1 /***************************************************************************/
5 /* Arithmetic computations (body). */
7 /* Copyright 1996-2006, 2008, 2012 by */
8 /* David Turner, Robert Wilhelm, and Werner Lemberg. */
10 /* This file is part of the FreeType project, and may only be used, */
11 /* modified, and distributed under the terms of the FreeType project */
12 /* license, LICENSE.TXT. By continuing to use, modify, or distribute */
13 /* this file you indicate that you have read the license and */
14 /* understand and accept it fully. */
16 /***************************************************************************/
18 /*************************************************************************/
20 /* Support for 1-complement arithmetic has been totally dropped in this */
21 /* release. You can still write your own code if you need it. */
23 /*************************************************************************/
25 /*************************************************************************/
27 /* Implementing basic computation routines. */
29 /* FT_MulDiv(), FT_MulFix(), FT_DivFix(), FT_RoundFix(), FT_CeilFix(), */
30 /* and FT_FloorFix() are declared in freetype.h. */
32 /*************************************************************************/
37 #include FT_INTERNAL_CALC_H
38 #include FT_INTERNAL_DEBUG_H
39 #include FT_INTERNAL_OBJECTS_H
41 #ifdef FT_MULFIX_INLINED
45 /* we need to define a 64-bits data type here */
49 typedef FT_INT64 FT_Int64;
53 typedef struct FT_Int64_
60 #endif /* FT_LONG64 */
63 /*************************************************************************/
65 /* The macro FT_COMPONENT is used in trace mode. It is an implicit */
66 /* parameter of the FT_TRACE() and FT_ERROR() macros, used to print/log */
67 /* messages during execution. */
70 #define FT_COMPONENT trace_calc
73 /* The following three functions are available regardless of whether */
74 /* FT_LONG64 is defined. */
76 /* documentation is in freetype.h */
78 FT_EXPORT_DEF( FT_Fixed )
79 FT_RoundFix( FT_Fixed a )
81 return ( a >= 0 ) ? ( a + 0x8000L ) & ~0xFFFFL
82 : -((-a + 0x8000L ) & ~0xFFFFL );
86 /* documentation is in freetype.h */
88 FT_EXPORT_DEF( FT_Fixed )
89 FT_CeilFix( FT_Fixed a )
91 return ( a >= 0 ) ? ( a + 0xFFFFL ) & ~0xFFFFL
92 : -((-a + 0xFFFFL ) & ~0xFFFFL );
96 /* documentation is in freetype.h */
98 FT_EXPORT_DEF( FT_Fixed )
99 FT_FloorFix( FT_Fixed a )
101 return ( a >= 0 ) ? a & ~0xFFFFL
102 : -((-a) & ~0xFFFFL );
106 #ifdef FT_CONFIG_OPTION_OLD_INTERNALS
108 /* documentation is in ftcalc.h */
110 FT_EXPORT_DEF( FT_Int32 )
111 FT_Sqrt32( FT_Int32 x )
113 FT_UInt32 val, root, newroot, mask;
117 mask = (FT_UInt32)0x40000000UL;
122 newroot = root + mask;
123 if ( newroot <= val )
126 root = newroot + mask;
132 } while ( mask != 0 );
137 #endif /* FT_CONFIG_OPTION_OLD_INTERNALS */
143 /* documentation is in freetype.h */
145 FT_EXPORT_DEF( FT_Long )
146 FT_MulDiv( FT_Long a,
155 if ( a < 0 ) { a = -a; s = -1; }
156 if ( b < 0 ) { b = -b; s = -s; }
157 if ( c < 0 ) { c = -c; s = -s; }
159 d = (FT_Long)( c > 0 ? ( (FT_Int64)a * b + ( c >> 1 ) ) / c
162 return ( s > 0 ) ? d : -d;
166 #ifdef TT_USE_BYTECODE_INTERPRETER
168 /* documentation is in ftcalc.h */
170 FT_BASE_DEF( FT_Long )
171 FT_MulDiv_No_Round( FT_Long a,
180 if ( a < 0 ) { a = -a; s = -1; }
181 if ( b < 0 ) { b = -b; s = -s; }
182 if ( c < 0 ) { c = -c; s = -s; }
184 d = (FT_Long)( c > 0 ? (FT_Int64)a * b / c
187 return ( s > 0 ) ? d : -d;
190 #endif /* TT_USE_BYTECODE_INTERPRETER */
