Input        Output
90           24
3000         430
9000         1117
4000000      283146           <--- input = 4*10^6
800000000    41146179         <--- input = 9*10^8
1100000000   55662470         <--- input = 1.1*10^9

Input        Output
1907000000   93875448         <--- input = 1.907*10^9
1337000000   66990613         <--- input = 1.337*10^9
1240000000   62366021         <--- input = 1.24*10^9
660000000    34286170         <--- input = 6.6*10^8
99820000     5751639          <--- input = 9.982*10^7
40550000     2465109          <--- input = 4.055*10^7
24850000     1557132          <--- input = 2.485*10^7
41500        4339

#!/bin/bash

a=(1907000000 1337000000 1240000000 660000000 99820000 40550000 24850000 41500)

echo DennisC
exec 2>> times/dennisc.txt
time for j in ${a[@]}; do ./dennisc $j; done >> /dev/null;

echo DennisPy
exec 2>> times/dennispy.txt
time for j in ${a[@]}; do pypy dennispy.py <<< $j; done >> /dev/null;

echo arjandelumens
exec 2>> times/arjandelumens.txt
time for j in ${a[@]}; do ./arjandelumens $j; done >> /dev/null;

echo orlp
exec 2>> times/orlp.txt
time for j in ${a[@]}; do ./orlp $j; done >> /dev/null;

# echo mwr247
# time node-v4.3.1-linux-x64/bin/node mwr247.js

# mwr247 using js seems a bit longer, so I am going to run the fastest
# and then come back to his.

# mwr247 provided a function, so I appended
# console.log( F( <argument> ) )
# to his code, for each argument.

$ ./timepi '-march=native -O3' pi 1000
pi(x) =  93875448    real time:  2774958 ns
pi(x) =  66990613    real time:  2158491 ns
pi(x) =  62366021    real time:  2023441 ns
pi(x) =  34286170    real time:  1233158 ns
pi(x) =   5751639    real time:   384284 ns
pi(x) =   2465109    real time:   239783 ns
pi(x) =   1557132    real time:   196248 ns
pi(x) =      4339    real time:    60597 ns

0.00572879 s

real    0m28.006s
user    0m15.703s
sys 0m14.319s

real    0m8.934s
user    0m8.795s
sys 0m0.150s

real    0m8.956s
user    0m8.818s
sys 0m0.150s

real    0m8.907s
user    0m8.775s
sys 0m0.145s

real    0m8.904s
user    0m8.775s
sys 0m0.141s

real    0m8.902s
user    0m8.783s
sys 0m0.132s

real    0m9.087s
user    0m8.923s
sys 0m0.176s

real    0m8.905s
user    0m8.778s
sys 0m0.140s

real    0m9.005s
user    0m8.859s
sys 0m0.158s

real    0m8.911s
user    0m8.789s
sys 0m0.135s

real    0m8.907s
user    0m8.781s
sys 0m0.138s

$ time for i in 1.907e9 1.337e9 1.24e9 6.6e8 9.982e7 4.055e7 2.485e7 41500
> do pypy pi.py <<< $i; done
93875448
66990613
62366021
34286170
5751639
2465109
1557132
4339

