RooFormula.cxx

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/*****************************************************************************
 * Project: RooFit                                                           *
 * Package: RooFitCore                                                       *
 * @(#)root/roofitcore:$Id$
 * Authors:                                                                  *
 *   WV, Wouter Verkerke, UC Santa Barbara, verkerke@slac.stanford.edu       *
 *   DK, David Kirkby,    UC Irvine,         dkirkby@uci.edu                 *
 *                                                                           *
 * Copyright (c) 2000-2005, Regents of the University of California          *
 *                          and Stanford University. All rights reserved.    *
 *                                                                           *
 * Redistribution and use in source and binary forms,                        *
 * with or without modification, are permitted according to the terms        *
 * listed in LICENSE (http://roofit.sourceforge.net/license.txt)             *
 *****************************************************************************/
 
/**
\file RooFormula.cxx
\class RooFormula
\ingroup Roofitcore
 
RooFormula internally uses ROOT's TFormula to compute user-defined expressions
of RooAbsArgs.
 
The string expression can be any valid TFormula expression referring to the
listed servers either by name or by their ordinal list position. These three are
forms equivalent:
```
  RooFormula("formula", "x*y",       RooArgList(x,y))  or
  RooFormula("formula", "@0*@1",     RooArgList(x,y))
  RooFormula("formula", "x[0]*x[1]", RooArgList(x,y))
```
Note that `x[i]` is an expression reserved for TFormula. If a variable with
the name `x` is given, the RooFormula interprets `x` as a variable name,
but `x[i]` as an index in the list of variables.
 
### Category expressions
State information of RooAbsCategories can be accessed using the '::' operator,
*i.e.*, `tagCat::Kaon` will resolve to the numerical value of
the `Kaon` state of the RooAbsCategory object named `tagCat`.
 
A formula to switch between lepton categories could look like this:
```
  RooFormula("formulaWithCat",
    "x * (leptonMulti == leptonMulti::one) + y * (leptonMulti == leptonMulti::two)",
    RooArgList(x, y, leptonMulti));
```
 
### Debugging a formula that won't compile
When the formula is preprocessed, RooFit can print information in the debug stream.
These can be retrieved by activating the RooFit::MsgLevel `RooFit::DEBUG`
and the RooFit::MsgTopic `RooFit::InputArguments`.
Check the tutorial rf506_msgservice.C for details.
**/
 
#include "RooFormula.h"
#include "BracketAdapters.h"
#include "RooAbsReal.h"
#include "RooAbsCategory.h"
#include "RooArgList.h"
#include "RooMsgService.h"
#include "RunContext.h"
#include "RooBatchCompute.h"
 
#include "TObjString.h"
#include "TClass.h"
 
#include <memory>
#include <regex>
#include <sstream>
#include <cctype>
 
using namespace std;
 
ClassImp(RooFormula);
 
namespace {
 
////////////////////////////////////////////////////////////////////////////////
/// Find all input arguments which are categories, and save this information in
/// with the names of the variables that are being used to evaluate it.
std::vector<bool> findCategoryServers(const RooAbsCollection& collection) {
  std::vector<bool> output;
  output.reserve(collection.size());
 
  for (unsigned int i = 0; i < collection.size(); ++i) {
    output.push_back(collection[i]->InheritsFrom(RooAbsCategory::Class()));
  }
 
  return output;
}
 
/// Convert `@i`-style references to `x[i]`.
void convertArobaseReferences(std::string &formula)
{
   bool match = false;
   for (std::size_t i = 0; i < formula.size(); ++i) {
      if (match && !isdigit(formula[i])) {
         formula.insert(formula.begin() + i, ']');
         i += 1;
         match = false;
      } else if (!match && formula[i] == '@') {
         formula[i] = 'x';
         formula.insert(formula.begin() + i + 1, '[');
         i += 1;
         match = true;
      }
   }
   if (match)
      formula += ']';
}
 
