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+/****************************************************************************
+ *
+ * ftcalc.c
+ *
+ * Arithmetic computations (body).
+ *
+ * Copyright (C) 1996-2023 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 <freetype/ftglyph.h>
+#include <freetype/fttrigon.h>
+#include <freetype/internal/ftcalc.h>
+#include <freetype/internal/ftdebug.h>
+#include <freetype/internal/ftobjs.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_INT64
+
+ typedef struct FT_Int64_
+ {
+ FT_UInt32 lo;
+ FT_UInt32 hi;
+
+ } FT_Int64;
+
+#endif /* !FT_INT64 */
+
+
+ /**************************************************************************
+ *
+ * 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 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_INT64 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_INT64
+
+
+ /* 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_INT64 */
+
+
+ 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_INT64 */
+
+
+ /* 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];
+ }
+
+ /* we only handle 32bit values */
+ if ( maxval > 0x7FFFFFFFL )
+ return 0;
+
+ 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 )
+ {
+ /* we silently ignore overflow errors since such large values */
+ /* lead to even more (harmless) rendering errors later on */
+
+#ifdef FT_INT64
+
+ FT_Int64 delta = SUB_INT64( MUL_INT64( in_x, out_y ),
+ MUL_INT64( in_y, out_x ) );
+
+
+ return ( delta > 0 ) - ( delta < 0 );
+
+#else
+
+ FT_Int result;
+
+
+ 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 );
+ }
+
+
+ FT_BASE_DEF( FT_Int32 )
+ FT_MulAddFix( FT_Fixed* s,
+ FT_Int32* f,
+ FT_UInt count )
+ {
+ FT_UInt i;
+ FT_Int64 temp;
+#ifndef FT_INT64
+ FT_Int64 halfUnit;
+#endif
+
+
+#ifdef FT_INT64
+ temp = 0;
+
+ for ( i = 0; i < count; ++i )
+ temp += (FT_Int64)s[i] * f[i];
+
+ return ( temp + 0x8000 ) >> 16;
+#else
+ temp.hi = 0;
+ temp.lo = 0;
+
+ for ( i = 0; i < count; ++i )
+ {
+ FT_Int64 multResult;
+
+ FT_Int sign = 1;
+ FT_UInt32 carry = 0;
+
+ FT_UInt32 scalar;
+ FT_UInt32 factor;
+
+
+ scalar = (FT_UInt32)s[i];
+ factor = (FT_UInt32)f[i];
+
+ FT_MOVE_SIGN( s[i], scalar, sign );
+ FT_MOVE_SIGN( f[i], factor, sign );
+
+ ft_multo64( scalar, factor, &multResult );
+
+ if ( sign < 0 )
+ {
+ /* Emulated `FT_Int64` negation. */
+ carry = ( multResult.lo == 0 );
+
+ multResult.lo = ~multResult.lo + 1;
+ multResult.hi = ~multResult.hi + carry;
+ }
+
+ FT_Add64( &temp, &multResult, &temp );
+ }
+
+ /* Round value. */
+ halfUnit.hi = 0;
+ halfUnit.lo = 0x8000;
+ FT_Add64( &temp, &halfUnit, &temp );
+
+ return (FT_Int32)( ( (FT_Int32)( temp.hi & 0xFFFF ) << 16 ) |
+ ( temp.lo >> 16 ) );
+
+#endif /* !FT_INT64 */
+
+ }
+
+
+/* END */
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