mirror of https://github.com/python/cpython
Issue #1811: Improve accuracy and consistency of true division for integers.
This commit is contained in:
Mark Dickinson
parent
13c2ef92f8
commit
4657283647
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@ -3,9 +3,52 @@ from __future__ import division
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# test_long.py instead. In the meantime, it's too obscure to try to
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# trick just part of test_long into using future division.
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import sys
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import random
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import unittest
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from test.test_support import run_unittest
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# decorator for skipping tests on non-IEEE 754 platforms
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requires_IEEE_754 = unittest.skipUnless(
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float.__getformat__("double").startswith("IEEE"),
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"test requires IEEE 754 doubles")
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DBL_MAX = sys.float_info.max
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DBL_MAX_EXP = sys.float_info.max_exp
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DBL_MIN_EXP = sys.float_info.min_exp
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DBL_MANT_DIG = sys.float_info.mant_dig
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DBL_MIN_OVERFLOW = 2**DBL_MAX_EXP - 2**(DBL_MAX_EXP - DBL_MANT_DIG - 1)
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# pure Python version of correctly-rounded true division
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def truediv(a, b):
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"""Correctly-rounded true division for integers."""
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negative = a^b < 0
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a, b = abs(a), abs(b)
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# exceptions: division by zero, overflow
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if not b:
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raise ZeroDivisionError("division by zero")
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if a >= DBL_MIN_OVERFLOW * b:
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raise OverflowError("int/int too large to represent as a float")
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# find integer d satisfying 2**(d - 1) <= a/b < 2**d
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d = a.bit_length() - b.bit_length()
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if d >= 0 and a >= 2**d * b or d < 0 and a * 2**-d >= b:
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d += 1
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# compute 2**-exp * a / b for suitable exp
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exp = max(d, DBL_MIN_EXP) - DBL_MANT_DIG
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a, b = a << max(-exp, 0), b << max(exp, 0)
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q, r = divmod(a, b)
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# round-half-to-even: fractional part is r/b, which is > 0.5 iff
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# 2*r > b, and == 0.5 iff 2*r == b.
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if 2*r > b or 2*r == b and q % 2 == 1:
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q += 1
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result = float(q) * 2.**exp
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return -result if negative else result
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class TrueDivisionTests(unittest.TestCase):
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def test(self):
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huge = 1L << 40000
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@ -45,6 +88,131 @@ class TrueDivisionTests(unittest.TestCase):
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with self.assertRaises(ZeroDivisionError):
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eval(zero, namespace)
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def check_truediv(self, a, b, skip_small=True):
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"""Verify that the result of a/b is correctly rounded, by
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comparing it with a pure Python implementation of correctly
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rounded division. b should be nonzero."""
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a, b = long(a), long(b)
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# skip check for small a and b: in this case, the current
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# implementation converts the arguments to float directly and
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# then applies a float division. This can give doubly-rounded
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# results on x87-using machines (particularly 32-bit Linux).
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if skip_small and max(abs(a), abs(b)) < 2**DBL_MANT_DIG:
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return
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try:
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# use repr so that we can distinguish between -0.0 and 0.0
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expected = repr(truediv(a, b))
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except OverflowError:
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expected = 'overflow'
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except ZeroDivisionError:
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expected = 'zerodivision'
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try:
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got = repr(a / b)
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except OverflowError:
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got = 'overflow'
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except ZeroDivisionError:
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got = 'zerodivision'
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if expected != got:
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self.fail("Incorrectly rounded division {}/{}: expected {!r}, "
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"got {!r}.".format(a, b, expected, got))
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@requires_IEEE_754
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def test_correctly_rounded_true_division(self):
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# more stringent tests than those above, checking that the
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# result of true division of ints is always correctly rounded.
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# This test should probably be considered CPython-specific.
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# Exercise all the code paths not involving Gb-sized ints.
