TF1.h

Go to the documentation of this file.

// @(#)root/hist:$Id$
// Author: Rene Brun   18/08/95

/*************************************************************************
 * Copyright (C) 1995-2000, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/
// ---------------------------------- F1.h

#ifndef ROOT_TF1
#define ROOT_TF1

//////////////////////////////////////////////////////////////////////////
//                                                                      //
// TF1                                                                  //
//                                                                      //
// The Parametric 1-D function                                          //
//                                                                      //
//////////////////////////////////////////////////////////////////////////

#include "RConfigure.h"
#include <functional>
#include <cassert>
#include "TFormula.h"
#include "TAttLine.h"
#include "TAttFill.h"
#include "TAttMarker.h"
#include "TF1AbsComposition.h"
#include "TMath.h"
#include "Math/Types.h"
#include "Math/ParamFunctor.h"

class TF1;
class TH1;
class TAxis;
class TMethodCall;

namespace ROOT {
   namespace Fit {
      class FitResult;
   }
}

class TF1Parameters {
public:
   TF1Parameters() {} // needed for the I/O
   TF1Parameters(Int_t npar) :
      fParameters(std::vector<Double_t>(npar)),
      fParNames(std::vector<std::string>(npar))
   {
      for (int i = 0; i < npar; ++i) {
         fParNames[i] = std::string(TString::Format("p%d", i).Data());
      }
   }
   // copy constructor
   TF1Parameters(const TF1Parameters &rhs) :
      fParameters(rhs.fParameters),
      fParNames(rhs.fParNames)
   {}
   // assignment
   TF1Parameters &operator=(const TF1Parameters &rhs)
   {
      if (&rhs == this) return *this;
      fParameters = rhs.fParameters;
      fParNames = rhs.fParNames;
      return *this;
   }
   virtual ~TF1Parameters() {}

   // getter methods
   Double_t GetParameter(Int_t iparam) const
   {
      return (CheckIndex(iparam)) ? fParameters[iparam] : 0;
   }
   Double_t GetParameter(const char *name) const
   {
      return GetParameter(GetParNumber(name));
   }
   const Double_t *GetParameters() const
   {
      return fParameters.data();
   }
   const std::vector<double> &ParamsVec() const
   {
      return fParameters;
   }

   Int_t GetParNumber(const char *name) const;

   const char *GetParName(Int_t iparam) const
   {
      return (CheckIndex(iparam)) ? fParNames[iparam].c_str() : "";
   }


   // setter methods
   void   SetParameter(Int_t iparam, Double_t value)
   {
      if (!CheckIndex(iparam)) return;
      fParameters[iparam] = value;
   }
   void  SetParameters(const Double_t *params)
   {
      std::copy(params, params + fParameters.size(), fParameters.begin());
   }
   void  SetParameters(Double_t p0, Double_t p1, Double_t p2 = 0, Double_t p3 = 0, Double_t p4 = 0,
                       Double_t p5 = 0, Double_t p6 = 0, Double_t p7 = 0, Double_t p8 = 0,
                       Double_t p9 = 0, Double_t p10 = 0);

   void   SetParameter(const char *name, Double_t value)
   {
      SetParameter(GetParNumber(name), value);
   }
   void   SetParName(Int_t iparam, const char *name)
   {
      if (!CheckIndex(iparam)) return;
      fParNames[iparam] = std::string(name);
   }
   void   SetParNames(const char *name0 = "p0", const char *name1 = "p1", const char *name2 = "p2",
                      const char *name3 = "p3", const char *name4 = "p4", const char *name5 = "p5",
                      const char *name6 = "p6", const char *name7 = "p7", const char *name8 = "p8",
                      const char *name9 = "p9", const char *name10 = "p10");



   ClassDef(TF1Parameters, 1)  // The Parameters of a parameteric function
private:

   bool CheckIndex(Int_t i) const
   {
      return (i >= 0 && i < int(fParameters.size()));
   }

   std::vector<Double_t> fParameters;    // parameter values
   std::vector<std::string> fParNames;   // parameter names
};

namespace ROOT {
   namespace Internal {
      /// Internal class used by TF1 for defining
      /// template specialization for different TF1 constructors
      template<class Func>
      struct TF1Builder {
         static void Build(TF1 *f, Func func);
      };

      template<class Func>
      struct TF1Builder<Func *> {
         static void Build(TF1 *f, Func *func);
      };

      // Internal class used by TF1 for obtaining the type from a functor
      // out of the set of valid operator() signatures.
      template<typename T>
      struct GetFunctorType {
      };

      template<typename F, typename T>
      struct GetFunctorType<T(F::*)(const T *, const double *)> {
         using type = T;
      };

      template<typename F, typename T>
      struct GetFunctorType<T(F::*)(const T *, const double *) const> {
         using type = T;
      };

      template<typename F, typename T>
      struct GetFunctorType<T(F::*)(T *, double *)> {
         using type = T;
      };

      template<typename F, typename T>
      struct GetFunctorType<T(F::*)(T *, double *) const> {
         using type = T;
      };

