Antoni Gual


« on: October 12, 2006, 08:56:19 AM » 

Make a program that calculates the value of the 1,000th, the 10,000th and the 100,000th prime. If FB is used the 1,000,000th prime must be calculated too. The first prime is 2, the second is 3 and so on... The number of primes up to the integer x can be aproximated by x/(log(x)1) (QB's log) The winner is who has the three primes right and faster. References: All you wanted to know about primes and never dared to ask http://primes.utm.edu/The idea is to generate primes and count them up to the required count. There are a lot of optimizations possible in this generation and counting so I hope we will get some interesing sources. We can open 3 categories according to the speed and memory limitations: Qbasic, Qb4.5 and FreeBASIC EDITED: Added an additional request for FB, without it even an unoptimized souce takes less than 0.5 second. EDIT2: For reference, the results are the 1000 th prime is 7919 the 10000 th prime is 104729 the 100000 th prime is 1299709 the 1000000 th prime is 15485863
of course your program must FIND these results!



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Antoni



nkk_kan
Member
Posts: 73


« Reply #1 on: October 12, 2006, 09:47:36 AM » 

err winer? :lol:



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Antoni Gual


« Reply #2 on: October 12, 2006, 09:51:30 AM » 

Thanks!



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Antoni



yetifoot


« Reply #3 on: October 13, 2006, 07:53:32 AM » 

Heres my simple FB prime program, using trial divison. It basically combines the only two rules i know, that theres no need to look at even numbers (except 2), and that you only need to check if n is divisible by any prime numbers smaller than sqr(n) It takes about 0.5 seconds to get the 100,000 prime, and about 12.5 to get the 1,000,000 on my P4 1.8Ghz I've been reading the link Antoni posted and may try again once i learn some different methods. Dim Shared prime_list(1 To 1000000) As uInteger Dim Shared prime_list_count As uInteger
Dim As uInteger prime_val, i, j, n Dim As Integer is_prime
prime_list(1) = 2 ' Setup the first two values prime_list(2) = 3 prime_list_count = 2 prime_val = 3
Do prime_val += 2 ' Add 2, we don't need to look at even numbers is_prime = 1 n = prime_val i = 1 j = Int(sqr(n)) + 1 ' Only need to check if n is divisible by any prime smaller than sqr(n) While prime_list(i) < j If (n mod prime_list(i)) = 0 Then is_prime = 0 Exit While End If i += 1 Wend If is_prime Then prime_list_count += 1 prime_list(prime_list_count) = prime_val End If Loop Until prime_list_count = 1000000
Print prime_list(1000) Print prime_list(10000) Print prime_list(100000) Print prime_list(1000000)



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Antoni Gual


« Reply #4 on: October 13, 2006, 10:41:55 AM » 

Great, we have one entry! But sieves are faster...



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Antoni



yetifoot


« Reply #5 on: October 13, 2006, 05:35:41 PM » 

Heres another one, a fairly standard sieve, using a bit array. I tried it just using a regular array which will take much more memory, and i though would be faster, but this bit based method won in the timings. I also tried to change IsGood to a macro, but that actually slowed it down, I think thats P4 weirdness. The only real optimization i did for this was the multiples of two, which because i used a bit array, i just iterated through, and masked out all multiple of two numbers. This one runs in about 3.5 to 4 seconds, so 1/3 of the time of my first attempt, and i'm sure there are better methods yet... Const MaxPrime = 1000000 ' The max value for P Const MaxVal = MaxPrime * 16 ' The max value for N (a bit of a cheat using * 16...) Const Max32 = (MaxVal \ 32) + 1 ' The number of 32bit vars needed to store bit array
Dim Shared BitsArray(0 To Max32) As uInteger
#macro MarkGood(n) Scope Dim As uInteger p = (n) \ 32, o = (n) mod 32 BitsArray(p) = BITRESET(BitsArray(p), o) End Scope #endmacro
#macro MarkBad(n) Scope Dim As uInteger p = (n) \ 32, o = (n) mod 32 BitsArray(p) = BITSET(BitsArray(p), o) End Scope #endmacro
Function IsGood(ByVal n As uInteger) As Integer Dim As uInteger p = n \ 32, o = n mod 32 Return NOT BIT(BitsArray(p), o) End Function
Dim As uInteger i, n1, n2, count, mask = &H55555555
For i = 0 To Max32  1 ' Mark off all multiples of two quickly using a mask of 10... BitsArray(i) = mask Next i
MarkGood(2) ' Restore 2 as a prime MarkBad(1) ' Make 1 not a prime
count = 1 ' start count offset at 1 to account for 2 being a prime
For n1 = 3 To MaxVal If IsGood(n1) Then count += 1 If count = 1000 Then Print n1 If count = 10000 Then Print n1 If count = 100000 Then Print n1 If count = 1000000 Then Print n1 Exit For ' We've found the 1 millionth prime, we can quit End If For n2 = (n1 + n1) To MaxVal Step n1 ' work from n+n to max marking off multiples MarkBad(n2) Next n2 End If Next n1



