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 // Basics.h is a part of the PYTHIA event generator.
 // Copyright (C) 2024 Torbjorn Sjostrand.
 // PYTHIA is licenced under the GNU GPL v2 or later, see COPYING for details.
 // Please respect the MCnet Guidelines, see GUIDELINES for details.
 
 // Header file for basic, often-used helper classes.
 // RndmEngine: base class for external random number generators.
 // Rndm: random number generator.
 // Vec4: simple four-vectors.
 // RotBstMatrix: matrices encoding rotations and boosts of Vec4 objects.
 // Hist: simple one-dimensional histograms.
 
 #ifndef Pythia8_Basics_H
 #define Pythia8_Basics_H
 
 #include "Pythia8/PythiaStdlib.h"
 #include "Pythia8/SharedPointers.h"
 
 namespace Pythia8 {
 
 //==========================================================================
 
 // Forward reference to RotBstMatrix class; needed in Vec4 class.
 class RotBstMatrix;
 
 //--------------------------------------------------------------------------
 
 // Vec4 class.
 // This class implements four-vectors, in energy-momentum space.
 // (But can equally well be used to hold space-time four-vectors.)
 
 class Vec4 {
 
 public:
 
   // Constructors.
   Vec4(double xIn = 0., double yIn = 0., double zIn = 0., double tIn = 0.)
     : xx(xIn), yy(yIn), zz(zIn), tt(tIn) { }
   Vec4(const Vec4& v) : xx(v.xx), yy(v.yy), zz(v.zz), tt(v.tt) { }
   Vec4& operator=(const Vec4& v) { if (this != &v) { xx = v.xx; yy = v.yy;
     zz = v.zz; tt = v.tt; } return *this; }
   Vec4& operator=(double value) { xx = value; yy = value; zz = value;
     tt = value; return *this; }
 
   // Member functions for input.
   void reset() {xx = 0.; yy = 0.; zz = 0.; tt = 0.;}
   void p(double xIn, double yIn, double zIn, double tIn)
     {xx = xIn; yy = yIn; zz = zIn; tt = tIn;}
   void p(Vec4 pIn) {xx = pIn.xx; yy = pIn.yy; zz = pIn.zz; tt = pIn.tt;}
   void px(double xIn) {xx = xIn;}
   void py(double yIn) {yy = yIn;}
   void pz(double zIn) {zz = zIn;}
   void e(double tIn) {tt = tIn;}
 
   // Member functions for output.
   double px() const {return xx;}
   double py() const {return yy;}
   double pz() const {return zz;}
   double e() const {return tt;}
   double& operator[](int i) {
     switch(i) {
       case 1: return xx;
       case 2: return yy;
       case 3: return zz;
       default: return tt;
     }
   }
   double mCalc() const {double temp = tt*tt - xx*xx - yy*yy - zz*zz;
     return (temp >= 0.) ? sqrt(temp) : -sqrt(-temp);}
   double m2Calc() const {return tt*tt - xx*xx - yy*yy - zz*zz;}
   double pT() const {return sqrt(xx*xx + yy*yy);}
   double pT2() const {return xx*xx + yy*yy;}
   double pAbs() const {return sqrt(xx*xx + yy*yy + zz*zz);}
   double pAbs2() const {return xx*xx + yy*yy + zz*zz;}
   double eT() const {double temp = xx*xx + yy*yy;
     return tt * sqrt( temp / (temp + zz*zz) );}
   double eT2() const {double temp = xx*xx + yy*yy;
     return tt*tt * temp / (temp + zz*zz);}
   double theta() const {return atan2(sqrt(xx*xx + yy*yy), zz);}
   double phi() const {return atan2(yy,xx);}
   double thetaXZ() const {return atan2(xx,zz);}
   double pPos() const {return tt + zz;}
   double pNeg() const {return tt - zz;}
   double rap() const {
     double txyz = (tt > 0.) ? tt : sqrt(xx*xx + yy*yy + zz*zz);
     if (zz >= txyz) return 20.;
     if (zz <= -txyz) return -20.;
     return 0.5 * log( (txyz + zz) / (txyz - zz) );}
   double eta() const {double xyz = sqrt(xx*xx + yy*yy + zz*zz);
     if (zz >= xyz) return 20.;
     if (zz <= -xyz) return -20.;
     return 0.5 * log( (xyz + zz) / (xyz - zz) );}
 
