0001-sin-cos-slow-paths-avoid-slow-paths-for-small-inputs.patch

From 3eaef825303efc4175751d3ca8cb50d873ddb081 Mon Sep 17 00:00:00 2001
From: Wilco Dijkstra <Wilco.Dijkstra@arm.com>
Date: Wed, 21 Mar 2018 17:50:07 +0000
Subject: [PATCH 1/7] sin/cos slow paths: avoid slow paths for small inputs

This series of patches removes the slow patchs from sin, cos and sincos.
Besides greatly simplifying the implementation, the new version is also much
faster for inputs up to PI (41% faster) and for large inputs needing range
reduction (27% faster).

ULP is ~0.55 with no errors found after testing 1.6 billion inputs across most
of the range with mpsin and mpcos.  The number of incorrectly rounded results
(ie. ULP >0.5) is at most ~2750 per million inputs between 0.125 and 0.5,
the average is ~850 per million between 0 and PI.

Tested on AArch64 and x86_64 with no regressions.

The first patch removes the slow paths for the cases where the input is small
and doesn't require range reduction.  Update ULP tables for sin, cos and sincos
on AArch64 and x86_64.

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

	* sysdeps/aarch64/libm-test-ulps: Update ULP for sin, cos, sincos.
	* sysdeps/ieee754/dbl-64/s_sin.c (__sin): Remove slow paths for small inputs.
	(__cos): Likewise.
	* sysdeps/x86_64/fpu/libm-test-ulps: Update ULP for sin, cos, sincos.
---
 sysdeps/aarch64/libm-test-ulps    |  6 ++++++
 sysdeps/ieee754/dbl-64/s_sin.c    | 40 ++++++++++++++-------------------------
 sysdeps/x86_64/fpu/libm-test-ulps |  6 ++++++
 3 files changed, 26 insertions(+), 26 deletions(-)

diff --git a/sysdeps/aarch64/libm-test-ulps b/sysdeps/aarch64/libm-test-ulps
index 1f469803be5..be06085154d 100644
--- a/sysdeps/aarch64/libm-test-ulps
+++ b/sysdeps/aarch64/libm-test-ulps
@@ -1012,7 +1012,9 @@ ildouble: 2
 ldouble: 2
 
 Function: "cos":
+double: 1
 float: 1
+idouble: 1
 ifloat: 1
 ildouble: 1
 ldouble: 1
@@ -1970,7 +1972,9 @@ ildouble: 2
 ldouble: 2
 
 Function: "sin":
+double: 1
 float: 1
+idouble: 1
 ifloat: 1
 ildouble: 1
 ldouble: 1
@@ -2000,7 +2004,9 @@ ildouble: 3
 ldouble: 3
 
 Function: "sincos":
+double: 1
 float: 1
+idouble: 1
 ifloat: 1
 ildouble: 1
 ldouble: 1
diff --git a/sysdeps/ieee754/dbl-64/s_sin.c b/sysdeps/ieee754/dbl-64/s_sin.c
index 8c589cbd4ab..0c16b728df1 100644
--- a/sysdeps/ieee754/dbl-64/s_sin.c
+++ b/sysdeps/ieee754/dbl-64/s_sin.c
@@ -448,7 +448,7 @@ SECTION
 #endif
 __sin (double x)
 {
-  double xx, res, t, cor;
+  double xx, t, cor;
   mynumber u;
   int4 k, m;
   double retval = 0;
@@ -471,26 +471,22 @@ __sin (double x)
       xx = x * x;
       /* Taylor series.  */
       t = POLYNOMIAL (xx) * (xx * x);
-      res = x + t;
-      cor = (x - res) + t;
-      retval = (res == res + 1.07 * cor) ? res : slow (x);
+      /* Max ULP of x + t is 0.535.  */
+      retval = x + t;
     }				/*  else  if (k < 0x3fd00000)    */
 /*---------------------------- 0.25<|x|< 0.855469---------------------- */
   else if (k < 0x3feb6000)
     {
-      res = do_sin (x, 0, &cor);
-      retval = (res == res + 1.096 * cor) ? res : slow1 (x);
-      retval = __copysign (retval, x);
+      /* Max ULP is 0.548.  */
+      retval = __copysign (do_sin (x, 0, &cor), x);
     }				/*   else  if (k < 0x3feb6000)    */
 
