TMath.h

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// @(#)root/mathcore:$Id$
// Authors: Rene Brun, Anna Kreshuk, Eddy Offermann, Fons Rademakers   29/07/95
 
/*************************************************************************
 * Copyright (C) 1995-2004, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/
 
#ifndef ROOT_TMath
#define ROOT_TMath
 
#include "TMathBase.h"
 
#include "TError.h"
#include "ROOT/RSpan.hxx"
 
#include <algorithm>
#include <limits>
#include <cmath>
#include <vector>
 
////////////////////////////////////////////////////////////////////////////////
///
/// TMath
///
/// Encapsulate most frequently used Math functions.
/// NB. The basic functions Min, Max, Abs and Sign are defined
/// in TMathBase.
 
namespace TMath {
 
////////////////////////////////////////////////////////////////////////////////
// Fundamental constants
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ \pi\f$
constexpr Double_t Pi()
{
   return 3.14159265358979323846;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ 2\pi\f$
constexpr Double_t TwoPi()
{
   return 2.0 * Pi();
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ \frac{\pi}{2} \f$
constexpr Double_t PiOver2()
{
   return Pi() / 2.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ \frac{\pi}{4} \f$
constexpr Double_t PiOver4()
{
   return Pi() / 4.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ \frac{1.}{\pi}\f$
constexpr Double_t InvPi()
{
   return 1.0 / Pi();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Conversion from radian to degree: \f$ \frac{180}{\pi} \f$
constexpr Double_t RadToDeg()
{
   return 180.0 / Pi();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Conversion from degree to radian: \f$ \frac{\pi}{180} \f$
constexpr Double_t DegToRad()
{
   return Pi() / 180.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ \sqrt{2} \f$
constexpr Double_t Sqrt2()
{
   return 1.4142135623730950488016887242097;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Base of natural log: \f$ e \f$
constexpr Double_t E()
{
   return 2.71828182845904523536;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Natural log of 10 (to convert log to ln)
constexpr Double_t Ln10()
{
   return 2.30258509299404568402;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Base-10 log of e  (to convert ln to log)
constexpr Double_t LogE()
{
   return 0.43429448190325182765;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Velocity of light in \f$ m s^{-1} \f$
constexpr Double_t C()
{
   return 2.99792458e8;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ cm s^{-1} \f$
constexpr Double_t Ccgs()
{
   return 100.0 * C();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Speed of light uncertainty.
constexpr Double_t CUncertainty()
{
   return 0.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Gravitational constant in: \f$ m^{3} kg^{-1} s^{-2} \f$
constexpr Double_t G()
{
   // use 2018 value from NIST  (https://physics.nist.gov/cgi-bin/cuu/Value?bg|search_for=G)
   return 6.67430e-11;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ cm^{3} g^{-1} s^{-2} \f$
constexpr Double_t Gcgs()
{
   return G() * 1000.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Gravitational constant uncertainty.
constexpr Double_t GUncertainty()
{
   // use 2018 value from NIST
   return 0.00015e-11;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ \frac{G}{\hbar C} \f$ in \f$ (GeV/c^{2})^{-2} \f$
constexpr Double_t GhbarC()
{
   // use new value from NIST (2018)
   return 6.70883e-39;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ \frac{G}{\hbar C} \f$ uncertainty.
constexpr Double_t GhbarCUncertainty()
{
   // use new value from NIST (2018)
   return 0.00015e-39;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Standard acceleration of gravity in \f$ m s^{-2} \f$
constexpr Double_t Gn()
{
   return 9.80665;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Standard acceleration of gravity uncertainty.
constexpr Double_t GnUncertainty()
{
   return 0.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Planck's constant in \f$ J s \f$: \f$ h \f$
constexpr Double_t H()
{
   return 6.62607015e-34;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ erg s \f$
constexpr Double_t Hcgs()
{
   return 1.0e7 * H();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Planck's constant uncertainty.
constexpr Double_t HUncertainty()
{
   // Planck constant is exact according to 2019 redefinition
   // (https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units)
   return 0.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ \hbar \f$ in \f$ J s \f$: \f$ \hbar = \frac{h}{2\pi} \f$
constexpr Double_t Hbar()
{
   return 1.054571817e-34;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ erg s \f$
constexpr Double_t Hbarcgs()
{
   return 1.0e7 * Hbar();
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ \hbar \f$ uncertainty.
constexpr Double_t HbarUncertainty()
{
   // hbar is an exact constant
   return 0.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ hc \f$ in \f$ J m \f$
constexpr Double_t HC()
{
   return H() * C();
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ erg cm \f$
constexpr Double_t HCcgs()
{
   return Hcgs() * Ccgs();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Boltzmann's constant in \f$ J K^{-1} \f$: \f$ k \f$
constexpr Double_t K()
{
   return 1.380649e-23;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \f$ erg K^{-1} \f$
constexpr Double_t Kcgs()
{
   return 1.0e7 * K();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Boltzmann's constant uncertainty.
constexpr Double_t KUncertainty()
{
   // constant is exact according to 2019 redefinition
   // (https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units)
   return 0.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Stefan-Boltzmann constant in \f$ W m^{-2} K^{-4}\f$: \f$ \sigma \f$
constexpr Double_t Sigma()
{
   return 5.670373e-8;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Stefan-Boltzmann constant uncertainty.
constexpr Double_t SigmaUncertainty()
{
   return 0.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Avogadro constant (Avogadro's Number) in \f$ mol^{-1} \f$
constexpr Double_t Na()
{
   return 6.02214076e+23;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Avogadro constant (Avogadro's Number) uncertainty.
constexpr Double_t NaUncertainty()
{
   // constant is exact according to 2019 redefinition
   // (https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units)
   return 0.0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// [Universal gas constant](http://scienceworld.wolfram.com/physics/UniversalGasConstant.html)
/// (\f$ Na K \f$) in \f$ J K^{-1} mol^{-1} \f$
//
constexpr Double_t R()
{
   return K() * Na();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Universal gas constant uncertainty.
constexpr Double_t RUncertainty()
{
   return R() * ((KUncertainty() / K()) + (NaUncertainty() / Na()));
}
 
////////////////////////////////////////////////////////////////////////////////
/// [Molecular weight of dry air 1976 US Standard Atmosphere](http://atmos.nmsu.edu/jsdap/encyclopediawork.html)
/// in \f$ kg kmol^{-1} \f$ or \f$ gm mol^{-1} \f$
constexpr Double_t MWair()
{
   return 28.9644;
}
 
