1 /***************************************************************************/ 2 /* */ 3 /* ftcalc.c */ 4 /* */ 5 /* Arithmetic computations (body). */ 6 /* */ 7 /* Copyright 1996-2017 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 #define FT_MOVE_SIGN( x, s ) \ 73 FT_BEGIN_STMNT \ 74 if ( x < 0 ) \ 75 { \ 76 x = -x; \ 77 s = -s; \ 78 } \ 79 FT_END_STMNT 80 81 /* The following three functions are available regardless of whether */ 82 /* FT_LONG64 is defined. */ 83 84 /* documentation is in freetype.h */ 85 86 FT_EXPORT_DEF( FT_Fixed ) 87 FT_RoundFix( FT_Fixed a ) 88 { 89 return ( a + 0x8000L - ( a < 0 ) ) & ~0xFFFFL; 90 } 91 92 93 /* documentation is in freetype.h */ 94 95 FT_EXPORT_DEF( FT_Fixed ) 96 FT_CeilFix( FT_Fixed a ) 97 { 98 return ( a + 0xFFFFL ) & ~0xFFFFL; 99 } 100 101 102 /* documentation is in freetype.h */ 103 104 FT_EXPORT_DEF( FT_Fixed ) 105 FT_FloorFix( FT_Fixed a ) 106 { 107 return a & ~0xFFFFL; 108 } 109 110 #ifndef FT_MSB 111 112 FT_BASE_DEF ( FT_Int ) 113 FT_MSB( FT_UInt32 z ) 114 { 115 FT_Int shift = 0; 116 117 118 /* determine msb bit index in `shift' */ 119 if ( z & 0xFFFF0000UL ) 120 { 121 z >>= 16; 122 shift += 16; 123 } 124 if ( z & 0x0000FF00UL ) 125 { 126 z >>= 8; 127 shift += 8; 128 } 129 if ( z & 0x000000F0UL ) 130 { 131 z >>= 4; 132 shift += 4; 133 } 134 if ( z & 0x0000000CUL ) 135 { 136 z >>= 2; 137 shift += 2; 138 } 139 if ( z & 0x00000002UL ) 140 { 141 /* z >>= 1; */ 142 shift += 1; 143 } 144 145 return shift; 146 } 147 148 #endif /* !FT_MSB */ 149 150 151 /* documentation is in ftcalc.h */ 152 153 FT_BASE_DEF( FT_Fixed ) 154 FT_Hypot( FT_Fixed x, 155 FT_Fixed y ) 156 { 157 FT_Vector v; 158 159 160 v.x = x; 161 v.y = y; 162 163 return FT_Vector_Length( &v ); 164 } 165 166 167 #ifdef FT_LONG64 168 169 170 /* documentation is in freetype.h */ 171 172 FT_EXPORT_DEF( FT_Long ) 173 FT_MulDiv( FT_Long a_, 174 FT_Long b_, 175 FT_Long c_ ) 176 { 177 FT_Int s = 1; 178 FT_UInt64 a, b, c, d; 179 FT_Long d_; 180 181 182 FT_MOVE_SIGN( a_, s ); 183 FT_MOVE_SIGN( b_, s ); 184 FT_MOVE_SIGN( c_, s ); 185 186 a = (FT_UInt64)a_; 187 b = (FT_UInt64)b_; 188 c = (FT_UInt64)c_; 189 190 d = c > 0 ? ( a * b + ( c >> 1 ) ) / c 191 : 0x7FFFFFFFUL; 192 193 d_ = (FT_Long)d; 194 195 return s < 0 ? -d_ : d_; 196 } 197 198 199 /* documentation is in ftcalc.h */ 200 201 FT_BASE_DEF( FT_Long ) 202 FT_MulDiv_No_Round( FT_Long a_, 203 FT_Long b_, 204 FT_Long c_ ) 205 { 206 FT_Int s = 1; 207 FT_UInt64 a, b, c, d; 208 FT_Long d_; 209 210 211 FT_MOVE_SIGN( a_, s ); 212 FT_MOVE_SIGN( b_, s ); 213 FT_MOVE_SIGN( c_, s ); 214 215 a = (FT_UInt64)a_; 216 b = (FT_UInt64)b_; 217 c = (FT_UInt64)c_; 218 219 d = c > 0 ? a * b / c 220 : 0x7FFFFFFFUL; 221 222 d_ = (FT_Long)d; 223 224 return s < 0 ? -d_ : d_; 225 } 226 227 228 /* documentation is in freetype.h */ 229 230 FT_EXPORT_DEF( FT_Long ) 231 FT_MulFix( FT_Long a_, 232 FT_Long b_ ) 233 { 234 #ifdef FT_MULFIX_ASSEMBLER 