1
2 /*
3 * Copyright 2008 The Android Open Source Project
4 *
5 * Use of this source code is governed by a BSD-style license that can be
6 * found in the LICENSE file.
7 */
8
9
10 #include "SkMath.h"
11 #include "SkCordic.h"
12 #include "SkFloatBits.h"
13 #include "SkFloatingPoint.h"
14 #include "Sk64.h"
15 #include "SkScalar.h"
16
17 #ifdef SK_SCALAR_IS_FLOAT
18 const uint32_t gIEEENotANumber = 0x7FFFFFFF;
19 const uint32_t gIEEEInfinity = 0x7F800000;
20 #endif
21
22 #define sub_shift(zeros, x, n) \
23 zeros -= n; \
24 x >>= n
25
SkCLZ_portable(uint32_t x)26 int SkCLZ_portable(uint32_t x) {
27 if (x == 0) {
28 return 32;
29 }
30
31 #ifdef SK_CPU_HAS_CONDITIONAL_INSTR
32 int zeros = 31;
33 if (x & 0xFFFF0000) {
34 sub_shift(zeros, x, 16);
35 }
36 if (x & 0xFF00) {
37 sub_shift(zeros, x, 8);
38 }
39 if (x & 0xF0) {
40 sub_shift(zeros, x, 4);
41 }
42 if (x & 0xC) {
43 sub_shift(zeros, x, 2);
44 }
45 if (x & 0x2) {
46 sub_shift(zeros, x, 1);
47 }
48 #else
49 int zeros = ((x >> 16) - 1) >> 31 << 4;
50 x <<= zeros;
51
52 int nonzero = ((x >> 24) - 1) >> 31 << 3;
53 zeros += nonzero;
54 x <<= nonzero;
55
56 nonzero = ((x >> 28) - 1) >> 31 << 2;
57 zeros += nonzero;
58 x <<= nonzero;
59
60 nonzero = ((x >> 30) - 1) >> 31 << 1;
61 zeros += nonzero;
62 x <<= nonzero;
63
64 zeros += (~x) >> 31;
65 #endif
66
67 return zeros;
68 }
69
SkMulDiv(int32_t numer1,int32_t numer2,int32_t denom)70 int32_t SkMulDiv(int32_t numer1, int32_t numer2, int32_t denom) {
71 SkASSERT(denom);
72
73 Sk64 tmp;
74 tmp.setMul(numer1, numer2);
75 tmp.div(denom, Sk64::kTrunc_DivOption);
76 return tmp.get32();
77 }
78
SkMulShift(int32_t a,int32_t b,unsigned shift)79 int32_t SkMulShift(int32_t a, int32_t b, unsigned shift) {
80 int sign = SkExtractSign(a ^ b);
81
82 if (shift > 63) {
83 return sign;
84 }
85
86 a = SkAbs32(a);
87 b = SkAbs32(b);
88
89 uint32_t ah = a >> 16;
90 uint32_t al = a & 0xFFFF;
91 uint32_t bh = b >> 16;
92 uint32_t bl = b & 0xFFFF;
93
94 uint32_t A = ah * bh;
95 uint32_t B = ah * bl + al * bh;
96 uint32_t C = al * bl;
97
98 /* [ A ]
99 [ B ]
100 [ C ]
101 */
102 uint32_t lo = C + (B << 16);
103 int32_t hi = A + (B >> 16) + (lo < C);
104
105 if (sign < 0) {
106 hi = -hi - Sk32ToBool(lo);
107 lo = 0 - lo;
108 }
109
110 if (shift == 0) {
111 #ifdef SK_DEBUGx
112 SkASSERT(((int32_t)lo >> 31) == hi);
113 #endif
114 return lo;
115 } else if (shift >= 32) {
116 return hi >> (shift - 32);
117 } else {
118 #ifdef SK_DEBUGx
119 int32_t tmp = hi >> shift;
120 SkASSERT(tmp == 0 || tmp == -1);
121 #endif
122 // we want (hi << (32 - shift)) | (lo >> shift) but rounded
123 int roundBit = (lo >> (shift - 1)) & 1;
124 return ((hi << (32 - shift)) | (lo >> shift)) + roundBit;
125 }
126 }
127
SkFixedMul_portable(SkFixed a,SkFixed b)128 SkFixed SkFixedMul_portable(SkFixed a, SkFixed b) {
129 #if 0
130 Sk64 tmp;
131
132 tmp.setMul(a, b);
133 tmp.shiftRight(16);
134 return tmp.fLo;
135 #elif defined(SkLONGLONG)
136 return static_cast<SkFixed>((SkLONGLONG)a * b >> 16);
137 #else
