math: Remove slow paths from atan2 [BZ #15267]

Remove slow paths from atan2. Add ULP annotations.

Reviewed-By: Paul Zimmermann <Paul.Zimmermann@inria.fr>

2 files changed, 40 insertions, 294 deletions

diff --git a/sysdeps/ieee754/dbl-64/atnat2.h b/sysdeps/ieee754/dbl-64/atnat2.h
index f1c0d23f9e..f9f8debd55 100644
--- a/sysdeps/ieee754/dbl-64/atnat2.h
+++ b/sysdeps/ieee754/dbl-64/atnat2.h
@@ -34,7 +34,7 @@
 #define MM 5
 #ifdef BIG_ENDI
 
-  static const number
+  static const mynumber
   /* polynomial I */
 /**/ d3             = {{0xbfd55555, 0x55555555} }, /* -0.333... */
 /**/ d5             = {{0x3fc99999, 0x999997fd} }, /*  0.199... */
@@ -96,7 +96,7 @@
 #else
 #ifdef LITTLE_ENDI
 
-  static const number
+  static const mynumber
   /* polynomial I */
 /**/ d3             = {{0x55555555, 0xbfd55555} }, /* -0.333... */
 /**/ d5             = {{0x999997fd, 0x3fc99999} }, /*  0.199... */
diff --git a/sysdeps/ieee754/dbl-64/e_atan2.c b/sysdeps/ieee754/dbl-64/e_atan2.c
index e6b98142fb..b77ad2dc6b 100644
--- a/sysdeps/ieee754/dbl-64/e_atan2.c
+++ b/sysdeps/ieee754/dbl-64/e_atan2.c
@@ -20,25 +20,14 @@
 /*  MODULE_NAME: atnat2.c                                               */
 /*                                                                      */
 /*  FUNCTIONS: uatan2                                                   */
-/*             atan2Mp                                                  */
 /*             signArctan2                                              */
-/*             normalized                                               */
 /*                                                                      */
-/*  FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat2.h                */
-/*                mpatan.c mpatan2.c mpsqrt.c                           */
+/*  FILES NEEDED: dla.h endian.h mydefs.h atnat2.h                      */
 /*                uatan.tbl                                             */
 /*                                                                      */
-/* An ultimate atan2() routine. Given two IEEE double machine numbers y,*/
-/* x it computes the correctly rounded (to nearest) value of atan2(y,x).*/
-/*                                                                      */
-/* Assumption: Machine arithmetic operations are performed in           */
-/* round to nearest mode of IEEE 754 standard.                          */
-/*                                                                      */
 /************************************************************************/
 
 #include <dla.h>
-#include "mpa.h"
-#include "MathLib.h"
 #include "mydefs.h"
 #include "uatan.tbl"
 #include "atnat2.h"
@@ -48,20 +37,15 @@
 #include <math-barriers.h>
 #include <math_private.h>
 #include <fenv_private.h>
-#include <stap-probe.h>
 #include <libm-alias-finite.h>
 
 #ifndef SECTION
 # define SECTION
 #endif
 
-/************************************************************************/
-/* An ultimate atan2 routine. Given two IEEE double machine numbers y,x */
-/* it computes the correctly rounded (to nearest) value of atan2(y,x).  */
-/* Assumption: Machine arithmetic operations are performed in           */
-/* round to nearest mode of IEEE 754 standard.                          */
-/************************************************************************/
-static double atan2Mp (double, double, const int[]);
+#define  TWO52     0x1.0p52
+#define  TWOM1022  0x1.0p-1022
+
   /* Fix the sign and return after stage 1 or stage 2 */
 static double
 signArctan2 (double y, double z)
@@ -69,18 +53,15 @@ signArctan2 (double y, double z)
   return copysign (z, y);
 }
 
