// Basics.cc is a part of the PYTHIA event generator.
// Copyright (C) 2017 Torbjorn Sjostrand.
// PYTHIA is licenced under the GNU GPL version 2, see COPYING for details.
// Please respect the MCnet Guidelines, see GUIDELINES for details.

// Function definitions (not found in the header) for the Rndm, Vec4,
// RotBstMatrix and Hist classes, and some related global functions.

#include "Pythia8/Basics.h"

// Access time information.
#include <ctime>
#include <limits>

namespace Pythia8 {

//==========================================================================

// Rndm class.
// This class handles random number generation according to the
// Marsaglia-Zaman-Tsang algorithm

//--------------------------------------------------------------------------

// Constants: could be changed here if desired, but normally should not.
// These are of technical nature, as described for each.

// The default seed, i.e. the Marsaglia-Zaman random number sequence.
const int Rndm::DEFAULTSEED     = 19780503;

//--------------------------------------------------------------------------

// Method to pass in pointer for external random number generation.

bool Rndm::rndmEnginePtr( RndmEngine* rndmEngPtrIn) {

  // Save pointer.
  if (rndmEngPtrIn == 0) return false;
  rndmEngPtr      = rndmEngPtrIn;
  useExternalRndm = true;

  // Done.
  return true;

}

//--------------------------------------------------------------------------

// Initialize, normally at construction or in first call.

void Rndm::init(int seedIn) {

  // Pick seed in convenient way. Assure it to be non-negative.
  int seed = seedIn;
  if (seedIn < 0) seed = DEFAULTSEED;
  else if (seedIn == 0) seed = int(time(0));
  if (seed < 0) seed = -seed;

  // Unpack seed.
  int ij = (seed/30082) % 31329;
  int kl = seed % 30082;
  int i  = (ij/177) % 177 + 2;
  int j  = ij % 177 + 2;
  int k  = (kl/169) % 178 + 1;
  int l  =  kl % 169;

  // Initialize random number array.
  for (int ii = 0; ii < 97; ++ii) {
    double s = 0.;
    double t = 0.5;
    for (int jj = 0; jj < 48; ++jj) {
      int m = (( (i*j)%179 )*k) % 179;
      i = j;
      j = k;
      k = m;
      l = (53*l+1) % 169;
      if ( (l*m) % 64 >= 32) s += t;
      t *= 0.5;
    }
    u[ii] = s;
  }

  // Initialize other variables.
  double twom24 = 1.;
  for (int i24 = 0; i24 < 24; ++i24) twom24 *= 0.5;
  c   = 362436. * twom24;
  cd  = 7654321. * twom24;
  cm  = 16777213. * twom24;
  i97 = 96;
  j97 = 32;

  // Finished.
  initRndm  = true;
  seedSave  = seed;
  sequence  = 0;

}

//--------------------------------------------------------------------------

// Generate next random number uniformly between 0 and 1.

double Rndm::flat() {

  // Use external random number generator if such has been linked.
  if (useExternalRndm) return rndmEngPtr->flat();

  // Ensure that already initialized.
  if (!initRndm) init(DEFAULTSEED);

  // Find next random number and update saved state.
  ++sequence;
  double uni;
  do {
    uni = u[i97] - u[j97];
    if (uni < 0.) uni += 1.;
    u[i97] = uni;
    if (--i97 < 0) i97 = 96;
    if (--j97 < 0) j97 = 96;
    c -= cd;
    if (c < 0.) c += cm;
    uni -= c;
    if(uni < 0.) uni += 1.;
   } while (uni <= 0. || uni >= 1.);
  return uni;

}

//--------------------------------------------------------------------------

// Pick one option among  vector of (positive) probabilities.

int Rndm::pick(const vector<double>& prob) {

  double work = 0.;
  for (int i = 0; i < int(prob.size()); ++i) work += prob[i];
  work *= flat();
  int index = -1;
  do work -= prob[++index];
  while (work > 0. && index < int(prob.size()));
  return index;

}

//--------------------------------------------------------------------------

// Save current state of the random number generator to a binary file.

bool Rndm::dumpState(string fileName) {

  // Open file as output stream.
  const char* fn = fileName.c_str();
  ofstream ofs(fn, ios::binary);

  if (!ofs.good()) {
    cout << " Rndm::dumpState: could not open output file" << endl;
    return false;
  }

  // Write the state of the generator on the file.
  ofs.write((char *) &seedSave, sizeof(int));
  ofs.write((char *) &sequence, sizeof(long));
  ofs.write((char *) &i97,      sizeof(int));
  ofs.write((char *) &j97,      sizeof(int));
  ofs.write((char *) &c,        sizeof(double));
  ofs.write((char *) &cd,       sizeof(double));
  ofs.write((char *) &cm,       sizeof(double));
  ofs.write((char *) &u,        sizeof(double) * 97);

  // Write confirmation on cout.
  cout << " PYTHIA Rndm::dumpState: seed = " << seedSave
       << ", sequence no = " << sequence << endl;
  return true;

}

//--------------------------------------------------------------------------

// Read in the state of the random number generator from a binary file.

bool Rndm::readState(string fileName) {

  // Open file as input stream.
  const char* fn = fileName.c_str();
  ifstream ifs(fn, ios::binary);

  if (!ifs.good()) {
    cout << " Rndm::readState: could not open input file" << endl;
    return false;
  }

  // Read the state of the generator from the file.
  ifs.read((char *) &seedSave, sizeof(int));
  ifs.read((char *) &sequence, sizeof(long));
  ifs.read((char *) &i97,      sizeof(int));
  ifs.read((char *) &j97,      sizeof(int));
  ifs.read((char *) &c,        sizeof(double));
  ifs.read((char *) &cd,       sizeof(double));
  ifs.read((char *) &cm,       sizeof(double));
  ifs.read((char *) &u,        sizeof(double) *97);

  // Write confirmation on cout.
  cout << " PYTHIA Rndm::readState: seed " << seedSave
       << ", sequence no = " << sequence << endl;
  return true;

}

//==========================================================================

// Vec4 class.
// This class implements four-vectors, in energy-momentum space.
// (But could also be used to hold space-time four-vectors.)

//--------------------------------------------------------------------------

// Constants: could be changed here if desired, but normally should not.
// These are of technical nature, as described for each.