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Antoni Gual


« Reply #6 on: October 13, 2006, 05:39:23 PM » 

This is an interesting source by Rich Geldreich anyone can find at ABC packets. It uses an original idea and it's probably the only way to do it in QBasic 1.1, because of the 160K memory limits. It ran in 14 secondes in QB1.1 and in 2 seconds compiled in QB4.5 'Prime tally using a moving window version of the Erathostenes' Sieve 'Antoni Gual 10/2006 for the comtest at QBN. Qbasic1.1 version ' 'A true bit sieve would be faster, but the memory sizes in QB1.1 'require bold ideas. This one was created for QB by Rich Geldreich in '1992 from an idea in Donald Knuth's TAOCP. ' 'In a normal sieve each prime found is used in turn to mark all its 'composites thus the complete sieve must be hold in memory tor the final 'tally. In Rich's version all primes found so far are used at the same 'time to mark composites in the same moving slice of ths sieve, the 'numbers left unmarked are primes,and they can be counted as the slice 'progresses. 'In fact there is no data representing the sieve slice...only a priority 'queue that keeps the primes and it's factors used in the present sieve 'slice. This queue has to be dimensioned to hold all primes up to the 'square root of the maximum prime, the present size of 4096 would allow 'for primes up to 2^31.
'Additional optimizations; ' Multiples of 2 and 3 are skipped ' A prime p starts to sieve at p*p, because p*a for a<p will be found ' by a. ' The heap is an udt but is kept in separate arrays for speed.
DEFINT AZ
DECLARE SUB PutPrime (a&) DECLARE FUNCTION GetPrime& ()
CONST heapsize = 4096
'Priority queue DIM heapq(1 TO heapsize) AS LONG DIM HeapQ1(1 TO heapsize) AS LONG DIM HeapQ2(1 TO heapsize) AS LONG
DIM SHARED n AS LONG DIM t AS LONG DIM Q AS LONG, Q1 AS LONG, Q2 AS LONG DIM TQ AS LONG, TQ1 AS LONG DIM u AS LONG, primepos AS LONG, cnt AS LONG
primepos = 1000
n = 5 d = 2 r = 1 t = 25 heapq(1) = 25 HeapQ1(1) = 10 HeapQ2(1) = 30
cnt = 2
DO DO Q = heapq(1) Q1 = HeapQ1(1) Q2 = HeapQ2(1)
TQ = Q + Q1 TQ1 = Q2  Q1
'***Insert Heap(1) into priority queue i = 1 DO j = i * 2 IF j <= r THEN IF j < r THEN IF heapq(j) > heapq(j + 1) THEN j = j + 1 END IF END IF
IF TQ > heapq(j) THEN heapq(i) = heapq(j) HeapQ1(i) = HeapQ1(j) HeapQ2(i) = HeapQ2(j) i = j ELSE EXIT DO END IF ELSE EXIT DO END IF LOOP heapq(i) = TQ HeapQ1(i) = TQ1 HeapQ2(i) = Q2 '***
LOOP UNTIL n <= Q
DO WHILE n < Q cnt = cnt + 1 IF cnt < heapsize THEN heapq(cnt  2) = n IF cnt = primepos THEN PRINT USING "The ####### th prime is ######### "; primepos; n IF primepos = 100000 THEN PRINT "Ended": SYSTEM primepos = primepos * 10 END IF n = n + d d = 6  d LOOP
IF n = t THEN u = heapq(r + 1) t = u * u
'***Find location for new entry j = r + 1 DO i = j \ 2 IF i = 0 THEN EXIT DO END IF IF heapq(i) <= t THEN EXIT DO END IF heapq(j) = heapq(i) HeapQ1(j) = HeapQ1(i) HeapQ2(j) = HeapQ2(i) j = i LOOP '*** heapq(j) = t IF (u MOD 3) = 2 THEN HeapQ1(j) = 2 * u ELSE HeapQ1(j) = 4 * u END IF HeapQ2(j) = 6 * u
r = r + 1 END IF n = n + d d = 6  d
LOOP



