1// Copyright 2009-2021 Intel Corporation
2// SPDX-License-Identifier: Apache-2.0
3
4#pragma once
5
6// Transcendental functions from "ispc": https://github.com/ispc/ispc/
7// Most of the transcendental implementations in ispc code come from
8// Solomon Boulos's "syrah": https://github.com/boulos/syrah/
9
10#include "../simd/simd.h"
11
12namespace embree
13{
14
15namespace fastapprox
16{
17
18template <typename T>
19__forceinline T sin(const T &v)
20{
21 static const float piOverTwoVec = 1.57079637050628662109375;
22 static const float twoOverPiVec = 0.636619746685028076171875;
23 auto scaled = v * twoOverPiVec;
24 auto kReal = floor(scaled);
25 auto k = toInt(kReal);
26
27 // Reduced range version of x
28 auto x = v - kReal * piOverTwoVec;
29 auto kMod4 = k & 3;
30 auto sinUseCos = (kMod4 == 1) | (kMod4 == 3);
31 auto flipSign = (kMod4 > 1);
32
33 // These coefficients are from sollya with fpminimax(sin(x)/x, [|0, 2,
34 // 4, 6, 8, 10|], [|single...|], [0;Pi/2]);
35 static const float sinC2 = -0.16666667163372039794921875;
36 static const float sinC4 = +8.333347737789154052734375e-3;
37 static const float sinC6 = -1.9842604524455964565277099609375e-4;
38 static const float sinC8 = +2.760012648650445044040679931640625e-6;
39 static const float sinC10 = -2.50293279435709337121807038784027099609375e-8;
40
41 static const float cosC2 = -0.5;
42 static const float cosC4 = +4.166664183139801025390625e-2;
43 static const float cosC6 = -1.388833043165504932403564453125e-3;
44 static const float cosC8 = +2.47562347794882953166961669921875e-5;
45 static const float cosC10 = -2.59630184018533327616751194000244140625e-7;
46
47 auto outside = select(sinUseCos, 1., x);
48 auto c2 = select(sinUseCos, T(cosC2), T(sinC2));
49 auto c4 = select(sinUseCos, T(cosC4), T(sinC4));
50 auto c6 = select(sinUseCos, T(cosC6), T(sinC6));
51 auto c8 = select(sinUseCos, T(cosC8), T(sinC8));
52 auto c10 = select(sinUseCos, T(cosC10), T(sinC10));
53
54 auto x2 = x * x;
55 auto formula = x2 * c10 + c8;
56 formula = x2 * formula + c6;
57 formula = x2 * formula + c4;
58 formula = x2 * formula + c2;
59 formula = x2 * formula + 1.;
60 formula *= outside;
61
62 formula = select(flipSign, -formula, formula);
63 return formula;
64}
65
66template <typename T>
67__forceinline T cos(const T &v)
68{
69 static const float piOverTwoVec = 1.57079637050628662109375;
70 static const float twoOverPiVec = 0.636619746685028076171875;
71 auto scaled = v * twoOverPiVec;
72 auto kReal = floor(scaled);
73 auto k = toInt(kReal);
74
75 // Reduced range version of x
76 auto x = v - kReal * piOverTwoVec;
77
78 auto kMod4 = k & 3;
79 auto cosUseCos = (kMod4 == 0) | (kMod4 == 2);
80 auto flipSign = (kMod4 == 1) | (kMod4 == 2);
81
82 const float sinC2 = -0.16666667163372039794921875;
83 const float sinC4 = +8.333347737789154052734375e-3;
84 const float sinC6 = -1.9842604524455964565277099609375e-4;
85 const float sinC8 = +2.760012648650445044040679931640625e-6;
86 const float sinC10 = -2.50293279435709337121807038784027099609375e-8;
87
88 const float cosC2 = -0.5;
89 const float cosC4 = +4.166664183139801025390625e-2;
90 const float cosC6 = -1.388833043165504932403564453125e-3;
91 const float cosC8 = +2.47562347794882953166961669921875e-5;
