Sieve Of Atkin, Improved

February 19, 2010

We examined the Sieve of Atkin for enumerating prime numbers in a recent exercise. Though the Sieve of Atkin has the potential to be faster than the Sieve of Eratosthenes, our implementation was not; the Sieve of Eratosthenes takes 1.219 seconds to enumerate the first million primes, our implementation of the Sieve of Atkins takes 22.954 seconds, nineteen times as long, to perform the same task.

The problem is that the naive implementation we used performs useless work. For instance, consider that you are sieving the numbers to a million, with the x and y variables of the naive implementation ranging from one to a thousand. In the first case, with 4x² + y² = k, 4·1000² + 1000² = 5000000, which is far beyond the end of the range being sieved. Reducing the limits means a lot less work and a much faster program.

Your task is to rewrite the naive Sieve of Atkins to eliminate useless work. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

Pages: 1 2

6 Responses to “Sieve Of Atkin, Improved”

  1. Mike said

    Here is my python version of an improved Sieve of Atkins. On my laptop it calculates the primes up to 1 million in .16 seconds, and primes to 10 million in about 1.5 seconds. That is 3.5 times faster than a basic Sieve of Eratosthenes and about 5.6 times faster than a naive Sieve of Atkins.

    To imptove the naive sieve, I first calculated appropriate loop limits. Second I replaced squaring of x and y with additions using the well known formula:(x+1)^2 = x^2 + (2x + 1), and changed the loop counters to return the difference between the squares (i.e., the 2x+1). Obviously, the factor on the x^2 term could also be included in the loop counter. For example, for the 4x^2 term goes 4 16 36 … , so the loop was set up to return the differences 4, 12, 20, ….

    Because of the way the loops counters incremented, I suspected there would be a pattern to the results of the n%12 calculations in the y^2 loops. After confirming the pattern, I changed the loop variables and split some loops in two, so that the loops only iterate over values of x and y for which the n%12 test would pass. This greatly reduced the number of loop iterations and eliminated the n%12 test. The start, stop, and step values for the loops were tricky to figure out.

    Although I plan further optimization of the 3x^2 – y^2 section when I have time, I thought I’d post what I have now. Also, I didn’t optimize memory usage at all, so I’m sure that can be reduced.

    def atkins13(limit=1000000):
        '''use sieve of atkins to find primes <= limit.'''
        primes = [0] * limit
    
        # n = 3x^2 + y^2 section
        xx3 = 3
        for dxx in xrange( 0, 12*int( sqrt( ( limit - 1 ) / 3 ) ), 24 ):
            xx3 += dxx
            y_limit = int(12*sqrt(limit - xx3) - 36)
            n = xx3 + 16
            for dn in xrange( -12, y_limit + 1, 72 ):
                n += dn
                primes[n] = not primes[n]
    
            n = xx3 + 4
            for dn in xrange( 12, y_limit + 1, 72 ):
                n += dn
                primes[n] = not primes[n]
    
        # n = 4x^2 + y^2 section
        xx4 = 0
        for dxx4 in xrange( 4, 8*int(sqrt((limit - 1 )/4)) + 4, 8 ):
            xx4 += dxx4
            n = xx4+1
    
            if xx4%3:
                for dn in xrange( 0, 4*int( sqrt( limit - xx4 ) ) - 3, 8 ):
                    n += dn
                    primes[n] = not primes[n]
    
            else:
                y_limit = 12 * int( sqrt( limit - xx4 ) ) - 36
    
                n = xx4 + 25
                for dn in xrange( -24, y_limit + 1, 72 ):
                    n += dn
                    primes[n] = not primes[n]
    
                n = xx4+1
                for dn in xrange( 24, y_limit + 1, 72 ):
                    n += dn
                    primes[n] = not primes[n]
    
