bpo-35588: Speed up mod, divmod and floordiv operations for Fraction type (#11322)

* bpo-35588: Implement mod and divmod operations for Fraction type by spelling out the numerator/denominator calculation, instead of instantiating and normalising Fractions along the way. This speeds up '%' and divmod() by 2-3x.

* bpo-35588: Also reimplement Fraction.__floordiv__() using integer operations to make it ~4x faster.

* Improve code formatting.

Co-Authored-By: scoder <stefan_ml@behnel.de>

* bpo-35588: Fix return type of divmod(): the result of the integer division should be an integer.

* bpo-35588: Further specialise __mod__() and inline the original helper function _flat_divmod() since it's no longer reused.

* bpo-35588: Add some tests with large numerators and/or denominators.

* bpo-35588: Use builtin "divmod()" function for implementing __divmod__() in order to simplify the implementation, even though performance results are mixed.

* Rremove accidentally added empty line.

* bpo-35588: Try to provide more informative output on test failures.

* bpo-35588: Improve wording in News entry.

Co-Authored-By: scoder <stefan_ml@behnel.de>

* Remove stray space.

Lib/fractions.py

@@ -429,14 +429,22 @@ def _div(a, b):
 
     def _floordiv(a, b):
         """a // b"""
-        return math.floor(a / b)
+        return (a.numerator * b.denominator) // (a.denominator * b.numerator)
 
     __floordiv__, __rfloordiv__ = _operator_fallbacks(_floordiv, operator.floordiv)
 
+    def _divmod(a, b):
+        """(a // b, a % b)"""
+        da, db = a.denominator, b.denominator
+        div, n_mod = divmod(a.numerator * db, da * b.numerator)
+        return div, Fraction(n_mod, da * db)
+
+    __divmod__, __rdivmod__ = _operator_fallbacks(_divmod, divmod)
+
     def _mod(a, b):
         """a % b"""
-        div = a // b
+        da, db = a.denominator, b.denominator
-        return a - b * div
+        return Fraction((a.numerator * db) % (b.numerator * da), da * db)
 
     __mod__, __rmod__ = _operator_fallbacks(_mod, operator.mod)
 

Lib/test/test_fractions.py

@@ -117,6 +117,11 @@ def assertTypedEquals(self, expected, actual):
         self.assertEqual(type(expected), type(actual))
         self.assertEqual(expected, actual)
 
+    def assertTypedTupleEquals(self, expected, actual):
+        """Asserts that both the types and values in the tuples are the same."""
+        self.assertTupleEqual(expected, actual)
+        self.assertListEqual(list(map(type, expected)), list(map(type, actual)))
+
     def assertRaisesMessage(self, exc_type, message,
                             callable, *args, **kwargs):
         """Asserts that callable(*args, **kwargs) raises exc_type(message)."""
@@ -349,7 +354,10 @@ def testArithmetic(self):
         self.assertEqual(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.assertEqual(F(5, 6), F(7, 3) % F(3, 2))
         self.assertEqual(F(2, 3), F(-7, 3) % F(3, 2))
+        self.assertEqual((F(1), F(5, 6)), divmod(F(7, 3), F(3, 2)))
+        self.assertEqual((F(-2), F(2, 3)), divmod(F(-7, 3), F(3, 2)))
         self.assertEqual(F(8, 27), F(2, 3) ** F(3))
         self.assertEqual(F(27, 8), F(2, 3) ** F(-3))
         self.assertTypedEquals(2.0, F(4) ** F(1, 2))
@@ -371,6 +379,40 @@ def testArithmetic(self):
         self.assertEqual(p.numerator, 4)
         self.assertEqual(p.denominator, 1)
 
+    def testLargeArithmetic(self):
+        self.assertTypedEquals(
+            F(10101010100808080808080808101010101010000000000000000,
+              1010101010101010101010101011111111101010101010101010101010101),
+            F(10**35+1, 10**27+1) % F(10**27+1, 10**35-1)
+        )
+        self.assertTypedEquals(
+            F(7, 1901475900342344102245054808064),
+            F(-2**100, 3) % F(5, 2**100)
+        )
+        self.assertTypedTupleEquals(
+            (9999999999999999,
+             F(10101010100808080808080808101010101010000000000000000,
+               1010101010101010101010101011111111101010101010101010101010101)),
+            divmod(F(10**35+1, 10**27+1), F(10**27+1, 10**35-1))
+        )
+        self.assertTypedEquals(
+            -2 ** 200 // 15,
+            F(-2**100, 3) // F(5, 2**100)
+        )
+        self.assertTypedEquals(
+            1,
+            F(5, 2**100) // F(3, 2**100)
+        )
+        self.assertTypedEquals(
+            (1, F(2, 2**100)),
+            divmod(F(5, 2**100), F(3, 2**100))
+        )
+        self.assertTypedTupleEquals(
+            (-2 ** 200 // 15,
+             F(7, 1901475900342344102245054808064)),
+            divmod(F(-2**100, 3), F(5, 2**100))
+        )
+
     def testMixedArithmetic(self):
         self.assertTypedEquals(F(11, 10), F(1, 10) + 1)
         self.assertTypedEquals(1.1, F(1, 10) + 1.0)
@@ -415,7 +457,14 @@ def testMixedArithmetic(self):
         self.assertTypedEquals(float('inf'), F(-1, 10) % float('inf'))
         self.assertTypedEquals(-0.1, F(-1, 10) % float('-inf'))
 
-        # No need for divmod since we don't override it.
+        self.assertTypedTupleEquals((0, F(1, 10)), divmod(F(1, 10), 1))
+        self.assertTypedTupleEquals(divmod(0.1, 1.0), divmod(F(1, 10), 1.0))
+        self.assertTypedTupleEquals((10, F(0)), divmod(1, F(1, 10)))
+        self.assertTypedTupleEquals(divmod(1.0, 0.1), divmod(1.0, F(1, 10)))
+        self.assertTypedTupleEquals(divmod(0.1, float('inf')), divmod(F(1, 10), float('inf')))
+        self.assertTypedTupleEquals(divmod(0.1, float('-inf')), divmod(F(1, 10), float('-inf')))
+        self.assertTypedTupleEquals(divmod(-0.1, float('inf')), divmod(F(-1, 10), float('inf')))
+        self.assertTypedTupleEquals(divmod(-0.1, float('-inf')), divmod(F(-1, 10), float('-inf')))
 
         # ** has more interesting conversion rules.
         self.assertTypedEquals(F(100, 1), F(1, 10) ** -2)

Misc/NEWS.d/next/Library/2018-12-26-10-55-59.bpo-35588.PSR6Ez.rst

@@ -0,0 +1,2 @@
+The floor division and modulo operations and the :func:`divmod` function on :class:`fractions.Fraction` types are 2--4x faster.
+Patch by Stefan Behnel.