TFormula.cxx

Go to the documentation of this file.

// @(#)root/hist:$Id$
// Author: Maciej Zimnoch 30/09/2013
 
/*************************************************************************
 * Copyright (C) 1995-2013, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/
 
#include "TROOT.h"
#include "TBuffer.h"
#include "TMethod.h"
#include "TF1.h"
#include "TMethodCall.h" 
#include "TError.h"
#include "TInterpreter.h"
#include "TInterpreterValue.h"
#include "TFormula.h"
#include "TRegexp.h"
 
#include "ROOT/StringUtils.hxx"
 
#include <functional>
#include <iomanip>
#include <iostream>
#include <limits>
#include <memory>
#include <set>
#include <stack>
#include <sstream>
#include <unordered_map>
 
using std::map, std::pair, std::make_pair, std::list, std::max, std::string;
 
#ifdef WIN32
#pragma optimize("",off)
#endif
#include "v5/TFormula.h"
 
 
namespace {
 
std::string doubleToString(double val)
{
   std::stringstream ss;
   ss << std::setprecision(std::numeric_limits<double>::max_digits10) << val;
   return ss.str();
}
 
bool areMatching(char opening, char closing)
{
   return (opening == '(' && closing == ')') || (opening == '{' && closing == '}') ||
          (opening == '[' && closing == ']') || (opening == '\"' && closing == '\"');
}
 
bool isBalanced(const TString &s)
{
   std::stack<char> i;
   auto sLength = s.Length();
   for (Ssiz_t c = 0; c < sLength; ++c) {
      if (s[c] == '[' || s[c] == '{' || s[c] == '(') {
         i.push(s[c]);
      } else if (s[c] == ']' || s[c] == '}' || s[c] == ')') {
         if (i.empty() || !areMatching(i.top(), s[c])) {
            Error("TFormula", "Found unbalanced char '%c' (expected closing of '%c') at index %d of '%s'", s[c], i.top(), c,
                  s.Data());
            return false;
         } else
            i.pop();
      } else if (s[c] == '\"') {
         if (i.empty())
            i.push(s[c]);
         else if (areMatching(i.top(), s[c]))
            i.pop();
         else {
            i.push(s[c]);
         }
      }
   }
   if (!i.empty()) {
      Error("TFormula", "String '%s' with %zu unbalanced chars.", s.Data(), i.size());
      return false;
   }
   return true;
}
 
// In the interpreter, we must match the SIMD width used by compiled ROOT.
// ROOT::Double_v aliases the best available native SIMD type, which may differ
// between compiled and interpreted contexts (e.g. with -march=native).
// Therefore, we explicitly select the fixed-size SIMD type corresponding to
// the native SIMD width used in compiled ROOT.
std::string vectorizedArgType()
{
#ifdef R__HAS_STD_EXPERIMENTAL_SIMD
   auto n = ROOT::Double_v::size();
   return "std::experimental::resize_simd_t<" + std::to_string(n) + ", ROOT::Double_v>";
#else
   // For other possible VecCore backends, we assume using the same type is fine.
   return "ROOT::Double_v";
#endif
}
 
} // namespace
 
/** \class TFormula  TFormula.h "inc/TFormula.h"
    \ingroup Hist
    The Formula class
 
    This is a new version of the TFormula class based on Cling.
    This class is not 100% backward compatible with the old TFormula class, which is still available in ROOT as
    `ROOT::v5::TFormula`. Some of the TFormula member functions available in version 5, such as
    `Analyze` and `AnalyzeFunction` are not available in the new TFormula.
    On the other hand formula expressions which were valid in version 5 are still valid in TFormula version 6
 
    This class has been implemented during Google Summer of Code 2013 by Maciej Zimnoch.
 
    ### Example of valid expressions:
 
    - `sin(x)/x`
    - `[0]*sin(x) + [1]*exp(-[2]*x)`
    - `x + y**2`
    - `x^2 + y^2`
    - `[0]*pow([1],4)`
    - `2*pi*sqrt(x/y)`
    - `gaus(0)*expo(3)  + ypol3(5)*x`
    - `gausn(0)*expo(3) + ypol3(5)*x`
    - `gaus(x, [0..2]) + expo(y, [3..4])`
 
    In the last examples above:
 
    - `gaus(0)` is a substitute for `[0]*exp(-0.5*((x-[1])/[2])**2)`
    and (0) means start numbering parameters at 0
    - `gausn(0)` is a substitute for `[0]*exp(-0.5*((x-[1])/[2])**2)/(sqrt(2*pi)*[2]))`
    and (0) means start numbering parameters at 0
    - `expo(3)` is a substitute for `exp([3]+[4]*x)`
    - `pol3(5)` is a substitute for `par[5]+par[6]*x+par[7]*x**2+par[8]*x**3`
    (`PolN` stands for Polynomial of degree N)
    - `gaus(x, [0..2])` is a more explicit way of writing `gaus(0)`
    - `expo(y, [3..4])` is a substitute for `exp([3]+[4]*y)`
 
    See below the [full list of predefined functions](\ref FormulaFuncs) which can be used as shortcuts in
    TFormula.
 
    `TMath` functions can be part of the expression, eg:
 
    - `TMath::Landau(x)*sin(x)`
    - `TMath::Erf(x)`
 
    Formula may contain constants, eg:
 
    - `sqrt2`
    - `e`
    - `pi`
    - `ln10`
    - `infinity`
 
    and more.
 
    Formulas may also contain other user-defined ROOT functions defined with a
    TFormula, eg, where `f1` is defined on one x-dimension and 2 parameters:
 
    - `f1(x, [omega], [phi])`
    - `f1([0..1])`
    - `f1([1], [0])`
    - `f1(y)`
 
    To replace only parameter names, the dimension variable can be dropped.
    Alternatively, to change only the dimension variable, the parameters can be
    dropped. Note that if a parameter is dropped or keeps its old name, its old
    value will be copied to the new function. The syntax used in the examples
    above also applies to the predefined parametrized functions like `gaus` and
    `expo`.
 
    Comparisons operators are also supported `(&&, ||, ==, <=, >=, !)`
 
    Examples:
 
    `sin(x*(x<0.5 || x>1))`
 
    If the result of a comparison is TRUE, the result is 1, otherwise 0.
 
    Already predefined names can be given. For example, if the formula
 
    `TFormula old("old",sin(x*(x<0.5 || x>1)))`
 
    one can assign a name to the formula. By default the name of the object = title = formula itself.
 
    `TFormula new("new","x*old")`
 
    is equivalent to:
 
    `TFormula new("new","x*sin(x*(x<0.5 || x>1))")`
 
    The class supports unlimited number of variables and parameters.
    By default the names which can be used for the variables are `x,y,z,t` or
    `x[0],x[1],x[2],x[3],....x[N]` for N-dimensional formulas.
 
    This class is not anymore the base class for the function classes `TF1`, but it has now
    a data member of TF1 which can be accessed via `TF1::GetFormula`.
 
    TFormula supports gradient and hessian calculations through clad.
    To calculate the gradient one needs to first declare a `CladStorage` of the
    same size as the number of parameters and then pass the variables and the
    created `CladStorage`:
 
    ```
    TFormula f("f", "x*[0] - y*[1]");
    Double_t p[] = {40, 30};
    Double_t x[] = {1, 2};
    f.SetParameters(p);
    TFormula::CladStorage grad(2);
    f.GradientPar(x, grad);
    ```
 
    The process is similar for hessians, except that the size of the created
    CladStorage should be the square of the number of parameters because
    `HessianPar` returns a flattened matrix:
 
    ```
    TFormula::CladStorage hess(4);
    f.HessianPar(x, hess);
    ```
 
    \anchor FormulaFuncs
    ### List of predefined functions
 
    The list of available predefined functions which can be used as shortcuts is the following:
    1. One Dimensional functions:
      - `gaus`  is a substitute for `[Constant]*exp(-0.5*((x-[Mean])/[Sigma])*((x-[Mean])/[Sigma]))`
      - `landau` is a substitute for `[Constant]*TMath::Landau (x,[MPV],[Sigma],false)`
      - `expo`  is a substitute for `exp([Constant]+[Slope]*x)`
      - `crystalball` is substitute for `[Constant]*ROOT::Math::crystalball_function (x,[Alpha],[N],[Sigma],[Mean])`
      - `breitwigner` is a substitute for `[p0]*ROOT::Math::breitwigner_pdf (x,[p2],[p1])`
      - `pol0,1,2,...N` is a substitute for a polynomial of degree `N` :
         `([p0]+[p1]*x+[p2]*pow(x,2)+....[pN]*pow(x,N)`
      - `cheb0,1,2,...N` is a substitute for a Chebyshev polynomial of degree `N`:
         `ROOT::Math::Chebyshev10(x,[p0],[p1],[p2],...[pN])`. Note the maximum N allowed here is 10.
    2. Two Dimensional functions:
      - `xygaus` is a substitute for `[Constant]*exp(-0.5*pow(((x-[MeanX])/[SigmaX]),2 )- 0.5*pow(((y-[MeanY])/[SigmaY]),2))`, a 2d Gaussian without correlation.
      - `bigaus` is a substitute for `[Constant]*ROOT::Math::bigaussian_pdf (x,y,[SigmaX],[SigmaY],[Rho],[MeanX],[MeanY])`, a 2d gaussian including a correlation parameter.
         Note the different order of parameters for bigaus: [Constant]=[0], [SigmaX]=[2], [SigmaY]=[4], [Rho]=[5], [MeanX]=[1], [MeanY]=[3]
    3. Three Dimensional functions:
      - `xyzgaus` is for a 3d Gaussians without correlations:
      `[Constant]*exp(-0.5*pow(((x-[MeanX])/[SigmaX]),2 )- 0.5*pow(((y-[MeanY])/[SigmaY]),2 )- 0.5*pow(((z-[MeanZ])/[SigmaZ]),2))`
 
 
    ### An expanded note on variables and parameters
 
    In a TFormula, a variable is a defined by a name `x`, `y`, `z` or `t` or an
    index like `x[0]`, `x[1]`, `x[2]`; that is `x[N]` where N is an integer.
 
    ```
    TFormula("", "x[0] * x[1] + 10")
    ```
 
    Parameters are similar and can take any name. It is specified using brackets
    e.g. `[expected_mass]` or `[0]`.
 
    ```
    TFormula("", "exp([expected_mass])-1")
    ```
 
    Variables and parameters can be combined in the same TFormula. Here we consider
    a very simple case where we have an exponential decay after some time t and a
    number of events with timestamps for which we want to evaluate this function.
 
    ```
    TFormula tf ("", "[0]*exp(-[1]*t)");
    tf.SetParameter(0, 1);
    tf.SetParameter(1, 0.5);
 
    for (auto & event : events) {
       tf.Eval(event.t);
    }
    ```
 
    The distinction between variables and parameters arose from the TFormula's
    application in fitting. There parameters are fitted to the data provided
    through variables. In other applications this distinction can go away.
 
    Parameter values can be provided dynamically using `TFormula::EvalPar`
    instead of `TFormula::Eval`. In this way parameters can be used identically
    to variables. See below for an example that uses only parameters to model a
    function.
 
    ```
    Int_t params[2] = {1, 2}; // {vel_x, vel_y}
    TFormula tf ("", "[vel_x]/sqrt(([vel_x + vel_y])**2)");
 
    tf.EvalPar(nullptr, params);
    ```
 
    ### A note on operators
 
    All operators of C/C++ are allowed in a TFormula with a few caveats.
 
    The operators `|`, `&`, `%` can be used but will raise an error if used in
    conjunction with a variable or a parameter. Variables and parameters are treated
    as doubles internally for which these operators are not defined.
    This means the following command will run successfully
       ```root -q -e TFormula("", "x+(10%3)").Eval(0)```
    but not
       ```root -q -e TFormula("", "x%10").Eval(0)```.
 
    The operator `^` is defined to mean exponentiation instead of the C/C++
    interpretation xor. `**` is added, also meaning exponentiation.
 
    The operators `++` and `@` are added, and are shorthand for the a linear
    function. That means the expression `x@2` will be expanded to
    ```[n]*x + [n+1]*2``` where n is the first previously unused parameter number.
 
    \class TFormulaFunction
    Helper class for TFormula
 
    \class TFormulaVariable
    Another helper class for TFormula
 
    \class TFormulaParamOrder
    Functor defining the parameter order
*/
 
// prefix used for function name passed to Cling
static const TString gNamePrefix = "TFormula__";
 
// static map of function pointers and expressions
//static std::unordered_map<std::string,  TInterpreter::CallFuncIFacePtr_t::Generic_t> gClingFunctions = std::unordered_map<TString,  TInterpreter::CallFuncIFacePtr_t::Generic_t>();
static std::unordered_map<std::string,  void *> gClingFunctions = std::unordered_map<std::string,  void * >();
 
static void R__v5TFormulaUpdater(Int_t nobjects, TObject **from, TObject **to)
{
   auto **fromv5 = (ROOT::v5::TFormula **)from;
   auto **target = (TFormula **)to;
 
   for (int i = 0; i < nobjects; ++i) {
      if (fromv5[i] && target[i]) {
         TFormula fnew(fromv5[i]->GetName(), fromv5[i]->GetExpFormula());
         *(target[i]) = fnew;
         target[i]->SetParameters(fromv5[i]->GetParameters());
      }
   }
}
 
using TFormulaUpdater_t = void (*)(Int_t nobjects, TObject **from, TObject **to);
bool R__SetClonesArrayTFormulaUpdater(TFormulaUpdater_t func);
 
int R__RegisterTFormulaUpdaterTrigger = R__SetClonesArrayTFormulaUpdater(R__v5TFormulaUpdater);
 
////////////////////////////////////////////////////////////////////////////////
Bool_t TFormula::IsOperator(const char c)
{
   // operator ":" must be handled separately
   const static std::set<char> ops {'+','^','-','/','*','<','>','|','&','!','=','?','%'};
   return ops.end() != ops.find(c);
}
 
////////////////////////////////////////////////////////////////////////////////
Bool_t TFormula::IsBracket(const char c)
{
   // Note that square brackets do not count as brackets here!!!
   char brackets[] = { ')','(','{','}'};
   Int_t bracketsLen = sizeof(brackets)/sizeof(char);
   for(Int_t i = 0; i < bracketsLen; ++i)
      if(brackets[i] == c)
         return true;
   return false;
}
 
////////////////////////////////////////////////////////////////////////////////
Bool_t TFormula::IsFunctionNameChar(const char c)
{
   return !IsBracket(c) && !IsOperator(c) && c != ',' && c != ' ';
}
 
////////////////////////////////////////////////////////////////////////////////
Bool_t TFormula::IsDefaultVariableName(const TString &name)
{
   return name == "x" || name == "z" || name == "y" || name == "t";
}
 
////////////////////////////////////////////////////////////////////////////////
Bool_t TFormula::IsScientificNotation(const TString & formula, int i)
{
   // check if the character at position i  is part of a scientific notation
   if ( (formula[i] == 'e' || formula[i] == 'E')  &&  (i > 0 && i <  formula.Length()-1) )  {
      // handle cases:  2e+3 2e-3 2e3 and 2.e+3
      if ( (isdigit(formula[i-1]) || formula[i-1] == '.') && ( isdigit(formula[i+1]) || formula[i+1] == '+' || formula[i+1] == '-' ) )
         return true;
   }
   return false;
}
 
////////////////////////////////////////////////////////////////////////////////
Bool_t TFormula::IsHexadecimal(const TString & formula, int i)
{
   // check if the character at position i  is part of a scientific notation
   if ( (formula[i] == 'x' || formula[i] == 'X')  &&  (i > 0 && i <  formula.Length()-1) && formula[i-1] == '0')  {
      if (isdigit(formula[i+1]) )
         return true;
      static char hex_values[12] = { 'a','A', 'b','B','c','C','d','D','e','E','f','F'};
      for (int jjj = 0; jjj < 12; ++jjj) {
         if (formula[i+1] == hex_values[jjj])
            return true;
      }
   }
   // else
   //    return false;
   //    // handle cases:  2e+3 2e-3 2e3 and 2.e+3
   //    if ( (isdigit(formula[i-1]) || formula[i-1] == '.') && ( isdigit(formula[i+1]) || formula[i+1] == '+' || formula[i+1] == '-' ) )
   //       return true;
   // }
   return false;
}
 
////////////////////////////////////////////////////////////////////////////
/// check if given position is in a parameter name i.e. within "[ ]"
Bool_t TFormula::IsAParameterName(const TString & formula, int pos) {
 
   Bool_t foundOpenParenthesis = false;
   if (pos == 0 || pos == formula.Length()-1) return false;
   for (int i = pos-1; i >=0; i--) {
      if (formula[i] == ']' ) return false;
      if (formula[i] == '[' ) {
         foundOpenParenthesis = true;
         break;
      }
   }
   if (!foundOpenParenthesis ) return false;
 
   // search after the position
   for (int i = pos+1; i < formula.Length(); i++) {
      if (formula[i] == ']' ) return true;
   }
   return false;
}
 
 
////////////////////////////////////////////////////////////////////////////////
bool TFormulaParamOrder::operator() (const TString& a, const TString& b) const {
   // implement comparison used to set parameter orders in TFormula
   // want p2 to be before p10
 
   // Returns true if (a < b), meaning a comes before b, and false if (a >= b)
 
   TRegexp numericPattern("p?[0-9]+");
   Ssiz_t len; // buffer to store length of regex match
 
   int patternStart = numericPattern.Index(a, &len);
   bool aNumeric = (patternStart == 0 && len == a.Length());
 
   patternStart = numericPattern.Index(b, &len);
   bool bNumeric = (patternStart == 0 && len == b.Length());
 
   if (aNumeric && !bNumeric)
      return true; // assume a (numeric) is always before b (not numeric)
   else if (!aNumeric && bNumeric)
      return false; // b comes before a
   else if (!aNumeric && !bNumeric)
      return a < b;
   else {
      int aInt = (a[0] == 'p') ? TString(a(1, a.Length())).Atoi() : a.Atoi();
      int bInt = (b[0] == 'p') ? TString(b(1, b.Length())).Atoi() : b.Atoi();
      return aInt < bInt;
   }
 