// Small number to avoid division by zero.
const double Vec4::TINY = 1e-20;

//--------------------------------------------------------------------------

// Rotation (simple).

void Vec4::rot(double thetaIn, double phiIn) {

  double cthe = cos(thetaIn);
  double sthe = sin(thetaIn);
  double cphi = cos(phiIn);
  double sphi = sin(phiIn);
  double tmpx =  cthe * cphi * xx -    sphi * yy + sthe * cphi * zz;
  double tmpy =  cthe * sphi * xx +    cphi * yy + sthe * sphi * zz;
  double tmpz = -sthe *        xx +                cthe *        zz;
  xx          = tmpx;
  yy          = tmpy;
  zz          = tmpz;

}

//--------------------------------------------------------------------------

// Azimuthal rotation phi around an arbitrary axis (nz, ny, nz).

void Vec4::rotaxis(double phiIn, double nx, double ny, double nz) {

  double norm = 1./sqrt(nx*nx + ny*ny + nz*nz);
  nx         *= norm;
  ny         *= norm;
  nz         *= norm;
  double cphi = cos(phiIn);
  double sphi = sin(phiIn);
  double comb = (nx * xx + ny * yy + nz * zz) * (1. - cphi);
  double tmpx = cphi * xx + comb * nx + sphi * (ny * zz - nz * yy);
  double tmpy = cphi * yy + comb * ny + sphi * (nz * xx - nx * zz);
  double tmpz = cphi * zz + comb * nz + sphi * (nx * yy - ny * xx);
  xx          = tmpx;
  yy          = tmpy;
  zz          = tmpz;

}

//--------------------------------------------------------------------------

// Azimuthal rotation phi around an arbitrary (3-vector component of) axis.

void Vec4::rotaxis(double phiIn, const Vec4& n) {

  double nx   = n.xx;
  double ny   = n.yy;
  double nz   = n.zz;
  double norm = 1./sqrt(nx*nx + ny*ny + nz*nz);
  nx         *= norm;
  ny          *=norm;
  nz          *=norm;
  double cphi = cos(phiIn);
  double sphi = sin(phiIn);
  double comb = (nx * xx + ny * yy + nz * zz) * (1. - cphi);
  double tmpx = cphi * xx + comb * nx + sphi * (ny * zz - nz * yy);
  double tmpy = cphi * yy + comb * ny + sphi * (nz * xx - nx * zz);
  double tmpz = cphi * zz + comb * nz + sphi * (nx * yy - ny * xx);
  xx          = tmpx;
  yy          = tmpy;
  zz          = tmpz;

}

//--------------------------------------------------------------------------

// Boost (simple).

void Vec4::bst(double betaX, double betaY, double betaZ) {

  double beta2 = betaX*betaX + betaY*betaY + betaZ*betaZ;
  double gamma = 1. / sqrt(1. - beta2);
  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
  xx += prod2 * betaX;
  yy += prod2 * betaY;
  zz += prod2 * betaZ;
  tt = gamma * (tt + prod1);

}

//--------------------------------------------------------------------------

// Boost (simple, given gamma).

void Vec4::bst(double betaX, double betaY, double betaZ, double gamma) {

  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
  xx += prod2 * betaX;
  yy += prod2 * betaY;
  zz += prod2 * betaZ;
  tt = gamma * (tt + prod1);

}

//--------------------------------------------------------------------------

// Boost given by a Vec4 p.

void Vec4::bst(const Vec4& pIn) {

  double betaX = pIn.xx / pIn.tt;
  double betaY = pIn.yy / pIn.tt;
  double betaZ = pIn.zz / pIn.tt;
  double beta2 = betaX*betaX + betaY*betaY + betaZ*betaZ;
  double gamma = 1. / sqrt(1. - beta2);
  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
  xx          += prod2 * betaX;
  yy          += prod2 * betaY;
  zz          += prod2 * betaZ;
  tt           = gamma * (tt + prod1);

}

//--------------------------------------------------------------------------

// Boost given by a Vec4 p and double m.

void Vec4::bst(const Vec4& pIn, double mIn) {

  double betaX = pIn.xx / pIn.tt;
  double betaY = pIn.yy / pIn.tt;
  double betaZ = pIn.zz / pIn.tt;
  double gamma = pIn.tt / mIn;
  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
  xx          += prod2 * betaX;
  yy          += prod2 * betaY;
  zz          += prod2 * betaZ;
  tt           = gamma * (tt + prod1);

}

//--------------------------------------------------------------------------

// Boost given by a Vec4 p; boost in opposite direction.

void Vec4::bstback(const Vec4& pIn) {

  double betaX = -pIn.xx / pIn.tt;
  double betaY = -pIn.yy / pIn.tt;
  double betaZ = -pIn.zz / pIn.tt;
  double beta2 = betaX*betaX + betaY*betaY + betaZ*betaZ;
  double gamma = 1. / sqrt(1. - beta2);
  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
  xx          += prod2 * betaX;
  yy          += prod2 * betaY;
  zz          += prod2 * betaZ;
  tt           = gamma * (tt + prod1);

}

//--------------------------------------------------------------------------

// Boost given by a Vec4 p and double m; boost in opposite direction.

void Vec4::bstback(const Vec4& pIn, double mIn) {

  double betaX = -pIn.xx / pIn.tt;
  double betaY = -pIn.yy / pIn.tt;
  double betaZ = -pIn.zz / pIn.tt;
  double gamma = pIn.tt / mIn;
  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
  xx          += prod2 * betaX;
  yy          += prod2 * betaY;
  zz          += prod2 * betaZ;
  tt           = gamma * (tt + prod1);

}

//--------------------------------------------------------------------------

// Arbitrary combination of rotations and boosts defined by 4 * 4 matrix.

void Vec4::rotbst(const RotBstMatrix& M) {

  double x = xx; double y = yy; double z = zz; double t = tt;
  tt = M.M[0][0] * t + M.M[0][1] * x + M.M[0][2] * y +  M.M[0][3] * z;
  xx = M.M[1][0] * t + M.M[1][1] * x + M.M[1][2] * y +  M.M[1][3] * z;
  yy = M.M[2][0] * t + M.M[2][1] * x + M.M[2][2] * y +  M.M[2][3] * z;
  zz = M.M[3][0] * t + M.M[3][1] * x + M.M[3][2] * y +  M.M[3][3] * z;