        # n = 3x^2 - y^2 section
        xx = 1
        for x in xrange( 3, int( sqrt( limit / 2 ) ) + 1, 2 ):
            xx += 4*x - 4
            n = 3*xx
    
            if n > limit:
                min_y = ((int(sqrt(n - limit))>>2)<<2)
                yy = min_y*min_y
                n -= yy
                s = 4*min_y + 4
    
            else:
                s = 4
    
            for dn in xrange( s, 4*x, 8 ):
                n -= dn
                if n <= limit and n%12 == 11:
                        primes[n] = not primes[n]
    
        xx = 0
        for x in xrange( 2, int( sqrt( limit / 2 ) ) + 1, 2 ):
            xx += 4*x - 4
            n = 3*xx
    
            if n > limit:
                min_y = ((int(sqrt(n - limit))>>2)<<2)-1
                yy = min_y*min_y
                n -= yy
                s = 4*min_y + 4
    
            else:
                n -= 1
                s = 0
    
            for dn in xrange( s, 4*x, 8 ):
                n -= dn
                if n <= limit and n%12 == 11:
                        primes[n] = not primes[n]
    
        # eliminate squares        
        for n in xrange(5,int(sqrt(limit))+1,2):
            if primes[n]:
                for k in range(n*n, limit, n*n):
                    primes[k] = False
    
        return [2,3] + filter(primes.__getitem__, xrange(5,limit,2))
    
  2. Garrett said

    I made the above code MUCH faster by simply replacing the line:

    primes = [0] * limit

    with:

    primes = numpy.zeros(limit, dtype = ‘bool’)

    Obviously this requires the module numpy. This also helps considerably with the “memory management” he was talking about I would think!

    I’m still trying to wrap my head around the rest of of the code (really trying to wrap my head around the sieve in genera). Will post when I find out more!

  3. geeksumm said

    It’s simple. We can cut the loop by using these condition:
    if (xx + yy >= limit) {
    break;
    }

    Using JavaScript running on Chrome’s V8, it took about 2.8 seconds to generate primes with limit of 10 million. Here’s the code:

    function sieveOfAtkin(limit){
    limitSqrt = Math.sqrt(limit);
    sieve = [];

    //prime start from 2, and 3
    sieve[2] = true;
    sieve[3] = true;

    for (x = 1; x <= limitSqrt; x++) {
    xx = x*x;
    for (y = 1; y = limit) {
    break;
    }
    // first quadratic using m = 12 and r in R1 = {r : 1, 5}
    n = (4 * xx) + (yy);
    if (n <= limit && (n % 12 == 1 || n % 12 == 5)) {
    sieve[n] = !sieve[n];
    }
    // second quadratic using m = 12 and r in R2 = {r : 7}
    n = (3 * xx) + (yy);
    if (n y && n <= limit && (n % 12 == 11)) {
    sieve[n] = !sieve[n];
    }
    }
    }

    for (n = 5; n <= limitSqrt; n++) {
    if (sieve[n]) {
    x = n * n;
    for (i = x; i <= limit; i += x) {
    sieve[i] = false;
    }
    }
    }

    // primes are the one which sieve[n] returns true
    return sieve;
    }

  4. Harsh said

    Could you please explain how the you got the limits?

  5. Willy Good said

    This is written in F# under the CLR virtual machine as I prefer the functional programming style; I have translated the codes to Scala under JVM and have gotten about the same results, which makes sense as both are under virtual machines using JIT compilation and both have built in array bounds checks. These versions have very fast bit counting functions to obtain the number of primes rather than using enumeration as the enumeration can take much longer than the actual sieving.

    On Gist (https://gist.github.com/anonymous/d17da4f1a8db2fadf10c) I have posted both a reference version of a maximally factorized Sieve of Eratosthenes just as Atkin and Berstein did, but my version is not “crippled” and this version of the SoA for comparison.

    On the same Gist link is a version of the SoA from the pseudo code of the updated Wkikpedia article with an added improvement of an inner loop to almost eliminate wasted time of loops that won’t pass the modulo checks, and with the added advantage that the tight innermost culling loops are almost as simple as those used for SoE.