      // Internal class used by TF1 to get the right operator() signature
      // from a Functor with several ones.
      template<typename T, typename F>
      auto GetTheRightOp(T(F::*opPtr)(const T *, const double *)) -> decltype(opPtr)
      {
         return opPtr;
      }

      template<typename T, typename F>
      auto GetTheRightOp(T(F::*opPtr)(const T *, const double *) const) -> decltype(opPtr)
      {
         return opPtr;
      }

      template<typename T, typename F>
      auto GetTheRightOp(T(F::*opPtr)(T *, double *)) -> decltype(opPtr)
      {
         return opPtr;
      }

      template<typename T, typename F>
      auto GetTheRightOp(T(F::*opPtr)(T *, double *) const) -> decltype(opPtr)
      {
         return opPtr;
      }
   }
}


class TF1 : public TNamed, public TAttLine, public TAttFill, public TAttMarker {

   template<class Func>
   friend struct ROOT::Internal::TF1Builder;

public:
   // Add to list behavior
   enum class EAddToList {
      kDefault,
      kAdd,
      kNo
   };

protected:

   struct TF1FunctorPointer {
      virtual  ~TF1FunctorPointer() {}
      virtual  TF1FunctorPointer * Clone() const = 0;
   };


   enum EFType {
      kFormula = 0,      // formula functions which can be stored,
      kPtrScalarFreeFcn, // pointer to scalar free function,
      kInterpreted,      // interpreted functions constructed by name,
      kTemplVec,         // vectorized free functions or TemplScalar functors evaluating on vectorized parameters,
      kTemplScalar,      // TemplScalar functors evaluating on scalar parameters
      kCompositionFcn
   }; // formula based on composition class (e.g. NSUM, CONV)

   Double_t    fXmin{-1111};        //Lower bounds for the range
   Double_t    fXmax{-1111};        //Upper bounds for the range
   Int_t       fNpar{};        //Number of parameters
   Int_t       fNdim{};        //Function dimension
   Int_t       fNpx{100};   //Number of points used for the graphical representation
   EFType      fType{EFType::kTemplScalar};
   Int_t       fNpfits{};    //Number of points used in the fit
   Int_t       fNDF{};       //Number of degrees of freedom in the fit
   Double_t    fChisquare{}; //Function fit chisquare
   Double_t    fMinimum{-1111};     //Minimum value for plotting
   Double_t    fMaximum{-1111};     //Maximum value for plotting
   std::vector<Double_t>    fParErrors;  //Array of errors of the fNpar parameters
   std::vector<Double_t>    fParMin;     //Array of lower limits of the fNpar parameters
   std::vector<Double_t>    fParMax;     //Array of upper limits of the fNpar parameters
   std::vector<Double_t>    fSave;       //Array of fNsave function values
   std::vector<Double_t>    fIntegral;   //!Integral of function binned on fNpx bins
   std::vector<Double_t>    fAlpha;      //!Array alpha. for each bin in x the deconvolution r of fIntegral
   std::vector<Double_t>    fBeta;       //!Array beta.  is approximated by x = alpha +beta*r *gamma*r**2
   std::vector<Double_t>    fGamma;      //!Array gamma.
   TObject     *fParent{nullptr};     //!Parent object hooking this function (if one)
   TH1         *fHistogram{nullptr};  //!Pointer to histogram used for visualisation
   TMethodCall *fMethodCall{nullptr}; //!Pointer to MethodCall in case of interpreted function
   Bool_t      fNormalized{false};    //Normalization option (false by default)
   Double_t    fNormIntegral{};        //Integral of the function before being normalized
   TF1FunctorPointer  *fFunctor{nullptr}; //! Functor object to wrap any C++ callable object
   TFormula    *fFormula{nullptr};        //Pointer to TFormula in case when user define formula
   TF1Parameters *fParams{nullptr};   //Pointer to Function parameters object (exists only for not-formula functions)
   std::unique_ptr<TF1AbsComposition> fComposition; //! Pointer to composition (NSUM or CONV)
   TF1AbsComposition *fComposition_ptr{nullptr};   // saved pointer (unique_ptr is transient)

   /// General constructor for TF1. Most of the other constructors delegate on it
   TF1(EFType functionType, const char *name, Double_t xmin, Double_t xmax, Int_t npar, Int_t ndim, EAddToList addToGlobList, TF1Parameters *params = nullptr, TF1FunctorPointer * functor = nullptr):
      TNamed(name, name), TAttLine(), TAttFill(), TAttMarker(), fXmin(xmin), fXmax(xmax), fNpar(npar), fNdim(ndim),
      fType(functionType), fParErrors(npar), fParMin(npar), fParMax(npar), fFunctor(functor), fParams(params)
   {
      DoInitialize(addToGlobList);
   };

private:
   // NSUM parsing helper functions
   void DefineNSUMTerm(TObjArray *newFuncs, TObjArray *coeffNames,
             TString &fullFormula,
             TString &formula, int termStart, int termEnd,
             Double_t xmin, Double_t xmax);
   int TermCoeffLength(TString &term);

protected:

   template <class T>
   struct TF1FunctorPointerImpl: TF1FunctorPointer {
      TF1FunctorPointerImpl(const ROOT::Math::ParamFunctorTempl<T> &func): fImpl(func) {};
      TF1FunctorPointerImpl(const std::function<T(const T *f, const Double_t *param)> &func) : fImpl(func){};
      virtual ~TF1FunctorPointerImpl() {}
      virtual  TF1FunctorPointer * Clone() const { return new TF1FunctorPointerImpl<T>(fImpl); }
      ROOT::Math::ParamFunctorTempl<T> fImpl;
   };




   static std::atomic<Bool_t> fgAbsValue;  //use absolute value of function when computing integral
   static Bool_t fgRejectPoint;  //True if point must be rejected in a fit
   static std::atomic<Bool_t> fgAddToGlobList; //True if we want to register the function in the global list
   static TF1   *fgCurrent;   //pointer to current function being processed


