#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <float.h>
#include <math.h>
/* Use long double to get a little more precision when we're calculating the
* math functions -- especially those calculated with a series. Long double
* is defined in C89 (ISO C), so it should be supported by any reasonable
* compiler we're using (seriously is your C compiler 20+ years out of date?).
* Noting that 'long double' on many platforms is no different than 'double'
* so it may buy us nothing. But it's worth trying.
*/
/* The C99 LD math functions are a clusterfrack. They're defined by C99, but
* NetBSD doesn't have them. You need them in both the headers and libraries,
* but there is no standard way to find out if the libraries have them. The
* best way (I believe) to deal with this is having the make system do test
* compiles. Barring that, we make limited guesses, and just give up
* precision on any system we don't recognize.
*/
#if _MSC_VER
/* MSVS has these as macros, and really doesn't want us defining them. */
#elif defined(__MATH_DECLARE_LDOUBLE) || \
defined(__LONG_DOUBLE_128__) || \
defined(__LONGDOUBLE128)
/* GLIBC */
extern long double powl(long double, long double);
extern long double expl(long double);
extern long double logl(long double);
extern long double fabsl(long double);
extern long double floorl(long double);
extern long double ceill(long double);
#else
#define powl(x, y) (long double) pow( (double) (x), (double) (y) )
#define expl(x) (long double) exp( (double) (x) )
#define logl(x) (long double) log( (double) (x) )
#define fabsl(x) (long double) fabs( (double) (x) )
#define floorl(x) (long double) floor( (double) (x) )
#define ceill(x) (long double) ceil( (double) (x) )
#endif
#ifdef LDBL_INFINITY
#undef INFINITY
#define INFINITY LDBL_INFINITY
#elif !defined(INFINITY)
#define INFINITY (DBL_MAX + DBL_MAX)
#endif
#ifndef LDBL_EPSILON
#define LDBL_EPSILON 1e-16
#endif
#ifndef LDBL_MAX
#define LDBL_MAX DBL_MAX
#endif
#define KAHAN_INIT(s) \
long double s ## _y, s ## _t; \
long double s ## _c = 0.0; \
long double s = 0.0;
#define KAHAN_SUM(s, term) \
do { \
s ## _y = (term) - s ## _c; \
s ## _t = s + s ## _y; \
s ## _c = (s ## _t - s) - s ## _y; \
s = s ## _t; \
} while (0)
#include "ptypes.h"
#define FUNC_isqrt 1
#define FUNC_icbrt 1
#define FUNC_lcm_ui 1
#define FUNC_ctz 1
#define FUNC_log2floor 1
#define FUNC_is_perfect_square
#define FUNC_next_prime_in_sieve 1
#define FUNC_prev_prime_in_sieve 1
#include "util.h"
#include "sieve.h"
#include "primality.h"
#include "cache.h"
#include "lmo.h"
#include "factor.h"
#include "mulmod.h"
#include "constants.h"
static int _verbose = 0;
void _XS_set_verbose(int v) { _verbose = v; }
int _XS_get_verbose(void) { return _verbose; }
static int _call_gmp = 0;
void _XS_set_callgmp(int v) { _call_gmp = v; }
int _XS_get_callgmp(void) { return _call_gmp; }
/* GCC 3.4 - 4.1 has broken 64-bit popcount.
* GCC 4.2+ can generate awful code when it doesn't have asm (GCC bug 36041).
* When the asm is present (e.g. compile with -march=native on a platform that
* has them, like Nahelem+), then it is almost as fast as the direct asm. */
#if BITS_PER_WORD == 64
#if defined(__POPCNT__) && defined(__GNUC__) && (__GNUC__> 4 || (__GNUC__== 4 && __GNUC_MINOR__> 1))
#define popcnt(b) __builtin_popcountll(b)
#else
static UV popcnt(UV b) {
b -= (b >> 1) & 0x5555555555555555;
b = (b & 0x3333333333333333) + ((b >> 2) & 0x3333333333333333);
b = (b + (b >> 4)) & 0x0f0f0f0f0f0f0f0f;
return (b * 0x0101010101010101) >> 56;
}
#endif
#endif
#if defined(__GNUC__)
#define word_unaligned(m,wordsize) ((uintptr_t)m & (wordsize-1))
#else /* uintptr_t is part of C99 */
#define word_unaligned(m,wordsize) ((unsigned int)m & (wordsize-1))
#endif
static const unsigned char byte_zeros[256] =
{8,7,7,6,7,6,6,5,7,6,6,5,6,5,5,4,7,6,6,5,6,5,5,4,6,5,5,4,5,4,4,3,
7,6,6,5,6,5,5,4,6,5,5,4,5,4,4,3,6,5,5,4,5,4,4,3,5,4,4,3,4,3,3,2,
7,6,6,5,6,5,5,4,6,5,5,4,5,4,4,3,6,5,5,4,5,4,4,3,5,4,4,3,4,3,3,2,
6,5,5,4,5,4,4,3,5,4,4,3,4,3,3,2,5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1,
7,6,6,5,6,5,5,4,6,5,5,4,5,4,4,3,6,5,5,4,5,4,4,3,5,4,4,3,4,3,3,2,
6,5,5,4,5,4,4,3,5,4,4,3,4,3,3,2,5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1,
6,5,5,4,5,4,4,3,5,4,4,3,4,3,3,2,5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1,
5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1,4,3,3,2,3,2,2,1,3,2,2,1,2,1,1,0};
static UV count_zero_bits(const unsigned char* m, UV nbytes)
{
UV count = 0;
#if BITS_PER_WORD == 64
if (nbytes >= 16) {
while ( word_unaligned(m,sizeof(UV)) && nbytes--)
count += byte_zeros[*m++];
if (nbytes >= 8) {
UV* wordptr = (UV*)m;
UV nwords = nbytes / 8;
UV nzeros = nwords * 64;
m += nwords * 8;
nbytes %= 8;
while (nwords--)
nzeros -= popcnt(*wordptr++);
count += nzeros;
}
}
#endif
while (nbytes--)
count += byte_zeros[*m++];
return count;
}
/* We'll use this little static sieve to quickly answer small values of
* is_prime, next_prime, prev_prime, prime_count
* for non-threaded Perl it's basically the same as getting the primary
* cache. It guarantees we'll have an answer with no waiting on any version.