}
 
////////////////////////////////////////////////////////////////////////////////
void TFormula::ReplaceAllNames(TString &formula, map<TString, TString> &substitutions)
{
   /// Apply the name substitutions to the formula, doing all replacements in one pass
 
   for (int i = 0; i < formula.Length(); i++) {
      // start of name
      // (a little subtle, since we want to match names like "{V0}" and "[0]")
      if (isalpha(formula[i]) || formula[i] == '{' || formula[i] == '[') {
         int j; // index to end of name
         for (j = i + 1;
              j < formula.Length() && (IsFunctionNameChar(formula[j]) // square brackets are function name chars
                                       || (formula[i] == '{' && formula[j] == '}'));
              j++)
            ;
         TString name = (TString)formula(i, j - i);
 
         // std::cout << "Looking for name: " << name << std::endl;
 
         // if we find the name, do the substitution
         if (substitutions.find(name) != substitutions.end()) {
            formula.Replace(i, name.Length(), "(" + substitutions[name] + ")");
            i += substitutions[name].Length() + 2 - 1; // +2 for parentheses
            // std::cout << "made substitution: " << name << " to " << substitutions[name] << std::endl;
         } else if (isalpha(formula[i])) {
            // if formula[i] is alpha, can skip to end of candidate name, otherwise, we'll just
            // move one character ahead and try again
            i += name.Length() - 1;
         }
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
TFormula::TFormula()
{
   fName = "";
   fTitle = "";
   fClingInput = "";
   fReadyToExecute = false;
   fClingInitialized = false;
   fAllParametersSetted = false;
   fNdim = 0;
   fNpar = 0;
   fNumber = 0;
   fClingName = "";
   fFormula = "";
   fLambdaPtr = nullptr;
}
 
////////////////////////////////////////////////////////////////////////////////
static bool IsReservedName(const char* name)
{
   if (strlen(name)!=1) return false;
   for (auto const & specialName : {"x","y","z","t"}){
      if (strcmp(name,specialName)==0) return true;
   }
   return false;
}
 
////////////////////////////////////////////////////////////////////////////////
TFormula::~TFormula()
{
 
   // N.B. a memory leak may happen if user set bit after constructing the object,
   // Setting of bit should be done only internally
   if (!TestBit(TFormula::kNotGlobal) && gROOT ) {
      R__LOCKGUARD(gROOTMutex);
      gROOT->GetListOfFunctions()->Remove(this);
   }
 
   int nLinParts = fLinearParts.size();
   if (nLinParts > 0) {
      for (int i = 0; i < nLinParts; ++i) delete fLinearParts[i];
   }
}
 
////////////////////////////////////////////////////////////////////////////////
TFormula::TFormula(const char *name, const char *formula, bool addToGlobList, bool vectorize)   :
   TNamed(name,formula),
   fClingInput(formula),fFormula(formula)
{
   fReadyToExecute = false;
   fClingInitialized = false;
   fNdim = 0;
   fNpar = 0;
   fNumber = 0;
   fLambdaPtr = nullptr;
   fVectorized = vectorize;
#ifndef R__HAS_STD_EXPERIMENTAL_SIMD
   fVectorized = false;
#endif
 
   FillDefaults();
 
 
   //fName = gNamePrefix + name;  // is this needed
 
   // do not process null formulas.
   if (!fFormula.IsNull() ) {
      PreProcessFormula(fFormula);
 
      bool ok = PrepareFormula(fFormula);
      // if the formula has been correctly initialized add to the list of global functions
      if (ok) {
         if (addToGlobList && gROOT) {
            TFormula *old = nullptr;
            R__LOCKGUARD(gROOTMutex);
            old = dynamic_cast<TFormula *>(gROOT->GetListOfFunctions()->FindObject(name));
            if (old)
               gROOT->GetListOfFunctions()->Remove(old);
            if (IsReservedName(name))
               Error("TFormula", "The name %s is reserved as a TFormula variable name.\n", name);
            else
               gROOT->GetListOfFunctions()->Add(this);
         }
         SetBit(kNotGlobal,!addToGlobList);
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Constructor from a full compile-able C++ expression
 
TFormula::TFormula(const char *name, const char *formula, int ndim, int npar, bool addToGlobList)   :
   TNamed(name,formula),
   fClingInput(formula),fFormula(formula)
{
   fReadyToExecute = false;
   fClingInitialized = false;
   fNpar = 0;
   fNumber = 0;
   fLambdaPtr = nullptr;
   fFuncPtr = nullptr;
   fGradFuncPtr = nullptr;
   fHessFuncPtr = nullptr;
 
 
   fNdim = ndim;
   for (int i = 0; i < npar; ++i) {
      DoAddParameter(TString::Format("p%d",i), 0, false);
   }
   fAllParametersSetted = true;
   assert (fNpar == npar);
 
   bool ret = InitLambdaExpression(formula);
 
   if (ret)  {
 
      SetBit(TFormula::kLambda);
 
      fReadyToExecute = true;
 
      if (addToGlobList && gROOT) {
         TFormula *old = nullptr;
         R__LOCKGUARD(gROOTMutex);
         old = dynamic_cast<TFormula*> ( gROOT->GetListOfFunctions()->FindObject(name) );
         if (old)
            gROOT->GetListOfFunctions()->Remove(old);
         if (IsReservedName(name))
            Error("TFormula","The name %s is reserved as a TFormula variable name.\n",name);
         else
            gROOT->GetListOfFunctions()->Add(this);
      }
      SetBit(kNotGlobal,!addToGlobList);
   }
   else
      Error("TFormula","Syntax error in building the lambda expression %s", formula );
}
 
////////////////////////////////////////////////////////////////////////////////
TFormula::TFormula(const TFormula &formula) :
   TNamed(formula.GetName(),formula.GetTitle())
{
   formula.TFormula::Copy(*this);
 
   if (!TestBit(TFormula::kNotGlobal) && gROOT ) {
      R__LOCKGUARD(gROOTMutex);
      TFormula *old = (TFormula*)gROOT->GetListOfFunctions()->FindObject(formula.GetName());
      if (old)
         gROOT->GetListOfFunctions()->Remove(old);
 
      if (IsReservedName(formula.GetName())) {
         Error("TFormula","The name %s is reserved as a TFormula variable name.\n",formula.GetName());
      } else
         gROOT->GetListOfFunctions()->Add(this);
   }
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// = operator.
 
TFormula& TFormula::operator=(const TFormula &rhs)
{
   if (this != &rhs)
      rhs.TFormula::Copy(*this);
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
Bool_t TFormula::InitLambdaExpression(const char * formula) {
 
   std::string lambdaExpression = formula;
 
   // check if formula exist already in the map
   {
      R__LOCKGUARD(gROOTMutex);
 
      auto funcit = gClingFunctions.find(lambdaExpression);
      if (funcit != gClingFunctions.end() ) {
         fLambdaPtr = funcit->second;
         fClingInitialized = true;
         return true;
      }
   }
 
   // to be sure the interpreter is initialized
   ROOT::GetROOT();
   R__ASSERT(gInterpreter);
 
   // set the cling name using hash of the static formulae map
   auto hasher = gClingFunctions.hash_function();
   TString lambdaName = TString::Format("lambda__id%zu", hasher(lambdaExpression) );
 
   //lambdaExpression = TString::Format("[&](double * x, double *){ return %s ;}",formula);
   //TString lambdaName = TString::Format("mylambda_%s",GetName() );
   TString lineExpr = TString::Format("std::function<double(double*,double*)> %s = %s ;",lambdaName.Data(), lambdaExpression.c_str() );
   gInterpreter->ProcessLine(lineExpr);
   fLambdaPtr = (void*) gInterpreter->ProcessLine(TString(lambdaName)+TString(";"));  // add ; to avoid printing
   if (fLambdaPtr != nullptr) {
      R__LOCKGUARD(gROOTMutex);
      gClingFunctions.insert ( std::make_pair ( lambdaExpression, fLambdaPtr) );
      fClingInitialized = true;
      return true;
   }
   fClingInitialized = false;
   return false;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compile the given expression with Cling
/// backward compatibility method to be used in combination with the empty constructor
/// if no expression is given , the current stored formula (retrieved with GetExpFormula()) or the title  is used.
/// return 0 if the formula compilation is successful
 
Int_t TFormula::Compile(const char *expression)
{
   TString formula = expression;
   if (formula.IsNull() ) {
      formula = fFormula;
      if (formula.IsNull() ) formula = GetTitle();
   }
 
   if (formula.IsNull() ) return -1;
 
   // do not re-process if it was done before
   if (IsValid() && formula == fFormula ) return 0;
 
   // clear if a formula was already existing
   if (!fFormula.IsNull() ) Clear();
 
   fFormula = formula;
 
   if (TestBit(TFormula::kLambda) ) {
      bool ret = InitLambdaExpression(fFormula);
      return (ret) ? 0 : 1;
   }
 
   if (fVars.empty() ) FillDefaults();
   // prepare the formula for Cling
   //printf("compile: processing formula %s\n",fFormula.Data() );
   PreProcessFormula(fFormula);
   // pass formula in CLing
   bool ret = PrepareFormula(fFormula);
 
   return (ret) ? 0 : 1;
}
 
////////////////////////////////////////////////////////////////////////////////
void TFormula::Copy(TObject &obj) const
{
   TNamed::Copy(obj);
   // need to copy also cling parameters
   TFormula & fnew = dynamic_cast<TFormula&>(obj);
 
   fnew.fClingParameters = fClingParameters;
   fnew.fClingVariables = fClingVariables;
 
   fnew.fFuncs = fFuncs;
   fnew.fVars = fVars;
   fnew.fParams = fParams;
   fnew.fConsts = fConsts;
   fnew.fFunctionsShortcuts = fFunctionsShortcuts;
   fnew.fFormula  = fFormula;
   fnew.fNdim = fNdim;
   fnew.fNpar = fNpar;
   fnew.fNumber = fNumber;
   fnew.fVectorized = fVectorized;
   fnew.SetParameters(GetParameters());
   // copy Linear parts (it is a vector of TFormula pointers) needs to be copied one by one
   // looping at all the elements
   // delete first previous elements
   int nLinParts = fnew.fLinearParts.size();
   if (nLinParts > 0) {
      for (int i = 0; i < nLinParts; ++i) delete fnew.fLinearParts[i];
      fnew.fLinearParts.clear();
   }
   // old size that needs to be copied
   nLinParts = fLinearParts.size();
   if (nLinParts > 0) {
      fnew.fLinearParts.reserve(nLinParts);
      for (int i = 0; i < nLinParts; ++i) {
         TFormula * linearNew = new TFormula();
         TFormula * linearOld = (TFormula*) fLinearParts[i];
         if (linearOld) {
            linearOld->Copy(*linearNew);
            fnew.fLinearParts.push_back(linearNew);
         }
         else
            Warning("Copy","Function %s - expr %s has a dummy linear part %d",GetName(),GetExpFormula().Data(),i);
      }
   }
 
   fnew.fClingInput = fClingInput;
   fnew.fReadyToExecute = fReadyToExecute;
   fnew.fClingInitialized = fClingInitialized.load();
   fnew.fAllParametersSetted = fAllParametersSetted;
   fnew.fClingName = fClingName;
   fnew.fSavedInputFormula = fSavedInputFormula;
   fnew.fLazyInitialization = fLazyInitialization;
 
   // case of function based on a C++  expression (lambda's) which is ready to be compiled
   if (fLambdaPtr && TestBit(TFormula::kLambda)) {
 
      bool ret = fnew.InitLambdaExpression(fnew.fFormula);
      if (ret)  {
         fnew.SetBit(TFormula::kLambda);
         fnew.fReadyToExecute = true;
      }
      else {
         Error("TFormula","Syntax error in building the lambda expression %s", fFormula.Data() );
         fnew.fReadyToExecute = false;
      }
   }
 
   // use copy-constructor of TMethodCall
   // if c++-14 could use std::make_unique
   TMethodCall *m = (fMethod) ? new TMethodCall(*fMethod) : nullptr;
   fnew.fMethod.reset(m);
 
   fnew.fFuncPtr = fFuncPtr;
   fnew.fGradGenerationInput = fGradGenerationInput;
   fnew.fHessGenerationInput = fHessGenerationInput;
   fnew.fGradFuncPtr = fGradFuncPtr;
   fnew.fHessFuncPtr = fHessFuncPtr;
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Clear the formula setting expression to empty and reset the variables and
/// parameters containers.
 
void TFormula::Clear(Option_t * )
{
   fNdim = 0;
   fNpar = 0;
   fNumber = 0;
   fFormula = "";
   fClingName = "";
 
   fMethod.reset();
 
   fClingVariables.clear();
   fClingParameters.clear();
   fReadyToExecute = false;
   fClingInitialized = false;
   fAllParametersSetted = false;
   fFuncs.clear();
   fVars.clear();
   fParams.clear();
   fConsts.clear();
   fFunctionsShortcuts.clear();
 
   // delete linear parts
   int nLinParts = fLinearParts.size();
   if (nLinParts > 0) {
      for (int i = 0; i < nLinParts; ++i) delete fLinearParts[i];
   }
   fLinearParts.clear();
 
}
 
// Returns nullptr on failure.
static std::unique_ptr<TMethodCall>
prepareMethod(bool HasParameters, bool HasVariables, const char* FuncName,
              bool IsVectorized, bool AddCladArrayRef = false) {
   std::unique_ptr<TMethodCall>
       Method = std::make_unique<TMethodCall>();
 
   TString prototypeArguments = "";
   if (HasVariables || HasParameters) {
      if (IsVectorized)
         prototypeArguments.Append(vectorizedArgType() + " const*");
      else
         prototypeArguments.Append("Double_t const*");
   }
   auto AddDoublePtrParam = [&prototypeArguments]() {
     prototypeArguments.Append(",");
     prototypeArguments.Append("Double_t*");
   };
   if (HasParameters)
      AddDoublePtrParam();
 
   // We need an extra Double_t* for the gradient return result.
   if (AddCladArrayRef) {
      prototypeArguments.Append(",");
      prototypeArguments.Append("Double_t*");
   }
 
   // Initialize the method call using real function name (cling name) defined
   // by ProcessFormula
   Method->InitWithPrototype(FuncName, prototypeArguments);
   if (!Method->IsValid()) {
      Error("prepareMethod",
            "Can't compile function %s prototype with arguments %s", FuncName,
            prototypeArguments.Data());
      return nullptr;
   }
 
   return Method;
}
 
static TInterpreter::CallFuncIFacePtr_t::Generic_t  prepareFuncPtr(TMethodCall *method) {
   if (!method) return nullptr;
   CallFunc_t *callfunc = method->GetCallFunc();
 
   if (!gCling->CallFunc_IsValid(callfunc)) {
      Error("prepareFuncPtr", "Callfunc retuned from Cling is not valid");
      return nullptr;
   }
 
   TInterpreter::CallFuncIFacePtr_t::Generic_t Result
      = gCling->CallFunc_IFacePtr(callfunc).fGeneric;
   if (!Result) {
      Error("prepareFuncPtr", "Compiled function pointer is null");
      return nullptr;
   }
   return Result;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Sets TMethodCall to function inside Cling environment.
/// TFormula uses it to execute function.
/// After call, TFormula should be ready to evaluate formula.
/// Returns false on failure.
 
bool TFormula::PrepareEvalMethod()
{
   if (!fMethod) {
      Bool_t hasParameters = (fNpar > 0);
      Bool_t hasVariables = (fNdim > 0);
      fMethod = prepareMethod(hasParameters, hasVariables, fClingName,fVectorized);
      if (!fMethod) return false;
      fFuncPtr = prepareFuncPtr(fMethod.get());
   }
   return fFuncPtr;
}
 
////////////////////////////////////////////////////////////////////////////////
///    Inputs formula, transferred to C++ code into Cling
 
void TFormula::InputFormulaIntoCling()
{
 
   if (!fClingInitialized && fReadyToExecute && fClingInput.Length() > 0) {
      // make sure the interpreter is initialized
      ROOT::GetROOT();
      R__ASSERT(gCling);
 
      // Trigger autoloading / autoparsing (ROOT-9840):
      TString triggerAutoparsing = "namespace ROOT_TFormula_triggerAutoParse {\n"; triggerAutoparsing += fClingInput + "\n}";
      gCling->ProcessLine(triggerAutoparsing);
 
      // add pragma for optimization of the formula
      fClingInput = TString("#pragma cling optimize(2)\n") + fClingInput;
 
