OpenVDB  7.2.1
Math.h
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1 // Copyright Contributors to the OpenVDB Project
2 // SPDX-License-Identifier: MPL-2.0
3 //
7 
8 #ifndef OPENVDB_MATH_HAS_BEEN_INCLUDED
9 #define OPENVDB_MATH_HAS_BEEN_INCLUDED
10 
11 #include <openvdb/Platform.h>
12 #include <openvdb/version.h>
13 #include <boost/numeric/conversion/conversion_traits.hpp>
14 #include <algorithm> // for std::max()
15 #include <cassert>
16 #include <cmath> // for std::ceil(), std::fabs(), std::pow(), std::sqrt(), etc.
17 #include <cstdlib> // for abs(int)
18 #include <random>
19 #include <string>
20 #include <type_traits> // for std::is_arithmetic
21 
22 
23 // Compile pragmas
24 
25 // Intel(r) compiler fires remark #1572: floating-point equality and inequality
26 // comparisons are unrealiable when == or != is used with floating point operands.
27 #if defined(__INTEL_COMPILER)
28  #define OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN \
29  _Pragma("warning (push)") \
30  _Pragma("warning (disable:1572)")
31  #define OPENVDB_NO_FP_EQUALITY_WARNING_END \
32  _Pragma("warning (pop)")
33 #elif defined(__clang__)
34  #define OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN \
35  PRAGMA(clang diagnostic push) \
36  PRAGMA(clang diagnostic ignored "-Wfloat-equal")
37  #define OPENVDB_NO_FP_EQUALITY_WARNING_END \
38  PRAGMA(clang diagnostic pop)
39 #else
40  // For GCC, #pragma GCC diagnostic ignored "-Wfloat-equal"
41  // isn't working until gcc 4.2+,
42  // Trying
43  // #pragma GCC system_header
44  // creates other problems, most notably "warning: will never be executed"
45  // in from templates, unsure of how to work around.
46  // If necessary, could use integer based comparisons for equality
47  #define OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN
48  #define OPENVDB_NO_FP_EQUALITY_WARNING_END
49 #endif
50 
51 namespace openvdb {
53 namespace OPENVDB_VERSION_NAME {
54 
59 template<typename T> inline T zeroVal() { return T(0); }
61 template<> inline std::string zeroVal<std::string>() { return ""; }
63 template<> inline bool zeroVal<bool>() { return false; }
64 
65 namespace math {
66 
68 
69 inline std::string operator+(const std::string& s, bool) { return s; }
72 inline std::string operator+(const std::string& s, int) { return s; }
73 inline std::string operator+(const std::string& s, float) { return s; }
74 inline std::string operator+(const std::string& s, double) { return s; }
76 
78 template<typename Type1, typename Type2>
79 inline auto cwiseAdd(const Type1& v, const Type2 s)
80 {
82  return v + s;
84 }
85 
87 template<typename Type1, typename Type2>
88 inline bool cwiseLessThan(const Type1& a, const Type2& b)
89 {
91  return a < b;
93 }
94 
96 template<typename Type1, typename Type2>
97 inline bool cwiseGreaterThan(const Type1& a, const Type2& b)
98 {
100  return a > b;
102 }
103 
104 
105 
108 template <typename T> inline constexpr T pi() { return 3.141592653589793238462643383279502884e+00; }
109 template <> inline constexpr float pi() { return 3.141592653589793238462643383279502884e+00F; }
110 template <> inline constexpr double pi() { return 3.141592653589793238462643383279502884e+00; }
111 template <> inline constexpr long double pi() { return 3.141592653589793238462643383279502884e+00L; }
112 
113 
117 template<typename T> inline T negative(const T& val)
118 {
119 // disable unary minus on unsigned warning
120 #if defined(_MSC_VER)
121 #pragma warning(push)
122 #pragma warning(disable:4146)
123 #endif
124  return T(-val);
125 #if defined(_MSC_VER)
126 #pragma warning(pop)
127 #endif
128 }
130 template<> inline bool negative(const bool& val) { return !val; }
132 template<> inline std::string negative(const std::string& val) { return val; }
133 
134 
136 template<typename T> struct Tolerance { static T value() { return zeroVal<T>(); } };
138 template<> struct Tolerance<float> { static float value() { return 1e-8f; } };
139 template<> struct Tolerance<double> { static double value() { return 1e-15; } };
141 
143 template<typename T> struct Delta { static T value() { return zeroVal<T>(); } };
145 template<> struct Delta<float> { static float value() { return 1e-5f; } };
146 template<> struct Delta<double> { static double value() { return 1e-9; } };
148 
149 
