ProbFuncMathCore.cxx

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// @(#)root/mathcore:$Id$
// Authors: L. Moneta, A. Zsenei   06/2005



#include "Math/Math.h"
#include "Math/Error.h"
#include "Math/ProbFuncMathCore.h"
#include "Math/SpecFuncMathCore.h"
#include <stdio.h>
#include <limits>
using namespace std;
namespace ROOT {
namespace Math {


   
   static const double kSqrt2 = 1.41421356237309515; // sqrt(2.)

   double beta_cdf_c(double x, double a, double b)
   {
      // use the fact that I(x,a,b) = 1. - I(1-x,b,a)
      return ROOT::Math::inc_beta(1-x, b, a);
   }


   double beta_cdf(double x, double a, double b )
   {
      return ROOT::Math::inc_beta(x, a, b);
   }


   double breitwigner_cdf_c(double x, double gamma, double x0)
   {
      return 0.5 - std::atan(2.0 * (x-x0) / gamma) / M_PI;
   }


   double breitwigner_cdf(double x, double gamma, double x0)
   {
      return 0.5 + std::atan(2.0 * (x-x0) / gamma) / M_PI;
   }


   double cauchy_cdf_c(double x, double b, double x0)
   {
      return 0.5 - std::atan( (x-x0) / b) / M_PI;
   }


   double cauchy_cdf(double x, double b, double x0)
   {
      return 0.5 + std::atan( (x-x0) / b) / M_PI;
   }


   double chisquared_cdf_c(double x, double r, double x0)
   {
      return ROOT::Math::inc_gamma_c ( 0.5 * r , 0.5* (x-x0) );
   }

   
   double chisquared_cdf(double x, double r, double x0)
   {
      return ROOT::Math::inc_gamma ( 0.5 * r , 0.5* (x-x0) );
   }

   
   double crystalball_cdf(double x, double alpha, double n, double sigma, double mean )
   {
      if (n <= 1.) {
         MATH_ERROR_MSG("crystalball_cdf","CrystalBall cdf not defined for n <=1");
         return std::numeric_limits<double>::quiet_NaN();
      }

      double abs_alpha = std::abs(alpha);
      double C = n/abs_alpha * 1./(n-1.) * std::exp(-alpha*alpha/2.);
      double D = std::sqrt(M_PI/2.)*(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
      double totIntegral = sigma*(C+D);

      double integral = crystalball_integral(x,alpha,n,sigma,mean); 
      return (alpha > 0) ? 1. - integral/totIntegral : integral/totIntegral; 
   }
   double crystalball_cdf_c(double x, double alpha, double n, double sigma, double mean )
   {
      if (n <= 1.) {
         MATH_ERROR_MSG("crystalball_cdf_c","CrystalBall cdf not defined for n <=1");
         return std::numeric_limits<double>::quiet_NaN();
      }
      double abs_alpha = std::abs(alpha);
      double C = n/abs_alpha * 1./(n-1.) * std::exp(-alpha*alpha/2.);
      double D = std::sqrt(M_PI/2.)*(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
      double totIntegral = sigma*(C+D);

      double integral = crystalball_integral(x,alpha,n,sigma,mean);      
      return (alpha > 0) ? integral/totIntegral : 1. - (integral/totIntegral); 
   }
   double crystalball_integral(double x, double alpha, double n, double sigma, double mean)
   {
      // compute the integral of the crystal ball function (ROOT::Math::crystalball_function)
      // If alpha > 0 the integral is the right tail integral.
      // If alpha < 0 is the left tail integrals which are always finite for finite x.     
      // parameters:
      // alpha : is non equal to zero, define the # of sigma from which it becomes a power-law function (from mean-alpha*sigma)
      // n > 1 : is integrer, is the power of the low  tail
      // add a value xmin for cases when n <=1 the integral diverges 
      if (sigma == 0)   return 0;
      if (alpha==0)
      {
         MATH_ERROR_MSG("crystalball_integral","CrystalBall function not defined at alpha=0");
         return 0.;
      }
      bool useLog = (n == 1.0); 
      if (n<=0)   MATH_WARN_MSG("crystalball_integral","No physical meaning when n<=0");

      double z = (x-mean)/sigma;
      if (alpha < 0 ) z = -z;
      
      double abs_alpha = std::abs(alpha);
      