138 int sa = SkExtractSign(a);
139 int sb = SkExtractSign(b);
140 // now make them positive
141 a = SkApplySign(a, sa);
142 b = SkApplySign(b, sb);
143
144 uint32_t ah = a >> 16;
145 uint32_t al = a & 0xFFFF;
146 uint32_t bh = b >> 16;
147 uint32_t bl = b & 0xFFFF;
148
149 uint32_t R = ah * b + al * bh + (al * bl >> 16);
150
151 return SkApplySign(R, sa ^ sb);
152 #endif
153 }
154
SkFractMul_portable(SkFract a,SkFract b)155 SkFract SkFractMul_portable(SkFract a, SkFract b) {
156 #if 0
157 Sk64 tmp;
158 tmp.setMul(a, b);
159 return tmp.getFract();
160 #elif defined(SkLONGLONG)
161 return static_cast<SkFract>((SkLONGLONG)a * b >> 30);
162 #else
163 int sa = SkExtractSign(a);
164 int sb = SkExtractSign(b);
165 // now make them positive
166 a = SkApplySign(a, sa);
167 b = SkApplySign(b, sb);
168
169 uint32_t ah = a >> 16;
170 uint32_t al = a & 0xFFFF;
171 uint32_t bh = b >> 16;
172 uint32_t bl = b & 0xFFFF;
173
174 uint32_t A = ah * bh;
175 uint32_t B = ah * bl + al * bh;
176 uint32_t C = al * bl;
177
178 /* [ A ]
179 [ B ]
180 [ C ]
181 */
182 uint32_t Lo = C + (B << 16);
183 uint32_t Hi = A + (B >>16) + (Lo < C);
184
185 SkASSERT((Hi >> 29) == 0); // else overflow
186
187 int32_t R = (Hi << 2) + (Lo >> 30);
188
189 return SkApplySign(R, sa ^ sb);
190 #endif
191 }
192
SkFixedMulCommon(SkFixed a,int b,int bias)193 int SkFixedMulCommon(SkFixed a, int b, int bias) {
194 // this function only works if b is 16bits
195 SkASSERT(b == (int16_t)b);
196 SkASSERT(b >= 0);
197
198 int sa = SkExtractSign(a);
199 a = SkApplySign(a, sa);
200 uint32_t ah = a >> 16;
201 uint32_t al = a & 0xFFFF;
202 uint32_t R = ah * b + ((al * b + bias) >> 16);
203 return SkApplySign(R, sa);
204 }
205
206 #ifdef SK_DEBUGx
207 #define TEST_FASTINVERT
208 #endif
209
SkFixedFastInvert(SkFixed x)210 SkFixed SkFixedFastInvert(SkFixed x) {
211 /* Adapted (stolen) from gglRecip()
212 */
213
214 if (x == SK_Fixed1) {
215 return SK_Fixed1;
216 }
217
218 int sign = SkExtractSign(x);
219 uint32_t a = SkApplySign(x, sign);
220
221 if (a <= 2) {
222 return SkApplySign(SK_MaxS32, sign);
223 }
224
225 #ifdef TEST_FASTINVERT
226 SkFixed orig = a;
227 uint32_t slow = SkFixedDiv(SK_Fixed1, a);
228 #endif
229
230 // normalize a
231 int lz = SkCLZ(a);
232 a = a << lz >> 16;
233
234 // compute 1/a approximation (0.5 <= a < 1.0)
235 uint32_t r = 0x17400 - a; // (2.90625 (~2.914) - 2*a) >> 1
236
237 // Newton-Raphson iteration:
238 // x = r*(2 - a*r) = ((r/2)*(1 - a*r/2))*4
239 r = ( (0x10000 - ((a*r)>>16)) * r ) >> 15;
240 r = ( (0x10000 - ((a*r)>>16)) * r ) >> (30 - lz);
241
242 #ifdef TEST_FASTINVERT
243 SkDebugf("SkFixedFastInvert(%x %g) = %x %g Slow[%x %g]\n",
244 orig, orig/65536.,
245 r, r/65536.,
246 slow, slow/65536.);
247 #endif
248
249 return SkApplySign(r, sign);
250 }
251
252 ///////////////////////////////////////////////////////////////////////////////
253
254 #define DIVBITS_ITER(n) \
255 case n: \
256 if ((numer = (numer << 1) - denom) >= 0) \
257 result |= 1 << (n - 1); else numer += denom