150 // ==========> Random Values <==================
151 
154 template<typename FloatType = double, typename EngineType = std::mt19937>
155 class Rand01
156 {
157 private:
158  EngineType mEngine;
159  std::uniform_real_distribution<FloatType> mRand;
160 
161 public:
162  using ValueType = FloatType;
163 
166  Rand01(const EngineType& engine): mEngine(engine) {}
167 
170  Rand01(unsigned int seed): mEngine(static_cast<typename EngineType::result_type>(seed)) {}
171 
173  void setSeed(unsigned int seed)
174  {
175  mEngine.seed(static_cast<typename EngineType::result_type>(seed));
176  }
177 
179  const EngineType& engine() const { return mEngine; }
180 
182  FloatType operator()() { return mRand(mEngine); }
183 };
184 
186 
187 
190 template<typename IntType = int, typename EngineType = std::mt19937>
191 class RandInt
192 {
193 private:
194  using Distr = std::uniform_int_distribution<IntType>;
195  EngineType mEngine;
196  Distr mRand;
197 
198 public:
202  RandInt(const EngineType& engine, IntType imin, IntType imax):
203  mEngine(engine),
204  mRand(std::min(imin, imax), std::max(imin, imax))
205  {}
206 
210  RandInt(unsigned int seed, IntType imin, IntType imax):
211  mEngine(static_cast<typename EngineType::result_type>(seed)),
212  mRand(std::min(imin, imax), std::max(imin, imax))
213  {}
214 
216  void setRange(IntType imin, IntType imax)
217  {
218  mRand = Distr(std::min(imin, imax), std::max(imin, imax));
219  }
220 
222  void setSeed(unsigned int seed)
223  {
224  mEngine.seed(static_cast<typename EngineType::result_type>(seed));
225  }
226 
228  const EngineType& engine() const { return mEngine; }
229 
231  IntType operator()() { return mRand(mEngine); }
232 
235  IntType operator()(IntType imin, IntType imax)
236  {
237  const IntType lo = std::min(imin, imax), hi = std::max(imin, imax);
238  return mRand(mEngine, typename Distr::param_type(lo, hi));
239  }
240 };
241 
243 
244 
245 // ==========> Clamp <==================
246 
248 template<typename Type>
249 inline Type
250 Clamp(Type x, Type min, Type max)
251 {
252  assert( !(min>max) );
253  return x > min ? x < max ? x : max : min;
254 }
255 
256 
258 template<typename Type>
259 inline Type
260 Clamp01(Type x) { return x > Type(0) ? x < Type(1) ? x : Type(1) : Type(0); }
261 
262 
264 template<typename Type>
265 inline bool
266 ClampTest01(Type &x)
267 {
268  if (x >= Type(0) && x <= Type(1)) return false;
269  x = x < Type(0) ? Type(0) : Type(1);
270  return true;
271 }
272 
274 template<typename Type>
275 inline Type
277 {
278  return x > 0 ? x < 1 ? (3-2*x)*x*x : Type(1) : Type(0);
279 }
280 
283 template<typename Type>
284 inline Type
285 SmoothUnitStep(Type x, Type min, Type max)
286 {
287  assert(min < max);
288  return SmoothUnitStep((x-min)/(max-min));
289 }
290 
291 
292 // ==========> Absolute Value <==================
293 
294 
296 inline int32_t Abs(int32_t i) { return abs(i); }
298 inline int64_t Abs(int64_t i)
299 {
300 #ifdef _MSC_VER
301  return (i < int64_t(0) ? -i : i);
302 #else
303  return labs(i);
304 #endif
305 }
306 inline float Abs(float x) { return std::fabs(x); }
307 inline double Abs(double x) { return std::fabs(x); }
308 inline long double Abs(long double x) { return std::fabs(x); }
309 inline uint32_t Abs(uint32_t i) { return i; }
310 inline uint64_t Abs(uint64_t i) { return i; }
311 inline bool Abs(bool b) { return b; }
312 // On OSX size_t and uint64_t are different types
313 #if defined(__APPLE__) || defined(MACOSX)
314 inline size_t Abs(size_t i) { return i; }
315 #endif
316 
317 
318 
320 
321 
322 // ==========> Value Comparison <==================
323 
324 
326 template<typename Type>
327 inline bool
328 isZero(const Type& x)
329 {
331  return x == zeroVal<Type>();
333 }
334 
335 
338 template<typename Type>
339 inline bool
340 isApproxZero(const Type& x)
341 {
342  const Type tolerance = Type(zeroVal<Type>() + Tolerance<Type>::value());
343  return !(x > tolerance) && !(x < -tolerance);
344 }
345 
347 template<typename Type>
348 inline bool
349 isApproxZero(const Type& x, const Type& tolerance)
350 {
351  return !(x > tolerance) && !(x < -tolerance);
352 }
353 
354 
356 template<typename Type>
357 inline bool
358 isNegative(const Type& x) { return x < zeroVal<Type>(); }
359 
360 // Return false, since bool values are never less than zero.
