text: /***************************************************************************/
  /*                                                                         */
  /*  fttrigon.c                                                             */
  /*                                                                         */
  /*    FreeType trigonometric functions (body).                             */
  /*                                                                         */
  /*  Copyright 2001-2016 by                                                 */
  /*  David Turner, Robert Wilhelm, and Werner Lemberg.                      */
  /*                                                                         */
  /*  This file is part of the FreeType project, and may only be used,       */
  /*  modified, and distributed under the terms of the FreeType project      */
  /*  license, LICENSE.TXT.  By continuing to use, modify, or distribute     */
  /*  this file you indicate that you have read the license and              */
  /*  understand and accept it fully.                                        */
  /*                                                                         */
  /***************************************************************************/
  
    /*************************************************************************/
    /*                                                                       */
    /* 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 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_Angle
    ft_trig_arctan_table[] =
    {
      1740967L, 919879L, 466945L, 234379L, 117304L, 58666L, 29335L,
      14668L, 7334L, 3667L, 1833L, 917L, 458L, 229L, 115L,
      57L, 29L, 14L, 7L, 4L, 2L, 1L
    };
  
  
  #ifdef FT_LONG64
  
    /* multiply a given value by the CORDIC shrink factor */
    static FT_Fixed
    ft_trig_downscale( FT_Fixed  val )
    {
      FT_Int  s = 1;
  
  
      if ( val < 0 )
      {
         val = -val;
         s = -1;
      }
  
      /* 0x40000000 comes from regression analysis between true */
      /* and CORDIC hypotenuse, so it minimizes the error       */
      val = (FT_Fixed)(
              ( (FT_UInt64)val * FT_TRIG_SCALE + 0x40000000UL ) >> 32 );
  
      return s < 0 ? -val : val;
    }
  
  #else /* !FT_LONG64 */
  
    /* multiply a given value by the CORDIC shrink factor */
    static FT_Fixed
    ft_trig_downscale( FT_Fixed  val )
    {
      FT_Int     s = 1;
      FT_UInt32  lo1, hi1, lo2, hi2, lo, hi, i1, i2;
  
  
      if ( val < 0 )
      {
         val = -val;
         s = -1;
      }
  
      lo1 = (FT_UInt32)val & 0x0000FFFFU;
      hi1 = (FT_UInt32)val >> 16;
      lo2 = FT_TRIG_SCALE & 0x0000FFFFU;
      hi2 = FT_TRIG_SCALE >> 16;
  
      lo = lo1 * lo2;
      i1 = lo1 * hi2;
      i2 = lo2 * hi1;
      hi = hi1 * hi2;
  
      /* Check carry overflow of i1 + i2 */
      i1 += i2;
      hi += (FT_UInt32)( i1 < i2 ) << 16;
  
      hi += i1 >> 16;
      i1  = i1 << 16;
  
      /* 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 += 0x40000000UL;
      hi += ( lo < 0x40000000UL );
  
      val = (FT_Fixed)hi;
  
      return s < 0 ? -val : val;
    }
  
  #endif /* !FT_LONG64 */
  
  
    /* undefined and never called for zero vector */
    static FT_Int
    ft_trig_prenorm( FT_Vector*  vec )
    {
      FT_Pos  x, y;
      FT_Int  shift;
  
  
      x = vec->x;
      y = vec->y;
  
      shift = FT_MSB( (FT_UInt32)( FT_ABS( x ) | FT_ABS( y ) ) );
  
      if ( shift <= FT_TRIG_SAFE_MSB )
      {
        shift  = FT_TRIG_SAFE_MSB - shift;
        vec->x = (FT_Pos)( (FT_ULong)x << shift );
        vec->y = (FT_Pos)( (FT_ULong)y << shift );
      }
      else
      {
        shift -= FT_TRIG_SAFE_MSB;
        vec->x = x >> shift;
        vec->y = y >> shift;
        shift  = -shift;
      }
  
      return shift;
    }
  
  
    static void
    ft_trig_pseudo_rotate( FT_Vector*  vec,
                           FT_Angle    theta )
    {
      FT_Int           i;
      FT_Fixed         x, y, xtemp, b;
      const FT_Angle  *arctanptr;
  
  
      x = vec->x;
      y = vec->y;
  
