1 /***************************************************************************/ 2 /* */ 3 /* ftcalc.c */ 4 /* */ 5 /* Arithmetic computations (body). */ 6 /* */ 7 /* Copyright 1996-2018 by */ 8 /* David Turner, Robert Wilhelm, and Werner Lemberg. */ 9 /* */ 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. */ 15 /* */ 16 /***************************************************************************/ 17 18 /*************************************************************************/ 19 /* */ 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. */ 22 /* */ 23 /*************************************************************************/ 24 25 /*************************************************************************/ 26 /* */ 27 /* Implementing basic computation routines. */ 28 /* */ 29 /* FT_MulDiv(), FT_MulFix(), FT_DivFix(), FT_RoundFix(), FT_CeilFix(), */ 30 /* and FT_FloorFix() are declared in freetype.h. */ 31 /* */ 32 /*************************************************************************/ 33 34 35 #include <ft2build.h> 36 #include FT_GLYPH_H 37 #include FT_TRIGONOMETRY_H 38 #include FT_INTERNAL_CALC_H 39 #include FT_INTERNAL_DEBUG_H 40 #include FT_INTERNAL_OBJECTS_H 41 42 43 #ifdef FT_MULFIX_ASSEMBLER 44 #undef FT_MulFix 45 #endif 46 47 /* we need to emulate a 64-bit data type if a real one isn't available */ 48 49 #ifndef FT_LONG64 50 51 typedef struct FT_Int64_ 52 { 53 FT_UInt32 lo; 54 FT_UInt32 hi; 55 56 } FT_Int64; 57 58 #endif /* !FT_LONG64 */ 59 60 61 /*************************************************************************/ 62 /* */ 63 /* The macro FT_COMPONENT is used in trace mode. It is an implicit */ 64 /* parameter of the FT_TRACE() and FT_ERROR() macros, used to print/log */ 65 /* messages during execution. */ 66 /* */ 67 #undef FT_COMPONENT 68 #define FT_COMPONENT trace_calc 69 70 71 /* transfer sign, leaving a positive number; */ 72 /* we need an unsigned value to safely negate INT_MIN (or LONG_MIN) */ 73 #define FT_MOVE_SIGN( x, x_unsigned, s ) \ 74 FT_BEGIN_STMNT \ 75 if ( x < 0 ) \ 76 { \ 77 x_unsigned = 0U - (x_unsigned); \ 78 s = -s; \ 79 } \ 80 FT_END_STMNT 81 82 /* The following three functions are available regardless of whether */ 83 /* FT_LONG64 is defined. */ 84 85 /* documentation is in freetype.h */ 86 87 FT_EXPORT_DEF( FT_Fixed ) 88 FT_RoundFix( FT_Fixed a ) 89 { 90 return ( ADD_LONG( a, 0x8000L - ( a < 0 ) ) ) & ~0xFFFFL; 91 } 92 93 94 /* documentation is in freetype.h */ 95 96 FT_EXPORT_DEF( FT_Fixed ) 97 FT_CeilFix( FT_Fixed a ) 98 { 99 return ( ADD_LONG( a, 0xFFFFL ) ) & ~0xFFFFL; 100 } 101 102 103 /* documentation is in freetype.h */ 104 105 FT_EXPORT_DEF( FT_Fixed ) 106 FT_FloorFix( FT_Fixed a ) 107 { 108 return a & ~0xFFFFL; 109 } 110 111 #ifndef FT_MSB 112 113 FT_BASE_DEF ( FT_Int ) 114 FT_MSB( FT_UInt32 z ) 115 { 116 FT_Int shift = 0; 