Merged revisions 65258,65292,65299,65308-65309,65315,65326 via svnmerge from
svn+ssh://pythondev@svn.python.org/python/trunk ........ r65258 | mark.dickinson | 2008-07-27 08:15:29 +0100 (Sun, 27 Jul 2008) | 4 lines Remove math.sum tests related to overflow, special values, and behaviour near the extremes of the floating-point range. (The behaviour of math.sum should be regarded as undefined in these cases.) ........ r65292 | mark.dickinson | 2008-07-29 19:45:38 +0100 (Tue, 29 Jul 2008) | 4 lines More modifications to tests for math.sum: replace the Python version of msum by a version using a different algorithm, and use the new float.fromhex method to specify test results exactly. ........ r65299 | mark.dickinson | 2008-07-30 13:01:41 +0100 (Wed, 30 Jul 2008) | 5 lines Fix special-value handling for math.sum. Also minor cleanups to the code: fix tabbing, remove trailing whitespace, and reformat to fit into 80 columns. ........ r65308 | mark.dickinson | 2008-07-30 17:20:10 +0100 (Wed, 30 Jul 2008) | 2 lines Rename math.sum to math.fsum ........ r65309 | mark.dickinson | 2008-07-30 17:25:16 +0100 (Wed, 30 Jul 2008) | 3 lines Replace math.sum with math.fsum in a couple of comments that were missed by r65308 ........ r65315 | mark.dickinson | 2008-07-30 21:23:15 +0100 (Wed, 30 Jul 2008) | 2 lines Add note about problems with math.fsum on x86 hardware. ........ r65326 | mark.dickinson | 2008-07-31 15:48:32 +0100 (Thu, 31 Jul 2008) | 2 lines Rename testSum to testFsum and move it to proper place in test_math.py ........
This commit is contained in:
Mark Dickinson
parent
c57df32319
commit
aa7633ab94
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@ -76,6 +76,42 @@ Number-theoretic and representation functions:
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apart" the internal representation of a float in a portable way.
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.. function:: fsum(iterable)
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Return an accurate floating point sum of values in the iterable. Avoids
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loss of precision by tracking multiple intermediate partial sums. The
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algorithm's accuracy depends on IEEE-754 arithmetic guarantees and the
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typical case where the rounding mode is half-even.
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.. note::
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On platforms where arithmetic results are not correctly rounded,
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:func:`fsum` may occasionally produce incorrect results; these
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results should be no less accurate than those from the builtin
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:func:`sum` function, but nevertheless may have arbitrarily
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large relative error.
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In particular, this affects some older Intel hardware (for
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example Pentium and earlier x86 processors) that makes use of
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'extended precision' floating-point registers with 64 bits of
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precision instead of the 53 bits of precision provided by a C
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double. Arithmetic operations using these registers may be
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doubly rounded (rounded first to 64 bits, and then rerounded to
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53 bits), leading to incorrectly rounded results. To test
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whether your machine is one of those affected, try the following
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at a Python prompt::
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>>> 1e16 + 2.9999
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10000000000000002.0
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Machines subject to the double-rounding problem described above
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are likely to print ``10000000000000004.0`` instead of
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``10000000000000002.0``.
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.. versionadded:: 2.6
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.. function:: isinf(x)
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Checks if the float *x* is positive or negative infinite.
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@ -100,12 +136,6 @@ Number-theoretic and representation functions:
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Return the fractional and integer parts of *x*. Both results carry the sign of
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*x*, and both are floats.
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.. function:: sum(iterable)
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Return an accurate floating point sum of values in the iterable. Avoids
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loss of precision by tracking multiple intermediate partial sums. The
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algorithm's accuracy depends on IEEE-754 arithmetic guarantees and the
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typical case where the rounding mode is half-even.
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.. function:: trunc(x)
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@ -1537,7 +1537,7 @@ Here are all of the changes that Python 2.6 makes to the core Python language.
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* :func:`~math.factorial` computes the factorial of a number.
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(Contributed by Raymond Hettinger; :issue:`2138`.)
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* :func:`~math.sum` adds up the stream of numbers from an iterable,
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* :func:`~math.fsum` adds up the stream of numbers from an iterable,
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and is careful to avoid loss of precision by calculating partial sums.
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(Contributed by Jean Brouwers, Raymond Hettinger, and Mark Dickinson;
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:issue:`2819`.)