real    0m1.696s
user    0m1.360s
sys     0m0.332s

real    3m56.961s
user    3m38.802s
sys 0m18.512s

0000000: cafe babe 0000 0034 0030 0a00 0900 1709  .......4.0......
0000010: 0018 0019 0a00 1a00 1b0a 0008 001c 0a00  ................
0000020: 1d00 1e0a 0008 001f 0a00 2000 2107 0022  .......... .!.."
0000030: 0700 2301 0006 3c69 6e69 743e 0100 0328  ..#...<init>...(
0000040: 2956 0100 0443 6f64 6501 000f 4c69 6e65  )V...Code...Line
0000050: 4e75 6d62 6572 5461 626c 6501 0004 6d61  NumberTable...ma
0000060: 696e 0100 1628 5b4c 6a61 7661 2f6c 616e  in...([Ljava/lan
0000070: 672f 5374 7269 6e67 3b29 5601 0008 6e75  g/String;)V...nu
0000080: 6d50 7269 6d65 0100 0428 4929 4901 000d  mPrime...(I)I...
0000090: 5374 6163 6b4d 6170 5461 626c 6501 0007  StackMapTable...
00000a0: 6973 5072 696d 6501 0004 2849 295a 0100  isPrime...(I)Z..
00000b0: 0a53 6f75 7263 6546 696c 6501 0006 452e  .SourceFile...E.
00000c0: 6a61 7661 0c00 0a00 0b07 0024 0c00 2500  java.......$..%.
00000d0: 2607 0027 0c00 2800 290c 0010 0011 0700  &..'..(.).......
00000e0: 2a0c 002b 002c 0c00 1300 1407 002d 0c00  *..+.,.......-..
00000f0: 2e00 2f01 0001 4501 0010 6a61 7661 2f6c  ../...E...java/l
0000100: 616e 672f 4f62 6a65 6374 0100 106a 6176  ang/Object...jav
0000110: 612f 6c61 6e67 2f53 7973 7465 6d01 0003  a/lang/System...
0000120: 6f75 7401 0015 4c6a 6176 612f 696f 2f50  out...Ljava/io/P
0000130: 7269 6e74 5374 7265 616d 3b01 0011 6a61  rintStream;...ja
0000140: 7661 2f6c 616e 672f 496e 7465 6765 7201  va/lang/Integer.
0000150: 0008 7061 7273 6549 6e74 0100 1528 4c6a  ..parseInt...(Lj
0000160: 6176 612f 6c61 6e67 2f53 7472 696e 673b  ava/lang/String;
0000170: 2949 0100 136a 6176 612f 696f 2f50 7269  )I...java/io/Pri
0000180: 6e74 5374 7265 616d 0100 0770 7269 6e74  ntStream...print
0000190: 6c6e 0100 0428 4929 5601 000e 6a61 7661  ln...(I)V...java
00001a0: 2f6c 616e 672f 4d61 7468 0100 0473 7172  /lang/Math...sqr
00001b0: 7401 0004 2844 2944 0021 0008 0009 0000  t...(D)D.!......
00001c0: 0000 0004 0001 000a 000b 0001 000c 0000  ................
00001d0: 001d 0001 0001 0000 0005 2ab7 0001 b100  ..........*.....
00001e0: 0000 0100 0d00 0000 0600 0100 0000 0100  ................
00001f0: 0900 0e00 0f00 0100 0c00 0000 2c00 0300  ............,...
0000200: 0100 0000 10b2 0002 2a03 32b8 0003 b800  ........*.2.....
0000210: 04b6 0005 b100 0000 0100 0d00 0000 0a00  ................
0000220: 0200 0000 0300 0f00 0400 0a00 1000 1100  ................
0000230: 0100 0c00 0000 6600 0200 0300 0000 2003  ......f....... .
0000240: 3c03 3d1c 1aa2 0018 1b1c b800 0699 0007  <.=.............
0000250: 04a7 0004 0360 3c84 0201 a7ff e91b ac00  .....`<.........
0000260: 0000 0200 0d00 0000 1600 0500 0000 0600  ................
0000270: 0200 0700 0900 0800 1800 0700 1e00 0900  ................
0000280: 1200 0000 1800 04fd 0004 0101 5001 ff00  ............P...
0000290: 0000 0301 0101 0002 0101 fa00 0700 0a00  ................
00002a0: 1300 1400 0100 0c00 0000 9700 0300 0300  ................
00002b0: 0000 4c1a 05a2 0005 03ac 1a05 9f00 081a  ..L.............
00002c0: 06a0 0005 04ac 1a05 7099 0009 1a06 709a  ........p.....p.
00002d0: 0005 03ac 1a87 b800 078e 0460 3c10 063d  ...........`<..=
00002e0: 1c1b a300 1b1a 1c04 6470 9900 0b1a 1c04  ........dp......
00002f0: 6070 9a00 0503 ac84 0206 a7ff e604 ac00  `p..............
0000300: 0000 0200 0d00 0000 2200 0800 0000 0c00  ........".......
0000310: 0700 0d00 1300 0e00 2100 0f00 2a00 1000  ........!...*...
0000320: 3200 1100 4400 1000 4a00 1200 1200 0000  2...D...J.......
0000330: 1100 0907 0901 0b01 fd00 0b01 0114 01fa  ................
0000340: 0005 0001 0015 0000 0002 0016            ............

> time java E 41500;time java E 24850000;time java E 40550000;time java E 99820000;time java E 660000000;time java E 1240000000;time java E 1337000000;time java E 1907000000
4339

real    0m0.236s
user    0m0.112s
sys     0m0.024s
1557132

real    0m8.842s
user    0m8.784s
sys     0m0.060s
2465109

real    0m18.442s
user    0m18.348s
sys     0m0.116s
5751639

real    1m15.642s
user    1m8.772s
sys     0m0.252s
34286170

real    40m35.810s
user    16m5.240s
sys     0m5.820s
62366021

real    104m12.628s
user    39m32.348s
sys     0m13.584s
66990613

real    110m22.064s
user    42m28.092s
sys     0m11.320s
93875448

real    171m51.650s
user    68m39.968s
sys     0m14.916s

use std::env;

const EMPTY: usize = std::usize::MAX;
const MAX_X: usize = 800;

fn main() {
    let args: Vec<_> = env::args().collect();
    let x: usize = args[1].trim().parse().expect("expected a number");

    let root = (x as f64).sqrt() as usize;
    let y = (x as f64).powf(0.3333333333333) as usize + 1;

    let sieve_size = x / y + 2;
    let mut sieve = vec![true; sieve_size];
    let mut primes = vec![0; sieve_size];
    sieve[0] = false;
    sieve[1] = false;

    let mut a = 0;
    let mut num_primes = 1;

    let mut num_primes_smaller_root = 0;