/// Replace all occurences of `what` with `with` inside of `inout`.
void replaceAll(std::string &inout, std::string_view what, std::string_view with)
{
   for (std::string::size_type pos{}; inout.npos != (pos = inout.find(what.data(), pos, what.length()));
        pos += with.length()) {
      inout.replace(pos, what.length(), with.data(), with.length());
   }
}
 
/// Find the word boundaries with a static std::regex and return a bool vector
/// flagging their positions. The end of the string is considered a word
/// boundary.
std::vector<bool> getWordBoundaryFlags(std::string const &s)
{
   static const std::regex r{"\\b"};
   std::vector<bool> out(s.size() + 1);
 
   for (auto i = std::sregex_iterator(s.begin(), s.end(), r); i != std::sregex_iterator(); ++i) {
      std::smatch m = *i;
      out[m.position()] = true;
   }
 
   // The end of a string is also a word boundary
   out[s.size()] = true;
 
   return out;
}
 
/// Replace all named references with "x[i]"-style.
void replaceVarNamesWithIndexStyle(std::string &formula, RooArgList const &varList)
{
   std::vector<bool> isWordBoundary = getWordBoundaryFlags(formula);
   for (unsigned int i = 0; i < varList.size(); ++i) {
      std::string_view varName = varList[i].GetName();
 
      std::stringstream replacementStream;
      replacementStream << "x[" << i << "]";
      std::string replacement = replacementStream.str();
 
      for (std::string::size_type pos{}; formula.npos != (pos = formula.find(varName.data(), pos, varName.length()));
           pos += replacement.size()) {
 
         std::string::size_type next = pos + varName.length();
 
         // The matched variable name has to be surrounded by word boundaries
         // std::cout << pos << "   " << next << std::endl;
         if (!isWordBoundary[pos] || !isWordBoundary[next])
            continue;
 
         // Veto '[' and ']' as next characters. If the variable is called `x`
         // or `0`, this might otherwise replace `x[0]`.
         if (next < formula.size() && (formula[next] == '[' || formula[next] == ']')) {
            continue;
         }
 
         // As we replace substrings in the middle of the string, we also have
         // to update the word boundary flag vector. Note that we don't care
         // the word boundaries in the `x[i]` are correct, as it has already
         // been replaced.
         std::size_t nOld = varName.length();
         std::size_t nNew = replacement.size();
         auto wbIter = isWordBoundary.begin() + pos;
         if (nNew > nOld)
            isWordBoundary.insert(wbIter + nOld, nNew - nOld, false);
         else if (nNew < nOld)
            isWordBoundary.erase(wbIter + nNew, wbIter + nOld);
 
         // Do the actual replacment
         formula.replace(pos, varName.length(), replacement);
      }
 
      oocxcoutD(static_cast<TObject *>(nullptr), InputArguments)
         << "Preprocessing formula: replace named references: " << varName << " --> " << replacement << "\n\t"
         << formula << endl;
   }
}
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Default constructor
/// coverity[UNINIT_CTOR]
 
RooFormula::RooFormula() : TNamed()
{
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Construct a new formula.
/// \param[in] name Name of the formula.
/// \param[in] formula Formula to be evaluated. Parameters/observables are identified by name
/// or ordinal position in `varList`.
/// \param[in] varList List of variables to be passed to the formula.
/// \param[in] checkVariables Check that the variables being passed in the `varList` are used in
/// the formula expression.
RooFormula::RooFormula(const char* name, const char* formula, const RooArgList& varList,
    bool checkVariables) :
  TNamed(name, formula), _tFormula{nullptr}
{
  _origList.add(varList);
  _isCategory = findCategoryServers(_origList);
 
  installFormulaOrThrow(formula);
 
  RooArgList useList = usedVariables();
  if (checkVariables && _origList.size() != useList.size()) {
    coutI(InputArguments) << "The formula " << GetName() << " claims to use the variables " << _origList
        << " but only " << useList << " seem to be in use."
        << "\n  inputs:         " << formula << std::endl;
  }
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Copy constructor
RooFormula::RooFormula(const RooFormula& other, const char* name) :
  TNamed(name ? name : other.GetName(), other.GetTitle()), RooPrintable(other)
{
  _origList.add(other._origList);
  _isCategory = findCategoryServers(_origList);
 