   // Member functions that perform operations.
   void rescale3(double fac) {xx *= fac; yy *= fac; zz *= fac;}
   void rescale4(double fac) {xx *= fac; yy *= fac; zz *= fac; tt *= fac;}
   void flip3() {xx = -xx; yy = -yy; zz = -zz;}
   void flip4() {xx = -xx; yy = -yy; zz = -zz; tt = -tt;}
   void rot(double thetaIn, double phiIn);
   void rotaxis(double phiIn, double nx, double ny, double nz);
   void rotaxis(double phiIn, const Vec4& n);
   void bst(double betaX, double betaY, double betaZ);
   void bst(double betaX, double betaY, double betaZ, double gamma);
   void bst(const Vec4& pIn);
   void bst(const Vec4& pIn, double mIn);
   void bstback(const Vec4& pIn);
   void bstback(const Vec4& pIn, double mIn);
   void rotbst(const RotBstMatrix& M);
   double eInFrame(const Vec4& pIn) const;
 
   // Operator overloading with member functions
   inline Vec4 operator-() const {Vec4 tmp; tmp.xx = -xx; tmp.yy = -yy;
     tmp.zz = -zz; tmp.tt = -tt; return tmp;}
   inline Vec4& operator+=(const Vec4& v) {xx += v.xx; yy += v.yy; zz += v.zz;
     tt += v.tt; return *this;}
   inline Vec4& operator-=(const Vec4& v) {xx -= v.xx; yy -= v.yy; zz -= v.zz;
     tt -= v.tt; return *this;}
   inline Vec4& operator*=(double f) {xx *= f; yy *= f; zz *= f;
     tt *= f; return *this;}
   inline Vec4& operator/=(double f) {xx /= f; yy /= f; zz /= f;
     tt /= f; return *this;}
   inline Vec4 operator+(const Vec4& v) const {
     Vec4 tmp = *this; return tmp += v;}
   inline Vec4 operator-(const Vec4& v) const {
     Vec4 tmp = *this; return tmp -= v;}
   inline Vec4 operator*(double f) const {
     Vec4 tmp = *this; return tmp *= f;}
   inline Vec4 operator/(double f) const {
     Vec4 tmp = *this; return tmp /= f;}
   inline double operator*(const Vec4& v) const {
     return tt*v.tt - xx*v.xx - yy*v.yy - zz*v.zz;}
 
   // Operator overloading with friends.
   friend Vec4 operator*(double f, const Vec4& v1);
 
   // Print a four-vector.
   friend ostream& operator<<(ostream&, const Vec4& v) ;
 
   // Check if NaN, INF, or finite.
   friend inline bool isnan(const Vec4 &v) {
     return isnan(v.tt) || isnan(v.xx) || isnan(v.yy) || isnan(v.zz);}
   friend inline bool isinf(const Vec4 &v) {
     return isinf(v.tt) || isinf(v.xx) || isinf(v.yy) || isinf(v.zz);}
   friend inline bool isfinite(const Vec4 &v) {
     return isfinite(v.tt) && isfinite(v.xx) && isfinite(v.yy)
       && isfinite(v.zz);}
 
   // Invariant mass and its square.
   friend double m(const Vec4& v1);
   friend double m(const Vec4& v1, const Vec4& v2);
   friend double m2(const Vec4& v1);
   friend double m2(const Vec4& v1, const Vec4& v2);
   friend double m2(const Vec4& v1, const Vec4& v2, const Vec4& v3);
   friend double m2(const Vec4& v1, const Vec4& v2, const Vec4& v3,
                    const Vec4& v4);
 
   // Scalar and cross product of 3-vector parts.
   friend double dot3(const Vec4& v1, const Vec4& v2);
   friend Vec4 cross3(const Vec4& v1, const Vec4& v2);
 
   // Cross-product of three 4-vectors ( p_i = epsilon_{iabc} p_a p_b p_c).
   friend Vec4 cross4(const Vec4& a, const Vec4& b, const Vec4& c);
 
   // theta is polar angle between v1 and v2.
   friend double theta(const Vec4& v1, const Vec4& v2);
   friend double costheta(const Vec4& v1, const Vec4& v2);
 
   // phi is azimuthal angle between v1 and v2 around z axis.
   friend double phi(const Vec4& v1, const Vec4& v2);
   friend double cosphi(const Vec4& v1, const Vec4& v2);
 
   // phi is azimuthal angle between v1 and v2 around n axis.
   friend double phi(const Vec4& v1, const Vec4& v2, const Vec4& n);
   friend double cosphi(const Vec4& v1, const Vec4& v2, const Vec4& n);
 
   // R is distance in cylindrical (y/eta, phi) coordinates.
   friend double RRapPhi(const Vec4& v1, const Vec4& v2);
   friend double REtaPhi(const Vec4& v1, const Vec4& v2);
 
   // Shift four-momenta within pair from old to new masses.
   friend bool pShift( Vec4& p1Move, Vec4& p2Move, double m1New, double m2New);
 
   // Create two vectors that are perpendicular to both input vectors.
   friend pair<Vec4,Vec4> getTwoPerpendicular(const Vec4& v1, const Vec4& v2);
 
 private:
 
   // Constants: could only be changed in the code itself.
   static const double TINY;
 
   // The four-vector data members.
   double xx, yy, zz, tt;
 
 };
 
 //--------------------------------------------------------------------------
 
 // Namespace function declarations; friends of Vec4 class.
 
