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Prime counting utility using primesieve (http://code.google.com/p/primesieve/). Includes counting via sieve, Legendre, Meissel, and Lehmer methods. Part of the Math::Prime::Util Perl module (https://github.com/danaj/Math-Prime-Util). All code is in C, but primesieve makes a C++ library, hence the C++ compilation.
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| #include <stdio.h> | |
| #include <stdlib.h> | |
| #include <string.h> | |
| #include <math.h> | |
| /* Below this size, just sieve. */ | |
| #define SIEVE_LIMIT 1000000 | |
| /***************************************************************************** | |
| * | |
| * Lehmer prime counting utility. Calculates pi(x), count of primes <= x. | |
| * | |
| * Copyright (c) 2012-2013 Dana Jacobsen ([email protected]). | |
| * This is free software; you can redistribute it and/or modify it under | |
| * the same terms as the Perl 5 programming language system itself. | |
| * | |
| * This file is part of the Math::Prime::Util Perl module, but also can be | |
| * compiled as a standalone UNIX program using the primesieve package. | |
| * | |
| * g++ -O3 -DPRIMESIEVE_STANDALONE lehmer.c -o prime_count -lprimesieve | |
| * | |
| * For faster prime counting in stage 4 with multiprocessor machines: | |
| * | |
| * g++ -O3 -DPRIMESIEVE_STANDALONE -DPRIMESIEVE_PARALLEL lehmer.c -o prime_count -lprimesieve -lgomp | |
| * | |
| * The phi(x,a) calculation is unique, to the best of my knowledge. It uses | |
| * two lists of all x values + signed counts for the given 'a' value, and walks | |
| * 'a' down until it is small enough to calculate directly (either with Mapes | |
| * or using a calculated table using the primorial/totient method). This | |
| * is relatively fast and low memory compared to many other solutions. As with | |
| * all Lehmer-Meissel-Legendre algorithms, memory use will be a constraint | |
| * with large values of x (see the table below). | |
| * | |
| * If you want something better, I highly recommend the paper "Computing | |
| * Pi(x): the combinatorial method" (2006) by Tomás Oliveira e Silva. His | |
| * implementation is certainly much faster and lower memory than this, but I | |
| * have not seen any working source code for one of the LMO methods so it is | |
| * difficult to compare. | |
| * | |
| * Using my sieve code with everything running in serial, calculating pi(10^12) | |
| * is done under 1 second on my computer. pi(10^14) takes under 30 seconds, | |
| * pi(10^16) in under 20 minutes. Compared with Thomas R. Nicely's pix4 | |
| * program, this one is 5x faster and uses 10x less memory. When compiled | |
| * with parallel primesieve it is over 10x faster. | |
| * pix4(10^16) takes 124 minutes, this code + primesieve takes < 4 minutes. | |
| * | |
| * Timings with Perl + MPU with all-serial computation. Using the standalone | |
| * program with parallel primesieve speeds up stage 4 a lot for large values. | |
| * The last column is the standalone time with 12-core parallel primesieve. | |
| * | |
| * n phi(x,a) mem/time | stage 4 mem/time | total time | pps time | |
| * 10^19 1979.41 | ~13GB | | 7h 26m | |
| * 10^18 5515MB 483.46 | 5390MB | | 87m 0s | |
| * 10^17 1698MB 109.56 | 1568MB 9684.1 | 163m 36 s | 17m 37s | |
| * 10^16 522MB 25.44 | 460MB 1066.3 | 18m 12 s | 3m 44s | |
| * 10^15 159MB 5.86 | 137MB 141.2 | 2m 28 s | 48.17 s | |
| * 10^14 48MB 1.34 | 41MB 22.55 | 23.58 s | 10.55 s | |
| * 10^13 14MB 0.304 | 12MB 3.87 | 4.16 s | 2.40 s | |