      // Now that all libraries and headers are loaded, Declare() a performant version
      // of the same code:
      gCling->Declare(fClingInput);
      fClingInitialized = PrepareEvalMethod();
      if (!fClingInitialized) Error("InputFormulaIntoCling","Error compiling formula expression in Cling");
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///    Fill structures with default variables, constants and function shortcuts
 
void TFormula::FillDefaults()
{
   const TString defvars[] = { "x","y","z","t"};
   const pair<TString, Double_t> defconsts[] = {{"pi", TMath::Pi()},
                                                {"sqrt2", TMath::Sqrt2()},
                                                {"infinity", TMath::Infinity()},
                                                {"e", TMath::E()},
                                                {"ln10", TMath::Ln10()},
                                                {"loge", TMath::LogE()},
                                                {"c", TMath::C()},
                                                {"g", TMath::G()},
                                                {"h", TMath::H()},
                                                {"k", TMath::K()},
                                                {"sigma", TMath::Sigma()},
                                                {"r", TMath::R()},
                                                {"eg", TMath::EulerGamma()},
                                                {"true", 1},
                                               {"false", 0}};
   // const pair<TString,Double_t> defconsts[] = { {"pi",TMath::Pi()}, {"sqrt2",TMath::Sqrt2()},
   //       {"infinity",TMath::Infinity()}, {"ln10",TMath::Ln10()},
   //       {"loge",TMath::LogE()}, {"true",1},{"false",0} };
   const std::pair<TString, TString> funShortcuts[] = {
      {"sign", "TMath::Sign"}, {"binomial", "TMath::Binomial"}, {"sq", "TMath::Sq"}};
 
   std::vector<TString> defvars2(10);
   for (int i = 0; i < 9; ++i)
      defvars2[i] = TString::Format("x[%d]",i);
 
   for (const auto &var : defvars) {
      int pos = fVars.size();
      fVars[var] = TFormulaVariable(var, 0, pos);
      fClingVariables.push_back(0);
   }
   // add also the variables defined like x[0],x[1],x[2],...
   // support up to x[9] - if needed extend that to higher value
   // const int maxdim = 10;
   // for (int i = 0; i < maxdim;  ++i) {
   //    TString xvar = TString::Format("x[%d]",i);
   //    fVars[xvar] =  TFormulaVariable(xvar,0,i);
   //    fClingVariables.push_back(0);
   // }
 
   for (auto con : defconsts) {
      fConsts[con.first] = con.second;
   }
   for (auto fun : funShortcuts) {
      fFunctionsShortcuts[fun.first] = fun.second;
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// \brief Handling Polynomial Notation (polN)
///
/// This section describes how polynomials are handled in the code.
///
/// - If any name appears before 'pol', it is treated as a variable used in the polynomial.
///   - Example: `varpol2(5)` will be replaced with `[5] + [6]*var + [7]*var^2`.
///   - Empty name is treated like variable `x`.
///
/// - The extended format allows direct variable specification:
///   - Example: `pol2(x, 0)` or `pol(x, [A])`.
///
/// - Traditional syntax: `polN` represents a polynomial of degree `N`:
///   - `ypol1` → `[p0] + [p1] * y`
///   - `pol1(x, 0)` → `[p0] + [p1] * x`
///   - `pol2(y, 1)` → `[p1] + [p2] * y + [p3] * TMath::Sq(y)`
////////////////////////////////////////////////////////////////////////////////
 
void TFormula::HandlePolN(TString &formula)
{
 
   Int_t polPos = formula.Index("pol");
   while (polPos != kNPOS && !IsAParameterName(formula, polPos)) {
      Bool_t defaultVariable = false;
      TString variable;
      Int_t openingBracketPos = formula.Index('(', polPos);
      Bool_t defaultCounter = openingBracketPos == kNPOS;
      Bool_t defaultDegree = true;
      Int_t degree = 1, counter = 0;
      TString sdegree;
      std::vector<TString> paramNames;
 
      // check for extended format
      Bool_t isNewFormat = false;
      if (!defaultCounter) {
         TString args = formula(openingBracketPos + 1, formula.Index(')', polPos) - openingBracketPos - 1);
         if (args.Contains(",")) {
            isNewFormat = true;
            sdegree = formula(polPos + 3, openingBracketPos - polPos - 3);
            if (!sdegree.IsDigit())
               sdegree = "1";
            degree = sdegree.Atoi();
 
            std::vector<TString> tokens;
            std::stringstream ss(args.Data());
            std::string token;
            while (std::getline(ss, token, ',')) { // spliting string at ,
               tokens.push_back(TString(token));
            }
 
            if (!tokens.empty()) {
               variable = tokens[0];
               variable.Strip(TString::kBoth);
 
               // extract counter if provided as a numeric value, without this it was not working with [A], [B]
               if (tokens.size() > 1) {
                  TString counterStr = tokens[1];
                  counterStr.Strip(TString::kBoth);
                  if (counterStr.IsDigit()) {
                     counter = counterStr.Atoi();
                  }
               }
 
               // store parameter names for later use
               for (size_t i = 1; i < tokens.size(); i++) {
                  TString paramStr = tokens[i];
                  paramStr.Strip(TString::kBoth);
                  if (paramStr.BeginsWith("[") && paramStr.EndsWith("]")) {
                     paramStr = paramStr(1, paramStr.Length() - 2);
                     paramNames.push_back(paramStr);
 
                     if (fParams.find(paramStr) == fParams.end()) {
                        fParams[paramStr] = fNpar;
                        fClingParameters.push_back(0.0);
                        fNpar++;
                     }
                  }
               }
            }
         }
      }
 
      // handle original format if not new format
      if (!isNewFormat) {
         if (!defaultCounter) {
            sdegree = formula(polPos + 3, openingBracketPos - polPos - 3);
            if (!sdegree.IsDigit())
               defaultCounter = true;
         }
         if (!defaultCounter) {
            degree = sdegree.Atoi();
            counter = TString(formula(openingBracketPos + 1, formula.Index(')', polPos) - openingBracketPos)).Atoi();
         } else {
            Int_t temp = polPos + 3;
            while (temp < formula.Length() && isdigit(formula[temp])) {
               defaultDegree = false;
               temp++;
            }
            degree = TString(formula(polPos + 3, temp - polPos - 3)).Atoi();
         }
 
         if (polPos - 1 < 0 || !IsFunctionNameChar(formula[polPos - 1]) || formula[polPos - 1] == ':') {
            variable = "x";
            defaultVariable = true;
         } else {
            Int_t tmp = polPos - 1;
            while (tmp >= 0 && IsFunctionNameChar(formula[tmp]) && formula[tmp] != ':') {
               tmp--;
            }
            variable = formula(tmp + 1, polPos - (tmp + 1));
         }
      }
 
      // build replacement string (modified)
      TString replacement;
      if (isNewFormat && !paramNames.empty()) {
         for (Int_t i = 0; i <= degree && i < static_cast<Int_t>(paramNames.size()); i++) {
            if (i == 0) {
               replacement = TString::Format("[%s]", paramNames[i].Data());
            } else if (i == 1) {
               replacement.Append(TString::Format("+[%s]*%s", paramNames[i].Data(), variable.Data()));
            } else {
               replacement.Append(TString::Format("+[%s]*%s^%d", paramNames[i].Data(), variable.Data(), i));
            }
         }
      } else {
         replacement = TString::Format("[%d]", counter);
         for (Int_t i = 1; i <= degree; i++) {
            if (i == 1) {
               replacement.Append(TString::Format("+[%d]*%s", counter + i, variable.Data()));
            } else {
               replacement.Append(TString::Format("+[%d]*%s^%d", counter + i, variable.Data(), i));
            }
         }
      }
 
      if (degree > 0) {
         replacement.Insert(0, '(');
         replacement.Append(')');
      }
 
      // create patern based on format either new or old
      TString pattern;
      if (isNewFormat) {
         pattern = formula(polPos, formula.Index(')', polPos) - polPos + 1);
      } else if (defaultCounter && !defaultDegree) {
         pattern = TString::Format("%spol%d", (defaultVariable ? "" : variable.Data()), degree);
      } else if (defaultCounter && defaultDegree) {
         pattern = TString::Format("%spol", (defaultVariable ? "" : variable.Data()));
      } else {
         pattern = TString::Format("%spol%d(%d)", (defaultVariable ? "" : variable.Data()), degree, counter);
      }
 
      if (!formula.Contains(pattern)) {
         Error("HandlePolN", "Error handling polynomial function - expression is %s - trying to replace %s with %s ",
               formula.Data(), pattern.Data(), replacement.Data());
         break;
      }
 
      if (formula == pattern) {
         SetBit(kLinear, true);
         fNumber = 300 + degree;
      }
 
      formula.ReplaceAll(pattern, replacement);
      polPos = formula.Index("pol");
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///    Handling parametrized functions
///    Function can be normalized, and have different variable then x.
///    Variables should be placed in brackets after function name.
///    No brackets are treated like [x].
///    Normalized function has char 'n' after name, eg.
///    gausn[var](0) will be replaced with [0]*exp(-0.5*((var-[1])/[2])^2)/(sqrt(2*pi)*[2])
///
///    Adding function is easy, just follow these rules, and add to
///    `TFormula::FillParametrizedFunctions` defined further below:
///
///    - Key for function map is pair of name and dimension of function
///    - value of key is a pair function body and normalized function body
///    - {Vn} is a place where variable appear, n represents n-th variable from variable list.
///      Count starts from 0.
///    - [num] stands for parameter number.
///      If user pass to function argument 5, num will stand for (5 + num) parameter.
///
 
void TFormula::HandleParametrizedFunctions(TString &formula)
{
   // define all parametrized functions
   map< pair<TString,Int_t> ,pair<TString,TString> > functions;
   FillParametrizedFunctions(functions);
 
   map<TString,Int_t> functionsNumbers;
   functionsNumbers["gaus"] = 100;
   functionsNumbers["bigaus"] = 102;
   functionsNumbers["landau"] = 400;
   functionsNumbers["expo"] = 200;
   functionsNumbers["crystalball"] = 500;
 
   // replace old names xygaus -> gaus[x,y]
   formula.ReplaceAll("xyzgaus","gaus[x,y,z]");
   formula.ReplaceAll("xygausn","gausn[x,y]");
   formula.ReplaceAll("xygaus","gaus[x,y]");
   formula.ReplaceAll("xgaus","gaus[x]");
   formula.ReplaceAll("ygaus","gaus[y]");
   formula.ReplaceAll("zgaus","gaus[z]");
   formula.ReplaceAll("xyexpo","expo[x,y]");
   formula.ReplaceAll("xexpo","expo[x]");
   formula.ReplaceAll("yexpo","expo[y]");
   formula.ReplaceAll("zexpo","expo[z]");
   formula.ReplaceAll("xylandaun","landaun[x,y]");
   formula.ReplaceAll("xylandau","landau[x,y]");
   // at the moment pre-defined functions have no more than 3 dimensions
   const char * defaultVariableNames[] = { "x","y","z"};
 
   for (map<pair<TString, Int_t>, pair<TString, TString>>::iterator it = functions.begin(); it != functions.end();
        ++it) {
 
      TString funName = it->first.first;
      Int_t funDim = it->first.second;
      Int_t funPos = formula.Index(funName);
 
      // std::cout << formula << " ---- " << funName << "  " << funPos << std::endl;
      while (funPos != kNPOS && !IsAParameterName(formula, funPos)) {
 
         // should also check that function is not something else (e.g. exponential - parse the expo)
         Int_t lastFunPos = funPos + funName.Length();
 
         // check that first and last character is not a special character
         Int_t iposBefore = funPos - 1;
         // std::cout << "looping on  funpos is " << funPos << " formula is " << formula << " function " << funName <<
         // std::endl;
         if (iposBefore >= 0) {
            assert(iposBefore < formula.Length());
            //if (isalpha(formula[iposBefore])) {
            if (IsFunctionNameChar(formula[iposBefore])) {
               // std::cout << "previous character for function " << funName << " is " << formula[iposBefore] << "- skip
               // " << std::endl;
               funPos = formula.Index(funName, lastFunPos);
               continue;
            }
         }
 
         Bool_t isNormalized = false;
         if (lastFunPos < formula.Length()) {
            // check if function is normalized by looking at "n" character after function name (e.g. gausn)
            isNormalized = (formula[lastFunPos] == 'n');
            if (isNormalized)
               lastFunPos += 1;
            if (lastFunPos < formula.Length()) {
               char c = formula[lastFunPos];
               // check if also last character is not alphanumeric or is not an operator and not a parenthesis ( or [.
               // Parenthesis [] are used to express the variables
               if (isalnum(c) || (!IsOperator(c) && c != '(' && c != ')' && c != '[' && c != ']')) {
                  // std::cout << "last character for function " << funName << " is " << c << " skip .." <<  std::endl;
                  funPos = formula.Index(funName, lastFunPos);
                  continue;
               }
            }
         }
 
         if (isNormalized) {
            SetBit(kNormalized, true);
         }
         std::vector<TString> variables;
         Int_t dim = 0;
         TString varList = "";
         Bool_t defaultVariables = false;
 
         // check if function has specified the [...] e.g. gaus[x,y]
         Int_t openingBracketPos = funPos + funName.Length() + (isNormalized ? 1 : 0);
         Int_t closingBracketPos = kNPOS;
         if (openingBracketPos >= formula.Length() || formula[openingBracketPos] != '[') {
            dim = funDim;
            variables.resize(dim);
            for (Int_t idim = 0; idim < dim; ++idim)
               variables[idim] = defaultVariableNames[idim];
            defaultVariables = true;
         } else {
            // in case of [..] found, assume they specify all the variables. Use it to get function dimension
            closingBracketPos = formula.Index(']', openingBracketPos);
            varList = formula(openingBracketPos + 1, closingBracketPos - openingBracketPos - 1);
            dim = varList.CountChar(',') + 1;
            variables.resize(dim);
            Int_t Nvar = 0;
            TString varName = "";
            for (Int_t i = 0; i < varList.Length(); ++i) {
               if (IsFunctionNameChar(varList[i])) {
                  varName.Append(varList[i]);
               }
               if (varList[i] == ',') {
                  variables[Nvar] = varName;
                  varName = "";
                  Nvar++;
               }
            }
            if (varName != "") // we will miss last variable
            {
               variables[Nvar] = varName;
            }
         }
         // check if dimension obtained from [...] is compatible with what is defined in existing pre-defined functions
         // std::cout << " Found dim = " << dim  << " and function dimension is " << funDim << std::endl;
         if (dim != funDim) {
            pair<TString, Int_t> key = make_pair(funName, dim);
            if (functions.find(key) == functions.end()) {
               Error("PreProcessFormula", "Dimension of function %s is detected to be of dimension %d and is not "
                                          "compatible with existing pre-defined function which has dim %d",
                     funName.Data(), dim, funDim);
               return;
            }
            // skip the particular function found - we might find later on the corresponding pre-defined function
            funPos = formula.Index(funName, lastFunPos);
            continue;
         }
         // look now for the (..) brackets to get the parameter counter (e.g. gaus(0) + gaus(3) )
         // need to start for a position
         Int_t openingParenthesisPos = (closingBracketPos == kNPOS) ? openingBracketPos : closingBracketPos + 1;
         bool defaultCounter = (openingParenthesisPos > formula.Length() || formula[openingParenthesisPos] != '(');
 
         // Int_t openingParenthesisPos = formula.Index('(',funPos);
         // Bool_t defaultCounter = (openingParenthesisPos == kNPOS);
         Int_t counter;
         if (defaultCounter) {
            counter = 0;
         } else {
            // Check whether this is just a number in parentheses. If not, leave
            // it to `HandleFunctionArguments` to be parsed
 
            TRegexp counterPattern("([0-9]+)");
            Ssiz_t len;
            if (counterPattern.Index(formula, &len, openingParenthesisPos) == -1) {
               funPos = formula.Index(funName, funPos + 1);
               continue;
            } else {
               counter =
                  TString(formula(openingParenthesisPos + 1, formula.Index(')', funPos) - openingParenthesisPos - 1))
                     .Atoi();
            }
         }
         // std::cout << "openingParenthesisPos  " << openingParenthesisPos << " counter is " << counter <<  std::endl;
 
         TString body = (isNormalized ? it->second.second : it->second.first);
         if (isNormalized && body == "") {
            Error("PreprocessFormula", "%d dimension function %s has no normalized form.", it->first.second,
                  funName.Data());
            break;
         }
         for (int i = 0; i < body.Length(); ++i) {
            if (body[i] == '{') {
               // replace {Vn} with variable names
               i += 2; // skip '{' and 'V'
               Int_t num = TString(body(i, body.Index('}', i) - i)).Atoi();
               TString variable = variables[num];
               TString pattern = TString::Format("{V%d}", num);
               i -= 2; // restore original position
               body.Replace(i, pattern.Length(), variable, variable.Length());
               i += variable.Length() - 1; // update i to reflect change in body string
            } else if (body[i] == '[') {
               // update parameter counters in case of many functions (e.g. gaus(0)+gaus(3) )
               Int_t tmp = i;
               while (tmp < body.Length() && body[tmp] != ']') {
                  tmp++;
               }
               Int_t num = TString(body(i + 1, tmp - 1 - i)).Atoi();
               num += counter;
               TString replacement = TString::Format("%d", num);
 
               body.Replace(i + 1, tmp - 1 - i, replacement, replacement.Length());
               i += replacement.Length() + 1;
            }
         }
         TString pattern;
         if (defaultCounter && defaultVariables) {
            pattern = TString::Format("%s%s", funName.Data(), (isNormalized ? "n" : ""));
         }
         if (!defaultCounter && defaultVariables) {
            pattern = TString::Format("%s%s(%d)", funName.Data(), (isNormalized ? "n" : ""), counter);
         }
         if (defaultCounter && !defaultVariables) {
            pattern = TString::Format("%s%s[%s]", funName.Data(), (isNormalized ? "n" : ""), varList.Data());
         }
         if (!defaultCounter && !defaultVariables) {
            pattern =
               TString::Format("%s%s[%s](%d)", funName.Data(), (isNormalized ? "n" : ""), varList.Data(), counter);
         }
         TString replacement = body;
 
         // set the number (only in case a function exists without anything else
         if (fNumber == 0 && formula.Length() <= (pattern.Length() - funPos) + 1) { // leave 1 extra
            fNumber = functionsNumbers[funName] + 10 * (dim - 1);
         }
 
         // std::cout << " replace " << pattern << " with " << replacement << std::endl;
 
         formula.Replace(funPos, pattern.Length(), replacement, replacement.Length());
 
         funPos = formula.Index(funName);
      }
      // std::cout << " End loop of " << funName << " formula is now " << formula << std::endl;
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///   Handling parameter ranges, in the form of [1..5]
void TFormula::HandleParamRanges(TString &formula)
{
   TRegexp rangePattern("\\[[0-9]+\\.\\.[0-9]+\\]");
   Ssiz_t len;
   int matchIdx = 0;
   while ((matchIdx = rangePattern.Index(formula, &len, matchIdx)) != -1) {
      int startIdx = matchIdx + 1;
      int endIdx = formula.Index("..", startIdx) + 2; // +2 for ".."
      int startCnt = TString(formula(startIdx, formula.Length())).Atoi();
      int endCnt = TString(formula(endIdx, formula.Length())).Atoi();
 
      if (endCnt <= startCnt)
         Error("HandleParamRanges", "End parameter (%d) <= start parameter (%d) in parameter range", endCnt, startCnt);
 
      TString newString = "[";
      for (int cnt = startCnt; cnt < endCnt; cnt++)
         newString += TString::Format("%d],[", cnt);
      newString += TString::Format("%d]", endCnt);
 
