Code:

function GetDays(ADate: TDate): Extended;

var

  FirstOfYear: TDateTime;

begin

  FirstOfYear := EncodeDate(StrToInt(FormatDateTime('yyyy', now)) - 1, 12, 31);

  Result      := ADate - FirstOfYear;

end;

procedure TForm1.Button1Click(Sender: TObject);

begin

  label1.Caption := 'Today is the ' + FloatToStr(GetDays(Date)) + '. day of the year';

end;