freetype2/src/base/ftcalc.c

/****************************************************************************
 *
 * ftcalc.c
 *
 *   Arithmetic computations (body).
 *
 * Copyright 1996-2018 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.
 *
 */

  /**************************************************************************
   *
   * Support for 1-complement arithmetic has been totally dropped in this
   * release.  You can still write your own code if you need it.
   *
   */

  /**************************************************************************
   *
   * Implementing basic computation routines.
   *
   * FT_MulDiv(), FT_MulFix(), FT_DivFix(), FT_RoundFix(), FT_CeilFix(),
   * and FT_FloorFix() are declared in freetype.h.
   *
   */


#include <ft2build.h>
#include FT_GLYPH_H
#include FT_TRIGONOMETRY_H
#include FT_INTERNAL_CALC_H
#include FT_INTERNAL_DEBUG_H
#include FT_INTERNAL_OBJECTS_H


#ifdef FT_MULFIX_ASSEMBLER
#undef FT_MulFix
#endif

/* we need to emulate a 64-bit data type if a real one isn't available */

#ifndef FT_LONG64

  typedef struct  FT_Int64_
  {
    FT_UInt32  lo;
    FT_UInt32  hi;

  } FT_Int64;

#endif /* !FT_LONG64 */


  /**************************************************************************
   *
   * The macro FT_COMPONENT is used in trace mode.  It is an implicit
   * parameter of the FT_TRACE() and FT_ERROR() macros, used to print/log
   * messages during execution.
   */
#undef  FT_COMPONENT
#define FT_COMPONENT  trace_calc


  /* transfer sign, leaving a positive number;                        */
  /* we need an unsigned value to safely negate INT_MIN (or LONG_MIN) */
#define FT_MOVE_SIGN( x, x_unsigned, s ) \
  FT_BEGIN_STMNT                         \
    if ( x < 0 )                         \
    {                                    \
      x_unsigned = 0U - (x_unsigned);    \
      s          = -s;                   \
    }                                    \
  FT_END_STMNT

  /* The following three functions are available regardless of whether */
  /* FT_LONG64 is defined.                                             */

  /* documentation is in freetype.h */

  FT_EXPORT_DEF( FT_Fixed )
  FT_RoundFix( FT_Fixed  a )
  {
    return ( ADD_LONG( a, 0x8000L - ( a < 0 ) ) ) & ~0xFFFFL;
  }


  /* documentation is in freetype.h */

  FT_EXPORT_DEF( FT_Fixed )
  FT_CeilFix( FT_Fixed  a )
  {
    return ( ADD_LONG( a, 0xFFFFL ) ) & ~0xFFFFL;
  }


  /* documentation is in freetype.h */

  FT_EXPORT_DEF( FT_Fixed )
  FT_FloorFix( FT_Fixed  a )
  {
    return a & ~0xFFFFL;
  }

#ifndef FT_MSB

  FT_BASE_DEF ( FT_Int )
  FT_MSB( FT_UInt32 z )
  {
    FT_Int  shift = 0;


    /* determine msb bit index in `shift' */
    if ( z & 0xFFFF0000UL )
    {
      z     >>= 16;
      shift  += 16;
    }
    if ( z & 0x0000FF00UL )
    {
      z     >>= 8;
      shift  += 8;
    }
    if ( z & 0x000000F0UL )
    {
      z     >>= 4;
      shift  += 4;
    }
    if ( z & 0x0000000CUL )
    {
      z     >>= 2;
      shift  += 2;
    }
    if ( z & 0x00000002UL )
    {
   /* z     >>= 1; */
      shift  += 1;
    }

    return shift;
  }

#endif /* !FT_MSB */


  /* documentation is in ftcalc.h */

  FT_BASE_DEF( FT_Fixed )
  FT_Hypot( FT_Fixed  x,
            FT_Fixed  y )
  {
    FT_Vector  v;


    v.x = x;
    v.y = y;

    return FT_Vector_Length( &v );
  }


#ifdef FT_LONG64


  /* documentation is in freetype.h */

  FT_EXPORT_DEF( FT_Long )
  FT_MulDiv( FT_Long  a_,
             FT_Long  b_,
             FT_Long  c_ )
  {
    FT_Int     s = 1;
    FT_UInt64  a, b, c, d;
    FT_Long    d_;


    a = (FT_UInt64)a_;
    b = (FT_UInt64)b_;
    c = (FT_UInt64)c_;

