#include <bits/stdc++.h>
#include <experimental/array>
#define mar experimental::make_array
using namespace std;
array<array<long double, 100>, 50> table {};
array<long double, 50> val {};
int n{};
int mfd=4;
long double h=0.1;
long double y {1.683635};
array<long double, 3> arX{0.151316, 0.815359, 0.281586};
void finDiff(int k);
long double intBegTable(long double x);
long double intEndTable(long double x);
long double intCenterTable(long double x);
long double revIntTable(long double y);
void crTable();
void solveForX();
void solveForY();
long double findT(long double y, int ind);
long double fact(int a);
int main(){
  freopen("in","r",stdin);
  while(cin >> val[n] >> table[n<<1][0] && ++n){}
  crTable();
  solveForX();
  solveForY();
  return 0;
}
void solveForX(){
  for(int i=0; i<3; i++){
    double res=-1337;
    for(int j=0; j<n; j++){
      if (arX[i]<val[j]){
        if (j<n/3)
          res=intBegTable(arX[i]);
        if (j>=n/3 && j<2*n/3)
          res=intCenterTable(arX[i]);
        if (j>=2*n/3)
          res=intEndTable(arX[i]);
        break;
      }
    }
    cout << "\np("<< arX[i] << ") ~=~ "  << setprecision(7)<< res << "\n\n";
    if(res!=-1337)
      continue;
    res=intEndTable(arX[i]);
  }
  double res{};
  res=intBegTable(arX[2]);
  cout << "p("<< arX[2] << ") ~=~ "  << setprecision(7)<< res << "\n\n";
  res=intEndTable(arX[2]);
  cout << "\np("<< arX[2] << ") ~=~ " << setprecision(7) << res << "\n\n";
}
void solveForY(){
  double res{},
         resX{};
  res=revIntTable(y);
  cout << "\n\np^-1("<< y << ") ~=~ " << setprecision(7) << res << "\n\n";
  cout << "Check: \n";
  for(int j=0; j<n; j++){
    if (res<val[j]){
      if (j<n/3)
        resX=intBegTable(res);
      if (j>=n/3 && j<2*n/3)
        resX=intCenterTable(res);
      if (j>=2*n/3)
        resX=intEndTable(res);
      break;
    }
  }
  cout << "\np("<< res << ") ~=~ " << resX << "\n\n";
  cout << y << " == " <<  resX << "?\n\n\n\n";
}
void finDiff(int k){
  for(int i=k; i<(n<<1)-k; i++)
    table[i][k]=table[i+1][k-1]-table[i-1][k-1];
}
void crTable(){
  mfd=min(mfd, n-1);
  cout << "x\t\t px\t ";
  for(int i=1; i<=mfd; i++)
    finDiff(i),
      cout << "d^{" << i << "}" << mar("\t", "\n\n")[i==mfd];
  for(int i=0; i<(n<<1)-1; i++){
    cout << mar("\t", to_string(val[i>>1]))[i%2==0] << " |\t";
    for(int j=0; j<=mfd; j++)
      cout << " \a"[table[i][j]<0] << mar("\t", to_string(table[i][j]))[table[i][j]!=0] << "\a\n"[j==mfd];
  }
}
long double intBegTable(long double x){
  int ind{};
  for(ind=0; ind<n; ind++)
    if (val[ind]>x){
      ind--;
      break;
    }
  long double t=(x-val[ind])/h,
       res=0,
       tt=1;
  cout << "\n\nInterp BEGIN table start, x=" << x << ", x_0="<<val[ind] << ", t=" << t << endl;
  for(int i=0; i<5; i++){
    //cout << table[(ind<<1)+i][i] << ' ' << tt << endl;
    res+=tt*table[(ind<<1)+i][i]/fact(i), tt=tt*(t-i);
  }
  return res;
}
long double intEndTable(long double x){
  int ind{};
  for(ind=n-1; ind>=0; ind--)
    if (val[ind]<x){
      ind++;
      break;
    }
  long double t=(x-val[ind])/h,
       res=0,
       tt=1;
  cout << "Interp END table start, x=" << x << ", x_0="<<val[ind] << ", t=" << t << endl;
  for(int i=0; i<5; i++){
    //cout << table[(ind<<1)-i][i] << ' ' << tt << endl;
    res+=tt*table[(ind<<1)-i][i]/fact(i), tt=tt*(t+i);
  }
  return res;
}
long double intCenterTable(long double x){
  int ind{};
  for(ind=0; ind<n; ind++)
    if (val[ind]>x){
      if (fabs(val[ind]-x)>fabs(val[ind-1]-x))
        ind--;
      break;
    }
  long double t=(x-val[ind])/h,
       res=0,
       tt=1;
  cout << "Interp CENTER table start, x=" << x << ", x_0="<<val[ind] << ", t=" << t << endl;
  for(int i=0; i<5; i++){
    //cout << table[(ind<<1)+1*(i%2==1)][i] << ' ' << tt << endl;
    res+=tt*table[(ind<<1)+1*(i%2==1)][i]/fact(i), tt=tt*(t+(((i+1)>>1))*pow(-1,i));
  }
  return res;
}
long double revIntTable(long double y){
  int ind{};
  for(ind=0; ind<(n<<1)-1; ind++)
    if (table[ind<<1][0]>y){
      ind--;
      break;
    }
  long double res=y,
       tt=1,
       t{};
  t=findT(y, ind);
  res=t*h+val[ind];
  return res;
}
long double findT(long double y, int ind){
  long double res{-1},
       t{1},
       tt;
  while(fabs(res-t)>1e-4){
    t=res;
    cout << "t=" << t << endl;
    res=0;
    tt=1;
    res=y-table[ind<<1][0]-t*(t-1)/2.*table[(ind<<1)+2][2]-t*(t-1)*(t-2)/6.*table[(ind<<1)+3][3];
    res/=table[(ind<<1)+1][1];
  }
  return res;
}
long double fact(int a){
  if (a<2)
    return 1.;
  return a*fact(a-1);
}