/// Calculate the number of primes below x using the
/// Lagarias-Miller-Odlyzko algorithm.
/// Run time: O(x^(2/3) / log x) operations, O(x^(1/3) * (log x)^2) space.
///
int64_t pi_lmo_parallel3(int64_t x, int threads)
{
if (x < 2)
return 0;
double alpha = get_alpha_lmo(x);
int64_t x13 = iroot<3>(x);
int64_t y = (int64_t) (x13 * alpha);
int64_t z = x / y;
int64_t c = PhiTiny::get_c(y);
print("");
print("=== pi_lmo_parallel3(x) ===");
print("pi(x) = S1 + S2 + pi(y) - 1 - P2");
print(x, y, z, c, alpha, threads);
int64_t p2 = P2(x, y, threads);
vector<int32_t> mu = generate_moebius(y);
vector<int32_t> lpf = generate_least_prime_factors(y);
vector<int32_t> primes = generate_primes(y);
int64_t pi_y = primes.size() - 1;
int64_t s1 = S1(x, y, c, threads);
int64_t s2_approx = S2_approx(x, pi_y, p2, s1);
int64_t s2 = S2(x, y, c, s2_approx, primes, lpf, mu, threads);
int64_t phi = s1 + s2;
int64_t sum = phi + pi_y - 1 - p2;
return sum;
}
/// Calculate the number of primes below x using the
/// Lagarias-Miller-Odlyzko algorithm.
/// Run time: O(x^(2/3)) operations, O(x^(1/3) * (log x)^2) space.
///
int64_t pi_lmo3(int64_t x)
{
if (x < 2)
return 0;
double alpha = get_alpha_lmo(x);
int64_t x13 = iroot<3>(x);
int64_t y = (int64_t) (x13 * alpha);
int64_t c = PhiTiny::get_c(y);
int64_t p2 = P2(x, y, 1);
vector<int32_t> mu = generate_moebius(y);
vector<int32_t> lpf = generate_least_prime_factors(y);
vector<int32_t> primes = generate_primes(y);
int64_t pi_y = primes.size() - 1;
int64_t s1 = S1(x, y, c, 1);
int64_t s2 = S2(x, y, c, primes, lpf, mu);
int64_t phi = s1 + s2;
int64_t sum = phi + pi_y - 1 - p2;
return sum;
}