235 236 return FT_MULFIX_ASSEMBLER( (FT_Int32)a_, (FT_Int32)b_ ); 237 238 #else 239 240 FT_Int64 ab = (FT_Int64)a_ * (FT_Int64)b_; 241 242 /* this requires arithmetic right shift of signed numbers */ 243 return (FT_Long)( ( ab + 0x8000L - ( ab < 0 ) ) >> 16 ); 244 245 #endif /* FT_MULFIX_ASSEMBLER */ 246 } 247 248 249 /* documentation is in freetype.h */ 250 251 FT_EXPORT_DEF( FT_Long ) 252 FT_DivFix( FT_Long a_, 253 FT_Long b_ ) 254 { 255 FT_Int s = 1; 256 FT_UInt64 a, b, q; 257 FT_Long q_; 258 259 260 FT_MOVE_SIGN( a_, s ); 261 FT_MOVE_SIGN( b_, s ); 262 263 a = (FT_UInt64)a_; 264 b = (FT_UInt64)b_; 265 266 q = b > 0 ? ( ( a << 16 ) + ( b >> 1 ) ) / b 267 : 0x7FFFFFFFUL; 268 269 q_ = (FT_Long)q; 270 271 return s < 0 ? -q_ : q_; 272 } 273 274 275 #else /* !FT_LONG64 */ 276 277 278 static void 279 ft_multo64( FT_UInt32 x, 280 FT_UInt32 y, 281 FT_Int64 *z ) 282 { 283 FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2; 284 285 286 lo1 = x & 0x0000FFFFU; hi1 = x >> 16; 287 lo2 = y & 0x0000FFFFU; hi2 = y >> 16; 288 289 lo = lo1 * lo2; 290 i1 = lo1 * hi2; 291 i2 = lo2 * hi1; 292 hi = hi1 * hi2; 293 294 /* Check carry overflow of i1 + i2 */ 295 i1 += i2; 296 hi += (FT_UInt32)( i1 < i2 ) << 16; 297 298 hi += i1 >> 16; 299 i1 = i1 << 16; 300 301 /* Check carry overflow of i1 + lo */ 302 lo += i1; 303 hi += ( lo < i1 ); 304 305 z->lo = lo; 306 z->hi = hi; 307 } 308 309 310 static FT_UInt32 311 ft_div64by32( FT_UInt32 hi, 312 FT_UInt32 lo, 313 FT_UInt32 y ) 314 { 315 FT_UInt32 r, q; 316 FT_Int i; 317 318 319 if ( hi >= y ) 320 return (FT_UInt32)0x7FFFFFFFL; 321 322 /* We shift as many bits as we can into the high register, perform */ 323 /* 32-bit division with modulo there, then work through the remaining */ 324 /* bits with long division. This optimization is especially noticeable */ 325 /* for smaller dividends that barely use the high register. */ 326 327 i = 31 - FT_MSB( hi ); 328 r = ( hi << i ) | ( lo >> ( 32 - i ) ); lo <<= i; /* left 64-bit shift */ 329 q = r / y; 330 r -= q * y; /* remainder */ 331 332 i = 32 - i; /* bits remaining in low register */ 333 do 334 { 335 q <<= 1; 336 r = ( r << 1 ) | ( lo >> 31 ); lo <<= 1; 337 338 if ( r >= y ) 339 { 340 r -= y; 341 q |= 1; 342 } 343 } while ( --i ); 344 345 return q; 346 } 347 348 349 static void 350 FT_Add64( FT_Int64* x, 351 FT_Int64* y, 352 FT_Int64 *z ) 353 { 354 FT_UInt32 lo, hi; 355 356 357 lo = x->lo + y->lo; 358 hi = x->hi + y->hi + ( lo < x->lo ); 359 360 z->lo = lo; 361 z->hi = hi; 362 } 363 364 365 /* The FT_MulDiv function has been optimized thanks to ideas from */ 366 /* Graham Asher and Alexei Podtelezhnikov. The trick is to optimize */ 367 /* a rather common case when everything fits within 32-bits. */ 368 /* */ 369 /* We compute 'a*b+c/2', then divide it by 'c' (all positive values). */ 370 /* */ 371 /* The product of two positive numbers never exceeds the square of */ 372 /* its mean values. Therefore, we