bpo-29962: add math.remainder (#950)
* Implement math.remainder. * Fix markup for arguments; use double spaces after period. * Mark up function reference in what's new entry. * Add comment explaining the calculation in the final branch. * Fix out-of-order entry in whatsnew. * Add comment explaining why it's good enough to compare m with c, in spite of possible rounding error.
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Mark Dickinson
GitHub
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a0ce375e10
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@ -175,6 +175,27 @@ Number-theoretic and representation functions
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of *x* and are floats.
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.. function:: remainder(x, y)
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Return the IEEE 754-style remainder of *x* with respect to *y*. For
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finite *x* and finite nonzero *y*, this is the difference ``x - n*y``,
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where ``n`` is the closest integer to the exact value of the quotient ``x /
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y``. If ``x / y`` is exactly halfway between two consecutive integers, the
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nearest *even* integer is used for ``n``. The remainder ``r = remainder(x,
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y)`` thus always satisfies ``abs(r) <= 0.5 * abs(y)``.
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Special cases follow IEEE 754: in particular, ``remainder(x, math.inf)`` is
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*x* for any finite *x*, and ``remainder(x, 0)`` and
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``remainder(math.inf, x)`` raise :exc:`ValueError` for any non-NaN *x*.
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If the result of the remainder operation is zero, that zero will have
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the same sign as *x*.
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On platforms using IEEE 754 binary floating-point, the result of this
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operation is always exactly representable: no rounding error is introduced.
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.. versionadded:: 3.7
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.. function:: trunc(x)
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Return the :class:`~numbers.Real` value *x* truncated to an
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@ -110,6 +110,12 @@ Added another argument *monetary* in :meth:`format_string` of :mod:`locale`.
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If *monetary* is true, the conversion uses monetary thousands separator and
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grouping strings. (Contributed by Garvit in :issue:`10379`.)
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math
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----
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New :func:`~math.remainder` function, implementing the IEEE 754-style remainder
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operation. (Contributed by Mark Dickinson in :issue:`29962`.)
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os
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--
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@ -1000,6 +1000,135 @@ class MathTests(unittest.TestCase):
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self.ftest('radians(-45)', math.radians(-45), -math.pi/4)
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self.ftest('radians(0)', math.radians(0), 0)
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@requires_IEEE_754
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def testRemainder(self):
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from fractions import Fraction
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def validate_spec(x, y, r):
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"""
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Check that r matches remainder(x, y) according to the IEEE 754
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specification. Assumes that x, y and r are finite and y is nonzero.
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"""
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fx, fy, fr = Fraction(x), Fraction(y), Fraction(r)
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# r should not exceed y/2 in absolute value
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self.assertLessEqual(abs(fr), abs(fy/2))
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# x - r should be an exact integer multiple of y
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n = (fx - fr) / fy
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self.assertEqual(n, int(n))
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if abs(fr) == abs(fy/2):
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# If |r| == |y/2|, n should be even.
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self.assertEqual(n/2, int(n/2))
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# triples (x, y, remainder(x, y)) in hexadecimal form.
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testcases = [
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# Remainders modulo 1, showing the ties-to-even behaviour.
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'-4.0 1 -0.0',
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'-3.8 1 0.8',
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'-3.0 1 -0.0',
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'-2.8 1 -0.8',
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'-2.0 1 -0.0',
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'-1.8 1 0.8',
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'-1.0 1 -0.0',
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'-0.8 1 -0.8',
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'-0.0 1 -0.0',
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' 0.0 1 0.0',
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' 0.8 1 0.8',
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' 1.0 1 0.0',
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' 1.8 1 -0.8',
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' 2.0 1 0.0',
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' 2.8 1 0.8',
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' 3.0 1 0.0',
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' 3.8 1 -0.8',
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' 4.0 1 0.0',
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# Reductions modulo 2*pi
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'0x0.0p+0 0x1.921fb54442d18p+2 0x0.0p+0',
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'0x1.921fb54442d18p+0 0x1.921fb54442d18p+2 0x1.921fb54442d18p+0',
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'0x1.921fb54442d17p+1 0x1.921fb54442d18p+2 0x1.921fb54442d17p+1',
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'0x1.921fb54442d18p+1 0x1.921fb54442d18p+2 0x1.921fb54442d18p+1',
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'0x1.921fb54442d19p+1 0x1.921fb54442d18p+2 -0x1.921fb54442d17p+1',
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'0x1.921fb54442d17p+2 0x1.921fb54442d18p+2 -0x0.0000000000001p+2',
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'0x1.921fb54442d18p+2 0x1.921fb54442d18p+2 0x0p0',
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'0x1.921fb54442d19p+2 0x1.921fb54442d18p+2 0x0.0000000000001p+2',
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'0x1.2d97c7f3321d1p+3 0x1.921fb54442d18p+2 0x1.921fb54442d14p+1',