361 template<> inline bool isNegative<bool>(const bool&) { return false; }
362 
363 
365 inline bool
366 isFinite(const float x) { return std::isfinite(x); }
367 
369 template<typename Type, typename std::enable_if<std::is_arithmetic<Type>::value, int>::type = 0>
370 inline bool
371 isFinite(const Type& x) { return std::isfinite(static_cast<double>(x)); }
372 
373 
375 inline bool
376 isInfinite(const float x) { return std::isinf(x); }
377 
379 template<typename Type, typename std::enable_if<std::is_arithmetic<Type>::value, int>::type = 0>
380 inline bool
381 isInfinite(const Type& x) { return std::isinf(static_cast<double>(x)); }
382 
383 
385 inline bool
386 isNan(const float x) { return std::isnan(x); }
387 
389 template<typename Type, typename std::enable_if<std::is_arithmetic<Type>::value, int>::type = 0>
390 inline bool
391 isNan(const Type& x) { return std::isnan(static_cast<double>(x)); }
392 
393 
395 template<typename Type>
396 inline bool
397 isApproxEqual(const Type& a, const Type& b, const Type& tolerance)
398 {
399  return !cwiseGreaterThan(Abs(a - b), tolerance);
400 }
401 
404 template<typename Type>
405 inline bool
406 isApproxEqual(const Type& a, const Type& b)
407 {
408  const Type tolerance = Type(zeroVal<Type>() + Tolerance<Type>::value());
409  return isApproxEqual(a, b, tolerance);
410 }
411 
412 #define OPENVDB_EXACT_IS_APPROX_EQUAL(T) \
413  template<> inline bool isApproxEqual<T>(const T& a, const T& b) { return a == b; } \
414  template<> inline bool isApproxEqual<T>(const T& a, const T& b, const T&) { return a == b; } \
415 
416 
419 
420 
423 template<typename Type>
424 inline bool
425 isApproxLarger(const Type& a, const Type& b, const Type& tolerance)
426 {
427  return (b - a < tolerance);
428 }
429 
430 
432 template<typename T0, typename T1>
433 inline bool
434 isExactlyEqual(const T0& a, const T1& b)
435 {
437  return a == b;
439 }
440 
441 
442 template<typename Type>
443 inline bool
444 isRelOrApproxEqual(const Type& a, const Type& b, const Type& absTol, const Type& relTol)
445 {
446  // First check to see if we are inside the absolute tolerance
447  // Necessary for numbers close to 0
448  if (!(Abs(a - b) > absTol)) return true;
449 
450  // Next check to see if we are inside the relative tolerance
451  // to handle large numbers that aren't within the abs tolerance
452  // but could be the closest floating point representation
453  double relError;
454  if (Abs(b) > Abs(a)) {
455  relError = Abs((a - b) / b);
456  } else {
457  relError = Abs((a - b) / a);
458  }
459  return (relError <= relTol);
460 }
461 
462 template<>
463 inline bool
464 isRelOrApproxEqual(const bool& a, const bool& b, const bool&, const bool&)
465 {
466  return (a == b);
467 }
468 
469 
470 // Avoid strict aliasing issues by using type punning
471 // http://cellperformance.beyond3d.com/articles/2006/06/understanding-strict-aliasing.html
472 // Using "casting through a union(2)"
473 inline int32_t
474 floatToInt32(const float aFloatValue)
475 {
476  union FloatOrInt32 { float floatValue; int32_t int32Value; };
477  const FloatOrInt32* foi = reinterpret_cast<const FloatOrInt32*>(&aFloatValue);
478  return foi->int32Value;
479 }
480 
481 
482 inline int64_t
483 doubleToInt64(const double aDoubleValue)
484 {
485  union DoubleOrInt64 { double doubleValue; int64_t int64Value; };
486  const DoubleOrInt64* dol = reinterpret_cast<const DoubleOrInt64*>(&aDoubleValue);
487  return dol->int64Value;
488 }
489 
490 
491 // aUnitsInLastPlace is the allowed difference between the least significant digits
492 // of the numbers' floating point representation
493 // Please read the reference paper before trying to use isUlpsEqual
494 // http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm
495 inline bool
496 isUlpsEqual(const double aLeft, const double aRight, const int64_t aUnitsInLastPlace)
497 {
498  int64_t longLeft = doubleToInt64(aLeft);
499  // Because of 2's complement, must restore lexicographical order
500  if (longLeft < 0) {
501  longLeft = INT64_C(0x8000000000000000) - longLeft;
502  }
503 
504  int64_t longRight = doubleToInt64(aRight);
505  // Because of 2's complement, must restore lexicographical order
506  if (longRight < 0) {
507  longRight = INT64_C(0x8000000000000000) - longRight;
508  }
509 
510  int64_t difference = labs(longLeft - longRight);