      /* Rotate inside [-PI/4,PI/4] sector */
      while ( theta < -FT_ANGLE_PI4 )
      {
        xtemp  =  y;
        y      = -x;
        x      =  xtemp;
        theta +=  FT_ANGLE_PI2;
      }
  
      while ( theta > FT_ANGLE_PI4 )
      {
        xtemp  = -y;
        y      =  x;
        x      =  xtemp;
        theta -=  FT_ANGLE_PI2;
      }
  
      arctanptr = ft_trig_arctan_table;
  
      /* Pseudorotations, with right shifts */
      for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ )
      {
        if ( theta < 0 )
        {
          xtemp  = x + ( ( y + b ) >> i );
          y      = y - ( ( x + b ) >> i );
          x      = xtemp;
          theta += *arctanptr++;
        }
        else
        {
          xtemp  = x - ( ( y + b ) >> i );
          y      = y + ( ( x + b ) >> i );
          x      = xtemp;
          theta -= *arctanptr++;
        }
      }
  
      vec->x = x;
      vec->y = y;
    }
  
  
    static void
    ft_trig_pseudo_polarize( FT_Vector*  vec )
    {
      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 [-PI/4,PI/4] sector */
      if ( y > x )
      {
        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
      {
        if ( y < -x )
        {
          theta = -FT_ANGLE_PI2;
          xtemp = -y;
          y     =  x;
          x     =  xtemp;
        }
        else
        {
          theta = 0;
        }
      }
  
      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 )
        {
          xtemp  = x + ( ( y + b ) >> i );
          y      = y - ( ( x + b ) >> i );
          x      = xtemp;
          theta += *arctanptr++;
        }
        else
        {
          xtemp  = x - ( ( y + b ) >> i );
          y      = y + ( ( x + b ) >> i );
          x      = xtemp;
          theta -= *arctanptr++;
        }
      }
  
      /* 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, 16 );
      else
        theta = -FT_PAD_ROUND( -theta, 16 );
  
      vec->x = x;
      vec->y = theta;
    }
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( FT_Fixed )
    FT_Cos( FT_Angle  angle )
    {
      FT_Vector  v;
  
  
      FT_Vector_Unit( &v, angle );
  
      return v.x;
    }
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( FT_Fixed )
    FT_Sin( FT_Angle  angle )
    {
      FT_Vector  v;
  
  
      FT_Vector_Unit( &v, angle );
  
      return v.y;
    }
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( FT_Fixed )
    FT_Tan( FT_Angle  angle )
    {
      FT_Vector  v;
  
  
      FT_Vector_Unit( &v, angle );
  
      return FT_DivFix( v.y, v.x );
    }
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( FT_Angle )
    FT_Atan2( FT_Fixed  dx,
              FT_Fixed  dy )
    {
      FT_Vector  v;
  
  
      if ( dx == 0 && dy == 0 )
        return 0;
  
      v.x = dx;
      v.y = dy;
      ft_trig_prenorm( &v );
      ft_trig_pseudo_polarize( &v );
  
      return v.y;
    }
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( void )
    FT_Vector_Unit( FT_Vector*  vec,
                    FT_Angle    angle )
    {
      if ( !vec )
        return;
  
      vec->x = FT_TRIG_SCALE >> 8;
      vec->y = 0;
      ft_trig_pseudo_rotate( vec, angle );
      vec->x = ( vec->x + 0x80L ) >> 8;
      vec->y = ( vec->y + 0x80L ) >> 8;
    }
  