      //double D = *(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
      //double N = 1./(sigma*(C+D));
      double intgaus = 0.;
      double intpow  = 0.;
 
      const double sqrtpiover2 = std::sqrt(M_PI/2.);
      const double sqrt2pi = std::sqrt( 2.*M_PI); 
      const double oneoversqrt2 = 1./sqrt(2.);
      if (z <= -abs_alpha)
      {
         double A = std::pow(n/abs_alpha,n) * std::exp(-0.5 * alpha*alpha);
         double B = n/abs_alpha - abs_alpha;

         if (!useLog) {
            double C = (n/abs_alpha) * (1./(n-1)) * std::exp(-alpha*alpha/2.);
            intpow  = C - A /(n-1.) * std::pow(B-z,-n+1) ;
         }
         else {
            // for n=1 the primitive of 1/x is log(x)
            intpow = -A * std::log( n / abs_alpha ) + A * std::log( B -z );
         }
         intgaus =  sqrtpiover2*(1.+ROOT::Math::erf(abs_alpha*oneoversqrt2));
      }
      else
      {
         intgaus = ROOT::Math::gaussian_cdf_c(z, 1);
         intgaus *= sqrt2pi;
         intpow  =  0;  
      }
      return sigma * (intgaus + intpow);
   }

   
   double exponential_cdf_c(double x, double lambda, double x0)
   {
      if ((x-x0) < 0)   return 1.0;
      else              return std::exp(- lambda * (x-x0));
   }


   double exponential_cdf(double x, double lambda, double x0)
   {
      if ((x-x0) < 0)   return 0.0;
      else              // use expm1 function to avoid errors at small x
                        return - ROOT::Math::expm1( - lambda * (x-x0) ) ;
   }


   double fdistribution_cdf_c(double x, double n, double m, double x0)
   {
      // f distribution  is defined only for both n and m > 0
      if (n < 0 || m < 0)              return std::numeric_limits<double>::quiet_NaN();

      double z = m/(m + n*(x-x0));
      // fox z->1 and large a and b IB looses precision use complement function
      if (z > 0.9 && n > 1 && m > 1)   return 1.-  fdistribution_cdf(x,n,m,x0);

      // for the complement use the fact that IB(x,a,b) = 1. - IB(1-x,b,a)
      return ROOT::Math::inc_beta(m/(m + n*(x-x0)), .5*m, .5*n);
   }


   double fdistribution_cdf(double x, double n, double m, double x0)
   {
      // f distribution  is defined only for both n and m > 0
      if (n < 0 || m < 0)
         return std::numeric_limits<double>::quiet_NaN();

      double z = n*(x-x0)/(m + n*(x-x0));
      // fox z->1 and large a and b IB looses precision use complement function
      if (z > 0.9 && n > 1 && m > 1)
         return 1. - fdistribution_cdf_c(x,n,m,x0);

      return ROOT::Math::inc_beta(z, .5*n, .5*m);
   }


   double gamma_cdf_c(double x, double alpha, double theta, double x0)
   {
      return ROOT::Math::inc_gamma_c(alpha, (x-x0)/theta);
   }


   double gamma_cdf(double x, double alpha, double theta, double x0)
   {
      return ROOT::Math::inc_gamma(alpha, (x-x0)/theta);
   }


   double lognormal_cdf_c(double x, double m, double s, double x0)
   {
      double z = (std::log((x-x0))-m)/(s*kSqrt2);
      if (z > 1.)  return 0.5*ROOT::Math::erfc(z);
      else         return 0.5*(1.0 - ROOT::Math::erf(z));
   }


   double lognormal_cdf(double x, double m, double s, double x0)
   {
      double z = (std::log((x-x0))-m)/(s*kSqrt2);
      if (z < -1.) return 0.5*ROOT::Math::erfc(-z);
      else         return 0.5*(1.0 + ROOT::Math::erf(z));
   }


   double normal_cdf_c(double x, double sigma, double x0)
   {
      double z = (x-x0)/(sigma*kSqrt2);
      if (z > 1.)  return 0.5*ROOT::Math::erfc(z);
      else         return 0.5*(1.-ROOT::Math::erf(z));
   }