    The test program to run each over a billion, only requiring the change of the “open ” state to run either is as follows:

    open SoA
    
    [<EntryPoint>]
    let main argv =
      printfn "%s" getName
      if argv = null || argv.Length = 0 then
        failwith "\r\nno command line argument for limit!!!"
    
      let test = primes 100UL // |> Seq.iter (printf "%A ")
    
      let topNum = System.UInt64.Parse argv.[0]
    
      let strt = System.DateTime.Now.Ticks
    
    //  let count = { 0 .. 3999 } |> Seq.map (fun i -> SoA.primes topNum) |> Seq.max // |> Seq.length
      let count = primes topNum // |> Seq.length
    
      let elpsd = System.DateTime.Now.Ticks - strt
    
      printfn "Found %A primes up to %A in %A milliseconds." count topNum (elpsd / 10000L)
    
      printf "\r\nPress any key to exit:"
      System.Console.ReadKey(true) |> ignore
      printfn ""
      0 // return an integer exit code
    

    With the results as follows, first SoEMWF:

    Uses the Sieve of Eratosthenes to calculate the primes to argument.
    
    Found 50847534UL primes up to 1000000000UL in 3714L milliseconds.
    

    and for the SoA:

    Uses the Sieve of Atkin to calculate the primes to argument.
    
    Found 50847534UL primes up to 1000000000UL in 3892L milliseconds.
    

    You can see that the SoEMWF is still slightly faster than the SoA at one billion, but testing shows that the SoA gradually passes it by about two billion on up until we run out of memory at about 60 billion; however, this is for the huge array model, it’s easy to provide a page segmented version for the SoE but very difficult for the SoA and the performance picture completely changes with the SoA losing terribly…

  6. Willy Good said

    Module codes are here:

    module SoEMWF =
      let getName = "\r\nUses the Sieve of Eratosthenes to calculate the primes to argument.\r\n"
      let private WRDBTS, FRSTBP = 16, 23UL
      let private BPRMS = [| 2UL; 3UL; 5UL; 7UL; 11UL; 13UL; 17UL; 19UL |]
      let private BWPS = [| //base wheel positions + 23
        23uy;29uy;31uy;37uy;41uy;43uy;47uy;53uy; 59uy;61uy;67uy;71uy;73uy;79uy;83uy;89uy;
        97uy;101uy;103uy;107uy;109uy;113uy;121uy;127uy; 131uy;137uy;139uy;143uy;149uy;151uy;157uy;163uy;
        167uy;169uy;173uy;179uy;181uy;187uy;191uy;193uy; 197uy;199uy;209uy;211uy;221uy;223uy;227uy;229uy |]
      let private GAPS = [| //wheel position deltas for each computation of next incremental cull position
        6uy; 2uy; 6uy; 4uy; 2uy; 4uy; 6uy; 6uy; 2uy; 6uy; 4uy; 2uy; 6uy; 4uy; 6uy; 8uy;
        4uy; 2uy; 4uy; 2uy; 4uy; 8uy; 6uy; 4uy; 6uy; 2uy; 4uy; 6uy; 2uy; 6uy; 6uy; 4uy;
        2uy; 4uy; 6uy; 2uy; 6uy; 4uy; 2uy; 4uy; 2uy; 10uy; 2uy; 10uy; 2uy; 4uy; 2uy; 4uy;
        6uy; 2uy; 6uy; 4uy; 2uy; 4uy; 6uy; 6uy; 2uy; 6uy; 4uy; 2uy; 6uy; 4uy; 6uy; 8uy; //2 rounds for overflow
        4uy; 2uy; 4uy; 2uy; 4uy; 8uy; 6uy; 4uy; 6uy; 2uy; 4uy; 6uy; 2uy; 6uy; 6uy; 4uy;
        2uy; 4uy; 6uy; 2uy; 6uy; 4uy; 2uy; 4uy; 2uy; 10uy; 2uy; 10uy; 2uy; 4uy; 2uy; 4uy |]
      let private WHTS, WCRC = BWPS.Length, (GAPS |> Seq.map uint64 |> Seq.sum) / 2UL //check: 48 and 210UL
      let private WHLNDXS = //look up table for (position - 23) mod WCRC - 0 to WCRC - 1, to wheel index - 0 to WHTS - 1
        (GAPS |> Seq.take WHTS |> Seq.mapi (fun i x -> i, x)
        |> Seq.collect (fun (i, n) -> seq { for n = 1uy to n do yield byte i })) |> Seq.toArray
     