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@ -359,6 +359,102 @@ class MathTests(unittest.TestCase):
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self.assertEquals(math.frexp(NINF)[0], NINF)
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self.assert_(math.isnan(math.frexp(NAN)[0]))
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def testFsum(self):
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# math.fsum relies on exact rounding for correct operation.
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# There's a known problem with IA32 floating-point that causes
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# inexact rounding in some situations, and will cause the
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# math.fsum tests below to fail; see issue #2937. On non IEEE
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# 754 platforms, and on IEEE 754 platforms that exhibit the
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# problem described in issue #2937, we simply skip the whole
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# test.
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if not float.__getformat__("double").startswith("IEEE"):
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return
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# on IEEE 754 compliant machines, both of the expressions
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# below should round to 10000000000000002.0.
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if 1e16+2.0 != 1e16+2.9999:
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return
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# Python version of math.fsum, for comparison. Uses a
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# different algorithm based on frexp, ldexp and integer
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# arithmetic.
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from sys import float_info
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mant_dig = float_info.mant_dig
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etiny = float_info.min_exp - mant_dig
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def msum(iterable):
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"""Full precision summation. Compute sum(iterable) without any
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intermediate accumulation of error. Based on the 'lsum' function
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at http://code.activestate.com/recipes/393090/
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"""
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tmant, texp = 0, 0
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for x in iterable:
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mant, exp = math.frexp(x)
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mant, exp = int(math.ldexp(mant, mant_dig)), exp - mant_dig
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if texp > exp:
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tmant <<= texp-exp
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texp = exp
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else:
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mant <<= exp-texp
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tmant += mant
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# Round tmant * 2**texp to a float. The original recipe
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# used float(str(tmant)) * 2.0**texp for this, but that's
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# a little unsafe because str -> float conversion can't be
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# relied upon to do correct rounding on all platforms.
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tail = max(len(bin(abs(tmant)))-2 - mant_dig, etiny - texp)
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if tail > 0:
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h = 1 << (tail-1)
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tmant = tmant // (2*h) + bool(tmant & h and tmant & 3*h-1)
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texp += tail
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return math.ldexp(tmant, texp)
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test_values = [
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([], 0.0),
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([0.0], 0.0),
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([1e100, 1.0, -1e100, 1e-100, 1e50, -1.0, -1e50], 1e-100),
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([2.0**53, -0.5, -2.0**-54], 2.0**53-1.0),
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([2.0**53, 1.0, 2.0**-100], 2.0**53+2.0),
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([2.0**53+10.0, 1.0, 2.0**-100], 2.0**53+12.0),
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([2.0**53-4.0, 0.5, 2.0**-54], 2.0**53-3.0),
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([1./n for n in range(1, 1001)],
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float.fromhex('0x1.df11f45f4e61ap+2')),
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([(-1.)**n/n for n in range(1, 1001)],
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float.fromhex('-0x1.62a2af1bd3624p-1')),
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([1.7**(i+1)-1.7**i for i in range(1000)] + [-1.7**1000], -1.0),
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([1e16, 1., 1e-16], 10000000000000002.0),
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([1e16-2., 1.-2.**-53, -(1e16-2.), -(1.-2.**-53)], 0.0),
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# exercise code for resizing partials array
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([2.**n - 2.**(n+50) + 2.**(n+52) for n in range(-1074, 972, 2)] +
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[-2.**1022],
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float.fromhex('0x1.5555555555555p+970')),
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]
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for i, (vals, expected) in enumerate(test_values):
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try:
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actual = math.fsum(vals)
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except OverflowError:
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self.fail("test %d failed: got OverflowError, expected %r "
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"for math.fsum(%.100r)" % (i, expected, vals))
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except ValueError:
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self.fail("test %d failed: got ValueError, expected %r "
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"for math.fsum(%.100r)" % (i, expected, vals))
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self.assertEqual(actual, expected)
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from random import random, gauss, shuffle
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for j in range(1000):
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vals = [7, 1e100, -7, -1e100, -9e-20, 8e-20] * 10
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s = 0
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for i in range(200):
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v = gauss(0, random()) ** 7 - s
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s += v
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vals.append(v)
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shuffle(vals)
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s = msum(vals)
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self.assertEqual(msum(vals), math.fsum(vals))
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def testHypot(self):
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self.assertRaises(TypeError, math.hypot)
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self.ftest('hypot(0,0)', math.hypot(0,0), 0)
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@ -641,158 +737,6 @@ class MathTests(unittest.TestCase):
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self.assertRaises(ValueError, math.sqrt, NINF)
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self.assert_(math.isnan(math.sqrt(NAN)))
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def testSum(self):
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# math.sum relies on exact rounding for correct operation.