92 const float cosC10 = -2.59630184018533327616751194000244140625e-7;
93
94 auto outside = select(cosUseCos, 1., x);
95 auto c2 = select(cosUseCos, T(cosC2), T(sinC2));
96 auto c4 = select(cosUseCos, T(cosC4), T(sinC4));
97 auto c6 = select(cosUseCos, T(cosC6), T(sinC6));
98 auto c8 = select(cosUseCos, T(cosC8), T(sinC8));
99 auto c10 = select(cosUseCos, T(cosC10), T(sinC10));
100
101 auto x2 = x * x;
102 auto formula = x2 * c10 + c8;
103 formula = x2 * formula + c6;
104 formula = x2 * formula + c4;
105 formula = x2 * formula + c2;
106 formula = x2 * formula + 1.;
107 formula *= outside;
108
109 formula = select(flipSign, -formula, formula);
110 return formula;
111}
112
113template <typename T>
114__forceinline void sincos(const T &v, T &sinResult, T &cosResult)
115{
116 const float piOverTwoVec = 1.57079637050628662109375;
117 const float twoOverPiVec = 0.636619746685028076171875;
118 auto scaled = v * twoOverPiVec;
119 auto kReal = floor(scaled);
120 auto k = toInt(kReal);
121
122 // Reduced range version of x
123 auto x = v - kReal * piOverTwoVec;
124 auto kMod4 = k & 3;
125 auto cosUseCos = ((kMod4 == 0) | (kMod4 == 2));
126 auto sinUseCos = ((kMod4 == 1) | (kMod4 == 3));
127 auto sinFlipSign = (kMod4 > 1);
128 auto cosFlipSign = ((kMod4 == 1) | (kMod4 == 2));
129
130 const float oneVec = +1.;
131 const float sinC2 = -0.16666667163372039794921875;
132 const float sinC4 = +8.333347737789154052734375e-3;
133 const float sinC6 = -1.9842604524455964565277099609375e-4;
134 const float sinC8 = +2.760012648650445044040679931640625e-6;
135 const float sinC10 = -2.50293279435709337121807038784027099609375e-8;
136
137 const float cosC2 = -0.5;
138 const float cosC4 = +4.166664183139801025390625e-2;
139 const float cosC6 = -1.388833043165504932403564453125e-3;
140 const float cosC8 = +2.47562347794882953166961669921875e-5;
141 const float cosC10 = -2.59630184018533327616751194000244140625e-7;
142
143 auto x2 = x * x;
144
145 auto sinFormula = x2 * sinC10 + sinC8;
146 auto cosFormula = x2 * cosC10 + cosC8;
147 sinFormula = x2 * sinFormula + sinC6;
148 cosFormula = x2 * cosFormula + cosC6;
149
150 sinFormula = x2 * sinFormula + sinC4;
151 cosFormula = x2 * cosFormula + cosC4;
152
153 sinFormula = x2 * sinFormula + sinC2;
154 cosFormula = x2 * cosFormula + cosC2;
155
156 sinFormula = x2 * sinFormula + oneVec;
157 cosFormula = x2 * cosFormula + oneVec;
158
159 sinFormula *= x;
160
161 sinResult = select(sinUseCos, cosFormula, sinFormula);
162 cosResult = select(cosUseCos, cosFormula, sinFormula);
163
164 sinResult = select(sinFlipSign, -sinResult, sinResult);
165 cosResult = select(cosFlipSign, -cosResult, cosResult);
166}
167
168template <typename T>
169__forceinline T tan(const T &v)
170{
171 const float piOverFourVec = 0.785398185253143310546875;
172 const float fourOverPiVec = 1.27323949337005615234375;
173
174 auto xLt0 = v < 0.;
175 auto y = select(xLt0, -v, v);
176 auto scaled = y * fourOverPiVec;
177
178 auto kReal = floor(scaled);
179 auto k = toInt(kReal);
180
181 auto x = y - kReal * piOverFourVec;
182
183 // If k & 1, x -= Pi/4
184 auto needOffset = (k & 1) != 0;
185 x = select(needOffset, x - piOverFourVec, x);
186
187 // If k & 3 == (0 or 3) let z = tan_In...(y) otherwise z = -cot_In0To...