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Antoni



yetifoot


« Reply #7 on: October 13, 2006, 06:18:50 PM » 

I've just been looking at some of the previous posts about primes, seeing what methods other people used. I found a lot by you Antoni!, and some other interesting things I may try and add to my program. I came across this too which made me laugh http://members.surfeu.fi/kklaine/primebear.html



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yetifoot


« Reply #8 on: October 14, 2006, 12:12:18 AM » 

Thats a nice one Antoni, works fast, i'm still trying to understand how it works. I improved my second one, its a bit faster now, but I still need to learn more to make it go even faster. Some of the code wasn't necessary, and i even forgot to ignore multiples of 2. Const MaxPrime = 1000000 ' The max value for P Const MaxVal = MaxPrime * 16 ' The max value for N (a bit of a cheat using * 16...) Const Max32 = (MaxVal \ 32) + 1 ' The number of 32bit vars needed to store bit array
Dim Shared BitsArray(0 To Max32) As uInteger Dim As uInteger n1, n2, count, p, o, n1x2, n1x3 Dim As uInteger steps(0 To 47) = { _ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, _ 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, _ 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10 _ } ' This lookup table is used to calculate the step, to avoid multiples of 2, 3, 5, and 7 ' any more that that and the table becomes very large (the one inc. 11 is 480 entrys) Dim As Integer curr_step Dim As Integer prime_to_find = 1000
count = 4 ' start count offset to account for 2, 3, 5 and 7 being prime
n1 = 11 ' start at 11 due to count starting at 4 While n1 <= MaxVal p = n1 shr 5 ' \ 32 'p is integer postition in bitarray o = n1 and 31 ' mod 32 'o is bit offset If NOT BIT(BitsArray(p), o) Then ' If the bit isn't set then it hasn't been struck out count += 1 If count = prime_to_find Then Print Using "###,###,###th prime  ###,###,###"; prime_to_find; n1 If prime_to_find = 1000000 Then Exit While prime_to_find *= 10 End If If n1 <= (sqr(MaxVal) + 1) Then ' Only strike out multiples of primes <= sqr(MaxVal) n1x2 = n1 + n1 ' (the +1 is just to account for any rounding, may not n1x3 = n1x2 + n1 ' be needed?) ' we don't need to step by n1, as that will wastefully look at even numbers (odd+odd=even) ' same goes for start pos, n1 is odd, so 2*n1 not needed, start at 3*n1 For n2 = n1x3 To MaxVal Step n1x2 ' work from 3n to max marking off multiples p = n2 shr 5 ' \ 32 'p is integer postition in bitarray o = n2 and 31 ' mod 32 'o is bit offset BitsArray(p) = BITSET(BitsArray(p), o) ' Set the bit to show its bad Next n2 End If End If ' we can step by set amounts, to avoid multiples of 2, 3 and 5 n1 += steps(curr_step) curr_step += 1 If curr_step = 48 Then curr_step = 0 Wend
EDIT: added a check for n1 <= sqr(maxval)



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Antoni Gual


« Reply #9 on: October 14, 2006, 06:13:17 AM » 

Ok,Arktinen Krokotiili Projekti is the winner! Let's close the contest, nothing more more can be done.. :rotfl: :rotfl: :rotfl: EDITED: Their javascript prime finding algorithm is a little slow.. function is_x_prime_number(x) { var limit=0; var div=3; var x_limit = Math.sqrt(x); while (x%div!=0 && div<x_limit)div+=2; is_prime = (x%div==0 && x!=div)*1 return is_prime; }
A simple trial division...