}

//--------------------------------------------------------------------------

// Print a four-vector: also operator overloading with friend.

ostream& operator<<(ostream& os, const Vec4& v) {
  os << fixed << setprecision(3) << " " << setw(9) << v.xx << " "
     << setw(9) << v.yy << " " << setw(9) << v.zz << " " << setw(9)
     << v.tt << " (" << setw(9) << v.mCalc() << ")\n";
  return os;
}

//--------------------------------------------------------------------------

// The invariant mass of two four-vectors.

double m(const Vec4& v1, const Vec4& v2) {
  double m2 = pow2(v1.tt + v2.tt) - pow2(v1.xx + v2.xx)
     - pow2(v1.yy + v2.yy) - pow2(v1.zz + v2.zz);
  return (m2 > 0.) ? sqrt(m2) : 0.;
}

//--------------------------------------------------------------------------

// The squared invariant mass of two four-vectors.

double m2(const Vec4& v1, const Vec4& v2) {
  double m2 = pow2(v1.tt + v2.tt) - pow2(v1.xx + v2.xx)
     - pow2(v1.yy + v2.yy) - pow2(v1.zz + v2.zz);
  return m2;
}

//--------------------------------------------------------------------------

// The scalar product of two three-vectors.

double dot3(const Vec4& v1, const Vec4& v2) {
  return v1.xx*v2.xx + v1.yy*v2.yy + v1.zz*v2.zz;
}

//--------------------------------------------------------------------------

// The cross product of two three-vectors.

Vec4 cross3(const Vec4& v1, const Vec4& v2) {
  Vec4 v;
  v.xx = v1.yy * v2.zz - v1.zz * v2.yy;
  v.yy = v1.zz * v2.xx - v1.xx * v2.zz;
  v.zz = v1.xx * v2.yy - v1.yy * v2.xx; return v;
}


//--------------------------------------------------------------------------

// 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) {
  Vec4 v(0.,0.,0.,0.);
  v.tt =   a.xx*b.yy*c.zz + a.yy*b.zz*c.xx + a.zz*b.xx*c.yy
         - a.xx*b.zz*c.yy - a.zz*b.yy*c.xx - a.yy*b.xx*c.zz;
  v.xx = -(- a.tt*b.yy*c.zz - a.yy*b.zz*c.tt - a.zz*b.tt*c.yy
           + a.tt*b.zz*c.yy + a.zz*b.yy*c.tt + a.yy*b.tt*c.zz);
  v.yy = -(- a.xx*b.tt*c.zz - a.tt*b.zz*c.xx - a.zz*b.xx*c.tt
           + a.xx*b.zz*c.tt + a.zz*b.tt*c.xx + a.tt*b.xx*c.zz);
  v.zz = -(- a.xx*b.yy*c.tt - a.yy*b.tt*c.xx - a.tt*b.xx*c.yy
           + a.xx*b.tt*c.yy + a.tt*b.yy*c.xx + a.yy*b.xx*c.tt);
  return v;
}

//--------------------------------------------------------------------------

// Opening angle between two three-vectors.

double theta(const Vec4& v1, const Vec4& v2) {
  double cthe = (v1.xx * v2.xx + v1.yy * v2.yy + v1.zz * v2.zz)
    / sqrt( (v1.xx*v1.xx + v1.yy*v1.yy + v1.zz*v1.zz)
    * (v2.xx*v2.xx + v2.yy*v2.yy + v2.zz*v2.zz) );
  cthe = max(-1., min(1., cthe));
  return acos(cthe);
}

//--------------------------------------------------------------------------

// Cosine of the opening angle between two three-vectors.

double costheta(const Vec4& v1, const Vec4& v2) {
  double cthe = (v1.xx * v2.xx + v1.yy * v2.yy + v1.zz * v2.zz)
    / sqrt( (v1.xx*v1.xx + v1.yy*v1.yy + v1.zz*v1.zz)
    * (v2.xx*v2.xx + v2.yy*v2.yy + v2.zz*v2.zz) );
  cthe = max(-1., min(1., cthe));
  return cthe;
}

//--------------------------------------------------------------------------

// Azimuthal angle between two three-vectors.

double phi(const Vec4& v1, const Vec4& v2) {
  double cphi = (v1.xx * v2.xx + v1.yy * v2.yy) / sqrt( max( Vec4::TINY,
    (v1.xx*v1.xx + v1.yy*v1.yy) * (v2.xx*v2.xx + v2.yy*v2.yy) ));
  cphi = max(-1., min(1., cphi));
  return acos(cphi);
}

//--------------------------------------------------------------------------

// Cosine of the azimuthal angle between two three-vectors.

double cosphi(const Vec4& v1, const Vec4& v2) {
  double cphi = (v1.xx * v2.xx + v1.yy * v2.yy) / sqrt( max( Vec4::TINY,
    (v1.xx*v1.xx + v1.yy*v1.yy) * (v2.xx*v2.xx + v2.yy*v2.yy) ));
  cphi = max(-1., min(1., cphi));
  return cphi;
}

//--------------------------------------------------------------------------

// Azimuthal angle between two three-vectors around a third.

double phi(const Vec4& v1, const Vec4& v2, const Vec4& n) {
  double nx = n.xx; double ny = n.yy; double nz = n.zz;
  double norm = 1. / sqrt(nx*nx + ny*ny + nz*nz);
  nx *= norm; ny *=norm; nz *=norm;
  double v1s = v1.xx * v1.xx + v1.yy * v1.yy + v1.zz * v1.zz;
  double v2s = v2.xx * v2.xx + v2.yy * v2.yy + v2.zz * v2.zz;
  double v1v2 = v1.xx * v2.xx + v1.yy * v2.yy + v1.zz * v2.zz;
  double v1n = v1.xx * nx + v1.yy * ny + v1.zz * nz;
  double v2n = v2.xx * nx + v2.yy * ny + v2.zz * nz;
  double cphi = (v1v2 - v1n * v2n) / sqrt( max( Vec4::TINY,
    (v1s - v1n*v1n) * (v2s - v2n*v2n) ));
  cphi = max(-1., min(1., cphi));
  return acos(cphi);
}