  TFormula* newTF = nullptr;
  if (other._tFormula) {
    newTF = new TFormula(*other._tFormula);
    newTF->SetName(GetName());
  }
 
  _tFormula.reset(newTF);
}
 
 
#ifndef _MSC_VER
#if !defined(__GNUC__) || defined(__clang__) || (__GNUC__ > 4) || ( __GNUC__ == 4 && __GNUC_MINOR__ > 8)
#define ROOFORMULA_HAVE_STD_REGEX
#endif //GCC < 4.9 Check
#endif //_MSC_VER
 
#ifdef ROOFORMULA_HAVE_STD_REGEX
////////////////////////////////////////////////////////////////////////////////
/// Process given formula by replacing all ordinal and name references by
/// `x[i]`, where `i` matches the position of the argument in `_origList`.
/// Further, references to category states such as `leptonMulti:one` are replaced
/// by the category index.
std::string RooFormula::processFormula(std::string formula) const {
 
  // WARNING to developers: people use RooFormula a lot via RooGenericPdf and
  // RooFormulaVar! Performance matters here. Avoid non-static std::regex,
  // because constructing these can become a bottleneck because of the regex
  // compilation.
 
  cxcoutD(InputArguments) << "Preprocessing formula step 1: find category tags (catName::catState) in "
      << formula << endl;
 
  // Step 1: Find all category tags and the corresponding index numbers
  static const std::regex categoryReg("(\\w+)::(\\w+)");
  std::map<std::string, int> categoryStates;
  for (sregex_iterator matchIt = sregex_iterator(formula.begin(), formula.end(), categoryReg);
       matchIt != sregex_iterator(); ++matchIt) {
    assert(matchIt->size() == 3);
    const std::string fullMatch = (*matchIt)[0];
    const std::string catName = (*matchIt)[1];
    const std::string catState = (*matchIt)[2];
 
    const auto catVariable = dynamic_cast<const RooAbsCategory*>(_origList.find(catName.c_str()));
    if (!catVariable) {
      cxcoutD(InputArguments) << "Formula " << GetName() << " uses '::' to reference a category state as '" << fullMatch
          << "' but a category '" << catName << "' cannot be found in the input variables." << endl;
      continue;
    }
 
    if (!catVariable->hasLabel(catState)) {
      coutE(InputArguments) << "Formula " << GetName() << " uses '::' to reference a category state as '" << fullMatch
          << "' but the category '" << catName << "' does not seem to have the state '" << catState << "'." << endl;
      throw std::invalid_argument(formula);
    }
    const int catNum = catVariable->lookupIndex(catState);
 
    categoryStates[fullMatch] = catNum;
    cxcoutD(InputArguments) << "\n\t" << fullMatch << "\tname=" << catName << "\tstate=" << catState << "=" << catNum;
  }
  cxcoutD(InputArguments) << "-- End of category tags --"<< endl;
 
  // Step 2: Replace all category tags
  for (const auto& catState : categoryStates) {
    replaceAll(formula, catState.first, std::to_string(catState.second));
  }
 
  cxcoutD(InputArguments) << "Preprocessing formula step 2: replace category tags\n\t" << formula << endl;
 
  // Step 3: Convert `@i`-style references to `x[i]`
  convertArobaseReferences(formula);
 
  cxcoutD(InputArguments) << "Preprocessing formula step 3: replace '@'-references\n\t" << formula << endl;
 
