bpo-29282: Add math.fma(): fused multiply-add function

Co-Authored-By: Mark Dickinson <[email protected]>

Author: vstinner

Doc/library/math.rst

@@ -82,6 +82,22 @@ Number-theoretic and representation functions
    :class:`~numbers.Integral` value.
 
 
+.. function:: fma(x, y, z)
+
+   Fused multiply-add operation. Return ``(x * y) + z``, computed as though with
+   infinite precision and range followed by a single round to the ``float``
+   format. This operation often provides better accuracy than the direct
+   expression ``(x * y) + z``.
+
+   This function follows the specification of the fusedMultiplyAdd operation
+   described in the IEEE 754-2008 standard. The standard leaves one case
+   implementation-defined, namely the result of ``fma(0, inf, nan)``
+   and ``fma(inf, 0, nan)``. In these cases, ``math.fma`` returns a NaN,
+   and does not raise any exception.
+
+   .. versionadded:: 3.9
+
+
 .. function:: fmod(x, y)
 
    Return ``fmod(x, y)``, as defined by the platform C library. Note that the

Doc/whatsnew/3.9.rst

@@ -188,6 +188,13 @@ Add :func:`math.ulp`: return the value of the least significant bit
 of a float.
 (Contributed by Victor Stinner in :issue:`39310`.)
 
+A new function :func:`~math.fma` for fused multiply-add operations has been
+added. This function computes ``x * y + z`` with only a single round, and so
+avoids any intermediate loss of precision. It wraps the ``fma`` function
+provided by C99, and follows the specification of the IEEE 754-2008
+"fusedMultiplyAdd" operation for special cases.
+(Contributed by Mark Dickinson in :issue:`29282`.)
+
 nntplib
 -------
 

Lib/test/test_math.py

@@ -2087,11 +2087,244 @@ def test_ulp(self):
                 self.assertEqual(math.ulp(-x), math.ulp(x))
 