always avoid the overflow by */ 373 /* imposing */ 374 /* */ 375 /* (a + b) / 2 <= sqrt(X - c/2) , */ 376 /* */ 377 /* where X = 2^32 - 1, the maximum unsigned 32-bit value, and using */ 378 /* unsigned arithmetic. Now we replace `sqrt' with a linear function */ 379 /* that is smaller or equal for all values of c in the interval */ 380 /* [0;X/2]; it should be equal to sqrt(X) and sqrt(3X/4) at the */ 381 /* endpoints. Substituting the linear solution and explicit numbers */ 382 /* we get */ 383 /* */ 384 /* a + b <= 131071.99 - c / 122291.84 . */ 385 /* */ 386 /* In practice, we should use a faster and even stronger inequality */ 387 /* */ 388 /* a + b <= 131071 - (c >> 16) */ 389 /* */ 390 /* or, alternatively, */ 391 /* */ 392 /* a + b <= 129894 - (c >> 17) . */ 393 /* */ 394 /* FT_MulFix, on the other hand, is optimized for a small value of */ 395 /* the first argument, when the second argument can be much larger. */ 396 /* This can be achieved by scaling the second argument and the limit */ 397 /* in the above inequalities. For example, */ 398 /* */ 399 /* a + (b >> 8) <= (131071 >> 4) */ 400 /* */ 401 /* covers the practical range of use. The actual test below is a bit */ 402 /* tighter to avoid the border case overflows. */ 403 /* */ 404 /* In the case of FT_DivFix, the exact overflow check */ 405 /* */ 406 /* a << 16 <= X - c/2 */ 407 /* */ 408 /* is scaled down by 2^16 and we use */ 409 /* */ 410 /* a <= 65535 - (c >> 17) . */ 411 412 /* documentation is in freetype.h */ 413 414 FT_EXPORT_DEF( FT_Long ) 415 FT_MulDiv( FT_Long a_, 416 FT_Long b_, 417 FT_Long c_ ) 418 { 419 FT_Int s = 1; 420 FT_UInt32 a, b, c; 421 422 423 /* XXX: this function does not allow 64-bit arguments */ 424 425 FT_MOVE_SIGN( a_, s ); 426 FT_MOVE_SIGN( b_, s ); 427 FT_MOVE_SIGN( c_, s ); 428 429 a = (FT_UInt32)a_; 430 b = (FT_UInt32)b_; 431 c = (FT_UInt32)c_; 432 433 if ( c == 0 ) 434 a = 0x7FFFFFFFUL; 435 436 else if ( a + b <= 129894UL - ( c >> 17 ) ) 437 a = ( a * b + ( c >> 1 ) ) / c; 438 439 else 440 { 441 FT_Int64 temp, temp2; 442 443 444 ft_multo64( a, b, &temp ); 445 446 temp2.hi = 0; 447 temp2.lo = c >> 1; 448 449 FT_Add64( &temp, &temp2, &temp ); 450 451 /* last attempt to ditch long division */ 452 a = ( temp.hi == 0 ) ? temp.lo / c 453 : ft_div64by32( temp.hi, temp.lo, c ); 454 } 455 456 a_ = (FT_Long)a; 457 458 return s < 0 ? -a_ : a_; 459 } 460 461 462 FT_BASE_DEF( FT_Long ) 463 FT_MulDiv_No_Round( FT_Long a_, 464 FT_Long b_, 465 FT_Long c_ ) 466 { 467 FT_Int s = 1; 468 FT_UInt32 a, b, c; 469 470 471 /* XXX: this function does not allow 64-bit arguments */ 472 473 FT_MOVE_SIGN( a_, s ); 474 FT_MOVE_SIGN( b_, s ); 475 FT_MOVE_SIGN( c_, s ); 476 477 a = (FT_UInt32)a_; 478 b = (FT_UInt32)b_; 479 c = (FT_UInt32)c_; 480 481 if ( c == 0 ) 482 a = 0x7FFFFFFFUL; 483 484 else if ( a + b <= 131071UL ) 485 a = a * b / c; 486 487 else 488 { 489 FT_Int64 temp; 490 491 492 