 // Implementation of operator overloading with friends.
 inline Vec4 operator*(double f, const Vec4& v1)
   {Vec4 v = v1; return v *= f;}
 
 // Invariant mass and its square.
 double m(const Vec4& v1);
 double m(const Vec4& v1, const Vec4& v2);
 double m2(const Vec4& v1);
 double m2(const Vec4& v1, const Vec4& v2);
 double m2(const Vec4& v1, const Vec4& v2, const Vec4& v3);
 double m2(const Vec4& v1, const Vec4& v2, const Vec4& v3,
           const Vec4& v4);
 
 // Scalar and cross product of 3-vector parts.
 double dot3(const Vec4& v1, const Vec4& v2);
 Vec4 cross3(const Vec4& v1, const Vec4& v2);
 
 // Cross-product of three 4-vectors ( p_i = epsilon_{iabc} p_a p_b p_c).
 Vec4 cross4(const Vec4& a, const Vec4& b, const Vec4& c);
 
 // theta is polar angle between v1 and v2.
 double theta(const Vec4& v1, const Vec4& v2);
 double costheta(const Vec4& v1, const Vec4& v2);
 double costheta(double e1, double e2, double m1, double m2, double s12);
 
 // phi is azimuthal angle between v1 and v2 around z axis.
 double phi(const Vec4& v1, const Vec4& v2);
 double cosphi(const Vec4& v1, const Vec4& v2);
 
 // phi is azimuthal angle between v1 and v2 around n axis.
 double phi(const Vec4& v1, const Vec4& v2, const Vec4& n);
 double cosphi(const Vec4& v1, const Vec4& v2, const Vec4& n);
 
 // R is distance in cylindrical (y/eta, phi) coordinates.
 double RRapPhi(const Vec4& v1, const Vec4& v2);
 double REtaPhi(const Vec4& v1, const Vec4& v2);
 
 // Print a four-vector.
 ostream& operator<<(ostream&, const Vec4& v) ;
 
 // Shift four-momenta within pair from old to new masses.
 bool pShift( Vec4& p1Move, Vec4& p2Move, double m1New, double m2New);
 
 // Create two vectors that are perpendicular to both input vectors.
 pair<Vec4,Vec4> getTwoPerpendicular(const Vec4& v1, const Vec4& v2);
 
 //==========================================================================
 
 // RotBstMatrix class.
 // This class implements 4 * 4 matrices that encode an arbitrary combination
 // of rotations and boosts, that can be applied to Vec4 four-vectors.
 
 class RotBstMatrix {
 
 public:
 
   // Constructors.
   RotBstMatrix() : M() {for (int i = 0; i < 4; ++i) {
     for (int j = 0; j < 4; ++j) { M[i][j] = (i==j) ? 1. : 0.; } } }
   RotBstMatrix(const RotBstMatrix& Min) : M() {
     for (int i = 0; i < 4; ++i) { for (int j = 0; j < 4; ++j) {
     M[i][j] = Min.M[i][j]; } } }
   RotBstMatrix& operator=(const RotBstMatrix& Min) {if (this != &Min) {
     for (int i = 0; i < 4; ++i) { for (int j = 0; j < 4; ++j) {
     M[i][j] = Min.M[i][j]; } } } return *this; }
 
   // Member functions.
   void rot(double = 0., double = 0.);
   void rot(const Vec4& p);
   void bst(double = 0., double = 0., double = 0.);
   void bst(const Vec4&);
   void bstback(const Vec4&);
   void bst(const Vec4&, const Vec4&);
   void toCMframe(const Vec4&, const Vec4&);
   void fromCMframe(const Vec4&, const Vec4&, bool flip = false);
   void toSameVframe(const Vec4&, const Vec4&);
   void fromSameVframe(const Vec4&, const Vec4&);
   void rotbst(const RotBstMatrix&);
   void invert();
   RotBstMatrix inverse() const { RotBstMatrix tmp = *this;
     tmp.invert(); return tmp; }
   void reset();
 
   // Return value of matrix element.
   double value(int i, int j) { return M[i][j];}
 
   // Crude estimate deviation from unit matrix.
   double deviation() const;
 