    FT_MOVE_SIGN( a_, a, s );
    FT_MOVE_SIGN( b_, b, s );
    FT_MOVE_SIGN( c_, c, s );

    d = c > 0 ? ( a * b + ( c >> 1 ) ) / c
              : 0x7FFFFFFFUL;

    d_ = (FT_Long)d;

    return s < 0 ? NEG_LONG( d_ ) : d_;
  }


  /* documentation is in ftcalc.h */

  FT_BASE_DEF( FT_Long )
  FT_MulDiv_No_Round( FT_Long  a_,
                      FT_Long  b_,
                      FT_Long  c_ )
  {
    FT_Int     s = 1;
    FT_UInt64  a, b, c, d;
    FT_Long    d_;


    a = (FT_UInt64)a_;
    b = (FT_UInt64)b_;
    c = (FT_UInt64)c_;

    FT_MOVE_SIGN( a_, a, s );
    FT_MOVE_SIGN( b_, b, s );
    FT_MOVE_SIGN( c_, c, s );

    d = c > 0 ? a * b / c
              : 0x7FFFFFFFUL;

    d_ = (FT_Long)d;

    return s < 0 ? NEG_LONG( d_ ) : d_;
  }


  /* documentation is in freetype.h */

  FT_EXPORT_DEF( FT_Long )
  FT_MulFix( FT_Long  a_,
             FT_Long  b_ )
  {
#ifdef FT_MULFIX_ASSEMBLER

    return FT_MULFIX_ASSEMBLER( (FT_Int32)a_, (FT_Int32)b_ );

#else

    FT_Int64  ab = (FT_Int64)a_ * (FT_Int64)b_;

    /* this requires arithmetic right shift of signed numbers */
    return (FT_Long)( ( ab + 0x8000L - ( ab < 0 ) ) >> 16 );

#endif /* FT_MULFIX_ASSEMBLER */
  }


  /* documentation is in freetype.h */

  FT_EXPORT_DEF( FT_Long )
  FT_DivFix( FT_Long  a_,
             FT_Long  b_ )
  {
    FT_Int     s = 1;
    FT_UInt64  a, b, q;
    FT_Long    q_;


    a = (FT_UInt64)a_;
    b = (FT_UInt64)b_;

    FT_MOVE_SIGN( a_, a, s );
    FT_MOVE_SIGN( b_, b, s );

    q = b > 0 ? ( ( a << 16 ) + ( b >> 1 ) ) / b
              : 0x7FFFFFFFUL;

    q_ = (FT_Long)q;

    return s < 0 ? NEG_LONG( q_ ) : q_;
  }


#else /* !FT_LONG64 */


  static void
  ft_multo64( FT_UInt32  x,
              FT_UInt32  y,
              FT_Int64  *z )
  {
    FT_UInt32  lo1, hi1, lo2, hi2, lo, hi, i1, i2;


    lo1 = x & 0x0000FFFFU;  hi1 = x >> 16;
    lo2 = y & 0x0000FFFFU;  hi2 = y >> 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 );

    z->lo = lo;
    z->hi = hi;
  }


  static FT_UInt32
  ft_div64by32( FT_UInt32  hi,
                FT_UInt32  lo,
                FT_UInt32  y )
  {
    FT_UInt32  r, q;
    FT_Int     i;


    if ( hi >= y )
      return (FT_UInt32)0x7FFFFFFFL;