    // find all primes up to x/y ~ x^2/3 aka sieve_size
    for i in 2..sieve_size {
        if sieve[i] {
            if i <= root {
                if i <= y {
                    a += 1;
                }
                num_primes_smaller_root += 1;
            }

            primes[num_primes] = i;
            num_primes += 1;
            let mut multiples = i;
            while multiples < sieve_size {
                sieve[multiples] = false;
                multiples += i;
            }
        }
    }

    let interesting_primes = primes[a + 1..num_primes_smaller_root + 1].iter();

    let p_2 =
        interesting_primes
        .map(|ip| primes.iter().take_while(|&&p| p <= x / ip).count())
        .enumerate()
        .map(|(i, v)| v - 1 - i - a)
        .fold(0, |acc, v| acc + v);

    let mut phi_results = vec![EMPTY; (a + 1) * MAX_X];
    println!("pi({}) = {}", x, phi(x, a, &primes, &mut phi_results) + a - 1 - p_2);
}

fn phi(x: usize, b: usize, primes: &[usize], phi_results: &mut [usize]) -> usize {
    if b == 0 {
        return x;
    }

    if x < MAX_X && phi_results[x + b * MAX_X] != EMPTY {
        return phi_results[x + b * MAX_X];
    }

    let value = phi(x, b - 1, primes, phi_results) - phi(x / primes[b], b - 1, primes, phi_results);
    if x < MAX_X {
        phi_results[x + b * MAX_X] = value;
    }
    value
}

#!/bin/bash

a=(1907000000 1337000000 1240000000 660000000 99820000 40550000 24850000 41500)

for i in {0..100}; do
    for j in ${a[@]}; do
        ./target/release/pi_n $j  > /dev/null;
    done;
done;

[me@localhost pi_n]$ time ./time.sh

real    0m37.011s
user    0m34.752s
sys 0m2.410s

#include <cmath>
#include <cstdint>
#include <iostream>
#include <vector>
#include <boost/dynamic_bitset.hpp>

uint64_t num_primes(uint64_t n) {
    // http://stackoverflow.com/questions/4643647/fast-prime-factorization-module
    uint64_t pi = (n >= 2) + (n >= 3);
    if (n < 5) return pi;

    n += 1;
    uint64_t correction = n % 6 > 1;
    uint64_t wheels[6] = { n, n - 1, n + 4, n + 3, n + 2, n + 1 };
    uint64_t limit = wheels[n % 6];

    boost::dynamic_bitset<> sieve(limit / 3);
    sieve.set();
    sieve[0] = false;

    for (uint64_t i = 0, upper = uint64_t(std::sqrt(limit))/3; i <= upper; ++i) {
        if (sieve[i]) {
            uint64_t k = (3*i + 1) | 1;
            for (uint64_t j = (k*k) / 3;                   j < limit/3; j += 2*k) sieve[j] = false;
            for (uint64_t j = (k*k + 4*k - 2*k*(i & 1))/3; j < limit/3; j += 2*k) sieve[j] = false;
        }
    }

    pi += sieve.count();
    for (uint64_t i = limit / 3 - correction; i < limit / 3; ++i) pi -= sieve[i];

    return pi;
}


int main(int argc, char** argv) {
    if (argc <= 1) {
        std::cout << "Usage: " << argv[0] << " nn";
        return 0;
    }

    std::cout << num_primes(std::stoi(argv[1])) << "n";
    return 0;
}

real    0m22.760s
user    0m22.704s
sys 0m0.080s

real    0m22.854s
user    0m22.800s
sys 0m0.077s

real    0m22.742s
user    0m22.700s
sys 0m0.066s

real    0m22.484s
user    0m22.450s
sys 0m0.059s

real    0m22.653s
user    0m22.597s
sys 0m0.080s

real    0m22.665s
user    0m22.602s
sys 0m0.088s

real    0m22.528s
user    0m22.489s
sys 0m0.062s

real    0m22.510s
user    0m22.474s
sys 0m0.060s

real    0m22.819s
user    0m22.759s
sys 0m0.084s

real    0m22.488s
user    0m22.459s
sys 0m0.053s

10^4 (10,000) => 0.001 seconds
10^5 (100,000) => 0.003 seconds
10^6 (1,000,000) => 0.009 seconds
10^7 (10,000,000) => 0.074 seconds
10^8 (100,000,000) => 1.193 seconds
10^9 (1,000,000,000) => 14.415 seconds