   //void CreateFromFunctor(const char *name, Int_t npar, Int_t ndim = 1);
   void DoInitialize(EAddToList addToGlobList);

   void IntegrateForNormalization();

   virtual Double_t GetMinMaxNDim(Double_t *x , Bool_t findmax, Double_t epsilon = 0, Int_t maxiter = 0) const;
   virtual void GetRange(Double_t *xmin, Double_t *xmax) const;
   virtual TH1 *DoCreateHistogram(Double_t xmin, Double_t xmax, Bool_t recreate = kFALSE);

public:

   // TF1 status bits
   enum EStatusBits {
      kNotGlobal   = BIT(10),  // don't register in global list of functions
      kNotDraw     = BIT(9)  // don't draw the function when in a TH1
   };

   TF1();
   TF1(const char *name, const char *formula, Double_t xmin = 0, Double_t xmax = 1, EAddToList addToGlobList = EAddToList::kDefault, bool vectorize = false);
   TF1(const char *name, const char *formula, Double_t xmin, Double_t xmax, Option_t * option);  // same as above but using a string for option
   TF1(const char *name, Double_t xmin, Double_t xmax, Int_t npar, Int_t ndim = 1, EAddToList addToGlobList = EAddToList::kDefault);
   TF1(const char *name, Double_t (*fcn)(Double_t *, Double_t *), Double_t xmin = 0, Double_t xmax = 1, Int_t npar = 0, Int_t ndim = 1, EAddToList addToGlobList = EAddToList::kDefault);
   TF1(const char *name, Double_t (*fcn)(const Double_t *, const Double_t *), Double_t xmin = 0, Double_t xmax = 1, Int_t npar = 0, Int_t ndim = 1, EAddToList addToGlobList = EAddToList::kDefault);

   template <class T>
   TF1(const char *name, std::function<T(const T *data, const Double_t *param)> &fcn, Double_t xmin = 0, Double_t xmax = 1, Int_t npar = 0, Int_t ndim = 1, EAddToList addToGlobList = EAddToList::kDefault):
      TF1(EFType::kTemplScalar, name, xmin, xmax, npar, ndim, addToGlobList, new TF1Parameters(npar), new TF1FunctorPointerImpl<T>(fcn))
   {
      fType = std::is_same<T, double>::value ? TF1::EFType::kTemplScalar : TF1::EFType::kTemplVec;
   }

   ////////////////////////////////////////////////////////////////////////////////
   /// Constructor using a pointer to function.
   ///
   /// \param npar is the number of free parameters used by the function
   ///
   /// This constructor creates a function of type C when invoked
   /// with the normal C++ compiler.
   ///
   ///
   /// WARNING! A function created with this constructor cannot be Cloned


   template <class T>
   TF1(const char *name, T(*fcn)(const T *, const Double_t *), Double_t xmin = 0, Double_t xmax = 1, Int_t npar = 0, Int_t ndim = 1, EAddToList addToGlobList = EAddToList::kDefault):
      TF1(EFType::kTemplVec, name, xmin, xmax, npar, ndim, addToGlobList, new TF1Parameters(npar), new TF1FunctorPointerImpl<T>(fcn))
   {}

   // Constructors using functors (compiled mode only)
   TF1(const char *name, ROOT::Math::ParamFunctor f, Double_t xmin = 0, Double_t xmax = 1, Int_t npar = 0, Int_t ndim = 1, EAddToList addToGlobList = EAddToList::kDefault);

   // Template constructors from any  C++ callable object,  defining  the operator() (double * , double *)
   // and returning a double.
   // The class name is not needed when using compile code, while it is required when using
   // interpreted code via the specialized constructor with void *.
   // An instance of the C++ function class or its pointer can both be used. The former is reccomended when using
   // C++ compiled code, but if CINT compatibility is needed, then a pointer to the function class must be used.
   // xmin and xmax specify the plotting range,  npar is the number of parameters.
   // See the tutorial math/exampleFunctor.C for an example of using this constructor
   template <typename Func>
   TF1(const char *name, Func f, Double_t xmin, Double_t xmax, Int_t npar, Int_t ndim = 1, EAddToList addToGlobList = EAddToList::kDefault) :
      TF1(EFType::kTemplScalar, name, xmin, xmax, npar, ndim, addToGlobList)
   {
      //actual fType set in TF1Builder
      ROOT::Internal::TF1Builder<Func>::Build(this, f);
   }

   // backward compatible interface
   template <typename Func>
   TF1(const char *name, Func f, Double_t xmin, Double_t xmax, Int_t npar, const char *, EAddToList addToGlobList = EAddToList::kDefault) :
      TF1(EFType::kTemplScalar, name, xmin, xmax, npar, 1, addToGlobList, new TF1Parameters(npar))
   {
      ROOT::Internal::TF1Builder<Func>::Build(this, f);
   }


