From 72186aa637bc88cd5f5e234803af64acab25994c Mon Sep 17 00:00:00 2001
From: Raymond Hettinger <rhettinger@users.noreply.github.com>
Date: Fri, 17 Mar 2023 14:06:52 -0500
Subject: Simplify and improve accuracy for subnormals in hypot() (GH-102785)

---
 Modules/mathmodule.c | 63 +++++++++++++++++++++++-----------------------------
 1 file changed, 28 insertions(+), 35 deletions(-)

diff --git a/Modules/mathmodule.c b/Modules/mathmodule.c
index ae9e321..48cd9a6 100644
--- a/Modules/mathmodule.c
+++ b/Modules/mathmodule.c
@@ -2498,7 +2498,7 @@ References:
 static inline double
 vector_norm(Py_ssize_t n, double *vec, double max, int found_nan)
 {
-    double x, h, scale, oldcsum, csum = 1.0, frac1 = 0.0, frac2 = 0.0;
+    double x, h, scale, csum = 1.0, frac1 = 0.0, frac2 = 0.0;
     DoubleLength pr, sm;
     int max_e;
     Py_ssize_t i;
@@ -2513,49 +2513,42 @@ vector_norm(Py_ssize_t n, double *vec, double max, int found_nan)
         return max;
     }
     frexp(max, &max_e);
-    if (max_e >= -1023) {
-        scale = ldexp(1.0, -max_e);
-        assert(max * scale >= 0.5);
-        assert(max * scale < 1.0);
+    if (max_e < -1023) {
+        /* When max_e < -1023, ldexp(1.0, -max_e) would overflow.
+           So we first perform lossless scaling from subnormals back to normals,
+           then recurse back to vector_norm(), and then finally undo the scaling.
+        */
         for (i=0 ; i < n ; i++) {
-            x = vec[i];
-            assert(Py_IS_FINITE(x) && fabs(x) <= max);
+            vec[i] /= DBL_MIN;
+        }
+        return DBL_MIN * vector_norm(n, vec, max / DBL_MIN, found_nan);
+    }
+    scale = ldexp(1.0, -max_e);
+    assert(max * scale >= 0.5);
+    assert(max * scale < 1.0);
+    for (i=0 ; i < n ; i++) {
+        x = vec[i];
+        assert(Py_IS_FINITE(x) && fabs(x) <= max);
 
-            x *= scale;
-            assert(fabs(x) < 1.0);
+        x *= scale;
+        assert(fabs(x) < 1.0);
 
-            pr = dl_mul(x, x);
-            assert(pr.hi <= 1.0);
+        pr = dl_mul(x, x);
+        assert(pr.hi <= 1.0);
 
-            sm = dl_fast_sum(csum, pr.hi);
-            csum = sm.hi;
-            frac1 += pr.lo;
-            frac2 += sm.lo;
-        }
-        h = sqrt(csum - 1.0 + (frac1 + frac2));
-        pr = dl_mul(-h, h);
         sm = dl_fast_sum(csum, pr.hi);
         csum = sm.hi;
         frac1 += pr.lo;
         frac2 += sm.lo;
-        x = csum - 1.0 + (frac1 + frac2);
-        return (h + x / (2.0 * h)) / scale;
     }
-    /* When max_e < -1023, ldexp(1.0, -max_e) overflows.
-       So instead of multiplying by a scale, we just divide by *max*.
-    */
-    for (i=0 ; i < n ; i++) {
-        x = vec[i];
-        assert(Py_IS_FINITE(x) && fabs(x) <= max);
-        x /= max;
-        x = x*x;
-        assert(x <= 1.0);
-        assert(fabs(csum) >= fabs(x));
-        oldcsum = csum;
-        csum += x;
-        frac1 += (oldcsum - csum) + x;
-    }
-    return max * sqrt(csum - 1.0 + frac1);
+    h = sqrt(csum - 1.0 + (frac1 + frac2));
+    pr = dl_mul(-h, h);
+    sm = dl_fast_sum(csum, pr.hi);
+    csum = sm.hi;
+    frac1 += pr.lo;
+    frac2 += sm.lo;
+    x = csum - 1.0 + (frac1 + frac2);
+    return (h + x / (2.0 * h)) / scale;
 }
 
 #define NUM_STACK_ELEMS 16
-- 
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