ft_multo64( a, b, &temp ); 493 494 /* last attempt to ditch long division */ 495 a = ( temp.hi == 0 ) ? temp.lo / c 496 : ft_div64by32( temp.hi, temp.lo, c ); 497 } 498 499 a_ = (FT_Long)a; 500 501 return s < 0 ? -a_ : a_; 502 } 503 504 505 /* documentation is in freetype.h */ 506 507 FT_EXPORT_DEF( FT_Long ) 508 FT_MulFix( FT_Long a_, 509 FT_Long b_ ) 510 { 511 #ifdef FT_MULFIX_ASSEMBLER 512 513 return FT_MULFIX_ASSEMBLER( a_, b_ ); 514 515 #elif 0 516 517 /* 518 * This code is nonportable. See comment below. 519 * 520 * However, on a platform where right-shift of a signed quantity fills 521 * the leftmost bits by copying the sign bit, it might be faster. 522 */ 523 524 FT_Long sa, sb; 525 FT_UInt32 a, b; 526 527 528 /* 529 * This is a clever way of converting a signed number `a' into its 530 * absolute value (stored back into `a') and its sign. The sign is 531 * stored in `sa'; 0 means `a' was positive or zero, and -1 means `a' 532 * was negative. (Similarly for `b' and `sb'). 533 * 534 * Unfortunately, it doesn't work (at least not portably). 535 * 536 * It makes the assumption that right-shift on a negative signed value 537 * fills the leftmost bits by copying the sign bit. This is wrong. 538 * According to K&R 2nd ed, section `A7.8 Shift Operators' on page 206, 539 * the result of right-shift of a negative signed value is 540 * implementation-defined. At least one implementation fills the 541 * leftmost bits with 0s (i.e., it is exactly the same as an unsigned 542 * right shift). This means that when `a' is negative, `sa' ends up 543 * with the value 1 rather than -1. After that, everything else goes 544 * wrong. 545 */ 546 sa = ( a_ >> ( sizeof ( a_ ) * 8 - 1 ) ); 547 a = ( a_ ^ sa ) - sa; 548 sb = ( b_ >> ( sizeof ( b_ ) * 8 - 1 ) ); 549 b = ( b_ ^ sb ) - sb; 550 551 a = (FT_UInt32)a_; 552 b = (FT_UInt32)b_; 553 554 if ( a + ( b >> 8 ) <= 8190UL ) 555 a = ( a * b + 0x8000U ) >> 16; 556 else 557 { 558 FT_UInt32 al = a & 0xFFFFUL; 559 560 561 a = ( a >> 16 ) * b + al * ( b >> 16 ) + 562 ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 ); 563 } 564 565 sa ^= sb; 566 a = ( a ^ sa ) - sa; 567 568 return (FT_Long)a; 569 570 #else /* 0 */ 571 572 FT_Int s = 1; 573 FT_UInt32 a, b; 574 575 576 /* XXX: this function does not allow 64-bit arguments */ 577 578 FT_MOVE_SIGN( a_, s ); 579 FT_MOVE_SIGN( b_, s ); 580 581 a = (FT_UInt32)a_; 582 b = (FT_UInt32)b_; 583 584 if ( a + ( b >> 8 ) <= 8190UL ) 585 a = ( a * b + 0x8000UL ) >> 16; 586 else 587 { 588 FT_UInt32 al = a & 0xFFFFUL; 589 590 591 a = ( a >> 16 ) * b + al * ( b >> 16 ) + 592 ( ( al * ( b & 0xFFFFUL ) + 0x8000UL ) >> 16 ); 593 } 594 595 a_ = (FT_Long)a; 596 597 return s < 0 ? -a_ : a_; 598 599 #endif /* 0 */ 600 601 } 602 603 604 /* documentation is in freetype.h */ 605 606 FT_EXPORT_DEF( FT_Long ) 607 FT_DivFix( FT_Long a_, 608 FT_Long b_ ) 609 { 610 FT_Int s = 1; 611 FT_UInt32 a, b, q; 612 FT_Long q_; 613 