   // Template constructors from a pointer to any C++ class of type PtrObj with a specific member function of type
   // MemFn.
   // The member function must have the signature of  (double * , double *) and returning a double.
   // The class name and the method name are not needed when using compile code
   // (the member function pointer is used in this case), while they are required when using interpreted
   // code via the specialized constructor with void *.
   // xmin and xmax specify the plotting range,  npar is the number of parameters.
   // See the tutorial math/exampleFunctor.C for an example of using this constructor
   template <class PtrObj, typename MemFn>
   TF1(const char *name, const  PtrObj &p, MemFn memFn, Double_t xmin, Double_t xmax, Int_t npar, Int_t ndim = 1, EAddToList addToGlobList = EAddToList::kDefault) :
      TF1(EFType::kTemplScalar, name, xmin, xmax, npar, ndim, addToGlobList, new TF1Parameters(npar), new TF1FunctorPointerImpl<double>(ROOT::Math::ParamFunctor(p, memFn)))
   {}

   // backward compatible interface
   template <class PtrObj, typename MemFn>
   TF1(const char *name, const  PtrObj &p, MemFn memFn, Double_t xmin, Double_t xmax, Int_t npar, const char *, const char *, EAddToList addToGlobList = EAddToList::kDefault) :
      TF1(EFType::kTemplScalar, name, xmin, xmax, npar, 1, addToGlobList, new TF1Parameters(npar), new TF1FunctorPointerImpl<double>(ROOT::Math::ParamFunctor(p, memFn)))
   {}