188 auto kMod4 = k & 3;
189 auto useCotan = (kMod4 == 1) | (kMod4 == 2);
190
191 const float oneVec = 1.0;
192
193 const float tanC2 = +0.33333075046539306640625;
194 const float tanC4 = +0.13339905440807342529296875;
195 const float tanC6 = +5.3348250687122344970703125e-2;
196 const float tanC8 = +2.46033705770969390869140625e-2;
197 const float tanC10 = +2.892402000725269317626953125e-3;
198 const float tanC12 = +9.500005282461643218994140625e-3;
199
200 const float cotC2 = -0.3333333432674407958984375;
201 const float cotC4 = -2.222204394638538360595703125e-2;
202 const float cotC6 = -2.11752182804048061370849609375e-3;
203 const float cotC8 = -2.0846328698098659515380859375e-4;
204 const float cotC10 = -2.548247357481159269809722900390625e-5;
205 const float cotC12 = -3.5257363606433500535786151885986328125e-7;
206
207 auto x2 = x * x;
208 T z;
209 if (any(useCotan))
210 {
211 auto cotVal = x2 * cotC12 + cotC10;
212 cotVal = x2 * cotVal + cotC8;
213 cotVal = x2 * cotVal + cotC6;
214 cotVal = x2 * cotVal + cotC4;
215 cotVal = x2 * cotVal + cotC2;
216 cotVal = x2 * cotVal + oneVec;
217 // The equation is for x * cot(x) but we need -x * cot(x) for the tan part.
218 cotVal /= -x;
219 z = cotVal;
220 }
221 auto useTan = !useCotan;
222 if (any(useTan))
223 {
224 auto tanVal = x2 * tanC12 + tanC10;
225 tanVal = x2 * tanVal + tanC8;
226 tanVal = x2 * tanVal + tanC6;
227 tanVal = x2 * tanVal + tanC4;
228 tanVal = x2 * tanVal + tanC2;
229 tanVal = x2 * tanVal + oneVec;
230 // Equation was for tan(x)/x
231 tanVal *= x;
232 z = select(useTan, tanVal, z);
233 }
234 return select(xLt0, -z, z);
235}
236
237template <typename T>
238__forceinline T asin(const T &x0)
239{
240 auto isneg = (x0 < 0.f);
241 auto x = abs(x0);
242 auto isnan = (x > 1.f);
243
244 // sollya
245 // fpminimax(((asin(x)-pi/2)/-sqrt(1-x)), [|0,1,2,3,4,5|],[|single...|],
246 // [1e-20;.9999999999999999]);
247 // avg error: 1.1105439e-06, max error 1.3187528e-06
248 auto v = 1.57079517841339111328125f +
249 x * (-0.21450997889041900634765625f +
250 x * (8.78556668758392333984375e-2f +
251 x * (-4.489909112453460693359375e-2f +
252 x * (1.928029954433441162109375e-2f +
253 x * (-4.3095736764371395111083984375e-3f)))));
254
255 v *= -sqrt(1.f - x);
256 v = v + 1.57079637050628662109375f;
257
258 v = select(v < 0.f, T(0.f), v);
259 v = select(isneg, -v, v);
260 v = select(isnan, T(cast_i2f(i: 0x7fc00000)), v);
261
262 return v;
263}
264
265template <typename T>
266__forceinline T acos(const T &v)
267{
268 return 1.57079637050628662109375f - asin(v);
269}
270
271template <typename T>
272__forceinline T atan(const T &v)
273{
274 const float piOverTwoVec = 1.57079637050628662109375;
275 // atan(-x) = -atan(x) (so flip from negative to positive first)
276 // If x > 1 -> atan(x) = Pi/2 - atan(1/x)
277 auto xNeg = v < 0.f;
278 auto xFlipped = select(xNeg, -v, v);
279
280 auto xGt1 = xFlipped > 1.;
281 auto x = select(xGt1, rcpSafe(xFlipped), xFlipped);
282
283 // These coefficients approximate atan(x)/x
284 const float atanC0 = +0.99999988079071044921875;
285 const float atanC2 = -0.3333191573619842529296875;
286 const float atanC4 = +0.199689209461212158203125;
287 const float atanC6 = -0.14015688002109527587890625;
288 const float atanC8 = +9.905083477497100830078125e-2;
289 const float atanC10 = -5.93664981424808502197265625e-2;
290 const float atanC12 = +2.417283318936824798583984375e-2;
291 const float atanC14 = -4.6721356920897960662841796875e-3;
292
293 auto x2 = x * x;
294 auto result = x2 * atanC14 + atanC12;
295 result = x2 * result + atanC10;
296 result = x2 * result + atanC8;
297 result = x2 * result + atanC6;
298 result = x2 * result + atanC4;
299 result = x2 * result + atanC2;
300 result = x2 * result + atanC0;
301 result *= x;
302
303 result = select(xGt1, piOverTwoVec - result, result);
304 result = select(xNeg, -result, result);
305 return result;
306}
307
308template <typename T>
309__forceinline T atan2(const T &y, const T &x)
310{
311 const float piVec = 3.1415926536;
312 // atan2(y, x) =
313 //
314 // atan2(y > 0, x = +-0) -> Pi/2
315 // atan2(y < 0, x = +-0) -> -Pi/2
316 // atan2(y = +-0, x < +0) -> +-Pi
317 // atan2(y = +-0, x >= +0) -> +-0
318 //
319 // atan2(y >= 0, x < 0) -> Pi + atan(y/x)
320 // atan2(y < 0, x < 0) -> -Pi + atan(y/x)
321 // atan2(y, x > 0) -> atan(y/x)
322 //
323 // and then a bunch of code for dealing with infinities.