////////////////////////////////////////////////////////////////////////////////
/// [Dry Air Gas Constant (R / MWair)](http://atmos.nmsu.edu/education_and_outreach/encyclopedia/gas_constant.htm)
/// in \f$ J kg^{-1} K^{-1} \f$
constexpr Double_t Rgair()
{
   return (1000.0 * R()) / MWair();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Euler-Mascheroni Constant.
constexpr Double_t EulerGamma()
{
   return 0.577215664901532860606512090082402431042;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Elementary charge in \f$ C \f$ .
constexpr Double_t Qe()
{
   return 1.602176634e-19;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Elementary charge uncertainty.
constexpr Double_t QeUncertainty()
{
   // constant is exact according to 2019 redefinition
   // (https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units)
   return 0.0;
}
 
////////////////////////////////////////////////////////////////////////////////
// Mathematical Functions
 
////////////////////////////////////////////////////////////////////////////////
// Trigonometrical Functions
 
inline Double_t Sin(Double_t);
inline Double_t Cos(Double_t);
inline Double_t Tan(Double_t);
inline Double_t SinH(Double_t);
inline Double_t CosH(Double_t);
inline Double_t TanH(Double_t);
inline Double_t ASin(Double_t);
inline Double_t ACos(Double_t);
inline Double_t ATan(Double_t);
inline Double_t ATan2(Double_t y, Double_t x);
Double_t ASinH(Double_t);
Double_t ACosH(Double_t);
Double_t ATanH(Double_t);
Double_t Hypot(Double_t x, Double_t y);
 
////////////////////////////////////////////////////////////////////////////////
// Elementary Functions
 
inline Double_t Ceil(Double_t x);
inline Int_t CeilNint(Double_t x);
inline Double_t Floor(Double_t x);
inline Int_t FloorNint(Double_t x);
template <typename T>
inline Int_t Nint(T x);
 
inline Double_t Sq(Double_t x);
inline Double_t Sqrt(Double_t x);
inline Double_t Exp(Double_t x);
inline Double_t Ldexp(Double_t x, Int_t exp);
Double_t Factorial(Int_t i);
inline LongDouble_t Power(LongDouble_t x, LongDouble_t y);
inline LongDouble_t Power(LongDouble_t x, Long64_t y);
inline LongDouble_t Power(Long64_t x, Long64_t y);
inline Double_t Power(Double_t x, Double_t y);
inline Double_t Power(Double_t x, Int_t y);
inline Double_t Log(Double_t x);
Double_t Log2(Double_t x);
inline Double_t Log10(Double_t x);
inline Int_t Finite(Double_t x);
inline Int_t Finite(Float_t x);
inline Bool_t IsNaN(Double_t x);
inline Bool_t IsNaN(Float_t x);
 
inline Double_t QuietNaN();
inline Double_t SignalingNaN();
inline Double_t Infinity();
 
template <typename T>
struct Limits {
   inline static T Min();
   inline static T Max();
   inline static T Epsilon();
   };
 
   // Some integer math
   Long_t   Hypot(Long_t x, Long_t y);
 
   /// Comparing floating points.
   /// Returns `kTRUE` if the absolute difference between `af` and `bf` is less than `epsilon`.
   inline Bool_t AreEqualAbs(Double_t af, Double_t bf, Double_t epsilon) {
      return af == bf || // shortcut for exact equality (necessary because otherwise (inf != inf))
             TMath::Abs(af-bf) < epsilon ||
             TMath::Abs(af - bf) < Limits<Double_t>::Min(); // handle 0 < 0 case
 
   }
   /// Comparing floating points.
   /// Returns `kTRUE` if the relative difference between `af` and `bf` is less than `relPrec`.
   inline Bool_t AreEqualRel(Double_t af, Double_t bf, Double_t relPrec) {
      return af == bf || // shortcut for exact equality (necessary because otherwise (inf != inf))
             TMath::Abs(af - bf) <= 0.5 * relPrec * (TMath::Abs(af) + TMath::Abs(bf)) ||
             TMath::Abs(af - bf) < Limits<Double_t>::Min(); // handle denormals
   }
 
   /////////////////////////////////////////////////////////////////////////////
   // Array Algorithms
 
   // Min, Max of an array
   template <typename T> T MinElement(Long64_t n, const T *a);
   template <typename T> T MaxElement(Long64_t n, const T *a);
 
   // Locate Min, Max element number in an array
   template <typename T> Long64_t  LocMin(Long64_t n, const T *a);
   template <typename Iterator> Iterator LocMin(Iterator first, Iterator last);
   template <typename T> Long64_t  LocMax(Long64_t n, const T *a);
   template <typename Iterator> Iterator LocMax(Iterator first, Iterator last);
 
   // Derivatives of an array
   template <typename T>
   T *Gradient(Long64_t n, T const *f, double h = 1);
   template <typename T>
   T *Laplacian(Long64_t n, T const *f, double h = 1);
 
   // Hashing
   ULong_t Hash(const void *txt, Int_t ntxt);
   ULong_t Hash(const char *str);
 
   void BubbleHigh(Int_t Narr, Double_t *arr1, Int_t *arr2);
   void BubbleLow (Int_t Narr, Double_t *arr1, Int_t *arr2);
 
   Bool_t   Permute(Int_t n, Int_t *a); // Find permutations
 
   /////////////////////////////////////////////////////////////////////////////
   // Geometrical Functions
 
   //Sample quantiles
   void      Quantiles(Int_t n, Int_t nprob, Double_t *x, Double_t *quantiles, Double_t *prob,
                       Bool_t isSorted=kTRUE, Int_t *index = nullptr, Int_t type=7);
 
   // IsInside
   template <typename T> Bool_t IsInside(T xp, T yp, Int_t np, T *x, T *y);
 
   // Calculate the Cross Product of two vectors
   template <typename T> T *Cross(const T v1[3],const T v2[3], T out[3]);
 
   Float_t   Normalize(Float_t v[3]);  // Normalize a vector
   Double_t  Normalize(Double_t v[3]); // Normalize a vector
 
   //Calculate the Normalized Cross Product of two vectors
   template <typename T> inline T NormCross(const T v1[3],const T v2[3],T out[3]);
 
   // Calculate a normal vector of a plane
   template <typename T> T *Normal2Plane(const T v1[3],const T v2[3],const T v3[3], T normal[3]);
 
   /////////////////////////////////////////////////////////////////////////////
   // Polynomial Functions
 
   Bool_t    RootsCubic(const Double_t coef[4],Double_t &a, Double_t &b, Double_t &c);
 