    /* We shift as many bits as we can into the high register, perform     */
    /* 32-bit division with modulo there, then work through the remaining  */
    /* bits with long division. This optimization is especially noticeable */
    /* for smaller dividends that barely use the high register.            */

    i = 31 - FT_MSB( hi );
    r = ( hi << i ) | ( lo >> ( 32 - i ) ); lo <<= i; /* left 64-bit shift */
    q = r / y;
    r -= q * y;   /* remainder */

    i = 32 - i;   /* bits remaining in low register */
    do
    {
      q <<= 1;
      r   = ( r << 1 ) | ( lo >> 31 ); lo <<= 1;

      if ( r >= y )
      {
        r -= y;
        q |= 1;
      }
    } while ( --i );

    return q;
  }


  static void
  FT_Add64( FT_Int64*  x,
            FT_Int64*  y,
            FT_Int64  *z )
  {
    FT_UInt32  lo, hi;


    lo = x->lo + y->lo;
    hi = x->hi + y->hi + ( lo < x->lo );

    z->lo = lo;
    z->hi = hi;
  }


  /*  The FT_MulDiv function has been optimized thanks to ideas from     */
  /*  Graham Asher and Alexei Podtelezhnikov.  The trick is to optimize  */
  /*  a rather common case when everything fits within 32-bits.          */
  /*                                                                     */
  /*  We compute 'a*b+c/2', then divide it by 'c' (all positive values). */
  /*                                                                     */
  /*  The product of two positive numbers never exceeds the square of    */
  /*  its mean values.  Therefore, we always avoid the overflow by       */
  /*  imposing                                                           */
  /*                                                                     */
  /*    (a + b) / 2 <= sqrt(X - c/2)    ,                                */
  /*                                                                     */
  /*  where X = 2^32 - 1, the maximum unsigned 32-bit value, and using   */
  /*  unsigned arithmetic.  Now we replace `sqrt' with a linear function */
  /*  that is smaller or equal for all values of c in the interval       */
  /*  [0;X/2]; it should be equal to sqrt(X) and sqrt(3X/4) at the       */
  /*  endpoints.  Substituting the linear solution and explicit numbers  */
  /*  we get                                                             */
  /*                                                                     */
  /*    a + b <= 131071.99 - c / 122291.84    .                          */
  /*                                                                     */
  /*  In practice, we should use a faster and even stronger inequality   */
  /*                                                                     */
  /*    a + b <= 131071 - (c >> 16)                                      */
  /*                                                                     */
  /*  or, alternatively,                                                 */
  /*                                                                     */
  /*    a + b <= 129894 - (c >> 17)    .                                 */
  /*                                                                     */
  /*  FT_MulFix, on the other hand, is optimized for a small value of    */
  /*  the first argument, when the second argument can be much larger.   */
  /*  This can be achieved by scaling the second argument and the limit  */
  /*  in the above inequalities.  For example,                           */
  /*                                                                     */
  /*    a + (b >> 8) <= (131071 >> 4)                                    */
  /*                                                                     */
  /*  covers the practical range of use. The actual test below is a bit  */
  /*  tighter to avoid the border case overflows.                        */
  /*                                                                     */
  /*  In the case of FT_DivFix, the exact overflow check                 */
  /*                                                                     */
  /*    a << 16 <= X - c/2                                               */
  /*                                                                     */
  /*  is scaled down by 2^16 and we use                                  */
  /*                                                                     */
  /*    a <= 65535 - (c >> 17)    .                                      */

  /* documentation is in freetype.h */

  FT_EXPORT_DEF( FT_Long )
  FT_MulDiv( FT_Long  a_,
             FT_Long  b_,
             FT_Long  c_ )
  {
    FT_Int     s = 1;
    FT_UInt32  a, b, c;


    /* XXX: this function does not allow 64-bit arguments */

    a = (FT_UInt32)a_;
    b = (FT_UInt32)b_;
    c = (FT_UInt32)c_;