   double normal_cdf(double x, double sigma, double x0)
   {
      double z = (x-x0)/(sigma*kSqrt2);
      if (z < -1.) return 0.5*ROOT::Math::erfc(-z);
      else         return 0.5*(1.0 + ROOT::Math::erf(z));
   }


   double tdistribution_cdf_c(double x, double r, double x0)
   {
      double p    = x-x0;
      double sign = (p>0) ? 1. : -1;
      return .5 - .5*ROOT::Math::inc_beta(p*p/(r + p*p), .5, .5*r)*sign;
   }


   double tdistribution_cdf(double x, double r, double x0)
   {
      double p    = x-x0;
      double sign = (p>0) ? 1. : -1;
      return  .5 + .5*ROOT::Math::inc_beta(p*p/(r + p*p), .5, .5*r)*sign;
   }


   double uniform_cdf_c(double x, double a, double b, double x0)
   {
      if      ((x-x0) <  a)   return 1.0;
      else if ((x-x0) >= b)   return 0.0;
      else                    return (b-(x-x0))/(b-a);
   }


   double uniform_cdf(double x, double a, double b, double x0)
   {
      if      ((x-x0) <  a)   return 0.0;
      else if ((x-x0) >= b)   return 1.0;
      else                    return ((x-x0)-a)/(b-a);
   }

   /// discrete distributions

   double poisson_cdf_c(unsigned int n, double mu)
   {
      // mu must be >= 0  . Use poisson - gamma relation
      //  Pr ( x <= n) = Pr( y >= a)   where x is poisson and y is gamma distributed ( a = n+1)
      double a = (double) n + 1.0;
      return ROOT::Math::gamma_cdf(mu, a, 1.0);
   }

   
   double poisson_cdf(unsigned int n, double mu)
   {
      // mu must be >= 0  . Use poisson - gamma relation
      //  Pr ( x <= n) = Pr( y >= a)   where x is poisson and y is gamma distributed ( a = n+1)
      double a = (double) n + 1.0;
      return ROOT::Math::gamma_cdf_c(mu, a, 1.0);
   }

   
   double binomial_cdf_c(unsigned int k, double p, unsigned int n)
   {
      // use relation with in beta distribution
      // p must be 0 <=p <= 1
      if ( k >= n)   return 0;
      double a = (double) k + 1.0;
      double b = (double) n - k;
      return ROOT::Math::beta_cdf(p, a, b);
   }

   
   double binomial_cdf(unsigned int k, double p, unsigned int n)
   {
      // use relation with in beta distribution
      // p must be 0 <=p <= 1
      if ( k >= n) return 1.0;

      double a = (double) k + 1.0;
      double b = (double) n - k;
      return ROOT::Math::beta_cdf_c(p, a, b);
   }

   
   double negative_binomial_cdf(unsigned int k, double p, double n)
   {
      // use relation with in beta distribution
      // p must be 0 <=p <= 1
      if (n < 0)          return 0;
      if (p < 0 || p > 1) return 0;
      return ROOT::Math::beta_cdf(p, n, k+1.0);
   }

   
   double negative_binomial_cdf_c(unsigned int k, double p, double n)
   {
      // use relation with in beta distribution
      // p must be 0 <=p <= 1
      if ( n < 0)          return 0;
      if ( p < 0 || p > 1) return 0;
      return ROOT::Math::beta_cdf_c(p, n, k+1.0);
   }


   double landau_cdf(double x, double xi, double x0)
   {
      // implementation of landau distribution (from DISLAN)
      //The algorithm was taken from the Cernlib function dislan(G110)
      //Reference: K.S.Kolbig and B.Schorr, "A program package for the Landau
      //distribution", Computer Phys.Comm., 31(1984), 97-111

      static double p1[5] = {0.2514091491e+0,-0.6250580444e-1, 0.1458381230e-1,-0.2108817737e-2, 0.7411247290e-3};
      static double q1[5] = {1.0            ,-0.5571175625e-2, 0.6225310236e-1,-0.3137378427e-2, 0.1931496439e-2};