   /////////////////////////////////////////////////////////////////////////////
   // Statistic Functions
 
   Double_t Binomial(Int_t n,Int_t k);  // Calculate the binomial coefficient n over k
   Double_t BinomialI(Double_t p, Int_t n, Int_t k);
   Double_t BreitWigner(Double_t x, Double_t mean=0, Double_t gamma=1);
   Double_t BreitWignerRelativistic(Double_t x, Double_t median=0, Double_t gamma=1);
   Double_t CauchyDist(Double_t x, Double_t t=0, Double_t s=1);
   Double_t ChisquareQuantile(Double_t p, Double_t ndf);
   Double_t FDist(Double_t F, Double_t N, Double_t M);
   Double_t FDistI(Double_t F, Double_t N, Double_t M);
   Double_t Gaus(Double_t x, Double_t mean=0, Double_t sigma=1, Bool_t norm=kFALSE);
   Double_t KolmogorovProb(Double_t z);
   Double_t KolmogorovTest(Int_t na, const Double_t *a, Int_t nb, const Double_t *b, Option_t *option);
   Double_t Landau(Double_t x, Double_t mpv=0, Double_t sigma=1, Bool_t norm=kFALSE);
   Double_t LandauI(Double_t x);
   Double_t LaplaceDist(Double_t x, Double_t alpha=0, Double_t beta=1);
   Double_t LaplaceDistI(Double_t x, Double_t alpha=0, Double_t beta=1);
   Double_t LogNormal(Double_t x, Double_t sigma, Double_t theta=0, Double_t m=1);
   Double_t NormQuantile(Double_t p);
   Double_t Poisson(Double_t x, Double_t par);
   Double_t PoissonI(Double_t x, Double_t par);
   Double_t Prob(Double_t chi2,Int_t ndf);
   Double_t Student(Double_t T, Double_t ndf);
   Double_t StudentI(Double_t T, Double_t ndf);
   Double_t StudentQuantile(Double_t p, Double_t ndf, Bool_t lower_tail=kTRUE);
   Double_t Vavilov(Double_t x, Double_t kappa, Double_t beta2);
   Double_t VavilovI(Double_t x, Double_t kappa, Double_t beta2);
   Double_t Voigt(Double_t x, Double_t sigma, Double_t lg, Int_t r = 4);
 
   /////////////////////////////////////////////////////////////////////////////
   // Statistics over arrays
 
   // Mean, Geometric Mean, Median, RMS(sigma), ModeHalfSample
 
   template <typename T> Double_t Mean(Long64_t n, const T *a, const Double_t *w=nullptr);
   template <typename Iterator> Double_t Mean(Iterator first, Iterator last);
   template <typename Iterator, typename WeightIterator> Double_t Mean(Iterator first, Iterator last, WeightIterator wfirst);
 
   template <typename T> Double_t GeomMean(Long64_t n, const T *a);
   template <typename Iterator> Double_t GeomMean(Iterator first, Iterator last);
 
   template <typename T> Double_t RMS(Long64_t n, const T *a, const Double_t *w=nullptr);
   template <typename Iterator> Double_t RMS(Iterator first, Iterator last);
   template <typename Iterator, typename WeightIterator> Double_t RMS(Iterator first, Iterator last, WeightIterator wfirst);
 
   template <typename T> Double_t StdDev(Long64_t n, const T *a, const Double_t * w = nullptr) { return RMS<T>(n,a,w); } /// Same as RMS
   template <typename Iterator> Double_t StdDev(Iterator first, Iterator last) { return RMS<Iterator>(first,last); } /// Same as RMS
   template <typename Iterator, typename WeightIterator> Double_t StdDev(Iterator first, Iterator last, WeightIterator wfirst) { return RMS<Iterator,WeightIterator>(first,last,wfirst); } /// Same as RMS
 
   template <typename T> Double_t Median(Long64_t n, const T *a, const Double_t *w=nullptr, Long64_t *work=nullptr);
   template <typename T> Double_t ModeHalfSample(Long64_t n, const T *a, const Double_t *w = nullptr);
 
   //k-th order statistic
   template <class Element, typename Size> Element KOrdStat(Size n, const Element *a, Size k, Size *work = 0);
 
   /////////////////////////////////////////////////////////////////////////////
   // Special Functions
 
   Double_t Beta(Double_t p, Double_t q);
   Double_t BetaCf(Double_t x, Double_t a, Double_t b);
   Double_t BetaDist(Double_t x, Double_t p, Double_t q);
   Double_t BetaDistI(Double_t x, Double_t p, Double_t q);
   Double_t BetaIncomplete(Double_t x, Double_t a, Double_t b);
 
   // Bessel functions
   Double_t BesselI(Int_t n,Double_t x);  /// Integer order modified Bessel function I_n(x)
   Double_t BesselK(Int_t n,Double_t x);  /// Integer order modified Bessel function K_n(x)
   Double_t BesselI0(Double_t x);         /// Modified Bessel function I_0(x)
   Double_t BesselK0(Double_t x);         /// Modified Bessel function K_0(x)
   Double_t BesselI1(Double_t x);         /// Modified Bessel function I_1(x)
   Double_t BesselK1(Double_t x);         /// Modified Bessel function K_1(x)
   Double_t BesselJ0(Double_t x);         /// Bessel function J0(x) for any real x
   Double_t BesselJ1(Double_t x);         /// Bessel function J1(x) for any real x
   Double_t BesselY0(Double_t x);         /// Bessel function Y0(x) for positive x
   Double_t BesselY1(Double_t x);         /// Bessel function Y1(x) for positive x
   Double_t StruveH0(Double_t x);         /// Struve functions of order 0
   Double_t StruveH1(Double_t x);         /// Struve functions of order 1
   Double_t StruveL0(Double_t x);         /// Modified Struve functions of order 0
   Double_t StruveL1(Double_t x);         /// Modified Struve functions of order 1
 
   Double_t DiLog(Double_t x);
   Double_t Erf(Double_t x);
   Double_t ErfInverse(Double_t x);
   Double_t Erfc(Double_t x);
   Double_t ErfcInverse(Double_t x);
   Double_t Freq(Double_t x);
   Double_t Gamma(Double_t z);
   Double_t Gamma(Double_t a,Double_t x);
   Double_t GammaDist(Double_t x, Double_t gamma, Double_t mu=0, Double_t beta=1);
   Double_t LnGamma(Double_t z);
 
   void KNNDensity(std::span<const double> observations, std::span<const double> queries, std::span<double> result,
                   int k, double dmin = 0.0);
}
 