   // Print a transformation matrix.
   friend ostream& operator<<(ostream&, const RotBstMatrix&) ;
 
   // Private members to be accessible from Vec4.
   friend class Vec4;
 
   // Multiplication.
   Vec4 operator*(Vec4 p) const { p.rotbst(*this); return p; }
   RotBstMatrix operator*(RotBstMatrix R) const { R.rotbst(*this); return R; }
 
 private:
 
   // Constants: could only be changed in the code itself.
   static const double TINY;
 
   // The rotation-and-boost matrix data members.
   double M[4][4];
 
 };
 
 //--------------------------------------------------------------------------
 
 // Namespace function declaration; friend of RotBstMatrix class.
 
 // Print a transformation matrix.
 ostream& operator<<(ostream&, const RotBstMatrix&) ;
 
 // Get a RotBstMatrix to rest frame of p.
 inline RotBstMatrix toCMframe(const Vec4& p) {
   RotBstMatrix tmp; tmp.bstback(p); return tmp; }
 
 // Get a RotBstMatrix from rest frame of p.
 inline RotBstMatrix fromCMframe(const Vec4& p) {
   RotBstMatrix tmp; tmp.bst(p); return tmp; }
 
 // Get a RotBstMatrix to rest frame of p1 and p2, where p1 is along
 // the z-axis.
 inline RotBstMatrix toCMframe(const Vec4& p1, const Vec4& p2) {
   RotBstMatrix tmp; tmp.toCMframe(p1, p2); return tmp; }
 
 // Get a RotBstMatrix from rest frame of p1 and p2, where p1 is
 // assumed by default to be along the z-axis. The flip option
 // handles the case when p1 is along the negative z-axis.
 inline RotBstMatrix fromCMframe(const Vec4& p1, const Vec4& p2,
   bool flip = false) {
   RotBstMatrix tmp; tmp.fromCMframe(p1, p2, flip); return tmp; }
 
 // Get a RotBstMatrix to rest frame of ptot where pz is along the
 // z-axis and pxz is in the xz-plane with positive x.
 inline RotBstMatrix toCMframe(const Vec4& ptot, const Vec4& pz,
   const Vec4 & pxz) { RotBstMatrix tmp = toCMframe(ptot);
   Vec4 pzp = tmp*pz; tmp.rot(0.0, -pzp.phi()); tmp.rot(-pzp.theta());
   tmp.rot(0.0, -(tmp*pxz).phi()); return tmp; }
 
 // Get a RotBstMatrix from rest frame of ptot where pz is along the
 // z-axis and pxz is in the xz-plane with positive x.
 inline RotBstMatrix fromCMframe(const Vec4& ptot, const Vec4& pz,
   const Vec4 & pxz) { return toCMframe(ptot, pz, pxz).inverse(); }
 
 //==========================================================================
 
 // RndmEngine is the base class for external random number generators.
 // There is only one pure virtual method, that should do the generation.
 
 class RndmEngine {
 
 public:
 
   // Destructor.
   virtual ~RndmEngine() {}
 
   // A virtual method, wherein the derived class method
   // generates a random number uniformly distributed between 0 and 1.
   virtual double flat() {return 1;}
 
 };
 
 //==========================================================================
 
 // RndmState class.
 // This class describes the configuration of a Rndm object.
 
 struct RndmState {
   int    i97{}, j97{}, seed{0};
   long   sequence{0};
   double u[97]{}, c{}, cd{}, cm{};
 
   // Test whether two random states would generate the same random sequence.
   bool operator==(const RndmState& other) const;
 };
 
 //==========================================================================
 
 // Rndm class.
 // This class handles random number generation according to the
 // Marsaglia-Zaman-Tsang algorithm.
 
 class Rndm {
 
 public:
 
   // Constructors.
   Rndm() : initRndm(false), stateSave(), useExternalRndm(false) { }
   Rndm(int seedIn) : initRndm(false), stateSave(), useExternalRndm(false) {
     init(seedIn);}
 
   // Possibility to pass in pointer for external random number generation.
   bool rndmEnginePtr( RndmEnginePtr rndmEngPtrIn);
 
   // Initialize, normally at construction or in first call.
   void init(int seedIn = 0) ;
 
   // Generate next random number uniformly between 0 and 1.
   double flat() ;
 
   // Generate random numbers according to exp(-x).
   double exp() ;
 
   // Generate random numbers according to x * exp(-x).
   double xexp() { return -log(flat() * flat()) ;}
 
   // Generate random numbers according to exp(-x^2/2).
   double gauss() {return sqrt(-2. * log(flat())) * cos(M_PI * flat());}
 
