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twin_primes.c
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#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "ptypes.h"
#include "constants.h"
#include "cache.h"
#include "sieve.h"
#include "twin_primes.h"
#include "prime_counts.h"
#include "inverse_interpolate.h"
#include "real.h"
#include "mathl.h"
#include "util.h"
/******************************************************************************/
/* TWIN PRIMES */
/******************************************************************************/
/* Twin prime counts (X * 10^Y to (X+1) * 10^Y). */
#if BITS_PER_WORD < 64
static const UV twin_steps[] =
{58980,48427,45485,43861,42348,41457,40908,39984,39640,39222,
373059,353109,341253,332437,326131,320567,315883,312511,309244,
2963535,2822103,2734294,2673728,
};
static const unsigned int twin_num_exponents = 3;
static const unsigned int twin_last_mult = 4; /* 4000M */
#else
static const UV twin_steps[] =
{58980,48427,45485,43861,42348,41457,40908,39984,39640,39222,
373059,353109,341253,332437,326131,320567,315883,312511,309244,
2963535,2822103,2734294,2673728,2626243,2585752,2554015,2527034,2501469,
/* pi2(1e10,2e10) = 24096420; pi2(2e10,3e10) = 23046519; ... */
24096420,23046519,22401089,21946975,21590715,21300632,21060884,20854501,20665634,
199708605,191801047,186932018,183404596,180694619,178477447,176604059,174989299,173597482,
1682185723,1620989842,1583071291,1555660927,1534349481,1517031854,1502382532,1489745250, 1478662752,
14364197903,13879821868,13578563641,13361034187,13191416949,13053013447,12936030624,12835090276, 12746487898,
124078078589,120182602778,117753842540,115995331742,114622738809,113499818125,112551549250,111732637241,111012321565,
1082549061370,1050759497170,1030883829367,1016473645857,1005206830409,995980796683,988183329733,981441437376,975508027029,
9527651328494, 9264843314051, 9100153493509, 8980561036751, 8886953365929, 8810223086411, 8745329823109, 8689179566509, 8639748641098,
84499489470819, 82302056642520, 80922166953330, 79918799449753, 79132610984280, 78487688897426, 77941865286827, 77469296499217, 77053075040105,
754527610498466, 735967887462370, 724291736697048,
};
static const unsigned int twin_num_exponents = 12;
static const unsigned int twin_last_mult = 4; /* 4e18 */
#endif
UV twin_prime_count(UV n)
{
return (n < 3) ? 0 : twin_prime_count_range(0,n);
}
UV twin_prime_count_range(UV beg, UV end)
{
unsigned char* segment;
UV sum = 0;
/* First use the tables of #e# from 1e7 to 4e18. */
if (beg <= 3 && end >= 10000000) {
UV mult, exp, step = 0, base = 10000000;
for (exp = 0; exp < twin_num_exponents && end >= base; exp++) {
for (mult = 1; mult < 10 && end >= mult*base; mult++) {
sum += twin_steps[step++];
beg = mult*base;
if (exp == twin_num_exponents-1 && mult >= twin_last_mult) break;
}
base *= 10;
}
}
if (beg <= 3 && end >= 3) sum++;
if (beg <= 5 && end >= 5) sum++;
if (beg < 11) beg = 7;
if (beg <= end) {
/* Make end points odd */
beg |= 1;
end = (end-1) | 1;
/* Cheesy way of counting the partial-byte edges */
while ((beg % 30) != 1) {
if (is_prime(beg) && is_prime(beg+2) && beg <= end) sum++;
beg += 2;
}
while ((end % 30) != 29) {
if (is_prime(end) && is_prime(end+2) && beg <= end) sum++;
end -= 2; if (beg > end) break;
}
}
if (beg <= end) {
UV seg_base, seg_low, seg_high;
void* ctx = start_segment_primes(beg, end, &segment);
while (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
UV bytes = seg_high/30 - seg_low/30 + 1;
unsigned char s, x;
const unsigned char* sp = segment;