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# There's a known problem with IA32 floating-point that causes
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# inexact rounding in some situations, and will cause the
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# math.sum tests below to fail; see issue #2937. On non IEEE
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# 754 platforms, and on IEEE 754 platforms that exhibit the
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# problem described in issue #2937, we simply skip the whole
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# test.
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if not float.__getformat__("double").startswith("IEEE"):
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return
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# on IEEE 754 compliant machines, both of the expressions
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# below should round to 10000000000000002.0.
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if 1e16+2.999 != 1e16+2.9999:
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return
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# Python version of math.sum algorithm, for comparison
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def msum(iterable):
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"""Full precision sum of values in iterable. Returns the value of
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the sum, rounded to the nearest representable floating-point number
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using the round-half-to-even rule.
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"""
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# Stage 1: accumulate partials
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partials = []
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for x in iterable:
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i = 0
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for y in partials:
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if abs(x) < abs(y):
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x, y = y, x
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hi = x + y
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lo = y - (hi - x)
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if lo:
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partials[i] = lo
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i += 1
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x = hi
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partials[i:] = [x] if x else []
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# Stage 2: sum partials
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if not partials:
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return 0.0
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# sum from the top, stopping as soon as the sum is inexact.
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total = partials.pop()
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while partials:
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x = partials.pop()
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old_total, total = total, total + x
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error = x - (total - old_total)
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if error != 0.0:
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# adjust for correct rounding if necessary
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if partials and (partials[-1] > 0.0) == (error > 0.0) and \
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total + 2*error - total == 2*error:
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total += 2*error
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break
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return total
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from sys import float_info
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maxfloat = float_info.max
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twopow = 2.**(float_info.max_exp - 1)
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test_values = [
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([], 0.0),
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([0.0], 0.0),
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([1e100, 1.0, -1e100, 1e-100, 1e50, -1.0, -1e50], 1e-100),
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([1e308, 1e308, -1e308], OverflowError),
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([-1e308, 1e308, 1e308], 1e308),
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([1e308, -1e308, 1e308], 1e308),
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([2.0**1023, 2.0**1023, -2.0**1000], OverflowError),
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([twopow, twopow, twopow, twopow, -twopow, -twopow, -twopow],
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OverflowError),
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([2.0**53, -0.5, -2.0**-54], 2.0**53-1.0),
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([2.0**53, 1.0, 2.0**-100], 2.0**53+2.0),
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([2.0**53+10.0, 1.0, 2.0**-100], 2.0**53+12.0),