   // Generate two random numbers according to exp(-x^2/2-y^2/2).
   pair<double, double> gauss2() {double r = sqrt(-2. * log(flat()));
     double phi = 2. * M_PI * flat();
     return { r * sin(phi), r * cos(phi) };}
 
   // Generate a random number according to a Gamma-distribution.
   double gamma(double k0, double r0);
 
   // Generate two random vectors according to the phase space distribution
   pair<Vec4, Vec4> phaseSpace2(double eCM, double m1, double m2);
 
   // Pick one option among  vector of (positive) probabilities.
   int pick(const vector<double>& prob) ;
 
   // Randomly shuffle a vector, standard Fisher-Yates algorithm.
   template<typename T> void shuffle(vector<T>& vec);
 
   // Save or read current state to or from a binary file.
   bool dumpState(string fileName);
   bool readState(string fileName);
 
   // Get or set the state of the random number generator.
   RndmState getState() const {return stateSave;}
   void setState(const RndmState& state) {stateSave = state;}
 
   // The default seed, i.e. the Marsaglia-Zaman random number sequence.
   static constexpr int DEFAULTSEED = 19780503;
 
 #ifdef RNGDEBUG
   // Random number methods used for debugging only.
   double flatDebug(string loc, string file, int line);
   double xexpDebug(string loc, string file, int line);
   double gaussDebug(string loc, string file, int line);
   pair<double, double> gauss2Debug(string loc, string file, int line);
   double gammaDebug(string loc, string file, int line, double k0, double r0);
   pair<Vec4, Vec4> phaseSpace2Debug(string loc, string file, int line,
     double eCM, double m1, double m2);
 
   // Static members for debugging to print call file location or filter.
   static bool debugNow, debugLocation, debugIndex;
   static int debugPrecision, debugCall;
   static set<string> debugStarts, debugEnds, debugContains, debugMatches;
 #endif
 
 private:
 
   // State of the random number generator.
   bool      initRndm;
   RndmState stateSave;
 
   // Pointer for external random number generation.
   bool   useExternalRndm;
   RndmEnginePtr rndmEngPtr{};
 
 };
 
 //==========================================================================
 
 // Hist class.
 // This class handles a single histogram at a time.
 
 class Hist {
 
 public:
 
   // Constructors, including copy constructors.
   Hist() : titleSave(""), nBin(), nFill(), nNonFinite(), xMin(),
     xMax(), linX(), doStats(), dx(), under(), inside(), over(), sumxNw()
     { }
   Hist(string titleIn, int nBinIn = 100, double xMinIn = 0.,
     double xMaxIn = 1., bool logXIn = false, bool doStatsIn = false) :
     nBin(), nFill(), nNonFinite(), xMin(), xMax(), linX(), doStats(), dx(),
       under(), inside(), over(), sumxNw()
   { book(titleIn, nBinIn, xMinIn, xMaxIn, logXIn, doStatsIn); }
   Hist(const Hist& h)
     : titleSave(h.titleSave), nBin(h.nBin), nFill(h.nFill),
       nNonFinite(h.nNonFinite), xMin(h.xMin), xMax(h.xMax), linX(h.linX),
       doStats(h.doStats), dx(h.dx), under(h.under), inside(h.inside),
       over(h.over), res(h.res), res2(h.res2), sumxNw() {
     for (int i = 0; i < nMoments; ++i) sumxNw[i] = h.sumxNw[i];
   }
   Hist(string titleIn, const Hist& h)
     : titleSave(titleIn), nBin(h.nBin), nFill(h.nFill),
       nNonFinite(h.nNonFinite), xMin(h.xMin), xMax(h.xMax), linX(h.linX),
       doStats(h.doStats), dx(h.dx), under(h.under), inside(h.inside),
       over(h.over), res(h.res), res2(h.res2), sumxNw() {
     for (int i = 0; i < nMoments; ++i) sumxNw[i] = h.sumxNw[i];
   }
   Hist& operator=(const Hist& h) { if(this != &h) {
     nBin = h.nBin; nFill = h.nFill; nNonFinite = h.nNonFinite; xMin = h.xMin;
     xMax = h.xMax; linX = h.linX; doStats = h.doStats; dx = h.dx;
     under = h.under; inside = h.inside; over = h.over;
     for (int i = 0; i < nMoments; ++i) sumxNw[i] = h.sumxNw[i];
     res = h.res; res2 = h.res2; } return *this; }
 
   // Create a histogram that is the plot of the given function.
   static Hist plotFunc(function<double(double)> f, string titleIn,
     int nBinIn, double xMinIn, double xMaxIn, bool logXIn = false);
 
