[PATCH 3/7] sin/cos slow paths: remove slow paths from small range reduction

[Wilco Dijkstra <Wilco.Dijkstra@arm.com>]

This patch improves the accuracy of the range reduction.  When the input is
large (2^27) and very close to a multiple of PI/2, using 110 bits of PI is not
enough.  Improve range reduction accuracy to 136 bits.  As a result the special
checks for results close to zero can be removed.  The ULP of the polynomials is
at worst 0.55ULP, so there is no reason for the slow functions, and they can be
removed.

ChangeLog:
2018-03-20  Wilco Dijkstra  <wdijkstr@arm.com>

	* sysdeps/ieee754/dbl-64/s_sin.c (reduce_sincos_1): Rename to sincos_1,
	improve accuracy to 136 bits.
	(do_sincos_1): Rename to do_sincos, remove fallbacks to slow functions.
	(__sin): Use improved reduction and simplified do_sincos calculation.
	(__cos): Likewise.
	* sysdeps/ieee754/dbl-64/s_sincos.c (__sincos): Likewise.
--
diff --git a/sysdeps/ieee754/dbl-64/s_sin.c b/sysdeps/ieee754/dbl-64/s_sin.c
index c86fb9f2aa9f18418defc522830a7b8f85c1dfae..b8c366a6f05ef6b6632302fac96cd19af518f1fe 100644
--- a/sysdeps/ieee754/dbl-64/s_sin.c
+++ b/sysdeps/ieee754/dbl-64/s_sin.c
@@ -295,9 +295,13 @@ reduce_and_compute (double x, bool shift_quadrant)
   return retval;
 }
 
+/* Reduce range of x to within PI/2 with abs (x) < 105414350.  The high part
+   is written to *a, the low part to *da.  Range reduction is accurate to 136
+   bits so that when x is large and *a very close to zero, all 53 bits of *a
+   are correct.  */
 static inline int4
 __always_inline
-reduce_sincos_1 (double x, double *a, double *da)
+reduce_sincos (double x, double *a, double *da)
 {
   mynumber v;
 
@@ -306,62 +310,45 @@ reduce_sincos_1 (double x, double *a, double *da)
   v.x = t;
   double y = (x - xn * mp1) - xn * mp2;
   int4 n = v.i[LOW_HALF] & 3;
-  double db = xn * mp3;
-  double b = y - db;
-  db = (y - b) - db;
+
+  double b, db, t1, t2;
+  t1 = xn * pp3;
+  t2 = y - t1;
+  db = (y - t2) - t1;
+
+  t1 = xn * pp4;
+  b = t2 - t1;
+  db += (t2 - b) - t1;
 
   *a = b;
   *da = db;
-
   return n;
 }
 
-/* Compute sin (A + DA).  cos can be computed by passing SHIFT_QUADRANT as
-   true, which results in shifting the quadrant N clockwise.  */
+/* Compute sin or cos (A + DA) for the given quadrant N.  */
 static double
 __always_inline
-do_sincos_1 (double a, double da, double x, int4 n, bool shift_quadrant)
+do_sincos (double a, double da, int4 n)
 {
-  double xx, retval, res, cor;
-  double eps = fabs (x) * 1.2e-30;
+  double retval, cor;
 