////////////////////////////////////////////////////////////////////////////////
// Trig and other functions
 
#include <cfloat>
#include <cmath>
 
#if defined(R__WIN32)
#   ifndef finite
#      define finite _finite
#   endif
#endif
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the sine of an angle of `x` radians.
 
inline Double_t TMath::Sin(Double_t x)
   { return sin(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the cosine of an angle of `x` radians.
 
inline Double_t TMath::Cos(Double_t x)
   { return cos(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the tangent of an angle of `x` radians.
 
inline Double_t TMath::Tan(Double_t x)
   { return tan(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the hyperbolic sine of `x`.
 
inline Double_t TMath::SinH(Double_t x)
   { return sinh(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the hyperbolic cosine of `x`.
 
inline Double_t TMath::CosH(Double_t x)
   { return cosh(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the hyperbolic tangent of `x`.
 
inline Double_t TMath::TanH(Double_t x)
   { return tanh(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the principal value of the arc sine of `x`, expressed in radians.
 
inline Double_t TMath::ASin(Double_t x)
   {
     return asin(x);
   }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the principal value of the arc cosine of `x`, expressed in radians.
 
inline Double_t TMath::ACos(Double_t x)
   {
     return acos(x);
   }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the principal value of the arc tangent of `x`, expressed in radians.
 
inline Double_t TMath::ATan(Double_t x)
   { return atan(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the principal value of the arc tangent of `y/x`, expressed in radians.
 
inline Double_t TMath::ATan2(Double_t y, Double_t x)
   { if (x != 0) return  atan2(y, x);
     if (y == 0) return  0;
     if (y >  0) return  Pi()/2;
     else        return -Pi()/2;
   }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns `x*x`.
 
inline Double_t TMath::Sq(Double_t x)
   { return x*x; }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the square root of x.
 
inline Double_t TMath::Sqrt(Double_t x)
   { return sqrt(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Rounds `x` upward, returning the smallest integral value that is not less than `x`.
 
inline Double_t TMath::Ceil(Double_t x)
   { return ceil(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the nearest integer of `TMath::Ceil(x)`.
 
inline Int_t TMath::CeilNint(Double_t x)
   { return TMath::Nint(ceil(x)); }
 
////////////////////////////////////////////////////////////////////////////////
/// Rounds `x` downward, returning the largest integral value that is not greater than `x`.
 
inline Double_t TMath::Floor(Double_t x)
   { return floor(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the nearest integer of `TMath::Floor(x)`.
 
inline Int_t TMath::FloorNint(Double_t x)
   { return TMath::Nint(floor(x)); }
 
////////////////////////////////////////////////////////////////////////////////
/// Round to nearest integer. Rounds half integers to the nearest even integer.
 
template<typename T>
inline Int_t TMath::Nint(T x)
{
   int i;
   if (x >= 0) {
      i = int(x + 0.5);
      if ( i & 1 && x + 0.5 == T(i) ) i--;
   } else {
      i = int(x - 0.5);
      if ( i & 1 && x - 0.5 == T(i) ) i++;
   }
   return i;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the base-e exponential function of x, which is e raised to the power `x`.
 
inline Double_t TMath::Exp(Double_t x)
   { return exp(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the result of multiplying `x` (the significant) by 2 raised to the power of `exp` (the exponent).
 
inline Double_t TMath::Ldexp(Double_t x, Int_t exp)
   { return ldexp(x, exp); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns `x` raised to the power `y`.
 
inline LongDouble_t TMath::Power(LongDouble_t x, LongDouble_t y)
   { return std::pow(x,y); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns `x` raised to the power `y`.
 
inline LongDouble_t TMath::Power(LongDouble_t x, Long64_t y)
   { return std::pow(x,(LongDouble_t)y); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns `x` raised to the power `y`.
 
inline LongDouble_t TMath::Power(Long64_t x, Long64_t y)
   { return std::pow(x,y); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns `x` raised to the power `y`.
 
inline Double_t TMath::Power(Double_t x, Double_t y)
   { return pow(x, y); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns `x` raised to the power `y`.
 
inline Double_t TMath::Power(Double_t x, Int_t y) {
#ifdef R__ANSISTREAM
   return std::pow(x, y);
#else
   return pow(x, (Double_t) y);
#endif
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the natural logarithm of `x`.
 
inline Double_t TMath::Log(Double_t x)
   { return log(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the common (base-10) logarithm of `x`.
 
inline Double_t TMath::Log10(Double_t x)
   { return log10(x); }
 
////////////////////////////////////////////////////////////////////////////////
/// Check if it is finite with a mask in order to be consistent in presence of
/// fast math.
/// Inspired from the CMSSW FWCore/Utilities package
 
inline Int_t TMath::Finite(Double_t x)
#if defined(R__FAST_MATH)
 
{
   const unsigned long long mask = 0x7FF0000000000000LL;
   union { unsigned long long l; double d;} v;
   v.d =x;
   return (v.l&mask)!=mask;
}
#else
#  if defined(R__HPUX11)
   { return isfinite(x); }
#  elif defined(R__MACOSX)
#  ifdef isfinite
   // from math.h
   { return isfinite(x); }
#  else
   // from cmath
   { return std::isfinite(x); }
#  endif
#  else
   { return finite(x); }
#  endif
#endif
 
////////////////////////////////////////////////////////////////////////////////
/// Check if it is finite with a mask in order to be consistent in presence of
/// fast math.
/// Inspired from the CMSSW FWCore/Utilities package
 
inline Int_t TMath::Finite(Float_t x)
#if defined(R__FAST_MATH)
 
{
   const unsigned int mask =  0x7f800000;
   union { unsigned int l; float d;} v;
   v.d =x;
   return (v.l&mask)!=mask;
}
#else
{ return std::isfinite(x); }
#endif
 
// This namespace provides all the routines necessary for checking if a number
// is a NaN also in presence of optimisations affecting the behaviour of the
// floating point calculations.
// Inspired from the CMSSW FWCore/Utilities package
 
#if defined (R__FAST_MATH)
namespace ROOT {
namespace Internal {
namespace Math {
// abridged from GNU libc 2.6.1 - in detail from
//   math/math_private.h
//   sysdeps/ieee754/ldbl-96/math_ldbl.h
 
// part of this file:
   /*
    * ====================================================
    * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
    *
    * Developed at SunPro, a Sun Microsystems, Inc. business.
    * Permission to use, copy, modify, and distribute this
    * software is freely granted, provided that this notice
    * is preserved.
    * ====================================================
    */
 
