/* */
/* FreeType trigonometric functions (body). */
/* */
-/* Copyright 2001, 2002, 2003, 2004, 2005 by */
+/* Copyright 2001-2005, 2012-2014 by */
/* David Turner, Robert Wilhelm, and Werner Lemberg. */
/* */
/* This file is part of the FreeType project, and may only be used, */
/* */
/***************************************************************************/
+ /*************************************************************************/
+ /* */
+ /* This is a fixed-point CORDIC implementation of trigonometric */
+ /* functions as well as transformations between Cartesian and polar */
+ /* coordinates. The angles are represented as 16.16 fixed-point values */
+ /* in degrees, i.e., the angular resolution is 2^-16 degrees. Note that */
+ /* only vectors longer than 2^16*180/pi (or at least 22 bits) on a */
+ /* discrete Cartesian grid can have the same or better angular */
+ /* resolution. Therefore, to maintain this precision, some functions */
+ /* require an interim upscaling of the vectors, whereas others operate */
+ /* with 24-bit long vectors directly. */
+ /* */
+ /*************************************************************************/
#include <ft2build.h>
#include FT_INTERNAL_OBJECTS_H
+#include FT_INTERNAL_CALC_H
#include FT_TRIGONOMETRY_H
- /* the following is 0.2715717684432231 * 2^30 */
-#define FT_TRIG_COSCALE 0x11616E8EUL
+ /* the Cordic shrink factor 0.858785336480436 * 2^32 */
+#define FT_TRIG_SCALE 0xDBD95B16UL
+
+ /* the highest bit in overflow-safe vector components, */
+ /* MSB of 0.858785336480436 * sqrt(0.5) * 2^30 */
+#define FT_TRIG_SAFE_MSB 29
/* 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] =
+ static const FT_Angle
+ ft_trig_arctan_table[] =
{
- 4157273L, 2949120L, 1740967L, 919879L, 466945L, 234379L, 117304L,
- 58666L, 29335L, 14668L, 7334L, 3667L, 1833L, 917L, 458L, 229L, 115L,
+ 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 1166391785UL /* 0x4585BA38UL */
-
-#ifdef FT_CONFIG_HAS_INT64
+#ifdef FT_LONG64
/* multiply a given value by the CORDIC shrink factor */
static FT_Fixed
ft_trig_downscale( FT_Fixed val )
{
- FT_Fixed s;
- FT_Int64 v;
+ FT_Int s = 1;
- s = val;
- val = ( val >= 0 ) ? val : -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;
+ return s < 0 ? -val : val;
}
-#else /* !FT_CONFIG_HAS_INT64 */
+#else /* !FT_LONG64 */
/* multiply a given value by the CORDIC shrink factor */
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;
+
+ if ( val < 0 )
+ {
+ val = -val;
+ s = -1;
+ }
+
+ lo1 = val & 0x0000FFFFU;
+ hi1 = val >> 16;
+ lo2 = FT_TRIG_SCALE & 0x0000FFFFU;
+ hi2 = FT_TRIG_SCALE >> 16;
- s = val;
- val = ( val >= 0 ) ? val : -val;
+ lo = lo1 * lo2;
+ i1 = lo1 * hi2;
+ i2 = lo2 * hi1;
+ hi = hi1 * hi2;
- v1 = (FT_UInt32)val >> 16;
- v2 = (FT_UInt32)(val & 0xFFFFL);
+ /* Check carry overflow of i1 + i2 */
+ i1 += i2;
+ hi += (FT_UInt32)( i1 < i2 ) << 16;
- k1 = (FT_UInt32)FT_TRIG_SCALE >> 16; /* constant */
- k2 = (FT_UInt32)(FT_TRIG_SCALE & 0xFFFFL); /* constant */
+ hi += i1 >> 16;
+ i1 = i1 << 16;
- hi = k1 * v1;
- lo1 = k1 * v2 + k2 * v1; /* can't overflow */
+ /* Check carry overflow of i1 + lo */
+ lo += i1;
+ hi += ( lo < i1 );
- lo2 = ( k2 * v2 ) >> 16;
- lo3 = ( lo1 >= lo2 ) ? lo1 : lo2;
- lo1 += lo2;