324 auto yOverX = y * rcpSafe(x);
325 auto atanArg = atan(yOverX);
326 auto xLt0 = x < 0.f;
327 auto yLt0 = y < 0.f;
328 auto offset = select(xLt0,
329 select(yLt0, T(-piVec), T(piVec)), 0.f);
330 return offset + atanArg;
331}
332
333template <typename T>
334__forceinline T exp(const T &v)
335{
336 const float ln2Part1 = 0.6931457519;
337 const float ln2Part2 = 1.4286067653e-6;
338 const float oneOverLn2 = 1.44269502162933349609375;
339
340 auto scaled = v * oneOverLn2;
341 auto kReal = floor(scaled);
342 auto k = toInt(kReal);
343
344 // Reduced range version of x
345 auto x = v - kReal * ln2Part1;
346 x -= kReal * ln2Part2;
347
348 // These coefficients are for e^x in [0, ln(2)]
349 const float one = 1.;
350 const float c2 = 0.4999999105930328369140625;
351 const float c3 = 0.166668415069580078125;
352 const float c4 = 4.16539050638675689697265625e-2;
353 const float c5 = 8.378830738365650177001953125e-3;
354 const float c6 = 1.304379315115511417388916015625e-3;
355 const float c7 = 2.7555381529964506626129150390625e-4;
356
357 auto result = x * c7 + c6;
358 result = x * result + c5;
359 result = x * result + c4;
360 result = x * result + c3;
361 result = x * result + c2;
362 result = x * result + one;
363 result = x * result + one;
364
365 // Compute 2^k (should differ for float and double, but I'll avoid
366 // it for now and just do floats)
367 const int fpbias = 127;
368 auto biasedN = k + fpbias;
369 auto overflow = kReal > fpbias;
370 // Minimum exponent is -126, so if k is <= -127 (k + 127 <= 0)
371 // we've got underflow. -127 * ln(2) -> -88.02. So the most
372 // negative float input that doesn't result in zero is like -88.
373 auto underflow = kReal <= -fpbias;
374 const int infBits = 0x7f800000;
375 biasedN <<= 23;
376 // Reinterpret this thing as float
377 auto twoToTheN = asFloat(biasedN);
378 // Handle both doubles and floats (hopefully eliding the copy for float)
379 auto elemtype2n = twoToTheN;
380 result *= elemtype2n;
381 result = select(overflow, cast_i2f(i: infBits), result);
382 result = select(underflow, 0., result);
383 return result;
384}
385
386// Range reduction for logarithms takes log(x) -> log(2^n * y) -> n
387// * log(2) + log(y) where y is the reduced range (usually in [1/2, 1)).
388template <typename T, typename R>
389__forceinline void __rangeReduceLog(const T &input,
390 T &reduced,
391 R &exponent)
392{
393 auto intVersion = asInt(input);
394 // single precision = SEEE EEEE EMMM MMMM MMMM MMMM MMMM MMMM
395 // exponent mask = 0111 1111 1000 0000 0000 0000 0000 0000
396 // 0x7 0xF 0x8 0x0 0x0 0x0 0x0 0x0
397 // non-exponent = 1000 0000 0111 1111 1111 1111 1111 1111
398 // = 0x8 0x0 0x7 0xF 0xF 0xF 0xF 0xF
399
400 //const int exponentMask(0x7F800000)
401 static const int nonexponentMask = 0x807FFFFF;
402
403 // We want the reduced version to have an exponent of -1 which is
404 // -1 + 127 after biasing or 126
405 static const int exponentNeg1 = (126l << 23);
406 // NOTE(boulos): We don't need to mask anything out since we know
407 // the sign bit has to be 0. If it's 1, we need to return infinity/nan
408 // anyway (log(x), x = +-0 -> infinity, x < 0 -> NaN).