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# ... divisions involving zero
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self.check_truediv(123, 0)
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self.check_truediv(-456, 0)
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self.check_truediv(0, 3)
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self.check_truediv(0, -3)
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self.check_truediv(0, 0)
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# ... overflow or underflow by large margin
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self.check_truediv(671 * 12345 * 2**DBL_MAX_EXP, 12345)
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self.check_truediv(12345, 345678 * 2**(DBL_MANT_DIG - DBL_MIN_EXP))
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# ... a much larger or smaller than b
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self.check_truediv(12345*2**100, 98765)
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self.check_truediv(12345*2**30, 98765*7**81)
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# ... a / b near a boundary: one of 1, 2**DBL_MANT_DIG, 2**DBL_MIN_EXP,
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# 2**DBL_MAX_EXP, 2**(DBL_MIN_EXP-DBL_MANT_DIG)
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bases = (0, DBL_MANT_DIG, DBL_MIN_EXP,
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DBL_MAX_EXP, DBL_MIN_EXP - DBL_MANT_DIG)
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for base in bases:
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for exp in range(base - 15, base + 15):
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self.check_truediv(75312*2**max(exp, 0), 69187*2**max(-exp, 0))
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self.check_truediv(69187*2**max(exp, 0), 75312*2**max(-exp, 0))
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# overflow corner case
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for m in [1, 2, 7, 17, 12345, 7**100,
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-1, -2, -5, -23, -67891, -41**50]:
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for n in range(-10, 10):
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self.check_truediv(m*DBL_MIN_OVERFLOW + n, m)
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self.check_truediv(m*DBL_MIN_OVERFLOW + n, -m)
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# check detection of inexactness in shifting stage
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for n in range(250):
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# (2**DBL_MANT_DIG+1)/(2**DBL_MANT_DIG) lies halfway
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# between two representable floats, and would usually be
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# rounded down under round-half-to-even. The tiniest of
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# additions to the numerator should cause it to be rounded
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# up instead.
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self.check_truediv((2**DBL_MANT_DIG + 1)*12345*2**200 + 2**n,
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2**DBL_MANT_DIG*12345)
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# 1/2731 is one of the smallest division cases that's subject
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# to double rounding on IEEE 754 machines working internally with
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# 64-bit precision. On such machines, the next check would fail,
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# were it not explicitly skipped in check_truediv.
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self.check_truediv(1, 2731)
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# a particularly bad case for the old algorithm: gives an
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# error of close to 3.5 ulps.
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self.check_truediv(295147931372582273023, 295147932265116303360)
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for i in range(1000):
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self.check_truediv(10**(i+1), 10**i)
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self.check_truediv(10**i, 10**(i+1))
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# test round-half-to-even behaviour, normal result
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for m in [1, 2, 4, 7, 8, 16, 17, 32, 12345, 7**100,
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-1, -2, -5, -23, -67891, -41**50]:
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for n in range(-10, 10):
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self.check_truediv(2**DBL_MANT_DIG*m + n, m)
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# test round-half-to-even, subnormal result
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for n in range(-20, 20):
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self.check_truediv(n, 2**1076)
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# largeish random divisions: a/b where |a| <= |b| <=
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# 2*|a|; |ans| is between 0.5 and 1.0, so error should
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# always be bounded by 2**-54 with equality possible only
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# if the least significant bit of q=ans*2**53 is zero.
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for M in [10**10, 10**100, 10**1000]:
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for i in range(1000):
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a = random.randrange(1, M)
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b = random.randrange(a, 2*a+1)
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self.check_truediv(a, b)
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self.check_truediv(-a, b)
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self.check_truediv(a, -b)
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self.check_truediv(-a, -b)
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# and some (genuinely) random tests
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for _ in range(10000):
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a_bits = random.randrange(1000)
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b_bits = random.randrange(1, 1000)
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x = random.randrange(2**a_bits)
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y = random.randrange(1, 2**b_bits)
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self.check_truediv(x, y)
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self.check_truediv(x, -y)
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self.check_truediv(-x, y)
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self.check_truediv(-x, -y)
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def test_main():
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run_unittest(TrueDivisionTests)
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@ -12,6 +12,11 @@ What's New in Python 2.7 alpha 2?
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Core and Builtins
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-----------------
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- Issue #1811: improve accuracy and cross-platform consistency for
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true division of integers: the result of a/b is now correctly
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rounded for ints a and b (at least on IEEE 754 platforms), and in
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particular does not depend on the internal representation of a long.