+ /* 0x40000000 comes from regression analysis between true */
+ /* and CORDIC hypotenuse, so it minimizes the error */
- hi += lo1 >> 16;
- if ( lo1 < lo3 )
- hi += (FT_UInt32)0x10000UL;
+ /* Check carry overflow of lo + 0x40000000 */
+ lo += 0x40000000UL;
+ hi += ( lo < 0x40000000UL );
val = (FT_Fixed)hi;
- return ( s >= 0 ) ? val : -val;
+ return s < 0 ? -val : val;
}
-#endif /* !FT_CONFIG_HAS_INT64 */
+#endif /* !FT_LONG64 */
+ /* undefined and never called for zero vector */
static FT_Int
ft_trig_prenorm( FT_Vector* vec )
{
- FT_Fixed x, y, z;
- FT_Int shift;
+ FT_Pos x, y;
+ FT_Int shift;
x = vec->x;
y = vec->y;
- z = ( ( x >= 0 ) ? x : - x ) | ( (y >= 0) ? y : -y );
- shift = 0;
-
-#if 1
- /* determine msb bit index in `shift' */
- if ( z >= ( 1L << 16 ) )
- {
- z >>= 16;
- shift += 16;
- }
- if ( z >= ( 1L << 8 ) )
- {
- z >>= 8;
- shift += 8;
- }
- if ( z >= ( 1L << 4 ) )
- {
- z >>= 4;
- shift += 4;
- }
- if ( z >= ( 1L << 2 ) )
- {
- z >>= 2;
- shift += 2;
- }
- if ( z >= ( 1L << 1 ) )
- {
- z >>= 1;
- shift += 1;
- }
+ shift = FT_MSB( FT_ABS( x ) | FT_ABS( y ) );
- if ( shift <= 27 )
+ if ( shift <= FT_TRIG_SAFE_MSB )
{
- shift = 27 - shift;
- vec->x = x << shift;
- vec->y = y << shift;
+ shift = FT_TRIG_SAFE_MSB - shift;
+ vec->x = (FT_Pos)( (FT_ULong)x << shift );
+ vec->y = (FT_Pos)( (FT_ULong)y << shift );
}
else
{
- shift -= 27;
+ shift -= FT_TRIG_SAFE_MSB;
vec->x = x >> shift;
vec->y = y >> shift;
shift = -shift;
}
-#else /* 0 */
-
- if ( z < ( 1L << 27 ) )
- {
- do
- {
- shift++;
- z <<= 1;
- } while ( z < ( 1L << 27 ) );
- vec->x = x << shift;
- vec->y = y << shift;
- }
- else if ( z > ( 1L << 28 ) )
- {
- do
- {
- shift++;
- z >>= 1;
- } while ( z > ( 1L << 28 ) );
-
- vec->x = x >> shift;
- vec->y = y >> shift;
- shift = -shift;
- }
-
-#endif /* 0 */
-
return shift;
}
FT_Angle theta )
{
FT_Int i;
- FT_Fixed x, y, xtemp;
- const FT_Fixed *arctanptr;
+ FT_Fixed x, y, xtemp, b;
+ const FT_Angle *arctanptr;
x = vec->x;
y = vec->y;
- /* Get angle between -90 and 90 degrees */
- while ( theta <= -FT_ANGLE_PI2 )
+ /* Rotate inside [-PI/4,PI/4] sector */
+ while ( theta < -FT_ANGLE_PI4 )
{
- x = -x;
- y = -y;
- theta += FT_ANGLE_PI;
+ xtemp = y;
+ y = -x;
+ x = xtemp;
+ theta += FT_ANGLE_PI2;
}
- while ( theta > FT_ANGLE_PI2 )
+ while ( theta > FT_ANGLE_PI4 )
{
- x = -x;
- y = -y;
- theta -= FT_ANGLE_PI;
+ xtemp = -y;
+ y = x;
+ x = xtemp;
+ theta -= FT_ANGLE_PI2;
}
- /* 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 */
- i = 0;
- do
+ /* Pseudorotations, with right shifts */
+ for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ )
{
if ( theta < 0 )
{
- xtemp = x + ( y >> i );
- y = y - ( x >> i );
+ xtemp = x + ( ( y + b ) >> i );
+ y = y - ( ( x + b ) >> i );
x = xtemp;
theta += *arctanptr++;
}
else
{
- xtemp = x - ( y >> i );
- y = y + ( x >> i );
+ xtemp = x - ( ( y + b ) >> i );
+ y = y + ( ( x + b ) >> i );
x = xtemp;
theta -= *arctanptr++;
}
- } while ( ++i < FT_TRIG_MAX_ITERS );
+ }
vec->x = x;
vec->y = y;
static void
ft_trig_pseudo_polarize( FT_Vector* vec )
{
- FT_Fixed theta;
- FT_Fixed yi, i;
- FT_Fixed x, y;
- const FT_Fixed *arctanptr;
+ FT_Angle theta;
+ FT_Int i;
+ FT_Fixed x, y, xtemp, b;
+ const FT_Angle *arctanptr;
x = vec->x;