614 615 /* XXX: this function does not allow 64-bit arguments */ 616 617 FT_MOVE_SIGN( a_, s ); 618 FT_MOVE_SIGN( b_, s ); 619 620 a = (FT_UInt32)a_; 621 b = (FT_UInt32)b_; 622 623 if ( b == 0 ) 624 { 625 /* check for division by 0 */ 626 q = 0x7FFFFFFFUL; 627 } 628 else if ( a <= 65535UL - ( b >> 17 ) ) 629 { 630 /* compute result directly */ 631 q = ( ( a << 16 ) + ( b >> 1 ) ) / b; 632 } 633 else 634 { 635 /* we need more bits; we have to do it by hand */ 636 FT_Int64 temp, temp2; 637 638 639 temp.hi = a >> 16; 640 temp.lo = a << 16; 641 temp2.hi = 0; 642 temp2.lo = b >> 1; 643 644 FT_Add64( &temp, &temp2, &temp ); 645 q = ft_div64by32( temp.hi, temp.lo, b ); 646 } 647 648 q_ = (FT_Long)q; 649 650 return s < 0 ? -q_ : q_; 651 } 652 653 654 #endif /* !FT_LONG64 */ 655 656 657 /* documentation is in ftglyph.h */ 658 659 FT_EXPORT_DEF( void ) 660 FT_Matrix_Multiply( const FT_Matrix* a, 661 FT_Matrix *b ) 662 { 663 FT_Fixed xx, xy, yx, yy; 664 665 666 if ( !a || !b ) 667 return; 668 669 xx = FT_MulFix( a->xx, b->xx ) + FT_MulFix( a->xy, b->yx ); 670 xy = FT_MulFix( a->xx, b->xy ) + FT_MulFix( a->xy, b->yy ); 671 yx = FT_MulFix( a->yx, b->xx ) + FT_MulFix( a->yy, b->yx ); 672 yy = FT_MulFix( a->yx, b->xy ) + FT_MulFix( a->yy, b->yy ); 673 674 b->xx = xx; b->xy = xy; 675 b->yx = yx; b->yy = yy; 676 } 677 678 679 /* documentation is in ftglyph.h */ 680 681 FT_EXPORT_DEF( FT_Error ) 682 FT_Matrix_Invert( FT_Matrix* matrix ) 683 { 684 FT_Pos delta, xx, yy; 685 686 687 if ( !matrix ) 688 return FT_THROW( Invalid_Argument ); 689 690 /* compute discriminant */ 691 delta = FT_MulFix( matrix->xx, matrix->yy ) - 692 FT_MulFix( matrix->xy, matrix->yx ); 693 694 if ( !delta ) 695 return FT_THROW( Invalid_Argument ); /* matrix can't be inverted */ 696 697 matrix->xy = - FT_DivFix( matrix->xy, delta ); 698 matrix->yx = - FT_DivFix( matrix->yx, delta ); 699 700 xx = matrix->xx; 701 yy = matrix->yy; 702 703 matrix->xx = FT_DivFix( yy, delta ); 704 matrix->yy = FT_DivFix( xx, delta ); 705 706 return FT_Err_Ok; 707 } 708 709 710 /* documentation is in ftcalc.h */ 711 712 FT_BASE_DEF( void ) 713 FT_Matrix_Multiply_Scaled( const FT_Matrix* a, 714 FT_Matrix *b, 715 FT_Long scaling ) 716 { 717 FT_Fixed xx, xy, yx, yy; 718 719 FT_Long val = 0x10000L * scaling; 720 721 722 if ( !a || !b ) 723 return; 724 725 xx = FT_MulDiv( a->xx, b->xx, val ) + FT_MulDiv( a->xy, b->yx, val ); 726 xy = FT_MulDiv( a->xx, b->xy, val ) + FT_MulDiv( a->xy, b->yy, val ); 727 yx = FT_MulDiv( a->yx, b->xx, val ) + FT_MulDiv( a->yy, b->yx, val ); 728 yy = FT_MulDiv( a->yx, b->xy, val ) + FT_MulDiv( a->yy, b->yy, val ); 729 730 b->xx = xx; b->xy = xy; 731 b->yx = yx; b->yy = yy; 732 } 733 734 735 /* documentation is in ftcalc.h */ 736 737 FT_BASE_DEF( void ) 738 FT_Vector_Transform_Scaled( FT_Vector* vector, 739 const FT_Matrix* matrix, 740 FT_Long scaling ) 741 { 742 FT_Pos xz, yz; 743 744 FT_Long val = 0x10000L * scaling; 745 746 747 if ( !vector || !matrix ) 748 return; 749 750 xz = FT_MulDiv( vector->x, matrix->xx, val ) + 751 FT_MulDiv( vector->y, matrix->xy, val ); 752 753 yz = FT_MulDiv( vector->x, matrix->yx, val ) + 754 FT_MulDiv( vector->y, matrix->yy, val ); 755 756 vector->x = xz; 757 vector->y = yz; 758 } 759 760 761 /* documentation is in ftcalc.h */ 762 763 FT_BASE_DEF( FT_UInt32 ) 764 FT_Vector_NormLen( FT_Vector* vector ) 765 { 766 FT_Int32 x_ = vector->x; 767 FT_Int32 y_ = vector->y; 768 FT_Int32 b, z; 769 FT_UInt32 x, y, u, v, l; 770 FT_Int sx = 1, sy = 1, shift; 771 772 773 FT_MOVE_SIGN( x_, sx ); 774 FT_MOVE_SIGN( y_, sy ); 775 776 x = (FT_UInt32)x_; 777 y = (FT_UInt32)y_; 778 779 /* trivial cases */ 780 if ( x == 0 ) 781 { 782 if ( y > 0 ) 783 vector->y = sy * 0x10000; 784 return y; 785 } 786 else if ( y == 0 ) 787 { 788 if ( x > 0 ) 789 vector->x = sx * 0x10000; 790 return x; 791 } 792 793 /* Estimate length and prenormalize by shifting so that */ 794 /* the new approximate length is between 2/3 and 4/3. */ 795 /* The magic constant 0xAAAAAAAAUL (2/3 of 2^32) helps */ 796 /* achieve this in 16.16 fixed-point representation. */ 797 l = x > y ? x + ( y >> 1 ) 798 : y + ( x >> 1 ); 799 800 shift = 31 - FT_MSB( l ); 801 shift -= 15 + ( l >= ( 0xAAAAAAAAUL >> shift ) ); 802 803 if ( shift > 0 ) 804 { 805 x <<= shift; 806 y <<= shift; 807 808 /* re-estimate length for tiny vectors */ 809 l = x > y ? x + ( y >> 1 ) 810 : y + ( x >> 1 ); 811 } 812 else 813 { 814 x >>= -shift; 815 y >>= -shift; 816 l >>= -shift; 817 } 818 819 /* lower linear approximation for reciprocal length minus one */ 820 b = 0x10000 - (FT_Int32)l; 821 822 x_ = (FT_Int32)x; 823 y_ = (FT_Int32)y; 824 825 /* Newton's iterations */ 826 do 827 { 828 u = (FT_UInt32)( x_ + ( x_ * b >> 16 ) ); 829 v = (FT_UInt32)( y_ + ( y_ * b >> 16 ) ); 830 831 /* Normalized squared length in the parentheses approaches 2^32. */ 832 /* On two's complement systems, converting to signed gives the */ 833 /* difference with 2^32 even if the expression wraps around. */ 834 z = -(FT_Int32)( u * u + v * v ) / 0x200; 835 z = z * ( ( 0x10000 + b ) >> 8 ) / 0x10000; 836 837 b += z; 838 839 } while ( z > 0 ); 840 841 vector->x = sx < 0 ? -(FT_Pos)u : (FT_Pos)u; 842 vector->y = sy < 0 ? -(FT_Pos)v : (FT_Pos)v; 843 844 /* Conversion to signed helps to recover from likely wrap around */ 845 /* in calculating the prenormalized length, because it gives the */ 846 /* correct difference with 2^32 on two's complement systems. */ 847 l = (FT_UInt32)( 0x10000 + (FT_Int32)( u * x + v * y ) / 0x10000 ); 848 if ( shift > 0 ) 849 l = ( l + ( 1 << ( shift - 1 ) ) ) >> shift; 850 else 851 l <<= -shift; 852 853 return l; 854 } 855 856 857 #if 0 858 859 /* documentation is in ftcalc.h */ 860 861 FT_BASE_DEF( FT_Int32 ) 862 FT_SqrtFixed( FT_Int32 x ) 863 { 864 FT_UInt32 root, rem_hi, rem_lo, test_div; 865 FT_Int count; 866 867 868 root = 0; 869 