 /*----------------------- 0.855469  <|x|<2.426265  ----------------------*/
   else if (k < 0x400368fd)
     {
-
       t = hp0 - fabs (x);
-      res = do_cos (t, hp1, &cor);
-      retval = (res == res + 1.020 * cor) ? res : slow2 (x);
-      retval = __copysign (retval, x);
+      /* Max ULP is 0.51.  */
+      retval = __copysign (do_cos (t, hp1, &cor), x);
     }				/*   else  if (k < 0x400368fd)    */
 
 #ifndef IN_SINCOS
@@ -541,7 +537,7 @@ SECTION
 #endif
 __cos (double x)
 {
-  double y, xx, res, cor, a, da;
+  double y, xx, cor, a, da;
   mynumber u;
   int4 k, m;
 
@@ -561,8 +557,8 @@ __cos (double x)
 
   else if (k < 0x3feb6000)
     {				/* 2^-27 < |x| < 0.855469 */
-      res = do_cos (x, 0, &cor);
-      retval = (res == res + 1.020 * cor) ? res : cslow2 (x);
+      /* Max ULP is 0.51.  */
+      retval = do_cos (x, 0, &cor);
     }				/*   else  if (k < 0x3feb6000)    */
 
   else if (k < 0x400368fd)
@@ -571,20 +567,12 @@ __cos (double x)
       a = y + hp1;
       da = (y - a) + hp1;
       xx = a * a;
+      /* Max ULP is 0.501 if xx < 0.01588 or 0.518 otherwise.
+	 Range reduction uses 106 bits here which is sufficient.  */
       if (xx < 0.01588)
-	{
-	  res = TAYLOR_SIN (xx, a, da, cor);
-	  cor = 1.02 * cor + __copysign (1.0e-31, cor);
-	  retval = (res == res + cor) ? res : sloww (a, da, x, true);
-	}
+	retval = TAYLOR_SIN (xx, a, da, cor);
       else
-	{
-	  res = do_sin (a, da, &cor);
-	  cor = 1.035 * cor + __copysign (1.0e-31, cor);
-	  retval = ((res == res + cor) ? __copysign (res, a)
-		    : sloww1 (a, da, x, true));
-	}
-
+	retval = __copysign (do_sin (a, da, &cor), a);
     }				/*   else  if (k < 0x400368fd)    */
 
 
diff --git a/sysdeps/x86_64/fpu/libm-test-ulps b/sysdeps/x86_64/fpu/libm-test-ulps
index 48e53f7ef2c..bbb8a4d0754 100644
--- a/sysdeps/x86_64/fpu/libm-test-ulps
+++ b/sysdeps/x86_64/fpu/libm-test-ulps
@@ -1262,7 +1262,9 @@ ildouble: 1
 ldouble: 1
 
 Function: "cos":
+double: 1
 float128: 1
+idouble: 1
 ifloat128: 1
 ildouble: 1
 ldouble: 1
@@ -2528,7 +2530,9 @@ Function: "pow_vlen8_avx2":
 float: 3
 
 Function: "sin":
+double: 1
 float128: 1
+idouble: 1
 ifloat128: 1
 ildouble: 1
 ldouble: 1
@@ -2578,7 +2582,9 @@ Function: "sin_vlen8_avx2":
 float: 1
 
 Function: "sincos":
+double: 1
 float128: 1
+idouble: 1
 ifloat128: 1
 ildouble: 1
 ldouble: 1
-- 
2.16.2