| * 10^12 4MB 0.070 | 4MB 0.716 | 0.78 s | 0.527 | |
| * 10^11 1MB 0.015 | 0.135 | 0.158s | 0.124s | |
| * 10^10 0.003 | 0.029 | 0.028s | 0.036s | |
| * | |
| * Reference: Hans Riesel, "Prime Numbers and Computer Methods for | |
| * Factorization", 2nd edition, 1994. | |
| */ | |
| static int const verbose = 0; | |
| /* #define STAGE_TIMING 1 */ | |
| #ifdef STAGE_TIMING | |
| #include <sys/time.h> | |
| #define DECLARE_TIMING_VARIABLES struct timeval t0, t1; | |
| #define TIMING_START gettimeofday(&t0, 0); | |
| #define TIMING_END_PRINT(text) \ | |
| { unsigned long long t; \ | |
| gettimeofday(&t1, 0); \ | |
| t = (t1.tv_sec-t0.tv_sec) * 1000000 + (t1.tv_usec - t0.tv_usec); \ | |
| printf("%s: %10.5f\n", text, ((double)t) / 1000000); } | |
| #else | |
| #define DECLARE_TIMING_VARIABLES | |
| #define TIMING_START | |
| #define TIMING_END_PRINT(text) | |
| #endif | |
| #ifdef PRIMESIEVE_STANDALONE | |
| /* countPrimes seems to be pretty slow for small ranges, so sieve more small | |
| * primes and count using binary search. Uses a lot of memory though. For | |
| * big ranges, countPrimes is really fast. */ | |
| #define SIEVE_MULT 10 | |
| #include <limits.h> | |
| #include <sys/time.h> | |
| #ifdef PRIMESIEVE_PARALLEL | |
| #include <primesieve/soe/ParallelPrimeSieve.h> | |
| ParallelPrimeSieve ps; | |
| #define SET_PPS_PARALLEL ps.setNumThreads(ParallelPrimeSieve::getMaxThreads()) | |
| #define SET_PPS_SERIAL ps.setNumThreads(1) | |
| #else | |
| #include <primesieve/soe/PrimeSieve.h> | |
| PrimeSieve ps; | |
| #define SET_PPS_PARALLEL /* */ | |
| #define SET_PPS_SERIAL /* */ | |
| #endif | |
| /* Translations from Perl + Math::Prime::Util to C/C++ + primesieve */ | |
| typedef unsigned long UV; | |
| typedef signed long IV; | |
| #define UV_MAX ULONG_MAX | |
| #define UVCONST(x) ((unsigned long)x##UL) | |
| #define New(id, mem, size, type) mem = (type*) malloc((size)*sizeof(type)) | |
| #define Newz(id, mem, size, type) mem = (type*) calloc(size, sizeof(type)) | |
| #define Renew(mem, size, type) mem = (type*) realloc(mem,(size)*sizeof(type)) | |
| #define Safefree(mem) free((void*)mem) | |
| #define _XS_prime_count(a, b) ps.countPrimes(a, b) | |
| #define croak(fmt,...) { printf(fmt,##__VA_ARGS__); exit(1); } | |
| #define prime_precalc(n) /* */ | |
| #define BITS_PER_WORD ((ULONG_MAX <= 4294967295UL) ? 32 : 64) | |
| static UV isqrt(UV n) | |
| { | |
| UV root; | |
| if (sizeof(UV) == 8 && n >= 18446744065119617025UL) return 4294967295UL; | |
| if (sizeof(UV) == 4 && n >= 4294836225UL) return 65535UL; | |
| root = (UV) sqrt((double)n); | |
| while (root*root > n) root--; | |
| while ((root+1)*(root+1) <= n) root++; | |
| return root; | |
| } | |
| /* Callback used for creating an array of primes. */ | |
| static UV* sieve_array = 0; | |
| static UV sieve_k; | |
| static UV sieve_n; | |
| void primesieve_callback(uint64_t pk) { | |
| if (sieve_k <= sieve_n) { | |
| if (pk < sieve_array[sieve_k-1]) | |
| croak("Primes generated out of order. Switch to serial mode.\n"); | |
| sieve_array[sieve_k++] = pk; | |
| } | |
| } | |
| /* Generate an array of n small primes, where the kth prime is element p[k]. | |
| * Remember to free when done. */ | |
| static UV* generate_small_primes(UV n) | |
| { | |
| UV* primes; | |
| double fn = (double)n; | |
| double flogn = log(fn); | |
| double flog2n = log(flogn); | |
| UV nth_prime = /* Dusart 2010 for > 179k, custom for 18-179k */ | |
| (n >= 688383) ? (UV) ceil(fn*(flogn+flog2n-1.0+((flog2n-2.00)/flogn))) : | |
| (n >= 178974) ? (UV) ceil(fn*(flogn+flog2n-1.0+((flog2n-1.95)/flogn))) : | |