193 /* documentation is in freetype.h */
195 FT_EXPORT_DEF( FT_Long )
196 FT_MulFix( FT_Long a,
199 #ifdef FT_MULFIX_ASSEMBLER
201 return FT_MULFIX_ASSEMBLER( a, b );
221 c = (FT_Long)( ( (FT_Int64)a * b + 0x8000L ) >> 16 );
223 return ( s > 0 ) ? c : -c;
225 #endif /* FT_MULFIX_ASSEMBLER */
229 /* documentation is in freetype.h */
231 FT_EXPORT_DEF( FT_Long )
232 FT_DivFix( FT_Long a,
239 if ( a < 0 ) { a = -a; s = -1; }
240 if ( b < 0 ) { b = -b; s = -s; }
243 /* check for division by 0 */
246 /* compute result directly */
247 q = (FT_UInt32)( ( ( (FT_Int64)a << 16 ) + ( b >> 1 ) ) / b );
249 return ( s < 0 ? -(FT_Long)q : (FT_Long)q );
253 #else /* !FT_LONG64 */
257 ft_multo64( FT_UInt32 x,
261 FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2;
264 lo1 = x & 0x0000FFFFU; hi1 = x >> 16;
265 lo2 = y & 0x0000FFFFU; hi2 = y >> 16;
272 /* Check carry overflow of i1 + i2 */
274 hi += (FT_UInt32)( i1 < i2 ) << 16;
279 /* Check carry overflow of i1 + lo */
289 ft_div64by32( FT_UInt32 hi,
301 return (FT_UInt32)0x7FFFFFFFL;
323 FT_Add64( FT_Int64* x,
327 register FT_UInt32 lo, hi;
331 hi = x->hi + y->hi + ( lo < x->lo );
338 /* documentation is in freetype.h */
340 /* The FT_MulDiv function has been optimized thanks to ideas from */
341 /* Graham Asher. The trick is to optimize computation when everything */
342 /* fits within 32-bits (a rather common case). */
344 /* we compute 'a*b+c/2', then divide it by 'c'. (positive values) */
346 /* 46340 is FLOOR(SQRT(2^31-1)). */
348 /* if ( a <= 46340 && b <= 46340 ) then ( a*b <= 0x7FFEA810 ) */
350 /* 0x7FFFFFFF - 0x7FFEA810 = 0x157F0 */
352 /* if ( c < 0x157F0*2 ) then ( a*b+c/2 <= 0x7FFFFFFF ) */
354 /* and 2*0x157F0 = 176096 */
357 FT_EXPORT_DEF( FT_Long )
358 FT_MulDiv( FT_Long a,
365 /* XXX: this function does not allow 64-bit arguments */
366 if ( a == 0 || b == c )
369 s = a; a = FT_ABS( a );
370 s ^= b; b = FT_ABS( b );
371 s ^= c; c = FT_ABS( c );
373 if ( a <= 46340L && b <= 46340L && c <= 176095L && c > 0 )
374 a = ( a * b + ( c >> 1 ) ) / c;
376 else if ( (FT_Int32)c > 0 )
378 FT_Int64 temp, temp2;
381 ft_multo64( (FT_Int32)a, (FT_Int32)b, &temp );
384 temp2.lo = (FT_UInt32)(c >> 1);
385 FT_Add64( &temp, &temp2, &temp );
386 a = ft_div64by32( temp.hi, temp.lo, (FT_Int32)c );
391 return ( s < 0 ? -a : a );
395 #ifdef TT_USE_BYTECODE_INTERPRETER
397 FT_BASE_DEF( FT_Long )
398 FT_MulDiv_No_Round( FT_Long a,
405 if ( a == 0 || b == c )
408 s = a; a = FT_ABS( a );
409 s ^= b; b = FT_ABS( b );
410 s ^= c; c = FT_ABS( c );
412 if ( a <= 46340L && b <= 46340L && c > 0 )
415 else if ( (FT_Int32)c > 0 )
420 ft_multo64( (FT_Int32)a, (FT_Int32)b, &temp );
421 a = ft_div64by32( temp.hi, temp.lo, (FT_Int32)c );
426 return ( s < 0 ? -a : a );
429 #endif /* TT_USE_BYTECODE_INTERPRETER */
432 /* documentation is in freetype.h */
434 FT_EXPORT_DEF( FT_Long )
435 FT_MulFix( FT_Long a,
438 #ifdef FT_MULFIX_ASSEMBLER
440 return FT_MULFIX_ASSEMBLER( a, b );
445 * This code is nonportable. See comment below.
447 * However, on a platform where right-shift of a signed quantity fills
448 * the leftmost bits by copying the sign bit, it might be faster.
455 if ( a == 0 || b == 0x10000L )
459 * This is a clever way of converting a signed number `a' into its
460 * absolute value (stored back into `a') and its sign. The sign is
461 * stored in `sa'; 0 means `a' was positive or zero, and -1 means `a'
462 * was negative. (Similarly for `b' and `sb').
464 * Unfortunately, it doesn't work (at least not portably).