511  return (difference <= aUnitsInLastPlace);
512 }
513 
514 inline bool
515 isUlpsEqual(const float aLeft, const float aRight, const int32_t aUnitsInLastPlace)
516 {
517  int32_t intLeft = floatToInt32(aLeft);
518  // Because of 2's complement, must restore lexicographical order
519  if (intLeft < 0) {
520  intLeft = 0x80000000 - intLeft;
521  }
522 
523  int32_t intRight = floatToInt32(aRight);
524  // Because of 2's complement, must restore lexicographical order
525  if (intRight < 0) {
526  intRight = 0x80000000 - intRight;
527  }
528 
529  int32_t difference = abs(intLeft - intRight);
530  return (difference <= aUnitsInLastPlace);
531 }
532 
533 
535 
536 
537 // ==========> Pow <==================
538 
540 template<typename Type>
541 inline Type Pow2(Type x) { return x*x; }
542 
544 template<typename Type>
545 inline Type Pow3(Type x) { return x*x*x; }
546 
548 template<typename Type>
549 inline Type Pow4(Type x) { return Pow2(Pow2(x)); }
550 
552 template<typename Type>
553 Type
554 Pow(Type x, int n)
555 {
556  Type ans = 1;
557  if (n < 0) {
558  n = -n;
559  x = Type(1)/x;
560  }
561  while (n--) ans *= x;
562  return ans;
563 }
564 
566 inline float
568 Pow(float b, float e)
569 {
570  assert( b >= 0.0f && "Pow(float,float): base is negative" );
571  return powf(b,e);
572 }
573 
574 inline double
575 Pow(double b, double e)
576 {
577  assert( b >= 0.0 && "Pow(double,double): base is negative" );
578  return std::pow(b,e);
579 }
581 
582 
583 // ==========> Max <==================
584 
586 template<typename Type>
587 inline const Type&
588 Max(const Type& a, const Type& b)
589 {
590  return std::max(a,b);
591 }
592 
594 template<typename Type>
595 inline const Type&
596 Max(const Type& a, const Type& b, const Type& c)
597 {
598  return std::max(std::max(a,b), c);
599 }
600 
602 template<typename Type>
603 inline const Type&
604 Max(const Type& a, const Type& b, const Type& c, const Type& d)
605 {
606  return std::max(std::max(a,b), std::max(c,d));
607 }
608 
610 template<typename Type>
611 inline const Type&
612 Max(const Type& a, const Type& b, const Type& c, const Type& d, const Type& e)
613 {
614  return std::max(std::max(a,b), Max(c,d,e));
615 }
616 
618 template<typename Type>
619 inline const Type&
620 Max(const Type& a, const Type& b, const Type& c, const Type& d, const Type& e, const Type& f)
621 {
622  return std::max(Max(a,b,c), Max(d,e,f));
623 }
624 
626 template<typename Type>
627 inline const Type&
628 Max(const Type& a, const Type& b, const Type& c, const Type& d,
629  const Type& e, const Type& f, const Type& g)
630 {
631  return std::max(Max(a,b,c,d), Max(e,f,g));
632 }
633 
635 template<typename Type>
636 inline const Type&
637 Max(const Type& a, const Type& b, const Type& c, const Type& d,
638  const Type& e, const Type& f, const Type& g, const Type& h)
639 {
640  return std::max(Max(a,b,c,d), Max(e,f,g,h));
641 }
642 
643 
644 // ==========> Min <==================
645 
647 template<typename Type>
648 inline const Type&
649 Min(const Type& a, const Type& b) { return std::min(a, b); }
650 
652 template<typename Type>
653 inline const Type&
654 Min(const Type& a, const Type& b, const Type& c) { return std::min(std::min(a, b), c); }
655 
657 template<typename Type>
658 inline const Type&
659 Min(const Type& a, const Type& b, const Type& c, const Type& d)
660 {
661  return std::min(std::min(a, b), std::min(c, d));
662 }
663 
665 template<typename Type>
666 inline const Type&
667 Min(const Type& a, const Type& b, const Type& c, const Type& d, const Type& e)
668 {
669  return std::min(std::min(a,b), Min(c,d,e));
670 }
671 
673 template<typename Type>
674 inline const Type&
675 Min(const Type& a, const Type& b, const Type& c, const Type& d, const Type& e, const Type& f)
676 {
677  return std::min(Min(a,b,c), Min(d,e,f));
678 }
679 
681 template<typename Type>
682 inline const Type&
683 Min(const Type& a, const Type& b, const Type& c, const Type& d,
684  const Type& e, const Type& f, const Type& g)
685 {
686  return std::min(Min(a,b,c,d), Min(e,f,g));
687 }
688 
690 template<typename Type>
691 inline const Type&
692 Min(const Type& a, const Type& b, const Type& c, const Type& d,
693  const Type& e, const Type& f, const Type& g, const Type& h)
694 {
695  return std::min(Min(a,b,c,d), Min(e,f,g,h));
696 }
697 
698 
699 // ============> Exp <==================