-static double normalized (double, double, double, double);
-void __mpatan2 (mp_no *, mp_no *, mp_no *, int);
-
+/* atan2 with max ULP of ~0.524 based on random sampling.  */
 double
 SECTION
 __ieee754_atan2 (double y, double x)
 {
   int i, de, ux, dx, uy, dy;
-  static const int pr[MM] = { 6, 8, 10, 20, 32 };
-  double ax, ay, u, du, u9, ua, v, vv, dv, t1, t2, t3,
-	 z, zz, cor, s1, ss1, s2, ss2;
-  number num;
+  double ax, ay, u, du, v, vv, dv, t1, t2, t3,
+	 z, zz, cor;
+  mynumber num;
 
   static const int ep = 59768832,      /*  57*16**5   */
 		   em = -59768832;      /* -57*16**5   */
@@ -208,10 +189,8 @@ __ieee754_atan2 (double y, double x)
       if (x > 0)
 	{
 	  double ret;
-	  if ((z = ay / ax) < TWOM1022)
-	    ret = normalized (ax, ay, y, z);
-	  else
-	    ret = signArctan2 (y, z);
+	  z = ay / ax;
+	  ret = signArctan2 (y, z);
 	  if (fabs (ret) < DBL_MIN)
 	    {
 	      double vret = ret ? ret : DBL_MIN;
@@ -270,30 +249,12 @@ __ieee754_atan2 (double y, double x)
 						      + v * (d11.d
 							     + v * d13.d)))));
 
-	      if ((z = u + (zz - u1.d * u)) == u + (zz + u1.d * u))
-		return signArctan2 (y, z);
-
-	      MUL2 (u, du, u, du, v, vv, t1, t2);
-	      s1 = v * (f11.d + v * (f13.d
-				     + v * (f15.d + v * (f17.d + v * f19.d))));
-	      ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
-	      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	      ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
-	      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	      ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
-	      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	      ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
-	      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	      MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
-	      ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
-
-	      if ((z = s1 + (ss1 - u5.d * s1)) == s1 + (ss1 + u5.d * s1))
-		return signArctan2 (y, z);
-
-	      return atan2Mp (x, y, pr);
+	      z = u + zz;
+	      /* Max ULP is 0.504.  */
+	      return signArctan2 (y, z);
 	    }
 
-	  i = (TWO52 + TWO8 * u) - TWO52;
+	  i = (TWO52 + 256 * u) - TWO52;
 	  i -= 16;
 	  t3 = u - cij[i][0].d;
 	  EADD (t3, du, v, dv);
@@ -304,43 +265,9 @@ __ieee754_atan2 (double y, double x)
 				    + v * (cij[i][4].d
 					   + v * (cij[i][5].d
 						  + v * cij[i][6].d))));
-	  if (i < 112)
-	    {
-	      if (i < 48)
-		u9 = u91.d;     /* u < 1/4	*/
-	      else
-		u9 = u92.d;
-	    }           /* 1/4 <= u < 1/2 */
-	  else
-	    {
-	      if (i < 176)
-		u9 = u93.d;     /* 1/2 <= u < 3/4 */
-	      else
-		u9 = u94.d;
-	    }           /* 3/4 <= u <= 1  */
-	  if ((z = t1 + (zz - u9 * t1)) == t1 + (zz + u9 * t1))
-	    return signArctan2 (y, z);
-
-	  t1 = u - hij[i][0].d;
-	  EADD (t1, du, v, vv);
-	  s1 = v * (hij[i][11].d
-		    + v * (hij[i][12].d
-			   + v * (hij[i][13].d
-				  + v * (hij[i][14].d
-					 + v * hij[i][15].d))));
-	  ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
-
-	  if ((z = s2 + (ss2 - ub.d * s2)) == s2 + (ss2 + ub.d * s2))
-	    return signArctan2 (y, z);
-	  return atan2Mp (x, y, pr);
+	  z = t1 + zz;
+	  /* Max ULP is 0.56.  */
+	  return signArctan2 (y, z);
 	}
 
       /* (ii)  x>0, abs(x)<=abs(y):  pi/2-atan(ax/ay) */
@@ -355,31 +282,12 @@ __ieee754_atan2 (double y, double x)
 						    + v * d13.d)))));
 	  ESUB (hpi.d, u, t2, cor);
 	  t3 = ((hpi1.d + cor) - du) - zz;
-	  if ((z = t2 + (t3 - u2.d)) == t2 + (t3 + u2.d))
-	    return signArctan2 (y, z);
-
-	  MUL2 (u, du, u, du, v, vv, t1, t2);
-	  s1 = v * (f11.d
-		    + v * (f13.d
-			   + v * (f15.d + v * (f17.d + v * f19.d))));
-	  ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
-	  ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
-	  SUB2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
-
-	  if ((z = s2 + (ss2 - u6.d)) == s2 + (ss2 + u6.d))
-	    return signArctan2 (y, z);
-	  return atan2Mp (x, y, pr);
+	  z = t2 + t3;
+	  /* Max ULP is 0.501.  */
+	  return signArctan2 (y, z);
 	}
 