  // Step 4: Replace all named references with "x[i]"-style
  replaceVarNamesWithIndexStyle(formula, _origList);
 
  cxcoutD(InputArguments) << "Final formula:\n\t" << formula << endl;
 
  return formula;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Analyse internal formula to find out which variables are actually in use.
RooArgList RooFormula::usedVariables() const {
  RooArgList useList;
  if (_tFormula == nullptr)
    return useList;
 
  const std::string formula(_tFormula->GetTitle());
 
  std::set<unsigned int> matchedOrdinals;
  static const std::regex newOrdinalRegex("\\bx\\[([0-9]+)\\]");
  for (sregex_iterator matchIt = sregex_iterator(formula.begin(), formula.end(), newOrdinalRegex);
      matchIt != sregex_iterator(); ++matchIt) {
    assert(matchIt->size() == 2);
    std::stringstream matchString((*matchIt)[1]);
    unsigned int i;
    matchString >> i;
 
    matchedOrdinals.insert(i);
  }
 
  for (unsigned int i : matchedOrdinals) {
    useList.add(_origList[i]);
  }
 
  return useList;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// From the internal representation, construct a formula by replacing all index place holders
/// with the names of the variables that are being used to evaluate it.
std::string RooFormula::reconstructFormula(std::string internalRepr) const {
  for (unsigned int i = 0; i < _origList.size(); ++i) {
    const auto& var = _origList[i];
    std::stringstream regexStr;
    regexStr << "x\\[" << i << "\\]|@" << i;
    std::regex regex(regexStr.str());
 
    std::string replacement = std::string("[") + var.GetName() + "]";
    internalRepr = std::regex_replace(internalRepr, regex, replacement);
  }
 
  return internalRepr;
}
#endif //ROOFORMULA_HAVE_STD_REGEX
 
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Recompile formula with new expression. In case of error, the old formula is
/// retained.
bool RooFormula::reCompile(const char* newFormula)
{
  try {
    installFormulaOrThrow(newFormula);
  } catch (std::runtime_error& e) {
    coutE(InputArguments) << __func__ << ": new equation doesn't compile, formula unchanged."
        << "\n" << e.what() << endl;
    return true;
  }
 
  SetTitle(newFormula);
  return false;
}
 
void RooFormula::dump() const
{
  printMultiline(std::cout, 0);
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Change used variables to those with the same name in given list.
/// \param[in] newDeps New dependents to replace the old ones.
/// \param[in] mustReplaceAll Will yield an error if one dependent does not have a replacement.
/// \param[in] nameChange Passed down to RooAbsArg::findNewServer(const RooAbsCollection&, bool) const.
bool RooFormula::changeDependents(const RooAbsCollection& newDeps, bool mustReplaceAll, bool nameChange)
{
  //Change current servers to new servers with the same name given in list
  bool errorStat = false;
 
  for (const auto arg : _origList) {
    RooAbsReal* replace = (RooAbsReal*) arg->findNewServer(newDeps,nameChange) ;
    if (replace) {
      _origList.replace(*arg, *replace);
 
      if (arg->getStringAttribute("origName")) {
        replace->setStringAttribute("origName",arg->getStringAttribute("origName")) ;
      } else {
        replace->setStringAttribute("origName",arg->GetName()) ;
      }
 
    } else if (mustReplaceAll) {
      coutE(LinkStateMgmt) << __func__ << ": cannot find replacement for " << arg->GetName() << endl;
      errorStat = true;
    }
  }
 
  _isCategory = findCategoryServers(_origList);
 
  return errorStat;
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Evaluate the internal TFormula.
///
/// First, all variables serving this instance are evaluated given the normalisation set,
/// and then the formula is evaluated.
/// \param[in] nset Normalisation set passed to evaluation of arguments serving values.
/// \return The result of the evaluation.
double RooFormula::eval(const RooArgSet* nset) const
{
  if (!_tFormula) {
    coutF(Eval) << __func__ << " (" << GetName() << "): Formula didn't compile: " << GetTitle() << endl;
    std::string what = "Formula ";
    what += GetTitle();
    what += " didn't compile.";
    throw std::runtime_error(what);
  }
 
  std::vector<double> pars;
  pars.reserve(_origList.size());
  for (unsigned int i = 0; i < _origList.size(); ++i) {
    if (_isCategory[i]) {
      const auto& cat = static_cast<RooAbsCategory&>(_origList[i]);
      pars.push_back(cat.getCurrentIndex());
    } else {
      const auto& real = static_cast<RooAbsReal&>(_origList[i]);
      pars.push_back(real.getVal(nset));
    }
  }
 