   // Book a histogram.
   void book(string titleIn = "  ", int nBinIn = 100, double xMinIn = 0.,
     double xMaxIn = 1., bool logXIn = false, bool doStatsIn = false) ;
 
   // Set title of a histogram.
   void title(string titleIn = "  ") {titleSave = titleIn; }
 
   // Reset bin contents.
   void null() ;
 
   // Fill bin with weight.
   void fill(double x, double w = 1.) ;
 
   // Print a histogram with overloaded << operator.
   friend ostream& operator<<(ostream& os, const Hist& h) ;
 
   // Print histogram contents as a table (e.g. for Gnuplot, Rivet or Pyplot),
   // optionally with statistical errors.
   void table(ostream& os = cout, bool printOverUnder = false,
     bool xMidBin = true, bool printError = false) const ;
   void table(string fileName, bool printOverUnder = false,
     bool xMidBin = true, bool printError = false) const {
     ofstream streamName(fileName.c_str());
     table(streamName, printOverUnder, xMidBin, printError);}
   void rivetTable(ostream& os = cout, bool printError = true) const ;
   void rivetTable(string fileName, bool printError = true) const {
     ofstream streamName(fileName.c_str()); rivetTable(streamName, printError);}
   void pyplotTable(ostream& os = cout, bool isHist = true,
     bool printError = false) const ;
   void pyplotTable(string fileName, bool isHist = true,
     bool printError = false) const {ofstream streamName(fileName.c_str());
     pyplotTable(streamName, isHist, printError);}
 
   // Fill contents of a two-column (x,y) table, e.g. written by table() above.
   void fillTable(istream& is = cin);
   void fillTable(string fileName) { ifstream streamName(fileName.c_str());
     fillTable(streamName);}
 
   // Print a table out of two histograms with same x axis (no errors printed).
   friend void table(const Hist& h1, const Hist& h2, ostream& os,
     bool printOverUnder, bool xMidBin) ;
   friend void table(const Hist& h1, const Hist& h2, string fileName,
     bool printOverUnder, bool xMidBin) ;
 
   // Return title and size of histogram. Also if logarithmic x scale.
   string getTitle() const {return titleSave;}
   int    getBinNumber() const {return nBin;}
   int    getNonFinite() const {return nNonFinite;}
   bool   getLinX() const {return linX;}
 
   // Return min and max in x and y directions.
   double getXMin() const {return xMin;}
   double getXMax() const {return xMax;}
   double getYMin() const { if (nBin == 0) return 0.;
     double yMin = res[0];
     for (int ix = 1; ix < nBin; ++ix)
       if (res[ix] < yMin ) yMin = res[ix];
     return yMin;}
   double getYMax() const { if (nBin == 0) return 0.;
     double yMax = res[0];
     for (int ix = 1; ix < nBin; ++ix)
       if (res[ix] > yMax ) yMax = res[ix];
     return yMax;}
   double getYAbsMin() const { double yAbsMin = 1e20; double yAbs;
     for (int ix = 0; ix < nBin; ++ix) { yAbs = abs(res[ix]);
       if (yAbs > 1e-20 && yAbs < yAbsMin) yAbsMin = yAbs; }
     return yAbsMin;}
 
   // Return <X> and error on <X>, unbinned from saved weight sums (default)
   // or directly from the histogram bins (unbinned = false). In the latter
   // case, the error estimate includes the difference between the binned and
   // unbinned value summed in quadrature with the statistical error, as a
   // measure of bin granularity error.
   double getXMean(bool unbinned=true) const;
   double getXMeanErr(bool unbinned=true) const;
 
   // Return Median in X and its statistical error, ignoring underflow and
   // overflow (default) or including them (includeOverUnder = true). By
   // default, error includes granularity estimate obtained by comparing binned
   // vs unbinned mean value, but this can be switched off (unbinned = false).
   double getXMedian(bool includeOverUnder=false) const;
   double getXMedianErr(bool unbinned=true) const;
 
   // Return average <Y> value.
   double getYMean() const { return inside / nFill; }
 
   // Return RMS and equivalent n'th roots of n'th moments about the mean,
   // and their error estimates. Up to n = 6, both unbinned and binned moments
   // can be calculated. For n >= 7, and for all error estimates, only
   // binned values are available. Note that (unlike ROOT), the error estimates
   // do not assume normal distributions.
   // RMN(2) = RMS is the standard root-mean-square deviation from the mean.
   // RMN(3) is the cube root of the mean-cube deviation,
   //         cbrt(<(x - <x>)^3>). It is sensitive to single-sided tails,
   //         as are characteristic of many particle-physics distributions.
   // RMN(4) adds sensitivity to double-sided long tails (eg BW vs
   //         Gaussian), and further sensitivity to long single-sided ones.
   // Etc.
   double getXRMN(int n=2, bool unbinned=true) const;
   double getXRMS(bool unbinned=true) const {return getXRMN(2, unbinned);}
 
   double getXRMNErr(int n=2, bool unbinned=true) const;
   double getXRMSErr(bool unbinned=true) const {
     return getXRMNErr(2, unbinned);}
 
