1 /*
2 * Copyright (C) 2006-2008 The Android Open Source Project
3 *
4 * Licensed under the Apache License, Version 2.0 (the "License");
5 * you may not use this file except in compliance with the License.
6 * You may obtain a copy of the License at
7 *
8 * http://www.apache.org/licenses/LICENSE-2.0
9 *
10 * Unless required by applicable law or agreed to in writing, software
11 * distributed under the License is distributed on an "AS IS" BASIS,
12 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
13 * See the License for the specific language governing permissions and
14 * limitations under the License.
15 */
16
17 #include "SkMath.h"
18 #include "SkCordic.h"
19 #include "SkFloatBits.h"
20 #include "SkFloatingPoint.h"
21 #include "Sk64.h"
22 #include "SkScalar.h"
23
24 #ifdef SK_SCALAR_IS_FLOAT
25 const uint32_t gIEEENotANumber = 0x7FFFFFFF;
26 const uint32_t gIEEEInfinity = 0x7F800000;
27 #endif
28
29 #define sub_shift(zeros, x, n) \
30 zeros -= n; \
31 x >>= n
32
SkCLZ_portable(uint32_t x)33 int SkCLZ_portable(uint32_t x) {
34 if (x == 0) {
35 return 32;
36 }
37
38 #ifdef SK_CPU_HAS_CONDITIONAL_INSTR
39 int zeros = 31;
40 if (x & 0xFFFF0000) {
41 sub_shift(zeros, x, 16);
42 }
43 if (x & 0xFF00) {
44 sub_shift(zeros, x, 8);
45 }
46 if (x & 0xF0) {
47 sub_shift(zeros, x, 4);
48 }
49 if (x & 0xC) {
50 sub_shift(zeros, x, 2);
51 }
52 if (x & 0x2) {
53 sub_shift(zeros, x, 1);
54 }
55 #else
56 int zeros = ((x >> 16) - 1) >> 31 << 4;
57 x <<= zeros;
58
59 int nonzero = ((x >> 24) - 1) >> 31 << 3;
60 zeros += nonzero;
61 x <<= nonzero;
62
63 nonzero = ((x >> 28) - 1) >> 31 << 2;
64 zeros += nonzero;
65 x <<= nonzero;
66
67 nonzero = ((x >> 30) - 1) >> 31 << 1;
68 zeros += nonzero;
69 x <<= nonzero;
70
71 zeros += (~x) >> 31;
72 #endif
73
74 return zeros;
75 }
76
SkMulDiv(int32_t numer1,int32_t numer2,int32_t denom)77 int32_t SkMulDiv(int32_t numer1, int32_t numer2, int32_t denom) {
78 SkASSERT(denom);
79
80 Sk64 tmp;
81 tmp.setMul(numer1, numer2);
82 tmp.div(denom, Sk64::kTrunc_DivOption);
83 return tmp.get32();
84 }
85
SkMulShift(int32_t a,int32_t b,unsigned shift)86 int32_t SkMulShift(int32_t a, int32_t b, unsigned shift) {
87 int sign = SkExtractSign(a ^ b);
88
89 if (shift > 63) {
90 return sign;
91 }
92
93 a = SkAbs32(a);
94 b = SkAbs32(b);
95
96 uint32_t ah = a >> 16;
97 uint32_t al = a & 0xFFFF;
98 uint32_t bh = b >> 16;
99 uint32_t bl = b & 0xFFFF;
100
101 uint32_t A = ah * bh;
102 uint32_t B = ah * bl + al * bh;
103 uint32_t C = al * bl;
104
105 /* [ A ]
106 [ B ]
107 [ C ]
108 */
109 uint32_t lo = C + (B << 16);
110 int32_t hi = A + (B >> 16) + (lo < C);
111
112 if (sign < 0) {
113 hi = -hi - Sk32ToBool(lo);
114 lo = 0 - lo;
115 }
116
117 if (shift == 0) {
118 #ifdef SK_DEBUGx
119 SkASSERT(((int32_t)lo >> 31) == hi);
120 #endif
121 return lo;
122 } else if (shift >= 32) {
123 return hi >> (shift - 32);
124 } else {
125 #ifdef SK_DEBUGx
126 int32_t tmp = hi >> shift;
127 SkASSERT(tmp == 0 || tmp == -1);
128 #endif
129 // we want (hi << (32 - shift)) | (lo >> shift) but rounded
130 int roundBit = (lo >> (shift - 1)) & 1;