   TF1(const TF1 &f1);
   TF1 &operator=(const TF1 &rhs);
   virtual   ~TF1();
   virtual void     AddParameter(const TString &name, Double_t value)
   {
      if (fFormula) fFormula->AddParameter(name, value);
   }
   // virtual void     AddParameters(const pair<TString,Double_t> *pairs, Int_t size) { fFormula->AddParameters(pairs,size); }
   // virtual void     AddVariable(const TString &name, Double_t value = 0) { if (fFormula) fFormula->AddVariable(name,value); }
   // virtual void     AddVariables(const TString *vars, Int_t size) { if (fFormula) fFormula->AddVariables(vars,size); }
   virtual Bool_t   AddToGlobalList(Bool_t on = kTRUE);
   static  Bool_t   DefaultAddToGlobalList(Bool_t on = kTRUE);
   virtual void     Browse(TBrowser *b);
   virtual void     Copy(TObject &f1) const;
   TObject*         Clone(const char* newname=0) const;
   virtual Double_t Derivative(Double_t x, Double_t *params = 0, Double_t epsilon = 0.001) const;
   virtual Double_t Derivative2(Double_t x, Double_t *params = 0, Double_t epsilon = 0.001) const;
   virtual Double_t Derivative3(Double_t x, Double_t *params = 0, Double_t epsilon = 0.001) const;
   static  Double_t DerivativeError();
   virtual Int_t    DistancetoPrimitive(Int_t px, Int_t py);
   virtual void     Draw(Option_t *option = "");
   virtual TF1     *DrawCopy(Option_t *option = "") const;
   virtual TObject *DrawDerivative(Option_t *option = "al"); // *MENU*
   virtual TObject *DrawIntegral(Option_t *option = "al"); // *MENU*
   virtual void     DrawF1(Double_t xmin, Double_t xmax, Option_t *option = "");
   virtual Double_t Eval(Double_t x, Double_t y = 0, Double_t z = 0, Double_t t = 0) const;
   //template <class T> T Eval(T x, T y = 0, T z = 0, T t = 0) const;
   virtual Double_t EvalPar(const Double_t *x, const Double_t *params = 0);
   template <class T> T EvalPar(const T *x, const Double_t *params = 0);
   virtual Double_t operator()(Double_t x, Double_t y = 0, Double_t z = 0, Double_t t = 0) const;
   template <class T> T operator()(const T *x, const Double_t *params = nullptr);
   virtual void     ExecuteEvent(Int_t event, Int_t px, Int_t py);
   virtual void     FixParameter(Int_t ipar, Double_t value);
   bool      IsVectorized()
   {
      return (fType == EFType::kTemplVec) || (fType == EFType::kFormula && fFormula && fFormula->IsVectorized());
   }
   Double_t     GetChisquare() const
   {
      return fChisquare;
   }
   virtual TH1     *GetHistogram() const;
   virtual TH1     *CreateHistogram()
   {
      return DoCreateHistogram(fXmin, fXmax);
   }
   virtual TFormula *GetFormula()
   {
      return fFormula;
   }
   virtual const TFormula *GetFormula() const
   {
      return fFormula;
   }
   virtual TString  GetExpFormula(Option_t *option = "") const
   {
      return (fFormula) ? fFormula->GetExpFormula(option) : "";
   }
   virtual const TObject *GetLinearPart(Int_t i) const
   {
      return (fFormula) ? fFormula->GetLinearPart(i) : nullptr;
   }
   virtual Double_t GetMaximum(Double_t xmin = 0, Double_t xmax = 0, Double_t epsilon = 1.E-10, Int_t maxiter = 100, Bool_t logx = false) const;
   virtual Double_t GetMinimum(Double_t xmin = 0, Double_t xmax = 0, Double_t epsilon = 1.E-10, Int_t maxiter = 100, Bool_t logx = false) const;
   virtual Double_t GetMaximumX(Double_t xmin = 0, Double_t xmax = 0, Double_t epsilon = 1.E-10, Int_t maxiter = 100, Bool_t logx = false) const;
   virtual Double_t GetMinimumX(Double_t xmin = 0, Double_t xmax = 0, Double_t epsilon = 1.E-10, Int_t maxiter = 100, Bool_t logx = false) const;
   virtual Double_t GetMaximumStored() const
   {
      return fMaximum;
   }
   virtual Double_t GetMinimumStored() const
   {
      return fMinimum;
   }
   virtual Int_t    GetNpar() const
   {
      return fNpar;
   }
   virtual Int_t    GetNdim() const
   {
      return fNdim;
   }
   virtual Int_t    GetNDF() const;
   virtual Int_t    GetNpx() const
   {
      return fNpx;
   }
   TMethodCall    *GetMethodCall() const
   {
      return fMethodCall;
   }
   virtual Int_t    GetNumber() const
   {
      return (fFormula) ? fFormula->GetNumber() : 0;
   }
   virtual Int_t    GetNumberFreeParameters() const;
   virtual Int_t    GetNumberFitPoints() const
   {
      return fNpfits;
   }
   virtual char    *GetObjectInfo(Int_t px, Int_t py) const;
   TObject    *GetParent() const
   {
      return fParent;
   }
   virtual Double_t GetParameter(Int_t ipar) const
   {
      return (fFormula) ? fFormula->GetParameter(ipar) : fParams->GetParameter(ipar);
   }
   virtual Double_t GetParameter(const TString &name)  const
   {
      return (fFormula) ? fFormula->GetParameter(name) : fParams->GetParameter(name);
   }
   virtual Double_t *GetParameters() const
   {
      return (fFormula) ? fFormula->GetParameters() : const_cast<Double_t *>(fParams->GetParameters());
   }
   virtual void     GetParameters(Double_t *params)
   {
      if (fFormula) fFormula->GetParameters(params);
      else std::copy(fParams->ParamsVec().begin(), fParams->ParamsVec().end(), params);
   }
   virtual const char *GetParName(Int_t ipar) const
   {
      return (fFormula) ? fFormula->GetParName(ipar) : fParams->GetParName(ipar);
   }
   virtual Int_t    GetParNumber(const char *name) const
   {
      return (fFormula) ? fFormula->GetParNumber(name) : fParams->GetParNumber(name);
   }
   virtual Double_t GetParError(Int_t ipar) const;
   virtual const Double_t *GetParErrors() const
   {
      return fParErrors.data();
   }
   virtual void     GetParLimits(Int_t ipar, Double_t &parmin, Double_t &parmax) const;
   virtual Double_t GetProb() const;
   virtual Int_t    GetQuantiles(Int_t nprobSum, Double_t *q, const Double_t *probSum);
   virtual Double_t GetRandom();
   virtual Double_t GetRandom(Double_t xmin, Double_t xmax);
   virtual void     GetRange(Double_t &xmin, Double_t &xmax) const;
   virtual void     GetRange(Double_t &xmin, Double_t &ymin, Double_t &xmax, Double_t &ymax) const;
   virtual void     GetRange(Double_t &xmin, Double_t &ymin, Double_t &zmin, Double_t &xmax, Double_t &ymax, Double_t &zmax) const;
   virtual Double_t GetSave(const Double_t *x);
   virtual Double_t GetX(Double_t y, Double_t xmin = 0, Double_t xmax = 0, Double_t epsilon = 1.E-10, Int_t maxiter = 100, Bool_t logx = false) const;
   virtual Double_t GetXmin() const
   {
      return fXmin;
   }
   virtual Double_t GetXmax() const
   {
      return fXmax;
   }
   TAxis           *GetXaxis() const ;
   TAxis           *GetYaxis() const ;
   TAxis           *GetZaxis() const ;
   virtual Double_t GetVariable(const TString &name)
   {
      return (fFormula) ? fFormula->GetVariable(name) : 0;
   }
   virtual Double_t GradientPar(Int_t ipar, const Double_t *x, Double_t eps = 0.01);
   template <class T>
   T GradientPar(Int_t ipar, const T *x, Double_t eps = 0.01);
   template <class T>
   T GradientParTempl(Int_t ipar, const T *x, Double_t eps = 0.01);

   virtual void     GradientPar(const Double_t *x, Double_t *grad, Double_t eps = 0.01);
   template <class T>
   void GradientPar(const T *x, T *grad, Double_t eps = 0.01);
   template <class T>
   void GradientParTempl(const T *x, T *grad, Double_t eps = 0.01);