      // std::cout << "newString generated by HandleParamRanges is " << newString << std::endl;
      formula.Replace(matchIdx, formula.Index("]", matchIdx) + 1 - matchIdx, newString);
 
      matchIdx += newString.Length();
   }
 
   // std::cout << "final formula is now " << formula << std::endl;
}
 
////////////////////////////////////////////////////////////////////////////////
///   Handling user functions (and parametrized functions)
///   to take variables and optionally parameters as arguments
void TFormula::HandleFunctionArguments(TString &formula)
{
   // std::cout << "calling `HandleFunctionArguments` on " << formula << std::endl;
 
   // Define parametrized functions, in case we need to use them
   std::map<std::pair<TString, Int_t>, std::pair<TString, TString>> parFunctions;
   FillParametrizedFunctions(parFunctions);
 
   // loop through characters
   for (Int_t i = 0; i < formula.Length(); ++i) {
      // List of things to ignore (copied from `TFormula::ExtractFunctors`)
 
      // ignore things that start with square brackets
      if (formula[i] == '[') {
         while (i < formula.Length() && formula[i] != ']')
            i++;
         continue;
      }
      // ignore strings
      if (i < formula.Length() && formula[i] == '\"') {
         do
            i++;
         while (i < formula.Length() && formula[i] != '\"');
         continue;
      }
      // ignore numbers (scientific notation)
      if (IsScientificNotation(formula, i))
         continue;
      // ignore x in hexadecimal number
      if (IsHexadecimal(formula, i)) {
         while (i < formula.Length() && !IsOperator(formula[i]))
            i++;
         continue;
      }
 
      // investigate possible start of function name
      if (i < formula.Length() && isalpha(formula[i]) && !IsOperator(formula[i])) {
         // std::cout << "character : " << i << " " << formula[i] << " is not an operator and is alpha" << std::endl;
 
         int j; // index to end of name
         for (j = i; j < formula.Length() && IsFunctionNameChar(formula[j]); j++)
            ;
         TString name = (TString)formula(i, j - i);
         // std::cout << "parsed name " << name << std::endl;
 
         // Count arguments (careful about parentheses depth)
         // Make list of indices where each argument is separated
         int nArguments = 1;
         int depth = 1;
         std::vector<int> argSeparators;
         argSeparators.push_back(j); // opening parenthesis
         int k;                      // index for end of closing parenthesis
         for (k = j + 1; depth >= 1 && k < formula.Length(); k++) {
            if (formula[k] == ',' && depth == 1) {
               nArguments++;
               argSeparators.push_back(k);
            } else if (formula[k] == '(')
               depth++;
            else if (formula[k] == ')')
               depth--;
         }
         argSeparators.push_back(k - 1); // closing parenthesis
 
         // retrieve `f` (code copied from ExtractFunctors)
         TObject *obj = nullptr;
         {
            R__LOCKGUARD(gROOTMutex);
            obj = gROOT->GetListOfFunctions()->FindObject(name);
         }
         TFormula *f = dynamic_cast<TFormula *>(obj);
         if (!f) {
            // maybe object is a TF1
            TF1 *f1 = dynamic_cast<TF1 *>(obj);
            if (f1)
               f = f1->GetFormula();
         }
         // `f` should be found by now, if it was a user-defined function.
         // The other possibility we need to consider is that this is a
         // parametrized function (else case below)
 
         bool nameRecognized = (f != nullptr);
 
         // Get ndim, npar, and replacementFormula of function
         int ndim = 0;
         int npar = 0;
         TString replacementFormula;
         if (f) {
            ndim = f->GetNdim();
            npar = f->GetNpar();
            replacementFormula = f->GetExpFormula();
         } else {
            // otherwise, try to match default parametrized functions
 
            for (const auto &keyval : parFunctions) {
               // (name, ndim)
               const pair<TString, Int_t> &name_ndim = keyval.first;
               // (formula without normalization, formula with normalization)
               const pair<TString, TString> &formulaPair = keyval.second;
 
               // match names like gaus, gausn, breitwigner
               if (name == name_ndim.first)
                  replacementFormula = formulaPair.first;
               else if (name == name_ndim.first + "n" && formulaPair.second != "")
                  replacementFormula = formulaPair.second;
               else
                  continue;
 
               // set ndim
               ndim = name_ndim.second;
 
               // go through replacementFormula to find the number of parameters
               npar = 0;
               int idx = 0;
               while ((idx = replacementFormula.Index('[', idx)) != kNPOS) {
                  npar = max(npar, 1 + TString(replacementFormula(idx + 1, replacementFormula.Length())).Atoi());
                  idx = replacementFormula.Index(']', idx);
                  if (idx == kNPOS)
                     Error("HandleFunctionArguments", "Square brackets not matching in formula %s",
                           (const char *)replacementFormula);
               }
               // npar should be set correctly now
 
               // break if number of arguments is good (note: `gaus`, has two
               // definitions with different numbers of arguments, but it works
               // out so that it should be unambiguous)
               if (nArguments == ndim + npar || nArguments == npar || nArguments == ndim) {
                  nameRecognized = true;
                  break;
               }
            }
         }
         if (nameRecognized && ndim > 4)
            Error("HandleFunctionArguments", "Number of dimensions %d greater than 4. Cannot parse formula.", ndim);
 
         // if we have "recognizedName(...", then apply substitutions
         if (nameRecognized && j < formula.Length() && formula[j] == '(') {
            // std::cout << "naive replacement formula: " << replacementFormula << std::endl;
            // std::cout << "formula: " << formula << std::endl;
 
            // map to rename each argument in `replacementFormula`
            map<TString, TString> argSubstitutions;
 
            const char *defaultVariableNames[] = {"x", "y", "z", "t"};
 
            // check nArguments and add to argSubstitutions map as appropriate
            bool canReplace = false;
            if (nArguments == ndim + npar) {
               // loop through all variables and parameters, filling in argSubstitutions
               for (int argNr = 0; argNr < nArguments; argNr++) {
 
                  // Get new name (for either variable or parameter)
                  TString newName =
                     TString(formula(argSeparators[argNr] + 1, argSeparators[argNr + 1] - argSeparators[argNr] - 1));
                  PreProcessFormula(newName); // so that nesting works
 
                  // Get old name(s)
                  // and add to argSubstitutions map as appropriate
                  if (argNr < ndim) { // variable
                     TString oldName = (f) ? TString::Format("x[%d]", argNr) : TString::Format("{V%d}", argNr);
                     argSubstitutions[oldName] = newName;
 
                     if (f)
                        argSubstitutions[defaultVariableNames[argNr]] = newName;
 
                  } else { // parameter
                     int parNr = argNr - ndim;
                     TString oldName =
                        (f) ? TString::Format("[%s]", f->GetParName(parNr)) : TString::Format("[%d]", parNr);
                     argSubstitutions[oldName] = newName;
 
                     // If the name stays the same, keep the old value of the parameter
                     if (f && oldName == newName)
                        DoAddParameter(f->GetParName(parNr), f->GetParameter(parNr), false);
                  }
               }
 
               canReplace = true;
            } else if (nArguments == npar) {
               // Try to assume variables are implicit (need all arguments to be
               // parameters)
 
               // loop to check if all arguments are parameters
               bool varsImplicit = true;
               for (int argNr = 0; argNr < nArguments && varsImplicit; argNr++) {
                  int openIdx = argSeparators[argNr] + 1;
                  int closeIdx = argSeparators[argNr + 1] - 1;
 
                  // check brackets on either end
                  if (formula[openIdx] != '[' || formula[closeIdx] != ']' || closeIdx <= openIdx + 1)
                     varsImplicit = false;
 
                  // check that the middle is a single function-name
                  for (int idx = openIdx + 1; idx < closeIdx && varsImplicit; idx++)
                     if (!IsFunctionNameChar(formula[idx]))
                        varsImplicit = false;
 
                  if (!varsImplicit)
                     Warning("HandleFunctionArguments",
                             "Argument %d is not a parameter. Cannot assume variables are implicit.", argNr);
               }
 
               // loop to replace parameter names
               if (varsImplicit) {
                  // if parametrized function, still need to replace parameter names
                  if (!f) {
                     for (int dim = 0; dim < ndim; dim++) {
                        argSubstitutions[TString::Format("{V%d}", dim)] = defaultVariableNames[dim];
                     }
                  }
 
                  for (int argNr = 0; argNr < nArguments; argNr++) {
                     TString oldName =
                        (f) ? TString::Format("[%s]", f->GetParName(argNr)) : TString::Format("[%d]", argNr);
                     TString newName =
                        TString(formula(argSeparators[argNr] + 1, argSeparators[argNr + 1] - argSeparators[argNr] - 1));
 
                     // preprocess the formula so that nesting works
                     PreProcessFormula(newName);
                     argSubstitutions[oldName] = newName;
 
                     // If the name stays the same, keep the old value of the parameter
                     if (f && oldName == newName)
                        DoAddParameter(f->GetParName(argNr), f->GetParameter(argNr), false);
                  }
 
                  canReplace = true;
               }
            }
            if (!canReplace && nArguments == ndim) {
               // Treat parameters as implicit
 
               // loop to replace variable names
               for (int argNr = 0; argNr < nArguments; argNr++) {
                  TString oldName = (f) ? TString::Format("x[%d]", argNr) : TString::Format("{V%d}", argNr);
                  TString newName =
                     TString(formula(argSeparators[argNr] + 1, argSeparators[argNr + 1] - argSeparators[argNr] - 1));
 
                  // preprocess so nesting works
                  PreProcessFormula(newName);
                  argSubstitutions[oldName] = newName;
 
                  if (f) // x, y, z are not used in parametrized function definitions
                     argSubstitutions[defaultVariableNames[argNr]] = newName;
               }
 
               if (f) {
                  // keep old values of the parameters
                  for (int parNr = 0; parNr < npar; parNr++)
                     DoAddParameter(f->GetParName(parNr), f->GetParameter(parNr), false);
               }
 
               canReplace = true;
            }
 
            if (canReplace)
               ReplaceAllNames(replacementFormula, argSubstitutions);
            // std::cout << "after replacement, replacementFormula is " << replacementFormula << std::endl;
 
            if (canReplace) {
               // std::cout << "about to replace position " << i << " length " << k-i << " in formula : " << formula <<
               // std::endl;
               formula.Replace(i, k - i, replacementFormula);
               i += replacementFormula.Length() - 1; // skip to end of replacement
               // std::cout << "new formula is : " << formula << std::endl;
            } else {
               Warning("HandleFunctionArguments", "Unable to make replacement. Number of parameters doesn't work : "
                                                  "%d arguments, %d dimensions, %d parameters",
                       nArguments, ndim, npar);
               i = j;
            }
 
         } else {
            i = j; // skip to end of candidate "name"
         }
      }
   }
 
}
 
////////////////////////////////////////////////////////////////////////////////
///    Handling exponentiation
///    Can handle multiple carets, eg.
///    2^3^4 will be treated like 2^(3^4)
 
void TFormula::HandleExponentiation(TString &formula)
{
   Int_t caretPos = formula.Last('^');
   while (caretPos != kNPOS && !IsAParameterName(formula, caretPos)) {
 
      TString right, left;
      Int_t temp = caretPos;
      temp--;
      // get the expression in ( ) which has the operator^ applied
      if (formula[temp] == ')') {
         Int_t depth = 1;
         temp--;
         while (depth != 0 && temp > 0) {
            if (formula[temp] == ')')
               depth++;
            if (formula[temp] == '(')
               depth--;
            temp--;
         }
         if (depth == 0)
            temp++;
      }
      // this in case of someting like sin(x+2)^2
      do {
         temp--; // go down one
         // handle scientific notation cases (1.e-2 ^ 3 )
         if (temp >= 2 && IsScientificNotation(formula, temp - 1))
            temp -= 3;
      } while (temp >= 0 && !IsOperator(formula[temp]) && !IsBracket(formula[temp]));
 
      assert(temp + 1 >= 0);
      Int_t leftPos = temp + 1;
      left = formula(leftPos, caretPos - leftPos);
      // std::cout << "left to replace is " << left << std::endl;
 
      // look now at the expression after the ^ operator
      temp = caretPos;
      temp++;
      if (temp >= formula.Length()) {
         Error("HandleExponentiation", "Invalid position of operator ^");
         return;
         }
         if (formula[temp] == '(') {
            Int_t depth = 1;
            temp++;
            while (depth != 0 && temp < formula.Length()) {
               if (formula[temp] == ')')
                  depth--;
               if (formula[temp] == '(')
                  depth++;
               temp++;
            }
            temp--;
         } else {
            // handle case  first character is operator - or + continue
            if (formula[temp] == '-' || formula[temp] == '+')
               temp++;
            // handle cases x^-2 or x^+2
            // need to handle also cases x^sin(x+y)
            Int_t depth = 0;
            // stop right expression if is an operator or if is a ")" from a zero depth
            while (temp < formula.Length() && ((depth > 0) || !IsOperator(formula[temp]))) {
               temp++;
               // handle scientific notation cases (1.e-2 ^ 3 )
               if (temp >= 2 && IsScientificNotation(formula, temp))
                  temp += 2;
               // for internal parenthesis
               if (temp < formula.Length() && formula[temp] == '(')
                  depth++;
               if (temp < formula.Length() && formula[temp] == ')') {
                  if (depth > 0)
                     depth--;
                  else
                     break; // case of end of a previously started expression e.g. sin(x^2)
               }
            }
         }
         right = formula(caretPos + 1, (temp - 1) - caretPos);
         // std::cout << "right to replace is " << right << std::endl;
 
         TString pattern = TString::Format("%s^%s", left.Data(), right.Data());
         TString replacement = TString::Format("pow(%s,%s)", left.Data(), right.Data());
 
         // special case for square function
         if (right == "2"){
            replacement = TString::Format("TMath::Sq(%s)",left.Data());
         }
 
         // std::cout << "pattern : " << pattern << std::endl;
         // std::cout << "replacement : " << replacement << std::endl;
         formula.Replace(leftPos, pattern.Length(), replacement, replacement.Length());
 
         caretPos = formula.Last('^');
   }
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Handle linear functions defined with the operator ++ (@ is the shorthand).
 
void TFormula::HandleLinear(TString &formula)
{
   if(formula.Length() == 0) return;
   auto terms = ROOT::Split(formula.Data(), "@");
   if(terms.size() <= 1) return; // function is not linear
   // Handle Linear functions identified with "@" operator
   fLinearParts.reserve(terms.size());
   TString expandedFormula = "";
   int delimeterPos = 0;
   for(std::size_t iTerm = 0; iTerm < terms.size(); ++iTerm) {
      if (iTerm < terms.size() - 1) { // N terms, N - 1 @
         // determine the position of the "@" operator in the formula
         delimeterPos += terms[iTerm].size() + iTerm;
         if(IsAParameterName(formula, delimeterPos)) {
            // append the current term and the remaining formula unchanged to the expanded formula
            expandedFormula += terms[iTerm];
            expandedFormula += formula(delimeterPos, formula.Length() - delimeterPos);
            break;
         }
      }
      SetBit(kLinear, true);
      auto termName = std::string("__linear") + std::to_string(iTerm+1);
      fLinearParts.push_back(new TFormula(termName.c_str(), terms[iTerm].c_str(), false));
      std::stringstream ss;
      ss << "([" << iTerm << "]*(" << terms[iTerm] << "))";
      expandedFormula += ss.str();
      if(iTerm < terms.size() - 1) expandedFormula += "+";
   }
   formula = expandedFormula;
}
 
////////////////////////////////////////////////////////////////////////////////
///    Preprocessing of formula
///    Replace all ** by ^, all ++ by @, and removes spaces.
///    Handle also parametrized functions like polN,gaus,expo,landau
///    and exponentiation.
///    Similar functionality should be added here.
 
void TFormula::PreProcessFormula(TString &formula)
{
   formula.ReplaceAll("**","^");
   formula.ReplaceAll("++","@");  // for linear functions
   formula.ReplaceAll(" ","");
   HandlePolN(formula);
   HandleParametrizedFunctions(formula);
   HandleParamRanges(formula);
   HandleFunctionArguments(formula);
   HandleExponentiation(formula);
   // "++" (now @ since we just called ReplaceAll a few lines above) will be dealt with HandleLinear
   HandleLinear(formula);
   // special case for "--" and "++"
   // ("++" needs to be written with whitespace that is removed before but then we re-add it again
   formula.ReplaceAll("--","- -");
   formula.ReplaceAll("++","+ +");
}
 
////////////////////////////////////////////////////////////////////////////////
/// prepare the formula to be executed
/// normally is called with fFormula
 
Bool_t TFormula::PrepareFormula(TString &formula)
{
   fFuncs.clear();
   fReadyToExecute = false;
 
   // Check for balanced parentheses
   if (!isBalanced(formula)) {
      return false;
   }
 
   ExtractFunctors(formula);
 
   // update the expression with the new formula
   fFormula = formula;
   // save formula to parse variable and parameters for Cling
   fClingInput = formula;
   // replace all { and }
   fFormula.ReplaceAll("{","");
   fFormula.ReplaceAll("}","");
 
   // std::cout << "functors are extracted formula is " << std::endl;
   // std::cout << fFormula << std::endl << std::endl;
 
   fFuncs.sort();
   fFuncs.unique();
 
   // use inputFormula for Cling
   ProcessFormula(fClingInput);
 
   // for pre-defined functions (need after processing)
   if (fNumber != 0) SetPredefinedParamNames();
 
   return fReadyToExecute && fClingInitialized;
}
 
////////////////////////////////////////////////////////////////////////////////
///    Extracts functors from formula, and put them in fFuncs.
///    Simple grammar:
///  -  `<function>`  := name(arg1,arg2...)
///  -  `<variable>`  := name
///  -  `<parameter>` := [number]
///  -  `<name>`      := String containing lower and upper letters, numbers, underscores
///  -  `<number>`    := Integer number
///    Operators are omitted.
 
void TFormula::ExtractFunctors(TString &formula)
{
   // std::cout << "Commencing ExtractFunctors on " << formula << std::endl;
 
   TString name = "";
   TString body = "";
   // printf("formula is : %s \n",formula.Data() );
   for (Int_t i = 0; i < formula.Length(); ++i) {
 