   // A union which permits us to convert between a double and two 32 bit ints.
   typedef union {
      Double_t value;
      struct {
         UInt_t lsw;
         UInt_t msw;
      } parts;
   } ieee_double_shape_type;
 
#define EXTRACT_WORDS(ix0,ix1,d)                                    \
   do {                                                             \
      ieee_double_shape_type ew_u;                                  \
      ew_u.value = (d);                                             \
      (ix0) = ew_u.parts.msw;                                       \
      (ix1) = ew_u.parts.lsw;                                       \
   } while (0)
 
   inline Bool_t IsNaN(Double_t x)
   {
      UInt_t hx, lx;
 
      EXTRACT_WORDS(hx, lx, x);
 
      lx |= hx & 0xfffff;
      hx &= 0x7ff00000;
      return (hx == 0x7ff00000) && (lx != 0);
   }
 
   typedef union {
      Float_t value;
      UInt_t word;
   } ieee_float_shape_type;
 
#define GET_FLOAT_WORD(i,d)                                         \
    do {                                                            \
      ieee_float_shape_type gf_u;                                   \
      gf_u.value = (d);                                             \
      (i) = gf_u.word;                                              \
    } while (0)
 
   inline Bool_t IsNaN(Float_t x)
   {
      UInt_t wx;
      GET_FLOAT_WORD (wx, x);
      wx &= 0x7fffffff;
      return (Bool_t)(wx > 0x7f800000);
   }
} } } // end NS ROOT::Internal::Math
#endif // End R__FAST_MATH
 
#if defined(R__FAST_MATH)
   inline Bool_t TMath::IsNaN(Double_t x) { return ROOT::Internal::Math::IsNaN(x); }
   inline Bool_t TMath::IsNaN(Float_t x) { return ROOT::Internal::Math::IsNaN(x); }
#else
   inline Bool_t TMath::IsNaN(Double_t x) { return std::isnan(x); }
   inline Bool_t TMath::IsNaN(Float_t x) { return std::isnan(x); }
#endif
 
////////////////////////////////////////////////////////////////////////////////
// Wrapper to numeric_limits
 
////////////////////////////////////////////////////////////////////////////////
/// Returns a quiet NaN as [defined by IEEE 754](http://en.wikipedia.org/wiki/NaN#Quiet_NaN).
 
inline Double_t TMath::QuietNaN() {
 
   return std::numeric_limits<Double_t>::quiet_NaN();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns a signaling NaN [as defined by IEEE 754](http://en.wikipedia.org/wiki/NaN#Signaling_NaN).
 
inline Double_t TMath::SignalingNaN() {
   return std::numeric_limits<Double_t>::signaling_NaN();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns an infinity as defined by the IEEE standard.
 
inline Double_t TMath::Infinity() {
   return std::numeric_limits<Double_t>::infinity();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns maximum representation for type T.
 
template<typename T>
inline T TMath::Limits<T>::Min() {
   return (std::numeric_limits<T>::min)();    //N.B. use this signature to avoid clashes with macro min() on Windows
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns minimum double representation.
 
template<typename T>
inline T TMath::Limits<T>::Max() {
   return (std::numeric_limits<T>::max)();  //N.B. use this signature to avoid clashes with macro max() on Windows
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns minimum double representation.
 
template<typename T>
inline T TMath::Limits<T>::Epsilon() {
   return std::numeric_limits<T>::epsilon();
}
 
////////////////////////////////////////////////////////////////////////////////
// Advanced.
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Normalized Cross Product of two vectors.
 
template <typename T> inline T TMath::NormCross(const T v1[3],const T v2[3],T out[3])
{
   return Normalize(Cross(v1,v2,out));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns minimum of array a of length n.
 
template <typename T>
T TMath::MinElement(Long64_t n, const T *a) {
   return *std::min_element(a,a+n);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns maximum of array a of length n.
 
template <typename T>
T TMath::MaxElement(Long64_t n, const T *a) {
   return *std::max_element(a,a+n);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns index of array with the minimum element.
/// If more than one element is minimum returns first found.
///
/// Implement here since this one is found to be faster (mainly on 64 bit machines)
/// than stl generic implementation.
/// When performing the comparison,  the STL implementation needs to de-reference both the array iterator
/// and the iterator pointing to the resulting minimum location
 
template <typename T>
Long64_t TMath::LocMin(Long64_t n, const T *a) {
   if  (n <= 0 || !a) return -1;
   T xmin = a[0];
   Long64_t loc = 0;
   for  (Long64_t i = 1; i < n; i++) {
      if (xmin > a[i])  {
         xmin = a[i];
         loc = i;
      }
   }
   return loc;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns index of array with the minimum element.
/// If more than one element is minimum returns first found.
 
template <typename Iterator>
Iterator TMath::LocMin(Iterator first, Iterator last) {
 
   return std::min_element(first, last);
}
 
////////////////////////////////////////////////////////////////////////////////
/// \brief Calculate the one-dimensional finite gradient of an array with length n.
/// It is assumed that the values in the array are spaced uniformly in "x", which
/// would amount to taking the derivative w.r.t. x with an infinitesimal step size `h`.
///
/// The first value in the returned array is a forward difference,
/// the next n-2 values are central differences, and the last is a backward difference.
///
/// \param n the number of points in the array
/// \param f the array of points.  It is assumed that these are spaced uniformly in "x".
/// \param h the distance between the points in `f`.
/// \return an array of size n with the gradient. Returns nullptr if n < 2 or f empty. Ownership is transferred to the
/// caller.
 
template <typename T>
T *TMath::Gradient(const Long64_t n, T const *f, const double h)
{
   if (!f) {
      ::Error("TMath::Gradient", "Input parameter f is empty.");
      return nullptr;
   } else if (n < 2) {
      ::Error("TMath::Gradient", "Input parameter n=%lld is smaller than 2.", n);
      return nullptr;
   }
   Long64_t i = 1;
   T *result = new T[n];
 
   // Forward difference
   result[0] = (f[1] - f[0]) / h;
 
   // Central difference
   while (i < n - 1) {
      result[i] = (f[i + 1] - f[i - 1]) / (2 * h);
      i++;
   }
   // Backward difference
   result[i] = (f[i] - f[i - 1]) / h;
   return result;
}
 