258
SkDivBits(int32_t numer,int32_t denom,int shift_bias)259 int32_t SkDivBits(int32_t numer, int32_t denom, int shift_bias) {
260 SkASSERT(denom != 0);
261 if (numer == 0) {
262 return 0;
263 }
264
265 // make numer and denom positive, and sign hold the resulting sign
266 int32_t sign = SkExtractSign(numer ^ denom);
267 numer = SkAbs32(numer);
268 denom = SkAbs32(denom);
269
270 int nbits = SkCLZ(numer) - 1;
271 int dbits = SkCLZ(denom) - 1;
272 int bits = shift_bias - nbits + dbits;
273
274 if (bits < 0) { // answer will underflow
275 return 0;
276 }
277 if (bits > 31) { // answer will overflow
278 return SkApplySign(SK_MaxS32, sign);
279 }
280
281 denom <<= dbits;
282 numer <<= nbits;
283
284 SkFixed result = 0;
285
286 // do the first one
287 if ((numer -= denom) >= 0) {
288 result = 1;
289 } else {
290 numer += denom;
291 }
292
293 // Now fall into our switch statement if there are more bits to compute
294 if (bits > 0) {
295 // make room for the rest of the answer bits
296 result <<= bits;
297 switch (bits) {
298 DIVBITS_ITER(31); DIVBITS_ITER(30); DIVBITS_ITER(29);
299 DIVBITS_ITER(28); DIVBITS_ITER(27); DIVBITS_ITER(26);
300 DIVBITS_ITER(25); DIVBITS_ITER(24); DIVBITS_ITER(23);
301 DIVBITS_ITER(22); DIVBITS_ITER(21); DIVBITS_ITER(20);
302 DIVBITS_ITER(19); DIVBITS_ITER(18); DIVBITS_ITER(17);
303 DIVBITS_ITER(16); DIVBITS_ITER(15); DIVBITS_ITER(14);
304 DIVBITS_ITER(13); DIVBITS_ITER(12); DIVBITS_ITER(11);
305 DIVBITS_ITER(10); DIVBITS_ITER( 9); DIVBITS_ITER( 8);
306 DIVBITS_ITER( 7); DIVBITS_ITER( 6); DIVBITS_ITER( 5);
307 DIVBITS_ITER( 4); DIVBITS_ITER( 3); DIVBITS_ITER( 2);
308 // we merge these last two together, makes GCC make better ARM
309 default:
310 DIVBITS_ITER( 1);
311 }
312 }
313
314 if (result < 0) {
315 result = SK_MaxS32;
316 }
317 return SkApplySign(result, sign);
318 }
319
320 /* mod(float numer, float denom) seems to always return the sign
321 of the numer, so that's what we do too
322 */
SkFixedMod(SkFixed numer,SkFixed denom)323 SkFixed SkFixedMod(SkFixed numer, SkFixed denom) {
324 int sn = SkExtractSign(numer);
325 int sd = SkExtractSign(denom);
326
327 numer = SkApplySign(numer, sn);
328 denom = SkApplySign(denom, sd);
329
330 if (numer < denom) {
331 return SkApplySign(numer, sn);
332 } else if (numer == denom) {
333 return 0;
334 } else {
335 SkFixed div = SkFixedDiv(numer, denom);
336 return SkApplySign(SkFixedMul(denom, div & 0xFFFF), sn);
337 }
338 }
339
340 /* www.worldserver.com/turk/computergraphics/FixedSqrt.pdf
341 */
SkSqrtBits(int32_t x,int count)342 int32_t SkSqrtBits(int32_t x, int count) {
343 SkASSERT(x >= 0 && count > 0 && (unsigned)count <= 30);
344
345 uint32_t root = 0;
346 uint32_t remHi = 0;
347 uint32_t remLo = x;
348
349 do {
350 root <<= 1;
351
352 remHi = (remHi<<2) | (remLo>>30);
353 remLo <<= 2;
354
355 uint32_t testDiv = (root << 1) + 1;
356 if (remHi >= testDiv) {
357 remHi -= testDiv;
358 root++;
359 }
360 } while (--count >= 0);
361
362 return root;
363 }
364
SkCubeRootBits(int32_t value,int bits)365 int32_t SkCubeRootBits(int32_t value, int bits) {
366 SkASSERT(bits > 0);
367
368 int sign = SkExtractSign(value);
369 value = SkApplySign(value, sign);
370
371 uint32_t root = 0;
372 uint32_t curr = (uint32_t)value >> 30;
373 value <<= 2;
374
375 do {
376 root <<= 1;
377 uint32_t guess = root * root + root;
378 guess = (guess << 1) + guess; // guess *= 3
379 if (guess < curr) {
380 curr -= guess + 1;
381 root |= 1;
382 }
383 curr = (curr << 3) | ((uint32_t)value >> 29);
384 value <<= 3;
385 } while (--bits);
386
387 return SkApplySign(root, sign);
388 }
389
SkFixedMean(SkFixed a,SkFixed b)390 SkFixed SkFixedMean(SkFixed a, SkFixed b) {
391 Sk64 tmp;
392
393 tmp.setMul(a, b);
394 return tmp.getSqrt();
395 }
396
397 ///////////////////////////////////////////////////////////////////////////////
398
399 #ifdef SK_SCALAR_IS_FLOAT
SkScalarSinCos(float radians,float * cosValue)400 float SkScalarSinCos(float radians, float* cosValue) {
401 float sinValue = sk_float_sin(radians);
402
403 if (cosValue) {
404 *cosValue = sk_float_cos(radians);
405 if (SkScalarNearlyZero(*cosValue)) {
406 *cosValue = 0;
407 }
408 }
409
410 if (SkScalarNearlyZero(sinValue)) {
411 sinValue = 0;
412 }
413 return sinValue;
414 }
415 #endif
416
417 #define INTERP_SINTABLE
418 #define BUILD_TABLE_AT_RUNTIMEx
419
420 #define kTableSize 256
421
422 #ifdef BUILD_TABLE_AT_RUNTIME
423 static uint16_t gSkSinTable[kTableSize];
424
build_sintable(uint16_t table[])425 static void build_sintable(uint16_t table[]) {
426 for (int i = 0; i < kTableSize; i++) {
427 double rad = i * 3.141592653589793 / (2*kTableSize);
428 double val = sin(rad);
429 int ival = (int)(val * SK_Fixed1);
430 table[i] = SkToU16(ival);
431 }
432 }
433 #else
434 #include "SkSinTable.h"
435 #endif
436
437 #define SK_Fract1024SizeOver2PI 0x28BE60 /* floatToFract(1024 / 2PI) */
438
439 #ifdef INTERP_SINTABLE
interp_table(const uint16_t table[],int index,int partial255)440 static SkFixed interp_table(const uint16_t table[], int index, int partial255) {
441 SkASSERT((unsigned)index < kTableSize);
442 SkASSERT((unsigned)partial255 <= 255);
443
444 SkFixed lower = table[index];
445 SkFixed upper = (index == kTableSize - 1) ? SK_Fixed1 : table[index + 1];
446
447 SkASSERT(lower < upper);
448 SkASSERT(lower >= 0);
449 SkASSERT(upper <= SK_Fixed1);
450
451 partial255 += (partial255 >> 7);
452 return lower + ((upper - lower) * partial255 >> 8);
453 }
454 #endif
455
SkFixedSinCos(SkFixed radians,SkFixed * cosValuePtr)456 SkFixed SkFixedSinCos(SkFixed radians, SkFixed* cosValuePtr) {
457 SkASSERT(SK_ARRAY_COUNT(gSkSinTable) == kTableSize);
458
459 #ifdef BUILD_TABLE_AT_RUNTIME
460 static bool gFirstTime = true;
461 if (gFirstTime) {
462 build_sintable(gSinTable);
463 gFirstTime = false;
464 }
465 #endif
466
467 // make radians positive
468 SkFixed sinValue, cosValue;
469 int32_t cosSign = 0;
470 int32_t sinSign = SkExtractSign(radians);
471 radians = SkApplySign(radians, sinSign);
472 // scale it to 0...1023 ...
473
474 #ifdef INTERP_SINTABLE
475 radians = SkMulDiv(radians, 2 * kTableSize * 256, SK_FixedPI);
476 int findex = radians & (kTableSize * 256 - 1);
477 int index = findex >> 8;
478 int partial = findex & 255;
479 sinValue = interp_table(gSkSinTable, index, partial);
480
481 findex = kTableSize * 256 - findex - 1;
482 index = findex >> 8;
483 partial = findex & 255;
484 cosValue = interp_table(gSkSinTable, index, partial);
485
486 int quad = ((unsigned)radians / (kTableSize * 256)) & 3;
487 #else
488 radians = SkMulDiv(radians, 2 * kTableSize, SK_FixedPI);
489 int index = radians & (kTableSize - 1);
490
491 if (index == 0) {
492 sinValue = 0;
493 cosValue = SK_Fixed1;
494 } else {
495 sinValue = gSkSinTable[index];
496 cosValue = gSkSinTable[kTableSize - index];
497 }
498 int quad = ((unsigned)radians / kTableSize) & 3;
499 #endif
500
501 if (quad & 1) {
502 SkTSwap<SkFixed>(sinValue, cosValue);
503 }
504 if (quad & 2) {
505 sinSign = ~sinSign;
506 }
507 if (((quad - 1) & 2) == 0) {
508 cosSign = ~cosSign;
509 }
510
511 // restore the sign for negative angles
512 sinValue = SkApplySign(sinValue, sinSign);
513 cosValue = SkApplySign(cosValue, cosSign);
514
515 #ifdef SK_DEBUG
516 if (1) {
517 SkFixed sin2 = SkFixedMul(sinValue, sinValue);
518 SkFixed cos2 = SkFixedMul(cosValue, cosValue);
519 int diff = cos2 + sin2 - SK_Fixed1;
520 SkASSERT(SkAbs32(diff) <= 7);
521 }
522 #endif
523
524 if (cosValuePtr) {
525 *cosValuePtr = cosValue;
526 }
527 return sinValue;
528 }
529
530 ///////////////////////////////////////////////////////////////////////////////
531
SkFixedTan(SkFixed radians)532 SkFixed SkFixedTan(SkFixed radians) { return SkCordicTan(radians); }
SkFixedASin(SkFixed x)533 SkFixed SkFixedASin(SkFixed x) { return SkCordicASin(x); }
SkFixedACos(SkFixed x)534 SkFixed SkFixedACos(SkFixed x) { return SkCordicACos(x); }
SkFixedATan2(SkFixed y,SkFixed x)535 SkFixed SkFixedATan2(SkFixed y, SkFixed x) { return SkCordicATan2(y, x); }
SkFixedExp(SkFixed x)536 SkFixed SkFixedExp(SkFixed x) { return SkCordicExp(x); }
SkFixedLog(SkFixed x)537 SkFixed SkFixedLog(SkFixed x) { return SkCordicLog(x); }
538
539