117 118 119 /* determine msb bit index in `shift' */ 120 if ( z & 0xFFFF0000UL ) 121 { 122 z >>= 16; 123 shift += 16; 124 } 125 if ( z & 0x0000FF00UL ) 126 { 127 z >>= 8; 128 shift += 8; 129 } 130 if ( z & 0x000000F0UL ) 131 { 132 z >>= 4; 133 shift += 4; 134 } 135 if ( z & 0x0000000CUL ) 136 { 137 z >>= 2; 138 shift += 2; 139 } 140 if ( z & 0x00000002UL ) 141 { 142 /* z >>= 1; */ 143 shift += 1; 144 } 145 146 return shift; 147 } 148 149 #endif /* !FT_MSB */ 150 151 152 /* documentation is in ftcalc.h */ 153 154 FT_BASE_DEF( FT_Fixed ) 155 FT_Hypot( FT_Fixed x, 156 FT_Fixed y ) 157 { 158 FT_Vector v; 159 160 161 v.x = x; 162 v.y = y; 163 164 return FT_Vector_Length( &v ); 165 } 166 167 168 #ifdef FT_LONG64 169 170 171 /* documentation is in freetype.h */ 172 173 FT_EXPORT_DEF( FT_Long ) 174 FT_MulDiv( FT_Long a_, 175 FT_Long b_, 176 FT_Long c_ ) 177 { 178 FT_Int s = 1; 179 FT_UInt64 a, b, c, d; 180 FT_Long d_; 181 182 183 a = (FT_UInt64)a_; 184 b = (FT_UInt64)b_; 185 c = (FT_UInt64)c_; 186 187 FT_MOVE_SIGN( a_, a, s ); 188 FT_MOVE_SIGN( b_, b, s ); 189 FT_MOVE_SIGN( c_, c, s ); 190 191 d = c > 0 ? ( a * b + ( c >> 1 ) ) / c 192 : 0x7FFFFFFFUL; 193 194 d_ = (FT_Long)d; 195 196 return s < 0 ? NEG_LONG( d_ ) : d_; 197 } 198 199 200 /* documentation is in ftcalc.h */ 201 202 FT_BASE_DEF( FT_Long ) 203 FT_MulDiv_No_Round( FT_Long a_, 204 FT_Long b_, 205 FT_Long c_ ) 206 { 207 FT_Int s = 1; 208 FT_UInt64 a, b, c, d; 209 FT_Long d_; 210 211 212 a = (FT_UInt64)a_; 213 b = (FT_UInt64)b_; 214 c = (FT_UInt64)c_; 215 216 FT_MOVE_SIGN( a_, a, s ); 217 FT_MOVE_SIGN( b_, b, s ); 218 FT_MOVE_SIGN( c_, c, s ); 219 220 d = c > 0 ? a * b / c 221 : 0x7FFFFFFFUL; 222 223 d_ = (FT_Long)d; 224 225 return s < 0 ? NEG_LONG( d_ ) : d_; 226 } 227 228 229 /* documentation is in freetype.h */ 230 231 FT_EXPORT_DEF( FT_Long ) 232 FT_MulFix( FT_Long a_, 233 FT_Long b_ ) 234 { 235 #ifdef FT_MULFIX_ASSEMBLER 236 237 return FT_MULFIX_ASSEMBLER( (FT_Int32)a_, (FT_Int32)b_ ); 238 239 #else 240 241 FT_Int64 ab = (FT_Int64)a_ * (FT_Int64)b_; 242 243 /* this requires arithmetic right shift of signed numbers */ 244 return (FT_Long)( ( ab + 0x8000L - ( ab < 0 ) ) >> 16 ); 245 246 #endif /* FT_MULFIX_ASSEMBLER */ 247 } 248 249 250 /* documentation is in freetype.h */ 251 252 FT_EXPORT_DEF( FT_Long ) 253 FT_DivFix( FT_Long a_, 254 FT_Long b_ ) 255 { 256 FT_Int s = 1; 257 FT_UInt64 a, b, q; 258 FT_Long q_; 259 260 261 a = (FT_UInt64)a_; 262 b = (FT_UInt64)b_; 263 264 FT_MOVE_SIGN( a_, a, s ); 265 FT_MOVE_SIGN( b_, b, s ); 266 267 q = b > 0 ? ( ( a << 16 ) + ( b >> 1 ) ) / b 268 : 0x7FFFFFFFUL; 269 270 q_ = (FT_Long)q; 271 272 return s < 0 ? NEG_LONG( q_ ) : q_; 273 } 274 275 276 #else /* !FT_LONG64 */ 277 278 279 static void 280 ft_multo64( FT_UInt32 x, 281 FT_UInt32 y, 282 FT_Int64 *z ) 283 { 284 FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2; 285 286 287 lo1 = x & 0x0000FFFFU; hi1 = x >> 16; 288 lo2 = y & 0x0000FFFFU; hi2 = y >> 16; 289 290 lo = lo1 * lo2; 291 i1 = lo1 * hi2; 292 i2 = lo2 * hi1; 293 hi = hi1 * hi2; 294 295 /* Check carry overflow of i1 + i2 */ 296 i1 += i2; 297 hi += (FT_UInt32)( i1 < i2 ) << 16; 298 299 hi += i1 >> 16; 300 i1 = i1 << 16; 301 302 /* Check carry overflow of i1 + lo */ 303 lo += i1; 304 hi += ( lo < i1 ); 305 306 z->lo = lo; 307 z->hi = hi; 308 } 309 310 311 static FT_UInt32 312 ft_div64by32( FT_UInt32 hi, 313 FT_UInt32 lo, 314 FT_UInt32 y ) 315 { 316 FT_UInt32 r, q; 317 FT_Int i; 318 319 320 if ( hi >= y ) 321 return (FT_UInt32)0x7FFFFFFFL; 322 323 /* We shift as many bits as we can into the high register, perform */ 324 /* 32-bit division with modulo there, then work through the remaining */ 325 /* bits with long division. This optimization is especially noticeable */ 326 /* for smaller dividends that barely use the high register. */ 327 328 i = 31 - FT_MSB( hi ); 329 r = ( hi << i ) | ( lo >> ( 32 - i ) ); lo <<= i; /* left 64-bit shift */ 330 q = r / y; 331 r -= q * y; /* remainder */ 332 333 i = 32 - i; /* bits remaining in low register */ 334 do 335 { 336 q <<= 1; 337 r = ( r << 1 ) | ( lo >> 31 ); lo <<= 1; 338 339 if ( r >= y ) 340 { 341 r -= y; 342 q |= 1; 343 } 344 } while ( --i ); 345 346 return q; 347 } 348 349 350 static void 351 FT_Add64( FT_Int64* x, 352 FT_Int64* y, 353 FT_Int64 *z ) 354 { 355 FT_UInt32 lo, hi; 356 357 358 lo = x->lo + y->lo; 359 hi = x->hi + y->hi + ( lo < x->lo ); 360 361 z->lo = lo; 362 z->hi = hi; 363 } 364 365 366 /* The FT_MulDiv function has been optimized thanks to ideas from */ 367 /* Graham Asher and Alexei Podtelezhnikov. The trick is to optimize */ 368 /* a rather common case when everything fits within 32-bits. */ 369 /* */ 370 /* We compute 'a*b+c/2', then divide it by 'c' (all positive values). */ 371 /* */ 372 /* The product of two positive numbers never exceeds the square of */ 373 /* its mean values. Therefore, we always avoid the overflow by */ 374 /* imposing */ 375 /* */ 376 /* (a + b) / 2 <= sqrt(X - c/2) , */ 377 /* */ 378 /* where X = 2^32 - 1, the maximum unsigned 32-bit value, and using */ 379 /* unsigned arithmetic. Now we replace `sqrt' with a linear function */ 380 /* that is smaller or equal for all values of c in the interval */ 381 /* [0;X/2]; it should be equal to sqrt(X) and sqrt(3X/4) at the */ 382 /* endpoints. Substituting the linear solution and explicit numbers */ 383 /* we get */ 384 /* */ 385 /* a + b <= 131071.99 - c / 122291.84 . */ 386 /* */ 387 /* In practice, we should use a faster and even stronger inequality */ 388 /* */ 389 /* a + b <= 131071 - (c >> 16) */ 390 /* */ 391 /* or, alternatively, */ 392 /* */ 393 /* a + b <= 129894 - (c >> 17) . */ 394 /* */ 395 /* FT_MulFix, on the