700 
702 template<typename Type>
703 inline Type Exp(const Type& x) { return std::exp(x); }
704 
705 // ============> Sin <==================
706 
708 inline float Sin(const float& x) { return std::sin(x); }
710 
711 inline double Sin(const double& x) { return std::sin(x); }
713 
714 // ============> Cos <==================
715 
717 inline float Cos(const float& x) { return std::cos(x); }
719 
720 inline double Cos(const double& x) { return std::cos(x); }
722 
723 
725 
726 
728 template <typename Type>
729 inline int Sign(const Type &x) { return (zeroVal<Type>() < x) - (x < zeroVal<Type>()); }
730 
731 
734 template <typename Type>
735 inline bool
736 SignChange(const Type& a, const Type& b)
737 {
738  return ( (a<zeroVal<Type>()) ^ (b<zeroVal<Type>()) );
739 }
740 
741 
744 template <typename Type>
745 inline bool
746 ZeroCrossing(const Type& a, const Type& b)
747 {
748  return a * b <= zeroVal<Type>();
749 }
750 
751 
753 inline float Sqrt(float x) { return std::sqrt(x); }
755 inline double Sqrt(double x) { return std::sqrt(x); }
756 inline long double Sqrt(long double x) { return std::sqrt(x); }
758 
759 
761 inline float Cbrt(float x) { return std::cbrt(x); }
763 inline double Cbrt(double x) { return std::cbrt(x); }
764 inline long double Cbrt(long double x) { return std::cbrt(x); }
766 
767 
769 inline int Mod(int x, int y) { return (x % y); }
771 inline float Mod(float x, float y) { return std::fmod(x, y); }
772 inline double Mod(double x, double y) { return std::fmod(x, y); }
773 inline long double Mod(long double x, long double y) { return std::fmod(x, y); }
774 template<typename Type> inline Type Remainder(Type x, Type y) { return Mod(x, y); }
776 
777 
779 inline float RoundUp(float x) { return std::ceil(x); }
781 inline double RoundUp(double x) { return std::ceil(x); }
782 inline long double RoundUp(long double x) { return std::ceil(x); }
784 template<typename Type>
786 inline Type
787 RoundUp(Type x, Type base)
788 {
789  Type remainder = Remainder(x, base);
790  return remainder ? x-remainder+base : x;
791 }
792 
793 
795 inline float RoundDown(float x) { return std::floor(x); }
797 inline double RoundDown(double x) { return std::floor(x); }
798 inline long double RoundDown(long double x) { return std::floor(x); }
800 template<typename Type>
802 inline Type
803 RoundDown(Type x, Type base)
804 {
805  Type remainder = Remainder(x, base);
806  return remainder ? x-remainder : x;
807 }
808 
809 
811 inline float Round(float x) { return RoundDown(x + 0.5f); }
813 inline double Round(double x) { return RoundDown(x + 0.5); }
814 inline long double Round(long double x) { return RoundDown(x + 0.5l); }
816 
817 
820 template<typename Type>
821 inline Type
822 EuclideanRemainder(Type x) { return x - RoundDown(x); }
823 
824 
826 template<typename Type>
827 inline Type
828 IntegerPart(Type x)
829 {
830  return (x > 0 ? RoundDown(x) : RoundUp(x));
831 }
832 
834 template<typename Type>
835 inline Type
836 FractionalPart(Type x) { return Mod(x,Type(1)); }
837 
838 
840 inline int Floor(float x) { return int(RoundDown(x)); }
842 inline int Floor(double x) { return int(RoundDown(x)); }
843 inline int Floor(long double x) { return int(RoundDown(x)); }
845 
846 
848 inline int Ceil(float x) { return int(RoundUp(x)); }
850 inline int Ceil(double x) { return int(RoundUp(x)); }
851 inline int Ceil(long double x) { return int(RoundUp(x)); }
853 
854 
856 template<typename Type>
857 inline Type Chop(Type x, Type delta) { return (Abs(x) < delta ? zeroVal<Type>() : x); }
858 
859 
861 template<typename Type>
862 inline Type
863 Truncate(Type x, unsigned int digits)
864 {
865  Type tenth = Pow(10,digits);
866  return RoundDown(x*tenth+0.5)/tenth;
867 }
868 
870 
871 
874 template<typename T>
875 inline auto PrintCast(const T& val) -> typename std::enable_if<!std::is_same<T, int8_t>::value
876  && !std::is_same<T, uint8_t>::value, const T&>::type { return val; }
877 inline int32_t PrintCast(int8_t val) { return int32_t(val); }
878 inline uint32_t PrintCast(uint8_t val) { return uint32_t(val); }
879 
880 
882 
883 
885 template<typename Type>
886 inline Type
887 Inv(Type x)
888 {
889  assert(x);
890  return Type(1)/x;
891 }
892 
893 
894 enum Axis {
895  X_AXIS = 0,
896  Y_AXIS = 1,
897  Z_AXIS = 2
898 };
899 
900 // enum values are consistent with their historical mx analogs.