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'0x1.2d97c7f3321d2p+3 0x1.921fb54442d18p+2 -0x1.921fb54442d18p+1',
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'0x1.2d97c7f3321d3p+3 0x1.921fb54442d18p+2 -0x1.921fb54442d14p+1',
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'0x1.921fb54442d17p+3 0x1.921fb54442d18p+2 -0x0.0000000000001p+3',
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'0x1.921fb54442d18p+3 0x1.921fb54442d18p+2 0x0p0',
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'0x1.921fb54442d19p+3 0x1.921fb54442d18p+2 0x0.0000000000001p+3',
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'0x1.f6a7a2955385dp+3 0x1.921fb54442d18p+2 0x1.921fb54442d14p+1',
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'0x1.f6a7a2955385ep+3 0x1.921fb54442d18p+2 0x1.921fb54442d18p+1',
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'0x1.f6a7a2955385fp+3 0x1.921fb54442d18p+2 -0x1.921fb54442d14p+1',
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'0x1.1475cc9eedf00p+5 0x1.921fb54442d18p+2 0x1.921fb54442d10p+1',
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'0x1.1475cc9eedf01p+5 0x1.921fb54442d18p+2 -0x1.921fb54442d10p+1',
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# Symmetry with respect to signs.
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' 1 0.c 0.4',
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'-1 0.c -0.4',
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' 1 -0.c 0.4',
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'-1 -0.c -0.4',
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' 1.4 0.c -0.4',
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'-1.4 0.c 0.4',
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' 1.4 -0.c -0.4',
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'-1.4 -0.c 0.4',
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# Huge modulus, to check that the underlying algorithm doesn't
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# rely on 2.0 * modulus being representable.
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'0x1.dp+1023 0x1.4p+1023 0x0.9p+1023',
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'0x1.ep+1023 0x1.4p+1023 -0x0.ap+1023',
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'0x1.fp+1023 0x1.4p+1023 -0x0.9p+1023',
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]
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for case in testcases:
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with self.subTest(case=case):
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x_hex, y_hex, expected_hex = case.split()
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x = float.fromhex(x_hex)
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y = float.fromhex(y_hex)
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expected = float.fromhex(expected_hex)
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validate_spec(x, y, expected)
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actual = math.remainder(x, y)
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# Cheap way of checking that the floats are
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# as identical as we need them to be.
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self.assertEqual(actual.hex(), expected.hex())
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# Test tiny subnormal modulus: there's potential for
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# getting the implementation wrong here (for example,
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# by assuming that modulus/2 is exactly representable).
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tiny = float.fromhex('1p-1074') # min +ve subnormal
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for n in range(-25, 25):
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if n == 0:
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continue
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y = n * tiny
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for m in range(100):
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x = m * tiny
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actual = math.remainder(x, y)
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validate_spec(x, y, actual)
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actual = math.remainder(-x, y)
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validate_spec(-x, y, actual)
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# Special values.
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# NaNs should propagate as usual.
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for value in [NAN, 0.0, -0.0, 2.0, -2.3, NINF, INF]:
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self.assertIsNaN(math.remainder(NAN, value))
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self.assertIsNaN(math.remainder(value, NAN))
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# remainder(x, inf) is x, for non-nan non-infinite x.
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for value in [-2.3, -0.0, 0.0, 2.3]:
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self.assertEqual(math.remainder(value, INF), value)
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self.assertEqual(math.remainder(value, NINF), value)
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# remainder(x, 0) and remainder(infinity, x) for non-NaN x are invalid
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# operations according to IEEE 754-2008 7.2(f), and should raise.
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for value in [NINF, -2.3, -0.0, 0.0, 2.3, INF]:
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with self.assertRaises(ValueError):
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math.remainder(INF, value)
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with self.assertRaises(ValueError):
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math.remainder(NINF, value)
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with self.assertRaises(ValueError):
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math.remainder(value, 0.0)
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with self.assertRaises(ValueError):
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math.remainder(value, -0.0)
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def testSin(self):
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self.assertRaises(TypeError, math.sin)
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self.ftest('sin(0)', math.sin(0), 0)
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@ -1286,6 +1415,12 @@ class MathTests(unittest.TestCase):
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self.fail('Failures in test_mtestfile:\n ' +
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'\n '.join(failures))
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# Custom assertions.
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def assertIsNaN(self, value):
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if not math.isnan(value):
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self.fail("Expected a NaN, got {!r}.".format(value))
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class IsCloseTests(unittest.TestCase):
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isclose = math.isclose # sublcasses should override this
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@ -303,6 +303,9 @@ Extension Modules
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Library
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-------
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- bpo-29962: Add math.remainder operation, implementing remainder
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as specified in IEEE 754.