#include <cstdint>
#include <vector>
#include <iostream>
#include <limits>
#include <cmath>
#include <array>
// uses posix ffsll
#include <string.h>
#include <algorithm>
#include <thread>

constexpr uint64_t wheel_width = 2;
constexpr uint64_t buf_size = 1<<(10+6);
constexpr uint64_t dtype_width = 6;
constexpr uint64_t dtype_mask = 63;
constexpr uint64_t buf_len = ((buf_size*wheel_width)>>dtype_width);
constexpr uint64_t seg_len = 6*buf_size;
constexpr uint64_t limit_i_max = 0xfffffffe00000001ULL;

typedef std::vector<uint64_t> buf_type;

void mark_composite(buf_type& buf, uint64_t prime,
                    std::array<uint64_t, 2>& poff,
                    uint64_t seg_start, uint64_t max_j)
{
  const auto p = 2*prime;
  for(uint64_t k = 0; k < wheel_width; ++k)
  {
    for(uint64_t j = 2*poff[k]+(k==0); j < max_j; j += p)
    {
      buf[(j-seg_start)>>dtype_width] |= 1ULL << (j & dtype_mask);
      poff[k] += prime;
    }
  }
}

struct prime_counter
{
  buf_type buf;
  uint64_t n;
  uint64_t seg_a, seg_b;
  uint64_t nj;
  uint64_t store_max;
  uint64_t& store_res;

  prime_counter(uint64_t n, uint64_t seg_a, uint64_t seg_b, uint64_t nj, uint64_t store_max,
                uint64_t& store_res) :
    buf(buf_len), n(n), nj(nj), seg_a(seg_a), seg_b(seg_b),
    store_max(store_max), store_res(store_res)
  {}

  prime_counter(const prime_counter&) = default;
  prime_counter(prime_counter&&) = default;

  prime_counter& operator =(const prime_counter&) = default;
  prime_counter& operator =(prime_counter&&) = default;

  void operator()(uint64_t nsmall_segs,
                  const std::vector<uint64_t>& primes,
                  std::vector<std::array<uint64_t, 2> > poffs)
  {
    uint64_t res = 0;
    // no new prime added portion
    uint64_t seg_start = buf_size*wheel_width*seg_a;
    uint64_t seg_min = seg_len*seg_a+5;

    if(seg_a > nsmall_segs)
    {
      uint64_t max_j = buf_size*wheel_width*nsmall_segs+(seg_a-nsmall_segs)*(buf_len<<dtype_width);
      for(size_t k = 0; k < wheel_width; ++k)
      {
        for(uint64_t i = 0; i < poffs.size() && max_j >= (2*poffs[i][k]+(k==0)); ++i)
        {
          // adjust poffs
          // TODO: might be a more efficient way
          auto w = (max_j-(2*poffs[i][k]+(k==0)));
          poffs[i][k] += primes[i]*(w/(2*primes[i]));
          if(w % (2*primes[i]) != 0)
          {
            poffs[i][k]+=primes[i];// += primes[i]*(w/(2*primes[i])+1);
          }
          /*else
          {

          }*/
        }
      }
    }

    for(uint64_t seg = seg_a; seg < seg_b; ++seg)
    {
      std::fill(buf.begin(), buf.end(), 0);
      const uint64_t limit_i = std::min<uint64_t>((((seg_len+seg_min) >= limit_i_max) ?
                                                   std::numeric_limits<uint32_t>::max() :
                                                   ceil(sqrt(seg_len+seg_min))),
                                                  store_max);
      uint64_t max_j = std::min(seg_start+(buf_len<<dtype_width), nj);
      for(uint64_t i = 0; i < primes.size() && primes[i] <= limit_i; ++i)
      {
        mark_composite(buf, primes[i], poffs[i], seg_start, max_j);
      }
      // sieve
      uint64_t val;
      const uint64_t stop = std::min(seg_min+seg_len, n);
      for(uint64_t i = ffsll(~(buf[0]))-((~buf[0]) != 0)+64*((~buf[0]) == 0);
          (val = 6ULL*(i>>1)+seg_min+2ULL*(i&1ULL)) < stop;)
      {
        if(!(buf[i>>dtype_width] & (1ULL << (i & dtype_mask))))
        {
          ++res;
          ++i;
        }
        else
        {
          uint64_t mask = buf[i>>dtype_width]>>(i&dtype_mask);
          const int64_t inc = ffsll(~mask)-((~mask) != 0)+64*((~mask) == 0);
          i += inc;
        }
      }
      seg_min += seg_len;
      seg_start += buf_size*wheel_width;
    }
    store_res = res;
  }
};

uint64_t num_primes(uint64_t n)
{
  uint64_t res = (n >= 2) + (n >= 3);
  if(n >= 5)
  {
    buf_type buf(buf_len);
    // compute and store primes < sqrt(n)
    const uint64_t store_max = ceil(sqrt(n));