    FT_MOVE_SIGN( a_, a, s );
    FT_MOVE_SIGN( b_, b, s );
    FT_MOVE_SIGN( c_, c, s );

    if ( c == 0 )
      a = 0x7FFFFFFFUL;

    else if ( a + b <= 129894UL - ( c >> 17 ) )
      a = ( a * b + ( c >> 1 ) ) / c;

    else
    {
      FT_Int64  temp, temp2;


      ft_multo64( a, b, &temp );

      temp2.hi = 0;
      temp2.lo = c >> 1;

      FT_Add64( &temp, &temp2, &temp );

      /* last attempt to ditch long division */
      a = ( temp.hi == 0 ) ? temp.lo / c
                           : ft_div64by32( temp.hi, temp.lo, c );
    }

    a_ = (FT_Long)a;

    return s < 0 ? NEG_LONG( a_ ) : a_;
  }


  FT_BASE_DEF( FT_Long )
  FT_MulDiv_No_Round( FT_Long  a_,
                      FT_Long  b_,
                      FT_Long  c_ )
  {
    FT_Int     s = 1;
    FT_UInt32  a, b, c;


    /* XXX: this function does not allow 64-bit arguments */

    a = (FT_UInt32)a_;
    b = (FT_UInt32)b_;
    c = (FT_UInt32)c_;

    FT_MOVE_SIGN( a_, a, s );
    FT_MOVE_SIGN( b_, b, s );
    FT_MOVE_SIGN( c_, c, s );

    if ( c == 0 )
      a = 0x7FFFFFFFUL;

    else if ( a + b <= 131071UL )
      a = a * b / c;

    else
    {
      FT_Int64  temp;


      ft_multo64( a, b, &temp );

      /* last attempt to ditch long division */
      a = ( temp.hi == 0 ) ? temp.lo / c
                           : ft_div64by32( temp.hi, temp.lo, c );
    }

    a_ = (FT_Long)a;

    return s < 0 ? NEG_LONG( a_ ) : a_;
  }


  /* documentation is in freetype.h */

  FT_EXPORT_DEF( FT_Long )
  FT_MulFix( FT_Long  a_,
             FT_Long  b_ )
  {
#ifdef FT_MULFIX_ASSEMBLER

    return FT_MULFIX_ASSEMBLER( a_, b_ );

#elif 0

    /*
     * This code is nonportable.  See comment below.
     *
     * However, on a platform where right-shift of a signed quantity fills
     * the leftmost bits by copying the sign bit, it might be faster.
     */

    FT_Long    sa, sb;
    FT_UInt32  a, b;


    /*
     * This is a clever way of converting a signed number `a' into its
     * absolute value (stored back into `a') and its sign.  The sign is
     * stored in `sa'; 0 means `a' was positive or zero, and -1 means `a'
     * was negative.  (Similarly for `b' and `sb').
     *
     * Unfortunately, it doesn't work (at least not portably).
     *
     * It makes the assumption that right-shift on a negative signed value
     * fills the leftmost bits by copying the sign bit.  This is wrong.
     * According to K&R 2nd ed, section `A7.8 Shift Operators' on page 206,
     * the result of right-shift of a negative signed value is
     * implementation-defined.  At least one implementation fills the
     * leftmost bits with 0s (i.e., it is exactly the same as an unsigned
     * right shift).  This means that when `a' is negative, `sa' ends up
     * with the value 1 rather than -1.  After that, everything else goes
     * wrong.
     */
    sa = ( a_ >> ( sizeof ( a_ ) * 8 - 1 ) );
    a  = ( a_ ^ sa ) - sa;
    sb = ( b_ >> ( sizeof ( b_ ) * 8 - 1 ) );
    b  = ( b_ ^ sb ) - sb;

    a = (FT_UInt32)a_;
    b = (FT_UInt32)b_;

    if ( a + ( b >> 8 ) <= 8190UL )
      a = ( a * b + 0x8000U ) >> 16;
    else
    {
      FT_UInt32  al = a & 0xFFFFUL;


      a = ( a >> 16 ) * b + al * ( b >> 16 ) +
          ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 );
    }

    sa ^= sb;
    a   = ( a ^ sa ) - sa;

    return (FT_Long)a;