////////////////////////////////////////////////////////////////////////////////
/// \brief Calculate second-order finite differences of an array with length n at second-order accuracy.
/// It is assumed that the values in the array are spaced uniformly in "x", where
/// the spacing is represented by the argument `h`.
///
/// The first value in the returned array is a forward difference at 2nd order accuracy,
/// the next n-2 values are central differences, and the last is the backward difference.
///
/// \param n the number of points in the array
/// \param f the array of points. It is assumed that these are spaced uniformly in "x".
/// \param h the distance between the points in `f`.
/// \return an array of size n with the laplacian. Returns nullptr if n < 4 or f empty. Ownership is transferred to the
/// caller.
 
template <typename T>
T *TMath::Laplacian(const Long64_t n, T const *f, const double h)
{
   if (!f) {
      ::Error("TMath::Laplacian", "Input parameter f is empty.");
      return nullptr;
   } else if (n < 4) {
      ::Error("TMath::Laplacian", "Need at least four elements, got %lld", n);
      return nullptr;
   }
   Long64_t i = 1;
   T *result = new T[n];
 
   // Forward difference
   result[0] = (4 * f[2] + 2 * f[0] - 5 * f[1] - f[3]) / (h * h);
 
   // Central difference
   while (i < n - 1) {
      result[i] = (f[i + 1] + f[i - 1] - 2 * f[i]) / (h * h);
      i++;
   }
   // Backward difference
   result[i] = (2 * f[i] - 5 * f[i - 1] + 4 * f[i - 2] - f[i - 3]) / (h * h);
   return result;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns index of array with the maximum element.
/// If more than one element is maximum returns first found.
///
/// Implement here since it is faster (see comment in LocMin function)
 
template <typename T>
Long64_t TMath::LocMax(Long64_t n, const T *a) {
   if  (n <= 0 || !a) return -1;
   T xmax = a[0];
   Long64_t loc = 0;
   for  (Long64_t i = 1; i < n; i++) {
      if (xmax < a[i])  {
         xmax = a[i];
         loc = i;
      }
   }
   return loc;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns index of array with the maximum element.
/// If more than one element is maximum returns first found.
 
template <typename Iterator>
Iterator TMath::LocMax(Iterator first, Iterator last)
{
 
   return std::max_element(first, last);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the weighted mean of an array defined by the iterators.
 
template <typename Iterator>
Double_t TMath::Mean(Iterator first, Iterator last)
{
   Double_t sum = 0;
   Double_t sumw = 0;
   while ( first != last )
   {
      sum += *first;
      sumw += 1;
      first++;
   }
 
   return sum/sumw;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the weighted mean of an array defined by the first and
/// last iterators. The w iterator should point to the first element
/// of a vector of weights of the same size as the main array.
 
template <typename Iterator, typename WeightIterator>
Double_t TMath::Mean(Iterator first, Iterator last, WeightIterator w)
{
 
   Double_t sum = 0;
   Double_t sumw = 0;
   int i = 0;
   while ( first != last ) {
      if ( *w < 0) {
         ::Error("TMath::Mean","w[%d] = %.4e < 0 ?!",i,*w);
         return 0;
      }
      sum  += (*w) * (*first);
      sumw += (*w) ;
      ++w;
      ++first;
      ++i;
   }
   if (sumw <= 0) {
      ::Error("TMath::Mean","sum of weights == 0 ?!");
      return 0;
   }
 
   return sum/sumw;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the weighted mean of an array a with length n.
 
template <typename T>
Double_t TMath::Mean(Long64_t n, const T *a, const Double_t *w)
{
   if (w) {
      return TMath::Mean(a, a+n, w);
   } else {
      return TMath::Mean(a, a+n);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the geometric mean of an array defined by the iterators.
/// \f[ GeomMean = (\prod_{i=0}^{n-1} |a[i]|)^{1/n} \f]
 
template <typename Iterator>
Double_t TMath::GeomMean(Iterator first, Iterator last)
{
   Double_t logsum = 0.;
   Long64_t n = 0;
   while ( first != last ) {
      if (*first == 0) return 0.;
      Double_t absa = (Double_t) TMath::Abs(*first);
      logsum += TMath::Log(absa);
      ++first;
      ++n;
   }
 
   return TMath::Exp(logsum/n);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the geometric mean of an array a of size n.
/// \f[ GeomMean = (\prod_{i=0}^{n-1} |a[i]|)^{1/n} \f]
 
template <typename T>
Double_t TMath::GeomMean(Long64_t n, const T *a)
{
   return TMath::GeomMean(a, a+n);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the Standard Deviation of an array defined by the iterators.
/// Note that this function returns the sigma(standard deviation) and
/// not the root mean square of the array.
///
/// Use the two pass algorithm, which is slower (! a factor of 2) but much more
/// precise.  Since we have a vector the 2 pass algorithm is still faster than the
/// Welford algorithm. (See also ROOT-5545)
 
template <typename Iterator>
Double_t TMath::RMS(Iterator first, Iterator last)
{
 
   Double_t n = 0;
 
   Double_t tot = 0;
   Double_t mean = TMath::Mean(first,last);
   while ( first != last ) {
      Double_t x = Double_t(*first);
      tot += (x - mean)*(x - mean);
      ++first;
      ++n;
   }
   Double_t rms = (n > 1) ? TMath::Sqrt(tot/(n-1)) : 0.0;
   return rms;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the weighted Standard Deviation of an array defined by the iterators.
/// Note that this function returns the sigma(standard deviation) and
/// not the root mean square of the array.
///
/// As in the unweighted case use the two pass algorithm
 
template <typename Iterator, typename WeightIterator>
Double_t TMath::RMS(Iterator first, Iterator last, WeightIterator w)
{
   Double_t tot = 0;
   Double_t sumw = 0;
   Double_t sumw2 = 0;
   Double_t mean = TMath::Mean(first,last,w);
   while ( first != last ) {
      Double_t x = Double_t(*first);
      sumw += *w;
      sumw2 += (*w) * (*w);
      tot += (*w) * (x - mean)*(x - mean);
      ++first;
      ++w;
   }
   // use the correction neff/(neff -1) for the unbiased formula
   Double_t rms =  TMath::Sqrt(tot * sumw/ (sumw*sumw - sumw2) );
   return rms;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the Standard Deviation of an array a with length n.
/// Note that this function returns the sigma(standard deviation) and
/// not the root mean square of the array.
 