      // std::cout << "loop on character : " << i << " " << formula[i] << std::endl;
      // case of parameters
      if (formula[i] == '[') {
         Int_t tmp = i;
         i++;
         TString param = "";
         while (i < formula.Length() && formula[i] != ']') {
            param.Append(formula[i++]);
         }
         i++;
         // rename parameter name XX to pXX
         // std::cout << "examine parameters " << param << std::endl;
         int paramIndex = -1;
         if (param.IsDigit()) {
            paramIndex = param.Atoi();
            param.Insert(0, 'p'); // needed for the replacement
            if (paramIndex >= fNpar || fParams.find(param) == fParams.end()) {
               // add all parameters up to given index found
               for (int idx = 0; idx <= paramIndex; ++idx) {
                  TString pname = TString::Format("p%d", idx);
                  if (fParams.find(pname) == fParams.end())
                     DoAddParameter(pname, 0, false);
               }
            }
         } else {
            // handle whitespace characters in parname
            param.ReplaceAll("\\s", " ");
 
            // only add if parameter does not already exist, because maybe
            // `HandleFunctionArguments` already assigned a default value to the
            // parameter
            if (fParams.find(param) == fParams.end() || GetParNumber(param) < 0 ||
                (unsigned)GetParNumber(param) >= fClingParameters.size()) {
               // std::cout << "Setting parameter " << param << " to 0" << std::endl;
               DoAddParameter(param, 0, false);
            }
         }
         TString replacement = TString::Format("{[%s]}", param.Data());
         formula.Replace(tmp, i - tmp, replacement, replacement.Length());
         fFuncs.push_back(TFormulaFunction(param));
         // we need to change index i after replacing since string length changes
         // and we need to re-calculate i position
         int deltai = replacement.Length() - (i-tmp);
         i += deltai;
         // printf("found parameter %s \n",param.Data() );
         continue;
      }
      // case of strings
      if (formula[i] == '\"') {
         // look for next instance of "\"
         do {
            i++;
         } while (i < formula.Length() && formula[i] != '\"');
      }
      // case of e or E for numbers in exponential notation (e.g. 2.2e-3)
      if (IsScientificNotation(formula, i))
         continue;
      // case of x for hexadecimal numbers
      if (IsHexadecimal(formula, i)) {
         // find position of operator
         // do not check cases if character is not only a to f, but accept anything
         while (i < formula.Length() && !IsOperator(formula[i])) {
            i++;
         }
         continue;
      }
 
      // std::cout << "investigating character : " << i << " " << formula[i] << " of formula " << formula <<
      // std::endl;
      // look for variable and function names. They  start in C++ with alphanumeric characters
      if (isalpha(formula[i]) &&
          !IsOperator(formula[i])) // not really needed to check if operator (if isalpha is not an operator)
      {
         // std::cout << "character : " << i << " " << formula[i] << " is not an operator and is alpha " <<
         // std::endl;
 
         while (i < formula.Length() && IsFunctionNameChar(formula[i])) {
            // need special case for separating operator  ":" from scope operator "::"
            if (formula[i] == ':' && ((i + 1) < formula.Length())) {
               if (formula[i + 1] == ':') {
                  // case of :: (scopeOperator)
                  name.Append("::");
                  i += 2;
                  continue;
               } else
                  break;
            }
 
            name.Append(formula[i++]);
         }
         // printf(" build a name %s \n",name.Data() );
         if (formula[i] == '(') {
            i++;
            if (i < formula.Length() && formula[i] == ')') {
               fFuncs.push_back(TFormulaFunction(name, body, 0));
               name = body = "";
               continue;
            }
            Int_t depth = 1;
            Int_t args = 1; // we will miss first argument
            while (depth != 0 && i < formula.Length()) {
               switch (formula[i]) {
               case '(': depth++; break;
               case ')': depth--; break;
               case ',':
                  if (depth == 1)
                     args++;
                  break;
               }
               if (depth != 0) // we don't want last ')' inside body
               {
                  body.Append(formula[i++]);
               }
            }
            Int_t originalBodyLen = body.Length();
            ExtractFunctors(body);
            formula.Replace(i - originalBodyLen, originalBodyLen, body, body.Length());
            i += body.Length() - originalBodyLen;
            fFuncs.push_back(TFormulaFunction(name, body, args));
         } else {
 
            // std::cout << "check if character : " << i << " " << formula[i] << " from name " << name << "  is a
            // function " << std::endl;
 
            // check if function is provided by gROOT
            TObject *obj = nullptr;
            // exclude case function name is x,y,z,t
            if (!IsReservedName(name))
            {
               R__LOCKGUARD(gROOTMutex);
               obj = gROOT->GetListOfFunctions()->FindObject(name);
            }
            TFormula *f = dynamic_cast<TFormula *>(obj);
            if (!f) {
               // maybe object is a TF1
               TF1 *f1 = dynamic_cast<TF1 *>(obj);
               if (f1)
                  f = f1->GetFormula();
            }
            if (f) {
               // Replacing user formula the old way (as opposed to 'HandleFunctionArguments')
               // Note this is only for replacing functions that do
               // not specify variables and/or parameters in brackets
               // (the other case is done by `HandleFunctionArguments`)
 
               TString replacementFormula = f->GetExpFormula();
 
               // analyze expression string
               // std::cout << "formula to replace for " << f->GetName() << " is " << replacementFormula <<
               // std::endl;
               PreProcessFormula(replacementFormula);
               // we need to define different parameters if we use the unnamed default parameters ([0])
               // I need to replace all the terms in the functor for backward compatibility of the case
               // f1("[0]*x") f2("[0]*x") f1+f2 - it is weird but it is better to support
               // std::cout << "current number of parameter is " << fNpar << std::endl;
               int nparOffset = 0;
               // if (fParams.find("0") != fParams.end() ) {
               // do in any case if parameters are existing
               std::vector<TString> newNames;
               if (fNpar > 0) {
                  nparOffset = fNpar;
                  newNames.resize(f->GetNpar());
                  // start from higher number to avoid overlap
                  for (int jpar = f->GetNpar() - 1; jpar >= 0; --jpar) {
                     // parameters name have a "p" added in front
                     TString pj = TString(f->GetParName(jpar));
                     if (pj[0] == 'p' && TString(pj(1, pj.Length())).IsDigit()) {
                        TString oldName = TString::Format("[%s]", f->GetParName(jpar));
                        TString newName = TString::Format("[p%d]", nparOffset + jpar);
                        // std::cout << "replace - parameter " << f->GetParName(jpar) << " with " <<  newName <<
                        // std::endl;
                        replacementFormula.ReplaceAll(oldName, newName);
                        newNames[jpar] = newName;
                     } else
                        newNames[jpar] = f->GetParName(jpar);
                  }
                  // std::cout << "after replacing params " << replacementFormula << std::endl;
               }
               ExtractFunctors(replacementFormula);
               // std::cout << "after re-extracting functors " << replacementFormula << std::endl;
 
               // set parameter value from replacement formula
               for (int jpar = 0; jpar < f->GetNpar(); ++jpar) {
                  if (nparOffset > 0) {
                     // parameter have an offset- so take this into account
                     assert((int)newNames.size() == f->GetNpar());
                     SetParameter(newNames[jpar], f->GetParameter(jpar));
                  } else
                     // names are the same between current formula and replaced one
                     SetParameter(f->GetParName(jpar), f->GetParameter(jpar));
               }
               // need to add parenthesis at begin and end of replacementFormula
               replacementFormula.Insert(0, '(');
               replacementFormula.Insert(replacementFormula.Length(), ')');
               formula.Replace(i - name.Length(), name.Length(), replacementFormula, replacementFormula.Length());
               // move forward the index i of the main loop
               i += replacementFormula.Length() - name.Length();
 
               // we have extracted all the functor for "fname"
               // std::cout << "We have extracted all the functors for fname" << std::endl;
               // std::cout << " i = " << i << " f[i] = " << formula[i] << " - " << formula << std::endl;
               name = "";
 
               continue;
            }
 
            // add now functor in
            TString replacement = TString::Format("{%s}", name.Data());
            formula.Replace(i - name.Length(), name.Length(), replacement, replacement.Length());
            i += 2;
            fFuncs.push_back(TFormulaFunction(name));
         }
      }
      name = body = "";
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///    Iterates through functors in fFuncs and performs the appropriate action.
///    If functor has 0 arguments (has only name) can be:
///     - variable
///       * will be replaced with x[num], where x is an array containing value of this variable under num.
///     - pre-defined formula
///       * will be replaced with formulas body
///     - constant
///       * will be replaced with constant value
///     - parameter
///       * will be replaced with p[num], where p is an array containing value of this parameter under num.
///    If has arguments it can be :
///     - function shortcut, eg. sin
///       * will be replaced with fullname of function, eg. sin -> TMath::Sin
///     - function from cling environment, eg. TMath::BreitWigner(x,y,z)
///       * first check if function exists, and has same number of arguments, then accept it and set as found.
///    If all functors after iteration are matched with corresponding action,
///    it inputs C++ code of formula into cling, and sets flag that formula is ready to evaluate.
 
void TFormula::ProcessFormula(TString &formula)
{
   // std::cout << "Begin: formula is " << formula << " list of functors " << fFuncs.size() << std::endl;
 
   for (list<TFormulaFunction>::iterator funcsIt = fFuncs.begin(); funcsIt != fFuncs.end(); ++funcsIt) {
      TFormulaFunction &fun = *funcsIt;
 
      // std::cout << "fun is " << fun.GetName() << std::endl;
 
      if (fun.fFound)
         continue;
      if (fun.IsFuncCall()) {
         // replace with pre-defined functions
         map<TString, TString>::iterator it = fFunctionsShortcuts.find(fun.GetName());
         if (it != fFunctionsShortcuts.end()) {
            TString shortcut = it->first;
            TString full = it->second;
            // std::cout << " functor " << fun.GetName() << " found - replace " <<  shortcut << " with " << full << " in
            // " << formula << std::endl;
            // replace all functors
            Ssiz_t index = formula.Index(shortcut, 0);
            while (index != kNPOS) {
               // check that function is not in a namespace and is not in other characters
               // std::cout << "analyzing " << shortcut << " in " << formula << std::endl;
               Ssiz_t i2 = index + shortcut.Length();
               if ((index > 0) && (isalpha(formula[index - 1]) || formula[index - 1] == ':')) {
                  index = formula.Index(shortcut, i2);
                  continue;
               }
               if (i2 < formula.Length() && formula[i2] != '(') {
                  index = formula.Index(shortcut, i2);
                  continue;
               }
               // now replace the string
               formula.Replace(index, shortcut.Length(), full);
               Ssiz_t inext = index + full.Length();
               index = formula.Index(shortcut, inext);
               fun.fFound = true;
            }
         }
         // for functions we can live it to cling to decide if it is a valid function or NOT
         // We don't need to retrieve this information from the ROOT interpreter
         // we assume that the function is then found and all the following code does not need to be there
#ifdef TFORMULA_CHECK_FUNCTIONS
 
         if (fun.fName.Contains("::")) // add support for nested namespaces
         {
            // look for last occurrence of "::"
            std::string name(fun.fName.Data());
            size_t index = name.rfind("::");
            assert(index != std::string::npos);
            TString className = fun.fName(0, fun.fName(0, index).Length());
            TString functionName = fun.fName(index + 2, fun.fName.Length());
 
            Bool_t silent = true;
            TClass *tclass = TClass::GetClass(className, silent);
            // std::cout << "looking for class " << className << std::endl;
            const TList *methodList = tclass->GetListOfAllPublicMethods();
            TIter next(methodList);
            TMethod *p;
            while ((p = (TMethod *)next())) {
               if (strcmp(p->GetName(), functionName.Data()) == 0 &&
                   (fun.GetNargs() <= p->GetNargs() && fun.GetNargs() >= p->GetNargs() - p->GetNargsOpt())) {
                  fun.fFound = true;
                  break;
               }
            }
         }
         if (!fun.fFound) {
            // try to look into all the global functions in gROOT
            TFunction *f;
            {
               R__LOCKGUARD(gROOTMutex);
               f = (TFunction *)gROOT->GetListOfGlobalFunctions(true)->FindObject(fun.fName);
            }
            // if found a function with matching arguments
            if (f && fun.GetNargs() <= f->GetNargs() && fun.GetNargs() >= f->GetNargs() - f->GetNargsOpt()) {
               fun.fFound = true;
            }
         }
 
         if (!fun.fFound) {
            // ignore not found functions
            if (gDebug)
               Info("TFormula", "Could not find %s function with %d argument(s)", fun.GetName(), fun.GetNargs());
            fun.fFound = false;
         }
#endif
      } else {
         TFormula *old = nullptr;
         {
            R__LOCKGUARD(gROOTMutex);
            old = (TFormula *)gROOT->GetListOfFunctions()->FindObject(gNamePrefix + fun.fName);
         }
         if (old) {
            // we should not go here (this analysis is done before in ExtractFunctors)
            assert(false);
            fun.fFound = true;
            TString pattern = TString::Format("{%s}", fun.GetName());
            TString replacement = old->GetExpFormula();
            PreProcessFormula(replacement);
            ExtractFunctors(replacement);
            formula.ReplaceAll(pattern, replacement);
            continue;
         }
         // looking for default variables defined in fVars
 
         map<TString, TFormulaVariable>::iterator varsIt = fVars.find(fun.GetName());
         if (varsIt != fVars.end()) {
 
            TString name = (*varsIt).second.GetName();
            Double_t value = (*varsIt).second.fValue;
 
            AddVariable(name, value); // this set the cling variable
            if (!fVars[name].fFound) {
 
               fVars[name].fFound = true;
               int varDim = (*varsIt).second.fArrayPos; // variable dimensions (0 for x, 1 for y, 2, for z)
               if (varDim >= fNdim) {
                  fNdim = varDim + 1;
 
                  // we need to be sure that all other variables are added with position less
                  for (auto &v : fVars) {
                     if (v.second.fArrayPos < varDim && !v.second.fFound) {
                        AddVariable(v.first, v.second.fValue);
                        v.second.fFound = true;
                     }
                  }
               }
            }
            // remove the "{.. }" added around the variable
            TString pattern = TString::Format("{%s}", name.Data());
            TString replacement = TString::Format("x[%d]", (*varsIt).second.fArrayPos);
            formula.ReplaceAll(pattern, replacement);
 
            // std::cout << "Found an observable for " << fun.GetName()  << std::endl;
 
            fun.fFound = true;
            continue;
         }
         // check for observables defined as x[0],x[1],....
         // maybe could use a regular expression here
         // only in case match with defined variables is not successful
         TString funname = fun.GetName();
         if (funname.Contains("x[") && funname.Contains("]")) {
            TString sdigit = funname(2, funname.Index("]"));
            int digit = sdigit.Atoi();
            if (digit >= fNdim) {
               fNdim = digit + 1;
               // we need to add the variables in fVars all of them before x[n]
               for (int j = 0; j < fNdim; ++j) {
                  TString vname = TString::Format("x[%d]", j);
                  if (fVars.find(vname) == fVars.end()) {
                     fVars[vname] = TFormulaVariable(vname, 0, j);
                     fVars[vname].fFound = true;
                     AddVariable(vname, 0.);
                  }
               }
            }
            // std::cout << "Found matching observable for " << funname  << std::endl;
            fun.fFound = true;
            // remove the "{.. }" added around the variable
            TString pattern = TString::Format("{%s}", funname.Data());
            formula.ReplaceAll(pattern, funname);
            continue;
         }
         //}
 
         auto paramsIt = fParams.find(fun.GetName());
         if (paramsIt != fParams.end()) {
            // TString name = (*paramsIt).second.GetName();
            TString pattern = TString::Format("{[%s]}", fun.GetName());
            // std::cout << "pattern is " << pattern << std::endl;
            if (formula.Index(pattern) != kNPOS) {
               // TString replacement = TString::Format("p[%d]",(*paramsIt).second.fArrayPos);
               TString replacement = TString::Format("p[%d]", (*paramsIt).second);
               // std::cout << "replace pattern  " << pattern << " with " << replacement << std::endl;
               formula.ReplaceAll(pattern, replacement);
            }
            fun.fFound = true;
            continue;
         } else {
            // std::cout << "functor  " << fun.GetName() << " is not a parameter " << std::endl;
         }
 
         // looking for constants (needs to be done after looking at the parameters)
         map<TString, Double_t>::iterator constIt = fConsts.find(fun.GetName());
         if (constIt != fConsts.end()) {
            TString pattern = TString::Format("{%s}", fun.GetName());
            formula.ReplaceAll(pattern, doubleToString(constIt->second));
            fun.fFound = true;
            // std::cout << "constant with name " << fun.GetName() << " is found " << std::endl;
            continue;
         }
 
         fun.fFound = false;
      }
   }
   // std::cout << "End: formula is " << formula << std::endl;
 
   // ignore case of functors have been matched - try to pass it to Cling
   if (!fReadyToExecute) {
      fReadyToExecute = true;
      Bool_t hasVariables = (fNdim > 0);
      Bool_t hasParameters = (fNpar > 0);
      if (!hasParameters) {
         fAllParametersSetted = true;
      }
      // assume a function without variables is always 1-dimensional ???
      // if (hasParameters && !hasVariables) {
      //    fNdim = 1;
      //    AddVariable("x", 0);
      //    hasVariables = true;
      // }
      // does not make sense to vectorize function which is of FNDim=0
      if (!hasVariables) fVectorized=false;
      // when there are no variables but only parameter we still need to ad
      //Bool_t hasBoth = hasVariables && hasParameters;
      Bool_t inputIntoCling = (formula.Length() > 0);
      if (inputIntoCling) {
         // save copy of inputFormula in a std::strig for the unordered map
         // and also formula is same as FClingInput typically and it will be modified
         std::string inputFormula(formula.Data());
 
         // The name we really use for the unordered map will have a flag that
         // says whether the formula is vectorized
         std::string inputFormulaVecFlag = inputFormula;
         if (fVectorized)
            inputFormulaVecFlag += " (vectorized)";
 
         TString argType = fVectorized ? vectorizedArgType() : "Double_t";
 