466 * It makes the assumption that right-shift on a negative signed value
467 * fills the leftmost bits by copying the sign bit. This is wrong.
468 * According to K&R 2nd ed, section `A7.8 Shift Operators' on page 206,
469 * the result of right-shift of a negative signed value is
470 * implementation-defined. At least one implementation fills the
471 * leftmost bits with 0s (i.e., it is exactly the same as an unsigned
472 * right shift). This means that when `a' is negative, `sa' ends up
473 * with the value 1 rather than -1. After that, everything else goes
476 sa = ( a >> ( sizeof ( a ) * 8 - 1 ) );
478 sb = ( b >> ( sizeof ( b ) * 8 - 1 ) );
484 if ( ua <= 2048 && ub <= 1048576L )
485 ua = ( ua * ub + 0x8000U ) >> 16;
488 FT_ULong al = ua & 0xFFFFU;
491 ua = ( ua >> 16 ) * ub + al * ( ub >> 16 ) +
492 ( ( al * ( ub & 0xFFFFU ) + 0x8000U ) >> 16 );
496 ua = (FT_ULong)(( ua ^ sa ) - sa);
506 if ( a == 0 || b == 0x10000L )
509 s = a; a = FT_ABS( a );
510 s ^= b; b = FT_ABS( b );
515 if ( ua <= 2048 && ub <= 1048576L )
516 ua = ( ua * ub + 0x8000UL ) >> 16;
519 FT_ULong al = ua & 0xFFFFUL;
522 ua = ( ua >> 16 ) * ub + al * ( ub >> 16 ) +
523 ( ( al * ( ub & 0xFFFFUL ) + 0x8000UL ) >> 16 );
526 return ( s < 0 ? -(FT_Long)ua : (FT_Long)ua );
533 /* documentation is in freetype.h */
535 FT_EXPORT_DEF( FT_Long )
536 FT_DivFix( FT_Long a,
543 /* XXX: this function does not allow 64-bit arguments */
544 s = (FT_Int32)a; a = FT_ABS( a );
545 s ^= (FT_Int32)b; b = FT_ABS( b );
547 if ( (FT_UInt32)b == 0 )
549 /* check for division by 0 */
550 q = (FT_UInt32)0x7FFFFFFFL;
552 else if ( ( a >> 16 ) == 0 )
554 /* compute result directly */
555 q = (FT_UInt32)( ( a << 16 ) + ( b >> 1 ) ) / (FT_UInt32)b;
559 /* we need more bits; we have to do it by hand */
560 FT_Int64 temp, temp2;
563 temp.hi = (FT_Int32) ( a >> 16 );
564 temp.lo = (FT_UInt32)( a << 16 );
566 temp2.lo = (FT_UInt32)( b >> 1 );
567 FT_Add64( &temp, &temp2, &temp );
568 q = ft_div64by32( temp.hi, temp.lo, (FT_Int32)b );
571 return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
577 /* documentation is in ftcalc.h */
579 FT_EXPORT_DEF( void )
580 FT_MulTo64( FT_Int32 x,
587 s = x; x = FT_ABS( x );
588 s ^= y; y = FT_ABS( y );
590 ft_multo64( x, y, z );
594 z->lo = (FT_UInt32)-(FT_Int32)z->lo;
595 z->hi = ~z->hi + !( z->lo );
600 /* apparently, the second version of this code is not compiled correctly */
601 /* on Mac machines with the MPW C compiler.. tsk, tsk, tsk... */
605 FT_EXPORT_DEF( FT_Int32 )
606 FT_Div64by32( FT_Int64* x,
610 FT_UInt32 q, r, i, lo;
616 x->lo = (FT_UInt32)-(FT_Int32)x->lo;
617 x->hi = ~x->hi + !x->lo;
619 s ^= y; y = FT_ABS( y );
629 return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
635 if ( r >= (FT_UInt32)y ) /* we know y is to be treated as unsigned here */
636 return ( s < 0 ? 0x80000001UL : 0x7FFFFFFFUL );
637 /* Return Max/Min Int32 if division overflow. */
638 /* This includes division by zero! */
640 for ( i = 0; i < 32; i++ )
646 if ( r >= (FT_UInt32)y )
654 return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
659 FT_EXPORT_DEF( FT_Int32 )
660 FT_Div64by32( FT_Int64* x,
670 x->lo = (FT_UInt32)-(FT_Int32)x->lo;
671 x->hi = ~x->hi + !x->lo;
673 s ^= y; y = FT_ABS( y );
679 q = ( x->lo + ( y >> 1 ) ) / y;
683 return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