y = vec->y;
- /* Get the vector into the right half plane */
- theta = 0;
- if ( x < 0 )
- {
- x = -x;
- y = -y;
- theta = 2 * FT_ANGLE_PI2;
- }
-
- if ( y > 0 )
- theta = - theta;
-
- arctanptr = ft_trig_arctan_table;
-
- if ( y < 0 )
+ /* Get the vector into [-PI/4,PI/4] sector */
+ if ( y > x )
{
- /* Rotate positive */
- yi = y + ( x << 1 );
- x = x - ( y << 1 );
- y = yi;
- theta -= *arctanptr++; /* Subtract angle */
+ if ( y > -x )
+ {
+ theta = FT_ANGLE_PI2;
+ xtemp = y;
+ y = -x;
+ x = xtemp;
+ }
+ else
+ {
+ theta = y > 0 ? FT_ANGLE_PI : -FT_ANGLE_PI;
+ x = -x;
+ y = -y;
+ }
}
else
{
- /* Rotate negative */
- yi = y - ( x << 1 );
- x = x + ( y << 1 );
- y = yi;
- theta += *arctanptr++; /* Add angle */
+ if ( y < -x )
+ {
+ theta = -FT_ANGLE_PI2;
+ xtemp = -y;
+ y = x;
+ x = xtemp;
+ }
+ else
+ {
+ theta = 0;
+ }
}
- i = 0;
- do
+ arctanptr = ft_trig_arctan_table;
+
+ /* Pseudorotations, with right shifts */
+ for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ )
{
- if ( y < 0 )
+ if ( y > 0 )
{
- /* Rotate positive */
- yi = y + ( x >> i );
- x = x - ( y >> i );
- y = yi;
- theta -= *arctanptr++;
+ xtemp = x + ( ( y + b ) >> i );
+ y = y - ( ( x + b ) >> i );
+ x = xtemp;
+ theta += *arctanptr++;
}
else
{
- /* Rotate negative */
- yi = y - ( x >> i );
- x = x + ( y >> i );
- y = yi;
- theta += *arctanptr++;
+ xtemp = x - ( ( y + b ) >> i );
+ y = y + ( ( x + b ) >> i );
+ x = xtemp;
+ theta -= *arctanptr++;
}
- } while ( ++i < FT_TRIG_MAX_ITERS );
+ }
- /* 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;
FT_Vector v;
- v.x = FT_TRIG_COSCALE >> 2;
+ v.x = FT_TRIG_SCALE >> 8;
v.y = 0;
ft_trig_pseudo_rotate( &v, angle );
- return v.x / ( 1 << 12 );
+ return ( v.x + 0x80L ) >> 8;
}
FT_Vector v;
- v.x = FT_TRIG_COSCALE >> 2;
+ v.x = FT_TRIG_SCALE >> 8;
v.y = 0;
ft_trig_pseudo_rotate( &v, angle );
FT_Vector_Unit( FT_Vector* vec,
FT_Angle angle )
{
- vec->x = FT_TRIG_COSCALE >> 2;
+ if ( !vec )
+ return;
+
+ vec->x = FT_TRIG_SCALE >> 8;
vec->y = 0;
ft_trig_pseudo_rotate( vec, angle );
- vec->x >>= 12;
- vec->y >>= 12;
+ vec->x = ( vec->x + 0x80L ) >> 8;
+ vec->y = ( vec->y + 0x80L ) >> 8;
}
FT_Vector v;
+ if ( !vec )
+ return;
+
v.x = vec->x;
v.y = vec->y;
else
{
shift = -shift;
- vec->x = v.x << shift;
- vec->y = v.y << shift;
+ vec->x = (FT_Pos)( (FT_ULong)v.x << shift );
+ vec->y = (FT_Pos)( (FT_ULong)v.y << shift );
}
}
}
FT_Vector v;
+ if ( !vec )
+ return 0;
+
v = *vec;
/* handle trivial cases */
if ( v.x == 0 )
{
- return ( v.y >= 0 ) ? v.y : -v.y;
+ return FT_ABS( v.y );
}
else if ( v.y == 0 )
{
- return ( v.x >= 0 ) ? v.x : -v.x;
+ return FT_ABS( v.x );
}
/* general case */
if ( shift > 0 )
return ( v.x + ( 1 << ( shift - 1 ) ) ) >> shift;
- return v.x << -shift;
+ return (FT_Fixed)( (FT_UInt32)v.x << -shift );
}
FT_Vector v;
+ if ( !vec || !length || !angle )
+ return;
+
v = *vec;
if ( v.x == 0 && v.y == 0 )
v.x = ft_trig_downscale( v.x );
- *length = ( shift >= 0 ) ? ( v.x >> shift ) : ( v.x << -shift );
+ *length = shift >= 0 ? ( v.x >> shift )
+ : (FT_Fixed)( (FT_UInt32)v.x << -shift );
*angle = v.y;
}
FT_Fixed length,
FT_Angle angle )
{
+ if ( !vec )
+ return;
+
vec->x = length;
vec->y = 0;
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