//--------------------------------------------------------------------------

// Cosine of the azimuthal angle between two three-vectors around a third.

double cosphi(const Vec4& v1, const Vec4& v2, const Vec4& n) {
  double nx = n.xx; double ny = n.yy; double nz = n.zz;
  double norm = 1. / sqrt(nx*nx + ny*ny + nz*nz);
  nx *= norm; ny *=norm; nz *=norm;
  double v1s = v1.xx * v1.xx + v1.yy * v1.yy + v1.zz * v1.zz;
  double v2s = v2.xx * v2.xx + v2.yy * v2.yy + v2.zz * v2.zz;
  double v1v2 = v1.xx * v2.xx + v1.yy * v2.yy + v1.zz * v2.zz;
  double v1n = v1.xx * nx + v1.yy * ny + v1.zz * nz;
  double v2n = v2.xx * nx + v2.yy * ny + v2.zz * nz;
  double cphi = (v1v2 - v1n * v2n) / sqrt( max( Vec4::TINY,
    (v1s - v1n*v1n) * (v2s - v2n*v2n) ));
  cphi = max(-1., min(1., cphi));
  return cphi;
}

//--------------------------------------------------------------------------

// Distance in cylindrical (y, phi) coordinates.

double RRapPhi(const Vec4& v1, const Vec4& v2) {
  double dRap = abs(v1.rap() - v2.rap());
  double dPhi = abs(v1.phi() - v2.phi());
  if (dPhi > M_PI) dPhi = 2. * M_PI - dPhi;
  return sqrt(dRap*dRap + dPhi*dPhi);
}

//--------------------------------------------------------------------------

// Distance in cylindrical (eta, phi) coordinates.

double REtaPhi(const Vec4& v1, const Vec4& v2) {
  double dEta = abs(v1.eta() - v2.eta());
  double dPhi = abs(v1.phi() - v2.phi());
  if (dPhi > M_PI) dPhi = 2. * M_PI - dPhi;
  return sqrt(dEta*dEta + dPhi*dPhi);
}

//--------------------------------------------------------------------------

// Shift four-momenta within pair from old to new masses.
// Note that p1Move and p2Move change values during operation.

bool pShift( Vec4& p1Move, Vec4& p2Move, double m1New, double m2New) {

  // Standard kinematics variables.
  double sH  = (p1Move + p2Move).m2Calc();
  double r1  = p1Move.m2Calc() / sH;
  double r2  = p2Move.m2Calc() / sH;
  double r3  = m1New * m1New / sH;
  double r4  = m2New * m2New / sH;
  double l12 = sqrtpos(pow2(1. - r1 - r2) - 4. * r1 * r2);
  double l34 = sqrtpos(pow2(1. - r3 - r4) - 4. * r3 * r4);

  // Check that shift operation possible.
  if (sH <= pow2(m1New + m2New) || l12 < Vec4::TINY || l34 < Vec4::TINY)
    return false;

  // Calculate needed shift and apply it.
  double c1  = 0.5 * ( (1. - r1 + r2) * l34 / l12 - (1. - r3 + r4) );
  double c2  = 0.5 * ( (1. + r1 - r2) * l34 / l12 - (1. + r3 - r4) );
  Vec4   pSh = c1 * p1Move - c2 * p2Move;
  p1Move    += pSh;
  p2Move    -= pSh;
  return true;
}

//--------------------------------------------------------------------------

// Create two vectors that are perpendicular to the both input vectors.

pair<Vec4,Vec4> getTwoPerpendicular(const Vec4& v1, const Vec4& v2) {

  // One perpendicular vector from three-dimensional cross-product.
  Vec4 nPerp( cross3(v1,v2) );
  double TINY = std::numeric_limits<double>::epsilon();
  if ( abs(nPerp.pAbs()) < TINY) {
    Vec4 aux;
    if (v1.px() != 0.)      aux.p(v1.yy,v1.px(),v1.pz(),v1.e());
    else if (v1.py() != 0.) aux.p(v1.px(),v1.pz(),v1.py(),v1.e());
    else if (v1.pz() != 0.) aux.p(v1.pz(),v1.py(),v1.px(),v1.e());
    nPerp.p( cross3(v1,aux) );
  }
  nPerp /= abs(nPerp.pAbs());

  // Second perpendicular vector from four-dimensional cross-product.
  Vec4 lPerp( cross4(v1,v2,nPerp) );
  lPerp /= sqrt(abs(lPerp.m2Calc()));
  return make_pair(nPerp,lPerp);
}

//==========================================================================

// 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.

//--------------------------------------------------------------------------

// Constants: could be changed here if desired, but normally should not.
// These are of technical nature, as described for each.

// Small number to avoid division by zero.
const double RotBstMatrix::TINY = 1e-20;

//--------------------------------------------------------------------------

// Rotate by polar angle theta and azimuthal angle phi.

void RotBstMatrix::rot(double theta, double phi) {

  // Set up rotation matrix.
  double cthe = cos(theta);
  double sthe = sin(theta);
  double cphi = cos(phi);
  double sphi = sin(phi);
  double Mrot[4][4] = {
    {1.,           0.,         0.,          0.},
    {0.,  cthe * cphi,     - sphi, sthe * cphi},
    {0.,  cthe * sphi,       cphi, sthe * sphi},
    {0., -sthe,                0., cthe       } };

  // Rotate current matrix accordingly.
  double Mtmp[4][4];
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    Mtmp[i][j] = M[i][j];
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    M[i][j] = Mrot[i][0] * Mtmp[0][j] + Mrot[i][1] * Mtmp[1][j]
            + Mrot[i][2] * Mtmp[2][j] + Mrot[i][3] * Mtmp[3][j];

}

//--------------------------------------------------------------------------

// Rotate so that vector originally along z axis becomes parallel with p.

void RotBstMatrix::rot(const Vec4& p) {

  double theta = p.theta();
  double phi = p.phi();
  rot(0., -phi);
  rot(theta, phi);

}

//--------------------------------------------------------------------------

// Boost with velocity vector (betaX, betaY, betaZ).