910 };
911 
912 
913 template <typename S, typename T>
914 struct promote {
915  using type = typename boost::numeric::conversion_traits<S, T>::supertype;
916 };
917 
918 
926 template<typename Vec3T>
927 size_t
928 MinIndex(const Vec3T& v)
929 {
930  static const size_t hashTable[8] = { 2, 1, 9, 1, 2, 9, 0, 0 };//9 is a dummy value
931  const size_t hashKey =
932  ((v[0] < v[1]) << 2) + ((v[0] < v[2]) << 1) + (v[1] < v[2]);// ?*4+?*2+?*1
933  return hashTable[hashKey];
934 }
935 
936 
944 template<typename Vec3T>
945 size_t
946 MaxIndex(const Vec3T& v)
947 {
948  static const size_t hashTable[8] = { 2, 1, 9, 1, 2, 9, 0, 0 };//9 is a dummy value
949  const size_t hashKey =
950  ((v[0] > v[1]) << 2) + ((v[0] > v[2]) << 1) + (v[1] > v[2]);// ?*4+?*2+?*1
951  return hashTable[hashKey];
952 }
953 
954 } // namespace math
955 } // namespace OPENVDB_VERSION_NAME
956 } // namespace openvdb
957 
958 #endif // OPENVDB_MATH_MATH_HAS_BEEN_INCLUDED
openvdb::v7_2::math::promote
Definition: Math.h:914
OPENVDB_NO_TYPE_CONVERSION_WARNING_END
#define OPENVDB_NO_TYPE_CONVERSION_WARNING_END
Definition: Platform.h:188
openvdb::v7_2::math::Pow4
Type Pow4(Type x)
Return x4.
Definition: Math.h:549
openvdb::v7_2::math::ZXY_ROTATION
@ ZXY_ROTATION
Definition: Math.h:906
openvdb::v7_2::math::YZX_ROTATION
@ YZX_ROTATION
Definition: Math.h:905
openvdb::v7_2::math::Chop
Type Chop(Type x, Type delta)
Return x if it is greater or equal in magnitude than delta. Otherwise, return zero.
Definition: Math.h:857
openvdb::v7_2::math::operator+
std::string operator+(const std::string &s, double)
Definition: Math.h:74
openvdb::v7_2::math::RandInt::RandInt
RandInt(const EngineType &engine, IntType imin, IntType imax)
Initialize the generator.
Definition: Math.h:202
openvdb::v7_2::math::RandInt::RandInt
RandInt(unsigned int seed, IntType imin, IntType imax)
Initialize the generator.
Definition: Math.h:210
openvdb::v7_2::math::Cos
double Cos(const double &x)
Definition: Math.h:720
openvdb::v7_2::math::Remainder
Type Remainder(Type x, Type y)
Definition: Math.h:774
openvdb::v7_2::math::Tolerance< float >::value
static float value()
Definition: Math.h:138
openvdb::v7_2::math::Rand01< double, RandomGenerator >::ValueType
double ValueType
Definition: Math.h:162
openvdb::v7_2::math::isNegative< bool >
bool isNegative< bool >(const bool &)
Definition: Math.h:361
openvdb::v7_2::zeroVal< bool >
bool zeroVal< bool >()
Return the bool value that corresponds to zero.
Definition: Math.h:63
openvdb::v7_2::math::Rand01::Rand01
Rand01(const EngineType &engine)
Initialize the generator.
Definition: Math.h:166
openvdb::v7_2::math::isNan
bool isNan(const Type &x)
Return true if x is a NaN (Not-A-Number) value.
Definition: Math.h:391
openvdb::v7_2::math::negative
std::string negative(const std::string &val)
Return the "negation" of the given string.
Definition: Math.h:132
openvdb::v7_2::math::ZXZ_ROTATION
@ ZXZ_ROTATION
Definition: Math.h:909
openvdb::v7_2::math::Max
const Type & Max(const Type &a, const Type &b, const Type &c, const Type &d, const Type &e, const Type &f, const Type &g, const Type &h)
Return the maximum of eight values.
Definition: Math.h:637
openvdb::v7_2::tools::composite::min
const std::enable_if<!VecTraits< T >::IsVec, T >::type & min(const T &a, const T &b)
Definition: Composite.h:103
openvdb::v7_2::math::Floor
int Floor(long double x)
Definition: Math.h:843
openvdb::v7_2::math::Delta
Delta for small floating-point offsets.
Definition: Math.h:144
openvdb::v7_2::math::isApproxZero
bool isApproxZero(const Type &x, const Type &tolerance)
Return true if x is equal to zero to within the given tolerance.
Definition: Math.h:349
openvdb::v7_2::math::floatToInt32
int32_t floatToInt32(const float aFloatValue)
Definition: Math.h:474
openvdb::v7_2::math::RotationOrder
RotationOrder
Definition: Math.h:901
openvdb::v7_2::math::Clamp01
Type Clamp01(Type x)
Return x clamped to [0, 1].
Definition: Math.h:260
openvdb::v7_2::math::isZero
bool isZero(const Type &x)
Return true if x is exactly equal to zero.
Definition: Math.h:328
openvdb::v7_2::math::RandInt::setSeed
void setSeed(unsigned int seed)
Set the seed value for the random number generator.
Definition: Math.h:222
openvdb::v7_2::math::Delta< float >::value
static float value()
Definition: Math.h:145
version.h
Library and file format version numbers.
openvdb::v7_2::math::Inv
Type Inv(Type x)
Return the inverse of x.
Definition: Math.h:887
openvdb::v7_2::math::Tolerance< double >::value
static double value()
Definition: Math.h:139
openvdb::v7_2::math::Abs
bool Abs(bool b)
Definition: Math.h:311
openvdb::v7_2::math::isRelOrApproxEqual
bool isRelOrApproxEqual(const bool &a, const bool &b, const bool &, const bool &)
Definition: Math.h:464
Platform.h
openvdb::v7_2::math::XZY_ROTATION
@ XZY_ROTATION
Definition: Math.h:903
openvdb::v7_2::math::Rand01::engine
const EngineType & engine() const
Return a const reference to the random number generator.