686 q = ft_div64by32( x->hi, x->lo, y );
688 return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
696 #endif /* FT_LONG64 */
699 /* documentation is in ftglyph.h */
701 FT_EXPORT_DEF( void )
702 FT_Matrix_Multiply( const FT_Matrix* a,
705 FT_Fixed xx, xy, yx, yy;
711 xx = FT_MulFix( a->xx, b->xx ) + FT_MulFix( a->xy, b->yx );
712 xy = FT_MulFix( a->xx, b->xy ) + FT_MulFix( a->xy, b->yy );
713 yx = FT_MulFix( a->yx, b->xx ) + FT_MulFix( a->yy, b->yx );
714 yy = FT_MulFix( a->yx, b->xy ) + FT_MulFix( a->yy, b->yy );
716 b->xx = xx; b->xy = xy;
717 b->yx = yx; b->yy = yy;
721 /* documentation is in ftglyph.h */
723 FT_EXPORT_DEF( FT_Error )
724 FT_Matrix_Invert( FT_Matrix* matrix )
726 FT_Pos delta, xx, yy;
730 return FT_Err_Invalid_Argument;
732 /* compute discriminant */
733 delta = FT_MulFix( matrix->xx, matrix->yy ) -
734 FT_MulFix( matrix->xy, matrix->yx );
737 return FT_Err_Invalid_Argument; /* matrix can't be inverted */
739 matrix->xy = - FT_DivFix( matrix->xy, delta );
740 matrix->yx = - FT_DivFix( matrix->yx, delta );
745 matrix->xx = FT_DivFix( yy, delta );
746 matrix->yy = FT_DivFix( xx, delta );
752 /* documentation is in ftcalc.h */
755 FT_Matrix_Multiply_Scaled( const FT_Matrix* a,
759 FT_Fixed xx, xy, yx, yy;
761 FT_Long val = 0x10000L * scaling;
767 xx = FT_MulDiv( a->xx, b->xx, val ) + FT_MulDiv( a->xy, b->yx, val );
768 xy = FT_MulDiv( a->xx, b->xy, val ) + FT_MulDiv( a->xy, b->yy, val );
769 yx = FT_MulDiv( a->yx, b->xx, val ) + FT_MulDiv( a->yy, b->yx, val );
770 yy = FT_MulDiv( a->yx, b->xy, val ) + FT_MulDiv( a->yy, b->yy, val );
772 b->xx = xx; b->xy = xy;
773 b->yx = yx; b->yy = yy;
777 /* documentation is in ftcalc.h */
780 FT_Vector_Transform_Scaled( FT_Vector* vector,
781 const FT_Matrix* matrix,
786 FT_Long val = 0x10000L * scaling;
789 if ( !vector || !matrix )
792 xz = FT_MulDiv( vector->x, matrix->xx, val ) +
793 FT_MulDiv( vector->y, matrix->xy, val );
795 yz = FT_MulDiv( vector->x, matrix->yx, val ) +
796 FT_MulDiv( vector->y, matrix->yy, val );
803 /* documentation is in ftcalc.h */
805 FT_BASE_DEF( FT_Int32 )
806 FT_SqrtFixed( FT_Int32 x )
808 FT_UInt32 root, rem_hi, rem_lo, test_div;
821 rem_hi = ( rem_hi << 2 ) | ( rem_lo >> 30 );
824 test_div = ( root << 1 ) + 1;
826 if ( rem_hi >= test_div )
834 return (FT_Int32)root;
838 /* documentation is in ftcalc.h */
840 FT_BASE_DEF( FT_Int )
841 ft_corner_orientation( FT_Pos in_x,
846 FT_Long result; /* avoid overflow on 16-bit system */
849 /* deal with the trivial cases quickly */
857 else if ( in_x == 0 )
864 else if ( out_y == 0 )
871 else if ( out_x == 0 )
878 else /* general case */
882 FT_Int64 delta = (FT_Int64)in_x * out_y - (FT_Int64)in_y * out_x;
888 result = 1 - 2 * ( delta < 0 );
895 /* XXX: this function does not allow 64-bit arguments */
896 ft_multo64( (FT_Int32)in_x, (FT_Int32)out_y, &z1 );
897 ft_multo64( (FT_Int32)in_y, (FT_Int32)out_x, &z2 );
901 else if ( z1.hi < z2.hi )
903 else if ( z1.lo > z2.lo )
905 else if ( z1.lo < z2.lo )
913 /* XXX: only the sign of return value, +1/0/-1 must be used */
914 return (FT_Int)result;
918 /* documentation is in ftcalc.h */
920 FT_BASE_DEF( FT_Int )
921 ft_corner_is_flat( FT_Pos in_x,
929 FT_Pos d_in, d_out, d_corner;
954 return ( d_in + d_out - d_corner ) < ( d_corner >> 4 );