1 /***************************************************************************/
5 /* Arithmetic computations (body). */
7 /* Copyright 1996-2000 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(), and FT_DivFix() are declared in freetype.h. */
31 /*************************************************************************/
34 #include <freetype/internal/ftcalc.h>
35 #include <freetype/internal/ftdebug.h>
36 #include <freetype/internal/ftobjs.h> /* for ABS() */
39 /*************************************************************************/
41 /* The macro FT_COMPONENT is used in trace mode. It is an implicit */
42 /* parameter of the FT_TRACE() and FT_ERROR() macros, used to print/log */
43 /* messages during execution. */
46 #define FT_COMPONENT trace_calc
49 #ifdef FT_CONFIG_OPTION_OLD_CALCS
51 static const FT_Long ft_square_roots[63] =
53 1L, 1L, 2L, 3L, 4L, 5L, 8L, 11L,
54 16L, 22L, 32L, 45L, 64L, 90L, 128L, 181L,
55 256L, 362L, 512L, 724L, 1024L, 1448L, 2048L, 2896L,
56 4096L, 5892L, 8192L, 11585L, 16384L, 23170L, 32768L, 46340L,
58 65536L, 92681L, 131072L, 185363L, 262144L, 370727L,
59 524288L, 741455L, 1048576L, 1482910L, 2097152L, 2965820L,
60 4194304L, 5931641L, 8388608L, 11863283L, 16777216L, 23726566L,
62 33554432L, 47453132L, 67108864L, 94906265L,
63 134217728L, 189812531L, 268435456L, 379625062L,
64 536870912L, 759250125L, 1073741824L, 1518500250L,
70 /*************************************************************************/
76 /* Computes the square root of an Int32 integer (which will be */
77 /* handled as an unsigned long value). */
80 /* x :: The value to compute the root for. */
83 /* The result of `sqrt(x)'. */
85 FT_EXPORT_FUNC( FT_Int32 ) FT_Sqrt32( FT_Int32 x )
87 FT_ULong val, root, newroot, mask;
96 newroot = root + mask;
100 root = newroot + mask;
106 } while ( mask != 0 );
111 #endif /* FT_CONFIG_OPTION_OLD_CALCS */
116 /*************************************************************************/
122 /* A very simple function used to perform the computation `(a*b)/c' */
123 /* with maximal accuracy (it uses a 64-bit intermediate integer */
124 /* whenever necessary). */
126 /* This function isn't necessarily as fast as some processor specific */
127 /* operations, but is at least completely portable. */
130 /* a :: The first multiplier. */
131 /* b :: The second multiplier. */
132 /* c :: The divisor. */
135 /* The result of `(a*b)/c'. This function never traps when trying to */
136 /* divide by zero; it simply returns `MaxInt' or `MinInt' depending */
137 /* on the signs of `a' and `b'. */
139 FT_EXPORT_FUNC( FT_Long ) FT_MulDiv( FT_Long a,
147 if ( a < 0 ) { a = -a; s = -s; }
148 if ( b < 0 ) { b = -b; s = -s; }
149 if ( c < 0 ) { c = -c; s = -s; }
151 return s * ( c > 0 ? ( (FT_Int64)a * b + ( c >> 1 ) ) / c
156 /*************************************************************************/
162 /* A very simple function used to perform the computation */
163 /* `(a*b)/0x10000' with maximal accuracy. Most of the time this is */
164 /* used to multiply a given value by a 16.16 fixed float factor. */
167 /* a :: The first multiplier. */
168 /* b :: The second multiplier. Use a 16.16 factor here whenever */
169 /* possible (see note below). */
172 /* The result of `(a*b)/0x10000'. */
175 /* This function has been optimized for the case where the absolute */
176 /* value of `a' is less than 2048, and `b' is a 16.16 scaling factor. */
177 /* As this happens mainly when scaling from notional units to */
178 /* fractional pixels in FreeType, it resulted in noticeable speed */
179 /* improvements between versions 2.x and 1.x. */
181 /* As a conclusion, always try to place a 16.16 factor as the */
182 /* _second_ argument of this function; this can make a great */
185 FT_EXPORT_FUNC( FT_Long ) FT_MulFix( FT_Long a,
192 if ( a < 0 ) { a = -a; s = -s; }
193 if ( b < 0 ) { b = -b; s = -s; }
195 return s * (FT_Long)( ( (FT_Int64)a * b + 0x8000 ) >> 16 );
199 /*************************************************************************/
205 /* A very simple function used to perform the computation */
206 /* `(a*0x10000)/b' with maximal accuracy. Most of the time, this is */
207 /* used to divide a given value by a 16.16 fixed float factor. */