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- bpo-29649: Improve struct.pack_into() exception messages for problems
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with the buffer size and offset. Patch by Andrew Nester.
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@ -600,6 +600,102 @@ m_atan2(double y, double x)
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return atan2(y, x);
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}
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/* IEEE 754-style remainder operation: x - n*y where n*y is the nearest
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multiple of y to x, taking n even in the case of a tie. Assuming an IEEE 754
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binary floating-point format, the result is always exact. */
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static double
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m_remainder(double x, double y)
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{
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/* Deal with most common case first. */
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if (Py_IS_FINITE(x) && Py_IS_FINITE(y)) {
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double absx, absy, c, m, r;
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if (y == 0.0) {
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return Py_NAN;
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}
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absx = fabs(x);
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absy = fabs(y);
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m = fmod(absx, absy);
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/*
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Warning: some subtlety here. What we *want* to know at this point is
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whether the remainder m is less than, equal to, or greater than half
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of absy. However, we can't do that comparison directly because we
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can't be sure that 0.5*absy is representable (the mutiplication
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might incur precision loss due to underflow). So instead we compare
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m with the complement c = absy - m: m < 0.5*absy if and only if m <
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c, and so on. The catch is that absy - m might also not be
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representable, but it turns out that it doesn't matter:
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- if m > 0.5*absy then absy - m is exactly representable, by
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Sterbenz's lemma, so m > c
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- if m == 0.5*absy then again absy - m is exactly representable
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and m == c
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- if m < 0.5*absy then either (i) 0.5*absy is exactly representable,
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in which case 0.5*absy < absy - m, so 0.5*absy <= c and hence m <
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c, or (ii) absy is tiny, either subnormal or in the lowest normal
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binade. Then absy - m is exactly representable and again m < c.
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*/
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c = absy - m;
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if (m < c) {
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r = m;
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}
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else if (m > c) {
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r = -c;
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}
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else {
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/*
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Here absx is exactly halfway between two multiples of absy,
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and we need to choose the even multiple. x now has the form
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absx = n * absy + m
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for some integer n (recalling that m = 0.5*absy at this point).
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If n is even we want to return m; if n is odd, we need to
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return -m.
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So
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0.5 * (absx - m) = (n/2) * absy
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and now reducing modulo absy gives us:
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| m, if n is odd
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fmod(0.5 * (absx - m), absy) = |
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| 0, if n is even
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Now m - 2.0 * fmod(...) gives the desired result: m
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if n is even, -m if m is odd.
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Note that all steps in fmod(0.5 * (absx - m), absy)
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will be computed exactly, with no rounding error
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introduced.
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*/
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assert(m == c);
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r = m - 2.0 * fmod(0.5 * (absx - m), absy);
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}
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return copysign(1.0, x) * r;
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}
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/* Special values. */
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if (Py_IS_NAN(x)) {
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return x;
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}
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if (Py_IS_NAN(y)) {
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return y;
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}
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if (Py_IS_INFINITY(x)) {
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return Py_NAN;
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}
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assert(Py_IS_INFINITY(y));
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return x;
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}
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/*
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Various platforms (Solaris, OpenBSD) do nonstandard things for log(0),
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log(-ve), log(NaN). Here are wrappers for log and log10 that deal with
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@ -1072,6 +1168,12 @@ FUNC1(log1p, m_log1p, 0,
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"log1p($module, x, /)\n--\n\n"
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"Return the natural logarithm of 1+x (base e).\n\n"
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"The result is computed in a way which is accurate for x near zero.")
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FUNC2(remainder, m_remainder,
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"remainder($module, x, y, /)\n--\n\n"
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"Difference between x and the closest integer multiple of y.\n\n"
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"Return x - n*y where n*y is the closest integer multiple of y.\n"
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"In the case where x is exactly halfway between two multiples of\n"
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"y, the nearest even value of n is used. The result is always exact.")
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FUNC1(sin, sin, 0,
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"sin($module, x, /)\n--\n\n"
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"Return the sine of x (measured in radians).")
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@ -2258,6 +2360,7 @@ static PyMethodDef math_methods[] = {
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MATH_MODF_METHODDEF
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MATH_POW_METHODDEF
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MATH_RADIANS_METHODDEF
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{"remainder", math_remainder, METH_VARARGS, math_remainder_doc},
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{"sin", math_sin, METH_O, math_sin_doc},
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{"sinh", math_sinh, METH_O, math_sinh_doc},
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{"sqrt", math_sqrt, METH_O, math_sqrt_doc},
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