      static double p2[4] = {0.2868328584e+0, 0.3564363231e+0, 0.1523518695e+0, 0.2251304883e-1};
      static double q2[4] = {1.0            , 0.6191136137e+0, 0.1720721448e+0, 0.2278594771e-1};

      static double p3[4] = {0.2868329066e+0, 0.3003828436e+0, 0.9950951941e-1, 0.8733827185e-2};
      static double q3[4] = {1.0            , 0.4237190502e+0, 0.1095631512e+0, 0.8693851567e-2};

      static double p4[4] = {0.1000351630e+1, 0.4503592498e+1, 0.1085883880e+2, 0.7536052269e+1};
      static double q4[4] = {1.0            , 0.5539969678e+1, 0.1933581111e+2, 0.2721321508e+2};

      static double p5[4] = {0.1000006517e+1, 0.4909414111e+2, 0.8505544753e+2, 0.1532153455e+3};
      static double q5[4] = {1.0            , 0.5009928881e+2, 0.1399819104e+3, 0.4200002909e+3};

      static double p6[4] = {0.1000000983e+1, 0.1329868456e+3, 0.9162149244e+3,-0.9605054274e+3};
      static double q6[4] = {1.0            , 0.1339887843e+3, 0.1055990413e+4, 0.5532224619e+3};

      static double a1[4] = {0              ,-0.4583333333e+0, 0.6675347222e+0,-0.1641741416e+1};
      static double a2[4] = {0              , 1.0            ,-0.4227843351e+0,-0.2043403138e+1};
 
      double v = (x - x0)/xi;
      double u;
      double lan;

      if (v < -5.5)
      {
         u   = std::exp(v+1);
         lan = 0.3989422803*std::exp(-1./u)*std::sqrt(u)*(1+(a1[1]+(a1[2]+a1[3]*u)*u)*u);
      }
      else if (v < -1 )
      {
         u   = std::exp(-v-1);
         lan = (std::exp(-u)/std::sqrt(u))*(p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*v)*v)*v)*v)/
                                           (q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*v)*v)*v)*v);
      }
      else if (v < 1)
         lan = (p2[0]+(p2[1]+(p2[2]+p2[3]*v)*v)*v)/(q2[0]+(q2[1]+(q2[2]+q2[3]*v)*v)*v);
      
      else if (v < 4)
         lan = (p3[0]+(p3[1]+(p3[2]+p3[3]*v)*v)*v)/(q3[0]+(q3[1]+(q3[2]+q3[3]*v)*v)*v);
      
      else if (v < 12)
      {
         u   = 1./v;
         lan = (p4[0]+(p4[1]+(p4[2]+p4[3]*u)*u)*u)/(q4[0]+(q4[1]+(q4[2]+q4[3]*u)*u)*u);
      }
      else if (v < 50)
      {
         u   = 1./v;
         lan = (p5[0]+(p5[1]+(p5[2]+p5[3]*u)*u)*u)/(q5[0]+(q5[1]+(q5[2]+q5[3]*u)*u)*u);
      }
      else if (v < 300)
      {
         u   = 1./v;
         lan = (p6[0]+(p6[1]+(p6[2]+p6[3]*u)*u)*u)/(q6[0]+(q6[1]+(q6[2]+q6[3]*u)*u)*u);
      }
      else
      {
         u   = 1./(v-v*std::log(v)/(v+1));
         lan = 1-(a2[1]+(a2[2]+a2[3]*u)*u)*u;
      }
      return lan;
   }


   double landau_xm1(double x, double xi, double x0)
   {
      // implementation of first momentum of Landau distribution
      // translated from Cernlib (XM1LAN function) by Benno List

