[PATCH 1/7] sin/cos slow paths: avoid slow paths for small inputs

[PATCH 1/7] sin/cos slow paths: avoid slow paths for small inputs

[Wilco Dijkstra]

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.

--
diff --git a/sysdeps/aarch64/libm-test-ulps b/sysdeps/aarch64/libm-test-ulps
index 1f469803be59bb4813370d95c6d091de901e6129..be06085154db24c8fd6cf1bce417028a959aaa27 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 8c589cbd4ab7451a5889e9a474bf4bd36c49d498..0c16b728df127ad54039da3eec376e5f1fe4c852 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 48e53f7ef2cf814d71d5d0c9f2bb907f594aa7ef..bbb8a4d0754dbe6665682cd8a7f51f7319a14014 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

Re: [PATCH 1/7] sin/cos slow paths: avoid slow paths for small inputs

[Joseph Myers]

On Wed, 21 Mar 2018, Wilco Dijkstra wrote:

> 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.

OK.

-- 
Joseph S. Myers
joseph@codesourcery.com