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Antoni



Antoni Gual


« Reply #10 on: October 14, 2006, 06:50:38 AM » 

Now seriously: Provisional scores for FB:
my (Rich's) entry Execution time: 2.585 s yetifoot's second entry Execution time: 4.377 s yetifoot's second entry Execution time: 2.131 s
Rich Geldreich's method is not that slow after all....



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Antoni



yetifoot


« Reply #11 on: October 14, 2006, 09:18:08 AM » 

Indeed, I found rich geldreichs method fast, even considering the fact its aimed at qb 1.1 and has to account for that. I spotted another optimization for my second one, i've just edited the post, as it wasn't enough to warrent posting the whole code again. I wasn't checking n1 <= sqr(maxval), even though i knew about that trick, i hadn't been able to find where to add it until know. I think my entry is now as fast as i can get it, but i already thought that about 5 times already, so who knows? Is anyone else thinking of entering some code? Maybe its just me and you Antoni, i fear i'll be in last place if that is the case



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Antoni Gual


« Reply #12 on: October 14, 2006, 10:33:49 AM » 

Your modified last entry scores 1.27 seconds, it nearly doubles its previous speed...
Im' surprised other people has'nt showed up. Is someone coding something?



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Antoni



Antoni Gual


« Reply #13 on: October 14, 2006, 04:18:01 PM » 

I think i have killed the competition by myself... First I forgot QB1.1 can't do huge arrays so a sieve up to 1.6M to find the 100,000th prime is out of limits. Then I posted Rich Geldeich's solution for QB4.5, thinking a sieve was faster. In fact it is..in FreeBASIC. With all that huge arrays, index offseting and bit twiddling a sieve in QB4.5 is actually slower than Rich's solution. So perhaps this is the reason because no one else enters. Here is a sieve for QB 4.5 'Prime sieve for QB4.5 by Antoni Gual 10/2006 'Run QB4.5 with /AH DEFLNG AH DEFINT IM amaxp = 1300000 'index offset of 4000 so it does not reach the limit of 16737... REDIM p(4000 TO amaxp \ 64  4000) AS LONG
'to avoid bit rotations DIM powers2(31) AS LONG b = 1 FOR i = 0 TO 31 powers2(i) = b IF i = 30 THEN EXIT FOR b = b + b NEXT powers2(31) = &H80000000
ctarget = 1000 cnt = 1 asqrt = INT(SQR(amaxp))
FOR b = 3 TO amaxp STEP 2 IF (p((b \ 64)  4000) AND powers2((b AND 63) \ 2)) = 0 THEN cnt = cnt + 1 IF cnt = ctarget THEN PRINT ctarget; b IF ctarget = 100000 THEN SYSTEM ctarget = ctarget * 10 END IF IF b <= asqrt THEN FOR bb = b * b TO amaxp STEP 2 * b p((bb \ 64)  4000) = p((bb \ 64)  4000) OR powers2((bb AND 63) \ 2) NEXT END IF END IF NEXT



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Antoni



yetifoot


« Reply #14 on: October 15, 2006, 01:13:51 PM » 

There are a few optimizations in there that I hadn't thought of.
1. Using a lookup table for the powers, to make the bittwiddling faster.
2. Starting the inner loop at n * n, i had discovered that i only needed to start an 3n, but n ^ 2 is even better!
3. Makeing the bitarray \ 64, because theres no need to store data for even numbers, I had thought of this, but I never got it working in my program, that halves the memory overheads.
I added these to my program, but it was only a very very small speed increase (less than 5%), so I haven't updated my code.



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