| (n >= 18) ? (UV) ceil(fn*(flogn+flog2n-1.0+((flog2n+0.30)/flogn))) | |
| : 59; | |
| New(0, primes, n+1, UV); | |
| if (primes == 0) | |
| croak("Can not allocate small primes\n"); | |
| primes[0] = 0; | |
| sieve_array = primes; | |
| sieve_n = n; | |
| sieve_k = 1; | |
| SET_PPS_SERIAL; | |
| ps.generatePrimes(2, nth_prime, primesieve_callback); | |
| SET_PPS_PARALLEL; | |
| sieve_array = 0; | |
| return primes; | |
| } | |
| #else | |
| /* We will use pre-sieving to speed up counting for small ranges */ | |
| #define SIEVE_MULT 1 | |
| #include "lehmer.h" | |
| #include "util.h" | |
| #include "cache.h" | |
| #include "sieve.h" | |
| /* Generate an array of n small primes, where the kth prime is element p[k]. | |
| * Remember to free when done. */ | |
| static UV* generate_small_primes(UV n) | |
| { | |
| const unsigned char* sieve; | |
| UV* primes; | |
| UV i; | |
| double fn = (double)n; | |
| double flogn = log(fn); | |
| double flog2n = log(flogn); | |
| UV nth_prime = /* Dusart 2010 for > 179k, custom for 18-179k */ | |
| (n >= 688383) ? (UV) ceil(fn*(flogn+flog2n-1.0+((flog2n-2.00)/flogn))) : | |
| (n >= 178974) ? (UV) ceil(fn*(flogn+flog2n-1.0+((flog2n-1.95)/flogn))) : | |
| (n >= 18) ? (UV) ceil(fn*(flogn+flog2n-1.0+((flog2n+0.30)/flogn))) | |
| : 59; | |
| if (get_prime_cache(nth_prime, &sieve) < nth_prime) { | |
| release_prime_cache(sieve); | |
| croak("Could not generate sieve for %"UVuf, nth_prime); | |
| } | |
| New(0, primes, n+1, UV); | |
| if (primes == 0) | |
| croak("Can not allocate small primes\n"); | |
| primes[0] = 0; primes[1] = 2; primes[2] = 3; primes[3] = 5; | |
| i = 3; | |
| START_DO_FOR_EACH_SIEVE_PRIME( sieve, 7, nth_prime ) { | |
| if (i >= n) break; | |
| primes[++i] = p; | |
| } END_DO_FOR_EACH_SIEVE_PRIME | |
| release_prime_cache(sieve); | |
| if (i < n) | |
| croak("Did not generate enough small primes.\n"); | |
| if (verbose > 1) printf("generated %lu small primes, from 2 to %lu\n", i, primes[i]); | |
| return primes; | |
| } | |
| #endif | |
| static UV icbrt(UV n) | |
| { | |
| UV root = 0; | |
| /* int s = BITS_PER_WORD - (BITS_PER_WORD % 3); */ | |
| #if BITS_PER_WORD == 32 | |
| int s = 30; | |
| if (n >= UVCONST(4291015625)) return UVCONST(1625); | |
| #else | |
| int s = 63; | |
| if (n >= UVCONST(18446724184312856125)) return UVCONST(2642245); | |
| #endif | |
| #if 0 | |
| /* The integer cube root code is about 30% faster for me */ | |
| root = (UV) pow(n, 1.0/3.0); | |
| if (root*root*root > n) { | |
| root--; | |
| while (root*root*root > n) root--; | |
| } else { | |
| while ((root+1)*(root+1)*(root+1) <= n) root++; | |
| } | |
| #else | |
| for ( ; s >= 0; s -= 3) { | |
| UV b; | |
| root += root; | |
| b = 3*root*(root+1)+1; | |
| if ((n >> s) >= b) { | |
| n -= b << s; | |
| root++; | |
| } | |
| } | |
| #endif | |
| return root; | |
| } | |
| /* Given an array of primes[1..lastprime], return Pi(n) where n <= lastprime. | |
| * This is actually quite fast, and definitely faster than sieving. By using | |
| * this we can avoid caching prime counts and also skip most calls to the | |
| * segment siever. | |
| */ | |
| static UV bs_prime_count(UV n, UV const* const primes, UV lastprime) | |
| { | |
| UV i, j; | |
| if (n < 2) return 0; | |
| /* if (n > primes[lastprime]) return _XS_prime_count(2, n); */ | |
| if (n >= primes[lastprime]) { | |
| if (n == primes[lastprime]) return lastprime; | |
| croak("called bspc(%lu) with counts up to %lu\n", n, primes[lastprime]); | |
| } | |
| i = 1; | |
| j = lastprime; | |
| while (i < j) { | |
| UV mid = (i+j)/2; | |
| if (primes[mid] <= n) i = mid+1; | |
| else j = mid; | |
| } | |
| return i-1; | |
| } | |
| /* Use Mapes' method to calculate phi(x,a) for small a. This is really | |
| * convenient and a little Perl script will spit this code out for whatever | |
| * limit we select. It gets unwieldy with large a values. | |
| */ | |
| static UV mapes(UV x, UV a) | |
| { | |
| IV val; | |
| if (a == 0) return x; | |
| if (a == 1) return x-x/2; | |
| val = x-x/2-x/3+x/6; | |
| if (a >= 3) val += 0-x/5+x/10+x/15-x/30; | |
| if (a >= 4) val += 0-x/7+x/14+x/21-x/42+x/35-x/70-x/105+x/210; | |
| if (a >= 5) val += 0-x/11+x/22+x/33-x/66+x/55-x/110-x/165+x/330+x/77-x/154-x/231+x/462-x/385+x/770+x/1155-x/2310; | |
| if (a >= 6) val += 0-x/13+x/26+x/39-x/78+x/65-x/130-x/195+x/390+x/91-x/182-x/273+x/546-x/455+x/910+x/1365-x/2730+x/143-x/286-x/429+x/858-x/715+x/1430+x/2145-x/4290-x/1001+x/2002+x/3003-x/6006+x/5005-x/10010-x/15015+x/30030; | |
| if (a >= 7) val += 0-x/17+x/34+x/51-x/102+x/85-x/170-x/255+x/510+x/119-x/238-x/357+x/714-x/595+x/1190+x/1785-x/3570+x/187-x/374-x/561+x/1122-x/935+x/1870+x/2805-x/5610-x/1309+x/2618+x/3927-x/7854+x/6545-x/13090-x/19635+x/39270+x/221-x/442-x/663+x/1326-x/1105+x/2210+x/3315-x/6630-x/1547+x/3094+x/4641-x/9282+x/7735-x/15470-x/23205+x/46410-x/2431+x/4862+x/7293-x/14586+x/12155-x/24310-x/36465+x/72930+x/17017-x/34034-x/51051+x/102102-x/85085+x/170170+x/255255-x/510510; | |
| return (UV) val; | |
| } | |
| static UV mapes7(UV x) { /* A tiny bit faster setup for a=7 */ | |
| IV val = x-x/2-x/3-x/5+x/6-x/7+x/10-x/11-x/13+x/14+x/15-x/17+x/21+x/22+x/26 | |
| -x/30+x/33+x/34+x/35+x/39-x/42+x/51+x/55+x/65-x/66-x/70+x/77-x/78 | |
| +x/85+x/91-x/102-x/105-x/110+x/119-x/130+x/143-x/154-x/165-x/170 | |
| -x/182+x/187-x/195+x/210+x/221-x/231-x/238-x/255-x/273-x/286+x/330 | |
| -x/357-x/374-x/385+x/390-x/429-x/442-x/455+x/462+x/510+x/546-x/561 | |
| -x/595-x/663+x/714; | |
| if (x >= 715) { | |
| val += 0-x/715+x/770+x/858+x/910-x/935-x/1001-x/1105+x/1122+x/1155+x/1190 | |
| -x/1309+x/1326+x/1365+x/1430-x/1547+x/1785+x/1870+x/2002+x/2145 | |
| +x/2210-x/2310-x/2431+x/2618-x/2730+x/2805+x/3003+x/3094+x/3315 | |
| -x/3570+x/3927-x/4290+x/4641+x/4862+x/5005-x/5610-x/6006+x/6545 | |
| -x/6630+x/7293+x/7735-x/7854; | |
| if (x >= 9282) | |
| val += 0-x/9282-x/10010+x/12155-x/13090-x/14586-x/15015-x/15470+x/17017 | |
| -x/19635-x/23205-x/24310+x/30030-x/34034-x/36465+x/39270+x/46410 | |
| -x/51051+x/72930-x/85085+x/102102+x/170170+x/255255-x/510510; | |
| } | |
| return (UV) val; | |
| } | |
| /******************************************************************************/ | |
| /* In-order lists for manipulating our UV value / IV count pairs */ | |
| /******************************************************************************/ | |
| typedef struct { | |
| UV v; | |
| IV c; | |
| } vc_t; | |
| typedef struct { | |
| vc_t* a; | |
| UV size; | |
| UV n; | |
| } vcarray_t; | |
| static vcarray_t vcarray_create(void) | |
| { | |
| vcarray_t l; | |
| l.a = 0; | |
| l.size = 0; | |
| l.n = 0; | |
| return l; | |
| } | |
| static void vcarray_destroy(vcarray_t* l) | |
| { | |
| if (l->a != 0) { | |
| if (verbose > 2) printf("FREE list %p\n", l->a); | |
| Safefree(l->a); | |
| } | |
| l->size = 0; | |
| l->n = 0; | |
| } | |
| /* Insert a value/count pair. We do this indirection because about 80% of | |
| * the calls result in a merge with the previous entry. */ | |
| static void vcarray_insert(vcarray_t* l, UV val, IV count) | |
| { | |
| UV n = l->n; | |
| if (n > 0 && l->a[n-1].v < val) | |
| croak("Previous value was %lu, inserting %lu out of order\n", l->a[n-1].v, val); | |
| if (n >= l->size) { | |
| UV new_size; | |
| if (l->size == 0) { | |
| new_size = 20000; | |
| if (verbose>2) printf("ALLOCing list, size %lu\n", new_size); | |
| New(0, l->a, new_size, vc_t); | |
| } else { | |
| new_size = (UV) (1.5 * l->size); | |
| if (verbose>2) printf("REALLOCing list %p, new size %lu\n",l->a,new_size); | |
| Renew( l->a, new_size, vc_t ); | |
| } | |
| if (l->a == 0) croak("could not allocate list\n"); | |
| l->size = new_size; | |
| } | |
| /* printf(" inserting %lu %ld\n", val, count); */ | |
| l->a[n].v = val; | |
| l->a[n].c = count; | |
| l->n++; | |
| } | |
| /* Merge the two sorted lists A and B into A. Each list has no duplicates, | |
| * but they may have duplications between the two. We're quite interested | |
| * in saving memory, so first remove all the duplicates, then do an in-place | |
| * merge. */ | |
| static void vcarray_merge(vcarray_t* a, vcarray_t* b) | |
| { | |
| long ai, bi, bj, k, kn; | |
| long an = a->n; | |
| long bn = b->n; | |
| vc_t* aa = a->a; | |
| vc_t* ba = b->a; | |
| /* Merge anything in B that appears in A. */ | |
| for (ai = 0, bi = 0, bj = 0; bi < bn; bi++) { | |
| /* Skip forward in A until empty or aa[ai].v <= ba[bi].v */ | |
| UV bval = ba[bi].v; | |
| while (ai < an && aa[ai].v > bval) | |
| ai++; | |
| /* if A empty then copy the remaining elements */ | |
| if (ai >= an) { | |
| if (bi == bj) | |
| bj = bn; | |
| else | |
| while (bi < bn) | |
| ba[bj++] = ba[bi++]; | |
| break; | |
| } | |
| if (aa[ai].v == bval) | |
| aa[ai].c += ba[bi].c; | |
| else | |
| ba[bj++] = ba[bi]; | |
| } | |
| if (verbose>2) printf(" removed %lu duplicates from b\n", bn - bj); | |
| bn = bj; | |
| if (bn == 0) { /* In case they were all duplicates */ | |
| b->n = 0; | |
| return; | |
| } | |
| /* kn = the final merged size. All duplicates are gone, so this is exact. */ | |
| kn = an+bn; | |
| if ((long)a->size < kn) { /* Make A big enough to hold kn elements */ | |
| UV new_size = (UV) (1.2 * kn); | |
| if (verbose>2) printf("REALLOCing list %p, new size %lu\n", a->a, new_size); | |
| Renew( a->a, new_size, vc_t ); | |
| aa = a->a; /* this could have been changed by the realloc */ | |
| a->size = new_size; | |
| } | |
| /* merge A and B. Very simple using reverse merge. */ | |
| ai = an-1; | |
| bi = bn-1; | |
| for (k = kn-1; k >= 0; k--) { | |
| if (ai < 0) { /* A is exhausted, just filling in B */ | |
| if (bi < 0) croak("ran out of data during merge"); | |
| aa[k] = ba[bi--]; | |
| } else if (bi < 0) { /* We've caught up with A */ | |
| break; | |
| } else if (aa[ai].v < ba[bi].v) { | |
| aa[k] = aa[ai--]; | |
| } else { | |
| if (aa[ai].v == ba[bi].v) croak("deduplication error"); | |
| aa[k] = ba[bi--]; | |
| } | |
| } | |
| a->n = kn; /* A now has this many items */ | |
| b->n = 0; /* B is marked empty */ | |
| } | |
| /* | |
| * The main phi(x,a) algorithm. In this implementation, it takes under 10% | |
| * of the total time for the Lehmer algorithm, but is a big memory consumer. | |
| */ | |
| static UV phi(UV x, UV a) | |
| { | |
| UV i, val, sval; | |
| UV sum = 0; | |
| IV count; | |
| const UV* primes; | |
| vcarray_t a1, a2; | |
| vc_t* arr; | |
| if (a == 1) return ((x+1)/2); | |
| if (a <= 7) return mapes(x, a); | |
| primes = generate_small_primes(a+1); | |
| if (primes == 0) | |
| croak("Could not generate primes for phi(%lu,%lu)\n", x, a); | |
| if (x < primes[a+1]) { Safefree(primes); return (x > 0) ? 1 : 0; } | |
| a1 = vcarray_create(); | |
| a2 = vcarray_create(); | |
| vcarray_insert(&a1, x, 1); | |
| while (a > 7) { | |
| UV primea = primes[a]; | |
| UV sval_last = 0; | |
| IV sval_count = 0; | |
| arr = a1.a; | |
| for (i = 0; i < a1.n; i++) { | |
| count = arr[i].c; | |
| if (count == 0) continue; /* Skip if count = 0 */ | |
| val = arr[i].v; | |
| sval = val / primea; | |
| if (sval < primea) break; /* stop inserting into a2 if small */ | |
| if (sval != sval_last) { /* non-merged value. Insert into a2 */ | |
| if (sval_last != 0) | |
| vcarray_insert(&a2, sval_last, sval_count); | |
| sval_last = sval; | |
| sval_count = 0; | |
| } | |
| sval_count -= count; /* Accumulate count for this sval */ | |
| } | |
| if (sval_last != 0) /* Insert the last sval */ | |
| vcarray_insert(&a2, sval_last, sval_count); | |
| /* For each small sval, add up the counts */ | |
| for ( ; i < a1.n; i++) | |
| sum -= arr[i].c; | |
| /* Merge a1 and a2 into a1. a2 will be emptied. */ | |
| vcarray_merge(&a1, &a2); | |
| a--; | |
| } | |
| vcarray_destroy(&a2); | |
| if (a != 7) croak("final loop is set for a=7, a = %lu\n", a); | |
| arr = a1.a; | |
| for (i = 0; i < a1.n; i++) { | |
| count = arr[i].c; | |
| if (count != 0) | |
| sum += count * mapes7( arr[i].v ); | |
| } | |
| vcarray_destroy(&a1); | |
| Safefree(primes); | |
| return (UV) sum; | |
| } | |
| /* Legendre's method. Interesting and a good test for phi(x,a), but Lehmer's | |
| * method is much faster (Legendre: a = pi(n^.5), Lehmer: a = pi(n^.25)) */ | |
| UV _XS_legendre_pi(UV n) | |
| { | |
| UV a; | |
| if (n < SIEVE_LIMIT) | |
| return _XS_prime_count(2, n); | |
| a = _XS_legendre_pi(isqrt(n)); | |
| return phi(n, a) + a - 1; | |
| } | |
| /* Meissel's method. */ | |
| UV _XS_meissel_pi(UV n) | |
| { | |
| UV a, b, c, sum, i, lastprime, lastpc, lastw, lastwpc; | |
| const UV* primes = 0; /* small prime cache */ | |
| DECLARE_TIMING_VARIABLES; | |
| if (n < SIEVE_LIMIT) | |
| return _XS_prime_count(2, n); | |
| if (verbose > 0) printf("meissel %lu stage 1: calculate a,b,c \n", n); | |
| TIMING_START; | |
| a = _XS_meissel_pi(icbrt(n)); /* a = floor(n^1/3) */ | |
| b = _XS_meissel_pi(isqrt(n)); /* b = floor(n^1/2) */ | |
| c = a; /* c = a */ | |
| TIMING_END_PRINT("stage 1") | |
| if (verbose > 0) printf("meissel %lu stage 2: phi(x,a) (a=%lu b=%lu c=%lu)\n", n, a, b, c); | |
| TIMING_START; | |
| sum = phi(n, a) + ((b+a-2) * (b-a+1) / 2); | |
| if (verbose > 0) printf("phi(%lu,%lu) = %lu. sum = %lu\n", n, a, sum - ((b+a-2) * (b-a+1) / 2), sum); | |
| TIMING_END_PRINT("phi(x,a)") | |
| lastprime = b*SIEVE_MULT; | |
| if (verbose > 0) printf("meissel %lu stage 3: %lu small primes\n", n, lastprime); | |
| TIMING_START; | |
| primes = generate_small_primes(lastprime); | |
| if (primes == 0) croak("Error generating primes.\n"); | |
| lastpc = primes[lastprime]; | |
| TIMING_END_PRINT("small primes") | |
| prime_precalc(isqrt(n / primes[a+1])); | |
| prime_precalc( (UV) pow(n, 2.0/5.0) ); /* Sieve more for speed */ | |
| if (verbose > 0) printf("meissel %lu stage 4: loop %lu to %lu, pc to %lu\n", n, a+1, b, n/primes[a+1]); | |
| TIMING_START; | |
| /* Reverse the i loop so w increases. Count w in segments. */ | |
| lastw = 0; | |
| lastwpc = 0; | |
| for (i = b; i > a; i--) { | |
| UV w = n / primes[i]; | |
| lastwpc = (w <= lastpc) ? bs_prime_count(w, primes, lastprime) | |
| : lastwpc + _XS_prime_count(lastw+1, w); | |
| lastw = w; | |
| sum = sum - lastwpc; | |
| } | |
| TIMING_END_PRINT("stage 4") | |
| Safefree(primes); | |
| return sum; | |
| } | |
| /* Lehmer's method. This is basically Riesel's Lehmer function (page 22), | |
| * with some additional code to help optimize it. */ | |
| UV _XS_lehmer_pi(UV n) | |
| { | |
| UV z, a, b, c, sum, i, j, lastprime, lastpc, lastw, lastwpc; | |
| const UV* primes = 0; /* small prime cache, first b=pi(z)=pi(sqrt(n)) */ | |
| DECLARE_TIMING_VARIABLES; | |
| if (n < SIEVE_LIMIT) | |
| return _XS_prime_count(2, n); | |
| /* Protect against overflow. 2^32-1 and 2^64-1 are both divisible by 3. */ | |
| if (n == UV_MAX) { | |
| if ( (n%3) == 0 || (n%5) == 0 || (n%7) == 0 || (n%31) == 0 ) | |
| n--; | |
| else | |
| return _XS_prime_count(2,n); | |
| } | |
| if (verbose > 0) printf("lehmer %lu stage 1: calculate a,b,c \n", n); | |
| TIMING_START; | |
| z = isqrt(n); | |
| a = _XS_lehmer_pi(isqrt(z)); /* a = floor(n^1/4) */ | |
| b = _XS_lehmer_pi(z); /* b = floor(n^1/2) */ | |
| c = _XS_lehmer_pi(icbrt(n)); /* c = floor(n^1/3) */ | |
| TIMING_END_PRINT("stage 1") | |
| if (verbose > 0) printf("lehmer %lu stage 2: phi(x,a) (z=%lu a=%lu b=%lu c=%lu)\n", n, z, a, b, c); | |
| TIMING_START; | |
| sum = phi(n, a) + ((b+a-2) * (b-a+1) / 2); | |
| TIMING_END_PRINT("phi(x,a)") | |
| /* We get an array of the first b primes. This is used in stage 4. If we | |
| * get more than necessary, we can use them to speed up some. | |
| */ | |
| lastprime = b*SIEVE_MULT; | |
| if (verbose > 0) printf("lehmer %lu stage 3: %lu small primes\n", n, lastprime); | |
| TIMING_START; | |
| primes = generate_small_primes(lastprime); | |
| if (primes == 0) croak("Error generating primes.\n"); | |
| lastpc = primes[lastprime]; | |
| TIMING_END_PRINT("small primes") | |
| TIMING_START; | |
| /* Speed up all the prime counts by doing a big base sieve */ | |
| prime_precalc( (UV) pow(n, 3.0/5.0) ); | |
| /* Ensure we have the base sieve for big prime_count ( n/primes[i] ). */ | |
| /* This is about 75k for n=10^13, 421k for n=10^15, 2.4M for n=10^17 */ | |
| prime_precalc(isqrt(n / primes[a+1])); | |
| TIMING_END_PRINT("sieve precalc") | |
| if (verbose > 0) printf("lehmer %lu stage 4: loop %lu to %lu, pc to %lu\n", n, a+1, b, n/primes[a+1]); | |
| TIMING_START; | |
| /* Reverse the i loop so w increases. Count w in segments. */ | |
| lastw = 0; | |
| lastwpc = 0; | |
| for (i = b; i >= a+1; i--) { | |
| UV w = n / primes[i]; | |
| lastwpc = (w <= lastpc) ? bs_prime_count(w, primes, lastprime) | |
| : lastwpc + _XS_prime_count(lastw+1, w); | |
| lastw = w; | |
| sum = sum - lastwpc; | |
| if (i <= c) { | |
| UV bi = bs_prime_count( isqrt(w), primes, lastprime ); | |
| for (j = i; j <= bi; j++) { | |
| sum = sum - bs_prime_count(w / primes[j], primes, lastprime) + j - 1; | |
| } | |
| /* We could wrap the +j-1 in: sum += ((bi+1-i)*(bi+i))/2 - (bi-i+1); */ | |
| } | |
| } | |
| TIMING_END_PRINT("stage 4") | |
| Safefree(primes); | |
| return sum; | |
| } | |
| UV _XS_LMO_pi(UV n) | |
| { | |
| UV a, b, sum, i, lastprime, lastpc, lastw, lastwpc; | |
| UV n13, n12, n23; | |
| IV S1; | |
| UV S2, P2; | |
| const UV* primes = 0; /* small prime cache */ | |
| char* mu = 0; /* moebius to n^1/3 */ | |
| UV* lpf = 0; /* least prime factor to n^1/3 */ | |
| DECLARE_TIMING_VARIABLES; | |
| if (n < SIEVE_LIMIT) | |
| return _XS_prime_count(2, n); | |
| if (verbose > 0) printf("LMO %lu stage 1: calculate pi(n^1/3) \n", n); | |
| TIMING_START; | |
| n13 = icbrt(n); | |
| n12 = isqrt(n); | |
| n23 = (UV) (pow(n, 2.0/3.0)+0.01); | |
| a = _XS_lehmer_pi(n13); | |
| b = _XS_lehmer_pi(n12); | |
| TIMING_END_PRINT("stage 1") | |
| lastprime = b*SIEVE_MULT; | |
| if (verbose > 0) printf("LMO %lu stage 2: %lu small primes\n", n, lastprime); | |
| TIMING_START; | |
| primes = generate_small_primes(lastprime); | |
| if (primes == 0) croak("Error generating primes.\n"); | |
| lastpc = primes[lastprime]; | |
| TIMING_END_PRINT("small primes") | |
| if (verbose > 0) printf("LMO %lu stage 3: calculate mu/lpf to %lu\n", n, a); | |
| TIMING_START; | |
| /* We could call MPU's: | |
| * mu = _moebius_range(0, n13+1) | |
| * but (1) it's a bit slower (something to be addressed), and (2) we will | |
| * do the least prime factor calculation at the same time. | |
| */ | |
| New(0, mu, n13+1, char); | |
| memset(mu, 1, sizeof(char) * (n13+1)); | |
| New(0, lpf, n13+1, UV); | |
| memset(lpf, 0, sizeof(UV) * (n13+1)); | |
| mu[0] = 0; | |
| for (i = 1; i <= a; i++) { | |
| UV primei = primes[i]; | |
| UV j, isquared; | |
| for (j = primei; j <= n13; j += primei) { | |
| mu[j] = -mu[j]; | |
| if (lpf[j] == 0) lpf[j] = primei; | |
| } | |
| isquared = primei * primei; | |
| for (j = isquared; j <= n13; j += isquared) | |
| mu[j] = 0; | |
| } | |
| /* for (i = 0; i <= n13; i++) { printf("mu %lu %ld\n", i, (IV)mu[i]); } */ | |
| TIMING_END_PRINT("mu") | |
| if (verbose > 0) printf("LMO %lu stage 4: calculate S1 (%lu)\n", n, n13); | |
| TIMING_START; | |
| S1 = 0; | |
| for (i = 1; i <= n13; i++) | |
| if (mu[i] != 0) | |
| S1 += mu[i] * (IV) (n/i); | |
| TIMING_END_PRINT("S1") | |
| if (verbose > 0) printf("LMO %lu stage 4: S1 = %ld\n", n, S1); | |
| S2 = 0; | |
| /* TODO... */ | |
| Safefree(mu); | |
| Safefree(lpf); | |
| prime_precalc(isqrt(n / primes[a+1])); | |
| if (verbose > 0) printf("LMO %lu stage 5: P2 loop %lu to %lu, pc to %lu\n", n, a+1, b, n/primes[a+1]); | |
| TIMING_START; | |
| P2 = 0; | |
| /* Reverse the i loop so w increases. Count w in segments. */ | |
| lastw = 0; | |
| lastwpc = 0; | |
| for (i = b; i > a; i--) { | |
| UV w = n / primes[i]; | |
| lastwpc = (w <= lastpc) ? bs_prime_count(w, primes, lastprime) | |
| : lastwpc + _XS_prime_count(lastw+1, w); | |
| lastw = w; | |
| P2 += lastwpc; | |
| } | |
| P2 -= ((b+a-2) * (b-a+1) / 2) - a + 1; | |
| TIMING_END_PRINT("P2") | |
| if (verbose > 0) printf("LMO %lu stage 5: P2 = %lu\n", n, P2); | |
| Safefree(primes); | |
| sum = P2 + S1 + S2; | |
| return sum; | |
| } | |
| #ifdef PRIMESIEVE_STANDALONE | |
| int main(int argc, char *argv[]) | |
| { | |
| UV n, pi; | |
| double t; | |
| const char* method; | |
| struct timeval t0, t1; | |
| if (argc <= 1) { printf("usage: %s <n> [<method>]\n", argv[0]); return(1); } | |
| n = strtoul(argv[1], 0, 10); | |
| if (n < 2) { printf("Pi(%lu) = 0\n", n); return(0); } | |
| if (argc > 2) | |
| method = argv[2]; | |
| else | |
| method = "lehmer"; | |
| gettimeofday(&t0, 0); | |
| SET_PPS_PARALLEL; | |
| if (!strcasecmp(method, "lehmer")) { pi = _XS_lehmer_pi(n); } | |
| else if (!strcasecmp(method, "meissel")) { pi = _XS_meissel_pi(n); } | |
| else if (!strcasecmp(method, "legendre")) { pi = _XS_legendre_pi(n); } | |
| else if (!strcasecmp(method, "lmo")) { pi = _XS_LMO_pi(n); } | |
| else if (!strcasecmp(method, "sieve")) { pi = _XS_prime_count(2, n); } | |
| else { | |
| printf("method must be one of: lehmer, meissel, legendre, lmo, or sieve\n"); | |
| return(2); | |
| } | |
| gettimeofday(&t1, 0); | |
| t = (t1.tv_sec-t0.tv_sec); t *= 1000000.0; t += (t1.tv_usec - t0.tv_usec); | |
| printf("%8s Pi(%lu) = %lu in %10.5fs\n", method, n, pi, t / 1000000.0); | |
| return(0); | |
| } | |
| #endif |
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