const unsigned char* const spend = segment + bytes - 1;
for (s = x = *sp; sp++ < spend; s = x) {
x = *sp;
if (!(s & 0x0C)) sum++;
if (!(s & 0x30)) sum++;
if (!(s & 0x80) && !(x & 0x01)) sum++;
}
x = is_prime(seg_high+2) ? 0x00 : 0xFF;
if (!(s & 0x0C)) sum++;
if (!(s & 0x30)) sum++;
if (!(s & 0x80) && !(x & 0x01)) sum++;
}
end_segment_primes(ctx);
}
return sum;
}
/* See http://numbers.computation.free.fr/Constants/Primes/twin.pdf, page 5 */
/* Upper limit is in Wu, Acta Arith 114 (2004). 4.48857*x/(log(x)*log(x) */
/* Lichtman (2021) improved the limit: https://arxiv.org/pdf/2109.02851.pdf */
UV twin_prime_count_approx(UV n)
{
/* Best would be another estimate for n < ~ 5000 */
if (n < 2000) return twin_prime_count(n);
{
/* Sebah and Gourdon 2002 */
const long double two_C2 = 1.32032363169373914785562422L;
const long double two_over_log_two = 2.8853900817779268147198494L;
long double ln = (long double) n;
long double logn = logl(ln);
long double li2 = Ei(logn) + two_over_log_two-ln/logn;
/* Try to minimize MSE. */
/* We compromise to prevent discontinuities. */
if (n < 32000000) {
long double fm;
if (n < 4000) fm = 0.2952;
else if (n < 8000) fm = 0.3102;
else if (n < 16000) fm = 0.3090;
else if (n < 32000) fm = 0.3096;
else if (n < 64000) fm = 0.3097;
else if (n < 128000) fm = 0.3094;
else if (n < 256000) fm = 0.3099;
else if (n < 600000) fm = .3098 + (n-256000) * (.3056-.3098) / (600000-256000);
else if (n < 1000000) fm = .3062 + (n-600000) * (.3042-.3062) / (1000000-600000);
else if (n < 4000000) fm = .3067 + (n-1000000) * (.3041-.3067) / (4000000-1000000);
else if (n <16000000) fm = .3041 + (n-4000000) * (.2983-.3041) / (16000000-4000000);
else fm = .2983 + (n-16000000) * (.2961-.2983) / (32000000-16000000);
li2 *= fm * logl(12+logn);
}
return (UV) (two_C2 * li2 + 0.5L);
}
}
UV nth_twin_prime(UV n)
{
unsigned char* segment;
double dend;
UV nth = 0;
UV beg, end;
if (n < 6) {
switch (n) {
case 0: nth = 0; break;
case 1: nth = 3; break;
case 2: nth = 5; break;
case 3: nth =11; break;
case 4: nth =17; break;
case 5:
default: nth =29; break;
}
return nth;
}
end = UV_MAX - 16;
dend = 800.0 + 1.01L * (double)nth_twin_prime_approx(n);
if (dend < (double)end) end = (UV) dend;
beg = 2;
if (n > 58980) { /* Use twin_prime_count tables to accelerate if possible */
UV mult, exp, step = 0, base = 10000000;
for (exp = 0; exp < twin_num_exponents && end >= base; exp++) {
for (mult = 1; mult < 10 && n > twin_steps[step]; mult++) {
n -= twin_steps[step++];
beg = mult*base;
if (exp == twin_num_exponents-1 && mult >= twin_last_mult) break;
}
base *= 10;
}
}
if (beg == 2) { beg = 31; n -= 5; }
{
UV seg_base, seg_low, seg_high;
void* ctx = start_segment_primes(beg, end, &segment);
while (n && next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
UV p, bytes = seg_high/30 - seg_low/30 + 1;
UV s = ((UV)segment[0]) << 8;
for (p = 0; p < bytes; p++) {
s >>= 8;
if (p+1 < bytes) s |= (((UV)segment[p+1]) << 8);
else if (!is_prime(seg_high+2)) s |= 0xFF00;
if (!(s & 0x000C) && !--n) { nth=seg_base+p*30+11; break; }
if (!(s & 0x0030) && !--n) { nth=seg_base+p*30+17; break; }
if (!(s & 0x0180) && !--n) { nth=seg_base+p*30+29; break; }
}
}
end_segment_primes(ctx);
}
return nth;
}
UV nth_twin_prime_approx(UV n)
{
long double fn = (long double) n;
long double flogn = logl(n);
long double fnlog2n = fn * flogn * flogn;
UV lo, hi;
if (n < 6)
return nth_twin_prime(n);
/* Binary search on the TPC estimate.
* Good results require that the TPC estimate is both fast and accurate.
* These bounds are good for the actual nth_twin_prime values.
