Improve commutativity of math.hypot() and math.dist() (GH-8984)

Modules/mathmodule.c

@@ -2037,26 +2037,32 @@ where *max* is the largest value in the vector, compute:
 
     max * sqrt(sum((x / max) ** 2 for x in vec))
 
-When a maximum value is found, it is swapped to the end.  This
-lets us skip one loop iteration and just add 1.0 at the end.
-Saving the largest value for last also helps improve accuracy.
-
-Kahan summation is used to improve accuracy.  The *csum*
-variable tracks the cumulative sum and *frac* tracks
-fractional round-off error for the most recent addition.
-
 The value of the *max* variable must be present in *vec*
 or should equal to 0.0 when n==0.  Likewise, *max* will
 be INF if an infinity is present in the vec.
 
 The *found_nan* variable indicates whether some member of
 the *vec* is a NaN.
+
+To improve accuracy and to increase the number of cases where
+vector_norm() is commutative, we use a variant of Neumaier
+summation specialized to exploit that we always know that
+|csum| >= |x|.
+
+The *csum* variable tracks the cumulative sum and *frac* tracks
+the cumulative fractional errors at each step.  Since this
+variant assumes that |csum| >= |x| at each step, we establish
+the precondition by starting the accumulation from 1.0 which
+represents an entry equal to *max*.  This also provides a nice
+side benefit in that it lets us skip over a *max* entry (which
+is swapped into *last*) saving us one iteration through the loop.
+
 */
 
 static inline double
 vector_norm(Py_ssize_t n, double *vec, double max, int found_nan)
 {
-    double x, csum = 0.0, oldcsum, frac = 0.0, last;
+    double x, csum = 1.0, oldcsum, frac = 0.0, last;
     Py_ssize_t i;
 
     if (Py_IS_INFINITY(max)) {
@@ -2078,14 +2084,14 @@ vector_norm(Py_ssize_t n, double *vec, double max, int found_nan)
             last = max;
         }
         x /= max;
-        x = x*x - frac;
+        x = x*x;
+        assert(csum >= x);
         oldcsum = csum;
         csum += x;
-        frac = (csum - oldcsum) - x;
+        frac += (oldcsum - csum) + x;
     }
     assert(last == max);
-    csum += 1.0 - frac;
-    return max * sqrt(csum);
+    return max * sqrt(csum + frac);
 }
 
 #define NUM_STACK_ELEMS 16