GH-100425: Improve accuracy of builtin sum() for float inputs (GH-100426)

Doc/library/functions.rst

@@ -1733,6 +1733,10 @@ are always available.  They are listed here in alphabetical order.
    .. versionchanged:: 3.8
       The *start* parameter can be specified as a keyword argument.
 
+   .. versionchanged:: 3.12 Summation of floats switched to an algorithm
+      that gives higher accuracy on most builds.
+
+
 .. class:: super()
            super(type, object_or_type=None)
 

Doc/library/math.rst

@@ -108,12 +108,7 @@ Number-theoretic and representation functions
 .. function:: fsum(iterable)
 
    Return an accurate floating point sum of values in the iterable.  Avoids
-   loss of precision by tracking multiple intermediate partial sums:
-
-        >>> sum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
-        0.9999999999999999
-        >>> fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
-        1.0
+   loss of precision by tracking multiple intermediate partial sums.
 
    The algorithm's accuracy depends on IEEE-754 arithmetic guarantees and the
    typical case where the rounding mode is half-even.  On some non-Windows

Doc/tutorial/floatingpoint.rst

@@ -192,7 +192,7 @@ added onto a running total.  That can make a difference in overall accuracy
 so that the errors do not accumulate to the point where they affect the
 final total:
 
-   >>> sum([0.1] * 10) == 1.0
+   >>> 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 == 1.0
    False
    >>> math.fsum([0.1] * 10) == 1.0
    True

Lib/test/test_builtin.py

@@ -9,6 +9,7 @@ import fractions
 import gc
 import io
 import locale
+import math
 import os
 import pickle
 import platform
@@ -31,6 +32,7 @@ from test.support import (swap_attr, maybe_get_event_loop_policy)
 from test.support.os_helper import (EnvironmentVarGuard, TESTFN, unlink)
 from test.support.script_helper import assert_python_ok
 from test.support.warnings_helper import check_warnings
+from test.support import requires_IEEE_754
 from unittest.mock import MagicMock, patch
 try:
     import pty, signal
@@ -38,6 +40,12 @@ except ImportError:
     pty = signal = None
 
 
+# Detect evidence of double-rounding: sum() does not always
+# get improved accuracy on machines that suffer from double rounding.
+x, y = 1e16, 2.9999 # use temporary values to defeat peephole optimizer
+HAVE_DOUBLE_ROUNDING = (x + y == 1e16 + 4)
+
+
 class Squares:
 
     def __init__(self, max):
@@ -1617,6 +1625,8 @@ class BuiltinTest(unittest.TestCase):
         self.assertEqual(repr(sum([-0.0])), '0.0')
         self.assertEqual(repr(sum([-0.0], -0.0)), '-0.0')
         self.assertEqual(repr(sum([], -0.0)), '-0.0')
+        self.assertTrue(math.isinf(sum([float("inf"), float("inf")])))
+        self.assertTrue(math.isinf(sum([1e308, 1e308])))
 
         self.assertRaises(TypeError, sum)
         self.assertRaises(TypeError, sum, 42)
@@ -1641,6 +1651,14 @@ class BuiltinTest(unittest.TestCase):
         sum(([x] for x in range(10)), empty)
         self.assertEqual(empty, [])
 
+    @requires_IEEE_754
+    @unittest.skipIf(HAVE_DOUBLE_ROUNDING,
+                         "sum accuracy not guaranteed on machines with double rounding")
+    @support.cpython_only    # Other implementations may choose a different algorithm
+    def test_sum_accuracy(self):
+        self.assertEqual(sum([0.1] * 10), 1.0)
+        self.assertEqual(sum([1.0, 10E100, 1.0, -10E100]), 2.0)
+
     def test_type(self):
         self.assertEqual(type(''),  type('123'))
         self.assertNotEqual(type(''), type(()))

Misc/NEWS.d/next/Core and Builtins/2022-12-21-22-48-41.gh-issue-100425.U64yLu.rst

@@ -0,0 +1 @@
+Improve the accuracy of ``sum()`` with compensated summation.

Python/bltinmodule.c

@@ -2532,6 +2532,7 @@ builtin_sum_impl(PyObject *module, PyObject *iterable, PyObject *start)
 
     if (PyFloat_CheckExact(result)) {
         double f_result = PyFloat_AS_DOUBLE(result);
+        double c = 0.0;
         Py_SETREF(result, NULL);
         while(result == NULL) {
             item = PyIter_Next(iter);
@@ -2539,10 +2540,25 @@ builtin_sum_impl(PyObject *module, PyObject *iterable, PyObject *start)
                 Py_DECREF(iter);
                 if (PyErr_Occurred())
                     return NULL;
+                /* Avoid losing the sign on a negative result,
+                   and don't let adding the compensation convert
+                   an infinite or overflowed sum to a NaN. */
+                if (c && Py_IS_FINITE(c)) {
+                    f_result += c;
+                }
                 return PyFloat_FromDouble(f_result);
             }
             if (PyFloat_CheckExact(item)) {
-                f_result += PyFloat_AS_DOUBLE(item);
+                // Improved Kahan–Babuška algorithm by Arnold Neumaier
+                // https://www.mat.univie.ac.at/~neum/scan/01.pdf
+                double x = PyFloat_AS_DOUBLE(item);
+                double t = f_result + x;
+                if (fabs(f_result) >= fabs(x)) {
+                    c += (f_result - t) + x;
+                } else {
+                    c += (x - t) + f_result;
+                }
+                f_result = t;
                 _Py_DECREF_SPECIALIZED(item, _PyFloat_ExactDealloc);
                 continue;
             }
@@ -2556,6 +2572,9 @@ builtin_sum_impl(PyObject *module, PyObject *iterable, PyObject *start)
                     continue;
                 }
             }
+            if (c && Py_IS_FINITE(c)) {
+                f_result += c;
+            }
             result = PyFloat_FromDouble(f_result);
             if (result == NULL) {
                 Py_DECREF(item);