      let primes topNum =
        if topNum >= ((1UL <<< 30) / 3UL - 1UL) * uint64 WCRC + FRSTBP then failwith "Argument too large for algorithm!!!"
        let BFSZ = if topNum < FRSTBP then 6 else ((int ((topNum - FRSTBP) / uint64 WCRC) + 2) * 3) //includes topNum
     
        let cull (cmpsts: _ array) (bps: _ seq) =
          let topNdx = uint64 (cmpsts.Length * WRDBTS - 1)
          let topRng = topNdx / uint64 WHTS * WCRC + uint64 BWPS.[WHTS - 1] //from size of current array
          let kSqrtLim = int (sqrt (float (topRng - WCRC))) / int WCRC //base prime test limit value comp for extra
          bps |> Seq.takeWhile (((>=) kSqrtLim) << fst) |> Seq.iter (fun (k, i) -> //for each base prime less than limit
            let x = uint64 BWPS.[int i] in let p = (uint64 k * WCRC) + x
            let s = p * p - FRSTBP - x * p in let pm = 3 * int p
            { int i .. int i + WHTS - 1 } |> Seq.map (fun i ->
                let adj, ni = if i >= WHTS then WCRC, i - WHTS else 0UL, i
                let c0 = s + p * (uint64 BWPS.[ni] + adj)
                let c0d = c0 / WCRC in let c0n = WHLNDXS.[int (c0 - c0d * WCRC)]
                let c0dwtry = 3 * int c0d
                if c0n < 16uy then c0dwtry, c0n else if c0n < 32uy then c0dwtry + 1, c0n - 16uy
                                                                    else c0dwtry + 2, c0n - 32uy )
            |> Seq.takeWhile (((>) cmpsts.Length) << fst) //loop maximum of wheel hits times; run tight cull loop for each
            |> Seq.iter (fun (ccd, cm) -> //tight culling loops almost all the time, following faster when 32-bit mode
              let cmsk = 1us <<< int cm
              let inline setCmpst i = cmpsts.[i] <- cmpsts.[i] ||| cmsk
              let rec cullp c = 
                if c < cmpsts.Length then setCmpst c; cullp (c + pm) in cullp ccd) ) //fast for 64 bit mode
     
        let cmpsts = //first generate the small master pre-culled pattern array
          let mstr = Array.zeroCreate (11 * 13 * 17 * 19 * WHTS / WRDBTS)
          { WHTS - 4 .. WHTS - 1 } |> Seq.map (fun m -> (-1, byte m)) |> cull mstr //cull 11/13/17/19
          let arr = Array.zeroCreate BFSZ
          { 0 .. mstr.Length .. (arr.Length - 1) } //copy repetitions of pre-cull pattern array until full
          |> Seq.iter (fun i -> let len = if i + mstr.Length <= arr.Length
                                          then mstr.Length else arr.Length - i
                                System.Array.Copy(mstr, 0, arr, i, len))
          arr //has an slight excess of up to two wheels to come out even in int's/wheels
     
        let inline testCmptst ndx = cmpsts.[ndx >>> 4] &&& (1us <<< (ndx &&& 15)) <> 0us
     
        Seq.initInfinite id |> Seq.collect (fun k -> { 0 .. WHTS - 1 } |> Seq.map (fun i -> k, byte i))
        |> Seq.filter (fun (k, i) -> not (testCmptst (WHTS * k + int i))) |> cull cmpsts //cull all composites
     