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([2.0**53-4.0, 0.5, 2.0**-54], 2.0**53-3.0),
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([2.0**1023-2.0**970, -1.0, 2.0**1023], OverflowError),
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([maxfloat, maxfloat*2.**-54], maxfloat),
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([maxfloat, maxfloat*2.**-53], OverflowError),
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([1./n for n in range(1, 1001)], 7.4854708605503451),
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([(-1.)**n/n for n in range(1, 1001)], -0.69264743055982025),
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([1.7**(i+1)-1.7**i for i in range(1000)] + [-1.7**1000], -1.0),
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([INF, -INF, NAN], ValueError),
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([NAN, INF, -INF], ValueError),
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([INF, NAN, INF], ValueError),
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([INF, INF], OverflowError),
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([INF, -INF], ValueError),
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([-INF, 1e308, 1e308, -INF], OverflowError),
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([2.0**1023-2.0**970, 0.0, 2.0**1023], OverflowError),
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([2.0**1023-2.0**970, 1.0, 2.0**1023], OverflowError),
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([2.0**1023, 2.0**1023], OverflowError),
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([2.0**1023, 2.0**1023, -1.0], OverflowError),
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([twopow, twopow, twopow, twopow, -twopow, -twopow],
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OverflowError),
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([twopow, twopow, twopow, twopow, -twopow, twopow], OverflowError),
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([-twopow, -twopow, -twopow, -twopow], OverflowError),
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([2.**1023, 2.**1023, -2.**971], OverflowError),
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([2.**1023, 2.**1023, -2.**970], OverflowError),
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([-2.**970, 2.**1023, 2.**1023, -2.**-1074], OverflowError),
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([ 2.**1023, 2.**1023, -2.**970, 2.**-1074], OverflowError),
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([-2.**1023, 2.**971, -2.**1023], -maxfloat),
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([-2.**1023, -2.**1023, 2.**970], OverflowError),
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([-2.**1023, -2.**1023, 2.**970, 2.**-1074], OverflowError),
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([-2.**-1074, -2.**1023, -2.**1023, 2.**970], OverflowError),
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([2.**930, -2.**980, 2.**1023, 2.**1023, twopow, -twopow],
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OverflowError),
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([2.**1023, 2.**1023, -1e307], OverflowError),
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([1e16, 1., 1e-16], 10000000000000002.0),
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([1e16-2., 1.-2.**-53, -(1e16-2.), -(1.-2.**-53)], 0.0),
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]
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for i, (vals, s) in enumerate(test_values):
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if isinstance(s, type) and issubclass(s, Exception):
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try:
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m = math.sum(vals)
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except s:
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pass
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else:
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self.fail("test %d failed: got %r, expected %r "
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"for math.sum(%.100r)" %
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(i, m, s.__name__, vals))
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else:
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try:
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self.assertEqual(math.sum(vals), s)
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except OverflowError:
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self.fail("test %d failed: got OverflowError, expected %r "
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"for math.sum(%.100r)" % (i, s, vals))
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except ValueError:
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self.fail("test %d failed: got ValueError, expected %r "
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"for math.sum(%.100r)" % (i, s, vals))
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# compare with output of msum above, but only when
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# result isn't an IEEE special or an exception
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if not math.isinf(s) and not math.isnan(s):
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self.assertEqual(msum(vals), s)