409 auto biasedExponent = intVersion >> 23; // This number is [0, 255] but it means [-127, 128]
410
411 auto offsetExponent = biasedExponent + 1; // Treat the number as if it were 2^{e+1} * (1.m)/2
412 exponent = offsetExponent - 127; // get the real value
413
414 // Blend the offset_exponent with the original input (do this in
415 // int for now, until I decide if float can have & and &not)
416 auto blended = (intVersion & nonexponentMask) | (exponentNeg1);
417 reduced = asFloat(blended);
418}
419
420template <typename T> struct ExponentType { };
421template <int N> struct ExponentType<vfloat_impl<N>> { typedef vint<N> Ty; };
422template <> struct ExponentType<float> { typedef int Ty; };
423
424template <typename T>
425__forceinline T log(const T &v)
426{
427 T reduced;
428 typename ExponentType<T>::Ty exponent;
429
430 const int nanBits = 0x7fc00000;
431 const int negInfBits = 0xFF800000;
432 const float nan = cast_i2f(i: nanBits);
433 const float negInf = cast_i2f(i: negInfBits);
434 auto useNan = v < 0.;
435 auto useInf = v == 0.;
436 auto exceptional = useNan | useInf;
437 const float one = 1.0;
438
439 auto patched = select(exceptional, one, v);
440 __rangeReduceLog(patched, reduced, exponent);
441
442 const float ln2 = 0.693147182464599609375;
443
444 auto x1 = one - reduced;
445 const float c1 = +0.50000095367431640625;
446 const float c2 = +0.33326041698455810546875;
447 const float c3 = +0.2519190013408660888671875;
448 const float c4 = +0.17541764676570892333984375;
449 const float c5 = +0.3424419462680816650390625;
450 const float c6 = -0.599632322788238525390625;
451 const float c7 = +1.98442304134368896484375;
452 const float c8 = -2.4899270534515380859375;
453 const float c9 = +1.7491014003753662109375;
454
455 auto result = x1 * c9 + c8;
456 result = x1 * result + c7;
457 result = x1 * result + c6;
458 result = x1 * result + c5;
459 result = x1 * result + c4;
460 result = x1 * result + c3;
461 result = x1 * result + c2;
462 result = x1 * result + c1;
463 result = x1 * result + one;
464
465 // Equation was for -(ln(red)/(1-red))
466 result *= -x1;
467 result += toFloat(exponent) * ln2;
468
469 return select(exceptional,
470 select(useNan, T(nan), T(negInf)),
471 result);
472}
473
474template <typename T>
475__forceinline T pow(const T &x, const T &y)
476{
477 auto x1 = abs(x);
478 auto z = exp(y * log(x1));
479
480 // Handle special cases
481 const float twoOver23 = 8388608.0f;
482 auto yInt = y == round(y);
483 auto yOddInt = select(yInt, asInt(abs(y) + twoOver23) << 31, 0); // set sign bit
484
485 // x == 0
486 z = select(x == 0.0f,
487 select(y < 0.0f, T(inf) | signmsk(x),
488 select(y == 0.0f, T(1.0f), asFloat(yOddInt) & x)), z);
489
490 // x < 0
491 auto xNegative = x < 0.0f;
492 if (any(xNegative))
493 {
494 auto z1 = z | asFloat(yOddInt);
495 z1 = select(yInt, z1, std::numeric_limits<float>::quiet_NaN());
496 z = select(xNegative, z1, z);
497 }
498
499 auto xFinite = isfinite(x);
500 auto yFinite = isfinite(y);
501 if (all(xFinite & yFinite))
502 return z;
503
504 // x finite and y infinite
505 z = select(andn(xFinite, yFinite),
506 select(x1 == 1.0f, 1.0f,
507 select((x1 > 1.0f) ^ (y < 0.0f), inf, T(0.0f))), z);
508
509 // x infinite
510 z = select(xFinite, z,
511 select(y == 0.0f, 1.0f,
512 select(y < 0.0f, T(0.0f), inf) | (asFloat(yOddInt) & x)));
513
514 return z;
515}
516
517template <typename T>
518__forceinline T pow(const T &x, float y)
519{
520 return pow(x, T(y));
521}
522
523} // namespace fastapprox
524
525} // namespace embree
526

source code of qtquick3d/src/3rdparty/embree/common/math/transcendental.h