 
+class FMATests(unittest.TestCase):
+    """ Tests for math.fma. """
+
+    def test_fma_nan_results(self):
+        # Selected representative values.
+        values = [
+            -math.inf, -1e300, -2.3, -1e-300, -0.0,
+            0.0, 1e-300, 2.3, 1e300, math.inf, math.nan
+        ]
+
+        # If any input is a NaN, the result should be a NaN, too.
+        for a, b in itertools.product(values, repeat=2):
+            self.assertIsNaN(math.fma(math.nan, a, b))
+            self.assertIsNaN(math.fma(a, math.nan, b))
+            self.assertIsNaN(math.fma(a, b, math.nan))
+
+    def test_fma_infinities(self):
+        # Cases involving infinite inputs or results.
+        positives = [1e-300, 2.3, 1e300, math.inf]
+        finites = [-1e300, -2.3, -1e-300, -0.0, 0.0, 1e-300, 2.3, 1e300]
+        non_nans = [-math.inf, -2.3, -0.0, 0.0, 2.3, math.inf]
+
+        # ValueError due to inf * 0 computation.
+        for c in non_nans:
+            for infinity in [math.inf, -math.inf]:
+                for zero in [0.0, -0.0]:
+                    with self.assertRaises(ValueError):
+                        math.fma(infinity, zero, c)
+                    with self.assertRaises(ValueError):
+                        math.fma(zero, infinity, c)
+
+        # ValueError when a*b and c both infinite of opposite signs.
+        for b in positives:
+            with self.assertRaises(ValueError):
+                math.fma(math.inf, b, -math.inf)
+            with self.assertRaises(ValueError):
+                math.fma(math.inf, -b, math.inf)
+            with self.assertRaises(ValueError):
+                math.fma(-math.inf, -b, -math.inf)
+            with self.assertRaises(ValueError):
+                math.fma(-math.inf, b, math.inf)
+            with self.assertRaises(ValueError):
+                math.fma(b, math.inf, -math.inf)
+            with self.assertRaises(ValueError):
+                math.fma(-b, math.inf, math.inf)
+            with self.assertRaises(ValueError):
+                math.fma(-b, -math.inf, -math.inf)
+            with self.assertRaises(ValueError):
+                math.fma(b, -math.inf, math.inf)
+
+        # Infinite result when a*b and c both infinite of the same sign.
+        for b in positives:
+            self.assertEqual(math.fma(math.inf, b, math.inf), math.inf)
+            self.assertEqual(math.fma(math.inf, -b, -math.inf), -math.inf)
+            self.assertEqual(math.fma(-math.inf, -b, math.inf), math.inf)
+            self.assertEqual(math.fma(-math.inf, b, -math.inf), -math.inf)
+            self.assertEqual(math.fma(b, math.inf, math.inf), math.inf)
+            self.assertEqual(math.fma(-b, math.inf, -math.inf), -math.inf)
+            self.assertEqual(math.fma(-b, -math.inf, math.inf), math.inf)
+            self.assertEqual(math.fma(b, -math.inf, -math.inf), -math.inf)
+
+        # Infinite result when a*b finite, c infinite.
+        for a, b in itertools.product(finites, finites):
+            self.assertEqual(math.fma(a, b, math.inf), math.inf)
+            self.assertEqual(math.fma(a, b, -math.inf), -math.inf)
+
+        # Infinite result when a*b infinite, c finite.
+        for b, c in itertools.product(positives, finites):
+            self.assertEqual(math.fma(math.inf, b, c), math.inf)
+            self.assertEqual(math.fma(-math.inf, b, c), -math.inf)
+            self.assertEqual(math.fma(-math.inf, -b, c), math.inf)
+            self.assertEqual(math.fma(math.inf, -b, c), -math.inf)
+
+            self.assertEqual(math.fma(b, math.inf, c), math.inf)
+            self.assertEqual(math.fma(b, -math.inf, c), -math.inf)
+            self.assertEqual(math.fma(-b, -math.inf, c), math.inf)
+            self.assertEqual(math.fma(-b, math.inf, c), -math.inf)
+
+    def test_fma_zero_result(self):
+        nonnegative_finites = [0.0, 1e-300, 2.3, 1e300]
+