#else /* 0 */

    FT_Int     s = 1;
    FT_UInt32  a, b;


    /* XXX: this function does not allow 64-bit arguments */

    a = (FT_UInt32)a_;
    b = (FT_UInt32)b_;

    FT_MOVE_SIGN( a_, a, s );
    FT_MOVE_SIGN( b_, b, s );

    if ( a + ( b >> 8 ) <= 8190UL )
      a = ( a * b + 0x8000UL ) >> 16;
    else
    {
      FT_UInt32  al = a & 0xFFFFUL;


      a = ( a >> 16 ) * b + al * ( b >> 16 ) +
          ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 );
    }

    a_ = (FT_Long)a;

    return s < 0 ? NEG_LONG( a_ ) : a_;

#endif /* 0 */

  }


  /* documentation is in freetype.h */

  FT_EXPORT_DEF( FT_Long )
  FT_DivFix( FT_Long  a_,
             FT_Long  b_ )
  {
    FT_Int     s = 1;
    FT_UInt32  a, b, q;
    FT_Long    q_;


    /* XXX: this function does not allow 64-bit arguments */

    a = (FT_UInt32)a_;
    b = (FT_UInt32)b_;

    FT_MOVE_SIGN( a_, a, s );
    FT_MOVE_SIGN( b_, b, s );

    if ( b == 0 )
    {
      /* check for division by 0 */
      q = 0x7FFFFFFFUL;
    }
    else if ( a <= 65535UL - ( b >> 17 ) )
    {
      /* compute result directly */
      q = ( ( a << 16 ) + ( b >> 1 ) ) / b;
    }
    else
    {
      /* we need more bits; we have to do it by hand */
      FT_Int64  temp, temp2;


      temp.hi  = a >> 16;
      temp.lo  = a << 16;
      temp2.hi = 0;
      temp2.lo = b >> 1;

      FT_Add64( &temp, &temp2, &temp );
      q = ft_div64by32( temp.hi, temp.lo, b );
    }

    q_ = (FT_Long)q;

    return s < 0 ? NEG_LONG( q_ ) : q_;
  }


#endif /* !FT_LONG64 */


  /* documentation is in ftglyph.h */

  FT_EXPORT_DEF( void )
  FT_Matrix_Multiply( const FT_Matrix*  a,
                      FT_Matrix        *b )
  {
    FT_Fixed  xx, xy, yx, yy;


    if ( !a || !b )
      return;

    xx = ADD_LONG( FT_MulFix( a->xx, b->xx ),
                   FT_MulFix( a->xy, b->yx ) );
    xy = ADD_LONG( FT_MulFix( a->xx, b->xy ),
                   FT_MulFix( a->xy, b->yy ) );
    yx = ADD_LONG( FT_MulFix( a->yx, b->xx ),
                   FT_MulFix( a->yy, b->yx ) );
    yy = ADD_LONG( FT_MulFix( a->yx, b->xy ),
                   FT_MulFix( a->yy, b->yy ) );

    b->xx = xx;
    b->xy = xy;
    b->yx = yx;
    b->yy = yy;
  }


  /* documentation is in ftglyph.h */

  FT_EXPORT_DEF( FT_Error )
  FT_Matrix_Invert( FT_Matrix*  matrix )
  {
    FT_Pos  delta, xx, yy;


    if ( !matrix )
      return FT_THROW( Invalid_Argument );