void RotBstMatrix::bst(double betaX, double betaY, double betaZ) {

  // Set up boost matrix.
  double gm = 1. / sqrt( max( TINY, 1. - betaX*betaX - betaY*betaY
    - betaZ*betaZ ) );
  double gf = gm*gm / (1. + gm);
  double Mbst[4][4] = {
    { gm,           gm*betaX,           gm*betaY,          gm*betaZ },
    { gm*betaX, 1. + gf*betaX*betaX, gf*betaX*betaY, gf*betaX*betaZ },
    { gm*betaY, gf*betaY*betaX, 1. + gf*betaY*betaY, gf*betaY*betaZ },
    { gm*betaZ, gf*betaZ*betaX, gf*betaZ*betaY, 1. + gf*betaZ*betaZ } };

  // Boost current matrix correspondingly.
  double Mtmp[4][4];
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    Mtmp[i][j] = M[i][j];
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    M[i][j] = Mbst[i][0] * Mtmp[0][j] + Mbst[i][1] * Mtmp[1][j]
            + Mbst[i][2] * Mtmp[2][j] + Mbst[i][3] * Mtmp[3][j];

}

//--------------------------------------------------------------------------

// Boost so that vector originally at rest obtains same velocity as p.

void RotBstMatrix::bst(const Vec4& p) {
  double betaX = p.px() / p.e();
  double betaY = p.py() / p.e();
  double betaZ = p.pz() / p.e();
  bst(betaX, betaY, betaZ);
}

//--------------------------------------------------------------------------

// Boost so vector originally with same velocity as p is brought to rest.

void RotBstMatrix::bstback(const Vec4& p) {
  double betaX = -p.px() / p.e();
  double betaY = -p.py() / p.e();
  double betaZ = -p.pz() / p.e();
  bst(betaX, betaY, betaZ);
}

//--------------------------------------------------------------------------

// Boost that transforms p1 to p2, where p1^2 = p2^2 is assumed.

void RotBstMatrix::bst(const Vec4& p1, const Vec4& p2) {
  double eSum = p1.e() + p2.e();
  double betaX = (p2.px() - p1.px()) / eSum;
  double betaY = (p2.py() - p1.py()) / eSum;
  double betaZ = (p2.pz() - p1.pz()) / eSum;
  double fac = 2. / (1. + betaX*betaX + betaY*betaY + betaZ*betaZ);
  betaX *= fac; betaY *= fac; betaZ *= fac;
  bst(betaX, betaY, betaZ);
}

//--------------------------------------------------------------------------

// Boost and rotation that transforms from p1 and p2
// to their rest frame with p1 along +z axis.

void RotBstMatrix::toCMframe(const Vec4& p1, const Vec4& p2) {
  Vec4 pSum = p1 + p2;
  Vec4 dir  = p1;
  dir.bstback(pSum);
  double theta = dir.theta();
  double phi   = dir.phi();
  bstback(pSum);
  rot(0., -phi);
  rot(-theta, phi);
}

//--------------------------------------------------------------------------

// Rotation and boost that transforms from rest frame of p1 and p2
// with p1 along +z axis to actual frame of p1 and p2. (Inverse of above.)

void RotBstMatrix::fromCMframe(const Vec4& p1, const Vec4& p2) {
  Vec4 pSum = p1 + p2;
  Vec4 dir  = p1;
  dir.bstback(pSum);
  double theta = dir.theta();
  double phi   = dir.phi();
  rot(0., -phi);
  rot(theta, phi);
  bst(pSum);
}

//--------------------------------------------------------------------------

// Combine existing rotation/boost matrix with another one.

void RotBstMatrix::rotbst(const RotBstMatrix& Mrb) {
  double Mtmp[4][4];
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    Mtmp[i][j] = M[i][j];
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    M[i][j] = Mrb.M[i][0] * Mtmp[0][j] + Mrb.M[i][1] * Mtmp[1][j]
            + Mrb.M[i][2] * Mtmp[2][j] + Mrb.M[i][3] * Mtmp[3][j];
}

//--------------------------------------------------------------------------

// Invert the rotation and boost.

void RotBstMatrix::invert() {
  double Mtmp[4][4];
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    Mtmp[i][j] = M[i][j];
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    M[i][j] = ( (i == 0 && j > 0) || (i > 0 && j == 0) )
      ? - Mtmp[j][i] : Mtmp[j][i];
}

//--------------------------------------------------------------------------

// Reset to diagonal matrix.

void RotBstMatrix::reset() {
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    M[i][j] = (i==j) ? 1. : 0.;
}

//--------------------------------------------------------------------------

// Crude estimate deviation from unit matrix.

double RotBstMatrix::deviation() const {
  double devSum = 0.;
  for (int i = 0; i < 4; ++i)
  for (int j = 0; j < 4; ++j)
    devSum += (i==j) ? abs(M[i][j] - 1.) : abs(M[i][j]);
  return devSum;
}

//--------------------------------------------------------------------------

// Print a rotation and boost matrix: operator overloading with friend.

ostream& operator<<(ostream& os, const RotBstMatrix& M) {
  os << fixed << setprecision(5) << "    Rotation/boost matrix: \n";
  for (int i = 0; i <4; ++i)
    os << setw(10) << M.M[i][0] << setw(10) << M.M[i][1]
       << setw(10) << M.M[i][2] << setw(10) << M.M[i][3] << "\n";
  return os;
}

//==========================================================================

// Hist class.
// This class handles a single histogram at a time
// (or a vector of histograms).

//--------------------------------------------------------------------------

// Constants: could be changed here if desired, but normally should not.
// These are of technical nature, as described for each.