    // only primes >= 5
    std::vector<uint64_t> primes;
    std::vector<std::array<uint64_t, 2> > poffs;
    primes.reserve(ceil(1.25506*store_max/log(store_max)));
    poffs.reserve(ceil(1.25506*store_max/log(store_max)));
    uint64_t seg_start = 0;
    uint64_t seg_min = 5;
    const uint64_t num_segs = 1+(n-seg_min)/seg_len;
    const uint64_t nj = (n-seg_min)/3+1;
    // compute how many small segments there are
    const uint64_t nsmall_segs = 1+(store_max-seg_min)/seg_len;
    for(uint64_t seg = 0; seg < nsmall_segs; ++seg)
    {
      std::fill(buf.begin(), buf.end(), 0);
      // mark off small primes
      const uint64_t limit_i = std::min<uint64_t>((((seg_len+seg_min) >= limit_i_max) ?
                                                   std::numeric_limits<uint32_t>::max() :
                                                   ceil(sqrt(seg_len+seg_min))),
                                                  store_max);
      uint64_t max_j = std::min(seg_start+(buf_len<<dtype_width), nj);
      for(uint64_t i = 0; i < primes.size() && primes[i] <= limit_i; ++i)
      {
        mark_composite(buf, primes[i], poffs[i], seg_start, max_j);
      }
      // sieve
      uint64_t val;
      const uint64_t stop = std::min(seg_min+seg_len, n);
      for(uint64_t i = ffsll(~(buf[0]))-((~buf[0]) != 0)+64*((~buf[0]) == 0);
            (val = 6ULL*(i>>1)+seg_min+2ULL*(i&1ULL)) < stop;)
      {
        if(!(buf[i>>dtype_width] & (1ULL << (i & dtype_mask))))
        {
          if(val <= store_max)
          {
            // add prime and poffs
            primes.push_back(val);
            poffs.emplace_back();
            poffs.back()[0] = (val*val-1)/6-1;
            if(i&1)
            {
              // 6n+1 prime
              poffs.back()[1] = (val*val+4*val-5)/6;
            }
            else
            {
              // 6n+5 prime
              poffs.back()[1] = (val*val+2*val-5)/6;
            }
            // mark-off multiples
            mark_composite(buf, val, poffs.back(), seg_start, max_j);
          }
          ++res;
          ++i;
        }
        else
        {
          uint64_t mask = buf[i>>dtype_width]>>(i&dtype_mask);
          const int64_t inc = ffsll(~mask)-((~mask) != 0)+64*((~mask) == 0);
          i += inc;
        }
      }
      seg_min += seg_len;
      seg_start += buf_size*wheel_width;
    }
    // multi-threaded sieving for remaining segments
    std::vector<std::thread> workers;
    auto num_workers = std::min<uint64_t>(num_segs-nsmall_segs, std::thread::hardware_concurrency());
    std::vector<uint64_t> store_reses(num_workers);

    workers.reserve(num_workers);
    auto num_segs_pw = ceil((num_segs-nsmall_segs)/static_cast<double>(num_workers));
    for(size_t i = 0; i < num_workers; ++i)
    {
      workers.emplace_back(prime_counter(n, nsmall_segs+i*num_segs_pw,
                                         std::min<uint64_t>(nsmall_segs+(i+1)*num_segs_pw,
                                                            num_segs),
                                         nj, store_max, store_reses[i]),
                           nsmall_segs, primes, poffs);
    }
    for(size_t i = 0; i < num_workers; ++i)
    {
      workers[i].join();
      res += store_reses[i];
    }
  }
  return res;
}

int main(int argc, char** argv)
{
  if(argc <= 1)
  {
    std::cout << "usage: " << argv[0] << " nn";
    return -1;
  }
  std::cout << num_primes(std::stoll(argv[1])) << 'n';
}

g++ -std=c++11 -o sieve_mt -O3 -march=native -pthread sieve_mt.cpp

real    4m7.215s
user    23m54.086s
sys 0m1.239s

#include <stdlib.h>
#include <stdio.h>
#include <string.h>

unsigned char p[2000000001];

int main(int argc, char **argv)
{
        unsigned int n, c, i, j;

        n = atoi(argv[1]);
        memset(p, 1, n + 1);

        p[1] = p[0] = 0;

        for (i = 2, c = 0; i <= n; i++)
        {
                if (p[i])
                {
                        c++;
                        for (j = i + i; j <= n; j += i)
                                p[j] = 0;
                }
        }

        printf("%d: %dn", n, c);