template <typename T>
Double_t TMath::RMS(Long64_t n, const T *a, const Double_t * w)
{
   return (w) ? TMath::RMS(a, a+n, w) : TMath::RMS(a, a+n);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Cross Product of two vectors:
///         out = [v1 x v2]
 
template <typename T> T *TMath::Cross(const T v1[3],const T v2[3], T out[3])
{
   out[0] = v1[1] * v2[2] - v1[2] * v2[1];
   out[1] = v1[2] * v2[0] - v1[0] * v2[2];
   out[2] = v1[0] * v2[1] - v1[1] * v2[0];
 
   return out;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates a normal vector of a plane.
///
/// \param[in]  p1, p2,p3     3 3D points belonged the plane to define it.
/// \param[out] normal        Pointer to 3D normal vector (normalized)
 
template <typename T> T * TMath::Normal2Plane(const T p1[3],const T p2[3],const T p3[3], T normal[3])
{
   T v1[3], v2[3];
 
   v1[0] = p2[0] - p1[0];
   v1[1] = p2[1] - p1[1];
   v1[2] = p2[2] - p1[2];
 
   v2[0] = p3[0] - p1[0];
   v2[1] = p3[1] - p1[1];
   v2[2] = p3[2] - p1[2];
 
   NormCross(v1,v2,normal);
   return normal;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Function which returns kTRUE if point xp,yp lies inside the
/// polygon defined by the np points in arrays x and y, kFALSE otherwise.
/// Note that the polygon may be open or closed.
 
template <typename T> Bool_t TMath::IsInside(T xp, T yp, Int_t np, T *x, T *y)
{
   Int_t i, j = np-1 ;
   Bool_t oddNodes = kFALSE;
 
   for (i=0; i<np; i++) {
      if ((y[i]<yp && y[j]>=yp) || (y[j]<yp && y[i]>=yp)) {
         if (x[i]+(yp-y[i])/(y[j]-y[i])*(x[j]-x[i])<xp) {
            oddNodes = !oddNodes;
         }
      }
      j=i;
   }
 
   return oddNodes;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the median of the array a where each entry i has weight w[i] .
/// Both arrays have a length of at least n . The median is a number obtained
/// from the sorted array a through
///
/// median = (a[jl]+a[jh])/2.  where (using also the sorted index on the array w)
///
/// sum_i=0,jl w[i] <= sumTot/2
/// sum_i=0,jh w[i] >= sumTot/2
/// sumTot = sum_i=0,n w[i]
///
/// If w=0, the algorithm defaults to the median definition where it is
/// a number that divides the sorted sequence into 2 halves.
/// When n is odd or n > 1000, the median is kth element k = (n + 1) / 2.
/// when n is even and n < 1000the median is a mean of the elements k = n/2 and k = n/2 + 1.
///
/// If the weights are supplied (w not 0) all weights must be >= 0
///
/// If work is supplied, it is used to store the sorting index and assumed to be
/// >= n . If work=0, local storage is used, either on the stack if n < kWorkMax
/// or on the heap for n >= kWorkMax .
 
template <typename T> Double_t TMath::Median(Long64_t n, const T *a,  const Double_t *w, Long64_t *work)
{
 
   const Int_t kWorkMax = 100;
 
   if (n <= 0 || !a) return 0;
   Bool_t isAllocated = kFALSE;
   Double_t median;
   Long64_t *ind;
   Long64_t workLocal[kWorkMax];
 
   if (work) {
      ind = work;
   } else {
      ind = workLocal;
      if (n > kWorkMax) {
         isAllocated = kTRUE;
         ind = new Long64_t[n];
      }
   }
 
   if (w) {
      Double_t sumTot2 = 0;
      for (Int_t j = 0; j < n; j++) {
         if (w[j] < 0) {
            ::Error("TMath::Median","w[%d] = %.4e < 0 ?!",j,w[j]);
            if (isAllocated)  delete [] ind;
            return 0;
         }
         sumTot2 += w[j];
      }
 
      sumTot2 /= 2.;
 
      Sort(n, a, ind, kFALSE);
 
      Double_t sum = 0.;
      Int_t jl;
      for (jl = 0; jl < n; jl++) {
         sum += w[ind[jl]];
         if (sum >= sumTot2) break;
      }
 
      Int_t jh;
      sum = 2.*sumTot2;
      for (jh = n-1; jh >= 0; jh--) {
         sum -= w[ind[jh]];
         if (sum <= sumTot2) break;
      }
 
      median = 0.5*(a[ind[jl]]+a[ind[jh]]);
 
   } else {
 
      if (n%2 == 1)
         median = KOrdStat(n, a,n/2, ind);
      else {
         median = 0.5*(KOrdStat(n, a, n/2 -1, ind)+KOrdStat(n, a, n/2, ind));
      }
   }
 
   if (isAllocated)
      delete [] ind;
   return median;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the half-sample mode of the array a where each entry i has weight w[i].
/// Both arrays must have a length of n. The mode is a number obtained
/// according to the paper http://arxiv.org/ftp/math/papers/0505/0505419.pdf (page 19).
/// The algorithm searches for the smallest range of sorted a values that
/// contains ~n/2 values. Then it re-applies the same algorithm to
/// that range until the range has <= 3 elements in it.
///
/// \note See David R. Bickel: "On a Fast, Robust Estimator of the Mode"
/// http://arxiv.org/ftp/math/papers/0505/0505419.pdf (page 19)
/// Initial implementation suggestion by Jean-Fran\c{c}ois Caron, 2014 on JIRA-6849
///
/// \tparam T data type of the values array `a`
/// \param n Number of elements in array `a`
/// \param a Pointer to array of values (owned by user, preallocated), must have length of n, and none of them should be NaN. The array must not be necessarily sorted nor contain unique values.
/// \param w Pointer to array of weights (owned by user, preallocated), must have length of n. If the weights are supplied (w not nullptr) all weights must be >= 0 to properly work.
/// \return NaN if n <= 0, a[0] if n == 1, TMath::Mean(n,a,w) if n == 2,
/// the (weighted) mean of the closest two if n == 3; for the rest of the cases,
/// if all values of `a` are unique or have equal weights (or `w` is nullptr), then the Bickel half-sample-mode algorithm is applied; otherwise the
/// value associated with the highest sum of weights (if `w` is not nullptr)
/// is returned.
/// \note If in a given iteration step, there are several equally small subranges, then the first occurrence is chosen.
template <typename T> Double_t TMath::ModeHalfSample(Long64_t n, const T *a, const Double_t *w)
{
   if (n <= 0)
      return TMath::QuietNaN();
   else if (n == 1)
      return a[0];
   else if (n == 2)
      return TMath::Mean(n, a, w);
 
   if (w && std::adjacent_find(w, w + n, std::not_equal_to<>()) == w + n) // If all weights are equal, ignore those
      w = nullptr;
 