// Maximum number of bins in a histogram.
const int    Hist::NBINMAX   = 1000;

// Maximum number of columns that can be printed for a histogram.
const int    Hist::NCOLMAX   = 100;

// Maximum number of lines a histogram can use at output.
const int    Hist::NLINES    = 30;

// Tolerance in deviation of xMin and xMax between two histograms.
const double Hist::TOLERANCE = 0.001;

// Small and large numbers to avoid division by zero and overflow.
const double Hist::TINY      = 1e-20;
const double Hist::LARGE     = 1e20;

// When minbin/maxbin < SMALLFRAC the y scale goes down to zero.
const double Hist::SMALLFRAC = 0.1;

// Constants for printout: fixed steps on y scale; filling characters.
const double DYAC[] = {0.04, 0.05, 0.06, 0.08, 0.10,
  0.12, 0.15, 0.20, 0.25, 0.30};
const char NUMBER[] = {'0', '1', '2', '3', '4', '5',
  '6', '7', '8', '9', 'X' };

//--------------------------------------------------------------------------

// Book a histogram.

void Hist::book(string titleIn, int nBinIn, double xMinIn,
  double xMaxIn) {

  titleSave = titleIn;
  nBin  = nBinIn;
  if (nBinIn < 1) nBin = 1;
  if (nBinIn > NBINMAX) nBin = NBINMAX;
  xMin  = xMinIn;
  xMax  = xMaxIn;
  dx    = (xMax - xMin)/nBin;
  res.resize(nBin);
  null();

}

//--------------------------------------------------------------------------

// Reset bin contents.

void Hist::null() {

  nFill  = 0;
  under  = 0.;
  inside = 0.;
  over   = 0.;
  for (int ix = 0; ix < nBin; ++ix) res[ix] = 0.;

}

//--------------------------------------------------------------------------

// Fill bin with weight.

void Hist::fill(double x, double w) {

  ++nFill;
  if (x < xMin) {under += w; return;}
  if (x > xMax) {over  += w; return;}
  int iBin = int(floor((x - xMin)/dx));
  if      (iBin < 0)     under += w;
  else if (iBin >= nBin) over  += w;
  else                 {inside += w; res[iBin] += w; }

}

//--------------------------------------------------------------------------

// Print a histogram: also operator overloading with friend.

ostream& operator<<(ostream& os, const Hist& h) {

  // Do not print empty histograms.
  if (h.nFill <= 0) return os;

  // Write time and title.
  time_t t = time(0);
  char date[18];
  strftime(date,18,"%Y-%m-%d %H:%M",localtime(&t));
  os << "\n\n  " << date << "       " << h.titleSave << "\n\n";

  // Group bins, where required, to make printout have fewer columns.
  // Avoid overflow.
  int nGroup = 1 + (h.nBin - 1) / Hist::NCOLMAX;
  int nCol   = 1 + (h.nBin - 1) / nGroup;
  vector<double> resCol(nCol);
  for (int iCol = 0; iCol < nCol; ++iCol) {
    resCol[iCol] = 0.;
    for (int ix = nGroup * iCol; ix < min( h.nBin, nGroup * (iCol + 1)); ++ix)
      resCol[iCol] += h.res[ix];
    resCol[iCol] = max( -Hist::LARGE, min( Hist::LARGE, resCol[iCol] ) );
  }

  // Find minimum and maximum bin content.
  double yMin = resCol[0];
  double yMax = resCol[0];
  for (int iCol = 1; iCol < nCol; ++iCol) {
    if (resCol[iCol] < yMin) yMin = resCol[iCol];
    if (resCol[iCol] > yMax) yMax = resCol[iCol];
  }

  // Determine scale and step size for y axis.
  if (yMax - yMin > Hist::NLINES * DYAC[0] * 1e-9) {
    if (yMin > 0. && yMin < Hist::SMALLFRAC * yMax) yMin = 0.;
    if (yMax < 0. && yMax > Hist::SMALLFRAC * yMin) yMax = 0.;
    int iPowY = int(floor( log10(yMax - yMin) ));
    if (yMax - yMin < Hist::NLINES * DYAC[0] * pow(10.,iPowY))
      iPowY = iPowY - 1;
    if (yMax - yMin > Hist::NLINES * DYAC[9] * pow(10.,iPowY))
      iPowY = iPowY + 1;
    double nLinePow = Hist::NLINES * pow(10.,iPowY);
    double delY = DYAC[0];
    for (int idel = 0; idel < 9; ++idel)
      if (yMax - yMin >= nLinePow * DYAC[idel]) delY = DYAC[idel+1];
    double dy = delY * pow(10.,iPowY);

    // Convert bin contents to integer form; fractional fill in top row.
    vector<int> row(nCol);
    vector<int> frac(nCol);
    for (int iCol = 0; iCol < nCol ; ++iCol) {
      double cta = abs(resCol[iCol]) / dy;
      row[iCol] = int(cta + 0.95);
      if(resCol[iCol] < 0.) row[iCol] = - row[iCol];
      frac[iCol] = int(10. * (cta + 1.05 - floor(cta + 0.95)));
    }
    int rowMin = int(abs(yMin)/dy + 0.95);
    if ( yMin < 0) rowMin = - rowMin;
    int rowMax = int(abs(yMax)/dy + 0.95);
    if ( yMax < 0) rowMax = - rowMax;

    // Print histogram row by row.
    os << fixed << setprecision(2);
    for (int iRow = rowMax; iRow >= rowMin; iRow--) if (iRow != 0) {
      os << "  " << setw(10) << iRow*delY << "*10^"
         << setw(2) << iPowY << "  ";
      for (int iCol = 0; iCol < nCol ; ++iCol) {
        if (iRow == row[iCol])                  os << NUMBER[frac[iCol]];
        else if (iRow * (row[iCol] - iRow) > 0) os << NUMBER[10];
        else                                    os << " ";
      } os << "\n";
    } os << "\n";

    // Print sign and value of bin contents
    double maxim = log10(max(yMax, -yMin));
    int iPowBin = int(floor(maxim + 0.0001));
    os << "          Contents  ";
    for (int iCol = 0; iCol < nCol ; ++iCol) {
      if (resCol[iCol] < - pow(10., iPowBin - 4)) os << "-";
      else os << " ";
      row[iCol] = int(abs(resCol[iCol]) * pow(10., 3 - iPowBin) + 0.5);
    } os << "\n";
    for (int iRow = 3; iRow >= 0; iRow--) {
      os << "            *10^" << setw(2) << iPowBin + iRow - 3 << "  ";
      int mask = int( pow(10., iRow) + 0.5);
      for (int iCol = 0; iCol < nCol ; ++iCol) {
        os << NUMBER[(row[iCol] / mask) % 10];
      } os << "\n";
    } os << "\n";