*/
lo = (UV) (0.9 * fnlog2n);
hi = (UV) ( (n >= 1e16) ? (1.04 * fnlog2n) :
(n >= 1e13) ? (1.10 * fnlog2n) :
(n >= 1e7 ) ? (1.31 * fnlog2n) :
(n >= 1200) ? (1.70 * fnlog2n) :
(2.3 * fnlog2n + 5) );
if (hi <= lo) hi = UV_MAX;
return inverse_interpolate(lo, hi, n, &twin_prime_count_approx, 0);
}
#if 0
/* Generic cluster sieve. Works but not as fast as we'd like. */
#include "sieve_cluster.h"
UV range_twin_prime_sieve(UV** list, UV beg, UV end)
{
const uint32_t cl[2] = {0,2};
UV ntwin;
*list = sieve_cluster(beg, end, 2, cl, &ntwin);
return ntwin;
}
#endif
#if 0
/* Prime sieve and look for twins */
UV range_twin_prime_sieve(UV** list, UV beg, UV end)
{
UV nalloc, *L, ntwin;
if (end > MPU_MAX_TWIN_PRIME) end = MPU_MAX_TWIN_PRIME;
/* overshoot bounds, could also compare to 3*((end+29)/30 - beg/30) */
nalloc = prime_count_upper(end) - prime_count_lower(beg);
New(0, L, nalloc + 1 + 3, UV);
ntwin = 0;
if (beg <= 3 && end >= 3) L[ntwin++] = 3;
if (beg <= 5 && end >= 5) L[ntwin++] = 5;
if (beg < 11) beg = 7;
if (beg <= end) {
unsigned char* segment;
UV seg_base, seg_low, seg_high, lastp = 0;
void* ctx = start_segment_primes(beg, end+2, &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 )
if (lastp+2 == p)
L[ntwin++] = lastp;
lastp = p;
END_DO_FOR_EACH_SIEVE_PRIME
}
end_segment_primes(ctx);
}
*list = L;
return ntwin;
}
#endif
#if 1
/* Also just using the prime sieve and pulling out twins. */
UV range_twin_prime_sieve(UV** list, UV beg, UV end)
{
UV nalloc, *L, ntwin;
if (end > MPU_MAX_TWIN_PRIME) end = MPU_MAX_TWIN_PRIME;
/* overshoot bounds, could also compare to 3*((end+29)/30 - beg/30) */
nalloc = prime_count_upper(end) - prime_count_lower(beg);
New(0, L, nalloc + 1 + 3, UV);
ntwin = 0;
if (beg <= 3 && end >= 3) L[ntwin++] = 3;
if (beg <= 5 && end >= 5) L[ntwin++] = 5;
if (beg < 11) beg = 7;
if (beg <= end) {
/* Make end points odd */
beg |= 1;
end = (end-1) | 1;
while (1) { /* Get us to the start of a sieve byte. */
uint32_t beg30 = beg % 30;
if (beg30 == 1) break;
else if (beg30 <= 11) beg = beg-beg30+11;
else if (beg30 <= 17) beg = beg-beg30+17;
else if (beg30 <= 29) beg = beg-beg30+29;
if (beg <= end && is_prime(beg) && is_prime(beg+2)) L[ntwin++] = beg;
beg = (beg30 <= 11) ? beg+6 : (beg30 <= 17) ? beg+12 : beg+2;
}
}
if (beg <= end) {
unsigned char* segment;
UV seg_base, seg_low, seg_high;
void* ctx = start_segment_primes(beg, end, &segment);
while (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
UV bytes = seg_high/30 - seg_low/30 + 1;
UV pos = seg_base;
unsigned char s, x;
const unsigned char* sp = segment;
const unsigned char* const spend = segment + bytes - 1;
for (s = x = *sp; sp++ < spend; s = x) {
x = *sp;
if (!(s & 0x0C)) L[ntwin++] = pos+11;
if (!(s & 0x30)) L[ntwin++] = pos+17;
if (!(s & 0x80) && !(x & 0x01)) L[ntwin++] = pos+29;
pos += 30;
}
x = is_prime(seg_high+2) ? 0x00 : 0xFF;
if (!(s & 0x0C)) L[ntwin++] = pos+11;
if (!(s & 0x30)) L[ntwin++] = pos+17;
if (!(s & 0x80) && !(x & 0x01)) L[ntwin++] = pos+29;
}
end_segment_primes(ctx);
/* Remove anything from the end because we did full bytes. */
while (ntwin > 0 && L[ntwin-1] > end) ntwin--;
}
*list = L;
return ntwin;
}
#endif