RFC: Improve hypot performance

[Wilco Dijkstra]

Hi Adhemerval,

Here is an early version of a much faster hypot implementation. It uses fma
to significantly outperform the powerpc version both in throughput and
latency. It has a worst-case ULP of ~0.949 and passes the testsuite. The
powerpc version has a worst-case ULP of ~1.21 and several test failures.

It applies on top of your hypot patch series. I didn't optimize the non-fma
case since modern targets have fma. It'll be interesting to compare it on
Power. You'll need to correctly set FAST_FMINMAX to indicate support for
inlined fmin/fmax instructions (this will be added to math_private.h for
targets that have it), without it the code tries to use a conditional move
since using a branch here is really bad for performance.

Cheers,
Wilco

---

diff --git a/sysdeps/ieee754/dbl-64/e_hypot.c b/sysdeps/ieee754/dbl-64/e_hypot.c
index d20bc3e3657e350a1103a8f8477db35ee60399e0..3906711788cff5d66725e5879bb4d6c36dd24dc7 100644
--- a/sysdeps/ieee754/dbl-64/e_hypot.c
+++ b/sysdeps/ieee754/dbl-64/e_hypot.c
@@ -37,73 +37,40 @@
 #include <math-svid-compat.h>
 #include <errno.h>
 
+#define FAST_FMINMAX 1
+//#undef __FP_FAST_FMA
+
+#define SCALE     0x1p-600
+#define LARGE_VAL 0x1p+511
+#define TINY_VAL  0x1p-459
+#define EPS       0x1p-54
+
+
 static inline double
 handle_errno (double r)
 {
+  r = math_narrow_eval (r);
   if (isinf (r))
     __set_errno (ERANGE);
   return r;
 }
 
-/* sqrt (DBL_EPSILON / 2.0)  */
-#define SQRT_EPS_DIV_2     0x1.6a09e667f3bcdp-27
-/* DBL_MIN / (sqrt (DBL_EPSILON / 2.0))   */
-#define DBL_MIN_THRESHOLD  0x1.6a09e667f3bcdp-996
-/* eps (double) * sqrt (DBL_MIN))  */
-#define SCALE              0x1p-563
-/* 1 / eps (sqrt (DBL_MIN)  */
-#define INV_SCALE          0x1p+563
-/* sqrt (DBL_MAX)  */
-#define SQRT_DBL_MAX       0x1.6a09e667f3bccp+511
-/* sqrt (DBL_MIN)  */
-#define SQRT_DBL_MIN       0x1p-511
-
-double
-__hypot (double x, double y)
+static inline double
+kernel (double ax, double ay)
 {
-  if ((isinf (x) || isinf (y))
-      && !issignaling (x) && !issignaling (y))
-    return INFINITY;
-  if (isnan (x) || isnan (y))
-    return x + y;
-
-  double ax = fabs (x);
-  double ay = fabs (y);
-  if (ay > ax)
-    {
-      double tmp = ax;
-      ax = ay;
-      ay = tmp;
-    }
-
-  /* Widely varying operands.  The DBL_MIN_THRESHOLD check is used to avoid
-     a spurious underflow from the multiplication.  */
-  if (ax >= DBL_MIN_THRESHOLD && ay <= ax * SQRT_EPS_DIV_2)
-    return (ay == 0.0)
-	   ? ax
-	   : handle_errno (math_narrow_eval (ax + DBL_TRUE_MIN));
+  double t1, t2;
+#ifdef __FP_FAST_FMA
+  t1 = ay + ay;
+  t2 = ax - ay;
 
-  double scale = SCALE;
-  if (ax > SQRT_DBL_MAX)
-    {
-      ax *= scale;
-      ay *= scale;
-      scale = INV_SCALE;
-    }
-  else if (ay < SQRT_DBL_MIN)
-    {
-      ax /= scale;
-      ay /= scale;
-    }
+  if (t1 >= ax)
+    return sqrt (fma (t1, ax, t2 * t2));
   else
-    scale = 1.0;
-
+    return sqrt (fma (ax, ax, ay * ay));
+#else
   double h = sqrt (ax * ax + ay * ay);
 
-  double t1, t2;
-  if (h == 0.0)
-    return h;
-  else if (h <= 2.0 * ay)
+  if (h <= 2.0 * ay)
     {
       double delta = h - ay;
       t1 = ax * (2.0 * delta - ax);
@@ -112,14 +79,57 @@ __hypot (double x, double y)
   else
     {
       double delta = h - ax;
-      t1 = 2.0 * delta * (ax - 2 * ay);
+      t1 = 2.0 * delta * (ax - 2.0 * ay);
       t2 = (4.0 * delta - ay) * ay + delta * delta;
     }
   h -= (t1 + t2) / (2.0 * h);
-  h = math_narrow_eval (h * scale);
-  math_check_force_underflow_nonneg (h);
-  return handle_errno (h);
+  return h;
+#endif
+}
+
+
+double
+__hypot (double x, double y)
+{
+  if (!isfinite (x) || !isfinite (y))
+    {
+      if ((isinf (x) || isinf (y))
+	  && !issignaling_inline (x) && !issignaling_inline (y))
+	return INFINITY;
+      return x + y;
+    }
+
+  x = fabs (x);
+  y = fabs (y);
+
+  double ax = FAST_FMINMAX ? fmax (x, y) : (x < y ? y : x);
+  double ay = FAST_FMINMAX ? fmin (x, y) : (x < y ? x : y);
+
+  if (__glibc_unlikely (ax > LARGE_VAL))
+    {
+      if (__glibc_unlikely (ay <= ax * EPS))
+	return handle_errno (ax + ay);
+
+      return handle_errno (kernel (ax * SCALE, ay * SCALE) / SCALE);
+    }
+
+  if (__glibc_unlikely (ay < TINY_VAL))
+    {
+      if (__glibc_unlikely (ax >= ay / EPS))
+	return math_narrow_eval (ax + ay);
+
+      ax = math_narrow_eval (kernel (ax / SCALE, ay / SCALE) * SCALE);
+      math_check_force_underflow_nonneg (ax);
+      return ax;
+    }
+
+  /* Common case: ax is not huge and ay is not tiny.  */
+  if (__glibc_unlikely (ay <= ax * EPS))
+    return math_narrow_eval (ax + ay);
+
+  return math_narrow_eval (kernel (ax, ay));
 }
+
 strong_alias (__hypot, __ieee754_hypot)
 libm_alias_finite (__ieee754_hypot, __hypot)
 #if LIBM_SVID_COMPAT