    /* compute discriminant */
    delta = FT_MulFix( matrix->xx, matrix->yy ) -
            FT_MulFix( matrix->xy, matrix->yx );

    if ( !delta )
      return FT_THROW( Invalid_Argument );  /* matrix can't be inverted */

    matrix->xy = - FT_DivFix( matrix->xy, delta );
    matrix->yx = - FT_DivFix( matrix->yx, delta );

    xx = matrix->xx;
    yy = matrix->yy;

    matrix->xx = FT_DivFix( yy, delta );
    matrix->yy = FT_DivFix( xx, delta );

    return FT_Err_Ok;
  }


  /* documentation is in ftcalc.h */

  FT_BASE_DEF( void )
  FT_Matrix_Multiply_Scaled( const FT_Matrix*  a,
                             FT_Matrix        *b,
                             FT_Long           scaling )
  {
    FT_Fixed  xx, xy, yx, yy;

    FT_Long   val = 0x10000L * scaling;


    if ( !a || !b )
      return;

    xx = ADD_LONG( FT_MulDiv( a->xx, b->xx, val ),
                   FT_MulDiv( a->xy, b->yx, val ) );
    xy = ADD_LONG( FT_MulDiv( a->xx, b->xy, val ),
                   FT_MulDiv( a->xy, b->yy, val ) );
    yx = ADD_LONG( FT_MulDiv( a->yx, b->xx, val ),
                   FT_MulDiv( a->yy, b->yx, val ) );
    yy = ADD_LONG( FT_MulDiv( a->yx, b->xy, val ),
                   FT_MulDiv( a->yy, b->yy, val ) );

    b->xx = xx;
    b->xy = xy;
    b->yx = yx;
    b->yy = yy;
  }


  /* documentation is in ftcalc.h */

  FT_BASE_DEF( FT_Bool )
  FT_Matrix_Check( const FT_Matrix*  matrix )
  {
    FT_Matrix  m;
    FT_Fixed   val[4];
    FT_Fixed   nonzero_minval, maxval;
    FT_Fixed   temp1, temp2;
    FT_UInt    i;


    if ( !matrix )
      return 0;

    val[0] = FT_ABS( matrix->xx );
    val[1] = FT_ABS( matrix->xy );
    val[2] = FT_ABS( matrix->yx );
    val[3] = FT_ABS( matrix->yy );

    /*
     * To avoid overflow, we ensure that each value is not larger than
     *
     *   int(sqrt(2^31 / 4)) = 23170  ;
     *
     * we also check that no value becomes zero if we have to scale.
     */

    maxval         = 0;
    nonzero_minval = FT_LONG_MAX;

    for ( i = 0; i < 4; i++ )
    {
      if ( val[i] > maxval )
        maxval = val[i];
      if ( val[i] && val[i] < nonzero_minval )
        nonzero_minval = val[i];
    }

    if ( maxval > 23170 )
    {
      FT_Fixed  scale = FT_DivFix( maxval, 23170 );


      if ( !FT_DivFix( nonzero_minval, scale ) )
        return 0;    /* value range too large */

      m.xx = FT_DivFix( matrix->xx, scale );
      m.xy = FT_DivFix( matrix->xy, scale );
      m.yx = FT_DivFix( matrix->yx, scale );
      m.yy = FT_DivFix( matrix->yy, scale );
    }
    else
      m = *matrix;

    temp1 = FT_ABS( m.xx * m.yy - m.xy * m.yx );
    temp2 = m.xx * m.xx + m.xy * m.xy + m.yx * m.yx + m.yy * m.yy;

    if ( temp1 == 0         ||
         temp2 / temp1 > 50 )
      return 0;

    return 1;
  }


  /* documentation is in ftcalc.h */

  FT_BASE_DEF( void )
  FT_Vector_Transform_Scaled( FT_Vector*        vector,
                              const FT_Matrix*  matrix,
                              FT_Long           scaling )
  {
    FT_Pos   xz, yz;