+        # Zero results from exact zero inputs.
+        for b in nonnegative_finites:
+            self.assertIsPositiveZero(math.fma(0.0, b, 0.0))
+            self.assertIsPositiveZero(math.fma(0.0, b, -0.0))
+            self.assertIsNegativeZero(math.fma(0.0, -b, -0.0))
+            self.assertIsPositiveZero(math.fma(0.0, -b, 0.0))
+            self.assertIsPositiveZero(math.fma(-0.0, -b, 0.0))
+            self.assertIsPositiveZero(math.fma(-0.0, -b, -0.0))
+            self.assertIsNegativeZero(math.fma(-0.0, b, -0.0))
+            self.assertIsPositiveZero(math.fma(-0.0, b, 0.0))
+
+            self.assertIsPositiveZero(math.fma(b, 0.0, 0.0))
+            self.assertIsPositiveZero(math.fma(b, 0.0, -0.0))
+            self.assertIsNegativeZero(math.fma(-b, 0.0, -0.0))
+            self.assertIsPositiveZero(math.fma(-b, 0.0, 0.0))
+            self.assertIsPositiveZero(math.fma(-b, -0.0, 0.0))
+            self.assertIsPositiveZero(math.fma(-b, -0.0, -0.0))
+            self.assertIsNegativeZero(math.fma(b, -0.0, -0.0))
+            self.assertIsPositiveZero(math.fma(b, -0.0, 0.0))
+
+        # Exact zero result from nonzero inputs.
+        self.assertIsPositiveZero(math.fma(2.0, 2.0, -4.0))
+        self.assertIsPositiveZero(math.fma(2.0, -2.0, 4.0))
+        self.assertIsPositiveZero(math.fma(-2.0, -2.0, -4.0))
+        self.assertIsPositiveZero(math.fma(-2.0, 2.0, 4.0))
+
+        # Underflow to zero.
+        tiny = 1e-300
+        self.assertIsPositiveZero(math.fma(tiny, tiny, 0.0))
+        self.assertIsNegativeZero(math.fma(tiny, -tiny, 0.0))
+        self.assertIsPositiveZero(math.fma(-tiny, -tiny, 0.0))
+        self.assertIsNegativeZero(math.fma(-tiny, tiny, 0.0))
+        self.assertIsPositiveZero(math.fma(tiny, tiny, -0.0))
+        self.assertIsNegativeZero(math.fma(tiny, -tiny, -0.0))
+        self.assertIsPositiveZero(math.fma(-tiny, -tiny, -0.0))
+        self.assertIsNegativeZero(math.fma(-tiny, tiny, -0.0))
+
+        # Corner case where rounding the multiplication would
+        # give the wrong result.
+        x = float.fromhex('0x1p-500')
+        y = float.fromhex('0x1p-550')
+        z = float.fromhex('0x1p-1000')
+        self.assertIsNegativeZero(math.fma(x-y, x+y, -z))
+        self.assertIsPositiveZero(math.fma(y-x, x+y, z))
+        self.assertIsNegativeZero(math.fma(y-x, -(x+y), -z))
+        self.assertIsPositiveZero(math.fma(x-y, -(x+y), z))
+
+    def test_fma_overflow(self):
+        a = b = float.fromhex('0x1p512')
+        c = float.fromhex('0x1p1023')
+        # Overflow from multiplication.
+        with self.assertRaises(OverflowError):
+            math.fma(a, b, 0.0)
+        self.assertEqual(math.fma(a, b/2.0, 0.0), c)
+        # Overflow from the addition.
+        with self.assertRaises(OverflowError):
+            math.fma(a, b/2.0, c)
+        # No overflow, even though a*b overflows a float.
+        self.assertEqual(math.fma(a, b, -c), c)
+
+        # Extreme case: a * b is exactly at the overflow boundary, so the
+        # tiniest offset makes a difference between overflow and a finite
+        # result.
+        a = float.fromhex('0x1.ffffffc000000p+511')
+        b = float.fromhex('0x1.0000002000000p+512')
+        c = float.fromhex('0x0.0000000000001p-1022')
+        with self.assertRaises(OverflowError):
+            math.fma(a, b, 0.0)
+        with self.assertRaises(OverflowError):
+            math.fma(a, b, c)
+        self.assertEqual(math.fma(a, b, -c),
+                         float.fromhex('0x1.fffffffffffffp+1023'))
+
+        # Another extreme case: here a*b is about as large as possible subject
+        # to math.fma(a, b, c) being finite.
+        a = float.fromhex('0x1.ae565943785f9p+512')
+        b = float.fromhex('0x1.3094665de9db8p+512')
+        c = float.fromhex('0x1.fffffffffffffp+1023')
+        self.assertEqual(math.fma(a, b, -c), c)