        return 0;
}

real    2m42.657s
user    2m42.065s
sys 0m0.757s

real    2m42.947s
user    2m42.400s
sys 0m0.708s

real    2m42.827s
user    2m42.282s
sys 0m0.703s

real    2m42.800s
user    2m42.300s
sys 0m0.665s

real    2m42.562s
user    2m42.050s
sys 0m0.675s

real    2m42.788s
user    2m42.192s
sys 0m0.756s

real    2m42.631s
user    2m42.074s
sys 0m0.720s

real    2m42.658s
user    2m42.115s
sys 0m0.707s

real    2m42.710s
user    2m42.219s
sys 0m0.657s

real    2m42.674s
user    2m42.110s
sys 0m0.730s

real    1m21.227s
user    1m20.755s
sys 0m0.576s

real    1m20.944s
user    1m20.426s
sys 0m0.640s

real    1m21.052s
user    1m20.581s
sys 0m0.573s

real    1m21.328s
user    1m20.862s
sys 0m0.570s

real    1m21.253s
user    1m20.780s
sys 0m0.588s

real    1m20.925s
user    1m20.460s
sys 0m0.576s

real    1m21.011s
user    1m20.512s
sys 0m0.601s

real    1m21.011s
user    1m20.550s
sys 0m0.564s

real    1m20.875s
user    1m20.409s
sys 0m0.569s

real    1m21.703s
user    1m21.088s
sys 0m0.701s

public class PrimeCounter
{
public static final String START_CODE="=",
TEST_FORMAT="Input = %d , Output = %d , calculated in %f seconds%n",
PROMPT="Enter numbers to compute pi(x) for (Type ""+START_CODE+"" to start):%n",
WAIT="Calculating, please wait...%n",
WARNING="Probably won't work with values close to or more than 2^31%n",
TOTAL_OUTPUT_FORMAT="Total time for all inputs is %f seconds%n";
public static final int NUM_THREADS=16,LOW_LIM=1,HIGH_LIM=1<<28;
private static final Object LOCK=new Lock();
private static final class Lock{}
/**
 * Generates and counts primes using an optimized but naive iterative algorithm.
 * Uses MultiThreading for arguments above LOW_LIM
 * @param MAX : argument x for pi(x), the limit to which to generate numbers.
 */
public static long primeCount(long MAX){
    long ctr=1;
    if(MAX<1<<7){
        for(long i=3;i<=MAX;i+=2){
            if(isPrime(i))++ctr;
        }
    }else{
        long[] counts=new long[NUM_THREADS];
        for(int i=0;i<NUM_THREADS;++i){
            counts[i]=-1;
        }
        long range=Math.round((double)MAX/NUM_THREADS);
        for(int i=0;i<NUM_THREADS;++i){
            long start=(i==0)?3:i*range+1,end=(i==NUM_THREADS-1)?MAX:(i+1)*range;
            final int idx=i;
            new Thread(new Runnable(){
                    public void run(){
                        for(long j=start;j<=end;j+=2){
                            if(isPrime(j))++counts[idx];
                        }
                    }
                }).start();
        }
        synchronized(LOCK){
            while(!completed(counts)){
                try{
                    LOCK.wait(300);}catch(InterruptedException ie){}
            }
            LOCK.notifyAll();
        }
        for(long count:counts){
            ctr+=count;
        }
        ctr+=NUM_THREADS;
    }
    return ctr;
}

/**
 * Checks for completion of threads
 * @param array : The array containing the completion data
 */
private static boolean completed(long[] array){
    for(long i:array){
        if(i<0)return false;
    }return true;
}

/**
 * Checks if the parameter is prime or not.
 * 2,3,5,7 are hardcoded as factors.
 * @param n : the number to check for primality
 */
private static boolean isPrime(long n){
    if(n==2||n==3||n==5||n==7)return true;
    else if(n%2==0||n%3==0||n%5==0||n%7==0)return false;
    else{
        for(long i=11;i<n;i+=2){
            if(n%i==0)return false;
        }
        return true;
    }
}