   virtual void     InitArgs(const Double_t *x, const Double_t *params);
   static  void     InitStandardFunctions();
   virtual Double_t Integral(Double_t a, Double_t b, Double_t epsrel = 1.e-12);
   virtual Double_t IntegralOneDim(Double_t a, Double_t b, Double_t epsrel, Double_t epsabs, Double_t &err);
   virtual Double_t IntegralError(Double_t a, Double_t b, const Double_t *params = 0, const Double_t *covmat = 0, Double_t epsilon = 1.E-2);
   virtual Double_t IntegralError(Int_t n, const Double_t *a, const Double_t *b, const Double_t *params = 0, const Double_t *covmat = 0, Double_t epsilon = 1.E-2);
   // virtual Double_t IntegralFast(const TGraph *g, Double_t a, Double_t b, Double_t *params=0);
   virtual Double_t IntegralFast(Int_t num, Double_t *x, Double_t *w, Double_t a, Double_t b, Double_t *params = 0, Double_t epsilon = 1e-12);
   virtual Double_t IntegralMultiple(Int_t n, const Double_t *a, const Double_t *b, Int_t maxpts, Double_t epsrel, Double_t epsabs , Double_t &relerr, Int_t &nfnevl, Int_t &ifail);
   virtual Double_t IntegralMultiple(Int_t n, const Double_t *a, const Double_t *b, Int_t /*minpts*/, Int_t maxpts, Double_t epsrel, Double_t &relerr, Int_t &nfnevl, Int_t &ifail)
   {
      return  IntegralMultiple(n, a, b, maxpts, epsrel, epsrel, relerr, nfnevl, ifail);
   }
   virtual Double_t IntegralMultiple(Int_t n, const Double_t *a, const Double_t *b, Double_t epsrel, Double_t &relerr);
   virtual Bool_t   IsEvalNormalized() const
   {
      return fNormalized;
   }
   /// return kTRUE if the point is inside the function range
   virtual Bool_t   IsInside(const Double_t *x) const
   {
      return !((x[0] < fXmin) || (x[0] > fXmax));
   }
   virtual Bool_t   IsLinear() const
   {
      return (fFormula) ? fFormula->IsLinear() : false;
   }
   virtual Bool_t   IsValid() const;
   virtual void     Print(Option_t *option = "") const;
   virtual void     Paint(Option_t *option = "");
   virtual void     ReleaseParameter(Int_t ipar);
   virtual void     Save(Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Double_t zmin, Double_t zmax);
   virtual void     SavePrimitive(std::ostream &out, Option_t *option = "");
   virtual void     SetChisquare(Double_t chi2)
   {
      fChisquare = chi2;
   }
   virtual void     SetFitResult(const ROOT::Fit::FitResult &result, const Int_t *indpar = 0);
   template <class PtrObj, typename MemFn>
   void SetFunction(PtrObj &p, MemFn memFn);
   template <typename Func>
   void SetFunction(Func f);
   virtual void     SetMaximum(Double_t maximum = -1111); // *MENU*
   virtual void     SetMinimum(Double_t minimum = -1111); // *MENU*
   virtual void     SetNDF(Int_t ndf);
   virtual void     SetNumberFitPoints(Int_t npfits)
   {
      fNpfits = npfits;
   }
   virtual void     SetNormalized(Bool_t flag)
   {
      fNormalized = flag;
      Update();
   }
   virtual void     SetNpx(Int_t npx = 100); // *MENU*
   virtual void     SetParameter(Int_t param, Double_t value)
   {
      (fFormula) ? fFormula->SetParameter(param, value) : fParams->SetParameter(param, value);
      Update();
   }
   virtual void     SetParameter(const TString &name, Double_t value)
   {
      (fFormula) ? fFormula->SetParameter(name, value) : fParams->SetParameter(name, value);
      Update();
   }
   virtual void     SetParameters(const Double_t *params)
   {
      (fFormula) ? fFormula->SetParameters(params) : fParams->SetParameters(params);
      Update();
   }
   virtual void     SetParameters(Double_t p0, Double_t p1, Double_t p2 = 0, Double_t p3 = 0, Double_t p4 = 0,
                                  Double_t p5 = 0, Double_t p6 = 0, Double_t p7 = 0, Double_t p8 = 0,
                                  Double_t p9 = 0, Double_t p10 = 0)
   {
      if (fFormula) fFormula->SetParameters(p0, p1, p2, p3, p4, p5, p6, p7, p8, p9, p10);
      else          fParams->SetParameters(p0, p1, p2, p3, p4, p5, p6, p7, p8, p9, p10);
      Update();
   } // *MENU*
   virtual void     SetParName(Int_t ipar, const char *name);
   virtual void     SetParNames(const char *name0 = "p0", const char *name1 = "p1", const char *name2 = "p2",
                                const char *name3 = "p3", const char *name4 = "p4", const char *name5 = "p5",
                                const char *name6 = "p6", const char *name7 = "p7", const char *name8 = "p8",
                                const char *name9 = "p9", const char *name10 = "p10"); // *MENU*
   virtual void     SetParError(Int_t ipar, Double_t error);
   virtual void     SetParErrors(const Double_t *errors);
   virtual void     SetParLimits(Int_t ipar, Double_t parmin, Double_t parmax);
   virtual void     SetParent(TObject *p = 0)
   {
      fParent = p;
   }
   virtual void     SetRange(Double_t xmin, Double_t xmax); // *MENU*
   virtual void     SetRange(Double_t xmin, Double_t ymin,  Double_t xmax, Double_t ymax);
   virtual void     SetRange(Double_t xmin, Double_t ymin, Double_t zmin,  Double_t xmax, Double_t ymax, Double_t zmax);
   virtual void     SetSavedPoint(Int_t point, Double_t value);
   virtual void     SetTitle(const char *title = ""); // *MENU*
   virtual void     SetVectorized(Bool_t vectorized)
   {
      if (fType == EFType::kFormula && fFormula)
         fFormula->SetVectorized(vectorized);
      else
         Warning("SetVectorized", "Can only set vectorized flag on formula-based TF1");
   }
   virtual void     Update();

   static  TF1     *GetCurrent();
   static  void     AbsValue(Bool_t reject = kTRUE);
   static  void     RejectPoint(Bool_t reject = kTRUE);
   static  Bool_t   RejectedPoint();
   static  void     SetCurrent(TF1 *f1);