other hand, is optimized for a small value of */ 396 /* the first argument, when the second argument can be much larger. */ 397 /* This can be achieved by scaling the second argument and the limit */ 398 /* in the above inequalities. For example, */ 399 /* */ 400 /* a + (b >> 8) <= (131071 >> 4) */ 401 /* */ 402 /* covers the practical range of use. The actual test below is a bit */ 403 /* tighter to avoid the border case overflows. */ 404 /* */ 405 /* In the case of FT_DivFix, the exact overflow check */ 406 /* */ 407 /* a << 16 <= X - c/2 */ 408 /* */ 409 /* is scaled down by 2^16 and we use */ 410 /* */ 411 /* a <= 65535 - (c >> 17) . */ 412 413 /* documentation is in freetype.h */ 414 415 FT_EXPORT_DEF( FT_Long ) 416 FT_MulDiv( FT_Long a_, 417 FT_Long b_, 418 FT_Long c_ ) 419 { 420 FT_Int s = 1; 421 FT_UInt32 a, b, c; 422 423 424 /* XXX: this function does not allow 64-bit arguments */ 425 426 a = (FT_UInt32)a_; 427 b = (FT_UInt32)b_; 428 c = (FT_UInt32)c_; 429 430 FT_MOVE_SIGN( a_, a, s ); 431 FT_MOVE_SIGN( b_, b, s ); 432 FT_MOVE_SIGN( c_, c, s ); 433 434 if ( c == 0 ) 435 a = 0x7FFFFFFFUL; 436 437 else if ( a + b <= 129894UL - ( c >> 17 ) ) 438 a = ( a * b + ( c >> 1 ) ) / c; 439 440 else 441 { 442 FT_Int64 temp, temp2; 443 444 445 ft_multo64( a, b, &temp ); 446 447 temp2.hi = 0; 448 temp2.lo = c >> 1; 449 450 FT_Add64( &temp, &temp2, &temp ); 451 452 /* last attempt to ditch long division */ 453 a = ( temp.hi == 0 ) ? temp.lo / c 454 : ft_div64by32( temp.hi, temp.lo, c ); 455 } 456 457 a_ = (FT_Long)a; 458 459 return s < 0 ? NEG_LONG( a_ ) : a_; 460 } 461 462 463 FT_BASE_DEF( FT_Long ) 464 FT_MulDiv_No_Round( FT_Long a_, 465 FT_Long b_, 466 FT_Long c_ ) 467 { 468 FT_Int s = 1; 469 FT_UInt32 a, b, c; 470 471 472 /* XXX: this function does not allow 64-bit arguments */ 473 474 a = (FT_UInt32)a_; 475 b = (FT_UInt32)b_; 476 c = (FT_UInt32)c_; 477 478 FT_MOVE_SIGN( a_, a, s ); 479 FT_MOVE_SIGN( b_, b, s ); 480 FT_MOVE_SIGN( c_, c, s ); 481 482 if ( c == 0 ) 483 a = 0x7FFFFFFFUL; 484 485 else if ( a + b <= 131071UL ) 486 a = a * b / c; 487 488 else 489 { 490 FT_Int64 temp; 491 492 493 ft_multo64( a, b, &temp ); 494 495 /* last attempt to ditch long division */ 496 a = ( temp.hi == 0 ) ? temp.lo / c 497 : ft_div64by32( temp.hi, temp.lo, c ); 498 } 499 500 a_ = (FT_Long)a; 501 502 return s < 0 ? NEG_LONG( a_ ) : a_; 503 } 504 505 506 /* documentation is in freetype.h */ 507 508 FT_EXPORT_DEF( FT_Long ) 509 FT_MulFix( FT_Long a_, 510 FT_Long b_ ) 511 { 512 #ifdef FT_MULFIX_ASSEMBLER 513 514 return FT_MULFIX_ASSEMBLER( a_, b_ ); 515 516 #elif 0 517 518 /* 519 * This code is nonportable. See comment below. 520 * 521 * However, on a platform where right-shift of a signed quantity fills 522 * the leftmost bits by copying the sign bit, it might be faster. 