      static double p1[5] = {-0.8949374280E+0, 0.4631783434E+0,-0.4053332915E-1,
                              0.1580075560E-1,-0.3423874194E-2};
      static double q1[5] = { 1.0            , 0.1002930749E+0, 0.3575271633E-1,
                             -0.1915882099E-2, 0.4811072364E-4};
      static double p2[5] = {-0.8933384046E+0, 0.1161296496E+0, 0.1200082940E+0,
                              0.2185699725E-1, 0.2128892058E-2};
      static double q2[5] = { 1.0            , 0.4935531886E+0, 0.1066347067E+0,
                              0.1250161833E-1, 0.5494243254E-3};
      static double p3[5] = {-0.8933322067E+0, 0.2339544896E+0, 0.8257653222E-1,
                              0.1411226998E-1, 0.2892240953E-3};
      static double q3[5] = { 1.0            , 0.3616538408E+0, 0.6628026743E-1,
                              0.4839298984E-2, 0.5248310361E-4};
      static double p4[4] = { 0.9358419425E+0, 0.6716831438E+2,-0.6765069077E+3,
                              0.9026661865E+3};
      static double q4[4] = { 1.0            , 0.7752562854E+2,-0.5637811998E+3,
                             -0.5513156752E+3};
      static double p5[4] = { 0.9489335583E+0, 0.5561246706E+3, 0.3208274617E+5,
                             -0.4889926524E+5};
      static double q5[4] = { 1.0            , 0.6028275940E+3, 0.3716962017E+5,
                              0.3686272898E+5};
      static double a0[6] = {-0.4227843351E+0,-0.1544313298E+0, 0.4227843351E+0,
                              0.3276496874E+1, 0.2043403138E+1,-0.8681296500E+1};
      static double a1[4] = { 0,              -0.4583333333E+0, 0.6675347222E+0,
                             -0.1641741416E+1};
      static double a2[5] = { 0,              -0.1958333333E+1, 0.5563368056E+1,
                             -0.2111352961E+2, 0.1006946266E+3};

      double v = (x-x0)/xi;
      double xm1lan;
      if (v < -4.5)
      {
         double u = std::exp(v+1);
         xm1lan   = v-u*(1+(a2[1]+(a2[2]+(a2[3]+a2[4]*u)*u)*u)*u)/
                                  (1+(a1[1]+(a1[2]+a1[3]*u)*u)*u);
      }
      else if (v < -2)
      {
         xm1lan   = (p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*v)*v)*v)*v)/
                    (q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*v)*v)*v)*v);
      }
      else if (v < 2)
      {
         xm1lan   = (p2[0]+(p2[1]+(p2[2]+(p2[3]+p2[4]*v)*v)*v)*v)/
                    (q2[0]+(q2[1]+(q2[2]+(q2[3]+q2[4]*v)*v)*v)*v);
      }
      else if (v < 10)
      {
         xm1lan   = (p3[0]+(p3[1]+(p3[2]+(p3[3]+p3[4]*v)*v)*v)*v)/
                    (q3[0]+(q3[1]+(q3[2]+(q3[3]+q3[4]*v)*v)*v)*v);
      }
      else if (v < 40)
      {
         double u = 1/v;
         xm1lan   = std::log(v)*(p4[0]+(p4[1]+(p4[2]+p4[3]*u)*u)*u)/
                                (q4[0]+(q4[1]+(q4[2]+q4[3]*u)*u)*u);
      }
      else if (v < 200)
      {
         double u = 1/v;
         xm1lan   = std::log(v)*(p5[0]+(p5[1]+(p5[2]+p5[3]*u)*u)*u)/
                                (q5[0]+(q5[1]+(q5[2]+q5[3]*u)*u)*u);
      }
      else
      {
         double u = v-v*std::log(v)/(v+1);
         v        = 1/(u-u*(u+ std::log(u)-v)/(u+1));
         u        = -std::log(v);
         xm1lan   = (u+a0[0]+(-u+a0[1]+(a0[2]*u+a0[3]+(a0[4]*u+a0[5])*v)*v)*v)/
                  (1-(1-(a0[2]+a0[4]*v)*v)*v);
      }
      return xm1lan*xi + x0;
   }



   double landau_xm2(double x, double xi, double x0)
   {
      // implementation of second momentum of Landau distribution
      // translated from Cernlib (XM2LAN function) by Benno List