         // valid input formula - try to put into Cling (in case of no variables but only parameter we need to add the standard signature)
         TString argumentsPrototype = TString::Format("%s%s%s", ( (hasVariables || hasParameters) ? (argType + " const *x").Data() : ""),
                                                      (hasParameters ? "," : ""), (hasParameters ? "Double_t *p" : ""));
 
         // set the name for Cling using the hash_function
         fClingName = gNamePrefix;
 
         // check if formula exist already in the map
         R__LOCKGUARD(gROOTMutex);
 
         // std::cout << "gClingFunctions list" << std::endl;
         // for (auto thing : gClingFunctions)
         //     std::cout << "gClingFunctions : " << thing.first << std::endl;
 
         auto funcit = gClingFunctions.find(inputFormulaVecFlag);
 
         if (funcit != gClingFunctions.end()) {
            fFuncPtr = (TFormula::CallFuncSignature)funcit->second;
            fClingInitialized = true;
            inputIntoCling = false;
         }
 
 
 
         // set the cling name using hash of the static formulae map
         auto hasher = gClingFunctions.hash_function();
         fClingName = TString::Format("%s__id%zu", gNamePrefix.Data(), hasher(inputFormulaVecFlag));
 
         fClingInput = TString::Format("%s %s(%s){ return %s ; }", argType.Data(), fClingName.Data(),
                                       argumentsPrototype.Data(), inputFormula.c_str());
 
 
         // std::cout << "Input Formula " << inputFormula << " \t vec formula  :  " << inputFormulaVecFlag << std::endl;
         // std::cout << "Cling functions existing " << std::endl;
         // for (auto & ff : gClingFunctions)
         //    std::cout << ff.first << std::endl;
         // std::cout << "\n";
         // std::cout << fClingName << std::endl;
 
         // this is not needed (maybe can be re-added in case of recompilation of identical expressions
         // // check in case of a change if need to re-initialize
         // if (fClingInitialized) {
         //    if (oldClingInput == fClingInput)
         //       inputIntoCling = false;
         //    else
         //       fClingInitialized = false;
         // }
 
         if (inputIntoCling) {
            if (!fLazyInitialization) {
               InputFormulaIntoCling();
               if (fClingInitialized) {
                  // if Cling has been successfully initialized
                  // put function ptr in the static map
                  R__LOCKGUARD(gROOTMutex);
                  gClingFunctions.insert(std::make_pair(inputFormulaVecFlag, (void *)fFuncPtr));
               }
            }
            if (!fClingInitialized) {
               // needed in case of lazy initialization of failure compiling the expression
               fSavedInputFormula = inputFormulaVecFlag;
            }
 
         } else {
            fAllParametersSetted = true;
            fClingInitialized = true;
         }
      }
   }
 
   // In case of a Cling Error check components which are not found in Cling
   // check that all formula components are matched otherwise emit an error
   if (!fClingInitialized && !fLazyInitialization) {
      //Bool_t allFunctorsMatched = false;
      for (list<TFormulaFunction>::iterator it = fFuncs.begin(); it != fFuncs.end(); ++it) {
         // functions are now by default always not checked
         if (!it->fFound && !it->IsFuncCall()) {
            //allFunctorsMatched = false;
            if (it->GetNargs() == 0)
               Error("ProcessFormula", "\"%s\" has not been matched in the formula expression", it->GetName());
            else
               Error("ProcessFormula", "Could not find %s function with %d argument(s)", it->GetName(), it->GetNargs());
         }
      }
      Error("ProcessFormula","Formula \"%s\" is invalid !", GetExpFormula().Data() );
      fReadyToExecute = false;
   }
 
   // clean up un-used default variables in case formula is valid
   //if (fClingInitialized && fReadyToExecute) {
   //don't check fClingInitialized in case of lazy execution
   if (fReadyToExecute) {
       auto itvar = fVars.begin();
       // need this loop because after erase change iterators
       do {
         if (!itvar->second.fFound) {
            // std::cout << "Erase variable " << itvar->first << std::endl;
            itvar = fVars.erase(itvar);
         } else
            itvar++;
      } while (itvar != fVars.end());
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Fill map with parametrized functions
 
void TFormula::FillParametrizedFunctions(map<pair<TString, Int_t>, pair<TString, TString>> &functions)
{
   // map< pair<TString,Int_t> ,pair<TString,TString> > functions;
   functions.insert(
      make_pair(make_pair("gaus", 1), make_pair("[0]*exp(-0.5*(({V0}-[1])/[2])*(({V0}-[1])/[2]))",
                                                "[0]*exp(-0.5*(({V0}-[1])/[2])*(({V0}-[1])/[2]))/(sqrt(2*pi)*[2])")));
   functions.insert(make_pair(make_pair("landau", 1), make_pair("[0]*TMath::Landau({V0},[1],[2],false)",
                                                                "[0]*TMath::Landau({V0},[1],[2],true)")));
   functions.insert(make_pair(make_pair("expo", 1), make_pair("exp([0]+[1]*{V0})", "")));
   functions.insert(
      make_pair(make_pair("crystalball", 1), make_pair("[0]*ROOT::Math::crystalball_function({V0},[3],[4],[2],[1])",
                                                       "[0]*ROOT::Math::crystalball_pdf({V0},[3],[4],[2],[1])")));
   functions.insert(
      make_pair(make_pair("breitwigner", 1), make_pair("[0]*ROOT::Math::breitwigner_pdf({V0},[2],[1])",
                                                       "[0]*ROOT::Math::breitwigner_pdf({V0},[2],[4],[1])")));
   // chebyshev polynomial
   functions.insert(make_pair(make_pair("cheb0", 1), make_pair("ROOT::Math::Chebyshev0({V0},[0])", "")));
   functions.insert(make_pair(make_pair("cheb1", 1), make_pair("ROOT::Math::Chebyshev1({V0},[0],[1])", "")));
   functions.insert(make_pair(make_pair("cheb2", 1), make_pair("ROOT::Math::Chebyshev2({V0},[0],[1],[2])", "")));
   functions.insert(make_pair(make_pair("cheb3", 1), make_pair("ROOT::Math::Chebyshev3({V0},[0],[1],[2],[3])", "")));
   functions.insert(
      make_pair(make_pair("cheb4", 1), make_pair("ROOT::Math::Chebyshev4({V0},[0],[1],[2],[3],[4])", "")));
   functions.insert(
      make_pair(make_pair("cheb5", 1), make_pair("ROOT::Math::Chebyshev5({V0},[0],[1],[2],[3],[4],[5])", "")));
   functions.insert(
      make_pair(make_pair("cheb6", 1), make_pair("ROOT::Math::Chebyshev6({V0},[0],[1],[2],[3],[4],[5],[6])", "")));
   functions.insert(
      make_pair(make_pair("cheb7", 1), make_pair("ROOT::Math::Chebyshev7({V0},[0],[1],[2],[3],[4],[5],[6],[7])", "")));
   functions.insert(make_pair(make_pair("cheb8", 1),
                              make_pair("ROOT::Math::Chebyshev8({V0},[0],[1],[2],[3],[4],[5],[6],[7],[8])", "")));
   functions.insert(make_pair(make_pair("cheb9", 1),
                              make_pair("ROOT::Math::Chebyshev9({V0},[0],[1],[2],[3],[4],[5],[6],[7],[8],[9])", "")));
   functions.insert(
      make_pair(make_pair("cheb10", 1),
                make_pair("ROOT::Math::Chebyshev10({V0},[0],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10])", "")));
   // 2-dimensional functions
   functions.insert(
      make_pair(make_pair("gaus", 2), make_pair("[0]*exp(-0.5*(({V0}-[1])/[2])^2 - 0.5*(({V1}-[3])/[4])^2)", "")));
   functions.insert(
      make_pair(make_pair("landau", 2),
                make_pair("[0]*TMath::Landau({V0},[1],[2],false)*TMath::Landau({V1},[3],[4],false)", "TMath::Landau({V0},[0],[1],true)*TMath::Landau({V1},[2],[3],true)")));
   functions.insert(
      make_pair(make_pair("expo", 2),
                make_pair("exp([0]+[1]*{V0}+[2]*{V1})", "")));
   // 3-dimensional function
   functions.insert(
      make_pair(make_pair("gaus", 3), make_pair("[0]*exp(-0.5*(({V0}-[1])/[2])^2 - 0.5*(({V1}-[3])/[4])^2 - 0.5*(({V2}-[5])/[6])^2)", "")));
   // 2-d gaussian with correlations
   functions.insert(
      make_pair(make_pair("bigaus", 2), make_pair("[0]*ROOT::Math::bigaussian_pdf({V0},{V1},[2],[4],[5],[1],[3])",
                                                  "[0]*ROOT::Math::bigaussian_pdf({V0},{V1},[2],[4],[5],[1],[3])")));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set parameter names only in case of pre-defined functions.
 
void TFormula::SetPredefinedParamNames() {
 
   if (fNumber == 0) return;
 
   if (fNumber == 100) { // Gaussian
      SetParName(0,"Constant");
      SetParName(1,"Mean");
      SetParName(2,"Sigma");
      return;
   }
   if (fNumber == 110) {
      SetParName(0,"Constant");
      SetParName(1,"MeanX");
      SetParName(2,"SigmaX");
      SetParName(3,"MeanY");
      SetParName(4,"SigmaY");
      return;
   }
   if (fNumber == 120) {
      SetParName(0,"Constant");
      SetParName(1,"MeanX");
      SetParName(2,"SigmaX");
      SetParName(3,"MeanY");
      SetParName(4,"SigmaY");
      SetParName(5,"MeanZ");
      SetParName(6,"SigmaZ");
      return;
   }
   if (fNumber == 112) {  // bigaus
      SetParName(0,"Constant");
      SetParName(1,"MeanX");
      SetParName(2,"SigmaX");
      SetParName(3,"MeanY");
      SetParName(4,"SigmaY");
      SetParName(5,"Rho");
      return;
   }
   if (fNumber == 200) { // exponential
      SetParName(0,"Constant");
      SetParName(1,"Slope");
      return;
   }
   if (fNumber == 400) { // landau
      SetParName(0,"Constant");
      SetParName(1,"MPV");
      SetParName(2,"Sigma");
      return;
   }
   if (fNumber == 500) { // crystal-ball
      SetParName(0,"Constant");
      SetParName(1,"Mean");
      SetParName(2,"Sigma");
      SetParName(3,"Alpha");
      SetParName(4,"N");
      return;
   }
   if (fNumber == 600) { // breit-wigner
      SetParName(0,"Constant");
      SetParName(1,"Mean");
      SetParName(2,"Gamma");
      return;
   }
   // if formula is a polynomial (or chebyshev), set parameter names
   // not needed anymore (p0 is assigned by default)
   // if (fNumber == (300+fNpar-1) ) {
   //    for (int i = 0; i < fNpar; i++) SetParName(i,TString::Format("p%d",i));
   //    return;
   // }
 
   // // general case if parameters are digits (XX) change to pXX
   // auto paramMap = fParams;  // need to copy the map because SetParName is going to modify it
   // for ( auto & p : paramMap) {
   //    if (p.first.IsDigit() )
   //        SetParName(p.second,TString::Format("p%s",p.first.Data()));
   // }
 
   return;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return linear part.
 
const TObject* TFormula::GetLinearPart(Int_t i) const
{
   if (!fLinearParts.empty()) {
      int n = fLinearParts.size();
      if (i < 0 || i >= n ) {
         Error("GetLinearPart","Formula %s has only %d linear parts - requested %d",GetName(),n,i);
         return nullptr;
      }
      return fLinearParts[i];
   }
   return nullptr;
}
 
////////////////////////////////////////////////////////////////////////////////
///    Adds variable to known variables, and reprocess formula.
 
void TFormula::AddVariable(const TString &name, double value)
{
   if (fVars.find(name) != fVars.end()) {
      TFormulaVariable &var = fVars[name];
      var.fValue = value;
 
      // If the position is not defined in the Cling vectors, make space for it
      // but normally is variable is defined in fVars a slot should be also present in fClingVariables
      if (var.fArrayPos < 0) {
         var.fArrayPos = fVars.size();
      }
      if (var.fArrayPos >= (int)fClingVariables.size()) {
         fClingVariables.resize(var.fArrayPos + 1);
      }
      fClingVariables[var.fArrayPos] = value;
   } else {
      TFormulaVariable var(name, value, fVars.size());
      fVars[name] = var;
      fClingVariables.push_back(value);
      if (!fFormula.IsNull()) {
         // printf("process formula again - %s \n",fClingInput.Data() );
         ProcessFormula(fClingInput);
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///    Adds multiple variables.
///    First argument is an array of pairs<TString,Double>, where
///    first argument is name of variable,
///    second argument represents value.
///    size - number of variables passed in first argument
 
void TFormula::AddVariables(const TString *vars, const Int_t size)
{
   Bool_t anyNewVar = false;
   for (Int_t i = 0; i < size; ++i) {
 
      const TString &vname = vars[i];
 
      TFormulaVariable &var = fVars[vname];
      if (var.fArrayPos < 0) {
 
         var.fName = vname;
         var.fArrayPos = fVars.size();
         anyNewVar = true;
         var.fValue = 0;
         if (var.fArrayPos >= (int)fClingVariables.capacity()) {
            Int_t multiplier = 2;
            if (fFuncs.size() > 100) {
               multiplier = TMath::Floor(TMath::Log10(fFuncs.size()) * 10);
            }
            fClingVariables.reserve(multiplier * fClingVariables.capacity());
         }
         fClingVariables.push_back(0.0);
      }
      // else
      // {
      //    var.fValue = v.second;
      //    fClingVariables[var.fArrayPos] = v.second;
      // }
   }
   if (anyNewVar && !fFormula.IsNull()) {
      ProcessFormula(fClingInput);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set the name of the formula. We need to allow the list of function to
/// properly handle the hashes.
 
void TFormula::SetName(const char* name)
{
   if (IsReservedName(name)) {
      Error("SetName", "The name \'%s\' is reserved as a TFormula variable name.\n"
                       "\tThis function will not be renamed.",
            name);
   } else {
      // Here we need to remove and re-add to keep the hashes consistent with
      // the underlying names.
      auto listOfFunctions = gROOT->GetListOfFunctions();
      TObject* thisAsFunctionInList = nullptr;
      R__LOCKGUARD(gROOTMutex);
      if (listOfFunctions){
         thisAsFunctionInList = listOfFunctions->FindObject(this);
         if (thisAsFunctionInList) listOfFunctions->Remove(thisAsFunctionInList);
      }
      TNamed::SetName(name);
      if (thisAsFunctionInList) listOfFunctions->Add(thisAsFunctionInList);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///
///    Sets multiple variables.
///    First argument is an array of pairs<TString,Double>, where
///    first argument is name of variable,
///    second argument represents value.
///    size - number of variables passed in first argument
 
void TFormula::SetVariables(const pair<TString,Double_t> *vars, const Int_t size)
{
   for(Int_t i = 0; i < size; ++i)
      {
      auto &v = vars[i];
      if (fVars.find(v.first) != fVars.end()) {
         fVars[v.first].fValue = v.second;
         fClingVariables[fVars[v.first].fArrayPos] = v.second;
      } else {
         Error("SetVariables", "Variable %s is not defined.", v.first.Data());
      }
      }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns variable value.
 
Double_t TFormula::GetVariable(const char *name) const
{
   const auto nameIt = fVars.find(name);
   if (fVars.end() == nameIt) {
      Error("GetVariable", "Variable %s is not defined.", name);
      return -1;
   }
   return nameIt->second.fValue;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns variable number (positon in array) given its name.
 
Int_t TFormula::GetVarNumber(const char *name) const
{
   const auto nameIt = fVars.find(name);
   if (fVars.end() == nameIt) {
      Error("GetVarNumber", "Variable %s is not defined.", name);
      return -1;
   }
   return nameIt->second.fArrayPos;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns variable name given its position in the array.
 
TString TFormula::GetVarName(Int_t ivar) const
{
   if (ivar < 0 || ivar >= fNdim) return "";
 
   // need to loop on the map to find corresponding variable
   for ( auto & v : fVars) {
      if (v.second.fArrayPos == ivar) return v.first;
   }
   Error("GetVarName","Variable with index %d not found !!",ivar);
   //return TString::Format("x%d",ivar);
   return "";
}
 
////////////////////////////////////////////////////////////////////////////////
/// Sets variable value.
 
void TFormula::SetVariable(const TString &name, Double_t value)
{
   if (fVars.find(name) == fVars.end()) {
      Error("SetVariable", "Variable %s is not defined.", name.Data());
      return;
   }
   fVars[name].fValue = value;
   fClingVariables[fVars[name].fArrayPos] = value;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Adds parameter to known parameters.
/// User should use SetParameter, because parameters are added during initialization part,
/// and after that adding new will be pointless.
 
void TFormula::DoAddParameter(const TString &name, Double_t value, Bool_t processFormula)
{
   //std::cout << "adding parameter " << name << std::endl;
 
   // if parameter is already defined in fParams - just set the new value
   if(fParams.find(name) != fParams.end() )
      {
      int ipos = fParams[name];
      // TFormulaVariable & par = fParams[name];
      // par.fValue = value;
      if (ipos < 0) {
         ipos = fParams.size();
         fParams[name] = ipos;
      }
      //
      if (ipos >= (int)fClingParameters.size()) {
         if (ipos >= (int)fClingParameters.capacity())
            fClingParameters.reserve(TMath::Max(int(fParams.size()), ipos + 1));
         fClingParameters.insert(fClingParameters.end(), ipos + 1 - fClingParameters.size(), 0.0);
      }
      fClingParameters[ipos] = value;
   } else {
      // new parameter defined
      fNpar++;
      // TFormulaVariable(name,value,fParams.size());
      int pos = fParams.size();
      // fParams.insert(std::make_pair<TString,TFormulaVariable>(name,TFormulaVariable(name,value,pos)));
      auto ret = fParams.insert(std::make_pair(name, pos));
      // map returns a std::pair<iterator, bool>
      // use map the order for default position of parameters in the vector
      // (i.e use the alphabetic order)
      if (ret.second) {
         // a new element is inserted
         if (ret.first == fParams.begin())
            pos = 0;
         else {
            auto previous = (ret.first);
            --previous;
            pos = previous->second + 1;
         }
 
         if (pos < (int)fClingParameters.size())
            fClingParameters.insert(fClingParameters.begin() + pos, value);
         else {
            // this should not happen
            if (pos > (int)fClingParameters.size())
               Warning("inserting parameter %s at pos %d when vector size is  %d \n", name.Data(), pos,
                       (int)fClingParameters.size());
 
            if (pos >= (int)fClingParameters.capacity())
               fClingParameters.reserve(TMath::Max(int(fParams.size()), pos + 1));
            fClingParameters.insert(fClingParameters.end(), pos + 1 - fClingParameters.size(), 0.0);
            fClingParameters[pos] = value;
         }
 
         // need to adjust all other positions
         for (auto it = ret.first; it != fParams.end(); ++it) {
            it->second = pos;
            pos++;
         }
 
         // for (auto & p : fParams)
         //    std::cout << "Parameter " << p.first << " position " << p.second << " value " <<
         //    fClingParameters[p.second] << std::endl;
         // printf("inserted parameters size params %d size cling %d \n",fParams.size(), fClingParameters.size() );
      }
      if (processFormula) {
         // replace first in input parameter name with [name]
         fClingInput.ReplaceAll(name, TString::Format("[%s]", name.Data()));
         ProcessFormula(fClingInput);
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return parameter index given a name (return -1 for not existing parameters)
/// non need to print an error
 
Int_t TFormula::GetParNumber(const char * name) const {
   auto it = fParams.find(name);
   if (it == fParams.end()) {
      return -1;
   }
   return it->second;
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns parameter value given by string.
 