   // Return content of specific bin: 0 gives underflow and nBin+1 overflow.
   double getBinContent(int iBin) const;
 
   // Return the lower edge of the bin.
   double getBinEdge(int iBin) const;
 
   // Return the width of the bin.
   double getBinWidth(int iBin=1) const;
 
   // Return bin contents.
   vector<double> getBinContents() const;
 
   // Return bin edges.
   vector<double> getBinEdges() const;
 
   // Return number of entries.
   int getEntries(bool alsoNonFinite = true) const {
     return alsoNonFinite ? nNonFinite + nFill : nFill; }
 
   // Return sum of weights.
   double getWeightSum(bool alsoOverUnder = true) const {
     return alsoOverUnder ? inside + over + under : inside; }
 
   // Return effective entries (for weighted histograms = number
   // of equivalent unweighted events for same statistical power).
   double getNEffective() const {
     double sumw2 = 0.;
     for (int ix = 0; ix < nBin; ++ix) sumw2 += res2[ix];
     if (sumw2 <= Hist::TINY) return 0.;
     else return pow2(sumxNw[0]) / sumw2;
   }
 
   // Check whether another histogram has same size and limits.
   bool sameSize(const Hist& h) const ;
 
   // Take an arbitrary function of bin contents.
   void takeFunc(function<double(double)> func);
 
   // Take logarithm (base 10 or e) of bin contents.
   void takeLog(bool tenLog = true);
 
   // Take square root of bin contents.
   void takeSqrt();
 
   // Normalize sum of bin contents to value, with or without overflow bins.
   void normalize(double f = 1, bool overflow = true) ;
 
   // Normalize sum of bin areas to value, with or without overflow bins.
   void normalizeIntegral(double f = 1, bool overflow = true);
 
   // Scale each bin content by 1 / (wtSum * bin width).
   void normalizeSpectrum(double wtSum);
 
   // Operator overloading with member functions
   Hist& operator+=(const Hist& h) ;
   Hist& operator-=(const Hist& h) ;
   Hist& operator*=(const Hist& h) ;
   Hist& operator/=(const Hist& h) ;
   Hist& operator+=(double f) ;
   Hist& operator-=(double f) ;
   Hist& operator*=(double f) ;
   Hist& operator/=(double f) ;
   Hist operator+(double f) const;
   Hist operator+(const Hist& h2) const;
   Hist operator-(double f) const;
   Hist operator-(const Hist& h2) const;
   Hist operator*(double f) const;
   Hist operator*(const Hist& h2) const;
   Hist operator/(double f) const;
   Hist operator/(const Hist& h2) const;
 
   // Operator overloading with friends
   friend Hist operator+(double f, const Hist& h1);
   friend Hist operator-(double f, const Hist& h1);
   friend Hist operator*(double f, const Hist& h1);
   friend Hist operator/(double f, const Hist& h1);
 
 private:
 
   // Constants: could only be changed in the code itself.
   static const int    NBINMAX, NCOLMAX, NLINES;
   static const double TOLERANCE, TINY, LARGE, SMALLFRAC, DYAC[];
   static const char   NUMBER[];
 
   // Properties and contents of a histogram.
   string titleSave;
   int    nBin, nFill, nNonFinite;
   double xMin, xMax;
   bool   linX, doStats;
   double dx, under, inside, over;
   vector<double> res, res2;
 
   // Sum x^N w, for different powers N, for calculation of unbinned moments.
   static constexpr int nMoments = 7;
   double sumxNw[nMoments];
 
 };
 
 //--------------------------------------------------------------------------
 
 // Namespace function declarations; friends of Hist class.
 
