Is the timer of an arduino accurate enough to build a clock or do I need a RTC?

Is the timer of an arduino accurate enough to build a clock or do I need a RTC? I have some problems with the precision of my clock. I use a TFT display to show the date and the time, it works just fine but the clock is running too slow. has anyone a solution?  I'm going to use this in a project, the code is not complete yet but I have to get this part right first. Here is my code:

#include <Adafruit_GFX.h>
#include <MCUFRIEND_kbv.h>                                // bibliotheek voor het communiceren met het diplay 
#include "DHT.h"
#include <stdio.h>


MCUFRIEND_kbv tft;                                        // maak een display aan
DHT dht;


#if defined(__SAM3X8E__)
#undef __FlashStringHelper::F(string_literal)
#define F(string_literal) string_literal
#endif


#define BLACK       0x0000      /*   0,   0,   0 */
#define WHITE       0xFFFF      /* 255, 255, 255 */


uint8_t hh, mm, ss;
int date_month, date_day, date_year;
String date_month_string, date_day_string, date_year_string;
unsigned long time_val, old_time_val, old_time_val_1;


int text_size = 3;
int time_cursor_x = 10;
int time_cursor_y = 110;


uint8_t conv2d(const char* p)
{
  uint8_t v = 0;
  if ('0' <= *p && *p <= '9') v = *p - '0';
  return 10 * v + *++p - '0';
}
void setup() {
  // put your setup code here, to run once:
  Serial.begin(9600);                               // begin de seriële monitor
  while (!Serial) ;                                 // wacht tot deze is opgestard
  String date_string = __DATE__;


  tft.reset();                                      // reset het tft display
  tft.begin(0x7783);                                // begin de communicatie met het display met adres 0x7783
  tft.setRotation(1);                               // roteer het display zodat het in landscape modus staat
  tft.fillScreen(BLACK);


  hh = conv2d(__TIME__);
  mm = conv2d(__TIME__ + 3);
  ss = conv2d(__TIME__ + 6);
  date_month_string = date_string.substring(0, 3);
  date_day_string = date_string.substring(4, 6);
  date_year_string = date_string.substring(7, 11);


  date_year = date_year_string.toInt();
  date_day = date_day_string.toInt();
  date_month = month_string_to_int(date_month_string);
  OCR0A = 0xAF;
  TIMSK0 |= _BV(OCIE0A);
}
void loop() {
  // put your main code here, to run repeatedly:
  text_size = 3;
  time_cursor_x = 10;
  time_cursor_y = 110;
}
SIGNAL(TIMER0_COMPA_vect){
  time_val = millis();
  if ((time_val - old_time_val) >= 1000) {
    if (++ss > 59) {
      ss = 0;
      mm++;
      if (mm > 59) {
        mm = 0;
        hh++;
        if (hh > 23) {
          hh = 0;
          date_day++;
          if (date_day == 31 && (date_month == 1 || date_month == 3 || date_month == 5 || date_month == 7 || date_month == 8 || date_month == 10 || date_month == 12)) {
            date_day = 0;
            date_month++;
            if (date_month > 11) {
              date_month = 1;
              date_year++;
            }
          }
          if (date_day == 30 && (date_month == 4 || date_month == 6 || date_month == 9 || date_month == 11)) {
            date_day = 0;
            date_month++;
          }
          if (date_day == 28 && date_month == 2 && (date_year % 4) != 0) {
            date_day = 0;
            date_month++;
          }
          if (date_day == 29 && date_month == 2 && (date_year % 4) == 0) {
            date_day = 0;
            date_month++;
          }
        }
      }
    }
    tft.setTextColor(WHITE);
    tft.setTextSize(text_size);
    char buf[8];
    char buf_2[12];
    sprintf(buf_2, "%02i/%02i/%04i", date_day, date_month, date_year);
    sprintf(buf, "%02d:%02d:%02d", hh, mm, ss);
    tft.setCursor(time_cursor_x, time_cursor_y);
    tft.fillRect(time_cursor_x, time_cursor_y, 50 * text_size, 7 * text_size , BLACK);
    tft.print(buf);
    tft.setCursor(time_cursor_x, time_cursor_y + 8 * text_size);
    tft.fillRect(time_cursor_x, time_cursor_y + 8 * text_size, 60 * text_size, 7 * text_size , BLACK);
    tft.print(buf_2);
    Serial.println("tijd weergegeven");
    old_time_val = time_val;
  }
}
int month_string_to_int(String date_month_string) {
  if (date_month_string == "Jan")return 1;
  if (date_month_string == "Feb")return 2;
  if (date_month_string == "Mar") return 3;
  if (date_month_string == "Apr") return 4;
  if (date_month_string == "May") return 5;
  if (date_month_string == "Jun") return 6;
  if (date_month_string == "Jul") return 7;
  if (date_month_string == "Aug") return 8;
  if (date_month_string == "Sep") return 9;
  if (date_month_string == "Oct") return 10;
  if (date_month_string == "Nov") return 11;
  if (date_month_string == "Dec") return 12;
}