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- Issue #6108: unicode(exception) and str(exception) should return the same
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message when only __str__ (and not __unicode__) is overridden in the subclass.
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@ -645,16 +645,35 @@ int_classic_div(PyIntObject *x, PyIntObject *y)
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}
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static PyObject *
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int_true_divide(PyObject *v, PyObject *w)
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int_true_divide(PyIntObject *x, PyIntObject *y)
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{
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long xi, yi;
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/* If they aren't both ints, give someone else a chance. In
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particular, this lets int/long get handled by longs, which
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underflows to 0 gracefully if the long is too big to convert
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to float. */
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if (PyInt_Check(v) && PyInt_Check(w))
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return PyFloat_Type.tp_as_number->nb_true_divide(v, w);
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Py_INCREF(Py_NotImplemented);
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return Py_NotImplemented;
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CONVERT_TO_LONG(x, xi);
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CONVERT_TO_LONG(y, yi);
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if (yi == 0) {
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PyErr_SetString(PyExc_ZeroDivisionError,
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"division by zero");
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return NULL;
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}
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if (xi == 0)
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return PyFloat_FromDouble(yi < 0 ? -0.0 : 0.0);
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#define WIDTH_OF_ULONG (CHAR_BIT*SIZEOF_LONG)
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#if DBL_MANT_DIG < WIDTH_OF_ULONG
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if ((xi >= 0 ? 0UL + xi : 0UL - xi) >> DBL_MANT_DIG ||
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(yi >= 0 ? 0UL + yi : 0UL - yi) >> DBL_MANT_DIG)
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/* Large x or y. Use long integer arithmetic. */
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return PyLong_Type.tp_as_number->nb_true_divide(
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(PyObject *)x, (PyObject *)y);
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else
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#endif
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/* Both ints can be exactly represented as doubles. Do a
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floating-point division. */
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return PyFloat_FromDouble((double)xi / (double)yi);
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}
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static PyObject *
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@ -1355,7 +1374,7 @@ static PyNumberMethods int_as_number = {
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0, /*nb_inplace_xor*/
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0, /*nb_inplace_or*/
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(binaryfunc)int_div, /* nb_floor_divide */
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int_true_divide, /* nb_true_divide */
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(binaryfunc)int_true_divide, /* nb_true_divide */
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0, /* nb_inplace_floor_divide */
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0, /* nb_inplace_true_divide */
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(unaryfunc)int_int, /* nb_index */
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@ -3037,50 +3037,271 @@ long_classic_div(PyObject *v, PyObject *w)
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return (PyObject *)div;
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}
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/* PyLong/PyLong -> float, with correctly rounded result. */
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#define MANT_DIG_DIGITS (DBL_MANT_DIG / PyLong_SHIFT)
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#define MANT_DIG_BITS (DBL_MANT_DIG % PyLong_SHIFT)
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static PyObject *
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long_true_divide(PyObject *v, PyObject *w)
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{
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PyLongObject *a, *b;
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double ad, bd;
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int failed, aexp = -1, bexp = -1;
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PyLongObject *a, *b, *x;
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Py_ssize_t a_size, b_size, shift, extra_bits, diff, x_size, x_bits;
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digit mask, low;
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int inexact, negate, a_is_small, b_is_small;
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double dx, result;
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CONVERT_BINOP(v, w, &a, &b);
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ad = _PyLong_AsScaledDouble((PyObject *)a, &aexp);
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bd = _PyLong_AsScaledDouble((PyObject *)b, &bexp);
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failed = (ad == -1.0 || bd == -1.0) && PyErr_Occurred();
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Py_DECREF(a);
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Py_DECREF(b);
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if (failed)
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return NULL;
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/* 'aexp' and 'bexp' were initialized to -1 to silence gcc-4.0.x,
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but should really be set correctly after sucessful calls to
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_PyLong_AsScaledDouble() */
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assert(aexp >= 0 && bexp >= 0);
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if (bd == 0.0) {
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/*
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Method in a nutshell:
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0. reduce to case a, b > 0; filter out obvious underflow/overflow
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1. choose a suitable integer 'shift'
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2. use integer arithmetic to compute x = floor(2**-shift*a/b)
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3. adjust x for correct rounding
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4. convert x to a double dx with the same value
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5. return ldexp(dx, shift).
