// Simultaneous evaluation of a given polynomial and its first derivative at a given point 
// Simultane Berechnung des Wertes eines Polynoms n-ten Grades und seiner Ableitung 

type
   TPolynomArray = array of Double;

 procedure Horner(Polynom: TPolynomArray; X: Extended; var FX, derivation: Extended);
 var
   i: Integer;
   H: Integer;
 begin
   H := High(Polynom);
   FX := Polynom[H];
   derivation := 0.0;
   for i := H - 1 downto 0 do
   begin
     derivation := derivation * X + FX;
     FX := FX * X + Polynom[i];
   end;
 end;

 {Beispiel / Sample code }

 procedure TForm1.Button1Click(Sender: TObject);
 var
   X, FX, derivation: Extended;
 begin
  (* Horner''s scheme give an algorithm for the simultaneous evaluation
    of a given polynomial and its first derivative at a given point *)

  (* Hornerschema zur Berechnung eines Polynoms n-ten Grades an einem
     bestimmten Punkt *)

   (* f(x) = 3 x^5 + 4 x^4 + 13 x^3 - 59 x^2 + 19 x - 97 *)

   X := 2.5;
   Horner(VarArrayOf([-97, 19, - 59, 13, 4, 3]), X, FX, derivation);
   ShowMessage(Format('x = %n'#13#10'f(x) = %n'#13#10'f''(x) = %n'    , [X, FX, derivation]));
 end;