131 return ((hi << (32 - shift)) | (lo >> shift)) + roundBit;
132 }
133 }
134
SkFixedMul_portable(SkFixed a,SkFixed b)135 SkFixed SkFixedMul_portable(SkFixed a, SkFixed b) {
136 #if 0
137 Sk64 tmp;
138
139 tmp.setMul(a, b);
140 tmp.shiftRight(16);
141 return tmp.fLo;
142 #elif defined(SkLONGLONG)
143 return static_cast<SkFixed>((SkLONGLONG)a * b >> 16);
144 #else
145 int sa = SkExtractSign(a);
146 int sb = SkExtractSign(b);
147 // now make them positive
148 a = SkApplySign(a, sa);
149 b = SkApplySign(b, sb);
150
151 uint32_t ah = a >> 16;
152 uint32_t al = a & 0xFFFF;
153 uint32_t bh = b >> 16;
154 uint32_t bl = b & 0xFFFF;
155
156 uint32_t R = ah * b + al * bh + (al * bl >> 16);
157
158 return SkApplySign(R, sa ^ sb);
159 #endif
160 }
161
SkFractMul_portable(SkFract a,SkFract b)162 SkFract SkFractMul_portable(SkFract a, SkFract b) {
163 #if 0
164 Sk64 tmp;
165 tmp.setMul(a, b);
166 return tmp.getFract();
167 #elif defined(SkLONGLONG)
168 return static_cast<SkFract>((SkLONGLONG)a * b >> 30);
169 #else
170 int sa = SkExtractSign(a);
171 int sb = SkExtractSign(b);
172 // now make them positive
173 a = SkApplySign(a, sa);
174 b = SkApplySign(b, sb);
175
176 uint32_t ah = a >> 16;
177 uint32_t al = a & 0xFFFF;
178 uint32_t bh = b >> 16;
179 uint32_t bl = b & 0xFFFF;
180
181 uint32_t A = ah * bh;
182 uint32_t B = ah * bl + al * bh;
183 uint32_t C = al * bl;
184
185 /* [ A ]
186 [ B ]
187 [ C ]
188 */
189 uint32_t Lo = C + (B << 16);
190 uint32_t Hi = A + (B >>16) + (Lo < C);
191
192 SkASSERT((Hi >> 29) == 0); // else overflow
193
194 int32_t R = (Hi << 2) + (Lo >> 30);
195
196 return SkApplySign(R, sa ^ sb);
197 #endif
198 }
199
SkFixedMulCommon(SkFixed a,int b,int bias)200 int SkFixedMulCommon(SkFixed a, int b, int bias) {
201 // this function only works if b is 16bits
202 SkASSERT(b == (int16_t)b);
203 SkASSERT(b >= 0);
204
205 int sa = SkExtractSign(a);
206 a = SkApplySign(a, sa);
207 uint32_t ah = a >> 16;
208 uint32_t al = a & 0xFFFF;
209 uint32_t R = ah * b + ((al * b + bias) >> 16);
210 return SkApplySign(R, sa);
211 }
212
213 #ifdef SK_DEBUGx
214 #define TEST_FASTINVERT
215 #endif
216
SkFixedFastInvert(SkFixed x)217 SkFixed SkFixedFastInvert(SkFixed x) {
218 /* Adapted (stolen) from gglRecip()
219 */
220
221 if (x == SK_Fixed1) {
222 return SK_Fixed1;
223 }
224
225 int sign = SkExtractSign(x);
226 uint32_t a = SkApplySign(x, sign);
227
228 if (a <= 2) {
229 return SkApplySign(SK_MaxS32, sign);
230 }
231
232 #ifdef TEST_FASTINVERT
233 SkFixed orig = a;
234 uint32_t slow = SkFixedDiv(SK_Fixed1, a);
235 #endif
236
237 // normalize a
238 int lz = SkCLZ(a);
239 a = a << lz >> 16;
240
241 // compute 1/a approximation (0.5 <= a < 1.0)
242 uint32_t r = 0x17400 - a; // (2.90625 (~2.914) - 2*a) >> 1
243
244 // Newton-Raphson iteration:
245 // x = r*(2 - a*r) = ((r/2)*(1 - a*r/2))*4
246 r = ( (0x10000 - ((a*r)>>16)) * r ) >> 15;
247 r = ( (0x10000 - ((a*r)>>16)) * r ) >> (30 - lz);
248
249 #ifdef TEST_FASTINVERT
250 SkDebugf("SkFixedFastInvert(%x %g) = %x %g Slow[%x %g]\n",
251 orig, orig/65536.,
252 r, r/65536.,
253 slow, slow/65536.);
254 #endif
255
256 return SkApplySign(r, sign);
257 }
258