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In more detail:
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0. For any a, a/0 raises ZeroDivisionError; for nonzero b, 0/b
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returns either 0.0 or -0.0, depending on the sign of b. For a and
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b both nonzero, ignore signs of a and b, and add the sign back in
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at the end. Now write a_bits and b_bits for the bit lengths of a
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and b respectively (that is, a_bits = 1 + floor(log_2(a)); likewise
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for b). Then
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2**(a_bits - b_bits - 1) < a/b < 2**(a_bits - b_bits + 1).
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So if a_bits - b_bits > DBL_MAX_EXP then a/b > 2**DBL_MAX_EXP and
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so overflows. Similarly, if a_bits - b_bits < DBL_MIN_EXP -
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DBL_MANT_DIG - 1 then a/b underflows to 0. With these cases out of
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the way, we can assume that
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DBL_MIN_EXP - DBL_MANT_DIG - 1 <= a_bits - b_bits <= DBL_MAX_EXP.
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1. The integer 'shift' is chosen so that x has the right number of
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bits for a double, plus two or three extra bits that will be used
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in the rounding decisions. Writing a_bits and b_bits for the
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number of significant bits in a and b respectively, a
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straightforward formula for shift is:
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shift = a_bits - b_bits - DBL_MANT_DIG - 2
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This is fine in the usual case, but if a/b is smaller than the
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smallest normal float then it can lead to double rounding on an
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IEEE 754 platform, giving incorrectly rounded results. So we
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adjust the formula slightly. The actual formula used is:
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shift = MAX(a_bits - b_bits, DBL_MIN_EXP) - DBL_MANT_DIG - 2
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2. The quantity x is computed by first shifting a (left -shift bits
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if shift <= 0, right shift bits if shift > 0) and then dividing by
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b. For both the shift and the division, we keep track of whether
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the result is inexact, in a flag 'inexact'; this information is
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needed at the rounding stage.
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With the choice of shift above, together with our assumption that
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a_bits - b_bits >= DBL_MIN_EXP - DBL_MANT_DIG - 1, it follows
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that x >= 1.
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3. Now x * 2**shift <= a/b < (x+1) * 2**shift. We want to replace
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this with an exactly representable float of the form
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round(x/2**extra_bits) * 2**(extra_bits+shift).
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For float representability, we need x/2**extra_bits <
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2**DBL_MANT_DIG and extra_bits + shift >= DBL_MIN_EXP -
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DBL_MANT_DIG. This translates to the condition:
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extra_bits >= MAX(x_bits, DBL_MIN_EXP - shift) - DBL_MANT_DIG
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To round, we just modify the bottom digit of x in-place; this can
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end up giving a digit with value > PyLONG_MASK, but that's not a
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problem since digits can hold values up to 2*PyLONG_MASK+1.
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With the original choices for shift above, extra_bits will always
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be 2 or 3. Then rounding under the round-half-to-even rule, we
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round up iff the most significant of the extra bits is 1, and
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either: (a) the computation of x in step 2 had an inexact result,
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or (b) at least one other of the extra bits is 1, or (c) the least
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significant bit of x (above those to be rounded) is 1.
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4. Conversion to a double is straightforward; all floating-point
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operations involved in the conversion are exact, so there's no
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danger of rounding errors.
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5. Use ldexp(x, shift) to compute x*2**shift, the final result.
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The result will always be exactly representable as a double, except
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in the case that it overflows. To avoid dependence on the exact
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behaviour of ldexp on overflow, we check for overflow before
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applying ldexp. The result of ldexp is adjusted for sign before
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returning.
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*/
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/* Reduce to case where a and b are both positive. */
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a_size = ABS(Py_SIZE(a));
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b_size = ABS(Py_SIZE(b));
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negate = (Py_SIZE(a) < 0) ^ (Py_SIZE(b) < 0);
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if (b_size == 0) {
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PyErr_SetString(PyExc_ZeroDivisionError,
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"long division or modulo by zero");
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return NULL;
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"division by zero");
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goto error;
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}
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if (a_size == 0)
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goto underflow_or_zero;
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/* Fast path for a and b small (exactly representable in a double).