   // Sort the array and remove duplicates (if w != nullptr)
   std::vector<T> values;
   std::vector<Double_t> weights;
   values.reserve(n);
   weights.reserve(n);
   for (Long64_t i = 0; i < n; ++i) {
      const T value = a[i];
      const Double_t weight = w ? w[i] : 1;
      const auto vstart = values.begin();
      const auto vstop = values.end();
      const auto viter = std::lower_bound(vstart, vstop, value);
      const auto vidx = std::distance(vstart, viter);
      if (viter != vstop && w) {
         weights[vidx] += weight;
      } else {
         values.insert(viter, value);
         weights.insert(weights.begin() + vidx, weight);
      }
   }
 
   const size_t sn = values.size();
   if (sn <= 0)
      return TMath::QuietNaN();
   else if (sn == 1)
      return values[0];
   else if (sn == 2)
      return TMath::Mean(sn, values.data(), weights.data());
 
   if (sn == 3) {
      const double d1_0 = values[1] - values[0];
      const double d2_1 = values[2] - values[1];
      if (d2_1 < d1_0)
         return TMath::Mean(2, values.data() + 1, weights.data() + 1);
      else if (d2_1 > d1_0)
         return TMath::Mean(2, values.data(), weights.data());
      else
         return values[1];
   }
 
   const auto wstart = weights.begin();
   const auto wstop = weights.end();
   if (std::adjacent_find(wstart, wstop, std::not_equal_to<>()) != wstop) {
      // Get highest (sum of) weight element
      const auto wmaxiter = std::max_element(wstart, wstop);
      const auto wmaxidx = std::distance(wstart, wmaxiter);
      return values[wmaxidx];
   }
 
   // All elements have equal weights (and are unique if w != nullptr)
 
   // Initialize search
   n = sn;
   size_t jMin = 0;
 
   // Do recursive calls dividing each time the interval by two
   while (n > 3) {
      Double_t min_v_range = values[jMin + n - 1] - values[jMin];
      const size_t N = std::ceil(n * 0.5);
      const size_t start = jMin;
      const size_t stop = start + n - N + 1; // +1 since we use < and not <=
      // Find sequentially what v_range is smallest by sliding the half-window
      for (size_t i = start; i < stop; i++) {
         Double_t range = values[i + N - 1] - values[i];
         if (range < min_v_range) {
            min_v_range = range;
            jMin = i;
         }
      }
      // assert(min_v_range == values[N - 1 + start] - values[start]);
      n = N;
   }
   if (n == 3) {
      const double d1_0 = values[jMin + 1] - values[jMin + 0];
      const double d2_1 = values[jMin + 2] - values[jMin + 1];
      if (d2_1 < d1_0)
         return (values[jMin + 1] + values[jMin + 2]) * 0.5;
      else if (d2_1 > d1_0)
         return (values[jMin] + values[jMin + 1]) * 0.5;
      else
         return values[jMin + 1];
   } else if (n == 2) {
      return (values[jMin] + values[jMin + 1]) * 0.5;
   } else {
      Error("ModeHalfSample", "Error in recursive algorithm, returning NaN"); // this should not happen
      return TMath::QuietNaN();
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns k_th order statistic of the array a of size n
/// (k_th smallest element out of n elements).
///
/// C-convention is used for array indexing, so if you want
/// the second smallest element, call KOrdStat(n, a, 1).
///
/// If work is supplied, it is used to store the sorting index and
/// assumed to be >= n. If work=0, local storage is used, either on
/// the stack if n < kWorkMax or on the heap for n >= kWorkMax.
/// Note that the work index array will not contain the sorted indices but
/// all indices of the smaller element in arbitrary order in work[0,...,k-1] and
/// all indices of the larger element in arbitrary order in work[k+1,..,n-1]
/// work[k] will contain instead the index of the returned element.
///
/// Taken from "Numerical Recipes in C++" without the index array
/// implemented by Anna Khreshuk.
///
/// See also the declarations at the top of this file
 
template <class Element, typename Size>
Element TMath::KOrdStat(Size n, const Element *a, Size k, Size *work)
{
 
   const Int_t kWorkMax = 100;
 
   typedef Size Index;
 
   Bool_t isAllocated = kFALSE;
   Size i, ir, j, l, mid;
   Index arr;
   Index *ind;
   Index workLocal[kWorkMax];
   Index temp;
 
   if (work) {
      ind = work;
   } else {
      ind = workLocal;
      if (n > kWorkMax) {
         isAllocated = kTRUE;
         ind = new Index[n];
      }
   }
 
   for (Size ii=0; ii<n; ii++) {
      ind[ii]=ii;
   }
   Size rk = k;
   l=0;
   ir = n-1;
   for(;;) {
      if (ir<=l+1) { //active partition contains 1 or 2 elements
         if (ir == l+1 && a[ind[ir]]<a[ind[l]])
            {temp = ind[l]; ind[l]=ind[ir]; ind[ir]=temp;}
         Element tmp = a[ind[rk]];
         if (isAllocated)
            delete [] ind;
         return tmp;
      } else {
         mid = (l+ir) >> 1; //choose median of left, center and right
         {temp = ind[mid]; ind[mid]=ind[l+1]; ind[l+1]=temp;}//elements as partitioning element arr.
         if (a[ind[l]]>a[ind[ir]])  //also rearrange so that a[l]<=a[l+1]
            {temp = ind[l]; ind[l]=ind[ir]; ind[ir]=temp;}
 
         if (a[ind[l+1]]>a[ind[ir]])
            {temp=ind[l+1]; ind[l+1]=ind[ir]; ind[ir]=temp;}
 
         if (a[ind[l]]>a[ind[l+1]])
            {temp = ind[l]; ind[l]=ind[l+1]; ind[l+1]=temp;}
 
         i=l+1;        //initialize pointers for partitioning
         j=ir;
         arr = ind[l+1];
         for (;;){
            do i++; while (a[ind[i]]<a[arr]);
            do j--; while (a[ind[j]]>a[arr]);
            if (j<i) break;  //pointers crossed, partitioning complete
               {temp=ind[i]; ind[i]=ind[j]; ind[j]=temp;}
         }
         ind[l+1]=ind[j];
         ind[j]=arr;
         if (j>=rk) ir = j-1; //keep active the partition that
         if (j<=rk) l=i;      //contains the k_th element
      }
   }
}
 
#endif