 // Print a histogram with overloaded << operator.
 ostream& operator<<(ostream& os, const Hist& h) ;
 
 // Print a table out of two histograms with same x axis.
 void table(const Hist& h1, const Hist& h2, ostream& os = cout,
   bool printOverUnder = false, bool xMidBin = true) ;
 void table(const Hist& h1, const Hist& h2, string fileName,
   bool printOverUnder = false, bool xMidBin = true) ;
 
 // Operator overloading with friends
 Hist operator+(double f, const Hist& h1);
 Hist operator-(double f, const Hist& h1);
 Hist operator*(double f, const Hist& h1);
 Hist operator/(double f, const Hist& h1);
 
 //==========================================================================
 
 // HistPlot class.
 // Writes a Python program that can generate PDF plots from Hist histograms.
 
 class HistPlot {
 
 public:
 
   // Constructor requires name of Python program (and adds .py).
   HistPlot(string pythonName, bool useLegacyIn = false)
     : nFrame(), nTable(), useLegacy(useLegacyIn) {
     toPython.open( (pythonName + ".py").c_str() );
     toPython << "from matplotlib import pyplot as plt" << endl
              << "from matplotlib.backends.backend_pdf import PdfPages" << endl;
     nPDF = 0; }
 
   // Destructor should do final close.
   ~HistPlot() { toPython << "pp.close()" << endl; }
 
   // New plot frame, with title, x and y labels, x and y sizes.
   void frame( string frameIn, string titleIn = "", string xLabIn = "",
     string yLabIn = "", double xSizeIn = 8., double ySizeIn = 6.) {
     framePrevious = frameName; frameName = frameIn; title = titleIn;
     xLabel = xLabIn; yLabel = yLabIn; xSize = xSizeIn; ySize = ySizeIn;
     histos.clear(); styles.clear(); legends.clear(); files.clear();
     fileStyles.clear(); fileLegends.clear(); filexyerr.clear();}
 
   // Add a histogram to the current plot, with optional style and legend.
   void add( const Hist& histIn, string styleIn = "h",
     string legendIn = "void") {
     if (histIn.getBinNumber() == 0) {
       cout << " Error: histogram is not booked" << endl;
       return;
     }
     histos.push_back(histIn);
     styles.push_back(styleIn); legends.push_back(legendIn); }
 
   // Add a file of (x, y) values not from a histogram, e.g. data points.
   void addFile( string fileIn, string styleIn = "o",
     string legendIn = "void", string xyerrIn="") { files.push_back(fileIn);
     fileStyles.push_back(styleIn); fileLegends.push_back(legendIn);
     filexyerr.push_back(xyerrIn);}
 
   // Plot a frame given the information from the new and add calls.
   void plot( bool logY = false, bool logX = false, bool userBorders = false);
   void plot( double xMinUserIn, double xMaxUserIn,  double yMinUserIn,
      double yMaxUserIn, bool logY = false, bool logX = false) {
      xMinUser = xMinUserIn; xMaxUser = xMaxUserIn; yMinUser = yMinUserIn;
      yMaxUser = yMaxUserIn; plot( logY, logX, true);}
 
   //  Omnibus single call when only one histogram in the frame.
   void plotFrame( string frameIn, const Hist& histIn, string titleIn = "",
     string xLabIn = "", string yLabIn = "", string styleIn = "h",
     string legendIn = "void",  bool logY = false) {
     frame( frameIn, titleIn, xLabIn, yLabIn);
     add( histIn, styleIn, legendIn); plot( logY); }
 
 private:
 
   // Initialization code.
   void init( string pythonName);
 
   // Stored quantities.
   ofstream toPython;
   int      nPDF, nFrame, nTable;
   double   xSize, ySize, xMinUser, xMaxUser, yMinUser, yMaxUser;
   string   frameName, framePrevious, title, xLabel, yLabel, fileName, tmpFig;
   vector<Hist> histos;
   vector<string> styles, legends, files, fileStyles, fileLegends, filexyerr;
 
   // If true, use old linthreshy matplotlib parameter (removed in 3.5.0)
   bool useLegacy;
 
 };
 
 //==========================================================================
 
 // Methods used for debugging random number sequences.
 
 #ifdef RNGDEBUG
 #define flat() flatDebug(__METHOD_NAME__, __FILE__, __LINE__)
 #define xexp() xexpDebug(__METHOD_NAME__, __FILE__, __LINE__)
 #define gauss() gaussDebug(__METHOD_NAME__, __FILE__, __LINE__)
 #define gamma(...) gammaDebug(__METHOD_NAME__, __FILE__, __LINE__, __VA_ARGS__)
 #define phaseSpace2(...) phaseSpace2Debug(__METHOD_NAME__, __FILE__, __LINE__,\
     __VA_ARGS__)
 #endif
 
 //==========================================================================
 
 // Randomly shuffle a vector, standard Fisher-Yates algorithm.
 // This must be defined after possible RNG debugging.
 
 template<typename T> void Rndm::shuffle(vector<T>& vec) {
   for (int i = vec.size() - 1; i > 0; --i)
     swap(vec[i], vec[floor(flat() * (i + 1))]);
 }
 
 //==========================================================================
 
 } // end namespace Pythia8
 
 #endif // Pythia8_Basics_H

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