    FT_Long  val = 0x10000L * scaling;


    if ( !vector || !matrix )
      return;

    xz = ADD_LONG( FT_MulDiv( vector->x, matrix->xx, val ),
                   FT_MulDiv( vector->y, matrix->xy, val ) );
    yz = ADD_LONG( FT_MulDiv( vector->x, matrix->yx, val ),
                   FT_MulDiv( vector->y, matrix->yy, val ) );

    vector->x = xz;
    vector->y = yz;
  }


  /* documentation is in ftcalc.h */

  FT_BASE_DEF( FT_UInt32 )
  FT_Vector_NormLen( FT_Vector*  vector )
  {
    FT_Int32   x_ = vector->x;
    FT_Int32   y_ = vector->y;
    FT_Int32   b, z;
    FT_UInt32  x, y, u, v, l;
    FT_Int     sx = 1, sy = 1, shift;


    x = (FT_UInt32)x_;
    y = (FT_UInt32)y_;

    FT_MOVE_SIGN( x_, x, sx );
    FT_MOVE_SIGN( y_, y, sy );

    /* trivial cases */
    if ( x == 0 )
    {
      if ( y > 0 )
        vector->y = sy * 0x10000;
      return y;
    }
    else if ( y == 0 )
    {
      if ( x > 0 )
        vector->x = sx * 0x10000;
      return x;
    }

    /* Estimate length and prenormalize by shifting so that */
    /* the new approximate length is between 2/3 and 4/3.   */
    /* The magic constant 0xAAAAAAAAUL (2/3 of 2^32) helps  */
    /* achieve this in 16.16 fixed-point representation.    */
    l = x > y ? x + ( y >> 1 )
              : y + ( x >> 1 );

    shift  = 31 - FT_MSB( l );
    shift -= 15 + ( l >= ( 0xAAAAAAAAUL >> shift ) );

    if ( shift > 0 )
    {
      x <<= shift;
      y <<= shift;

      /* re-estimate length for tiny vectors */
      l = x > y ? x + ( y >> 1 )
                : y + ( x >> 1 );
    }
    else
    {
      x >>= -shift;
      y >>= -shift;
      l >>= -shift;
    }

    /* lower linear approximation for reciprocal length minus one */
    b = 0x10000 - (FT_Int32)l;

    x_ = (FT_Int32)x;
    y_ = (FT_Int32)y;

    /* Newton's iterations */
    do
    {
      u = (FT_UInt32)( x_ + ( x_ * b >> 16 ) );
      v = (FT_UInt32)( y_ + ( y_ * b >> 16 ) );

      /* Normalized squared length in the parentheses approaches 2^32. */
      /* On two's complement systems, converting to signed gives the   */
      /* difference with 2^32 even if the expression wraps around.     */
      z = -(FT_Int32)( u * u + v * v ) / 0x200;
      z = z * ( ( 0x10000 + b ) >> 8 ) / 0x10000;

      b += z;

    } while ( z > 0 );

    vector->x = sx < 0 ? -(FT_Pos)u : (FT_Pos)u;
    vector->y = sy < 0 ? -(FT_Pos)v : (FT_Pos)v;

    /* Conversion to signed helps to recover from likely wrap around */
    /* in calculating the prenormalized length, because it gives the */
    /* correct difference with 2^32 on two's complement systems.     */
    l = (FT_UInt32)( 0x10000 + (FT_Int32)( u * x + v * y ) / 0x10000 );
    if ( shift > 0 )
      l = ( l + ( 1 << ( shift - 1 ) ) ) >> shift;
    else
      l <<= -shift;

    return l;
  }


#if 0

  /* documentation is in ftcalc.h */

  FT_BASE_DEF( FT_Int32 )
  FT_SqrtFixed( FT_Int32  x )
  {
    FT_UInt32  root, rem_hi, rem_lo, test_div;
    FT_Int     count;


    root = 0;