523 */ 524 525 FT_Long sa, sb; 526 FT_UInt32 a, b; 527 528 529 /* 530 * This is a clever way of converting a signed number `a' into its 531 * absolute value (stored back into `a') and its sign. The sign is 532 * stored in `sa'; 0 means `a' was positive or zero, and -1 means `a' 533 * was negative. (Similarly for `b' and `sb'). 534 * 535 * Unfortunately, it doesn't work (at least not portably). 536 * 537 * It makes the assumption that right-shift on a negative signed value 538 * fills the leftmost bits by copying the sign bit. This is wrong. 539 * According to K&R 2nd ed, section `A7.8 Shift Operators' on page 206, 540 * the result of right-shift of a negative signed value is 541 * implementation-defined. At least one implementation fills the 542 * leftmost bits with 0s (i.e., it is exactly the same as an unsigned 543 * right shift). This means that when `a' is negative, `sa' ends up 544 * with the value 1 rather than -1. After that, everything else goes 545 * wrong. 546 */ 547 sa = ( a_ >> ( sizeof ( a_ ) * 8 - 1 ) ); 548 a = ( a_ ^ sa ) - sa; 549 sb = ( b_ >> ( sizeof ( b_ ) * 8 - 1 ) ); 550 b = ( b_ ^ sb ) - sb; 551 552 a = (FT_UInt32)a_; 553 b = (FT_UInt32)b_; 554 555 if ( a + ( b >> 8 ) <= 8190UL ) 556 a = ( a * b + 0x8000U ) >> 16; 557 else 558 { 559 FT_UInt32 al = a & 0xFFFFUL; 560 561 562 a = ( a >> 16 ) * b + al * ( b >> 16 ) + 563 ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 ); 564 } 565 566 sa ^= sb; 567 a = ( a ^ sa ) - sa; 568 569 return (FT_Long)a; 570 571 #else /* 0 */ 572 573 FT_Int s = 1; 574 FT_UInt32 a, b; 575 576 577 /* XXX: this function does not allow 64-bit arguments */ 578 579 a = (FT_UInt32)a_; 580 b = (FT_UInt32)b_; 581 582 FT_MOVE_SIGN( a_, a, s ); 583 FT_MOVE_SIGN( b_, b, s ); 584 585 if ( a + ( b >> 8 ) <= 8190UL ) 586 a = ( a * b + 0x8000UL ) >> 16; 587 else 588 { 589 FT_UInt32 al = a & 0xFFFFUL; 590 591 592 a = ( a >> 16 ) * b + al * ( b >> 16 ) + 593 ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 ); 594 } 595 596 a_ = (FT_Long)a; 597 598 return s < 0 ? NEG_LONG( a_ ) : a_; 599 600 #endif /* 0 */ 601 602 } 603 604 605 /* documentation is in freetype.h */ 606 607 FT_EXPORT_DEF( FT_Long ) 608 FT_DivFix( FT_Long a_, 609 FT_Long b_ ) 610 { 611 FT_Int s = 1; 612 FT_UInt32 a, b, q; 613 FT_Long q_; 614 615 616 /* XXX: this function does not allow 64-bit arguments */ 617 618 a = (FT_UInt32)a_; 619 b = (FT_UInt32)b_; 620 621 FT_MOVE_SIGN( a_, a, s ); 622 FT_MOVE_SIGN( b_, b, s ); 623 624 if ( b == 0 ) 625 { 626 /* check for division by 0 */ 627 q = 0x7FFFFFFFUL; 628 } 629 else if ( a <= 65535UL - ( b >> 17 ) ) 630 { 631 /* compute result directly */ 632 q = ( ( a << 16 ) + ( b >> 1 ) ) / b; 633 } 634 else 635 { 636 /* we need more bits; we have to do it by hand */ 637 FT_Int64 temp, temp2; 638 639 640 temp.hi = a >> 16; 641 temp.lo = a << 16; 642 temp2.hi = 0; 643 temp2.lo = b >> 