+
+    def test_fma_single_round(self):
+        a = float.fromhex('0x1p-50')
+        self.assertEqual(math.fma(a - 1.0, a + 1.0, 1.0), a*a)
+
+    def test_random(self):
+        # A collection of randomly generated inputs for which the naive FMA
+        # (with two rounds) gives a different result from a singly-rounded FMA.
+
+        # tuples (a, b, c, expected)
+        test_values = [
+            ('0x1.694adde428b44p-1', '0x1.371b0d64caed7p-1',
+             '0x1.f347e7b8deab8p-4', '0x1.19f10da56c8adp-1'),
+            ('0x1.605401ccc6ad6p-2', '0x1.ce3a40bf56640p-2',
+             '0x1.96e3bf7bf2e20p-2', '0x1.1af6d8aa83101p-1'),
+            ('0x1.e5abd653a67d4p-2', '0x1.a2e400209b3e6p-1',
+             '0x1.a90051422ce13p-1', '0x1.37d68cc8c0fbbp+0'),
+            ('0x1.f94e8efd54700p-2', '0x1.123065c812cebp-1',
+             '0x1.458f86fb6ccd0p-1', '0x1.ccdcee26a3ff3p-1'),
+            ('0x1.bd926f1eedc96p-1', '0x1.eee9ca68c5740p-1',
+             '0x1.960c703eb3298p-2', '0x1.3cdcfb4fdb007p+0'),
+            ('0x1.27348350fbccdp-1', '0x1.3b073914a53f1p-1',
+             '0x1.e300da5c2b4cbp-1', '0x1.4c51e9a3c4e29p+0'),
+            ('0x1.2774f00b3497bp-1', '0x1.7038ec336bff0p-2',
+             '0x1.2f6f2ccc3576bp-1', '0x1.99ad9f9c2688bp-1'),
+            ('0x1.51d5a99300e5cp-1', '0x1.5cd74abd445a1p-1',
+             '0x1.8880ab0bbe530p-1', '0x1.3756f96b91129p+0'),
+            ('0x1.73cb965b821b8p-2', '0x1.218fd3d8d5371p-1',
+             '0x1.d1ea966a1f758p-2', '0x1.5217b8fd90119p-1'),
+            ('0x1.4aa98e890b046p-1', '0x1.954d85dff1041p-1',
+             '0x1.122b59317ebdfp-1', '0x1.0bf644b340cc5p+0'),
+            ('0x1.e28f29e44750fp-1', '0x1.4bcc4fdcd18fep-1',
+             '0x1.fd47f81298259p-1', '0x1.9b000afbc9995p+0'),
+            ('0x1.d2e850717fe78p-3', '0x1.1dd7531c303afp-1',
+             '0x1.e0869746a2fc2p-2', '0x1.316df6eb26439p-1'),
+            ('0x1.cf89c75ee6fbap-2', '0x1.b23decdc66825p-1',
+             '0x1.3d1fe76ac6168p-1', '0x1.00d8ea4c12abbp+0'),
+            ('0x1.3265ae6f05572p-2', '0x1.16d7ec285f7a2p-1',
+             '0x1.0b8405b3827fbp-1', '0x1.5ef33c118a001p-1'),
+            ('0x1.c4d1bf55ec1a5p-1', '0x1.bc59618459e12p-2',
+             '0x1.ce5b73dc1773dp-1', '0x1.496cf6164f99bp+0'),
+            ('0x1.d350026ac3946p-1', '0x1.9a234e149a68cp-2',
+             '0x1.f5467b1911fd6p-2', '0x1.b5cee3225caa5p-1'),
+        ]
+        for a_hex, b_hex, c_hex, expected_hex in test_values:
+            a = float.fromhex(a_hex)
+            b = float.fromhex(b_hex)
+            c = float.fromhex(c_hex)
+            expected = float.fromhex(expected_hex)
+            self.assertEqual(math.fma(a, b, c), expected)
+            self.assertEqual(math.fma(b, a, c), expected)
+
+    # Custom assertions.
+    def assertIsNaN(self, value):
+        self.assertTrue(
+            math.isnan(value),
+            msg="Expected a NaN, got {!r}".format(value)
+        )
+
+    def assertIsPositiveZero(self, value):
+        self.assertTrue(
+            value == 0 and math.copysign(1, value) > 0,
+            msg="Expected a positive zero, got {!r}".format(value)
+        )
+
+    def assertIsNegativeZero(self, value):
+        self.assertTrue(
+            value == 0 and math.copysign(1, value) < 0,
+            msg="Expected a negative zero, got {!r}".format(value)
+        )
+
+
 def test_main():
     from doctest import DocFileSuite
     suite = unittest.TestSuite()
     suite.addTest(unittest.makeSuite(MathTests))
     suite.addTest(unittest.makeSuite(IsCloseTests))
+    suite.addTest(unittest.makeSuite(FMATests))
     suite.addTest(DocFileSuite("ieee754.txt"))
     run_unittest(suite)
 

Misc/NEWS.d/next/Library/2020-01-13-15-04-26.bpo-29282.lO827U.rst

@@ -0,0 +1,2 @@
+Added new :func:`math.fma` function, wrapping C99's ``fma()`` operation: fused
+multiply-add function.