        //an extremely fast bit counting routine by 16-bit words...
        //mask off all buffer composite bits above the top number:
        let topd = (topNum - FRSTBP) / WCRC in let topn = int WHLNDXS.[int (topNum - FRSTBP - topd * WCRC)]
        let trywrd = int topd * 3
        let wrd, wrdn = if topn < 16 then trywrd, topn else if topn < 32 then trywrd + 1, topn - 16
                                                                         else trywrd + 2, topn - 32
        let awrd = if topNum < FRSTBP then -1 else cmpsts.[wrd] <- cmpsts.[wrd] ||| (uint16 -2 <<< wrdn); wrd    
        { awrd + 1 .. cmpsts.Length - 1 } |> Seq.iter (fun w -> cmpsts.[w] <- uint16 -1)
        let rec count w acc = //tight count loop
          if w < cmpsts.Length then //magic int16 bit count algorithm - slightly slower than Look Up Table for int16's
            let a = cmpsts.[w] in let b = a - ((a >>> 1) &&& 0x5555us)
            let c = (b &&& 0x3333us) + ((b >>> 2) &&& 0x3333us)
            count (w + 1) (acc - uint64 ((((c + (c >>> 4)) &&& 0x0F0Fus) * 0x0101us) >>> 8))
          else acc in count 0 (uint64 (BPRMS |> Seq.takeWhile ((>=) topNum) |> Seq.length)) + //include base primes
                      uint64 cmpsts.Length * uint64 WRDBTS //counts bits not set
     
    //    seq { yield! BPRMS; |> Seq.takeWhile ((>=) topNum) //very slow but literal way of enumerating resulting primes:
    //          yield! Seq.initInfinite id
    //                 |> Seq.collect (fun k -> BWPS |> Seq.mapi (fun i x -> WCRC * uint64 k + uint64 x, WHTS * k + i))
    //                 |> Seq.takeWhile (((>=) topNum) << fst)
    //                 |> Seq.filter (not << testCmptst << snd) |> Seq.map (fst) }
     
    module SoA =
      let getName = "\r\nUses the Sieve of Atkin to calculate the primes to argument.\r\n"
      let private WRDBTS, FRSTBP = 16, 7UL
      let private BPRMS = [| 2UL; 3UL; 5UL |]
      let private BWPS = [| 7uy;11uy;13uy;17uy;19uy;23uy;29uy;31uy; //base wheel positions + 7
                            37uy;41uy;43uy;47uy;49uy;53uy;59uy;61uy |]
      let private GAPS = [| //wheel position deltas for each computation of next incremental cull position
        4uy; 2uy; 4uy; 2uy; 4uy; 6uy; 2uy; 6uy; 4uy; 2uy; 4uy; 2uy; 4uy; 6uy; 2uy; 6uy; //2 times for overflow
        4uy; 2uy; 4uy; 2uy; 4uy; 6uy; 2uy; 6uy; 4uy; 2uy; 4uy; 2uy; 4uy; 6uy; 2uy; 6uy |]
      let private WHTS, WCRC = BWPS.Length, (GAPS |> Seq.map uint64 |> Seq.sum) / 2UL //check: 16 and 60UL
      let private WHLNDXS = //look up table for (position - 23) mod WCRC - 0 to WCRC - 1, to wheel index - 0 to WHTS - 1
        (GAPS |> Seq.take WHTS |> Seq.mapi (fun i x -> i, x)
        |> Seq.collect (fun (i, n) -> seq { for n = 1uy to n do yield byte i })) |> Seq.toArray
     
      let primes topNum =
        if topNum >= ((1UL <<< 30) - 1UL) * uint64 WCRC + FRSTBP then failwith "Argument too large for algorithm!!!"
        let BFSZ = if topNum < FRSTBP then 2 else int ((topNum - FRSTBP) / WCRC) + 2 //includes topNum with spare
     
        let is_prms = //initialize the prime sieving array
          Array.zeroCreate BFSZ //has an slight excess of the rest of the last wheel to come out even in int's/wheels
     
        let setprms (prms: _ array) (bps: _ seq) =
          let topNdx = uint64 (prms.Length * WRDBTS - 1)
          let topRng = topNdx / uint64 WHTS * WCRC + uint64 BWPS.[WHTS - 1] //maximum value of current array
          let kSqrtLim = int (sqrt (float (topRng - WCRC))) / int WCRC //compensate for extra wheel
     