/**
 * Calculates primes using the atandard Sieve of Eratosthenes.
 * Uses 2,3,5,7 wheel factorization for elimination (hardcoded for performance reasons)
 * @param MAX : argument x for pi(x)
 * Will delegate to <code>primeCount(long)</code> for MAX<LOW_LIM and to <code>bitPrimeSieve(long)</code>
 * for MAX>HIGH_LIM, for performance reasons.
 */
public static long primeSieve(long MAX){
    if(MAX<=1)return 0;
    else if(LOW_LIM>0&&MAX<LOW_LIM){return primeCount(MAX);}
    else if(HIGH_LIM>0&&MAX>HIGH_LIM){return bitPrimeSieve(MAX);}
    int n=(int)MAX;
    int sn=(int)Math.sqrt(n),ctr=2;
    if(sn%2==0)--sn;
    boolean[]ps=new boolean[n+1];
    for(int i=2;i<=n;++i){
        if(i==2||i==3||i==5||i==7)ps[i]=true;
        else if(i%2!=0&&i%3!=0&&i%5!=0&&i%7!=0)ps[i]=true;
        else ++ctr;
    }
    for(int i=(n>10)?11:3;i<=sn;i+=2){
        if(ps[i]){
            for(int j=i*i;j<=n;j+=i){
                if(ps[j]){ ps[j]=false;++ctr;}
            }
        }
    }
    return (n+1-ctr);
}
/**
 * Calculates primes using bitmasked Sieve of Eratosthenes.
 * @param MAX : argument x for pi(x)
 */
public static long bitPrimeSieve(long MAX) {
    long SQRT_MAX = (long) Math.sqrt(MAX);
    if(SQRT_MAX%2==0)--SQRT_MAX;
    int MEMORY_SIZE = (int) ((MAX+1) >> 4);
    byte[] array = new byte[MEMORY_SIZE];
    for (long i = 3; i <= SQRT_MAX; i += 2) {
        if ((array[(int) (i >> 4)] & (byte) (1 << ((i >> 1) & 7))) == 0) {
            for(long j=i*i;j<=MAX;j+=i<<1) {
                if((array[(int) (j >> 4)] & (byte) (1 << ((j >> 1) & 7))) == 0){
                    array[(int) (j >> 4)] |= (byte) (1 << ((j >> 1) & 7));
                }
            }
        }
    }
    long pi = 1;
    for (long i = 3; i <= MAX; i += 2) {
        if ((array[(int) (i >> 4)] & (byte) (1 << ((i >> 1) & 7))) == 0) {
            ++pi;
        }
    }
    return pi;
}
/**
 * Private testing and timer function
 * @param MAX : input to be passed on to <code>primeSieve(long)</code>
 */
private static long sieveTest(long MAX){
    long start=System.nanoTime();
    long ps=primeSieve(MAX);
    long end=System.nanoTime();
    System.out.format(TEST_FORMAT,MAX,ps,((end-start)/1E9));
    return end-start;
}
/**
 * Main method: accepts user input and shows total execution time taken
 * @param args : The command-line arguments
 */
public static void main(String[]args){
    double total_time=0;
    java.util.Scanner sc=new java.util.Scanner(System.in);
    java.util.ArrayList<Long> numbers=new java.util.ArrayList<>();
    System.out.format(PROMPT+WARNING);
    String line=sc.nextLine();
    while(!line.equals(START_CODE)/*sc.hasNextLine()&&Character.isDigit(line.charAt(0))*/){
        numbers.add(Long.valueOf(line));
        line=sc.nextLine();
    }
    System.out.format(WAIT);
    for(long num:numbers){
        total_time+=sieveTest(num);
    }
    System.out.format(TOTAL_OUTPUT_FORMAT,total_time/1e9);
}
}

javac PrimeCounter.java
java PrimeCounter

Enter numbers to compute pi(x) for (Type "=" to start):
Probably won't work with values close to or more than 2^31
41500
24850000
40550000
99820000
660000000
1240000000
1337000000
1907000000
=
Calculating, please wait...
Input = 41500 , Output = 4339 , calculated in 0.002712 seconds
Input = 24850000 , Output = 1557132 , calculated in 0.304792 seconds
Input = 40550000 , Output = 2465109 , calculated in 0.523999 seconds
Input = 99820000 , Output = 5751639 , calculated in 1.326542 seconds
Input = 660000000 , Output = 34286170 , calculated in 4.750049 seconds
Input = 1240000000 , Output = 62366021 , calculated in 9.160406 seconds
Input = 1337000000 , Output = 66990613 , calculated in 9.989093 seconds
Input = 1907000000 , Output = 93875448 , calculated in 14.832107 seconds
Total time for all inputs is 40.889700 seconds

import sys

sys.setrecursionlimit(sys.maxsize)

n = int(sys.argv[-1])

if n < 4:
    print(0 if n < 2 else n-1)
    exit()

p = [0, 0] + [True] * n

i = 0
while i < pow(n, 0.5):
    if p[i]:
        j = pow(i, 2)
        while j < n:
            p[j] = False
            j += i
    i += 1

print(sum(p) - 2)

$ time python3 test.py 90
24

real    0m0.045s
user    0m0.031s
 sys    0m0.010s

#include <cstdint>
#include <vector>
#include <iostream>
#include <limits>
#include <cmath>
#include <array>
// uses posix ffsll
#include <string.h>
#include <algorithm>

constexpr uint64_t wheel_width = 2;
constexpr uint64_t buf_size = 1<<(10+6);
constexpr uint64_t dtype_width = 6;
constexpr uint64_t dtype_mask = 63;
constexpr uint64_t buf_len = ((buf_size*wheel_width)>>dtype_width);

typedef std::vector<uint64_t> buf_type;

void mark_composite(buf_type& buf, uint64_t prime,
                    std::array<uint64_t, 2>& poff,
                    uint64_t seg_start, uint64_t max_j)
{
  const auto p = 2*prime;
  for(uint64_t k = 0; k < wheel_width; ++k)
  {
    for(uint64_t j = 2*poff[k]+(k==0); j < max_j; j += p)
    {
      buf[(j-seg_start)>>dtype_width] |= 1ULL << (j & dtype_mask);
      poff[k] += prime;
    }
  }
}

uint64_t num_primes(uint64_t n)
{
  uint64_t res = (n >= 2) + (n >= 3);
  if(n >= 5)
  {
    buf_type buf(buf_len);
    // compute and store primes < sqrt(n)
    const uint64_t store_max = ceil(sqrt(n));

