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

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.

	* sysdeps/ieee754/dbl-64/s_sin.c (reduce_sincos_1): Rename to
	reduce_sincos, 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.

ChangeLog

@@ -1,3 +1,12 @@
+2018-04-03  Wilco Dijkstra  <wdijkstr@arm.com>
+
+	* sysdeps/ieee754/dbl-64/s_sin.c (reduce_sincos_1): Rename to
+	reduce_sincos, 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.
+
 2018-04-03  Wilco Dijkstra  <wdijkstr@arm.com>
 
 	* sysdeps/ieee754/dbl-64/s_sin.c (reduce_sincos_2): Remove function.

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;
+  double b, db, t1, t2;
-  db = (y - b) - db;
+  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
+/* Compute sin or cos (A + DA) for the given quadrant N.  */
-   true, which results in shifting the quadrant N clockwise.  */
 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 retval, cor;
-  double eps = fabs (x) * 1.2e-30;
 
-  int k1 = (n + shift_quadrant) & 3;
+  if (n & 1)
-  switch (k1)
+    /* Max ULP is 0.513.  */
-    {			/* quarter of unit circle */
+    retval = do_cos (a, da, &cor);
-    case 2:
-      a = -a;
-      da = -da;
-      /* Fall through.  */
-    case 0:
-      xx = a * a;
-      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);
-	}
   else
     {
-	  res = do_sin (a, da, &cor);
+      double xx = a * a;
-	  cor = 1.035 * cor + __copysign (eps, cor);
+      /* Max ULP is 0.501 if xx < 0.01588, otherwise ULP is 0.518.  */
-	  retval = ((res == res + cor) ? __copysign (res, a)
+      if (xx < 0.01588)
-		    : sloww1 (a, da, x, shift_quadrant));
+	retval = TAYLOR_SIN (xx, a, da, cor);
-	}
+      else
-      break;
+	retval = __copysign (do_sin (a, da, &cor), a);
-
-    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;
     }
 
-  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)
 {
+#ifndef IN_SINCOS
+  double xx, t, a, da, cor;
+  mynumber u;
+  int4 k, m, n;
+  double retval = 0;
+
+  SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
+#else
   double xx, t, cor;
   mynumber u;
   int4 k, m;
   double retval = 0;
-
-#ifndef IN_SINCOS
-  SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
 #endif
 
   u.x = x;
@@ -419,9 +411,8 @@ __sin (double x)
 /*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
   else if (k < 0x419921FB)
     {
-      double a, da;
+      n = reduce_sincos (x, &a, &da);
-      int4 n = reduce_sincos_1 (x, &a, &da);
+      retval = do_sincos (a, da, n);
-      retval = do_sincos_1 (a, da, x, n, false);
     }				/*   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;
+      n = reduce_sincos (x, &a, &da);
-      int4 n = reduce_sincos_1 (x, &a, &da);
+      retval = do_sincos (a, da, n + 1);
-      retval = do_sincos_1 (a, da, x, n, true);
     }				/*   else  if (k <  0x419921FB )    */
 
   /* 105414350 <|x| <2^1024 */

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);
+      *sinx = do_sincos (a, da, n);
-      *cosx = do_sincos_1 (a, da, x, n, true);
+      *cosx = do_sincos (a, da, n + 1);
 
       return;
     }