-  int k1 = (n + shift_quadrant) & 3;
-  switch (k1)
-    {			/* quarter of unit circle */
-    case 2:
-      a = -a;
-      da = -da;
-      /* Fall through.  */
-    case 0:
-      xx = a * a;
+  if (n & 1)
+    /* Max ULP is 0.513.  */
+    retval = do_cos (a, da, &cor);
+  else
+    {
+      double xx = a * a;
+      /* Max ULP is 0.501 if xx < 0.01588, otherwise ULP is 0.518.  */
       if (xx < 0.01588)
-	{
-	  /* Taylor series.  */
-	  res = TAYLOR_SIN (xx, a, da, cor);
-	  cor = 1.02 * cor + __copysign (eps, cor);
-	  retval = (res == res + cor) ? res : sloww (a, da, x, shift_quadrant);
-	}
+	retval = TAYLOR_SIN (xx, a, da, cor);
       else
-	{
-	  res = do_sin (a, da, &cor);
-	  cor = 1.035 * cor + __copysign (eps, cor);
-	  retval = ((res == res + cor) ? __copysign (res, a)
-		    : sloww1 (a, da, x, shift_quadrant));
-	}
-      break;
-
-    case 1:
-    case 3:
-      res = do_cos (a, da, &cor);
-      cor = 1.025 * cor + __copysign (eps, cor);
-      retval = ((res == res + cor) ? ((n & 2) ? -res : res)
-		: sloww2 (a, da, x, n));
-      break;
+	retval = __copysign (do_sin (a, da, &cor), a);
     }
 
-  return retval;
+  return (n & 2) ? -retval : retval;
 }
 
+
 /*******************************************************************/
 /* An ultimate sin routine. Given an IEEE double machine number x   */
 /* it computes the correctly rounded (to nearest) value of sin(x)  */
@@ -374,13 +361,18 @@ SECTION
 #endif
 __sin (double x)
 {
-  double xx, t, cor;
+#ifndef IN_SINCOS
+  double xx, t, a, da, cor;
   mynumber u;
-  int4 k, m;
+  int4 k, m, n;
   double retval = 0;
 
-#ifndef IN_SINCOS
   SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
+#else
+  double xx, t, cor;
+  mynumber u;
+  int4 k, m;
+  double retval = 0;
 #endif
 
   u.x = x;
@@ -419,9 +411,8 @@ __sin (double x)
 /*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
   else if (k < 0x419921FB)
     {
-      double a, da;
-      int4 n = reduce_sincos_1 (x, &a, &da);
-      retval = do_sincos_1 (a, da, x, n, false);
+      n = reduce_sincos (x, &a, &da);
+      retval = do_sincos (a, da, n);
     }				/*   else  if (k <  0x419921FB )    */
 
 /* --------------------105414350 <|x| <2^1024------------------------------*/
@@ -456,7 +447,11 @@ __cos (double x)
 {
   double y, xx, cor, a, da;
   mynumber u;
+#ifndef IN_SINCOS
+  int4 k, m, n;
+#else
   int4 k, m;
+#endif
 
   double retval = 0;
 
@@ -496,9 +491,8 @@ __cos (double x)
 #ifndef IN_SINCOS
   else if (k < 0x419921FB)
     {				/* 2.426265<|x|< 105414350 */
-      double a, da;
-      int4 n = reduce_sincos_1 (x, &a, &da);
-      retval = do_sincos_1 (a, da, x, n, true);
+      n = reduce_sincos (x, &a, &da);
+      retval = do_sincos (a, da, n + 1);
     }				/*   else  if (k <  0x419921FB )    */
 
   /* 105414350 <|x| <2^1024 */
diff --git a/sysdeps/ieee754/dbl-64/s_sincos.c b/sysdeps/ieee754/dbl-64/s_sincos.c
index a9af8ce526bfe78c06cfafa65de0815ec69585c5..4f032d2e42593ccde22169b374728386dd8fca8e 100644
--- a/sysdeps/ieee754/dbl-64/s_sincos.c
+++ b/sysdeps/ieee754/dbl-64/s_sincos.c
@@ -79,10 +79,10 @@ __sincos (double x, double *sinx, double *cosx)
   if (k < 0x419921FB)
     {
       double a, da;
-      int4 n = reduce_sincos_1 (x, &a, &da);
+      int4 n = reduce_sincos (x, &a, &da);
 
-      *sinx = do_sincos_1 (a, da, x, n, false);
-      *cosx = do_sincos_1 (a, da, x, n, true);
+      *sinx = do_sincos (a, da, n);
+      *cosx = do_sincos (a, da, n + 1);
 
       return;
     }