      static double p1[5] = { 0.1169837582E+1,-0.4834874539E+0, 0.4383774644E+0,
                              0.3287175228E-2, 0.1879129206E-1};
      static double q1[5] = { 1.0            , 0.1795154326E+0, 0.4612795899E-1,
                              0.2183459337E-2, 0.7226623623E-4};
      static double p2[5] = { 0.1157939823E+1,-0.3842809495E+0, 0.3317532899E+0,
                              0.3547606781E-1, 0.6725645279E-2};
      static double q2[5] = { 1.0            , 0.2916824021E+0, 0.5259853480E-1,
                              0.3840011061E-2, 0.9950324173E-4};
      static double p3[4] = { 0.1178191282E+1, 0.1011623342E+2,-0.1285585291E+2,
                              0.3641361437E+2};
      static double q3[4] = { 1.0            , 0.8614160194E+1, 0.3118929630E+2,
                              0.1514351300E+0};
      static double p4[5] = { 0.1030763698E+1, 0.1216758660E+3, 0.1637431386E+4,
                             -0.2171466507E+4, 0.7010168358E+4};
      static double q4[5] = { 1.0            , 0.1022487911E+3, 0.1377646350E+4,
                              0.3699184961E+4, 0.4251315610E+4};
      static double p5[4] = { 0.1010084827E+1, 0.3944224824E+3, 0.1773025353E+5,
                             -0.7075963938E+5};
      static double q5[4] = { 1.0            , 0.3605950254E+3, 0.1392784158E+5,
                             -0.1881680027E+5};
      static double a0[7] = {-0.2043403138E+1,-0.8455686702E+0,-0.3088626596E+0,
                              0.5821346754E+1, 0.4227843351E+0, 0.6552993748E+1,
                             -0.1076714945E+2};
      static double a1[4] = { 0.             ,-0.4583333333E+0, 0.6675347222E+0,
                             -0.1641741416E+1};
      static double a2[4] = {-0.1958333333E+1, 0.5563368056E+1,-0.2111352961E+2,
                              0.1006946266E+3};
      static double a3[4] = {-1.0            , 0.4458333333E+1,-0.2116753472E+2,
                              0.1163674359E+3};

      double v    = (x-x0)/xi;
      double xm2lan;
      if (v < -4.5)
      {
         double u = std::exp(v+1);
         xm2lan   = v*v-2*u*u*
                    (v/u+a2[0]*v+a3[0]+(a2[1]*v+a3[1]+(a2[2]*v+a3[2]+
                    (a2[3]*v+a3[3])*u)*u)*u)/
                    (1+(a1[1]+(a1[2]+a1[3]*u)*u)*u);
      }
      else if (v < -2)
      {
         xm2lan   = (p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*v)*v)*v)*v)/
                    (q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*v)*v)*v)*v);
      }
      else if (v < 2)
      {
         xm2lan   = (p2[0]+(p2[1]+(p2[2]+(p2[3]+p2[4]*v)*v)*v)*v)/
                    (q2[0]+(q2[1]+(q2[2]+(q2[3]+q2[4]*v)*v)*v)*v);
      }
      else if (v < 5)
      {
         double u = 1/v;
         xm2lan   = v*(p3[0]+(p3[1]+(p3[2]+p3[3]*u)*u)*u)/
                      (q3[0]+(q3[1]+(q3[2]+q3[3]*u)*u)*u);
      }
      else if (v < 50)
      {
         double u = 1/v;
         xm2lan   = v*(p4[0]+(p4[1]+(p4[2]+(p4[3]+p4[4]*u)*u)*u)*u)/
                      (q4[0]+(q4[1]+(q4[2]+(q4[3]+q4[4]*u)*u)*u)*u);
      }
      else if (v < 200)
      {
         double u = 1/v;
         xm2lan   = v*(p5[0]+(p5[1]+(p5[2]+p5[3]*u)*u)*u)/
                      (q5[0]+(q5[1]+(q5[2]+q5[3]*u)*u)*u);
      }
      else
      {
         double u = v-v*std::log(v)/(v+1);
         v        = 1/(u-u*(u+log(u)-v)/(u+1));
         u        = -std::log(v);
         xm2lan   = (1/v+u*u+a0[0]+a0[1]*u+(-u*u+a0[2]*u+a0[3]+
                    (a0[4]*u*u+a0[5]*u+a0[6])*v)*v)/(1-(1-a0[4]*v)*v);
      }
      if (x0 == 0) return xm2lan*xi*xi;
      double xm1lan = ROOT::Math::landau_xm1(x, xi, x0);
      return xm2lan*xi*xi + (2*xm1lan-x0)*x0;
   }

} // namespace Math
} // namespace ROOT