    // only primes >= 5
    std::vector<uint64_t> primes; // 5,7,11
    std::vector<std::array<uint64_t, 2> > poffs;// {{3,0},{0,5},{8,1}};
    primes.reserve(ceil(1.25506*store_max/log(store_max)));
    poffs.reserve(ceil(1.25506*store_max/log(store_max)));
    uint64_t seg_start = 0;
    uint64_t seg_min = 5;
    constexpr uint64_t seg_len = 6*buf_size;///wheel_width;
    constexpr uint64_t limit_i_max = 0xfffffffe00000001ULL;
    const uint64_t num_segs = 1+(n-seg_min)/seg_len;
    const uint64_t nj = (n-seg_min)/3+1;
    for(uint64_t seg = 0; seg < num_segs; ++seg)
    {
      std::fill(buf.begin(), buf.end(), 0);
      // mark off small primes
      const uint64_t limit_i = std::min<uint64_t>((((seg_len+seg_min) >= limit_i_max) ?
                                                   std::numeric_limits<uint32_t>::max() :
                                                   ceil(sqrt(seg_len+seg_min))),
                                                  store_max);
      uint64_t max_j = std::min(seg_start+(buf_len<<dtype_width), nj);
      for(uint64_t i = 0; i < primes.size() && primes[i] <= limit_i; ++i)
      {
        mark_composite(buf, primes[i], poffs[i], seg_start, max_j);
      }
      // sieve
      uint64_t val;
      const uint64_t stop = std::min(seg_min+seg_len, n);
      for(uint64_t i = ffsll(~(buf[0]))-((~buf[0]) != 0)+64*((~buf[0]) == 0);
            (val = 6ULL*(i>>1)+seg_min+2ULL*(i&1ULL)) < stop;)
      {
        if(!(buf[i>>dtype_width] & (1ULL << (i & dtype_mask))))
        {
          if(val <= store_max)
          {
            // add prime and poffs
            primes.push_back(val);
            poffs.emplace_back();
            poffs.back()[0] = (val*val-1)/6-1;
            if(i&1)
            {
              // 6n+1 prime
              poffs.back()[1] = (val*val+4*val-5)/6;
            }
            else
            {
              // 6n+5 prime
              poffs.back()[1] = (val*val+2*val-5)/6;
            }
            // mark-off multiples
            mark_composite(buf, val, poffs.back(), seg_start, max_j);
          }
          ++res;
          ++i;
        }
        else
        {
          uint64_t mask = buf[i>>dtype_width]>>(i&dtype_mask);
          const int64_t inc = ffsll(~mask)-((~mask) != 0)+64*((~mask) == 0);
          i += inc;
        }
      }
      seg_min += seg_len;
      seg_start += buf_size*wheel_width;
    }
  }
  return res;
}

int main(int argc, char** argv)
{
  if(argc <= 1)
  {
    std::cout << "usage: " << argv[0] << " nn";
    return -1;
  }
  std::cout << num_primes(std::stoll(argv[1])) << 'n';
}

g++ -std=c++11 -o sieve -O3 -march=native sieve.cpp

41500
4339

real    0m0.001s
user    0m0.000s
sys     0m0.000s

24850000
1557132

real    0m0.036s
user    0m0.032s
sys     0m0.000s

40550000
2465109

real    0m0.056s
user    0m0.052s
sys     0m0.000s

99820000
5751639

real    0m0.149s
user    0m0.144s
sys     0m0.000s

660000000
34286170

real    0m0.795s
user    0m0.788s
sys     0m0.000s

1240000000
62366021

real    0m1.468s
user    0m1.464s
sys     0m0.000s

1337000000
66990613

real    0m1.583s
user    0m1.576s
sys     0m0.004s

1907000000
93875448

real    0m2.363s
user    0m2.356s
sys     0m0.000s

real    0m9.415s
user    0m9.414s
sys 0m0.014s

real    0m9.315s
user    0m9.315s
sys 0m0.013s

real    0m9.307s
user    0m9.309s
sys 0m0.012s

real    0m9.333s
user    0m9.330s
sys 0m0.017s

real    0m9.288s
user    0m9.289s
sys 0m0.012s

real    0m9.319s
user    0m9.318s
sys 0m0.015s

real    0m9.285s
user    0m9.284s
sys 0m0.015s

real    0m9.342s
user    0m9.342s
sys 0m0.014s

real    0m9.305s
user    0m9.305s
sys 0m0.014s

real    0m9.312s
user    0m9.313s
sys 0m0.012s