259 ///////////////////////////////////////////////////////////////////////////////
260
261 #define DIVBITS_ITER(n) \
262 case n: \
263 if ((numer = (numer << 1) - denom) >= 0) \
264 result |= 1 << (n - 1); else numer += denom
265
SkDivBits(int32_t numer,int32_t denom,int shift_bias)266 int32_t SkDivBits(int32_t numer, int32_t denom, int shift_bias) {
267 SkASSERT(denom != 0);
268 if (numer == 0) {
269 return 0;
270 }
271
272 // make numer and denom positive, and sign hold the resulting sign
273 int32_t sign = SkExtractSign(numer ^ denom);
274 numer = SkAbs32(numer);
275 denom = SkAbs32(denom);
276
277 int nbits = SkCLZ(numer) - 1;
278 int dbits = SkCLZ(denom) - 1;
279 int bits = shift_bias - nbits + dbits;
280
281 if (bits < 0) { // answer will underflow
282 return 0;
283 }
284 if (bits > 31) { // answer will overflow
285 return SkApplySign(SK_MaxS32, sign);
286 }
287
288 denom <<= dbits;
289 numer <<= nbits;
290
291 SkFixed result = 0;
292
293 // do the first one
294 if ((numer -= denom) >= 0) {
295 result = 1;
296 } else {
297 numer += denom;
298 }
299
300 // Now fall into our switch statement if there are more bits to compute
301 if (bits > 0) {
302 // make room for the rest of the answer bits
303 result <<= bits;
304 switch (bits) {
305 DIVBITS_ITER(31); DIVBITS_ITER(30); DIVBITS_ITER(29);
306 DIVBITS_ITER(28); DIVBITS_ITER(27); DIVBITS_ITER(26);
307 DIVBITS_ITER(25); DIVBITS_ITER(24); DIVBITS_ITER(23);
308 DIVBITS_ITER(22); DIVBITS_ITER(21); DIVBITS_ITER(20);
309 DIVBITS_ITER(19); DIVBITS_ITER(18); DIVBITS_ITER(17);
310 DIVBITS_ITER(16); DIVBITS_ITER(15); DIVBITS_ITER(14);
311 DIVBITS_ITER(13); DIVBITS_ITER(12); DIVBITS_ITER(11);
312 DIVBITS_ITER(10); DIVBITS_ITER( 9); DIVBITS_ITER( 8);
313 DIVBITS_ITER( 7); DIVBITS_ITER( 6); DIVBITS_ITER( 5);
314 DIVBITS_ITER( 4); DIVBITS_ITER( 3); DIVBITS_ITER( 2);
315 // we merge these last two together, makes GCC make better ARM
316 default:
317 DIVBITS_ITER( 1);
318 }
319 }
320
321 if (result < 0) {
322 result = SK_MaxS32;
323 }
324 return SkApplySign(result, sign);
325 }
326
327 /* mod(float numer, float denom) seems to always return the sign
328 of the numer, so that's what we do too
329 */
SkFixedMod(SkFixed numer,SkFixed denom)330 SkFixed SkFixedMod(SkFixed numer, SkFixed denom) {
331 int sn = SkExtractSign(numer);
332 int sd = SkExtractSign(denom);
333
334 numer = SkApplySign(numer, sn);
335 denom = SkApplySign(denom, sd);
336
337 if (numer < denom) {
338 return SkApplySign(numer, sn);
339 } else if (numer == denom) {
340 return 0;
341 } else {
342 SkFixed div = SkFixedDiv(numer, denom);
343 return SkApplySign(SkFixedMul(denom, div & 0xFFFF), sn);
344 }
345 }
346
347 /* www.worldserver.com/turk/computergraphics/FixedSqrt.pdf
348 */
SkSqrtBits(int32_t x,int count)349 int32_t SkSqrtBits(int32_t x, int count) {
350 SkASSERT(x >= 0 && count > 0 && (unsigned)count <= 30);
351
352 uint32_t root = 0;
353 uint32_t remHi = 0;
354 uint32_t remLo = x;
355
356 do {
357 root <<= 1;
358
359 remHi = (remHi<<2) | (remLo>>30);
360 remLo <<= 2;
361
362 uint32_t testDiv = (root << 1) + 1;
363 if (remHi >= testDiv) {
364 remHi -= testDiv;
365 root++;
366 }
367 } while (--count >= 0);
368
369 return root;
370 }
371
SkCubeRootBits(int32_t value,int bits)372 int32_t SkCubeRootBits(int32_t value, int bits) {
373 SkASSERT(bits > 0);
374
375 int sign = SkExtractSign(value);
376 value = SkApplySign(value, sign);
377
378 uint32_t root = 0;
379 uint32_t curr = (uint32_t)value >> 30;
380 value <<= 2;
381
382 do {
383 root <<= 1;
384 uint32_t guess = root * root + root;
385 guess = (guess << 1) + guess; // guess *= 3
386 if (guess < curr) {
387 curr -= guess + 1;
388 root |= 1;
389 }
390 curr = (curr << 3) | ((uint32_t)value >> 29);
391 value <<= 3;
392 } while (--bits);
393
394 return SkApplySign(root, sign);
395 }
396
SkFixedMean(SkFixed a,SkFixed b)397 SkFixed SkFixedMean(SkFixed a, SkFixed b) {
398 Sk64 tmp;
399
400 tmp.setMul(a, b);
401 return tmp.getSqrt();
402 }
403
404 ///////////////////////////////////////////////////////////////////////////////
405
406 #ifdef SK_SCALAR_IS_FLOAT
SkScalarSinCos(float radians,float * cosValue)407 float SkScalarSinCos(float radians, float* cosValue) {
408 float sinValue = sk_float_sin(radians);
409
410 if (cosValue) {
411 *cosValue = sk_float_cos(radians);
412 if (SkScalarNearlyZero(*cosValue)) {
413 *cosValue = 0;
414 }
415 }
416
417 if (SkScalarNearlyZero(sinValue)) {
418 sinValue = 0;
419 }
420 return sinValue;
421 }
422 #endif
423
424 #define INTERP_SINTABLE
425 #define BUILD_TABLE_AT_RUNTIMEx
426
427 #define kTableSize 256
428
429 #ifdef BUILD_TABLE_AT_RUNTIME
430 static uint16_t gSkSinTable[kTableSize];
431
build_sintable(uint16_t table[])432 static void build_sintable(uint16_t table[]) {
433 for (int i = 0; i < kTableSize; i++) {
434 double rad = i * 3.141592653589793 / (2*kTableSize);
435 double val = sin(rad);
436 int ival = (int)(val * SK_Fixed1);
437 table[i] = SkToU16(ival);
438 }
439 }
440 #else
441 #include "SkSinTable.h"
442 #endif
443
444 #define SK_Fract1024SizeOver2PI 0x28BE60 /* floatToFract(1024 / 2PI) */
445
446 #ifdef INTERP_SINTABLE
interp_table(const uint16_t table[],int index,int partial255)447 static SkFixed interp_table(const uint16_t table[], int index, int partial255) {
448 SkASSERT((unsigned)index < kTableSize);
449 SkASSERT((unsigned)partial255 <= 255);
450
451 SkFixed lower = table[index];
452 SkFixed upper = (index == kTableSize - 1) ? SK_Fixed1 : table[index + 1];
453
454 SkASSERT(lower < upper);
455 SkASSERT(lower >= 0);
456 SkASSERT(upper <= SK_Fixed1);
457
458 partial255 += (partial255 >> 7);
459 return lower + ((upper - lower) * partial255 >> 8);
460 }
461 #endif
462
SkFixedSinCos(SkFixed radians,SkFixed * cosValuePtr)463 SkFixed SkFixedSinCos(SkFixed radians, SkFixed* cosValuePtr) {
464 SkASSERT(SK_ARRAY_COUNT(gSkSinTable) == kTableSize);
465
466 #ifdef BUILD_TABLE_AT_RUNTIME
467 static bool gFirstTime = true;
468 if (gFirstTime) {
469 build_sintable(gSinTable);
470 gFirstTime = false;
471 }
472 #endif
473
474 // make radians positive
475 SkFixed sinValue, cosValue;
476 int32_t cosSign = 0;
477 int32_t sinSign = SkExtractSign(radians);
478 radians = SkApplySign(radians, sinSign);
479 // scale it to 0...1023 ...