210 /* a :: The first multiplier. */
211 /* b :: The second multiplier. Use a 16.16 factor here whenever */
212 /* possible (see note below). */
215 /* The result of `(a*0x10000)/b'. */
218 /* The optimization for FT_DivFix() is simple: If (a << 16) fits in */
219 /* 32 bits, then the division is computed directly. Otherwise, we */
220 /* use a specialized version of the old FT_MulDiv64(). */
222 FT_EXPORT_FUNC( FT_Long ) FT_DivFix( FT_Long a,
233 /* check for division by 0 */
236 /* compute result directly */
237 q = ( (FT_Int64)a << 16 ) / b;
239 return (FT_Int32)( s < 0 ? -q : q );
243 #ifdef FT_CONFIG_OPTION_OLD_CALCS
245 /* a helper function for FT_Sqrt64() */
248 int ft_order64( FT_Int64 z )
255 z = (unsigned FT_INT64)z >> 1;
262 /*************************************************************************/
268 /* Computes the square root of a 64-bit value. That sounds stupid, */
269 /* but it is needed to obtain maximal accuracy in the TrueType */
270 /* bytecode interpreter. */
273 /* l :: A 64-bit integer. */
276 /* The 32-bit square-root. */
278 FT_EXPORT_FUNC( FT_Int32 ) FT_Sqrt64( FT_Int64 l )
283 if ( l <= 0 ) return 0;
284 if ( l == 1 ) return 1;
286 r = ft_square_roots[ft_order64( l )];
291 r = ( r + l / r ) >> 1;
293 } while ( r > s || r * r > l );
298 #endif /* FT_CONFIG_OPTION_OLD_CALCS */
301 #else /* FT_LONG64 */
304 /*************************************************************************/
310 /* A very simple function used to perform the computation `(a*b)/c' */
311 /* with maximal accuracy (it uses a 64-bit intermediate integer */
312 /* whenever necessary). */
314 /* This function isn't necessarily as fast as some processor specific */
315 /* operations, but is at least completely portable. */
318 /* a :: The first multiplier. */
319 /* b :: The second multiplier. */
320 /* c :: The divisor. */
323 /* The result of `(a*b)/c'. This function never traps when trying to */
324 /* divide by zero; it simply returns `MaxInt' or `MinInt' depending */
325 /* on the signs of `a' and `b'. */
328 /* The FT_MulDiv() function has been optimized thanks to ideas from */
329 /* Graham Asher. The trick is to optimize computation if everything */
330 /* fits within 32 bits (a rather common case). */
332 /* We compute `a*b+c/2', then divide it by `c' (positive values). */
334 /* 46340 is FLOOR(SQRT(2^31-1)). */
336 /* if ( a <= 46340 && b <= 46340 ) then ( a*b <= 0x7FFEA810 ) */
338 /* 0x7FFFFFFF - 0x7FFEA810 = 0x157F0 */
340 /* if ( c < 0x157F0*2 ) then ( a*b+c/2 <= 0x7FFFFFFF ) */
342 /* and 2*0x157F0 = 176096. */
344 FT_EXPORT_FUNC( FT_Long ) FT_MulDiv( FT_Long a,
351 if ( a == 0 || b == c )
355 s ^= b; b = ABS( b );
356 s ^= c; c = ABS( c );
358 if ( a <= 46340 && b <= 46340 && c <= 176095L && c > 0 )
360 a = ( a * b + ( c >> 1 ) ) / c;
364 FT_Int64 temp, temp2;
367 FT_MulTo64( a, b, &temp );
368 temp2.hi = (FT_Int32)( c >> 31 );
369 temp2.lo = (FT_UInt32)( c / 2 );
370 FT_Add64( &temp, &temp2, &temp );
371 a = FT_Div64by32( &temp, c );
376 return ( s < 0 ? -a : a );
380 /*************************************************************************/
386 /* A very simple function used to perform the computation */
387 /* `(a*b)/0x10000' with maximal accuracy. Most of the time, this is */
388 /* used to multiply a given value by a 16.16 fixed float factor. */
391 /* a :: The first multiplier. */
392 /* b :: The second multiplier. Use a 16.16 factor here whenever */
393 /* possible (see note below). */
396 /* The result of `(a*b)/0x10000'. */
399 /* The optimization for FT_MulFix() is different. We could simply be */
400 /* happy by applying the same principles as with FT_MulDiv(), because */
402 /* c = 0x10000 < 176096 */
404 /* However, in most cases, we have a `b' with a value around 0x10000 */
405 /* which is greater than 46340. */
407 /* According to some testing, most cases have `a' < 2048, so a good */
408 /* idea is to use bounds like 2048 and 1048576 (=floor((2^31-1)/2048) */
409 /* for `a' and `b', respectively. */
411 FT_EXPORT_FUNC( FT_Long ) FT_MulFix( FT_Long a,
418 if ( a == 0 || b == 0x10000L )
427 if ( ua <= 2048 && ub <= 1048576L )
429 ua = ( ua * ub + 0x8000 ) >> 16;
433 FT_ULong al = ua & 0xFFFF;