   //Moments
   virtual Double_t Moment(Double_t n, Double_t a, Double_t b, const Double_t *params = 0, Double_t epsilon = 0.000001);
   virtual Double_t CentralMoment(Double_t n, Double_t a, Double_t b, const Double_t *params = 0, Double_t epsilon = 0.000001);
   virtual Double_t Mean(Double_t a, Double_t b, const Double_t *params = 0, Double_t epsilon = 0.000001)
   {
      return Moment(1, a, b, params, epsilon);
   }
   virtual Double_t Variance(Double_t a, Double_t b, const Double_t *params = 0, Double_t epsilon = 0.000001)
   {
      return CentralMoment(2, a, b, params, epsilon);
   }

   //some useful static utility functions to compute sampling points for Integral
   //static  void     CalcGaussLegendreSamplingPoints(TGraph *g, Double_t eps=3.0e-11);
   //static  TGraph  *CalcGaussLegendreSamplingPoints(Int_t num=21, Double_t eps=3.0e-11);
   static  void     CalcGaussLegendreSamplingPoints(Int_t num, Double_t *x, Double_t *w, Double_t eps = 3.0e-11);

private:
   template <class T>
   T EvalParTempl(const T *data, const Double_t *params = 0);

#ifdef R__HAS_VECCORE
   inline double EvalParVec(const Double_t *data, const Double_t *params);
#endif

   ClassDef(TF1, 10) // The Parametric 1-D function
};

namespace ROOT {
   namespace Internal {

      template<class Func>
      void TF1Builder<Func>::Build(TF1 *f, Func func)
      {
         using Fnc_t = typename ROOT::Internal::GetFunctorType<decltype(ROOT::Internal::GetTheRightOp(&Func::operator()))>::type;
         f->fType = std::is_same<Fnc_t, double>::value? TF1::EFType::kTemplScalar : TF1::EFType::kTemplVec;
         f->fFunctor = new TF1::TF1FunctorPointerImpl<Fnc_t>(ROOT::Math::ParamFunctorTempl<Fnc_t>(func));
         f->fParams = new TF1Parameters(f->fNpar);
      }

      template<class Func>
      void TF1Builder<Func *>::Build(TF1 *f, Func *func)
      {
         using Fnc_t = typename ROOT::Internal::GetFunctorType<decltype(ROOT::Internal::GetTheRightOp(&Func::operator()))>::type;
         f->fType = std::is_same<Fnc_t, double>::value? TF1::EFType::kTemplScalar : TF1::EFType::kTemplVec;
         f->fFunctor = new TF1::TF1FunctorPointerImpl<Fnc_t>(ROOT::Math::ParamFunctorTempl<Fnc_t>(func));
         f->fParams = new TF1Parameters(f->fNpar);
      }

      /// TF1 building from a string
      /// used to build a TFormula based on a lambda function
      template<>
      struct TF1Builder<const char *> {
         static void Build(TF1 *f, const char *formula)
         {
            f->fType = TF1::EFType::kFormula;
            f->fFormula = new TFormula("tf1lambda", formula, f->fNdim, f->fNpar, false);
            TString formulaExpression(formula);
            Ssiz_t first = formulaExpression.Index("return") + 7;
            Ssiz_t last  = formulaExpression.Last(';');
            TString title = formulaExpression(first, last - first);
            f->SetTitle(title);
         }
      };
   }
}

inline Double_t TF1::operator()(Double_t x, Double_t y, Double_t z, Double_t t) const
{
   return Eval(x, y, z, t);
}

template <class T>
inline T TF1::operator()(const T *x, const Double_t *params)
{
   return EvalPar(x, params);
}

////////////////////////////////////////////////////////////////////////////////
///   EvalPar for vectorized
template <class T>
T TF1::EvalPar(const T *x, const Double_t *params)
{
  if (fType == EFType::kTemplVec || fType == EFType::kTemplScalar) {
     return EvalParTempl(x, params);
  } else if (fType == EFType::kFormula) {
     return fFormula->EvalPar(x, params);
  } else
     return TF1::EvalPar((double *)x, params);
}

////////////////////////////////////////////////////////////////////////////////
///   Eval for vectorized functions
// template <class T>
// T TF1::Eval(T x, T y, T z, T t) const
// {
//    if (fType == EFType::kFormula)
//       return fFormula->Eval(x, y, z, t);

//    T xx[] = {x, y, z, t};
//    Double_t *pp = (Double_t *)fParams->GetParameters();
//    return ((TF1 *)this)->EvalPar(xx, pp);
// }