480
481 #ifdef INTERP_SINTABLE
482 radians = SkMulDiv(radians, 2 * kTableSize * 256, SK_FixedPI);
483 int findex = radians & (kTableSize * 256 - 1);
484 int index = findex >> 8;
485 int partial = findex & 255;
486 sinValue = interp_table(gSkSinTable, index, partial);
487
488 findex = kTableSize * 256 - findex - 1;
489 index = findex >> 8;
490 partial = findex & 255;
491 cosValue = interp_table(gSkSinTable, index, partial);
492
493 int quad = ((unsigned)radians / (kTableSize * 256)) & 3;
494 #else
495 radians = SkMulDiv(radians, 2 * kTableSize, SK_FixedPI);
496 int index = radians & (kTableSize - 1);
497
498 if (index == 0) {
499 sinValue = 0;
500 cosValue = SK_Fixed1;
501 } else {
502 sinValue = gSkSinTable[index];
503 cosValue = gSkSinTable[kTableSize - index];
504 }
505 int quad = ((unsigned)radians / kTableSize) & 3;
506 #endif
507
508 if (quad & 1) {
509 SkTSwap<SkFixed>(sinValue, cosValue);
510 }
511 if (quad & 2) {
512 sinSign = ~sinSign;
513 }
514 if (((quad - 1) & 2) == 0) {
515 cosSign = ~cosSign;
516 }
517
518 // restore the sign for negative angles
519 sinValue = SkApplySign(sinValue, sinSign);
520 cosValue = SkApplySign(cosValue, cosSign);
521
522 #ifdef SK_DEBUG
523 if (1) {
524 SkFixed sin2 = SkFixedMul(sinValue, sinValue);
525 SkFixed cos2 = SkFixedMul(cosValue, cosValue);
526 int diff = cos2 + sin2 - SK_Fixed1;
527 SkASSERT(SkAbs32(diff) <= 7);
528 }
529 #endif
530
531 if (cosValuePtr) {
532 *cosValuePtr = cosValue;
533 }
534 return sinValue;
535 }
536
537 ///////////////////////////////////////////////////////////////////////////////
538
SkFixedTan(SkFixed radians)539 SkFixed SkFixedTan(SkFixed radians) { return SkCordicTan(radians); }
SkFixedASin(SkFixed x)540 SkFixed SkFixedASin(SkFixed x) { return SkCordicASin(x); }
SkFixedACos(SkFixed x)541 SkFixed SkFixedACos(SkFixed x) { return SkCordicACos(x); }
SkFixedATan2(SkFixed y,SkFixed x)542 SkFixed SkFixedATan2(SkFixed y, SkFixed x) { return SkCordicATan2(y, x); }
SkFixedExp(SkFixed x)543 SkFixed SkFixedExp(SkFixed x) { return SkCordicExp(x); }
SkFixedLog(SkFixed x)544 SkFixed SkFixedLog(SkFixed x) { return SkCordicLog(x); }
545
546