    // Print sign and value of lower bin edge.
    maxim = log10( max( -h.xMin, h.xMax - h.dx));
    int iPowExp = int(floor(maxim + 0.0001));
    os << "          Low edge  ";
    for (int iCol = 0; iCol < nCol ; ++iCol) {
      if (h.xMin + iCol * nGroup * h.dx < - pow(10., iPowExp - 3)) os << "-";
      else os << " ";
      row[iCol] = int(abs(h.xMin + iCol * nGroup * h.dx)
        * pow(10., 2 - iPowExp) + 0.5);
    } os << "\n";
    for (int iRow = 2; iRow >= 0; iRow--) {
      os << "            *10^" << setw(2) << iPowExp + iRow - 2 << "  ";
      int mask = int( pow(10., iRow) + 0.5);
      for (int iCol = 0; iCol < nCol ; ++iCol)
        os << NUMBER[(row[iCol] / mask) % 10];
      os << "\n";
    } os << "\n";

  // Print explanation if histogram cannot be shown.
  } else os << "     Histogram not shown since lowest value" << scientific
       << setprecision(4) << setw(12) << yMin << " and highest value"
       << setw(12) << yMax << " are too close \n \n";

  // Calculate and print statistics.
  double cSum   = 0.;
  double cxSum  = 0.;
  double cxxSum = 0.;
  for (int ix = 0; ix < h.nBin ; ++ix) {
    double cta = abs(h.res[ix]);
    double x = h.xMin + (ix + 0.5) * h.dx;
    cSum   = cSum   + cta;
    cxSum  = cxSum  + cta * x;
    cxxSum = cxxSum + cta * x * x;
  }
  double xmean = cxSum / max(cSum, Hist::TINY);
  double rms = sqrtpos( cxxSum / max(cSum, Hist::TINY) - xmean*xmean );
  os << scientific << setprecision(4)
     << "   Entries  =" << setw(12) << h.nFill
     << "    Mean =" << setw(12) << xmean
     << "    Underflow =" << setw(12) << h.under
     << "    Low edge  =" << setw(12) << h.xMin << "\n"
     << "   All chan =" << setw(12) << h.inside
     << "    Rms  =" << setw(12) << rms
     << "    Overflow  =" << setw(12) << h.over
     << "    High edge =" << setw(12) << h.xMax << endl;
  return os;
}

//--------------------------------------------------------------------------

// Print histogram contents as a table (e.g. for Gnuplot).

void Hist::table(ostream& os, bool printOverUnder, bool xMidBin) const {

  // Print histogram vector bin by bin, with mean x as first column.
  os << scientific << setprecision(4);
  double xBeg = (xMidBin) ? xMin + 0.5 * dx : xMin;
  if (printOverUnder)
    os << setw(12) << xBeg - dx << setw(12) << under << "\n";
  for (int ix = 0; ix < nBin; ++ix)
    os << setw(12) << xBeg + ix * dx << setw(12) << res[ix] << "\n";
  if (printOverUnder)
    os << setw(12) << xBeg + nBin * dx << setw(12) << over << "\n";

}

//--------------------------------------------------------------------------

// Print histogram contents as a table, in Rivet's *.dat style.

void Hist::rivetTable(ostream& os, bool printError) const {

  // Print histogram vector bin by bin, with x range in first two columns
  // and +- error in last two (assuming that contents is number of events).
  os << scientific << setprecision(4);
  double xBeg = xMin;
  double xEnd = xMin+dx;
  for (int ix = 0; ix < nBin; ++ix) {
    double err = (printError) ? sqrtpos(res[ix]) : 0.0;
    os << setw(12) << xBeg + ix * dx << setw(12) << xEnd + ix * dx
       << setw(12) << res[ix] << setw(12) << err << setw(12) << err << "\n";
  }

}

//--------------------------------------------------------------------------

// Print a table out of two histograms with same x axis  (e.g. for Gnuplot).

void table(const Hist& h1, const Hist& h2, ostream& os, bool printOverUnder,
  bool xMidBin) {

  // Require histogram x axes to agree.
  if (h1.nBin != h2.nBin || abs(h1.xMin - h2.xMin) > Hist::TOLERANCE * h1.dx
    || abs(h1.xMax - h2.xMax) > Hist::TOLERANCE * h1.dx) return;

  // Print histogram vectors bin by bin, with mean x as first column.
  os << scientific << setprecision(4);
  double xBeg = (xMidBin) ? h1.xMin + 0.5 * h1.dx : h1.xMin;
  if (printOverUnder)
    os << setw(12) << xBeg - h1.dx << setw(12) << h1.under
       << setw(12) << h2.under << "\n";
  for (int ix = 0; ix < h1.nBin; ++ix)
    os << setw(12) << xBeg + ix * h1.dx << setw(12) << h1.res[ix]
       << setw(12) << h2.res[ix] << "\n";
  if (printOverUnder)
    os << setw(12) << xBeg + h1.nBin * h1.dx << setw(12) << h1.over
       << setw(12) << h2.over << "\n";

}

void table(const Hist& h1, const Hist& h2, string fileName,
  bool printOverUnder, bool xMidBin) {

  ofstream streamName(fileName.c_str());
  table( h1, h2, streamName, printOverUnder, xMidBin);

}

//--------------------------------------------------------------------------

// Get content of specific bin.
// Special values are bin 0 for underflow and bin nBin+1 for overflow.
// All other bins outside proper histogram range return 0.

double Hist::getBinContent(int iBin) const {

  if (iBin > 0 && iBin <= nBin) return res[iBin - 1];
  else if (iBin == 0)           return under;
  else if (iBin == nBin + 1)    return over;
  else                          return 0.;

}

//--------------------------------------------------------------------------

// Check whether another histogram has same size and limits.

bool Hist::sameSize(const Hist& h) const {

  if (nBin == h.nBin && abs(xMin - h.xMin) < TOLERANCE * dx &&
    abs(xMax - h.xMax) < TOLERANCE * dx) return true;
  else return false;

}

//--------------------------------------------------------------------------

// Take 10-logarithm or natural logarithm of contents bin by bin.

void Hist::takeLog(bool tenLog) {

  // Find smallest positive bin content, and put min a bit below.
  double yMin = Hist::LARGE;
  for (int ix = 0; ix < nBin; ++ix)
    if (res[ix] > Hist::TINY && res[ix] < yMin ) yMin = res[ix];
  yMin *= 0.8;