Definition: Math.h:179
openvdb::v7_2::math::XYZ_ROTATION
@ XYZ_ROTATION
Definition: Math.h:902
openvdb::v7_2::tools::composite::max
const std::enable_if<!VecTraits< T >::IsVec, T >::type & max(const T &a, const T &b)
Definition: Composite.h:107
openvdb::v7_2::math::ZeroCrossing
bool ZeroCrossing(const Type &a, const Type &b)
Return true if the interval [a, b] includes zero, i.e., if either a or b is zero or if they have diff...
Definition: Math.h:746
openvdb::v7_2::math::pi
constexpr float pi()
Definition: Math.h:109
openvdb::v7_2::math::PrintCast
uint32_t PrintCast(uint8_t val)
Definition: Math.h:878
openvdb::v7_2::math::Tolerance
Tolerance for floating-point comparison.
Definition: Math.h:137
openvdb::v7_2::math::RoundDown
Type RoundDown(Type x, Type base)
Return x rounded down to the nearest multiple of base.
Definition: Math.h:803
openvdb::v7_2::math::isApproxEqual
bool isApproxEqual(const Type &a, const Type &b)
Return true if a is equal to b to within the default floating-point comparison tolerance.
Definition: Math.h:406
openvdb::v7_2::math::Z_AXIS
@ Z_AXIS
Definition: Math.h:897
openvdb::v7_2::math::RandInt::setRange
void setRange(IntType imin, IntType imax)
Change the range over which integers are distributed to [imin, imax].
Definition: Math.h:216
openvdb::v7_2::math::Pow2
Type Pow2(Type x)
Return x2.
Definition: Math.h:541
openvdb::v7_2::math::Axis
Axis
Definition: Math.h:894
openvdb::v7_2::math::Rand01
Simple generator of random numbers over the range [0, 1)
Definition: Math.h:156
openvdb::v7_2::math::Ceil
int Ceil(long double x)
Definition: Math.h:851
openvdb::v7_2::math::Truncate
Type Truncate(Type x, unsigned int digits)
Return x truncated to the given number of decimal digits.
Definition: Math.h:863
openvdb::v7_2::math::RoundUp
Type RoundUp(Type x, Type base)
Return x rounded up to the nearest multiple of base.
Definition: Math.h:787
openvdb::v7_2::math::Rand01::operator()
FloatType operator()()
Return a uniformly distributed random number in the range [0, 1).
Definition: Math.h:182
openvdb::v7_2::math::FractionalPart
Type FractionalPart(Type x)
Return the fractional part of x.
Definition: Math.h:836
openvdb::v7_2::math::cwiseAdd
auto cwiseAdd(const Type1 &v, const Type2 s)
Componentwise adder for POD types.
Definition: Math.h:79
OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN
#define OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN
Definition: Math.h:47
openvdb::v7_2::math::Pow3
Type Pow3(Type x)
Return x3.
Definition: Math.h:545
openvdb::v7_2::math::isFinite
bool isFinite(const Type &x)
Return true if x is finite.
Definition: Math.h:371
openvdb::v7_2::math::Rand01::setSeed
void setSeed(unsigned int seed)
Set the seed value for the random number generator.
Definition: Math.h:173
openvdb::v7_2::math::cwiseLessThan
bool cwiseLessThan(const Type1 &a, const Type2 &b)
Componentwise less than for POD types.
Definition: Math.h:88
openvdb::v7_2::math::Sin
double Sin(const double &x)
Definition: Math.h:711
openvdb::v7_2::math::MaxIndex
size_t MaxIndex(const Vec3T &v)
Return the index [0,1,2] of the largest value in a 3D vector.
Definition: Math.h:946
openvdb::v7_2::math::isInfinite
bool isInfinite(const Type &x)
Return true if x is an infinity value (either positive infinity or negative infinity).
Definition: Math.h:381
openvdb::v7_2::math::isNegative
bool isNegative(const Type &x)
Return true if x is less than zero.
Definition: Math.h:358
openvdb::v7_2::math::isApproxLarger
bool isApproxLarger(const Type &a, const Type &b, const Type &tolerance)
Return true if a is larger than b to within the given tolerance, i.e., if b - a < tolerance.
Definition: Math.h:425
openvdb::v7_2::math::MinIndex
size_t MinIndex(const Vec3T &v)
Return the index [0,1,2] of the smallest value in a 3D vector.