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Relies on floating-point division being correctly rounded; results
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may be subject to double rounding on x86 machines that operate with
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the x87 FPU set to 64-bit precision. */
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a_is_small = a_size <= MANT_DIG_DIGITS ||
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(a_size == MANT_DIG_DIGITS+1 &&
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a->ob_digit[MANT_DIG_DIGITS] >> MANT_DIG_BITS == 0);
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b_is_small = b_size <= MANT_DIG_DIGITS ||
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(b_size == MANT_DIG_DIGITS+1 &&
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b->ob_digit[MANT_DIG_DIGITS] >> MANT_DIG_BITS == 0);
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if (a_is_small && b_is_small) {
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double da, db;
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da = a->ob_digit[--a_size];
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while (a_size > 0)
|
||||
da = da * PyLong_BASE + a->ob_digit[--a_size];
|
||||
db = b->ob_digit[--b_size];
|
||||
while (b_size > 0)
|
||||
db = db * PyLong_BASE + b->ob_digit[--b_size];
|
||||
result = da / db;
|
||||
goto success;
|
||||
}
|
||||
|
||||
/* True value is very close to ad/bd * 2**(PyLong_SHIFT*(aexp-bexp)) */
|
||||
ad /= bd; /* overflow/underflow impossible here */
|
||||
aexp -= bexp;
|
||||
if (aexp > INT_MAX / PyLong_SHIFT)
|
||||
/* Catch obvious cases of underflow and overflow */
|
||||
diff = a_size - b_size;
|
||||
if (diff > PY_SSIZE_T_MAX/PyLong_SHIFT - 1)
|
||||
/* Extreme overflow */
|
||||
goto overflow;
|
||||
else if (aexp < -(INT_MAX / PyLong_SHIFT))
|
||||
return PyFloat_FromDouble(0.0); /* underflow to 0 */
|
||||
errno = 0;
|
||||
ad = ldexp(ad, aexp * PyLong_SHIFT);
|
||||
if (Py_OVERFLOWED(ad)) /* ignore underflow to 0.0 */
|
||||
else if (diff < 1 - PY_SSIZE_T_MAX/PyLong_SHIFT)
|
||||
/* Extreme underflow */
|
||||
goto underflow_or_zero;
|
||||
/* Next line is now safe from overflowing a Py_ssize_t */
|
||||
diff = diff * PyLong_SHIFT + bits_in_digit(a->ob_digit[a_size - 1]) -
|
||||
bits_in_digit(b->ob_digit[b_size - 1]);
|
||||
/* Now diff = a_bits - b_bits. */
|
||||
if (diff > DBL_MAX_EXP)
|
||||
goto overflow;
|
||||
return PyFloat_FromDouble(ad);
|
||||
else if (diff < DBL_MIN_EXP - DBL_MANT_DIG - 1)
|
||||
goto underflow_or_zero;
|
||||
|
||||
overflow:
|
||||
/* Choose value for shift; see comments for step 1 above. */
|
||||
shift = MAX(diff, DBL_MIN_EXP) - DBL_MANT_DIG - 2;
|
||||
|
||||
inexact = 0;
|
||||
|
||||
/* x = abs(a * 2**-shift) */
|
||||
if (shift <= 0) {
|
||||
Py_ssize_t i, shift_digits = -shift / PyLong_SHIFT;
|
||||
digit rem;
|
||||
/* x = a << -shift */
|
||||
if (a_size >= PY_SSIZE_T_MAX - 1 - shift_digits) {
|
||||
/* In practice, it's probably impossible to end up
|
||||
here. Both a and b would have to be enormous,
|
||||
using close to SIZE_T_MAX bytes of memory each. */
|
||||
PyErr_SetString(PyExc_OverflowError,
|
||||