  return _tFormula->EvalPar(pars.data());
}
 
 
RooSpan<double> RooFormula::evaluateSpan(const RooAbsReal* dataOwner, RooBatchCompute::RunContext& inputData, const RooArgSet* nset) const {
  if (!_tFormula) {
    coutF(Eval) << __func__ << " (" << GetName() << "): Formula didn't compile: " << GetTitle() << endl;
    std::string what = "Formula ";
    what += GetTitle();
    what += " didn't compile.";
    throw std::runtime_error(what);
  }
 
  std::vector<RooBatchCompute::BracketAdapterWithMask> valueAdapters;
  std::vector<RooSpan<const double>> inputSpans;
  size_t nData=1;
  for (const auto arg : _origList) {
    auto realArg = static_cast<const RooAbsReal*>(arg);
    auto batch = realArg->getValues(inputData, nset);
    assert(!batch.empty());
    nData = std::max(nData, batch.size());
    valueAdapters.emplace_back(batch[0], batch);
    inputSpans.push_back(std::move(batch));
  }
 
  auto output = inputData.makeBatch(dataOwner, nData);
  std::vector<double> pars(_origList.size());
 
 
  for (std::size_t i=0; i < nData; ++i) {
    for (unsigned int j=0; j < _origList.size(); ++j) {
      if (_isCategory[j]) {
        // TODO: As long as category states cannot be passed in the RunContext,
        // the current state has to be used.
        const auto& cat = static_cast<RooAbsCategory&>(_origList[j]);
        pars[j] = cat.getCurrentIndex();
      } else {
        pars[j] = valueAdapters[j][i];
      }
    }
 
    output[i] = _tFormula->EvalPar(pars.data());
  }
 
  return output;
}
 
void RooFormula::computeBatch(cudaStream_t*, double* output, size_t nEvents, RooFit::Detail::DataMap const& dataMap) const
{
  const int nPars=_origList.size();
  std::vector<RooSpan<const double>> inputSpans(nPars);
  for (int i=0; i<nPars; i++)
    inputSpans[i] = dataMap.at( static_cast<const RooAbsReal*>(&_origList[i]) );
 
  std::vector<double> pars(nPars);
  for (size_t i=0; i<nEvents; i++)
  {
    for (int j=0; j<nPars; j++) pars[j] = inputSpans[j].size()>1 ? inputSpans[j][i] : inputSpans[j][0];
    output[i] = _tFormula->EvalPar( pars.data() );
  }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Printing interface
 
void RooFormula::printMultiline(ostream& os, Int_t /*contents*/, bool /*verbose*/, TString indent) const
{
  os << indent << "--- RooFormula ---" << endl;
  os << indent << " Formula:        '" << GetTitle() << "'" << endl;
  os << indent << " Interpretation: '" << reconstructFormula(GetTitle()) << "'" << endl;
  indent.Append("  ");
  os << indent << "Servers: " << _origList << "\n";
  os << indent << "In use : " << actualDependents() << endl;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Print value of formula
 
void RooFormula::printValue(ostream& os) const
{
  os << const_cast<RooFormula*>(this)->eval(0) ;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Print name of formula
 
void RooFormula::printName(ostream& os) const
{
  os << GetName() ;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Print title of formula
 
void RooFormula::printTitle(ostream& os) const
{
  os << GetTitle() ;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Print class name of formula
 
void RooFormula::printClassName(ostream& os) const
{
  os << IsA()->GetName() ;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Print arguments of formula, i.e. dependents that are actually used
 
void RooFormula::printArgs(ostream& os) const
{
  os << "[ actualVars=";
  for (const auto arg : usedVariables()) {
     os << " " << arg->GetName();
  }
  os << " ]";
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Check that the formula compiles, and also fulfills the assumptions.
///
void RooFormula::installFormulaOrThrow(const std::string& formula) {
  const std::string processedFormula = processFormula(formula);
 
  cxcoutD(InputArguments) << "RooFormula '" << GetName() << "' will be compiled as "
      << "\n\t" << processedFormula
      << "\n  and used as"
      << "\n\t" << reconstructFormula(processedFormula)
      << "\n  with the parameters " << _origList << endl;
 