870 if ( x > 0 ) 871 { 872 rem_hi = 0; 873 rem_lo = (FT_UInt32)x; 874 count = 24; 875 do 876 { 877 rem_hi = ( rem_hi << 2 ) | ( rem_lo >> 30 ); 878 rem_lo <<= 2; 879 root <<= 1; 880 test_div = ( root << 1 ) + 1; 881 882 if ( rem_hi >= test_div ) 883 { 884 rem_hi -= test_div; 885 root += 1; 886 } 887 } while ( --count ); 888 } 889 890 return (FT_Int32)root; 891 } 892 893 #endif /* 0 */ 894 895 896 /* documentation is in ftcalc.h */ 897 898 FT_BASE_DEF( FT_Int ) 899 ft_corner_orientation( FT_Pos in_x, 900 FT_Pos in_y, 901 FT_Pos out_x, 902 FT_Pos out_y ) 903 { 904 #ifdef FT_LONG64 905 906 FT_Int64 delta = (FT_Int64)in_x * out_y - (FT_Int64)in_y * out_x; 907 908 909 return ( delta > 0 ) - ( delta < 0 ); 910 911 #else 912 913 FT_Int result; 914 915 916 if ( (FT_ULong)FT_ABS( in_x ) + (FT_ULong)FT_ABS( out_y ) <= 131071UL && 917 (FT_ULong)FT_ABS( in_y ) + (FT_ULong)FT_ABS( out_x ) <= 131071UL ) 918 { 919 FT_Long z1 = in_x * out_y; 920 FT_Long z2 = in_y * out_x; 921 922 923 if ( z1 > z2 ) 924 result = +1; 925 else if ( z1 < z2 ) 926 result = -1; 927 else 928 result = 0; 929 } 930 else /* products might overflow 32 bits */ 931 { 932 FT_Int64 z1, z2; 933 934 935 /* XXX: this function does not allow 64-bit arguments */ 936 ft_multo64( (FT_UInt32)in_x, (FT_UInt32)out_y, &z1 ); 937 ft_multo64( (FT_UInt32)in_y, (FT_UInt32)out_x, &z2 ); 938 939 if ( z1.hi > z2.hi ) 940 result = +1; 941 else if ( z1.hi < z2.hi ) 942 result = -1; 943 else if ( z1.lo > z2.lo ) 944 result = +1; 945 else if ( z1.lo < z2.lo ) 946 result = -1; 947 else 948 result = 0; 949 } 950 951 /* XXX: only the sign of return value, +1/0/-1 must be used */ 952 return result; 953 954 #endif 955 } 956 957 958 /* documentation is in ftcalc.h */ 959 960 FT_BASE_DEF( FT_Int ) 961 ft_corner_is_flat( FT_Pos in_x, 962 FT_Pos in_y, 963 FT_Pos out_x, 964 FT_Pos out_y ) 965 { 966 FT_Pos ax = in_x + out_x; 967 FT_Pos ay = in_y + out_y; 968 969 FT_Pos d_in, d_out, d_hypot; 970 971 972 /* The idea of this function is to compare the length of the */ 973 /* hypotenuse with the `in' and `out' length. The `corner' */ 974 /* represented by `in' and `out' is flat if the hypotenuse's */ 975 /* length isn't too large. */ 976 /* */ 977 /* This approach has the advantage that the angle between */ 978 /* `in' and `out' is not checked. In case one of the two */ 979 /* vectors is `dominant', this is, much larger than the */ 980 /* other vector, we thus always have a flat corner. */ 981 /* */ 982 /* hypotenuse */ 983 /* x---------------------------x */ 984 /* \ / */ 985 /* \ / */ 986 /* in \ / out */ 987 /* \ / */ 988 /* o */ 989 /* Point */ 990 991 d_in = FT_HYPOT( in_x, in_y ); 992 d_out = FT_HYPOT( out_x, out_y ); 993 d_hypot = FT_HYPOT( ax, ay ); 994 995 /* now do a simple length comparison: */ 996 /* */ 997 /* d_in + d_out < 17/16 d_hypot */ 998 999 return ( d_in + d_out - d_hypot ) < ( d_hypot >> 4 ); 1000 } 1001 1002 1003 /* END */ 1004