*/
static const unsigned char prime_sieve30[] =
{0x01,0x20,0x10,0x81,0x49,0x24,0xc2,0x06,0x2a,0xb0,0xe1,0x0c,0x15,0x59,0x12,
0x61,0x19,0xf3,0x2c,0x2c,0xc4,0x22,0xa6,0x5a,0x95,0x98,0x6d,0x42,0x87,0xe1,
0x59,0xa9,0xa9,0x1c,0x52,0xd2,0x21,0xd5,0xb3,0xaa,0x26,0x5c,0x0f,0x60,0xfc,
0xab,0x5e,0x07,0xd1,0x02,0xbb,0x16,0x99,0x09,0xec,0xc5,0x47,0xb3,0xd4,0xc5,
0xba,0xee,0x40,0xab,0x73,0x3e,0x85,0x4c,0x37,0x43,0x73,0xb0,0xde,0xa7,0x8e,
0x8e,0x64,0x3e,0xe8,0x10,0xab,0x69,0xe5,0xf7,0x1a,0x7c,0x73,0xb9,0x8d,0x04,
0x51,0x9a,0x6d,0x70,0xa7,0x78,0x2d,0x6d,0x27,0x7e,0x9a,0xd9,0x1c,0x5f,0xee,
0xc7,0x38,0xd9,0xc3,0x7e,0x14,0x66,0x72,0xae,0x77,0xc1,0xdb,0x0c,0xcc,0xb2,
0xa5,0x74,0xe3,0x58,0xd5,0x4b,0xa7,0xb3,0xb1,0xd9,0x09,0xe6,0x7d,0x23,0x7c,
0x3c,0xd3,0x0e,0xc7,0xfd,0x4a,0x32,0x32,0xfd,0x4d,0xb5,0x6b,0xf3,0xa8,0xb3,
0x85,0xcf,0xbc,0xf4,0x0e,0x34,0xbb,0x93,0xdb,0x07,0xe6,0xfe,0x6a,0x57,0xa3,
0x8c,0x15,0x72,0xdb,0x69,0xd4,0xaf,0x59,0xdd,0xe1,0x3b,0x2e,0xb7,0xf9,0x2b,
0xc5,0xd0,0x8b,0x63,0xf8,0x95,0xfa,0x77,0x40,0x97,0xea,0xd1,0x9f,0xaa,0x1c,
0x48,0xae,0x67,0xf7,0xeb,0x79,0xa5,0x55,0xba,0xb2,0xb6,0x8f,0xd8,0x2d,0x6c,
0x2a,0x35,0x54,0xfd,0x7c,0x9e,0xfa,0xdb,0x31,0x78,0xdd,0x3d,0x56,0x52,0xe7,
0x73,0xb2,0x87,0x2e,0x76,0xe9,0x4f,0xa8,0x38,0x9d,0x5d,0x3f,0xcb,0xdb,0xad,
0x51,0xa5,0xbf,0xcd,0x72,0xde,0xf7,0xbc,0xcb,0x49,0x2d,0x49,0x26,0xe6,0x1e,
0x9f,0x98,0xe5,0xc6,0x9f,0x2f,0xbb,0x85,0x6b,0x65,0xf6,0x77,0x7c,0x57,0x8b,
0xaa,0xef,0xd8,0x5e,0xa2,0x97,0xe1,0xdc,0x37,0xcd,0x1f,0xe6,0xfc,0xbb,0x8c,
0xb7,0x4e,0xc7,0x3c,0x19,0xd5,0xa8,0x9e,0x67,0x4a,0xe3,0xf5,0x97,0x3a,0x7e,
0x70,0x53,0xfd,0xd6,0xe5,0xb8,0x1c,0x6b,0xee,0xb1,0x9b,0xd1,0xeb,0x34,0xc2,
0x23,0xeb,0x3a,0xf9,0xef,0x16,0xd6,0x4e,0x7d,0x16,0xcf,0xb8,0x1c,0xcb,0xe6,
0x3c,0xda,0xf5,0xcf};
#define NPRIME_SIEVE30 (sizeof(prime_sieve30)/sizeof(prime_sieve30[0]))
/* Return of 2 if n is prime, 0 if not. Do it fast. */
int _XS_is_prime(UV n)
{
if (n <= 10)
return (n == 2 || n == 3 || n == 5 || n == 7) ? 2 : 0;
if (n < UVCONST(200000000)) {
UV d = n/30;
UV m = n - d*30;
unsigned char mtab = masktab30[m]; /* Bitmask in mod30 wheel */
const unsigned char* sieve;
int isprime;
/* Return 0 if a multiple of 2, 3, or 5 */
if (mtab == 0)
return 0;
/* Check static tiny sieve */
if (d < NPRIME_SIEVE30)
return (prime_sieve30[d] & mtab) ? 0 : 2;
if (!(n%7) || !(n%11) || !(n%13)) return 0;
/* Check primary cache */
isprime = -1;
if (n <= get_prime_cache(0, &sieve))
isprime = 2*((sieve[d] & mtab) == 0);
release_prime_cache(sieve);
if (isprime >= 0)
return isprime;
}
return is_prob_prime(n);
}
UV next_prime(UV n)
{
UV m, sieve_size, next;
const unsigned char* sieve;
if (n < 30*NPRIME_SIEVE30) {
next = next_prime_in_sieve(prime_sieve30, n,30*NPRIME_SIEVE30);
if (next != 0) return next;
}
if (n >= MPU_MAX_PRIME) return 0; /* Overflow */
sieve_size = get_prime_cache(0, &sieve);
next = (n < sieve_size) ? next_prime_in_sieve(sieve, n, sieve_size) : 0;
release_prime_cache(sieve);
if (next != 0) return next;
m = n % 30;
do { /* Move forward one. */
n += wheeladvance30[m];
m = nextwheel30[m];
} while (!is_prob_prime(n));
return n;
}
UV prev_prime(UV n)
{
const unsigned char* sieve;
UV m, prev;
if (n < 30*NPRIME_SIEVE30)
return prev_prime_in_sieve(prime_sieve30, n);
if (n < get_prime_cache(0, &sieve)) {
prev = prev_prime_in_sieve(sieve, n);
release_prime_cache(sieve);
return prev;
}
release_prime_cache(sieve);
m = n % 30;
do { /* Move back one. */
n -= wheelretreat[m];
m = prevwheel30[m];
} while (!is_prob_prime(n));
return n;
}
/* Given a sieve of size nbytes, walk it counting zeros (primes) until:
*
* (1) we counted them all: return the count, which will be less than maxcount.