  auto theFormula = std::make_unique<TFormula>(GetName(), processedFormula.c_str(), false);
 
  if (!theFormula || !theFormula->IsValid()) {
    std::stringstream msg;
    msg << "RooFormula '" << GetName() << "' did not compile or is invalid."
        << "\nInput:\n\t" << formula
        << "\nPassed over to TFormula:\n\t" << processedFormula << std::endl;
    coutF(InputArguments) << msg.str();
    throw std::runtime_error(msg.str());
  }
 
  if (theFormula && theFormula->GetNdim() != 1) {
    // TFormula thinks that we have a multi-dimensional formula, e.g. with variables x,y,z,t.
    // We have to check now that this is not the case, as RooFit only uses the syntax x[0], x[1], x[2], ...
    bool haveProblem = false;
    std::stringstream msg;
    msg << "TFormula interprets the formula " << formula << " as " << theFormula->GetNdim() << "-dimensional with the variable(s) {";
    for (int i=1; i < theFormula->GetNdim(); ++i) {
      const TString varName = theFormula->GetVarName(i);
      if (varName.BeginsWith("x[") && varName[varName.Length()-1] == ']')
        continue;
 
      haveProblem = true;
      msg << theFormula->GetVarName(i) << ",";
    }
    if (haveProblem) {
      msg << "}, which could not be supplied by RooFit."
          << "\nThe formula must be modified, or those variables must be supplied in the list of variables." << std::endl;
      coutF(InputArguments) << msg.str();
      throw std::invalid_argument(msg.str());
    }
  }
 
  _tFormula = std::move(theFormula);
}
 
 
#ifndef ROOFORMULA_HAVE_STD_REGEX
/*
 * g++ 4.8 doesn't support the std::regex. It has headers, but no implementations of the standard, leading to linker
 * errors. As long as centos 7 needs to be supported, this forces us to have a legacy implementation.
 */
 
#include "TPRegexp.h"
 
////////////////////////////////////////////////////////////////////////////////
/// Process given formula by replacing all ordinal and name references by
/// `x[i]`, where `i` matches the position of the argument in `_origList`.
/// Further, references to category states such as `leptonMulti:one` are replaced
/// by the category index.
std::string RooFormula::processFormula(std::string formula) const {
  TString formulaTString = formula.c_str();
 
  cxcoutD(InputArguments) << "Preprocessing formula step 1: find category tags (catName::catState) in "
      << formulaTString.Data() << endl;
 
  // Step 1: Find all category tags and the corresponding index numbers
  TPRegexp categoryReg("(\\w+)::(\\w+)");
  std::map<std::string, int> categoryStates;
  int offset = 0;
  do {
    std::unique_ptr<TObjArray> matches(categoryReg.MatchS(formulaTString, "", offset, 3));
    if (matches->GetEntries() == 0)
      break;
 
    std::string fullMatch = static_cast<TObjString*>(matches->At(0))->GetString().Data();
    std::string catName = static_cast<TObjString*>(matches->At(1))->GetString().Data();
    std::string catState = static_cast<TObjString*>(matches->At(2))->GetString().Data();
    offset = formulaTString.Index(categoryReg, offset) + fullMatch.size();
 
    const auto catVariable = dynamic_cast<const RooAbsCategory*>(_origList.find(catName.c_str()));
    if (!catVariable) {
      cxcoutD(InputArguments) << "Formula " << GetName() << " uses '::' to reference a category state as '" << fullMatch
          << "' but a category '" << catName << "' cannot be found in the input variables." << endl;
      continue;
    }
 
    const RooCatType* catType = catVariable->lookupType(catState.c_str(), false);
    if (!catType) {
      coutE(InputArguments) << "Formula " << GetName() << " uses '::' to reference a category state as '" << fullMatch
          << "' but the category '" << catName << "' does not seem to have the state '" << catState << "'." << endl;
      throw std::invalid_argument(formula);
    }
    const int catNum = catType->getVal();
 
    categoryStates[fullMatch] = catNum;
    cxcoutD(InputArguments) << "\n\t" << fullMatch << "\tname=" << catName << "\tstate=" << catState << "=" << catNum;
  } while (offset != -1);
  cxcoutD(InputArguments) << "-- End of category tags --"<< endl;
 