// Internal to TF1. Evaluates Templated interfaces
template <class T>
inline T TF1::EvalParTempl(const T *data, const Double_t *params)
{
   assert(fType == EFType::kTemplScalar || fType == EFType::kTemplVec);
   if (!params) params = (Double_t *)fParams->GetParameters();
   if (fFunctor)
      return ((TF1FunctorPointerImpl<T> *)fFunctor)->fImpl(data, params);

   // this should throw an error
   // we nned to implement a vectorized GetSave(x)
   return TMath::SignalingNaN();
}

#ifdef R__HAS_VECCORE
// Internal to TF1. Evaluates Vectorized TF1 on data of type Double_v
inline double TF1::EvalParVec(const Double_t *data, const Double_t *params)
{
   assert(fType == EFType::kTemplVec);
   std::vector<ROOT::Double_v> d(fNdim);
   ROOT::Double_v res;

   for(auto i=0; i<fNdim; i++) {
      d[i] = ROOT::Double_v(data[i]);
   }

   if (fFunctor) {
      res = ((TF1FunctorPointerImpl<ROOT::Double_v> *) fFunctor)->fImpl(d.data(), params);
   } else {
      //    res = GetSave(x);
      return TMath::SignalingNaN();
   }
   return vecCore::Get<ROOT::Double_v>(res, 0);
}
#endif

inline void TF1::SetRange(Double_t xmin, Double_t,  Double_t xmax, Double_t)
{
   TF1::SetRange(xmin, xmax);
}
inline void TF1::SetRange(Double_t xmin, Double_t, Double_t,  Double_t xmax, Double_t, Double_t)
{
   TF1::SetRange(xmin, xmax);
}

template <typename Func>
void TF1::SetFunction(Func f)
{
   // set function from a generic C++ callable object
   fType = EFType::kPtrScalarFreeFcn;
   fFunctor = new TF1::TF1FunctorPointerImpl<double>(ROOT::Math::ParamFunctor(f));
}
template <class PtrObj, typename MemFn>
void TF1::SetFunction(PtrObj &p, MemFn memFn)
{
   // set from a pointer to a member function
   fType = EFType::kPtrScalarFreeFcn;
   fFunctor = new TF1::TF1FunctorPointerImpl<double>(ROOT::Math::ParamFunctor(p, memFn));
}

template <class T>
inline T TF1::GradientPar(Int_t ipar, const T *x, Double_t eps)
{
   if (fType == EFType::kTemplVec || fType == EFType::kTemplScalar) {
      return GradientParTempl<T>(ipar, x, eps);
   } else
      return GradientParTempl<Double_t>(ipar, (const Double_t *)x, eps);
}

template <class T>
inline T TF1::GradientParTempl(Int_t ipar, const T *x, Double_t eps)
{
   if (GetNpar() == 0)
      return 0;

   if (eps < 1e-10 || eps > 1) {
      Warning("Derivative", "parameter esp=%g out of allowed range[1e-10,1], reset to 0.01", eps);
      eps = 0.01;
   }
   Double_t h;
   TF1 *func = (TF1 *)this;
   Double_t *parameters = GetParameters();

   // Copy parameters for thread safety
   std::vector<Double_t> parametersCopy(parameters, parameters + GetNpar());
   parameters = parametersCopy.data();

   Double_t al, bl, h2;
   T f1, f2, g1, g2, d0, d2;

   ((TF1 *)this)->GetParLimits(ipar, al, bl);
   if (al * bl != 0 && al >= bl) {
      // this parameter is fixed
      return 0;
   }

   // check if error has been computer (is not zero)
   if (func->GetParError(ipar) != 0)
      h = eps * func->GetParError(ipar);
   else
      h = eps;

   // save original parameters
   Double_t par0 = parameters[ipar];

   parameters[ipar] = par0 + h;
   f1 = func->EvalPar(x, parameters);
   parameters[ipar] = par0 - h;
   f2 = func->EvalPar(x, parameters);
   parameters[ipar] = par0 + h / 2;
   g1 = func->EvalPar(x, parameters);
   parameters[ipar] = par0 - h / 2;
   g2 = func->EvalPar(x, parameters);

   // compute the central differences
   h2 = 1 / (2. * h);
   d0 = f1 - f2;
   d2 = 2 * (g1 - g2);

   T grad = h2 * (4 * d2 - d0) / 3.;

   // restore original value
   parameters[ipar] = par0;

   return grad;
}

template <class T>
inline void TF1::GradientPar(const T *x, T *grad, Double_t eps)
{
   if (fType == EFType::kTemplVec || fType == EFType::kTemplScalar) {
      GradientParTempl<T>(x, grad, eps);
   } else
      GradientParTempl<Double_t>((const Double_t *)x, (Double_t *)grad, eps);
}

template <class T>
inline void TF1::GradientParTempl(const T *x, T *grad, Double_t eps)
{
   if (eps < 1e-10 || eps > 1) {
      Warning("Derivative", "parameter esp=%g out of allowed range[1e-10,1], reset to 0.01", eps);
      eps = 0.01;
   }

   for (Int_t ipar = 0; ipar < GetNpar(); ipar++) {
      grad[ipar] = GradientParTempl<T>(ipar, x, eps);
   }
}

#endif