Double_t TFormula::GetParameter(const char * name) const
{
   const int i = GetParNumber(name);
   if (i  == -1) {
      Error("GetParameter","Parameter %s is not defined.",name);
      return TMath::QuietNaN();
   }
 
   return GetParameter( i );
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return parameter value given by integer.
 
Double_t TFormula::GetParameter(Int_t param) const
{
   //TString name = TString::Format("%d",param);
   if(param >=0 && param < (int) fClingParameters.size())
      return fClingParameters[param];
   Error("GetParameter","wrong index used - use GetParameter(name)");
   return TMath::QuietNaN();
}
 
////////////////////////////////////////////////////////////////////////////////
///    Return parameter name given by integer.
 
const char * TFormula::GetParName(Int_t ipar) const
{
   if (ipar < 0 || ipar >= fNpar) return "";
 
   // need to loop on the map to find corresponding parameter
   for ( auto & p : fParams) {
      if (p.second == ipar) return p.first.Data();
   }
   Error("GetParName","Parameter with index %d not found !!",ipar);
   //return TString::Format("p%d",ipar);
   return "";
}
 
////////////////////////////////////////////////////////////////////////////////
Double_t* TFormula::GetParameters() const
{
   if(!fClingParameters.empty())
      return const_cast<Double_t*>(&fClingParameters[0]);
   return nullptr;
}
 
void TFormula::GetParameters(Double_t *params) const
{
   for (Int_t i = 0; i < fNpar; ++i) {
      if (Int_t(fClingParameters.size()) > i)
         params[i] = fClingParameters[i];
      else
         params[i] = -1;
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Sets parameter value.
 
void TFormula::SetParameter(const char *name, Double_t value)
{
   SetParameter( GetParNumber(name), value);
 
   // do we need this ???
#ifdef OLDPARAMS
   if (fParams.find(name) == fParams.end()) {
      Error("SetParameter", "Parameter %s is not defined.", name.Data());
      return;
   }
   fParams[name].fValue = value;
   fParams[name].fFound = true;
   fClingParameters[fParams[name].fArrayPos] = value;
   fAllParametersSetted = true;
   for (map<TString, TFormulaVariable>::iterator it = fParams.begin(); it != fParams.end(); ++it) {
      if (!it->second.fFound) {
         fAllParametersSetted = false;
         break;
      }
   }
#endif
}
 
#ifdef OLDPARAMS
 
////////////////////////////////////////////////////////////////////////////////
/// Set multiple parameters.
/// First argument is an array of pairs<TString,Double>, where
/// first argument is name of parameter,
/// second argument represents value.
/// size - number of params passed in first argument
 
void TFormula::SetParameters(const pair<TString,Double_t> *params,const Int_t size)
{
   for(Int_t i = 0 ; i < size ; ++i)
      {
      pair<TString, Double_t> p = params[i];
      if (fParams.find(p.first) == fParams.end()) {
         Error("SetParameters", "Parameter %s is not defined", p.first.Data());
         continue;
      }
      fParams[p.first].fValue = p.second;
      fParams[p.first].fFound = true;
      fClingParameters[fParams[p.first].fArrayPos] = p.second;
   }
   fAllParametersSetted = true;
   for (map<TString, TFormulaVariable>::iterator it = fParams.begin(); it != fParams.end(); ++it) {
      if (!it->second.fFound) {
         fAllParametersSetted = false;
         break;
      }
   }
}
#endif
 
////////////////////////////////////////////////////////////////////////////////
void TFormula::DoSetParameters(const Double_t *params, Int_t size)
{
   if(!params || size < 0 || size > fNpar) return;
   // reset vector of cling parameters
   if (size != (int) fClingParameters.size() ) {
      Warning("SetParameters","size is not same of cling parameter size %d - %d",size,int(fClingParameters.size()) );
      for (Int_t i = 0; i < size; ++i) {
         TString name = TString::Format("%d", i);
         SetParameter(name, params[i]);
      }
      return;
   }
   fAllParametersSetted = true;
   std::copy(params, params+size, fClingParameters.begin() );
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set a vector of parameters value.
/// Order in the vector is by default the alphabetic order given to the parameters
/// apart if the users has defined explicitly the parameter names
 
void TFormula::SetParameters(const Double_t *params)
{
   DoSetParameters(params,fNpar);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set a parameter given a parameter index.
/// The parameter index is by default the alphabetic order given to the parameters,
/// apart if the users has defined explicitly the parameter names.
 
void TFormula::SetParameter(Int_t param, Double_t value)
{
   if (param < 0 || param >= fNpar) return;
   assert(int(fClingParameters.size()) == fNpar);
   fClingParameters[param] = value;
   // TString name = TString::Format("%d",param);
   // SetParameter(name,value);
}
 
////////////////////////////////////////////////////////////////////////////////
void TFormula::SetParName(Int_t ipar, const char * name)
{
 
   if (ipar < 0 || ipar > fNpar) {
      Error("SetParName","Wrong Parameter index %d ",ipar);
      return;
   }
   TString oldName;
   // find parameter with given index
   for ( auto &it : fParams) {
      if (it.second  == ipar) {
         oldName =  it.first;
         fParams.erase(oldName);
         fParams.insert(std::make_pair(name, ipar) );
         break;
      }
   }
   if (oldName.IsNull() ) {
      Error("SetParName","Parameter %d is not existing.",ipar);
      return;
   }
 
   //replace also parameter name in formula expression in case is not a lambda
   if (! TestBit(TFormula::kLambda))  ReplaceParamName(fFormula, oldName, name);
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Replace in Formula expression the parameter name.
 
void TFormula::ReplaceParamName(TString & formula, const TString & oldName, const TString & name){
   if (!formula.IsNull() ) {
      bool found = false;
      for(list<TFormulaFunction>::iterator it = fFuncs.begin(); it != fFuncs.end(); ++it)
         {
         if (oldName == it->GetName()) {
            found = true;
            it->fName = name;
            break;
         }
      }
      if (!found) {
         Error("SetParName", "Parameter %s is not defined.", oldName.Data());
         return;
      }
      // change whitespace to \s to avoid problems in parsing
      TString newName = name;
      newName.ReplaceAll(" ", "\\s");
      TString pattern = TString::Format("[%s]", oldName.Data());
      TString replacement = TString::Format("[%s]", newName.Data());
      formula.ReplaceAll(pattern, replacement);
   }
 
}
 
////////////////////////////////////////////////////////////////////////////////
void TFormula::SetVectorized(Bool_t vectorized)
{
#ifdef R__HAS_STD_EXPERIMENTAL_SIMD
   if (fNdim == 0) {
      Info("SetVectorized","Cannot vectorized a function of zero dimension");
      return;
   }
   if (vectorized != fVectorized) {
      if (!fFormula)
         Error("SetVectorized", "Cannot set vectorized to %d -- Formula is missing", vectorized);
 
      fVectorized = vectorized;
      // no need to JIT a new signature in case of zero dimension
      //if (fNdim== 0) return;
      fClingInitialized = false;
      fReadyToExecute = false;
      fClingName = "";
      fClingInput = fFormula;
 
      fMethod.reset();
 
      PreProcessFormula(fFormula);
      PrepareFormula(fFormula);
   }
#else
   if (vectorized)
      Warning("SetVectorized", "Cannot set vectorized -- try building with C++20 on Linux");
#endif
}
 
////////////////////////////////////////////////////////////////////////////////
Double_t TFormula::EvalPar(const Double_t *x,const Double_t *params) const
{
   if (!fVectorized)
      return DoEval(x, params);
 
#ifdef R__HAS_STD_EXPERIMENTAL_SIMD
 
   if (fNdim == 0 || !x) {
      ROOT::Double_v ret =  DoEvalVec(nullptr, params);
      return ret[0];
   }
 
    // otherwise, regular Double_t inputs on a vectorized function
 
   // convert our input into vectors then convert back
   if (gDebug)
      Info("EvalPar", "Function is vectorized - converting Double_t into ROOT::Double_v and back");
 
   if (fNdim < 5) {
      const int maxDim = 4;
      std::array<ROOT::Double_v, maxDim> xvec;
      for (int i = 0; i < fNdim; i++)
         xvec[i] = x[i];
 
      ROOT::Double_v ans = DoEvalVec(xvec.data(), params);
      return ans[0];
   }
   // allocating a vector is much slower (we do only for dim > 4)
   std::vector<ROOT::Double_v> xvec(fNdim);
   for (int i = 0; i < fNdim; i++)
      xvec[i] = x[i];
 
   ROOT::Double_v ans = DoEvalVec(xvec.data(), params);
   return ans[0];
 
#else
   // this should never happen, because fVectorized can only be set true with
   // R__HAS_STD_EXPERIMENTAL_SIMD, but just in case:
   Error("EvalPar", "Formula is vectorized (even though vectorizaton is not supported!)");
   return TMath::QuietNaN();
#endif
}
 
bool TFormula::fIsCladRuntimeIncluded = false;
 
static bool functionExists(const string &Name) {
   return gInterpreter->GetFunction(/*cl*/nullptr, Name.c_str());
}
 
static void IncludeCladRuntime(Bool_t &IsCladRuntimeIncluded) {
#ifdef ROOT_SUPPORT_CLAD
   if (!IsCladRuntimeIncluded) {
      IsCladRuntimeIncluded = true;
      gInterpreter->Declare("#include <Math/CladDerivator.h>\n#pragma clad OFF");
   }
#else
   IsCladRuntimeIncluded = false;
#endif
}
 
static bool
DeclareGenerationInput(std::string FuncName, std::string CladStatement,
                       std::string &GenerationInput) {
   std::string ReqFuncName = FuncName + "_req";
   // We want to call clad::differentiate(TFormula_id);
   GenerationInput = std::string("#pragma cling optimize(2)\n") +
       "#pragma clad ON\n" +
       "void " + ReqFuncName + "() {\n" +
       CladStatement + "\n " +
       "}\n" +
       "#pragma clad OFF";
 
   return gInterpreter->Declare(GenerationInput.c_str());
}
 
static TInterpreter::CallFuncIFacePtr_t::Generic_t
GetFuncPtr(std::string FuncName, Int_t Npar, Int_t Ndim, Bool_t Vectorized) {
   Bool_t hasParameters = (Npar > 0);
   Bool_t hasVariables = (Ndim > 0);
   std::unique_ptr<TMethodCall>
       method = prepareMethod(hasParameters, hasVariables, FuncName.c_str(),
                              Vectorized, /*AddCladArrayRef*/ true);
   return prepareFuncPtr(method.get());
}
 
static void CallCladFunction(TInterpreter::CallFuncIFacePtr_t::Generic_t FuncPtr, const Double_t *vars,
                             const Double_t *pars, Double_t *result, const Int_t /*result_size*/)
{
   void *args[3];
   args[0] = &vars;
   if (!pars) {
      // __attribute__((used)) extern "C" void __cf_0(void* obj, int nargs, void** args, void* ret)
      // {
      //    if (ret) {
      //       new (ret) (double) (((double (&)(double*))TFormula____id)(*(double**)args[0]));
      //       return;
      //    } else {
      //       ((double (&)(double*))TFormula____id)(*(double**)args[0]);
      //       return;
      //    }
      // }
      args[1] = &result;
      (*FuncPtr)(nullptr, 2, args, /*ret*/ nullptr); // We do not use ret in a return-void func.
   } else {
      // __attribute__((used)) extern "C" void __cf_0(void* obj, int nargs, void** args, void* ret)
      // {
      //    ((void (&)(double*, double*, double*))TFormula____id_grad_1)(*(double**)args[0],
      //                                                                 *(double**)args[1],
      //                                                                 *(double**)args[2]);
      //    return;
      // }
      args[1] = &pars;
      args[2] = &result;
      (*FuncPtr)(nullptr, 3, args, /*ret*/nullptr); // We do not use ret in a return-void func.
   }
}
 
/// returns true on success.
bool TFormula::GenerateGradientPar() {
   // We already have generated the gradient.
   if (fGradFuncPtr)
      return true;
 
   if (HasGradientGenerationFailed())
      return false;
 
   IncludeCladRuntime(fIsCladRuntimeIncluded);
   if (!fIsCladRuntimeIncluded)
      return false;
    
   // Check if the gradient request was made as part of another TFormula.
   // This can happen when we create multiple TFormula objects with the same
   // formula. In that case, the hasher will give identical id and we can
   // reuse the already generated gradient function.
   if (!functionExists(GetGradientFuncName())) {
      std::string GradientCall
          ("clad::gradient(" + std::string(fClingName.Data()) + ", \"p\");");
      if (!DeclareGenerationInput(GetGradientFuncName(),
                                  GradientCall,
                                  fGradGenerationInput))
         return false;
   }
 
   fGradFuncPtr = GetFuncPtr(GetGradientFuncName(), fNpar, fNdim, fVectorized);
   return true;
}
 
// Compute the gradient with respect to the parameter passing
/// a CladStorageObject, i.e. a std::vector, which has the size as the nnumber of parameters.
/// Note that the result buffer needs to be initialized to zero before passing it to this function.
void TFormula::GradientPar(const Double_t *x, TFormula::CladStorage& result)
{
   if (DoEval(x) == TMath::QuietNaN())
      return;
 
   if (!fClingInitialized) {
      Error("GradientPar", "Could not initialize the formula!");
      return;
   }
 
   if (!GenerateGradientPar()) {
      Error("GradientPar", "Could not generate a gradient for the formula %s!",
            fClingName.Data());
      return;
   }
 
   if ((int)result.size() < fNpar) {
      Warning("GradientPar",
              "The size of gradient result is %zu but %d is required. Resizing.",
              result.size(), fNpar);
      result.resize(fNpar);
   }
   GradientPar(x, result.data());
}
/// Compute the gradient with respect to the parameter passing
/// a buffer with a size at least equal to the number of parameters.
/// Note that the result buffer needs to be initialized to zero before passed to this function.
void TFormula::GradientPar(const Double_t *x, Double_t *result) {
   const Double_t *vars = (x) ? x : fClingVariables.data();
   const Double_t *pars = (fNpar <= 0) ? nullptr : fClingParameters.data();
   CallCladFunction(fGradFuncPtr, vars, pars, result, fNpar);
}
 
/// returns true on success.
bool TFormula::GenerateHessianPar()
{
   // We already have generated the hessian.
   if (fHessFuncPtr)
      return true;
 
   if (HasHessianGenerationFailed())
      return false;
 
   IncludeCladRuntime(fIsCladRuntimeIncluded);
   if (!fIsCladRuntimeIncluded)
      return false;
 
   // Check if the hessian request was made as part of another TFormula.
   // This can happen when we create multiple TFormula objects with the same
   // formula. In that case, the hasher will give identical id and we can
   // reuse the already generated hessian function.
   if (!functionExists(GetHessianFuncName())) {
      std::string indexes = (fNpar - 1 == 0) ? "0" : std::string("0:")
          + std::to_string(fNpar - 1);
      std::string HessianCall
          ("clad::hessian(" + std::string(fClingName.Data()) + ", \"p["
               + indexes + "]\" );");
      if (!DeclareGenerationInput(GetHessianFuncName(), HessianCall,
                                  fHessGenerationInput))
         return false;
   }
 
   fHessFuncPtr = GetFuncPtr(GetHessianFuncName(), fNpar, fNdim, fVectorized);
   return true;
}
 
void TFormula::HessianPar(const Double_t *x, TFormula::CladStorage& result)
{
   if (DoEval(x) == TMath::QuietNaN())
      return;
 
   if (!fClingInitialized) {
      Error("HessianPar", "Could not initialize the formula!");
      return;
   }
 
   if (!GenerateHessianPar()) {
      Error("HessianPar", "Could not generate a hessian for the formula %s!",
            fClingName.Data());
      return;
   }
 
   if ((int)result.size() < fNpar) {
      Warning("HessianPar",
              "The size of hessian result is %zu but %d is required. Resizing.",
              result.size(), fNpar * fNpar);
      result.resize(fNpar * fNpar);
   }
   HessianPar(x, result.data());
}
 
void TFormula::HessianPar(const Double_t *x, Double_t *result) {
   const Double_t *vars = (x) ? x : fClingVariables.data();
   const Double_t *pars = (fNpar <= 0) ? nullptr : fClingParameters.data();
   CallCladFunction(fHessFuncPtr, vars, pars, result, fNpar * fNpar);
}
 
////////////////////////////////////////////////////////////////////////////////
#ifdef R__HAS_STD_EXPERIMENTAL_SIMD
// ROOT::Double_v TFormula::Eval(ROOT::Double_v x, ROOT::Double_v y, ROOT::Double_v z, ROOT::Double_v t) const
// {
//    ROOT::Double_v xxx[] = {x, y, z, t};
//    return EvalPar(xxx, nullptr);
// }
 
ROOT::Double_v TFormula::EvalParVec(const ROOT::Double_v *x, const Double_t *params) const
{
   if (fVectorized)
      return DoEvalVec(x, params);
 
   if (fNdim == 0 || !x)
      return DoEval(nullptr, params); // automatic conversion to vectorized
 
   // otherwise, trying to input vectors into a scalar function
 
   if (gDebug)
      Info("EvalPar", "Function is not vectorized - converting ROOT::Double_v into Double_t and back");
 
   const int vecSize = ROOT::Double_v::size();
   std::vector<Double_t>  xscalars(vecSize*fNdim);
 
   for (int i = 0; i < vecSize; i++)
      for (int j = 0; j < fNdim; j++)
         xscalars[i * fNdim + j] = x[j][i];
 
   ROOT::Double_v answers(0.);
   for (int i = 0; i < vecSize; i++)
      answers[i] = DoEval(&xscalars[i * fNdim], params);
 
   return answers;
}
#endif
 
////////////////////////////////////////////////////////////////////////////////
/// Evaluate formula.
/// If formula is not ready to execute(missing parameters/variables),
/// print these which are not known.
/// If parameter has default value, and has not been set, appropriate warning is shown.
 