*
* (2) we hit maxcount: set position to the index of the maxcount'th prime
* and return count (which will be equal to maxcount).
*/
static UV count_segment_maxcount(const unsigned char* sieve, UV base, UV nbytes, UV maxcount, UV* pos)
{
UV count = 0;
UV byte = 0;
const unsigned char* sieveptr = sieve;
const unsigned char* maxsieve = sieve + nbytes;
MPUassert(sieve != 0, "count_segment_maxcount incorrect args");
MPUassert(pos != 0, "count_segment_maxcount incorrect args");
*pos = 0;
if ( (nbytes == 0) || (maxcount == 0) )
return 0;
/* Do fixed-length word counts to start, with possible overcounting */
while ((count+64) < maxcount && sieveptr < maxsieve) {
UV top = base + 3*maxcount;
UV div = (top < 8000) ? 8 : /* 8 cannot overcount */
(top < 1000000) ? 4 :
(top < 10000000) ? 3 : 2;
UV minbytes = (maxcount-count)/div;
if (minbytes > (UV)(maxsieve-sieveptr)) minbytes = maxsieve-sieveptr;
count += count_zero_bits(sieveptr, minbytes);
sieveptr += minbytes;
}
/* Count until we reach the end or >= maxcount */
while ( (sieveptr < maxsieve) && (count < maxcount) )
count += byte_zeros[*sieveptr++];
/* If we went too far, back up. */
while (count >= maxcount)
count -= byte_zeros[*--sieveptr];
/* We counted this many bytes */
byte = sieveptr - sieve;
MPUassert(count < maxcount, "count_segment_maxcount wrong count");
if (byte == nbytes)
return count;
/* The result is somewhere in the next byte */
START_DO_FOR_EACH_SIEVE_PRIME(sieve, byte*30+1, nbytes*30-1)
if (++count == maxcount) { *pos = p; return count; }
END_DO_FOR_EACH_SIEVE_PRIME;
MPUassert(0, "count_segment_maxcount failure");
return 0;
}
/* Given a sieve of size nbytes, counting zeros (primes) but excluding the
* areas outside lowp and highp.
*/
static UV count_segment_ranged(const unsigned char* sieve, UV nbytes, UV lowp, UV highp)
{
UV count, hi_d, lo_d, lo_m;
MPUassert( sieve != 0, "count_segment_ranged incorrect args");
if (nbytes == 0) return 0;
count = 0;
hi_d = highp/30;
if (hi_d >= nbytes) {
hi_d = nbytes-1;
highp = hi_d*30+29;
}
if (highp < lowp)
return 0;
#if 0
/* Dead simple way */
START_DO_FOR_EACH_SIEVE_PRIME(sieve, lowp, highp)
count++;
END_DO_FOR_EACH_SIEVE_PRIME;
return count;
#endif
lo_d = lowp/30;
lo_m = lowp - lo_d*30;
/* Count first fragment */
if (lo_m > 1) {
UV upper = (highp <= (lo_d*30+29)) ? highp : (lo_d*30+29);
START_DO_FOR_EACH_SIEVE_PRIME(sieve, lowp, upper)
count++;
END_DO_FOR_EACH_SIEVE_PRIME;
lowp = upper+2;
lo_d = lowp/30;
}
if (highp < lowp)
return count;
/* Count bytes in the middle */
{
UV hi_m = highp - hi_d*30;
UV count_bytes = hi_d - lo_d + (hi_m == 29);
if (count_bytes > 0) {
count += count_zero_bits(sieve+lo_d, count_bytes);
lowp += 30*count_bytes;
}
}
if (highp < lowp)
return count;
/* Count last fragment */
START_DO_FOR_EACH_SIEVE_PRIME(sieve, lowp, highp)
count++;
END_DO_FOR_EACH_SIEVE_PRIME;
return count;
}
/*
* The pi(x) prime count functions. prime_count(x) gives an exact number,
* but requires determining all the primes up to x, so will be much slower.
*
* prime_count_lower(x) and prime_count_upper(x) give lower and upper limits,
* which will bound the exact value. These bounds should be fairly tight.
*
* pi_upper(x) - pi(x) pi_lower(x) - pi(x)
* < 10 for x < 5_371 < 10 for x < 9_437
* < 50 for x < 295_816 < 50 for x < 136_993
* < 100 for x < 1_761_655 < 100 for x < 909_911
* < 200 for x < 9_987_821 < 200 for x < 8_787_901
* < 400 for x < 34_762_891 < 400 for x < 30_332_723
* < 1000 for x < 372_748_528 < 1000 for x < 233_000_533
* < 5000 for x < 1_882_595_905 < 5000 for x < over 4300M
*
* The average of the upper and lower bounds is within 9 for all x < 15809, and
* within 50 for all x < 1_763_367.
*
* It is common to use the following Chebyshev inequality for x >= 17:
* 1*x/logx <-> 1.25506*x/logx
* but this gives terribly loose bounds.
*
* Rosser and Schoenfeld's bound for x >= 67 of
* x/(logx-1/2) <-> x/(logx-3/2)
* is much tighter. These bounds can be tightened even more.
*
* The formulas of Dusart for higher x are better yet. I recommend the paper
* by Burde for further information. Dusart's thesis is also a good resource.
*
* I have tweaked the bounds formulas for small (under 70_000M) numbers so they
* are tighter. These bounds are verified via trial. The Dusart bounds
* (1.8 and 2.51) are used for larger numbers since those are proven.
*
*/
#define USE_PC_TABLES 1
#if USE_PC_TABLES
/* These tables let us have fast answers up to 3000M for the cost of ~1.4k of
* static data/code. We can get a 4 to 100x speedup here. We don't want to
* push this idea too far because Lehmer's method should be faster. */
/* mpu '$step=30_000; $pc=prime_count(5); print "$pc\n", join(",", map { $spc=$pc; $pc=prime_count($_*$step); $pc-$spc; } 1..200), "\n"' */
static const unsigned short step_counts_30k[] = /* starts at 7 */
{3242,2812,2656,2588,2547,2494,2465,2414,2421,2355,2407,2353,2310,2323,2316,
2299,2286,2281,2247,2279,2243,2223,2251,2214,2209,2230,2215,2207,2205,2179,
2200,2144,2159,2193,2164,2136,2180,2152,2162,2174,2113,2131,2150,2101,2111,
2146,2115,2123,2119,2108,2124,2097,2075,2089,2094,2119,2084,2065,2069,2101,
2094,2083,2089,2076,2088,2027,2109,2073,2061,2033,2079,2078,2036,2025,2058,
2083,2037,2005,2048,2048,2024,2045,2027,2025,2039,2049,2022,2034,2046,2032,
2019,2000,2014,2069,2042,1980,2021,2014,1995,2017,1992,1985,2045,2007,1990,
2008,2052,2033,1988,1984,2010,1943,2024,2005,2027,1937,1955,1956,1993,1976,
2048,1940,2002,2007,1994,1954,1972,2002,1973,1993,1984,1969,1940,1960,2026,
1966,1981,1912,1994,1971,1977,1952,1932,1977,1932,1954,1938,2018,1987,1967,
1937,1938,1963,1973,1947,1947,1963,1959,1941,1923,1943,1957,1974,1964,1958,
1984,1933,1935,1935,1949,1928,1943,1917,1956,1970,1932,1937,1929,1932,1947,
1927,1944,1915,1913,1918,1925,1931,1919,1900,1952,1934,1922,1891,1926,1925,
1903,1970,1962,1905,1905};
#define NSTEP_COUNTS_30K (sizeof(step_counts_30k)/sizeof(step_counts_30k[0]))
/* mpu '$step=300_000; $pc=prime_count(20*$step); print "$pc\n", join(",", map { $spc=$pc; $pc=prime_count($_*$step); $pc-$spc; } 21..212), "\n"' */
static const unsigned short step_counts_300k[] = /* starts at 6M */
{19224,19086,19124,19036,18942,18893,18870,18853,18837,18775,18688,18674,
18594,18525,18639,18545,18553,18424,18508,18421,18375,18366,18391,18209,
18239,18298,18209,18294,18125,18138,18147,18115,18126,18021,18085,18068,
18094,17963,18041,18003,17900,17881,17917,17888,17880,17852,17892,17779,
17823,17764,17806,17762,17780,17716,17633,17758,17746,17678,17687,17613,
17709,17628,17634,17556,17528,17598,17604,17532,17606,17548,17493,17576,
17456,17468,17555,17452,17407,17472,17415,17500,17508,17418,17463,17240,
17345,17351,17380,17394,17379,17330,17322,17335,17354,17113,17210,17231,
17238,17305,17268,17219,17281,17235,17119,17292,17161,17212,17166,17277,
17137,17260,17228,17197,17154,17097,17195,17136,17067,17058,17041,17045,
17187,17034,17029,17037,17090,16985,17054,17017,17106,17001,17095,17125,
17027,16948,16969,17031,16916,17031,16905,16937,16881,16952,16919,16938,
17028,16963,16902,16922,16944,16901,16847,16969,16900,16876,16841,16874,
16894,16861,16761,16886,16778,16820,16727,16921,16817,16845,16847,16824,
16844,16809,16859,16783,16713,16752,16762,16857,16760,16626,16784,16784,
16718,16745,16871,16635,16714,16630,16779,16709,16660,16730,16715,16724};
#define NSTEP_COUNTS_300K (sizeof(step_counts_300k)/sizeof(step_counts_300k[0]))
static const unsigned int step_counts_30m[] = /* starts at 60M */
{1654839,1624694,1602748,1585989,1571241,1559918,1549840,1540941,1533150,
1525813,1519922,1513269,1508559,1503386,1497828,1494129,1489905,1486417,
1482526,1478941,1475577,1472301,1469133,1466295,1464711,1461223,1458478,
1455327,1454218,1451883,1449393,1447612,1445029,1443285,1442268,1438511,
1437688,1435603,1433623,1432638,1431158,1429158,1427934,1426191,1424449,
1423146,1421898,1421628,1419519,1417646,1416274,1414828,1414474,1412536,
1412147,1410149,1409474,1408847,1406619,1405863,1404699,1403820,1402802,
1402215,1401459,1399972,1398687,1397968,1397392,1396025,1395311,1394081,
1393614,1393702,1391745,1390950,1389856,1389245,1388381,1387557,1387087,
1386285,1386089,1385355,1383659,1383030,1382174,1382128,1380556,1379940,
1379988,1379181,1378300,1378033,1376974,1376282,1375646,1374445,1373813};
#define NSTEP_COUNTS_30M (sizeof(step_counts_30m)/sizeof(step_counts_30m[0]))
#endif
UV _XS_prime_count(UV low, UV high)
{
const unsigned char* cache_sieve;
unsigned char* segment;
UV segment_size, low_d, high_d;
UV count = 0;
if ((low <= 2) && (high >= 2)) count++;
if ((low <= 3) && (high >= 3)) count++;
if ((low <= 5) && (high >= 5)) count++;
if (low < 7) low = 7;
if (low > high) return count;
if (low == 7 && high <= 30*NPRIME_SIEVE30) {
count += count_segment_ranged(prime_sieve30, NPRIME_SIEVE30, low, high);
return count;
}
#if USE_PC_TABLES
if (low == 7 && high >= 30000) {
UV i, maxi;
if (high < (30000*(NSTEP_COUNTS_30K+1))) {
low = 0;
maxi = high/30000;
for (i = 0; i < maxi && i < NSTEP_COUNTS_30K; i++) {
count += step_counts_30k[i];
low += 30000;
}
} else if (high < (6000000 + 300000*(NSTEP_COUNTS_300K+1))) {
count = 412849;
low = 6000000;
maxi = (high-6000000)/300000;
for (i = 0; i < maxi && i < NSTEP_COUNTS_300K; i++) {
count += step_counts_300k[i];
low += 300000;
}
} else {
count = 3562115;
low = 60000000;
maxi = (high-60000000)/30000000;
for (i = 0; i < maxi && i < NSTEP_COUNTS_30M; i++) {
count += step_counts_30m[i];
low += 30000000;
}
}
}
#endif
low_d = low/30;
high_d = high/30;
/* Count full bytes only -- no fragments from primary cache */
segment_size = get_prime_cache(0, &cache_sieve) / 30;
if (segment_size < high_d) {
/* Expand sieve to sqrt(n) */
UV endp = (high_d >= (UV_MAX/30)) ? UV_MAX-2 : 30*high_d+29;
release_prime_cache(cache_sieve);
segment_size = get_prime_cache( isqrt(endp) + 1 , &cache_sieve) / 30;
}
if ( (segment_size > 0) && (low_d <= segment_size) ) {
/* Count all the primes in the primary cache in our range */
count += count_segment_ranged(cache_sieve, segment_size, low, high);
if (high_d < segment_size) {
release_prime_cache(cache_sieve);
return count;
}
low_d = segment_size;
if (30*low_d > low) low = 30*low_d;
}
release_prime_cache(cache_sieve);
/* More primes needed. Repeatedly segment sieve. */
{
void* ctx = start_segment_primes(low, high, &segment);
UV seg_base, seg_low, seg_high;
while (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
segment_size = seg_high/30 - seg_low/30 + 1;
count += count_segment_ranged(segment, segment_size, seg_low-seg_base, seg_high-seg_base);
}
end_segment_primes(ctx);
}
return count;
}
UV prime_count_approx(UV n)
{
if (n < 3000000) return _XS_prime_count(2, n);
return (UV) (_XS_RiemannR( (long double) n ) + 0.5 );
}
/* See http://numbers.computation.free.fr/Constants/Primes/twin.pdf, page 5 */
UV twin_prime_count_approx(UV n)
{
/* Best would be another estimate for n < ~ 5000 */
if (n < 2000) return twin_prime_count(3,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 = _XS_ExponentialIntegral(logn) + two_over_log_two-ln/logn;
/* try to minimize MSE */
if (n < 4000) li2 *= 1.2035 * logl(logl(logl(ln)));
else if (n < 8000) li2 *= 0.9439 * logl(logl(logl(ln*1000)));
else if (n < 32000) li2 *= 0.8967 * logl(logl(logl(ln*4000)));
else if (n < 200000) li2 *= 0.8937 * logl(logl(logl(ln*4000)));
else if (n < 1000000) li2 *= 0.8640 * logl(logl(logl(ln*16000)));
else if (n < 4000000) li2 *= 0.8627 * logl(logl(logl(ln*16000)));
else if (n <10000000) li2 *= 0.8536 * logl(logl(logl(ln*16000)));
return (UV) (two_C2 * li2 + 0.5L);
}
}
UV prime_count_lower(UV n)
{
long double fn, flogn, lower, a;
if (n < 33000) return _XS_prime_count(2, n);
fn = (long double) n;
flogn = logl(n);
if (n < 176000) a = 1.80;
else if (n < 315000) a = 2.10;
else if (n < 1100000) a = 2.20;
else if (n < 4500000) a = 2.31;
else if (n <233000000) a = 2.36;
#if BITS_PER_WORD == 32
else a = 2.32;
#else
else if (n < UVCONST( 5433800000)) a = 2.32;
else if (n < UVCONST(60000000000)) a = 2.15;
else a = 2.00;
#endif
lower = fn/flogn * (1.0 + 1.0/flogn + a/(flogn*flogn));
return (UV) floorl(lower);
}
typedef struct {
UV thresh;
float aval;
} thresh_t;
static const thresh_t _upper_thresh[] = {
{ 59000, 2.48 },
{ 350000, 2.52 },
{ 355991, 2.54 },
{ 356000, 2.51 },
{ 3550000, 2.50 },
{ 3560000, 2.49 },
{ 5000000, 2.48 },
{ 8000000, 2.47 },
{ 13000000, 2.46 },
{ 18000000, 2.45 },
{ 31000000, 2.44 },
{ 41000000, 2.43 },
{ 48000000, 2.42 },
{ 119000000, 2.41 },
{ 182000000, 2.40 },
{ 192000000, 2.395 },
{ 213000000, 2.390 },
{ 271000000, 2.385 },
{ 322000000, 2.380 },
{ 400000000, 2.375 },
{ 510000000, 2.370 },
{ 682000000, 2.367 },
{ UVCONST(2953652287), 2.362 }
};
#define NUPPER_THRESH (sizeof(_upper_thresh)/sizeof(_upper_thresh[0]))
UV prime_count_upper(UV n)
{
int i;
long double fn, flogn, upper, a;
if (n < 33000) return _XS_prime_count(2, n);
for (i = 0; i < (int)NUPPER_THRESH; i++)
if (n < _upper_thresh[i].thresh)
break;
if (i < (int)NUPPER_THRESH) a = _upper_thresh[i].aval;
else a = 2.334; /* Dusart 2010, page 2 */
fn = (long double) n;
flogn = logl(n);
upper = fn/flogn * (1.0 + 1.0/flogn + a/(flogn*flogn));
return (UV) ceill(upper);
}
static const unsigned short primes_small[] =
{0,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,
193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,
293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,
409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499};
#define NPRIMES_SMALL (sizeof(primes_small)/sizeof(primes_small[0]))
/* The nth prime will be less or equal to this number */
UV nth_prime_upper(UV n)
{
long double fn, flogn, flog2n, upper;
if (n < NPRIMES_SMALL)
return primes_small[n];
fn = (long double) n;
flogn = logl(n);
flog2n = logl(flogn); /* Note distinction between log_2(n) and log^2(n) */
if (n >= 688383) /* Dusart 2010 page 2 */
upper = fn * (flogn + flog2n - 1.0 + ((flog2n-2.00)/flogn));
else if (n >= 178974) /* Dusart 2010 page 7 */
upper = fn * (flogn + flog2n - 1.0 + ((flog2n-1.95)/flogn));
else if (n >= 39017) /* Dusart 1999 page 14 */
upper = fn * (flogn + flog2n - 0.9484);
else if (n >= 6) /* Modified from Robin 1983 for 50-39016 _only_ */
upper = fn * ( flogn + 0.5982 * flog2n ) - 5;
else
upper = fn * ( flogn + flog2n );
/* For all three analytical functions, it is possible that for a given valid
* input, we will not be able to return an output that fits in the UV type.
* For example, if they ask for the 203280222nd prime, we should return
* 4294967311. But in 32-bit, that overflows. What we do is calculate our
* double precision value. If that would overflow, then we look at the input
* and if it is <= the index of the last representable prime, then we return
* the last representable prime. Otherwise, we croak an overflow message.
* This should maintain the invariant:
* nth_prime_lower(n) <= nth_prime(n) <= nth_prime_upper(n)
*/
/* Watch out for overflow */
if (upper >= (long double)UV_MAX) {
if (n <= MPU_MAX_PRIME_IDX) return MPU_MAX_PRIME;
croak("nth_prime_upper(%"UVuf") overflow", n);
}
return (UV) ceill(upper);
}
/* The nth prime will be greater than or equal to this number */
UV nth_prime_lower(UV n)
{
long double fn, flogn, flog2n, lower;
if (n < NPRIMES_SMALL)
return primes_small[n];
fn = (long double) n;
flogn = logl(n);
flog2n = logl(flogn); /* Note distinction between log_2(n) and log^2(n) */
/* Dusart 2010 page 2, for all n >= 3 */
lower = fn * (flogn + flog2n - 1.0 + ((flog2n-2.10)/flogn));
/* Tighten small values */
if (n < 2679) lower = 1.003 * lower + 23;
else if (n < 14353) lower = 1.001 * lower + 21;
return (UV) floorl(lower);
}
UV nth_prime_approx(UV n)
{
long double fn, flogn;
UV lo, hi;
if (n < NPRIMES_SMALL)
return primes_small[n];
/* Binary search for inverse Riemann R */
fn = (long double) n;
flogn = logl(n);
lo = (UV) (fn * flogn);
hi = (UV) (fn * flogn * 2 + 2);
if (hi <= lo) hi = UV_MAX;
while (lo < hi) {
UV mid = lo + (hi-lo)/2;
if (_XS_RiemannR(mid) < fn) lo = mid+1;
else hi = mid;
}
return lo;
}
UV nth_prime(UV n)
{
const unsigned char* cache_sieve;
unsigned char* segment;
UV upper_limit, segbase, segment_size;
UV p = 0;
UV target = n-3;
UV count = 0;
/* If very small, return the table entry */
if (n < NPRIMES_SMALL)
return primes_small[n];
/* Determine a bound on the nth prime. We know it comes before this. */
upper_limit = nth_prime_upper(n);
MPUassert(upper_limit > 0, "nth_prime got an upper limit of 0");
/* For relatively small values, generate a sieve and count the results.
*
* For larger values, compute an approximate low estimate, use our fast
* prime count, then segment sieve forwards or backwards for the rest.
*/
if (upper_limit <= get_prime_cache(0, 0) || upper_limit <= 32*1024*30) {
/* Generate a sieve and count. */
segment_size = get_prime_cache(upper_limit, &cache_sieve) / 30;
/* Count up everything in the cached sieve. */
if (segment_size > 0)
count += count_segment_maxcount(cache_sieve, 0, segment_size, target, &p);
release_prime_cache(cache_sieve);
} else {
/* A binary search on RiemannR is nice, but ends up either often being
* being higher (requiring going backwards) or biased and then far too
* low. Using the inverse Li is easier and more consistent. */
UV lower_limit = _XS_Inverse_Li(n);
/* For even better performance, add in half the usual correction, which
* will get us even closer, so even less sieving required. However, it
* is now possible to get a result higher than the value, so we'll need
* to handle that case. It still ends up being a better deal than R,
* given that we don't have a fast backward sieve. */
lower_limit += _XS_Inverse_Li(isqrt(n))/4;
segment_size = lower_limit / 30;
lower_limit = 30 * segment_size - 1;
count = _XS_LMO_pi(lower_limit);
/* printf("We've estimated %lu too %s.\n", (count>n)?count-n:n-count, (count>n)?"FAR":"little"); */
/* printf("Our limit %lu %s a prime\n", lower_limit, _XS_is_prime(lower_limit) ? "is" : "is not"); */
if (count >= n) { /* Too far. Walk backwards */
if (_XS_is_prime(lower_limit)) count--;
for (p = 0; p <= (count-n); p++)
lower_limit = prev_prime(lower_limit);
return lower_limit;
}
count -= 3;
/* Make sure the segment siever won't have to keep resieving. */
prime_precalc(isqrt(upper_limit));
}
if (count == target)
return p;
/* Start segment sieving. Get memory to sieve into. */
segbase = segment_size;
segment = get_prime_segment(&segment_size);
while (count < target) {
/* Limit the segment size if we know the answer comes earlier */
if ( (30*(segbase+segment_size)+29) > upper_limit )
segment_size = (upper_limit - segbase*30 + 30) / 30;
/* Do the actual sieving in the range */
sieve_segment(segment, segbase, segbase + segment_size-1);
/* Count up everything in this segment */
count += count_segment_maxcount(segment, 30*segbase, segment_size, target-count, &p);
if (count < target)
segbase += segment_size;
}
release_prime_segment(segment);
MPUassert(count == target, "nth_prime got incorrect count");
return ( (segbase*30) + p );
}
#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,
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; /* 4e19 */
#endif
UV twin_prime_count(UV beg, UV end)
{
unsigned char* segment;
UV sum = 0;
/* First use the tables of #e# from 1e7 to 2e16. */
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 (_XS_is_prime(beg) && _XS_is_prime(beg+2) && beg <= end) sum++;
beg += 2;
}
while ((end % 30) != 29) {
if (_XS_is_prime(end) && _XS_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-seg_low+29)/30;
unsigned char s;
const unsigned char* sp = segment;
const unsigned char* const spend = segment + bytes - 1;
while (sp < spend) {
s = *sp++;
if (!(s & 0x0C)) sum++;
if (!(s & 0x30)) sum++;
if (!(s & 0x80) && !(*sp & 0x01)) sum++;
}
s = *sp;
if (!(s & 0x0C)) sum++;
if (!(s & 0x30)) sum++;
if (!(s & 0x80) && _XS_is_prime(seg_high+2)) sum++;
}
end_segment_primes(ctx);
}
return sum;
}
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 (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
UV p, bytes = (seg_high-seg_low+29)/30;
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 (!_XS_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; }
}
if (n == 0) 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) (1.2 * fnlog2n);
hi = (UV) ( (n >= 1200) ? (1.7 * fnlog2n) : (2.3 * fnlog2n + 5) );
if (hi <= lo) hi = UV_MAX;
while (lo < hi) {
UV mid = lo + (hi-lo)/2;
if (twin_prime_count_approx(mid) < fn) lo = mid+1;
else hi = mid;
}
return lo;
}
/* Return a char array with lo-hi+1 elements. mu[k-lo] = µ(k) for k = lo .. hi.
* It is the callers responsibility to call Safefree on the result. */
#define PGTLO(p,lo) ((p) >= lo) ? (p) : ((p)*(lo/(p)) + ((lo%(p))?(p):0))
#define P2GTLO(pinit, p, lo) \
((pinit) >= lo) ? (pinit) : ((p)*(lo/(p)) + ((lo%(p))?(p):0))
signed char* _moebius_range(UV lo, UV hi)
{
signed char* mu;
UV i;
UV sqrtn = isqrt(hi);
/* Kuznetsov indicates that the Deléglise & Rivat (1996) method can be
* modified to work on logs, which allows us to operate with no
* intermediate memory at all. Same time as the D&R method, less memory. */
unsigned char logp;
UV nextlog;
Newz(0, mu, hi-lo+1, signed char);
if (sqrtn*sqrtn != hi) sqrtn++; /* ceil sqrtn */
logp = 1; nextlog = 3; /* 2+1 */
START_DO_FOR_EACH_PRIME(2, sqrtn) {
UV p2 = p*p;
if (p > nextlog) {
logp += 2; /* logp is 1 | ceil(log(p)/log(2)) */
nextlog = ((nextlog-1)*4)+1;
}
for (i = PGTLO(p, lo); i <= hi; i += p)
mu[i-lo] += logp;
for (i = PGTLO(p2, lo); i <= hi; i += p2)
mu[i-lo] |= 0x80;
} END_DO_FOR_EACH_PRIME
logp = log2floor(lo);
nextlog = 2UL << logp;
for (i = lo; i <= hi; i++) {
unsigned char a = mu[i-lo];
if (i >= nextlog) { logp++; nextlog *= 2; } /* logp is log(p)/log(2) */
if (a & 0x80) { a = 0; }
else if (a >= logp) { a = 1 - 2*(a&1); }
else { a = -1 + 2*(a&1); }
mu[i-lo] = a;
}
if (lo == 0) mu[0] = 0;
return mu;
}
UV* _totient_range(UV lo, UV hi) {
UV* totients;
UV i, seg_base, seg_low, seg_high;
unsigned char* segment;
void* ctx;
if (hi < lo) croak("_totient_range error hi %"UVuf" < lo %"UVuf"\n", hi, lo);
New(0, totients, hi-lo+1, UV);
/* Do via factoring if very small or if we have a small range */
if (hi < 100 || (hi-lo) < 10 || hi/(hi-lo+1) > 1000) {
for (i = lo; i <= hi; i++)
totients[i-lo] = totient(i);
return totients;
}
/* If doing a full sieve, do it monolithic. Faster. */
if (lo == 0) {
UV* prime;
double loghi = log(hi);
UV max_index = (hi < 67) ? 18
: (hi < 355991) ? 15+(hi/(loghi-1.09))
: (hi/loghi) * (1.0+1.0/loghi+2.51/(loghi*loghi));
UV j, index, nprimes = 0;
New(0, prime, max_index, UV); /* could use prime_count_upper(hi) */
memset(totients, 0, (hi-lo+1) * sizeof(UV));
for (i = 2; i <= hi/2; i++) {
index = 2*i;
if ( !(i&1) ) {
if (i == 2) { totients[2] = 1; prime[nprimes++] = 2; }
totients[index] = totients[i]*2;
} else {
if (totients[i] == 0) {
totients[i] = i-1;
prime[nprimes++] = i;
}
for (j=0; j < nprimes && index <= hi; index = i*prime[++j]) {
if (i % prime[j] == 0) {
totients[index] = totients[i]*prime[j];
break;
} else {
totients[index] = totients[i]*(prime[j]-1);
}
}
}
}
Safefree(prime);
/* All totient values have been filled in except the primes. Mark them. */
for (i = ((hi/2) + 1) | 1; i <= hi; i += 2)
if (totients[i] == 0)
totients[i] = i-1;
totients[1] = 1;
totients[0] = 0;
return totients;
}
for (i = lo; i <= hi; i++) {
UV v = i;
if (i % 2 == 0) v -= v/2;
if (i % 3 == 0) v -= v/3;
if (i % 5 == 0) v -= v/5;
totients[i-lo] = v;
}
ctx = start_segment_primes(7, hi/2, &segment);
while (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
START_DO_FOR_EACH_SIEVE_PRIME( segment, seg_low - seg_base, seg_high - seg_base ) {
p += seg_base;
for (i = P2GTLO(2*p,p,lo); i <= hi; i += p)
totients[i-lo] -= totients[i-lo]/p;
} END_DO_FOR_EACH_SIEVE_PRIME
}
end_segment_primes(ctx);
/* Fill in all primes */
for (i = lo | 1; i <= hi; i += 2)
if (totients[i-lo] == i)
totients[i-lo]--;
if (lo <= 1) totients[1-lo] = 1;
return totients;
}
IV mertens(UV n) {
/* See Deléglise and Rivat (1996) for O(n^2/3 log(log(n))^1/3) algorithm.
* This implementation uses their lemma 2.1 directly, so is ~ O(n).
* In serial it is quite a bit faster than segmented summation of mu
* ranges, though the latter seems to be a favored method for GPUs.
