Simplify vector_norm() by eliminating special cases in the main loop (GH-9006)

The *max* value is no longer treated as a special case in the main loop.  Besides making the main loop simpler and branchless, this also lets us relax the input restriction of *vec* to contain only non-negative values.

Modules/mathmodule.c

@@ -2032,14 +2032,14 @@ math_fmod_impl(PyObject *module, double x, double y)
 }
 
 /*
-Given an *n* length *vec* of non-negative values
-where *max* is the largest value in the vector, compute:
+Given an *n* length *vec* of values and a value *max*, compute:
 
     max * sqrt(sum((x / max) ** 2 for x in vec))
 
-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 value of the *max* variable must be non-negative and
+at least equal to the absolute value of the largest magnitude
+entry in the vector.  If n==0, then *max* should be 0.0.
+If an infinity is present in the vec, *max* should be INF.
 
 The *found_nan* variable indicates whether some member of
 the *vec* is a NaN.
@@ -2053,16 +2053,19 @@ 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.
+represents the largest possible value of (x/max)**2.
+
+After the loop is finished, the initial 1.0 is subtracted out
+for a net zero effect on the final sum.  Since *csum* will be
+greater than 1.0, the subtraction of 1.0 will not cause
+fractional digits to be dropped from *csum*.
 
 */
 
 static inline double
 vector_norm(Py_ssize_t n, double *vec, double max, int found_nan)
 {
-    double x, csum = 1.0, oldcsum, frac = 0.0, last;
+    double x, csum = 1.0, oldcsum, frac = 0.0;
     Py_ssize_t i;
 
     if (Py_IS_INFINITY(max)) {
@@ -2071,27 +2074,20 @@ vector_norm(Py_ssize_t n, double *vec, double max, int found_nan)
     if (found_nan) {
         return Py_NAN;
     }
-    if (max == 0.0) {
-        return 0.0;
+    if (max == 0.0 || n == 1) {
+        return max;
     }
-    assert(n > 0);
-    last = vec[n-1];
-    for (i=0 ; i < n-1 ; i++) {
+    for (i=0 ; i < n ; i++) {
         x = vec[i];
-        assert(Py_IS_FINITE(x) && x >= 0.0 && x <= max);
-        if (x == max) {
-            x = last;
-            last = max;
-        }
+        assert(Py_IS_FINITE(x) && fabs(x) <= max);
         x /= max;
         x = x*x;
-        assert(csum >= x);
         oldcsum = csum;
         csum += x;
+        assert(csum >= x);
         frac += (oldcsum - csum) + x;
     }
-    assert(last == max);
-    return max * sqrt(csum + frac);
+    return max * sqrt(csum - 1.0 + frac);
 }
 
 #define NUM_STACK_ELEMS 16