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Merged revisions 77062 via svnmerge from
svn+ssh://pythondev@svn.python.org/python/trunk ........ r77062 | mark.dickinson | 2009-12-27 14:55:57 +0000 (Sun, 27 Dec 2009) | 2 lines Issue #1811: Improve accuracy and consistency of true division for integers. ........
This commit is contained in:
Mark Dickinson
parent
99b2c8f811
commit
cbb62745ac
3 changed files with 418 additions and 33 deletions
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@ -14,6 +14,11 @@ class Frm(object):
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def __str__(self):
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return self.format % self.args
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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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# SHIFT should match the value in longintrepr.h for best testing.
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SHIFT = sys.int_info.bits_per_digit
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BASE = 2 ** SHIFT
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@ -35,6 +40,43 @@ del p2
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# add complements & negations
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special += [~x for x in special] + [-x for x in special]
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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 LongTest(unittest.TestCase):
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# Get quasi-random long consisting of ndigits digits (in base BASE).
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@ -306,10 +348,6 @@ class LongTest(unittest.TestCase):
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@unittest.skipUnless(float.__getformat__("double").startswith("IEEE"),
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"test requires IEEE 754 doubles")
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def test_float_conversion(self):
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import sys
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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_MANT_DIG = sys.float_info.mant_dig
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exact_values = [0, 1, 2,
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2**53-3,
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@ -614,6 +652,128 @@ class LongTest(unittest.TestCase):
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for zero in ["huge / 0", "mhuge / 0"]:
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self.assertRaises(ZeroDivisionError, 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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# 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_small_ints(self):
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for i in range(-5, 257):
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