436 ua = ( ua >> 16 ) * ub +
438 ( al * ( ub & 0xFFFF ) >> 16 );
441 return ( s < 0 ? -(FT_Long)ua : ua );
445 /*************************************************************************/
451 /* A very simple function used to perform the computation */
452 /* `(a*0x10000)/b' with maximal accuracy. Most of the time, this is */
453 /* used to divide a given value by a 16.16 fixed float factor. */
456 /* a :: The first multiplier. */
457 /* b :: The second multiplier. Use a 16.16 factor here whenever */
458 /* possible (see note below). */
461 /* The result of `(a*0x10000)/b'. */
464 /* The optimization for FT_DivFix() is simple: If (a << 16) fits into */
465 /* 32 bits, then the division is computed directly. Otherwise, we */
466 /* use a specialized version of the old FT_MulDiv64(). */
468 FT_EXPORT_FUNC( FT_Long ) FT_DivFix( FT_Long a,
480 /* check for division by 0 */
483 else if ( ( a >> 16 ) == 0 )
485 /* compute result directly */
486 q = (FT_UInt32)( a << 16 ) / (FT_UInt32)b;
490 /* we need more bits; we have to do it by hand */
497 /* we must compute C*0x10000/B: we simply shift C and B so */
498 /* C becomes smaller than 16 bits */
505 q += ( c << 16 ) / b;
508 return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
512 /*************************************************************************/
518 /* Add two Int64 values. */
521 /* x :: A pointer to the first value to be added. */
522 /* y :: A pointer to the second value to be added. */
525 /* z :: A pointer to the result of `x + y'. */
528 /* Will be wrapped by the ADD_64() macro. */
530 FT_EXPORT_FUNC( void ) FT_Add64( FT_Int64* x,
534 register FT_UInt32 lo, hi;
538 hi = x->hi + y->hi + ( lo < x->lo );
545 /*************************************************************************/
551 /* Multiplies two Int32 integers. Returns an Int64 integer. */
554 /* x :: The first multiplier. */
555 /* y :: The second multiplier. */
558 /* z :: A pointer to the result of `x * y'. */
561 /* Will be wrapped by the MUL_64() macro. */
563 FT_EXPORT_FUNC( void ) FT_MulTo64( FT_Int32 x,
571 s ^= y; y = ABS( y );
574 FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2;
577 lo1 = x & 0x0000FFFF; hi1 = x >> 16;
578 lo2 = y & 0x0000FFFF; hi2 = y >> 16;
585 /* Check carry overflow of i1 + i2 */
593 /* Check carry overflow of i1 + lo */
603 z->lo = (FT_UInt32)-(FT_Int32)z->lo;
604 z->hi = ~z->hi + !( z->lo );
609 /*************************************************************************/
615 /* Divides an Int64 value by an Int32 value. Returns an Int32 */
619 /* x :: A pointer to the dividend. */
620 /* y :: The divisor. */
623 /* The result of `x / y'. */
626 /* Will be wrapped by the DIV_64() macro. */
628 FT_EXPORT_FUNC( FT_Int32 ) FT_Div64by32( FT_Int64* x,
632 FT_UInt32 q, r, i, lo;
638 x->lo = (FT_UInt32)-(FT_Int32)x->lo;
639 x->hi = ~x->hi + !( x->lo );
641 s ^= y; y = ABS( y );
651 return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
657 if ( r >= (FT_UInt32)y ) /* we know y is to be treated as unsigned here */
658 return ( s < 0 ? 0x80000001UL : 0x7FFFFFFFUL );
659 /* Return Max/Min Int32 if division overflow. */
660 /* This includes division by zero! */
662 for ( i = 0; i < 32; i++ )
668 if ( r >= (FT_UInt32)y )
676 return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
680 #ifdef FT_CONFIG_OPTION_OLD_CALCS
683 /* two helper functions for FT_Sqrt64() */
686 void FT_Sub64( FT_Int64* x,
690 register FT_UInt32 lo, hi;
694 hi = x->hi - y->hi - ( (FT_Int32)lo < 0 );
702 int ft_order64( FT_Int64* z )
725 /*************************************************************************/
731 /* Computes the square root of a 64-bits value. That sounds stupid, */
732 /* but it is needed to obtain maximal accuracy in the TrueType */
733 /* bytecode interpreter. */
736 /* z :: A pointer to a 64-bit integer. */
739 /* The 32-bit square-root. */
741 FT_EXPORT_FUNC( FT_Int32 ) FT_Sqrt64( FT_Int64* l )
747 if ( (FT_Int32)l->hi < 0 ||
748 ( l->hi == 0 && l->lo == 0 ) )
755 r = ft_square_roots[s];
759 r = ( r + FT_Div64by32( l, r ) ) >> 1;
760 FT_MulTo64( r, r, &l2 );
761 FT_Sub64 ( l, &l2, &l2 );
763 } while ( r > s || (FT_Int32)l2.hi < 0 );
768 #endif /* FT_CONFIG_OPTION_OLD_CALCS */
770 #endif /* FT_LONG64 */