(4, 2))
+        self.assertTrue(F(5, 2) < 3)
+        self.assertFalse(F(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))
+        self.assertTrue(F(1, 2) < 0.6)
+        self.assertFalse(F(1, 2) < 0.4)
+        self.assertTrue(0.4 < F(1, 2))
+        self.assertFalse(0.5 < F(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)
+        self.assertTrue(0.5 <= F(1, 2))
+        self.assertFalse(0.6 <= F(1, 2))
+        self.assertTrue(F(1, 2) <= 0.5)
+        self.assertFalse(F(1, 2) <= 0.4)
+        self.assertTrue(2 <= F(4, 2))
+        self.assertFalse(2 <= F(3, 2))
+        self.assertTrue(F(4, 2) <= 2)
+        self.assertFalse(F(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))
+        self.assertFalse(F(10**23) == float(10**23))
         # The first test demonstrates why these are important.
-        self.assertFalse(1e23 < float(R(math.trunc(1e23) + 1)))
-        self.assertTrue(1e23 < R(math.trunc(1e23) + 1))
-        self.assertFalse(1e23 <= R(math.trunc(1e23) - 1))
-        self.assertTrue(1e23 > R(math.trunc(1e23) - 1))
-        self.assertFalse(1e23 >= R(math.trunc(1e23) + 1))
+        self.assertFalse(1e23 < float(F(math.trunc(1e23) + 1)))
+        self.assertTrue(1e23 < F(math.trunc(1e23) + 1))
+        self.assertFalse(1e23 <= F(math.trunc(1e23) - 1))
+        self.assertTrue(1e23 > F(math.trunc(1e23) - 1))
+        self.assertFalse(1e23 >= F(math.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))
+        self.assertFalse(F(10**23) == complex(10**23))
+        self.assertTrue(F(10**23) > complex(10**23))
+        self.assertFalse(F(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)
+        self.assertTrue(0.5 == F(1, 2))
+        self.assertFalse(0.6 == F(1, 2))
+        self.assertTrue(F(1, 2) == 0.5)
+        self.assertFalse(F(1, 2) == 0.4)
+        self.assertTrue(2 == F(4, 2))
+        self.assertFalse(2 == F(3, 2))
+        self.assertTrue(F(4, 2) == 2)
+        self.assertFalse(F(5, 2) == 2)
 
     def testStringification(self):
-        self.assertEquals("Fraction(7,3)", repr(R(7, 3)))
-        self.assertEquals("7/3", str(R(7, 3)))
-        self.assertEquals("7", str(R(7, 1)))
+        self.assertEquals("Fraction(7, 3)", repr(F(7, 3)))
+        self.assertEquals("7/3", str(F(7, 3)))
+        self.assertEquals("7", str(F(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)))
+        self.assertEquals(hash(2.5), hash(F(5, 2)))
+        self.assertEquals(hash(10**50), hash(F(10**50)))
+        self.assertNotEquals(hash(float(10**23)), hash(F(10**23)))
 
     def testApproximatePi(self):
         # Algorithm borrowed from
         # http://docs.python.org/lib/decimal-recipes.html
-        three = R(3)
+        three = F(3)
         lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
-        while abs(s - lasts) > R(1, 10**9):
+        while abs(s - lasts) > F(1, 10**9):
             lasts = s
             n, na = n+na, na+8
             d, da = d+da, da+32
@@ -403,9 +388,9 @@ class FractionTest(unittest.TestCase):
     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):
+        x = F(1)
+        i, lasts, s, fact, num, sign = 0, 0, F(1), 1, 1, 1
+        while abs(s - lasts) > F(1, 10**9):
             lasts = s
             i += 2
             fact *= i * (i-1)
@@ -415,7 +400,7 @@ class FractionTest(unittest.TestCase):
         self.assertAlmostEquals(math.cos(1), s)
 
     def test_copy_deepcopy_pickle(self):
-        r = R(13, 7)
+        r = F(13, 7)
         self.assertEqual(r, loads(dumps(r)))
         self.assertEqual(id(r), id(copy(r)))
         self.assertEqual(id(r), id(deepcopy(r)))