  // Step 2: Replace all category tags
  for (const auto& catState : categoryStates) {
    std::stringstream replacement;
    replacement << catState.second;
    formulaTString.ReplaceAll(catState.first.c_str(), replacement.str().c_str());
  }
 
  cxcoutD(InputArguments) << "Preprocessing formula step 2: replace category tags\n\t" << formulaTString.Data() << endl;
 
  // Step 3: Convert `@i`-style references to `x[i]`
  TPRegexp ordinalRegex("@([0-9]+)");
  int nsub = 0;
  do {
    nsub = ordinalRegex.Substitute(formulaTString, "x[$1]");
  } while (nsub > 0);
 
  cxcoutD(InputArguments) << "Preprocessing formula step 3: replace '@'-references\n\t" << formulaTString.Data() << endl;
 
  // Step 4: Replace all named references with "x[i]"-style
  for (unsigned int i = 0; i < _origList.size(); ++i) {
    const auto& var = _origList[i];
    TString regex = "\\b";
    regex += var.GetName();
    regex += "\\b([^\\[\\]]|$)"; //Negative lookahead. If the variable is called `x` or `0`, this might otherwise replace `x[0]`.
    TPRegexp findParameterRegex(regex);
 
    std::stringstream replacement;
    replacement << "x[" << i << "]$1";
    int nsub2 = 0;
    do {
      nsub2 = findParameterRegex.Substitute(formulaTString, replacement.str().c_str());
    } while (nsub2 > 0);
 
    cxcoutD(InputArguments) << "Preprocessing formula step 4: replace named references: "
        << var.GetName() << " --> " << replacement.str()
        << "\n\t" << formulaTString.Data() << endl;
  }
 
  cxcoutD(InputArguments) << "Final formula:\n\t" << formulaTString << endl;
 
  return formulaTString.Data();
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Analyse internal formula to find out which variables are actually in use.
RooArgList RooFormula::usedVariables() const {
  RooArgList useList;
  if (_tFormula == nullptr)
    return useList;
 
  const TString formulaTString = _tFormula->GetTitle();
 
  std::set<unsigned int> matchedOrdinals;
  TPRegexp newOrdinalRegex("\\bx\\[([0-9]+)\\]");
  int offset = 0;
  do {
    std::unique_ptr<TObjArray> matches(newOrdinalRegex.MatchS(formulaTString, "", offset, 2));
    if (matches->GetEntries() == 0)
      break;
 
    std::string fullMatch = static_cast<TObjString*>(matches->At(0))->GetString().Data();
    std::string ordinal   = static_cast<TObjString*>(matches->At(1))->GetString().Data();
    offset = formulaTString.Index(newOrdinalRegex, offset) + fullMatch.size();
 
    std::stringstream matchString(ordinal.c_str());
    unsigned int i;
    matchString >> i;
 
    matchedOrdinals.insert(i);
  } while (offset != -1);
 
  for (unsigned int i : matchedOrdinals) {
    useList.add(_origList[i]);
  }
 
  return useList;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// From the internal representation, construct a formula by replacing all index place holders
/// with the names of the variables that are being used to evaluate it.
std::string RooFormula::reconstructFormula(std::string internalRepr) const {
  TString internalReprT = internalRepr.c_str();
 
  for (unsigned int i = 0; i < _origList.size(); ++i) {
    const auto& var = _origList[i];
    std::stringstream regexStr;
    regexStr << "x\\[" << i << "\\]|@" << i;
    TPRegexp regex(regexStr.str().c_str());
 
    std::string replacement = std::string("[") + var.GetName() + "]";
    regex.Substitute(internalReprT, replacement.c_str());
  }
 
  return internalReprT.Data();
}
#endif //ROOFORMULA_HAVE_STD_REGEX