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from random import random, gauss, shuffle
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for j in range(1000):
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vals = [7, 1e100, -7, -1e100, -9e-20, 8e-20] * 10
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s = 0
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for i in range(200):
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v = gauss(0, random()) ** 7 - s
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s += v
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vals.append(v)
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shuffle(vals)
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s = msum(vals)
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self.assertEqual(msum(vals), math.sum(vals))
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def testTan(self):
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self.assertRaises(TypeError, math.tan)
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self.ftest('tan(0)', math.tan(0), 0)
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@ -5,7 +5,7 @@ import random
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import time
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import pickle
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import warnings
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from math import log, exp, sqrt, pi, sum as msum
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from math import log, exp, sqrt, pi, fsum as msum
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from test import support
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class TestBasicOps(unittest.TestCase):
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@ -396,7 +396,7 @@ FUNC1(tanh, tanh, 0,
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Note 4: A similar implementation is in Modules/cmathmodule.c.
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Be sure to update both when making changes.
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Note 5: The signature of math.sum() differs from __builtin__.sum()
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Note 5: The signature of math.fsum() differs from __builtin__.sum()
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because the start argument doesn't make sense in the context of
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accurate summation. Since the partials table is collapsed before
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returning a result, sum(seq2, start=sum(seq1)) may not equal the
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@ -407,7 +407,7 @@ FUNC1(tanh, tanh, 0,
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/* Extend the partials array p[] by doubling its size. */
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static int /* non-zero on error */
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_sum_realloc(double **p_ptr, Py_ssize_t n,
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_fsum_realloc(double **p_ptr, Py_ssize_t n,
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double *ps, Py_ssize_t *m_ptr)
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{
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void *v = NULL;
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@ -425,7 +425,7 @@ _sum_realloc(double **p_ptr, Py_ssize_t n,
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v = PyMem_Realloc(p, sizeof(double) * m);
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}
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if (v == NULL) { /* size overflow or no memory */
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PyErr_SetString(PyExc_MemoryError, "math sum partials");
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PyErr_SetString(PyExc_MemoryError, "math.fsum partials");
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return 1;
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}
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*p_ptr = (double*) v;
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@ -464,18 +464,19 @@ _sum_realloc(double **p_ptr, Py_ssize_t n,
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*/
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static PyObject*
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math_sum(PyObject *self, PyObject *seq)
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math_fsum(PyObject *self, PyObject *seq)
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{
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PyObject *item, *iter, *sum = NULL;
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Py_ssize_t i, j, n = 0, m = NUM_PARTIALS;
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double x, y, t, ps[NUM_PARTIALS], *p = ps;
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double xsave, special_sum = 0.0, inf_sum = 0.0;
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volatile double hi, yr, lo;
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iter = PyObject_GetIter(seq);
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if (iter == NULL)
|
||||
return NULL;
|
||||
|
||||
PyFPE_START_PROTECT("sum", Py_DECREF(iter); return NULL)
|
||||
PyFPE_START_PROTECT("fsum", Py_DECREF(iter); return NULL)
|
||||
|
||||
for(;;) { /* for x in iterable */