"intermediate overflow during division");
|
||||
goto error;
|
||||
}
|
||||
x = _PyLong_New(a_size + shift_digits + 1);
|
||||
if (x == NULL)
|
||||
goto error;
|
||||
for (i = 0; i < shift_digits; i++)
|
||||
x->ob_digit[i] = 0;
|
||||
rem = v_lshift(x->ob_digit + shift_digits, a->ob_digit,
|
||||
a_size, -shift % PyLong_SHIFT);
|
||||
x->ob_digit[a_size + shift_digits] = rem;
|
||||
}
|
||||
else {
|
||||
Py_ssize_t shift_digits = shift / PyLong_SHIFT;
|
||||
digit rem;
|
||||
/* x = a >> shift */
|
||||
assert(a_size >= shift_digits);
|
||||
x = _PyLong_New(a_size - shift_digits);
|
||||
if (x == NULL)
|
||||
goto error;
|
||||
rem = v_rshift(x->ob_digit, a->ob_digit + shift_digits,
|
||||
a_size - shift_digits, shift % PyLong_SHIFT);
|
||||
/* set inexact if any of the bits shifted out is nonzero */
|
||||
if (rem)
|
||||
inexact = 1;
|
||||
while (!inexact && shift_digits > 0)
|
||||
if (a->ob_digit[--shift_digits])
|
||||
inexact = 1;
|
||||
}
|
||||
long_normalize(x);
|
||||
x_size = Py_SIZE(x);
|
||||
|
||||
/* x //= b. If the remainder is nonzero, set inexact. We own the only
|
||||
reference to x, so it's safe to modify it in-place. */
|
||||
if (b_size == 1) {
|
||||
digit rem = inplace_divrem1(x->ob_digit, x->ob_digit, x_size,
|
||||
b->ob_digit[0]);
|
||||
long_normalize(x);
|
||||
if (rem)
|
||||
inexact = 1;
|
||||
}
|
||||
else {
|
||||
PyLongObject *div, *rem;
|
||||
div = x_divrem(x, b, &rem);
|
||||
Py_DECREF(x);
|
||||
x = div;
|
||||
if (x == NULL)
|
||||
goto error;
|
||||
if (Py_SIZE(rem))
|
||||
inexact = 1;
|
||||
Py_DECREF(rem);
|
||||
}
|
||||
x_size = ABS(Py_SIZE(x));
|
||||
assert(x_size > 0); /* result of division is never zero */
|
||||
x_bits = (x_size-1)*PyLong_SHIFT+bits_in_digit(x->ob_digit[x_size-1]);
|
||||
|
||||
/* The number of extra bits that have to be rounded away. */
|
||||
extra_bits = MAX(x_bits, DBL_MIN_EXP - shift) - DBL_MANT_DIG;
|
||||
assert(extra_bits == 2 || extra_bits == 3);
|
||||
|
||||
/* Round by directly modifying the low digit of x. */
|
||||
mask = (digit)1 << (extra_bits - 1);
|
||||
low = x->ob_digit[0] | inexact;
|
||||
if (low & mask && low & (3*mask-1))
|
||||
low += mask;
|
||||
x->ob_digit[0] = low & ~(mask-1U);
|
||||
|
||||
/* Convert x to a double dx; the conversion is exact. */
|
||||
dx = x->ob_digit[--x_size];
|
||||
while (x_size > 0)
|
||||
dx = dx * PyLong_BASE + x->ob_digit[--x_size];
|
||||
Py_DECREF(x);
|
||||
|
||||
/* Check whether ldexp result will overflow a double. */
|
||||
if (shift + x_bits >= DBL_MAX_EXP &&
|
||||
(shift + x_bits > DBL_MAX_EXP || dx == ldexp(1.0, x_bits)))
|
||||
goto overflow;
|
||||
result = ldexp(dx, shift);
|
||||
|
||||
success:
|
||||
Py_DECREF(a);
|
||||
Py_DECREF(b);
|
||||
return PyFloat_FromDouble(negate ? -result : result);
|
||||
|
||||
underflow_or_zero:
|
||||
Py_DECREF(a);
|
||||
Py_DECREF(b);
|
||||
return PyFloat_FromDouble(negate ? -0.0 : 0.0);
|
||||
|
||||
overflow:
|
||||
PyErr_SetString(PyExc_OverflowError,
|
||||
"long/long too large for a float");
|
||||
"integer division result too large for a float");
|
||||
error:
|
||||
Py_DECREF(a);
|
||||
Py_DECREF(b);
|
||||
return NULL;
|
||||
|
||||
}
|
||||
|
||||
static PyObject *
|
||||
|
|
|
|||
Loading…
Reference in New Issue