1; 644 645 FT_Add64( &temp, &temp2, &temp ); 646 q = ft_div64by32( temp.hi, temp.lo, b ); 647 } 648 649 q_ = (FT_Long)q; 650 651 return s < 0 ? NEG_LONG( q_ ) : q_; 652 } 653 654 655 #endif /* !FT_LONG64 */ 656 657 658 /* documentation is in ftglyph.h */ 659 660 FT_EXPORT_DEF( void ) 661 FT_Matrix_Multiply( const FT_Matrix* a, 662 FT_Matrix *b ) 663 { 664 FT_Fixed xx, xy, yx, yy; 665 666 667 if ( !a || !b ) 668 return; 669 670 xx = ADD_LONG( FT_MulFix( a->xx, b->xx ), 671 FT_MulFix( a->xy, b->yx ) ); 672 xy = ADD_LONG( FT_MulFix( a->xx, b->xy ), 673 FT_MulFix( a->xy, b->yy ) ); 674 yx = ADD_LONG( FT_MulFix( a->yx, b->xx ), 675 FT_MulFix( a->yy, b->yx ) ); 676 yy = ADD_LONG( FT_MulFix( a->yx, b->xy ), 677 FT_MulFix( a->yy, b->yy ) ); 678 679 b->xx = xx; 680 b->xy = xy; 681 b->yx = yx; 682 b->yy = yy; 683 } 684 685 686 /* documentation is in ftglyph.h */ 687 688 FT_EXPORT_DEF( FT_Error ) 689 FT_Matrix_Invert( FT_Matrix* matrix ) 690 { 691 FT_Pos delta, xx, yy; 692 693 694 if ( !matrix ) 695 return FT_THROW( Invalid_Argument ); 696 697 /* compute discriminant */ 698 delta = FT_MulFix( matrix->xx, matrix->yy ) - 699 FT_MulFix( matrix->xy, matrix->yx ); 700 701 if ( !delta ) 702 return FT_THROW( Invalid_Argument ); /* matrix can't be inverted */ 703 704 matrix->xy = - FT_DivFix( matrix->xy, delta ); 705 matrix->yx = - FT_DivFix( matrix->yx, delta ); 706 707 xx = matrix->xx; 708 yy = matrix->yy; 709 710 matrix->xx = FT_DivFix( yy, delta ); 711 matrix->yy = FT_DivFix( xx, delta ); 712 713 return FT_Err_Ok; 714 } 715 716 717 /* documentation is in ftcalc.h */ 718 719 FT_BASE_DEF( void ) 720 FT_Matrix_Multiply_Scaled( const FT_Matrix* a, 721 FT_Matrix *b, 722 FT_Long scaling ) 723 { 724 FT_Fixed xx, xy, yx, yy; 725 726 FT_Long val = 0x10000L * scaling; 727 728 729 if ( !a || !b ) 730 return; 731 732 xx = ADD_LONG( FT_MulDiv( a->xx, b->xx, val ), 733 FT_MulDiv( a->xy, b->yx, val ) ); 734 xy = ADD_LONG( FT_MulDiv( a->xx, b->xy, val ), 735 FT_MulDiv( a->xy, b->yy, val ) ); 736 yx = ADD_LONG( FT_MulDiv( a->yx, b->xx, val ), 737 FT_MulDiv( a->yy, b->yx, val ) ); 738 yy = ADD_LONG( FT_MulDiv( a->yx, b->xy, val ), 739 FT_MulDiv( a->yy, b->yy, val ) ); 740 741 b->xx = xx; 742 b->xy = xy; 743 b->yx = yx; 744 b->yy = yy; 745 } 746 747 748 /* documentation is in ftcalc.h */ 749 750 FT_BASE_DEF( void ) 751 FT_Vector_Transform_Scaled( FT_Vector* vector, 752 const FT_Matrix* matrix, 753 FT_Long scaling ) 754 { 755 FT_Pos xz, yz; 756 757 FT_Long val = 0x10000L * scaling; 758 759 760 if ( !vector || !matrix ) 761 return; 762 763 xz = ADD_LONG( FT_MulDiv( vector->x, matrix->xx, val ), 764 FT_MulDiv( vector->y, matrix->xy, val ) ); 765 yz = ADD_LONG( FT_MulDiv( vector->x, matrix->yx, val ), 766 FT_MulDiv( vector->y, matrix->yy, val ) ); 767 768 vector->x = xz; 769 vector->y = yz; 770 } 771 772 773 /* documentation