  
    /* these macros return 0 for positive numbers,
       and -1 for negative ones */
  #define FT_SIGN_LONG( x )   ( (x) >> ( FT_SIZEOF_LONG * 8 - 1 ) )
  #define FT_SIGN_INT( x )    ( (x) >> ( FT_SIZEOF_INT * 8 - 1 ) )
  #define FT_SIGN_INT32( x )  ( (x) >> 31 )
  #define FT_SIGN_INT16( x )  ( (x) >> 15 )
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( void )
    FT_Vector_Rotate( FT_Vector*  vec,
                      FT_Angle    angle )
    {
      FT_Int     shift;
      FT_Vector  v;
  
  
      if ( !vec || !angle )
        return;
  
      v = *vec;
  
      if ( v.x == 0 && v.y == 0 )
        return;
  
      shift = ft_trig_prenorm( &v );
      ft_trig_pseudo_rotate( &v, angle );
      v.x = ft_trig_downscale( v.x );
      v.y = ft_trig_downscale( v.y );
  
      if ( shift > 0 )
      {
        FT_Int32  half = (FT_Int32)1L << ( shift - 1 );
  
  
        vec->x = ( v.x + half + FT_SIGN_LONG( v.x ) ) >> shift;
        vec->y = ( v.y + half + FT_SIGN_LONG( v.y ) ) >> shift;
      }
      else
      {
        shift  = -shift;
        vec->x = (FT_Pos)( (FT_ULong)v.x << shift );
        vec->y = (FT_Pos)( (FT_ULong)v.y << shift );
      }
    }
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( FT_Fixed )
    FT_Vector_Length( FT_Vector*  vec )
    {
      FT_Int     shift;
      FT_Vector  v;
  
  
      if ( !vec )
        return 0;
  
      v = *vec;
  
      /* handle trivial cases */
      if ( v.x == 0 )
      {
        return FT_ABS( v.y );
      }
      else if ( v.y == 0 )
      {
        return FT_ABS( v.x );
      }
  
      /* general case */
      shift = ft_trig_prenorm( &v );
      ft_trig_pseudo_polarize( &v );
  
      v.x = ft_trig_downscale( v.x );
  
      if ( shift > 0 )
        return ( v.x + ( 1L << ( shift - 1 ) ) ) >> shift;
  
      return (FT_Fixed)( (FT_UInt32)v.x << -shift );
    }
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( void )
    FT_Vector_Polarize( FT_Vector*  vec,
                        FT_Fixed   *length,
                        FT_Angle   *angle )
    {
      FT_Int     shift;
      FT_Vector  v;
  
  
      if ( !vec || !length || !angle )
        return;
  
      v = *vec;
  
      if ( v.x == 0 && v.y == 0 )
        return;
  
      shift = ft_trig_prenorm( &v );
      ft_trig_pseudo_polarize( &v );
  
      v.x = ft_trig_downscale( v.x );
  
      *length = shift >= 0 ?                      ( v.x >>  shift )
                           : (FT_Fixed)( (FT_UInt32)v.x << -shift );
      *angle  = v.y;
    }
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( void )
    FT_Vector_From_Polar( FT_Vector*  vec,
                          FT_Fixed    length,
                          FT_Angle    angle )
    {
      if ( !vec )
        return;
  
      vec->x = length;
      vec->y = 0;
  
      FT_Vector_Rotate( vec, angle );
    }
  
  
    /* documentation is in fttrigon.h */
  
    FT_EXPORT_DEF( FT_Angle )
    FT_Angle_Diff( FT_Angle  angle1,
                   FT_Angle  angle2 )
    {
      FT_Angle  delta = angle2 - angle1;
  
  
      while ( delta <= -FT_ANGLE_PI )
        delta += FT_ANGLE_2PI;
  
      while ( delta > FT_ANGLE_PI )
        delta -= FT_ANGLE_2PI;
  
      return delta;
    }
  
  
  /* END */
  