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[base] CORDIC improvements.

The scaling between the hypotenuse and its CORDIC approximation is
based on regression analysis. The smaller padding for `theta' is
justifed by its maximum error of less than 6.

* src/base/fttrigon.c (ft_trig_downscale): Borrow code from
./ftcalc.c (ft_multo64), change linear intercept.
(ft_trig_pseudo_polarize): Decrease `theta' padding.

git/fs: mount .git/fs: mount/attach disallowed

--- a/ChangeLog

+++ b/ChangeLog

@@ -1,3 +1,15 @@

+2014-11-10  Alexei Podtelezhnikov  <[email protected]>

+

+	[base] CORDIC improvements.

+

+	The scaling between the hypotenuse and its CORDIC approximation is

+	based on regression analysis. The smaller padding for `theta' is

+	justifed by its maximum error of less than 6.

+

+	* src/base/fttrigon.c (ft_trig_downscale): Borrow code from

+	./ftcalc.c (ft_multo64), change linear intercept.

+	(ft_trig_pseudo_polarize): Decrease `theta' padding.

+

 2014-11-09  Werner Lemberg  <[email protected]>

 	* src/base/ftstroke.c (ft_stroker_inside): Fix border intersections.

--- a/src/base/fttrigon.c

+++ b/src/base/fttrigon.c

@@ -60,15 +60,18 @@

   static FT_Fixed

   ft_trig_downscale( FT_Fixed  val )

   {

-    FT_Fixed  s;

-    FT_Int64  v;

+    FT_Int  s = 1;

-    s   = val;

-    val = FT_ABS( val );

+    if ( val < 0 )

+    {

+       val = -val;

+       s = -1;

+    }

-    v   = ( val * (FT_Int64)FT_TRIG_SCALE ) + 0x100000000UL;

-    val = (FT_Fixed)( v >> 32 );

+    /* 0x40000000 comes from regression analysis between true */

+    /* and CORDIC hypotenuse, so it minimizes the error       */

+    val = (FT_Fixed)( ( (FT_Int64)val * FT_TRIG_SCALE + 0x40000000UL ) >> 32 );

     return ( s >= 0 ) ? val : -val;

   }

@@ -79,30 +82,44 @@

   static FT_Fixed

   ft_trig_downscale( FT_Fixed  val )

   {

-    FT_Fixed   s;

-    FT_UInt32  v1, v2, k1, k2, hi, lo1, lo2, lo3;

+    FT_Int     s = 1;

+    FT_UInt32  lo1, hi1, lo2, hi2, lo, hi, i1, i2;

-    s   = val;

-    val = FT_ABS( val );

+    if ( val < 0 )

+    {

+       val = -val;

+       s = -1;

+    }

-    v1 = (FT_UInt32)val >> 16;

-    v2 = (FT_UInt32)( val & 0xFFFFL );

+    lo1 = val & 0x0000FFFFU;

+    hi1 = val >> 16;

+    lo2 = FT_TRIG_SCALE & 0x0000FFFFU;

+    hi2 = FT_TRIG_SCALE >> 16;

-    k1 = (FT_UInt32)FT_TRIG_SCALE >> 16;           /* constant */

-    k2 = (FT_UInt32)( FT_TRIG_SCALE & 0xFFFFL );   /* constant */

+    lo = lo1 * lo2;

+    i1 = lo1 * hi2;

+    i2 = lo2 * hi1;

+    hi = hi1 * hi2;

-    hi   = k1 * v1;

-    lo1  = k1 * v2 + k2 * v1;       /* can't overflow */

+    /* Check carry overflow of i1 + i2 */

+    i1 += i2;

+    hi += (FT_UInt32)( i1 < i2 ) << 16;

-    lo2  = ( k2 * v2 ) >> 16;

-    lo3  = FT_MAX( lo1, lo2 );

-    lo1 += lo2;

+    hi += i1 >> 16;

+    i1  = i1 << 16;

-    hi  += lo1 >> 16;

-    if ( lo1 < lo3 )

-      hi += (FT_UInt32)0x10000UL;

+    /* Check carry overflow of i1 + lo */

+    lo += i1;

+    hi += ( lo < i1 );

+    /* 0x40000000 comes from regression analysis between true */

+    /* and CORDIC hypotenuse, so it minimizes the error       */

+

+    /* Check carry overflow of lo + 0x40000000 */

+    lo += 0x40000000U;

+    hi += ( lo < 0x40000000U );

+

     val  = (FT_Fixed)hi;

     return ( s >= 0 ) ? val : -val;

@@ -111,7 +128,7 @@

 #endif /* !FT_LONG64 */

-  /* undefined and never called for zero vector */  

+  /* undefined and never called for zero vector */

   static FT_Int

   ft_trig_prenorm( FT_Vector*  vec )

   {

@@ -262,11 +279,12 @@

       }

     }

-    /* round theta */

+    /* round theta to acknowledge its error that mostly comes */

+    /* from accumulated rounding errors in the arctan table   */

     if ( theta >= 0 )

-      theta = FT_PAD_ROUND( theta, 32 );

+      theta = FT_PAD_ROUND( theta, 16 );

     else

-      theta = -FT_PAD_ROUND( -theta, 32 );

+      theta = -FT_PAD_ROUND( -theta, 16 );

     vec->x = x;

     vec->y = theta;