Double_t TFormula::DoEval(const double * x, const double * params) const
{
   if(!fReadyToExecute)
      {
      Error("Eval", "Formula is invalid and not ready to execute ");
      for (auto it = fFuncs.begin(); it != fFuncs.end(); ++it) {
         TFormulaFunction fun = *it;
         if (!fun.fFound) {
            printf("%s is unknown.\n", fun.GetName());
         }
      }
      return TMath::QuietNaN();
   }
 
   // Lazy initialization is set and needed when reading from a file
   if (!fClingInitialized && fLazyInitialization) {
      // try recompiling the formula. We need to lock because this is not anymore thread safe
      R__LOCKGUARD(gROOTMutex);
      // check again in case another thread has initialized the formula (see ROOT-10994)
      if (!fClingInitialized) {
         auto thisFormula = const_cast<TFormula*>(this);
         thisFormula->ReInitializeEvalMethod();
      }
      if (!fClingInitialized) {
         Error("DoEval", "Formula has error and  it is not properly initialized ");
         return TMath::QuietNaN();
      }
   }
 
   if (fLambdaPtr && TestBit(TFormula::kLambda)) {// case of lambda functions
      std::function<double(double *, double *)> & fptr = * ( (std::function<double(double *, double *)> *) fLambdaPtr);
      assert(x);
      //double * v = (x) ? const_cast<double*>(x) : const_cast<double*>(fClingVariables.data());
      double * v = const_cast<double*>(x);
      double * p = (params) ? const_cast<double*>(params) : const_cast<double*>(fClingParameters.data());
      return fptr(v, p);
   }
 
 
   Double_t result = 0;
   void* args[2];
   double * vars = (x) ? const_cast<double*>(x) : const_cast<double*>(fClingVariables.data());
   args[0] = &vars;
   if (fNpar <= 0) {
      (*fFuncPtr)(nullptr, 1, args, &result);
   } else {
      double *pars = (params) ? const_cast<double *>(params) : const_cast<double *>(fClingParameters.data());
      args[1] = &pars;
      (*fFuncPtr)(nullptr, 2, args, &result);
   }
   return result;
}
 
////////////////////////////////////////////////////////////////////////////////
// Copied from DoEval, but this is the vectorized version
#ifdef R__HAS_STD_EXPERIMENTAL_SIMD
ROOT::Double_v TFormula::DoEvalVec(const ROOT::Double_v *x, const double *params) const
{
   if (!fReadyToExecute) {
      Error("Eval", "Formula is invalid and not ready to execute ");
      for (auto it = fFuncs.begin(); it != fFuncs.end(); ++it) {
         TFormulaFunction fun = *it;
         if (!fun.fFound) {
            printf("%s is unknown.\n", fun.GetName());
         }
      }
      return TMath::QuietNaN();
   }
   // todo maybe save lambda ptr stuff for later
 
   if (!fClingInitialized && fLazyInitialization) {
      // try recompiling the formula. We need to lock because this is not anymore thread safe
      R__LOCKGUARD(gROOTMutex);
      // check again in case another thread has initialized the formula (see ROOT-10994)
      if (!fClingInitialized) {
         auto thisFormula = const_cast<TFormula*>(this);
         thisFormula->ReInitializeEvalMethod();
      }
      if (!fClingInitialized) {
         Error("DoEval", "Formula has error and  it is not properly initialized ");
         ROOT::Double_v res = TMath::QuietNaN();
         return res;
      }
   }
 
   ROOT::Double_v result = 0;
   void *args[2];
 
   ROOT::Double_v *vars = const_cast<ROOT::Double_v *>(x);
   args[0] = &vars;
   if (fNpar <= 0) {
      (*fFuncPtr)(0, 1, args, &result);
   }else {
      double *pars = (params) ? const_cast<double *>(params) : const_cast<double *>(fClingParameters.data());
      args[1] = &pars;
      (*fFuncPtr)(0, 2, args, &result);
   }
   return result;
}
#endif // R__HAS_STD_EXPERIMENTAL_SIMD
 
//////////////////////////////////////////////////////////////////////////////
/// Re-initialize eval method
///
/// This function is called by DoEval and DoEvalVector in case of a previous failure
///  or in case of reading from a file
////////////////////////////////////////////////////////////////////////////////
void TFormula::ReInitializeEvalMethod() {
 
 
   if (TestBit(TFormula::kLambda) ) {
      Info("ReInitializeEvalMethod","compile now lambda expression function using Cling");
      InitLambdaExpression(fFormula);
      fLazyInitialization = false;
      return;
   }
   fMethod.reset();
 
   if (!fLazyInitialization)   Warning("ReInitializeEvalMethod", "Formula is NOT properly initialized - try calling again TFormula::PrepareEvalMethod");
   //else  Info("ReInitializeEvalMethod", "Compile now the formula expression using Cling");
 
   // check first if formula exists in the global map
   {
 
      R__LOCKGUARD(gROOTMutex);
 
      // std::cout << "gClingFunctions list" << std::endl;
      //  for (auto thing : gClingFunctions)
      //     std::cout << "gClingFunctions : " << thing.first << std::endl;
 
      auto funcit = gClingFunctions.find(fSavedInputFormula);
 
      if (funcit != gClingFunctions.end()) {
         fFuncPtr = (TFormula::CallFuncSignature)funcit->second;
         fClingInitialized = true;
         fLazyInitialization = false;
         return;
      }
   }
   // compile now formula using cling
   InputFormulaIntoCling();
   if (fClingInitialized && !fLazyInitialization) Info("ReInitializeEvalMethod", "Formula is now properly initialized !!");
   fLazyInitialization = false;
 
   // add function pointer in global map
   if (fClingInitialized) {
      R__ASSERT(!fSavedInputFormula.empty());
      // if Cling has been successfully initialized
      // put function ptr in the static map
      R__LOCKGUARD(gROOTMutex);
      gClingFunctions.insert(std::make_pair(fSavedInputFormula, (void *)fFuncPtr));
   }
 
 
   return;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return the expression formula.
///
///  - If option = "P" replace the parameter names with their values
///  - If option = "CLING" return the actual expression used to build the function passed to Cling
///  - If option = "CLINGP" replace in the Cling expression the parameter with their values
 
TString TFormula::GetExpFormula(Option_t *option) const
{
   TString opt(option);
   if (opt.IsNull() || TestBit(TFormula::kLambda) ) return fFormula;
   opt.ToUpper();
 
   //  if (opt.Contains("N") ) {
   //    TString formula = fFormula;
   //    ReplaceParName(formula, ....)
   // }
 
   if (opt.Contains("CLING") ) {
      std::string clingFunc = fClingInput.Data();
      std::size_t found = clingFunc.find("return");
      std::size_t found2 = clingFunc.rfind(';');
      if (found == std::string::npos || found2 == std::string::npos) {
         Error("GetExpFormula","Invalid Cling expression - return default formula expression");
         return fFormula;
      }
      TString clingFormula = fClingInput(found+7,found2-found-7);
      // to be implemented
      if (!opt.Contains("P")) return clingFormula;
      // replace all "p[" with "[parname"
      int i = 0;
      while (i < clingFormula.Length()-2 ) {
         // look for p[number
         if (clingFormula[i] == 'p' && clingFormula[i+1] == '[' && isdigit(clingFormula[i+2]) ) {
            int j = i+3;
            while ( j < clingFormula.Length() && isdigit(clingFormula[j]) ) { j++;}
            if ( j >= clingFormula.Length() || clingFormula[j] != ']') {
               Error("GetExpFormula","Parameters not found - invalid expression - return default cling formula");
               return clingFormula;
            }
            TString parNumbName = clingFormula(i+2,j-i-2);
            int parNumber = parNumbName.Atoi();
            assert(parNumber < fNpar);
            std::stringstream ss;
            std::string replacement = doubleToString(GetParameter(parNumber));
            clingFormula.Replace(i,j-i+1, replacement);
            i += replacement.size();
         }
         i++;
      }
      return clingFormula;
   }
   if (opt.Contains("P") ) {
      // replace parameter names with their values
      TString expFormula = fFormula;
      int i = 0;
      while (i < expFormula.Length()-2 ) {
         // look for [parName]
         if (expFormula[i] == '[') {
            int j = i+1;
            while ( j < expFormula.Length() && expFormula[j] != ']' ) { j++;}
            if ( j >= expFormula.Length() || expFormula[j] != ']') {
               Error("GetExpFormula","Parameter names not found - invalid expression - return default formula");
               return expFormula;
            }
            TString parName = expFormula(i+1,j-i-1);
            std::stringstream ss;
            std::string replacement = doubleToString(GetParameter(parName));
            expFormula.Replace(i,j-i+1, replacement);
            i += replacement.size();
         }
         i++;
      }
      return expFormula;
   }
   Warning("GetExpFormula","Invalid option - return default formula expression");
   return fFormula;
}
 
TString TFormula::GetGradientFormula() const {
   std::unique_ptr<TInterpreterValue> v = gInterpreter->MakeInterpreterValue();
   std::string s("(void (&)(Double_t const *, Double_t *, Double_t *)) ");
   s += GetGradientFuncName();
   gInterpreter->Evaluate(s.c_str(), *v);
   return v->ToString();
}
 
TString TFormula::GetHessianFormula() const {
   std::unique_ptr<TInterpreterValue> v = gInterpreter->MakeInterpreterValue();
   gInterpreter->Evaluate(GetHessianFuncName().c_str(), *v);
   return v->ToString();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Print the formula and its attributes.
 
void TFormula::Print(Option_t *option) const
{
   printf(" %20s : %s Ndim= %d, Npar= %d, Number= %d \n",GetName(),GetTitle(), fNdim,fNpar,fNumber);
   printf(" Formula expression: \n");
   printf("\t%s \n",fFormula.Data() );
   TString opt(option);
   opt.ToUpper();
   // do an evaluation as a cross-check
   //if (fReadyToExecute) Eval();
 
   if (opt.Contains("V") ) {
      if (fNdim > 0 && !TestBit(TFormula::kLambda)) {
         printf("List of  Variables: \n");
         assert(int(fClingVariables.size()) >= fNdim);
         for ( int ivar = 0; ivar < fNdim ; ++ivar) {
            printf("Var%4d %20s =  %10f \n",ivar,GetVarName(ivar).Data(), fClingVariables[ivar]);
         }
      }
      if (fNpar > 0) {
         printf("List of  Parameters: \n");
         if ( int(fClingParameters.size()) < fNpar)
            Error("Print","Number of stored parameters in vector %zu in map %zu is different than fNpar %d",fClingParameters.size(), fParams.size(), fNpar);
         assert(int(fClingParameters.size()) >= fNpar);
         // print with order passed to Cling function
         for ( int ipar = 0; ipar < fNpar ; ++ipar) {
            printf("Par%4d %20s =  %10f \n",ipar,GetParName(ipar), fClingParameters[ipar] );
         }
      }
      printf("Expression passed to Cling:\n");
      printf("\t%s\n",fClingInput.Data() );
      if (fGradFuncPtr) {
         printf("Generated Gradient:\n");
         printf("%s\n", fGradGenerationInput.c_str());
         printf("%s\n", GetGradientFormula().Data());
      }
      if(fHessFuncPtr) {
         printf("Generated Hessian:\n");
         printf("%s\n", fHessGenerationInput.c_str());
         printf("%s\n", GetHessianFormula().Data());
      }
   }
   if(!fReadyToExecute)
      {
      Warning("Print", "Formula is not ready to execute. Missing parameters/variables");
      for (list<TFormulaFunction>::const_iterator it = fFuncs.begin(); it != fFuncs.end(); ++it) {
         TFormulaFunction fun = *it;
         if (!fun.fFound) {
            printf("%s is unknown.\n", fun.GetName());
         }
      }
   }
   if (!fAllParametersSetted) {
      // we can skip this
      // Info("Print","Not all parameters are set.");
      // for(map<TString,TFormulaVariable>::const_iterator it = fParams.begin(); it != fParams.end(); ++it)
      // {
      //    pair<TString,TFormulaVariable> param = *it;
      //    if(!param.second.fFound)
      //    {
      //       printf("%s has default value %lf\n",param.first.Data(),param.second.GetInitialValue());
      //    }
      // }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Stream a class object.
 
void TFormula::Streamer(TBuffer &b)
{
   if (b.IsReading() ) {
      UInt_t R__s, R__c;
      Version_t v = b.ReadVersion(&R__s, &R__c);
      //std::cout << "version " << v << std::endl;
      if (v <= 8 && v > 3 && v != 6) {
         // old TFormula class
         ROOT::v5::TFormula * fold = new ROOT::v5::TFormula();
         // read old TFormula class
         fold->Streamer(b, v,  R__s, R__c, TFormula::Class());
         //std::cout << "read old tformula class " << std::endl;
         TFormula fnew(fold->GetName(), fold->GetExpFormula() );
 
         *this = fnew;
 
         //          printf("copying content in a new TFormula \n");
         SetParameters(fold->GetParameters() );
         if (!fReadyToExecute ) {
            Error("Streamer","Old formula read from file is NOT valid");
            Print("v");
         }
         delete fold;
         return;
      }
      else if (v > 8) {
         // new TFormula class
         b.ReadClassBuffer(TFormula::Class(), this, v, R__s, R__c);
 
         //std::cout << "reading npar = " << GetNpar() << std::endl;
 
         // initialize the formula
         // need to set size of fClingVariables which is transient
         //fClingVariables.resize(fNdim);
 
         // case of formula contains only parameters
         if (fFormula.IsNull() ) return;
 
 
         // store parameter values, names and order
         std::vector<double> parValues = fClingParameters;
         auto paramMap = fParams;
         fNpar = fParams.size();
 
         fLazyInitialization = true;   // when reading we initialize the formula later to avoid problem of recursive Jitting
 
         if (!TestBit(TFormula::kLambda) ) {
 
            // save dimension read from the file (stored for V >=12)
            // and we check after initializing if it is the same
            int ndim = fNdim;
            fNdim = 0;
 
            //std::cout << "Streamer::Reading preprocess the formula " << fFormula << " ndim = " << fNdim << " npar = " << fNpar << std::endl;
            // for ( auto &p : fParams)
            //    std::cout << "parameter " << p.first << " index " << p.second << std::endl;
 
            fClingParameters.clear();  // need to be reset before re-initializing it
 
            FillDefaults();
 
 
            PreProcessFormula(fFormula);
 
            //std::cout << "Streamer::after pre-process the formula " << fFormula << " ndim = " << fNdim << " npar = " << fNpar << std::endl;
 
            PrepareFormula(fFormula);
 
            //std::cout << "Streamer::after prepared " << fClingInput << " ndim = " << fNdim << " npar = " << fNpar << std::endl;
 
 
            // restore parameter values
            if (fNpar != (int) parValues.size() ) {
               Error("Streamer","number of parameters computed (%d) is not same as the stored parameters (%d)",fNpar,int(parValues.size()) );
               Print("v");
            }
            if (v > 11 && fNdim != ndim) {
               Error("Streamer","number of dimension computed (%d) is not same as the stored value (%d)",fNdim, ndim );
               Print("v");
            }
         }
         else {
            // we also delay the initialization of lamda expressions
            if (!fLazyInitialization) {
               bool ret = InitLambdaExpression(fFormula);
               if (ret) {
                  fClingInitialized  = true;
               }
            }else {
               fReadyToExecute  = true;
            }
         }
         assert(fNpar == (int) parValues.size() );
         std::copy( parValues.begin(), parValues.end(), fClingParameters.begin() );
         // restore parameter names and order
         if (fParams.size() != paramMap.size() ) {
            Warning("Streamer","number of parameters list found (%zu) is not same as the stored one (%zu) - use re-created list",fParams.size(),paramMap.size()) ;
            //Print("v");
         }
         else
            //assert(fParams.size() == paramMap.size() );
            fParams = paramMap;
 
         // input formula into Cling
         // need to replace in cling the name of the pointer of this object
         // TString oldClingName = fClingName;
         // fClingName.Replace(fClingName.Index("_0x")+1,fClingName.Length(), TString::Format("%p",this) );
         // fClingInput.ReplaceAll(oldClingName, fClingName);
         // InputFormulaIntoCling();
 
         if (!TestBit(kNotGlobal)) {
            R__LOCKGUARD(gROOTMutex);
            gROOT->GetListOfFunctions()->Add(this);
         }
         if (!fReadyToExecute ) {
            Error("Streamer","Formula read from file is NOT ready to execute");
            Print("v");
         }
         //std::cout << "reading 2 npar = " << GetNpar() << std::endl;
 
         return;
      }
      else {
         Error("Streamer","Reading version %d is not supported",v);
         return;
      }
   }
   else {
      // case of writing
      b.WriteClassBuffer(TFormula::Class(), this);
      // std::cout << "writing npar = " << GetNpar() << std::endl;
   }
}