          //run the 4 * x * x + y * y quadratic prime generating sequences = all x's and odd y's
          Seq.initInfinite (uint64 << ((+) 1)) |> Seq.map (fun x -> 4UL * x * x + 1UL)
          |> Seq.takeWhile ((>=) topRng)
          |> Seq.collect (fun st -> { 1UL .. 2UL .. 29UL } |> Seq.map (fun midy -> st + midy * midy - 1UL, int midy + 15)
                                    |> Seq.takeWhile (((>=) topRng) << fst) )
          |> Seq.iter (fun (n, myp15) ->
            if n < FRSTBP then () else //no action for n = 5 - not one of the modulos for this quadratic
              let cbd = (n - FRSTBP) / WCRC in let cr = n - cbd * WCRC
              let nm = int (if cr < WCRC then cr else cr - WCRC)
              if 1UL <<< nm &&& 0x22022020022002UL = 0UL then () else //if nm in {1,13,17,29,37,41,49,53}
                let cm = 1us <<< (int WHLNDXS.[int (cr - FRSTBP)]) //following is the tight loop that does it
                let rec tgl c inc = if c < prms.Length then prms.[c] <- prms.[c] ^^^ cm; tgl (c + inc) (inc + 30)
                tgl (int cbd) myp15)
     
          //run the 3 * x * x + y * y quadratic prime generating sequences - odd x's and even y's
          Seq.initInfinite (uint64 << ((+) 1) << ((*) 2)) |> Seq.map (fun x -> 3UL * x * x + 4UL)
          |> Seq.takeWhile ((>=) topRng)
          |> Seq.collect (fun st -> { 2UL .. 2UL .. 30UL } |> Seq.map (fun midy -> st + midy * midy - 4UL, int midy + 15)
                                    |> Seq.takeWhile (((>=) topRng) << fst) )
          |> Seq.iter (fun (n, myp15) ->
            let cbd = (n - FRSTBP) / WCRC in let cr = n - cbd * WCRC
            let nm = int (if cr < WCRC then cr else cr - WCRC)
            if 1UL <<< nm &&& 0x80080080080UL = 0UL then () else //if cm in {7,19,31,43}
              let cm = 1us <<< (int WHLNDXS.[int (cr - FRSTBP)]) //following is the tight loop that does it
              let rec tgl c inc = if c < prms.Length then prms.[c] <- prms.[c] ^^^ cm; tgl (c + inc) (inc + 30)
              tgl (int cbd) myp15)
     
          //run the 3 * x * x - y * y quadratic prime generating sequences - both even/odd and odd/even combos in x's/y's
          Seq.initInfinite (uint64 << ((+) 2)) |> Seq.map (fun x -> 2UL * x * x + 2UL * x - 1UL, x - 1UL)
          |> Seq.takeWhile (((>=) topRng) << fst) //reverse order ok in the following
          |> Seq.collect (fun (st, by) -> Seq.unfold (fun x -> if x + 28UL < by || x < 1UL then None
                                                               else if x < 2UL then Some(x, 0UL) else Some(x, x - 2UL)) by
                                          |> Seq.map (fun midy -> st - midy * midy + by * by, int midy - 15)
                                          |> Seq.takeWhile (((>=) topRng) << fst))
          |> Seq.iter (fun (n, mym15) ->
            let cbd = (n - FRSTBP) / WCRC in let cr = n - cbd * WCRC
            let nm = int (if cr < WCRC then cr else cr - WCRC)
            if 1UL <<< nm &&& 0x800800000800800UL = 0UL then () else //if cm in {11,23,47,59}
              let cm = 1us <<< (int WHLNDXS.[int (cr - FRSTBP)]) //following is the tight loop that does it
              let rec tgl c inc = if inc > -15 && c < prms.Length then
                                    prms.[c] <- prms.[c] ^^^ cm; tgl (c + inc) (inc - 30) in tgl (int cbd) mym15)
     
          //run the prime square-free culling operations:
          bps |> Seq.takeWhile (((>=) kSqrtLim) << fst) |> Seq.iter (fun (k, i) -> //for each base prime less than limit
            let x = uint64 BWPS.[int i] in let p = (uint64 k * WCRC) + x in let sndx = BWPS.Length - 1
            let sqr = p * p in let s = sqr - FRSTBP - sqr * uint64 BWPS.[sndx]
            { sndx .. sndx + WHTS - 1 } |> Seq.map (fun i ->
                let adj, ni = if i >= WHTS then WCRC, i - WHTS else 0UL, i
                let c0 = s + sqr * (uint64 BWPS.[ni] + adj)
                let c0d = c0 / WCRC in let c0n = WHLNDXS.[int (c0 - c0d * WCRC)]
                (c0d, c0n) )
            |> Seq.takeWhile (((>) (prms.Length - 1)) << int << fst) //loop maximum of wheel hits times; run tight cull loop for each
            |> Seq.iter (fun (ccd, cm) -> //tight culling loops almost all the time, following faster when 32-bit mode
              let cmsk = ~~~(1us <<< int cm)
              let rec cullp c = //tight prime square culling loop
                if c < uint64 prms.Length then prms.[int c] <- prms.[int c] &&& cmsk; cullp (c + sqr) in cullp ccd) )
     
        let inline testPrm ndx = is_prms.[int (ndx >>> 4)] &&& (1us <<< (int ndx &&& 15)) <> 0us
     
        Seq.initInfinite id |> Seq.collect (fun k -> { 0 .. WHTS - 1 } |> Seq.map (fun i -> k, byte i))
        |> Seq.filter (fun (k, i) -> testPrm (WHTS * int k + int i)) |> setprms is_prms //find all primes
     
        //an extremely fast bit counting routine by 16-bit words => number of primes...
        //mask off all buffer prime bits above the top number:
        let wrd = (topNum - FRSTBP) / WCRC in let wrdn = int WHLNDXS.[int (topNum - FRSTBP - wrd * WCRC)]
        let awrd = if topNum < FRSTBP then -1 else
                   is_prms.[int wrd] <- is_prms.[int wrd] &&& ~~~(uint16 -2 <<< wrdn); int wrd
        { awrd + 1 .. is_prms.Length - 1 } |> Seq.iter (fun w -> is_prms.[w] <- 0us)
        let rec count w acc = //tight count loop
          if w < is_prms.Length then //magic int16 bit count algorithm - slightly slower than Look Up Table for int16's
            let a = is_prms.[w] in let b = a - ((a >>> 1) &&& 0x5555us)
            let c = (b &&& 0x3333us) + ((b >>> 2) &&& 0x3333us)
            count (w + 1) (acc + uint64 ((((c + (c >>> 4)) &&& 0x0F0Fus) * 0x0101us) >>> 8))
          else acc in count 0 (uint64 (BPRMS |> Seq.takeWhile ((>=) topNum) |> Seq.length)) //counts set bits incl. base
     
    //    seq { yield! BPRMS |> Seq.takeWhile ((>=) topNum); //very slow but literal way of enumerating resulting primes:
    //          yield! Seq.initInfinite id
    //                 |> Seq.collect (fun k -> BWPS |> Seq.mapi (fun i x -> WCRC * uint64 k + uint64 x, WHTS * k + i))
    //                 |> Seq.takeWhile (((>=) topNum) << fst)
    //                 |> Seq.filter (testPrm << snd) |> Seq.map (fst) }
    open SoA
     
    [<EntryPoint>]
    let main argv =
      printfn "%s" getName
      if argv = null || argv.Length = 0 then
        failwith "\r\nno command line argument for limit!!!"
     
      let test = primes 100UL // |> Seq.iter (printf "%A ")
     
      let topNum = System.UInt64.Parse argv.[0]
     
      let strt = System.DateTime.Now.Ticks
     
    //  let count = { 0 .. 3999 } |> Seq.map (fun i -> SoA.primes topNum) |> Seq.max // |> Seq.length
      let count = primes topNum // |> Seq.length
     
      let elpsd = System.DateTime.Now.Ticks - strt
     
      printfn "Found %A primes up to %A in %A milliseconds." count topNum (elpsd / 10000L)
     
      printf "\r\nPress any key to exit:"
      System.Console.ReadKey(true) |> ignore
      printfn ""
      0 // return an integer exit code
    

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 677 other followers

%d bloggers like this: