[base] Improve trigonometric core.

FreeType used to rely on a 24-step iteration CORDIC algorithm to
calculate trigonometric functions and rotate vectors. It turns out
that once the vector is in the right half-plane, the initial rotation
by 63 degrees is not necessary. The algorithm is perfectly capable
to converge to any angle starting from the second 45 degree rotation.
This patch removes the first rotation and makes it a 23-step CORDIC
algorithm.

* src/base/fttrigon.c (FT_TRIG_SCALE, FT_TRIG_COSCALE): Update macro
values.
(ft_trig_pseudo_rotate, ft_trig_pseudo_polarize): Remove initial
rotation.

ChangeLog

@@ -1,3 +1,20 @@
+2012-12-20  Alexei Podtelezhnikov  <apodtele@gmail.com>
+
+	[base] Improve trigonometric core.
+
+	FreeType used to rely on a 24-step iteration CORDIC algorithm to
+	calculate trigonometric functions and rotate vectors. It turns out
+	that once the vector is in the right half-plane, the initial rotation
+	by 63 degrees is not necessary. The algorithm is perfectly capable
+	to converge to any angle starting from the second 45 degree rotation.
+	This patch removes the first rotation and makes it a 23-step CORDIC
+	algorithm.
+
+	* src/base/fttrigon.c (FT_TRIG_SCALE, FT_TRIG_COSCALE): Update macro
+	values.
+	(ft_trig_pseudo_rotate, ft_trig_pseudo_polarize): Remove initial
+	rotation.
+
 2012-12-19  Werner Lemberg  <wl@gnu.org>
 
 	* src/base/ftobjs.c (ft_property_do): Fix compiler warning.

src/base/fttrigon.c

@@ -26,23 +26,23 @@
 #endif
 
 
-  /* the following is 0.2715717684432231 * 2^30 */
-#define FT_TRIG_COSCALE  0x11616E8EUL
+  /* the Cordic shrink factor 0.607252935008887 * 2^32 */
+#define FT_TRIG_SCALE    0x9B74EDA8UL
+
+  /* the following is 0.607252935008887 * 2^30 */
+#define FT_TRIG_COSCALE  0x26DD3B6AUL
 
   /* this table was generated for FT_PI = 180L << 16, i.e. degrees */
 #define FT_TRIG_MAX_ITERS  23
 
   static const FT_Fixed
-  ft_trig_arctan_table[24] =
+  ft_trig_arctan_table[23] =
   {
-    4157273L, 2949120L, 1740967L, 919879L, 466945L, 234379L, 117304L,
-    58666L, 29335L, 14668L, 7334L, 3667L, 1833L, 917L, 458L, 229L, 115L,
+    2949120L, 1740967L, 919879L, 466945L, 234379L, 117304L, 58666L,
+    29335L, 14668L, 7334L, 3667L, 1833L, 917L, 458L, 229L, 115L,
     57L, 29L, 14L, 7L, 4L, 2L, 1L
   };
 
-  /* the Cordic shrink factor, multiplied by 2^32 */
-#define FT_TRIG_SCALE    1166391864UL  /* 0x4585BA38UL */
-
 
 #ifdef FT_LONG64
 
@@ -214,25 +214,9 @@
       theta -= FT_ANGLE_PI;
     }
 
-    /* Initial pseudorotation, with left shift */
     arctanptr = ft_trig_arctan_table;
 
-    if ( theta < 0 )
-    {
-      xtemp  = x + ( y << 1 );
-      y      = y - ( x << 1 );
-      x      = xtemp;
-      theta += *arctanptr++;
-    }
-    else
-    {
-      xtemp  = x - ( y << 1 );
-      y      = y + ( x << 1 );
-      x      = xtemp;
-      theta -= *arctanptr++;
-    }
-
-    /* Subsequent pseudorotations, with right shifts */
+    /* Pseudorotations, with right shifts */
     i = 0;
     do
     {
@@ -283,22 +267,7 @@
 
     arctanptr = ft_trig_arctan_table;
 
-    if ( y > 0 )
-    {
-      xtemp  = x + ( y << 1 );
-      y      = y - ( x << 1 );
-      x      = xtemp;
-      theta += *arctanptr++;
-    }
-    else
-    {
-      xtemp  = x - ( y << 1 );
-      y      = y + ( x << 1 );
-      x      = xtemp;
-      theta -= *arctanptr++;
-    }
-
-    /* Subsequent pseudorotations, with right shifts */
+    /* Pseudorotations, with right shifts */
     i = 0;
     do
     {