    if ( x > 0 )
    {
      rem_hi = 0;
      rem_lo = (FT_UInt32)x;
      count  = 24;
      do
      {
        rem_hi   = ( rem_hi << 2 ) | ( rem_lo >> 30 );
        rem_lo <<= 2;
        root   <<= 1;
        test_div = ( root << 1 ) + 1;

        if ( rem_hi >= test_div )
        {
          rem_hi -= test_div;
          root   += 1;
        }
      } while ( --count );
    }

    return (FT_Int32)root;
  }

#endif /* 0 */


  /* documentation is in ftcalc.h */

  FT_BASE_DEF( FT_Int )
  ft_corner_orientation( FT_Pos  in_x,
                         FT_Pos  in_y,
                         FT_Pos  out_x,
                         FT_Pos  out_y )
  {
#ifdef FT_LONG64

    FT_Int64  delta = (FT_Int64)in_x * out_y - (FT_Int64)in_y * out_x;


    return ( delta > 0 ) - ( delta < 0 );

#else

    FT_Int  result;


    /* we silently ignore overflow errors, since such large values */
    /* lead to even more (harmless) rendering errors later on      */
    if ( ADD_LONG( FT_ABS( in_x ), FT_ABS( out_y ) ) <= 131071L &&
         ADD_LONG( FT_ABS( in_y ), FT_ABS( out_x ) ) <= 131071L )
    {
      FT_Long  z1 = MUL_LONG( in_x, out_y );
      FT_Long  z2 = MUL_LONG( in_y, out_x );


      if ( z1 > z2 )
        result = +1;
      else if ( z1 < z2 )
        result = -1;
      else
        result = 0;
    }
    else /* products might overflow 32 bits */
    {
      FT_Int64  z1, z2;


      /* XXX: this function does not allow 64-bit arguments */
      ft_multo64( (FT_UInt32)in_x, (FT_UInt32)out_y, &z1 );
      ft_multo64( (FT_UInt32)in_y, (FT_UInt32)out_x, &z2 );

      if ( z1.hi > z2.hi )
        result = +1;
      else if ( z1.hi < z2.hi )
        result = -1;
      else if ( z1.lo > z2.lo )
        result = +1;
      else if ( z1.lo < z2.lo )
        result = -1;
      else
        result = 0;
    }

    /* XXX: only the sign of return value, +1/0/-1 must be used */
    return result;

#endif
  }


  /* documentation is in ftcalc.h */

  FT_BASE_DEF( FT_Int )
  ft_corner_is_flat( FT_Pos  in_x,
                     FT_Pos  in_y,
                     FT_Pos  out_x,
                     FT_Pos  out_y )
  {
    FT_Pos  ax = in_x + out_x;
    FT_Pos  ay = in_y + out_y;

    FT_Pos  d_in, d_out, d_hypot;


    /* The idea of this function is to compare the length of the */
    /* hypotenuse with the `in' and `out' length.  The `corner'  */
    /* represented by `in' and `out' is flat if the hypotenuse's */
    /* length isn't too large.                                   */
    /*                                                           */
    /* This approach has the advantage that the angle between    */
    /* `in' and `out' is not checked.  In case one of the two    */
    /* vectors is `dominant', this is, much larger than the      */
    /* other vector, we thus always have a flat corner.          */
    /*                                                           */
    /*                hypotenuse                                 */
    /*       x---------------------------x                       */
    /*        \                      /                           */
    /*         \                /                                */
    /*      in  \          /  out                                */
    /*           \    /                                          */
    /*            o                                              */
    /*              Point                                        */

    d_in    = FT_HYPOT(  in_x,  in_y );
    d_out   = FT_HYPOT( out_x, out_y );
    d_hypot = FT_HYPOT(    ax,    ay );

    /* now do a simple length comparison: */
    /*                                    */
    /*   d_in + d_out < 17/16 d_hypot     */

    return ( d_in + d_out - d_hypot ) < ( d_hypot >> 4 );
  }


/* END */