Definition: Math.h:928
openvdb::v7_2::math::doubleToInt64
int64_t doubleToInt64(const double aDoubleValue)
Definition: Math.h:483
openvdb::v7_2::math::EuclideanRemainder
Type EuclideanRemainder(Type x)
Definition: Math.h:822
openvdb::v7_2::math::YXZ_ROTATION
@ YXZ_ROTATION
Definition: Math.h:904
openvdb::v7_2::math::X_AXIS
@ X_AXIS
Definition: Math.h:895
openvdb::v7_2::math::promote::type
typename boost::numeric::conversion_traits< S, T >::supertype type
Definition: Math.h:915
OPENVDB_EXACT_IS_APPROX_EQUAL
#define OPENVDB_EXACT_IS_APPROX_EQUAL(T)
Definition: Math.h:412
OPENVDB_USE_VERSION_NAMESPACE
#define OPENVDB_USE_VERSION_NAMESPACE
Definition: version.h:147
openvdb::v7_2::zeroVal
T zeroVal()
Return the value of type T that corresponds to zero.
Definition: Math.h:59
openvdb::v7_2::math::IntegerPart
Type IntegerPart(Type x)
Return the integer part of x.
Definition: Math.h:828
openvdb::v7_2::math::Delta< double >::value
static double value()
Definition: Math.h:146
openvdb::v7_2::math::Min
const Type & Min(const Type &a, const Type &b, const Type &c, const Type &d, const Type &e, const Type &f, const Type &g, const Type &h)
Return the minimum of eight values.
Definition: Math.h:692
openvdb::v7_2::math::XZX_ROTATION
@ XZX_ROTATION
Definition: Math.h:908
openvdb::v7_2::math::Sign
int Sign(const Type &x)
Return the sign of the given value as an integer (either -1, 0 or 1).
Definition: Math.h:729
std
Definition: Coord.h:587
OPENVDB_NO_TYPE_CONVERSION_WARNING_BEGIN
#define OPENVDB_NO_TYPE_CONVERSION_WARNING_BEGIN
Bracket code with OPENVDB_NO_TYPE_CONVERSION_WARNING_BEGIN/_END, to inhibit warnings about type conve...
Definition: Platform.h:187
openvdb::v7_2::math::Round
long double Round(long double x)
Definition: Math.h:814
openvdb::v7_2::math::RandInt::operator()
IntType operator()()
Return a randomly-generated integer in the current range.
Definition: Math.h:231
openvdb::v7_2::math::SignChange
bool SignChange(const Type &a, const Type &b)
Return true if a and b have different signs.
Definition: Math.h:736
openvdb::v7_2::math::Y_AXIS
@ Y_AXIS
Definition: Math.h:896
openvdb::v7_2::math::RandInt
Simple random integer generator.
Definition: Math.h:192
openvdb::v7_2::math::Exp
Type Exp(const Type &x)
Return ex.
Definition: Math.h:703
openvdb::v7_2::math::Rand01::Rand01
Rand01(unsigned int seed)
Initialize the generator.
Definition: Math.h:170
openvdb::v7_2::math::RandInt::engine
const EngineType & engine() const
Return a const reference to the random number generator.
Definition: Math.h:228
openvdb::v7_2::math::Cbrt
long double Cbrt(long double x)
Definition: Math.h:764
OPENVDB_VERSION_NAME
#define OPENVDB_VERSION_NAME
The version namespace name for this library version.
Definition: version.h:95
openvdb::v7_2::math::Sqrt
long double Sqrt(long double x)
Definition: Math.h:756
openvdb::v7_2::math::Pow
double Pow(double b, double e)
Definition: Math.h:575
openvdb::v7_2::math::ClampTest01
bool ClampTest01(Type &x)
Return true if x is outside [0,1].
Definition: Math.h:266
openvdb::v7_2::math::isExactlyEqual
bool isExactlyEqual(const T0 &a, const T1 &b)
Return true if a is exactly equal to b.
Definition: Math.h:434
openvdb::v7_2::math::SmoothUnitStep
Type SmoothUnitStep(Type x, Type min, Type max)
Return 0 if x < min, 1 if x > max or else (3 − 2 t) t², where t = (x − min)/(max − min).
Definition: Math.h:285
openvdb
Definition: openvdb/Exceptions.h:13
openvdb::v7_2::math::isUlpsEqual
bool isUlpsEqual(const float aLeft, const float aRight, const int32_t aUnitsInLastPlace)
Definition: Math.h:515
openvdb::v7_2::math::ZYX_ROTATION
@ ZYX_ROTATION
Definition: Math.h:907
openvdb::v7_2::math::Mod
long double Mod(long double x, long double y)
Definition: Math.h:773
openvdb::v7_2::math::cwiseGreaterThan
bool cwiseGreaterThan(const Type1 &a, const Type2 &b)
Componentwise greater than for POD types.
Definition: Math.h:97
openvdb::v7_2::math::RandInt::operator()
IntType operator()(IntType imin, IntType imax)
Return a randomly-generated integer in the new range [imin, imax], without changing the current range...
Definition: Math.h:235
openvdb::v7_2::math::Clamp
Type Clamp(Type x, Type min, Type max)
Return x clamped to [min, max].
Definition: Math.h:250
OPENVDB_NO_FP_EQUALITY_WARNING_END
#define OPENVDB_NO_FP_EQUALITY_WARNING_END
Definition: Math.h:48