|
||||
assert(0 <= n && n <= m);
|
||||
|
|
@ -485,18 +486,19 @@ math_sum(PyObject *self, PyObject *seq)
|
|||
item = PyIter_Next(iter);
|
||||
if (item == NULL) {
|
||||
if (PyErr_Occurred())
|
||||
goto _sum_error;
|
||||
goto _fsum_error;
|
||||
break;
|
||||
}
|
||||
x = PyFloat_AsDouble(item);
|
||||
Py_DECREF(item);
|
||||
if (PyErr_Occurred())
|
||||
goto _sum_error;
|
||||
goto _fsum_error;
|
||||
|
||||
xsave = x;
|
||||
for (i = j = 0; j < n; j++) { /* for y in partials */
|
||||
y = p[j];
|
||||
if (fabs(x) < fabs(y)) {
|
||||
t = x; x = y; y = t;
|
||||
t = x; x = y; y = t;
|
||||
}
|
||||
hi = x + y;
|
||||
yr = hi - x;
|
||||
|
|
@ -505,59 +507,73 @@ math_sum(PyObject *self, PyObject *seq)
|
|||
p[i++] = lo;
|
||||
x = hi;
|
||||
}
|
||||
|
||||
n = i; /* ps[i:] = [x] */
|
||||
|
||||
n = i; /* ps[i:] = [x] */
|
||||
if (x != 0.0) {
|
||||
/* If non-finite, reset partials, effectively
|
||||
adding subsequent items without roundoff
|
||||
and yielding correct non-finite results,
|
||||
provided IEEE 754 rules are observed */
|
||||
if (! Py_IS_FINITE(x))
|
||||
if (! Py_IS_FINITE(x)) {
|
||||
/* a nonfinite x could arise either as
|
||||
a result of intermediate overflow, or
|
||||
as a result of a nan or inf in the
|
||||
summands */
|
||||
if (Py_IS_FINITE(xsave)) {
|
||||
PyErr_SetString(PyExc_OverflowError,
|
||||
"intermediate overflow in fsum");
|
||||
goto _fsum_error;
|
||||
}
|
||||
if (Py_IS_INFINITY(xsave))
|
||||
inf_sum += xsave;
|
||||
special_sum += xsave;
|
||||
/* reset partials */
|
||||
n = 0;
|
||||
else if (n >= m && _sum_realloc(&p, n, ps, &m))
|
||||
goto _sum_error;
|
||||
p[n++] = x;
|
||||
}
|
||||
else if (n >= m && _fsum_realloc(&p, n, ps, &m))
|
||||
goto _fsum_error;
|
||||
else
|
||||
p[n++] = x;
|
||||
}
|
||||
}
|
||||
|
||||
if (special_sum != 0.0) {
|
||||
if (Py_IS_NAN(inf_sum))
|
||||
PyErr_SetString(PyExc_ValueError,
|
||||
"-inf + inf in fsum");
|
||||
else
|
||||
sum = PyFloat_FromDouble(special_sum);
|
||||
goto _fsum_error;
|
||||
}
|
||||
|
||||
hi = 0.0;
|
||||
if (n > 0) {
|
||||
hi = p[--n];
|
||||
if (Py_IS_FINITE(hi)) {
|
||||
/* sum_exact(ps, hi) from the top, stop when the sum becomes inexact. */
|
||||
while (n > 0) {
|
||||
x = hi;
|
||||
y = p[--n];
|
||||
assert(fabs(y) < fabs(x));
|
||||
hi = x + y;
|
||||
yr = hi - x;
|
||||
lo = y - yr;
|
||||
if (lo != 0.0)
|
||||
break;
|
||||
}
|
||||
/* Make half-even rounding work across multiple partials. Needed
|
||||
so that sum([1e-16, 1, 1e16]) will round-up the last digit to
|
||||
two instead of down to zero (the 1e-16 makes the 1 slightly
|
||||
closer to two). With a potential 1 ULP rounding error fixed-up,
|
||||
math.sum() can guarantee commutativity. */
|
||||
if (n > 0 && ((lo < 0.0 && p[n-1] < 0.0) ||
|
||||
(lo > 0.0 && p[n-1] > 0.0))) {
|
||||
y = lo * 2.0;
|
||||
x = hi + y;
|
||||
yr = x - hi;
|
||||
if (y == yr)
|
||||
hi = x;
|
||||
}
|
||||
/* sum_exact(ps, hi) from the top, stop when the sum becomes
|
||||
inexact. */
|
||||
while (n > 0) {
|
||||
x = hi;
|
||||
y = p[--n];
|
||||
assert(fabs(y) < fabs(x));
|
||||
hi = x + y;
|
||||
yr = hi - x;
|
||||
lo = y - yr;
|
||||
if (lo != 0.0)
|
||||
break;
|
||||
}
|
||||
else { /* raise exception corresponding to a special value */
|
||||
errno = Py_IS_NAN(hi) ? EDOM : ERANGE;
|
||||
if (is_error(hi))
|
||||
goto _sum_error;
|
||||
/* Make half-even rounding work across multiple partials.
|
||||
Needed so that sum([1e-16, 1, 1e16]) will round-up the last
|
||||
digit to two instead of down to zero (the 1e-16 makes the 1
|
||||
slightly closer to two). With a potential 1 ULP rounding
|
||||
error fixed-up, math.fsum() can guarantee commutativity. */
|
||||
if (n > 0 && ((lo < 0.0 && p[n-1] < 0.0) ||
|
||||
(lo > 0.0 && p[n-1] > 0.0))) {
|
||||
y = lo * 2.0;
|
||||
x = hi + y;
|
||||
yr = x - hi;
|
||||
if (y == yr)
|
||||
hi = x;
|
||||
}
|
||||
}
|
||||
sum = PyFloat_FromDouble(hi);
|
||||
|
||||
_sum_error:
|
||||
_fsum_error:
|
||||
PyFPE_END_PROTECT(hi)
|
||||
Py_DECREF(iter);
|
||||
if (p != ps)
|
||||
|
|
@ -567,7 +583,7 @@ _sum_error:
|
|||
|
||||
#undef NUM_PARTIALS
|
||||
|
||||
PyDoc_STRVAR(math_sum_doc,
|
||||
PyDoc_STRVAR(math_fsum_doc,
|
||||
"sum(iterable)\n\n\
|
||||
Return an accurate floating point sum of values in the iterable.\n\
|
||||
Assumes IEEE-754 floating point arithmetic.");
|
||||
|
|
@ -1078,6 +1094,7 @@ static PyMethodDef math_methods[] = {
|
|||
{"floor", math_floor, METH_O, math_floor_doc},
|
||||
{"fmod", math_fmod, METH_VARARGS, math_fmod_doc},
|
||||
{"frexp", math_frexp, METH_O, math_frexp_doc},
|
||||
{"fsum", math_fsum, METH_O, math_fsum_doc},
|
||||
{"hypot", math_hypot, METH_VARARGS, math_hypot_doc},
|
||||
{"isinf", math_isinf, METH_O, math_isinf_doc},
|
||||
{"isnan", math_isnan, METH_O, math_isnan_doc},
|
||||
|
|
@ -1091,10 +1108,9 @@ static PyMethodDef math_methods[] = {
|
|||
{"sin", math_sin, METH_O, math_sin_doc},
|
||||
{"sinh", math_sinh, METH_O, math_sinh_doc},
|
||||
{"sqrt", math_sqrt, METH_O, math_sqrt_doc},
|
||||
{"sum", math_sum, METH_O, math_sum_doc},
|
||||
{"tan", math_tan, METH_O, math_tan_doc},
|
||||
{"tanh", math_tanh, METH_O, math_tanh_doc},
|
||||
{"trunc", math_trunc, METH_O, math_trunc_doc},
|
||||
{"trunc", math_trunc, METH_O, math_trunc_doc},
|
||||
{NULL, NULL} /* sentinel */
|
||||
};
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue