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semi_primes.c
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#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "ptypes.h"
#include "constants.h"
#define FUNC_isqrt 1
#define FUNC_icbrt 1
#define FUNC_ctz 1
#include "util.h"
#include "cache.h"
#include "sieve.h"
#include "prime_counts.h"
#include "inverse_interpolate.h"
#include "semi_primes.h"
#define SP_SIEVE_THRESH 100 /* When to sieve vs. iterate */
/******************************************************************************/
/* SEMI PRIMES */
/******************************************************************************/
#if 0
static const unsigned char _semiprimelist[] =
{0,4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,55,57,58,62,65,69,74,
77,82,85,86,87,91,93,94,95,106,111,115,118,119,121,122,123,129,133,134,141,
142,143,145,146,155,158,159,161,166,169,177,178,183,185,187,194,201,202,
203,205,206,209,213,214,215,217,218,219,221,226,235,237,247,249,253,254};
#else
static const unsigned short _semiprimelist[] =
{0,4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,55,57,58,62,65,69,74,
77,82,85,86,87,91,93,94,95,106,111,115,118,119,121,122,123,129,133,134,141,
142,143,145,146,155,158,159,161,166,169,177,178,183,185,187,194,201,202,
203,205,206,209,213,214,215,217,218,219,221,226,235,237,247,249,253,254,
259,262,265,267,274,278,287,289,291,295,298,299,301,302,303,305,309,314,
319,321,323,326,327,329,334,335,339,341,346,355,358,361,362,365,371,377,
381,382,386,391,393,394,395,398,403,407,411,413,415,417,422,427,437,445,
446,447,451,453,454,458,466,469,471,473,478,481,482,485,489,493,497,501,
502,505,511,514,515,517,519,526,527,529,533,535,537,538,542,543,545,551,
553,554,559,562,565,566,573,579,581,583,586,589,591,597,611,614,622,623};
#endif
#define NSEMIPRIMELIST (sizeof(_semiprimelist)/sizeof(_semiprimelist[0]))
#if 1
static UV _bs_count(UV n, UV const* const primes, UV lastidx)
{
UV i = 0, j = lastidx; /* primes may not start at 0 */
MPUassert(n >= primes[0] && n < primes[lastidx], "prime count via binary search out of range");
while (i < j) {
UV mid = i + (j-i)/2;
if (primes[mid] <= n) i = mid+1;
else j = mid;
}
return i-1;
}
UV semiprime_count(UV n)
{
UV pc = 0, sum = 0, sqrtn = prev_prime(isqrt(n)+1);
UV xbeg = 0, xend = 0, xlim = 0, xoff = 0, xsize, *xarr = 0;
UV const xmax = 200000000UL;
if (n > 1000000) { /* Upfront work to speed up the many small calls */
UV nprecalc = (UV) pow(n, .75);
if (nprecalc > _MPU_LMO_CROSSOVER) nprecalc = _MPU_LMO_CROSSOVER;
prime_precalc(nprecalc);
/* Make small calls even faster using binary search on a list */
xlim = (UV) pow(n, 0.70);
}
if (sqrtn >= 2) sum += prime_count(n/2) - pc++;
if (sqrtn >= 3) sum += prime_count(n/3) - pc++;
if (sqrtn >= 5) sum += prime_count(n/5) - pc++;
if (sqrtn >= 7) {
unsigned char* segment;
UV seg_base, seg_low, seg_high, np, cnt;
void* ctx = start_segment_primes(7, sqrtn, &segment);
while (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
START_DO_FOR_EACH_SIEVE_PRIME( segment, seg_base, seg_low, seg_high )
np = n/p;
if (np < xlim) {
if (xarr == 0 || np < xbeg) {
if (xarr != 0) { Safefree(xarr); xarr = 0; }
xend = np;
xbeg = n/sqrtn;
if (xend - xbeg > xmax) xbeg = xend - xmax;
xbeg = prev_prime(xbeg);
xend = next_prime(xend);
xoff = prime_count(xbeg);
xsize = range_prime_sieve(&xarr, xbeg, xend);
xend = xarr[xsize-1];
}
cnt = xoff + _bs_count(np, xarr, xsize-1);
} else {
cnt = prime_count(np);
}
sum += cnt - pc++;
END_DO_FOR_EACH_SIEVE_PRIME
}
if (xarr != 0) { Safefree(xarr); xarr = 0; }
end_segment_primes(ctx);
}
return sum;
}
#else
/* This is much cleaner, but ends up being a little slower. */
#include "prime_count_cache.h"
#define CACHED_PC(cache,n) prime_count_cache_lookup(cache,n)
UV semiprime_count(UV n)
{
UV sum = 0, sqrtn = prev_prime(isqrt(n)+1), pc_sqrtn;
void *cache = prime_count_cache_create( (UV)pow(n,0.70) );
if (sqrtn >= 2) sum += CACHED_PC(cache,n/2);
if (sqrtn >= 3) sum += CACHED_PC(cache,n/3);
if (sqrtn >= 5) sum += CACHED_PC(cache,n/5);
if (sqrtn >= 7) {
unsigned char* segment;
UV seg_base, seg_low, seg_high;
void* ctx = start_segment_primes(7, sqrtn, &segment);
while (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
START_DO_FOR_EACH_SIEVE_PRIME( segment, seg_base, seg_low, seg_high )
sum += CACHED_PC(cache, n/p);
END_DO_FOR_EACH_SIEVE_PRIME
}
end_segment_primes(ctx);
}
pc_sqrtn = CACHED_PC(cache, sqrtn);
sum -= (pc_sqrtn * pc_sqrtn - pc_sqrtn) / 2;
prime_count_cache_destroy(cache);
return sum;
}
#endif
/* TODO: This overflows, see p=3037000507,lo=10739422018595509581.
* p2 = 9223372079518257049 => 9223372079518257049 + 9223372079518257049
* Also with lo=18446744073709551215,hi=18446744073709551515.
* Using P_GT_LO_0 might help, but the biggest issue is 2*p*p overflows.
*/
#define MARKSEMI(p,arr,lo,hi) \
do { UV i, p2=(p)*(p); \
for (i = P_GT_LO(p2, p, lo); i >= lo && i <= hi; i += p) arr[i-lo]++; \
for (i = P_GT_LO(2*p2, p2, lo); i >= lo && i <= hi; i += p2) arr[i-lo]++; \
} while (0);
UV range_semiprime_sieve(UV** semis, UV lo, UV hi)
{
UV *S, i, count = 0;
if (lo < 4) lo = 4;
if (hi > MPU_MAX_SEMI_PRIME) hi = MPU_MAX_SEMI_PRIME;
if (hi <= _semiprimelist[NSEMIPRIMELIST-1]) {
if (semis == 0) {
for (i = 1; i < NSEMIPRIMELIST && _semiprimelist[i] <= hi; i++)
if (_semiprimelist[i] >= lo)
count++;
} else {
Newz(0, S, NSEMIPRIMELIST+1, UV);
for (i = 1; i < NSEMIPRIMELIST && _semiprimelist[i] <= hi; i++)
if (_semiprimelist[i] >= lo)
S[count++] = _semiprimelist[i];
*semis = S;
}
} else {
unsigned char* nfacs;
UV cutn, sqrtn = isqrt(hi);
Newz(0, nfacs, hi-lo+1, unsigned char);
if (sqrtn*sqrtn < hi && sqrtn < (UVCONST(1)<<(BITS_PER_WORD/2))-1) sqrtn++;
cutn = (sqrtn > 30000) ? 30000 : sqrtn;
START_DO_FOR_EACH_PRIME(2, cutn) {
MARKSEMI(p,nfacs,lo,hi);
} END_DO_FOR_EACH_PRIME
if (cutn < sqrtn) {
unsigned char* segment;
UV seg_base, seg_low, seg_high;
void* ctx = start_segment_primes(cutn, sqrtn, &segment);
while (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
START_DO_FOR_EACH_SIEVE_PRIME( segment, seg_base, seg_low, seg_high )
MARKSEMI(p,nfacs,lo,hi);
END_DO_FOR_EACH_SIEVE_PRIME
}
end_segment_primes(ctx);
}
if (semis == 0) {
for (i = lo; i <= hi; i++)
if (nfacs[i-lo] == 1)
count++;
} else {
UV cn = 50 + 1.01 * (semiprime_count_approx(hi) - semiprime_count_approx(lo));
New(0, S, cn, UV);
for (i = lo; i <= hi; i++) {
if (nfacs[i-lo] == 1) {
if (count >= cn)
Renew(S, cn += 4000, UV);
S[count++] = i;
}
}
*semis = S;
}
Safefree(nfacs);
}
return count;
}
static UV _range_semiprime_count_iterate(UV lo, UV hi)
{
UV sum = 0;
for (; lo < hi; lo++) /* TODO: We should walk composites */
if (is_semiprime(lo))
sum++;
if (is_semiprime(hi))
sum++;
return sum;
}
#if 0
static UV _range_semiprime_selection(UV** semis, UV lo, UV hi)
{
UV *S = 0, *pr, cn = 0, count = 0;
UV i, xsize, lim = hi/2 + 1000, sqrtn = isqrt(hi);
if (lo < 4) lo = 4;
if (hi > MPU_MAX_SEMI_PRIME) hi = MPU_MAX_SEMI_PRIME;
if (semis != 0) {
cn = 50 + 1.01 * (semiprime_count_approx(hi) - semiprime_count_approx(lo));
New(0, S, cn, UV);
}
xsize = range_prime_sieve(&pr, 0, lim);
for (i = 0; pr[i] <= sqrtn; i++) {
UV const pi = pr[i], jlo = (lo+pi-1)/pi, jhi = hi/pi;
UV skip, j = i;
if (pr[j] < jlo)
for (skip = 2048; skip > 0; skip >>= 1)
while (j+skip-1 < xsize && pr[j+skip-1] < jlo)
j += skip;
if (semis == 0) {
while (pr[j++] <= jhi)
count++;
} else {
for (; pr[j] <= jhi; j++) {
if (count >= cn)
Renew(S, cn += 4000, UV);
S[count++] = pi * pr[j];
}
}
}
Safefree(pr);
if (semis != 0) {
sort_uv_array(S, count);
*semis = S;
}
return count;
}
#endif
UV semiprime_count_range(UV lo, UV hi)
{
if (lo > hi || hi < 4)
return 0;
/* tiny sizes fastest with the sieving code */
if (hi <= 400) return range_semiprime_sieve(0, lo, hi);
/* Large sizes best with the prime count method */
if (lo <= 4) return semiprime_count(hi);
/* Now it gets interesting. lo > 4, hi > 400. */
if ((hi-lo+1) < hi / (isqrt(hi)*200)) {
MPUverbose(2, "semiprimes %"UVuf"-%"UVuf" via iteration\n", lo, hi);
return _range_semiprime_count_iterate(lo,hi);
}
/* TODO: Determine when _range_semiprime_selection(0,lo,hi) is better */
if ((hi-lo+1) < hi / (isqrt(hi)/4)) {
MPUverbose(2, "semiprimes %"UVuf"-%"UVuf" via sieving\n", lo, hi);
return range_semiprime_sieve(0, lo, hi);
}
MPUverbose(2, "semiprimes %"UVuf"-%"UVuf" via prime count\n", lo, hi);
return semiprime_count(hi) - semiprime_count(lo-1);
}
UV semiprime_count_approx(UV n) {
UV i;
if (n <= _semiprimelist[NSEMIPRIMELIST-1]) {
for (i = 0; i < NSEMIPRIMELIST-1 && n >= _semiprimelist[i+1]; i++)
;
return i;
} else {
/* Crisan and Erban (2020) https://arxiv.org/abs/2006.16491 */
UV L, res;
double logn = log(n), loglogn = log(logn);
double series = 0, den = 1, mc;
static const double C[19] = {
0.26149721284764278375L,
-2.0710850628855780875L,
-7.6972777412176108802L,
-35.345660320564161516L,
-206.71503925406509339L,
-1511.1997871316530251L,
-13546.323682845914021L,
-146229.10675883565523L,
-1867579.6280076650637L,
-27733045.258413542557L,
-470983423.57703294361L,
/*
* Values for C_11+ are not exact, but that's ok here.
* \p 80
* zetald(n) = { zeta'(n) / zeta(n) }
* zetalim(n) = { derivnum(s = 1-1e-40, zetald(s) + 1/(s-1), n-1) }
* B(n,x=100) = { if(n==0,return(0.2614972128476427837554268386086958590516)); (-1)^n * (sum(i=2, x, moebius(i) * i^(n-1) * derivnum(X=i,zetald(X),n-1)) + zetalim(n)) }
* BN = vector(20,n,B(n-1,500));
* C(n) = { n!*(sum(i=0,n,BN[i+1]/i!) - sum(i=1,n,1/i)) }
*/
-9011500984.0L,
-191744069149.0L,
-4487573459711.0L,
-114472069580580.0L,
-3158610502077136.0L,
-93682567786528912.0L,
-2970838770257639696.0L,
-100274471240063911725.0L }; /* ~ C_18 */
/* We will use C[0] to C[L-1]. Hence L must be 19 or less. */
static const double CROSS[15] =
{ 632, 9385, 136411, 4610076, 66358000, 440590000, 2557200000, 53032001000.0, 1151076796431.0L, 20416501389724.0L,
165815501587300.0L, /* Below this L = 13, Above this L = 14 */
953038830319448.0L, /* Cross from L = 14 to 15 */
20019396133340433.0L, /* Cross from L = 15 to 16 */
192558867109258424.0L, /* Cross from L = 16 to 17 */
1757883874953032448.0L }; /* Cross from L = 17 to 18 */
static const double mincount[16] =
{ 82, 195, 2485, 31446, 906319, 11741185, 72840337, 398702652, 7538564737, 150382042176.0L, 2482510001499.0L, 19204997230933.0L, 106211451717048.0L, 2094735089989940.0L, 19282342825922188.0L, 168996486318315136.0L };
/* Pick truncation point, note L can be one higher than the value below*/
for (L = 3; L <= 17 && (double)n >= CROSS[L-3]; L++) ;
/* Calculate truncated asymptotic value */
for (i = 1; i <= L; i++) {
series += factorial(i-1) * (loglogn / den);
series += C[i-1] / den;
den *= logn;
}
res = (UV) ( (n / logn) * series + 0.5L );
/* Check for overflow */
if (res >= MPU_MAX_SEMI_PRIME_IDX) return MPU_MAX_SEMI_PRIME_IDX;
/* Ensure monotonic using simple clamping */
mc = mincount[L-3];
/* mc = (L == 3) ? 82 : semiprime_count_approx(CROSS[L-4]-1); */
if ((double)res < mc) return mc;
return res;
}
}
UV nth_semiprime_approx(UV n) {
double logn,log2n,log3n,log4n, err_lo, err_md, err_hi, err_factor, est;
UV lo, hi;
if (n < NSEMIPRIMELIST)
return _semiprimelist[n];
if (n > MPU_MAX_SEMI_PRIME_IDX)
return 0;
/* Piecewise with blending. Hacky and maybe overkill. It makes a good
* estimator by itself, but our count approximation is even better, so we
* use this as an excellent initial estimate, then use inverse binary
* search to lower the error another order of magnitude.
*
* Interp Range Crossover to next
* lo 2^8 - 2^28 2^26 - 2^27
* md 2^25 - 2^48 2^46 - 2^47
* hi 2^45 - 2^64
*/
logn = log(n); log2n = log(logn); log3n = log(log2n); log4n=log(log3n);
err_lo = 1.000 - 0.00018216088*logn + 0.18099609886*log2n - 0.51962474356*log3n - 0.01136143381*log4n;
err_md = 0.968 - 0.00073297945*logn + 0.09731690314*log2n - 0.25212500749*log3n - 0.01366795346*log4n;
err_hi = 0.968 - 0.00008034109*logn + 0.01522628393*log2n - 0.04020257367*log3n - 0.01266447175*log4n;
if (n <= (1UL<<26)) {
err_factor = err_lo;
} else if (n < (1UL<<27)) { /* Linear interpolate the two in the blend area */
double x = (n - 67108864.0L) / 67108864.0L;
err_factor = ((1.0L-x) * err_lo) + (x * err_md);
} else if (logn <= 31.88477030575) {
err_factor = err_md;
} else if (logn < 32.57791748632) {
double x = (n - 70368744177664.0L) / 70368744177664.0L;
err_factor = ((1.0L-x) * err_md) + (x * err_hi);
} else {
err_factor = err_hi;
}
est = err_factor * n * logn / log2n;
if (est >= MPU_MAX_SEMI_PRIME) return MPU_MAX_SEMI_PRIME;
/* Use inverse interpolation to improve the result. */
lo = 0.979 * est - 5;
hi = 1.03 * est;
return inverse_interpolate(lo, hi, n, &semiprime_count_approx, 0);
}
static UV _next_semiprime(UV n) {
while (!is_semiprime(++n))
;
return n;
}
static UV _prev_semiprime(UV n) {
while (!is_semiprime(--n))
;
return n;
}
UV nth_semiprime(UV n)
{
UV guess, spcnt, sptol, gn, ming = 0, maxg = UV_MAX;
if (n < NSEMIPRIMELIST)
return _semiprimelist[n];
guess = nth_semiprime_approx(n); /* Initial guess */
sptol = 16*icbrt(n); /* Guess until within this many SPs */
MPUverbose(2, " using exact counts until within %"UVuf"\n",sptol);
/* Make successive interpolations until small enough difference */
for (gn = 2; gn < 20; gn++) {
IV adjust;
while (!is_semiprime(guess)) guess++; /* Guess is a semiprime */
MPUverbose(2, " %"UVuf"-th semiprime is around %"UVuf" ... ", n, guess);
/* Compute exact count at our nth-semiprime guess */
spcnt = semiprime_count(guess);
MPUverbose(2, "(%"IVdf")\n", (IV)(n-spcnt));
/* Stop guessing if within our tolerance */
if (n==spcnt || (n>spcnt && n-spcnt < sptol) || (n<spcnt && spcnt-n < sptol)) break;
/* Determine how far off we think we are */
adjust = (IV) (nth_semiprime_approx(n) - nth_semiprime_approx(spcnt));
/* When computing new guess, ensure we don't overshoot. Rarely used. */
if (spcnt <= n && guess > ming) ming = guess; /* Previous guesses */
if (spcnt >= n && guess < maxg) maxg = guess;
guess += adjust;
if (guess <= ming || guess >= maxg) MPUverbose(2, " fix min/max for %"UVuf"\n",n);
if (guess <= ming) guess = ming + sptol - 1;
if (guess >= maxg) guess = maxg - sptol + 1;
}
/* If we have far enough to go, sieve for semiprimes */
if (n > spcnt && (n-spcnt) > SP_SIEVE_THRESH) { /* sieve forwards */
UV *S, count, i, range;
while (n > spcnt) {
range = nth_semiprime_approx(n) - nth_semiprime_approx(spcnt);
range = 1.10 * range + 100;
if (range > guess) range = guess; /* just in case */
if (range > 125000000) range = 125000000; /* Not too many at a time */
/* Get a bunch of semiprimes */
MPUverbose(2, " sieving forward %"UVuf"\n", range);
count = range_semiprime_sieve(&S, guess+1, guess+range);
if (spcnt+count <= n) {
guess = S[count-1];
spcnt += count;
} else { /* Walk forwards */
for (i = 0; i < count && spcnt < n; i++) {
guess = S[i];
spcnt++;
}
}
Safefree(S);
}
} else if (n < spcnt && (spcnt-n) > SP_SIEVE_THRESH) { /* sieve backwards */
UV *S, count, range;
while (n < spcnt) {
range = nth_semiprime_approx(spcnt) - nth_semiprime_approx(n);
range = 1.10 * range + 100;
if (range > guess) range = guess; /* just in case */
if (range > 125000000) range = 125000000; /* Not too many at a time */
/* Get a bunch of semiprimes */
MPUverbose(2, " sieving backward %"UVuf"\n", range);
count = range_semiprime_sieve(&S, guess-range, guess-1);
if (spcnt-count >= n) {
guess = S[0];
spcnt -= count;
} else { /* Walk backwards */
while (count > 0 && n < spcnt) {
guess = S[--count];
spcnt--;
}
}
Safefree(S);
}
}
/* Finally, iterate over semiprimes until we hit the exact spot */
for (; spcnt > n; spcnt--)
guess = _prev_semiprime(guess);
for (; spcnt < n; spcnt++)
guess = _next_semiprime(guess);
return guess;
}