-      i = (TWO52 + TWO8 * u) - TWO52;
+      i = (TWO52 + 256 * u) - TWO52;
       i -= 16;
       v = (u - cij[i][0].d) + du;
 
@@ -389,36 +297,9 @@ __ieee754_atan2 (double y, double x)
 				       + v * (cij[i][5].d
 					      + v * cij[i][6].d))));
       t1 = hpi.d - cij[i][1].d;
-      if (i < 112)
-	ua = ua1.d;	/* w <  1/2 */
-      else
-	ua = ua2.d;	/* w >= 1/2 */
-      if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
-	return signArctan2 (y, z);
-
-      t1 = u - hij[i][0].d;
-      EADD (t1, du, v, vv);
-
-      s1 = v * (hij[i][11].d
-		+ v * (hij[i][12].d
-		       + v * (hij[i][13].d
-			      + v * (hij[i][14].d
-				     + v * hij[i][15].d))));
-
-      ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
-      SUB2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
-
-      if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
-	return signArctan2 (y, z);
-      return atan2Mp (x, y, pr);
+      z = t1 + zz;
+      /* Max ULP is 0.503.  */
+      return signArctan2 (y, z);
     }
 
   /* (iii) x<0, abs(x)< abs(y):  pi/2+atan(ax/ay) */
@@ -434,30 +315,12 @@ __ieee754_atan2 (double y, double x)
 					     + v * (d11.d + v * d13.d)))));
 	  EADD (hpi.d, u, t2, cor);
 	  t3 = ((hpi1.d + cor) + du) + zz;
-	  if ((z = t2 + (t3 - u3.d)) == t2 + (t3 + u3.d))
-	    return signArctan2 (y, z);
-
-	  MUL2 (u, du, u, du, v, vv, t1, t2);
-	  s1 = v * (f11.d
-		    + v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
-	  ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
-	  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-	  MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
-	  ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
-	  ADD2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
-
-	  if ((z = s2 + (ss2 - u7.d)) == s2 + (ss2 + u7.d))
-	    return signArctan2 (y, z);
-	  return atan2Mp (x, y, pr);
+	  z = t2 + t3;
+	  /* Max ULP is 0.501.  */
+	  return signArctan2 (y, z);
 	}
 
-      i = (TWO52 + TWO8 * u) - TWO52;
+      i = (TWO52 + 256 * u) - TWO52;
       i -= 16;
       v = (u - cij[i][0].d) + du;
       zz = hpi1.d + v * (cij[i][2].d
@@ -466,34 +329,9 @@ __ieee754_atan2 (double y, double x)
 				       + v * (cij[i][5].d
 					      + v * cij[i][6].d))));
       t1 = hpi.d + cij[i][1].d;
-      if (i < 112)
-	ua = ua1.d;	/* w <  1/2 */
-      else
-	ua = ua2.d;	/* w >= 1/2 */
-      if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
-	return signArctan2 (y, z);
-
-      t1 = u - hij[i][0].d;
-      EADD (t1, du, v, vv);
-      s1 = v * (hij[i][11].d
-		+ v * (hij[i][12].d
-		       + v * (hij[i][13].d
-			      + v * (hij[i][14].d
-				     + v * hij[i][15].d))));
-      ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
-      ADD2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
-
-      if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
-	return signArctan2 (y, z);
-      return atan2Mp (x, y, pr);
+      z = t1 + zz;
+      /* Max ULP is 0.503.  */
+      return signArctan2 (y, z);
     }
 
   /* (iv)  x<0, abs(y)<=abs(x):  pi-atan(ax/ay) */
@@ -506,29 +344,12 @@ __ieee754_atan2 (double y, double x)
 				  + v * (d9.d + v * (d11.d + v * d13.d)))));
       ESUB (opi.d, u, t2, cor);
       t3 = ((opi1.d + cor) - du) - zz;
-      if ((z = t2 + (t3 - u4.d)) == t2 + (t3 + u4.d))
-	return signArctan2 (y, z);
-
-      MUL2 (u, du, u, du, v, vv, t1, t2);
-      s1 = v * (f11.d + v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
-      ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
-      MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-      MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
-      ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
-      SUB2 (opi.d, opi1.d, s1, ss1, s2, ss2, t1, t2);
-
-      if ((z = s2 + (ss2 - u8.d)) == s2 + (ss2 + u8.d))
-	return signArctan2 (y, z);
-      return atan2Mp (x, y, pr);
+      z = t2 + t3;
+      /* Max ULP is 0.501.  */
+      return signArctan2 (y, z);
     }
 
-  i = (TWO52 + TWO8 * u) - TWO52;
+  i = (TWO52 + 256 * u) - TWO52;
   i -= 16;
   v = (u - cij[i][0].d) + du;
   zz = opi1.d - v * (cij[i][2].d
@@ -536,86 +357,11 @@ __ieee754_atan2 (double y, double x)
 			    + v * (cij[i][4].d
 				   + v * (cij[i][5].d + v * cij[i][6].d))));
   t1 = opi.d - cij[i][1].d;
-  if (i < 112)
-    ua = ua1.d;	/* w <  1/2 */
-  else
-    ua = ua2.d;	/* w >= 1/2 */
-  if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
-    return signArctan2 (y, z);
-
-  t1 = u - hij[i][0].d;
-
-  EADD (t1, du, v, vv);
-
-  s1 = v * (hij[i][11].d
-	    + v * (hij[i][12].d
-		   + v * (hij[i][13].d
-			  + v * (hij[i][14].d + v * hij[i][15].d))));
-
-  ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
-  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-  ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
-  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-  ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
-  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-  ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
-  MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
-  ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
-  SUB2 (opi.d, opi1.d, s2, ss2, s1, ss1, t1, t2);
-
-  if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
-    return signArctan2 (y, z);
-  return atan2Mp (x, y, pr);
+  z = t1 + zz;
+  /* Max ULP is 0.502.  */
+  return signArctan2 (y, z);
 }
 
 #ifndef __ieee754_atan2
 libm_alias_finite (__ieee754_atan2, __atan2)
 #endif
-
-/* Treat the Denormalized case */
-static double
-SECTION
-normalized (double ax, double ay, double y, double z)
-{
-  int p;
-  mp_no mpx, mpy, mpz, mperr, mpz2, mpt1;
-  p = 6;
-  __dbl_mp (ax, &mpx, p);
-  __dbl_mp (ay, &mpy, p);
-  __dvd (&mpy, &mpx, &mpz, p);
-  __dbl_mp (ue.d, &mpt1, p);
-  __mul (&mpz, &mpt1, &mperr, p);
-  __sub (&mpz, &mperr, &mpz2, p);
-  __mp_dbl (&mpz2, &z, p);
-  return signArctan2 (y, z);
-}
-
-/* Stage 3: Perform a multi-Precision computation */
-static double
-SECTION
-atan2Mp (double x, double y, const int pr[])
-{
-  double z1, z2;
-  int i, p;
-  mp_no mpx, mpy, mpz, mpz1, mpz2, mperr, mpt1;
-  for (i = 0; i < MM; i++)
-    {
-      p = pr[i];
-      __dbl_mp (x, &mpx, p);
-      __dbl_mp (y, &mpy, p);
-      __mpatan2 (&mpy, &mpx, &mpz, p);
-      __dbl_mp (ud[i].d, &mpt1, p);
-      __mul (&mpz, &mpt1, &mperr, p);
-      __add (&mpz, &mperr, &mpz1, p);
-      __sub (&mpz, &mperr, &mpz2, p);
-      __mp_dbl (&mpz1, &z1, p);
-      __mp_dbl (&mpz2, &z2, p);
-      if (z1 == z2)
-	{
-	  LIBC_PROBE (slowatan2, 4, &p, &x, &y, &z1);
-	  return z1;
-	}
-    }
-  LIBC_PROBE (slowatan2_inexact, 4, &p, &x, &y, &z1);
-  return z1;			/*if impossible to do exact computing */
-}