template <typename T>
inline constexpr T div_ceil(T a, T b)
{
  return (a / b) + ((a % b) != 0);
}

static_assert(div_ceil<int>(2, 2) == 1);
static_assert(div_ceil<int>(3, 2) == 2);
static_assert(div_ceil<int>(55, 8) == 7);
static_assert(div_ceil<int>(55, 11) == 5);

template <typename T>
inline constexpr T sqrt_floor(T a)
{
  if (a == 0) return 0;
  constexpr int num_bits = (sizeof (T)) * 8;
  T x = 1 << div_ceil<T>(num_bits, 2);
  while (true) {
    T y = (x + (a / x)) / 2;
    if (y >= x) return x;
    x = y;
  }
}

static_assert(sqrt_floor<int>(4) == 2);
static_assert(sqrt_floor<int>(9) == 3);
static_assert(sqrt_floor<int>(7056) == 84);
static_assert(sqrt_floor<int>(7211) == 84);
static_assert(sqrt_floor<int>(7225) == 85);
static_assert(sqrt_floor<int>(7226) == 85);

template <typename T>
inline constexpr T sqrt_ceil(T a)
{
  T q = sqrt_floor<T>(a);
  if (q * q != a) q += 1;
  return q;
}

static_assert(sqrt_ceil<int>(7056) == 84);
static_assert(sqrt_ceil<int>(7211) == 85);
static_assert(sqrt_ceil<int>(7225) == 85);
static_assert(sqrt_ceil<int>(7226) == 86);