  // Take 10-logarithm bin by bin, but ensure positivity.
  if (tenLog) {
    for (int ix = 0; ix < nBin; ++ix)
      res[ix] = log10( max( yMin, res[ix]) );
    under  =  log10( max( yMin, under) );
    inside =  log10( max( yMin, inside) );
    over   =  log10( max( yMin, over) );

  // Take natural logarithm bin by bin, but ensure positivity.
  } else {
    for (int ix = 0; ix < nBin; ++ix)
      res[ix] = log( max( yMin, res[ix]) );
    under  =  log( max( yMin, under) );
    inside =  log( max( yMin, inside) );
    over   =  log( max( yMin, over) );
  }

}

//--------------------------------------------------------------------------

// Take square root of contents bin by bin; set 0 for negative content.

void Hist::takeSqrt() {

  for (int ix = 0; ix < nBin; ++ix) res[ix] = sqrtpos(res[ix]);
  under  = sqrtpos(under);
  inside = sqrtpos(inside);
  over   = sqrtpos(over);

}

//--------------------------------------------------------------------------

// Add histogram to existing one.

Hist& Hist::operator+=(const Hist& h) {
  if (!sameSize(h)) return *this;
  nFill  += h.nFill;
  under  += h.under;
  inside += h.inside;
  over += h.over;
  for (int ix = 0; ix < nBin; ++ix) res[ix] += h.res[ix];
  return *this;
}

//--------------------------------------------------------------------------

// Subtract histogram from existing one.

Hist& Hist::operator-=(const Hist& h) {
  if (!sameSize(h)) return *this;
  nFill  += h.nFill;
  under  -= h.under;
  inside -= h.inside;
  over -= h.over;
  for (int ix = 0; ix < nBin; ++ix) res[ix] -= h.res[ix];
  return *this;
}

//--------------------------------------------------------------------------

// Multiply existing histogram by another one.

Hist& Hist::operator*=(const Hist& h) {
  if (!sameSize(h)) return *this;
  nFill   += h.nFill;
  under  *= h.under;
  inside *= h.inside;
  over *= h.over;
  for (int ix = 0; ix < nBin; ++ix) res[ix] *= h.res[ix];
  return *this;
}

//--------------------------------------------------------------------------

// Divide existing histogram by another one.

Hist& Hist::operator/=(const Hist& h) {
  if (!sameSize(h)) return *this;
  nFill += h.nFill;
  under  = (abs(h.under) < Hist::TINY) ? 0. : under/h.under;
  inside = (abs(h.inside) < Hist::TINY) ? 0. : inside/h.inside;
  over  = (abs(h.over) < Hist::TINY) ? 0. : over/h.over;
  for (int ix = 0; ix < nBin; ++ix)
    res[ix] = (abs(h.res[ix]) < Hist::TINY) ? 0. : res[ix]/h.res[ix];
  return *this;
}

//--------------------------------------------------------------------------

// Add constant offset to histogram.

Hist& Hist::operator+=(double f) {
  under  += f;
  inside += nBin * f;
  over   += f;
  for (int ix = 0; ix < nBin; ++ix) res[ix] += f;
  return *this;
}

//--------------------------------------------------------------------------

// Subtract constant offset from histogram.

Hist& Hist::operator-=(double f) {
  under  -= f;
  inside -= nBin * f;
  over   -= f;
  for (int ix = 0; ix < nBin; ++ix) res[ix] -= f;
  return *this;
}

//--------------------------------------------------------------------------

// Multiply histogram by constant.

Hist& Hist::operator*=(double f) {
  under  *= f;
  inside *= f;
  over   *= f;
  for (int ix = 0; ix < nBin; ++ix) res[ix] *= f;
  return *this;
}

//--------------------------------------------------------------------------

// Divide histogram by constant.

Hist& Hist::operator/=(double f) {
  if (abs(f) > Hist::TINY) {
    under  /= f;
    inside /= f;
    over   /= f;
    for (int ix = 0; ix < nBin; ++ix) res[ix] /= f;
  // Set empty contents when division by zero.
  } else {
    under  = 0.;
    inside = 0.;
    over   = 0.;
    for (int ix = 0; ix < nBin; ++ix) res[ix] = 0.;
  }
  return *this;
}

Hist Hist::operator+(double f) const {
  Hist h = *this; return h += f;}

Hist Hist::operator+(const Hist& h2) const {
  Hist h = *this; return h += h2;}

Hist Hist::operator-(double f) const {
  Hist h = *this; return h -= f;}

Hist Hist::operator-(const Hist& h2) const {
  Hist h = *this; return h -= h2;}

Hist Hist::operator*(double f) const {
  Hist h = *this; return h *= f;}

Hist Hist::operator*(const Hist& h2) const {
  Hist h = *this; return h *= h2;}

Hist Hist::operator/(double f) const {
  Hist h = *this; return h /= f;}

Hist Hist::operator/(const Hist& h2) const {
  Hist h = *this; return h /= h2;}

//--------------------------------------------------------------------------

// Implementation of operator overloading with friends.

Hist operator+(double f, const Hist& h1) {
  Hist h = h1; return h += f;}

Hist operator-(double f, const Hist& h1) {
  Hist h   = h1;
  h.under  = f - h1.under;
  h.inside = h1.nBin * f - h1.inside;
  h.over   = f - h1.over;
  for (int ix = 0; ix < h1.nBin; ++ix) h.res[ix] = f - h1.res[ix];
  return h;}

Hist operator*(double f, const Hist& h1) {
  Hist h = h1; return h *= f;}

Hist operator/(double f, const Hist& h1) {
  Hist h = h1;
  h.under  = (abs(h1.under)  < Hist::TINY) ? 0. :  f/h1.under;
  h.inside = (abs(h1.inside) < Hist::TINY) ? 0. :  f/h1.inside;
  h.over   = (abs(h1.over)   < Hist::TINY) ? 0. :  f/h1.over;
  for (int ix = 0; ix < h1.nBin; ++ix)
    h.res[ix] = (abs(h1.res[ix]) < Hist::TINY) ? 0. : f/h1.res[ix];
  return h;
}

//==========================================================================

} // end namespace Pythia8