is in ftcalc.h */ 774 775 FT_BASE_DEF( FT_UInt32 ) 776 FT_Vector_NormLen( FT_Vector* vector ) 777 { 778 FT_Int32 x_ = vector->x; 779 FT_Int32 y_ = vector->y; 780 FT_Int32 b, z; 781 FT_UInt32 x, y, u, v, l; 782 FT_Int sx = 1, sy = 1, shift; 783 784 785 x = (FT_UInt32)x_; 786 y = (FT_UInt32)y_; 787 788 FT_MOVE_SIGN( x_, x, sx ); 789 FT_MOVE_SIGN( y_, y, sy ); 790 791 /* trivial cases */ 792 if ( x == 0 ) 793 { 794 if ( y > 0 ) 795 vector->y = sy * 0x10000; 796 return y; 797 } 798 else if ( y == 0 ) 799 { 800 if ( x > 0 ) 801 vector->x = sx * 0x10000; 802 return x; 803 } 804 805 /* Estimate length and prenormalize by shifting so that */ 806 /* the new approximate length is between 2/3 and 4/3. */ 807 /* The magic constant 0xAAAAAAAAUL (2/3 of 2^32) helps */ 808 /* achieve this in 16.16 fixed-point representation. */ 809 l = x > y ? x + ( y >> 1 ) 810 : y + ( x >> 1 ); 811 812 shift = 31 - FT_MSB( l ); 813 shift -= 15 + ( l >= ( 0xAAAAAAAAUL >> shift ) ); 814 815 if ( shift > 0 ) 816 { 817 x <<= shift; 818 y <<= shift; 819 820 /* re-estimate length for tiny vectors */ 821 l = x > y ? x + ( y >> 1 ) 822 : y + ( x >> 1 ); 823 } 824 else 825 { 826 x >>= -shift; 827 y >>= -shift; 828 l >>= -shift; 829 } 830 831 /* lower linear approximation for reciprocal length minus one */ 832 b = 0x10000 - (FT_Int32)l; 833 834 x_ = (FT_Int32)x; 835 y_ = (FT_Int32)y; 836 837 /* Newton's iterations */ 838 do 839 { 840 u = (FT_UInt32)( x_ + ( x_ * b >> 16 ) ); 841 v = (FT_UInt32)( y_ + ( y_ * b >> 16 ) ); 842 843 /* Normalized squared length in the parentheses approaches 2^32. */ 844 /* On two's complement systems, converting to signed gives the */ 845 /* difference with 2^32 even if the expression wraps around. */ 846 z = -(FT_Int32)( u * u + v * v ) / 0x200; 847 z = z * ( ( 0x10000 + b ) >> 8 ) / 0x10000; 848 849 b += z; 850 851 } while ( z > 0 ); 852 853 vector->x = sx < 0 ? -(FT_Pos)u : (FT_Pos)u; 854 vector->y = sy < 0 ? -(FT_Pos)v : (FT_Pos)v; 855 856 /* Conversion to signed helps to recover from likely wrap around */ 857 /* in calculating the prenormalized length, because it gives the */ 858 /* correct difference with 2^32 on two's complement systems. */ 859 l = (FT_UInt32)( 0x10000 + (FT_Int32)( u * x + v * y ) / 0x10000 ); 860 if ( shift > 0 ) 861 l = ( l + ( 1 << ( shift - 1 ) ) ) >> shift; 862 else 863 l <<= -shift; 864 865 return l; 866 } 867 868 869 #if 0 870 871 /* documentation is in ftcalc.h */ 872 873 FT_BASE_DEF( FT_Int32 ) 874 FT_SqrtFixed( FT_Int32 x ) 875 { 876 FT_UInt32 root, rem_hi, rem_lo, test_div; 877 FT_Int count; 878 879 880 root = 0; 881 882 if ( x > 0 ) 883 { 884 rem_hi = 0; 885 rem_lo = (FT_UInt32)x; 886 count = 24; 887 do 888 { 889 rem_hi = ( rem_hi << 2 ) | ( rem_lo >> 30 ); 890 rem_lo <<= 2; 891 root <<= 1; 892 test_div = ( root << 1 ) + 1; 893 894 if ( rem_hi >= test_div ) 895 { 896 rem_hi -= test_div; 897 root += 1; 898 } 899 } while ( --count ); 900 } 901 902 return (FT_Int32)root; 903 } 904 905 #endif /* 0 */ 906 907 908 /* documentation is in ftcalc.h */ 909 910 FT_BASE_DEF( FT_Int ) 911 ft_corner_orientation( FT_Pos in_x, 912 FT_Pos in_y, 913 FT_Pos out_x, 914 FT_Pos out_y ) 915 { 916 #ifdef FT_LONG64 917 918 FT_Int64 delta = (FT_Int64)in_x * out_y - (FT_Int64)in_y * out_x; 919 920 921 return ( delta > 0 ) - ( delta < 0 ); 922 923 #else 924 925 FT_Int result; 926 927 928 /* we silently ignore overflow errors, since such large values */ 929 /* lead to even more (harmless) rendering errors later on */ 930 if ( ADD_LONG( FT_ABS( in_x ), FT_ABS( out_y ) ) <= 131071L && 931 ADD_LONG( FT_ABS( in_y ), FT_ABS( out_x ) ) <= 131071L ) 932 { 933 FT_Long z1 = MUL_LONG( in_x, out_y ); 934 FT_Long z2 = MUL_LONG( in_y, out_x ); 935 936 937 if ( z1 > z2 ) 938 result = +1; 939 else if ( z1 < z2 ) 940 result = -1; 941 else 942 result = 0; 943 } 944 else /* products might overflow 32 bits */ 945 { 946 FT_Int64 z1, z2; 947 948 949 /* XXX: this function does not allow 64-bit arguments */ 950 ft_multo64( (FT_UInt32)in_x, (FT_UInt32)out_y, &z1 ); 951 ft_multo64( (FT_UInt32)in_y, (FT_UInt32)out_x, &z2 ); 952 953 if ( z1.hi > z2.hi ) 954 result = +1; 955 else if ( z1.hi < z2.hi ) 956 result = -1; 957 else if ( z1.lo > z2.lo ) 958 result = +1; 959 else if ( z1.lo < z2.lo ) 960 result = -1; 961 else 962 result = 0; 963 } 964 965 /* XXX: only the sign of return value, +1/0/-1 must be used */ 966 return result; 967 968 #endif 969 } 970 971 972 /* documentation is in ftcalc.h */ 973 974 FT_BASE_DEF( FT_Int ) 975 ft_corner_is_flat( FT_Pos in_x, 976 FT_Pos in_y, 977 FT_Pos out_x, 978 FT_Pos out_y ) 979 { 980 FT_Pos ax = in_x + out_x; 981 FT_Pos ay = in_y + out_y; 982 983 FT_Pos d_in, d_out, d_hypot; 984 985 986 /* The idea of this function is to compare the length of the */ 987 /* hypotenuse with the `in' and `out' length. The `corner' */ 988 /* represented by `in' and `out' is flat if the hypotenuse's */ 989 /* length isn't too large. */ 990 /* */ 991 /* This approach has the advantage that the angle between */ 992 /* `in' and `out' is not checked. In case one of the two */ 993 /* vectors is `dominant', this is, much larger than the */ 994 /* other vector, we thus always have a flat corner. */ 995 /* */ 996 /* hypotenuse */ 997 /* x---------------------------x */ 998 /* \ / */ 999 /* \ / */ 1000 /* in \ / out */ 1001 /* \ / */ 1002 /* o */ 1003 /* Point */ 1004 1005 d_in = FT_HYPOT( in_x, in_y ); 1006 d_out = FT_HYPOT( out_x, out_y ); 1007 d_hypot = FT_HYPOT( ax, ay ); 1008 1009 /* now do a simple length comparison: */ 1010 /* */ 1011 /* d_in + d_out < 17/16 d_hypot */ 1012 1013 return ( d_in + d_out - d_hypot ) < ( d_hypot >> 4 ); 1014 } 1015 1016 1017 /* END */ 1018