*/
UV u, i, m, nmk, maxmu;
signed char* mu;
IV* M;
IV sum;
if (n <= 1) return n;
u = isqrt(n);
maxmu = (n/(u+1)); /* maxmu lets us handle u < sqrt(n) */
if (maxmu < u) maxmu = u;
mu = _moebius_range(0, maxmu);
New(0, M, maxmu+1, IV);
M[0] = 0;
for (i = 1; i <= maxmu; i++)
M[i] = M[i-1] + mu[i];
sum = M[u];
for (m = 1; m <= u; m++) {
if (mu[m] != 0) {
IV inner_sum = 0;
UV lower = (u/m) + 1;
UV last_nmk = n/(m*lower);
UV this_k = 0;
UV next_k = n/(m*1);
UV nmkm = m * 2;
for (nmk = 1; nmk <= last_nmk; nmk++, nmkm += m) {
this_k = next_k;
next_k = n/nmkm;
inner_sum += M[nmk] * (this_k - next_k);
}
sum -= mu[m] * inner_sum;
}
}
Safefree(M);
Safefree(mu);
return sum;
}
/* There are at least 4 ways to do this, plus hybrids.
* 1) use a table. Great for 32-bit, too big for 64-bit.
* 2) Use pow() to check. Relatively slow and FP is always dangerous.
* 3) factor or trial factor. Slow for 64-bit.
* 4) Dietzfelbinger algorithm 2.3.5. Quite slow.
* This currently uses a hybrid of 1 and 2.
*/
int powerof(UV n) {
int ib;
const int iblast = (n > UVCONST(4294967295)) ? 6 : 4;
if ((n <= 3) || (n == UV_MAX)) return 1;
if ((n & (n-1)) == 0) return ctz(n); /* powers of 2 */
if (is_perfect_square(n)) return 2 * powerof(isqrt(n));
{ UV cb = icbrt(n); if (cb*cb*cb==n) return 3 * powerof(cb); }
for (ib = 3; ib <= iblast; ib++) { /* prime exponents from 5 to 7-or-13 */
UV k, pk, root, b = primes_small[ib];
root = (UV) ( pow(n, 1.0 / b ) + 0.01 );
pk = root * root * root * root * root;
for (k = 5; k < b; k++)
pk *= root;
if (n == pk) return b * powerof(root);
}
if (n > 177146) {
switch (n) { /* Check for powers of 11, 13, 17, 19 within 32 bits */
case 177147: case 48828125: case 362797056: case 1977326743: return 11;
case 1594323: case 1220703125: return 13;
case 129140163: return 17;
case 1162261467: return 19;
default: break;
}
#if BITS_PER_WORD == 64
if (n > UVCONST(4294967295)) {
switch (n) {
case UVCONST(762939453125):
case UVCONST(16926659444736):
case UVCONST(232630513987207):
case UVCONST(100000000000000000):
case UVCONST(505447028499293771):
case UVCONST(2218611106740436992):
case UVCONST(8650415919381337933): return 17;
case UVCONST(19073486328125):
case UVCONST(609359740010496):
case UVCONST(11398895185373143):
case UVCONST(10000000000000000000): return 19;
case UVCONST(94143178827):
case UVCONST(11920928955078125):
case UVCONST(789730223053602816): return 23;
case UVCONST(68630377364883): return 29;
case UVCONST(617673396283947): return 31;
case UVCONST(450283905890997363): return 37;
default: break;
}
}
#endif
}
return 1;
}
int is_power(UV n, UV a)
{
int ret;
if (a > 0) {
if (a == 1 || n <= 1) return 1;
if ((a % 2) == 0)
return !is_perfect_square(n) ? 0 : (a == 2) ? 1 : is_power(isqrt(n),a>>1);
if ((a % 3) == 0)
{ UV cb = icbrt(n);
return (cb*cb*cb != n) ? 0 : (a == 3) ? 1 : is_power(cb, a/3); }
if ((a % 5) == 0)
{ UV r5 = (UV)(pow(n,0.2) + 0.0001);
return (r5*r5*r5*r5*r5 != n) ? 0 : (a == 5) ? 1 : is_power(r5, a/5); }
}
ret = powerof(n);
if (a != 0) return !(ret % a); /* Is the max power divisible by a? */
return (ret == 1) ? 0 : ret;
}
UV valuation(UV n, UV k)
{
UV v = 0;
UV kpower = k;
if (k < 2 || n < 2) return 0;
if (k == 2) return ctz(n);
while ( !(n % kpower) ) {
kpower *= k;
v++;
}
return v;
}
/* How many times does 2 divide n? */
#define padic2(n) ctz(n)
#define IS_MOD8_3OR5(x) (((x)&7)==3 || ((x)&7)==5)
static int kronecker_uu_sign(UV a, UV b, int s) {
while (a) {
int r = padic2(a);
if (r) {
if ((r&1) && IS_MOD8_3OR5(b)) s = -s;
a >>= r;
}
if (a & b & 2) s = -s;
{ UV t = b % a; b = a; a = t; }
}
return (b == 1) ? s : 0;
}
int kronecker_uu(UV a, UV b) {
int r, s;
if (b & 1) return kronecker_uu_sign(a, b, 1);
if (!(a&1)) return 0;
s = 1;
r = padic2(b);
if (r) {
if ((r&1) && IS_MOD8_3OR5(a)) s = -s;
b >>= r;
}
return kronecker_uu_sign(a, b, s);
}
int kronecker_su(IV a, UV b) {
int r, s;
if (a >= 0) return kronecker_uu(a, b);
if (b == 0) return (a == 1 || a == -1) ? 1 : 0;
s = 1;
r = padic2(b);
if (r) {
if (!(a&1)) return 0;
if ((r&1) && IS_MOD8_3OR5(a)) s = -s;
b >>= r;
}
a %= (IV) b;
if (a < 0) a += b;
return kronecker_uu_sign(a, b, s);
}
int kronecker_ss(IV a, IV b) {
if (a >= 0 && b >= 0)
return (b & 1) ? kronecker_uu_sign(a, b, 1) : kronecker_uu(a,b);
if (b >= 0)
return kronecker_su(a, b);
return kronecker_su(a, -b) * ((a < 0) ? -1 : 1);
}
/* Thanks to MJD and RosettaCode */
UV binomial(UV n, UV k) {
UV d, g, r = 1;
if (k == 0) return 1;
if (k == 1) return n;
if (k >= n) return (k == n);
if (k > n/2) k = n-k;
for (d = 1; d <= k; d++) {
if (r >= UV_MAX/n) { /* Possible overflow */
UV nr, dr; /* reduced numerator / denominator */
g = gcd_ui(n, d); nr = n/g; dr = d/g;
g = gcd_ui(r, dr); r = r/g; dr = dr/g;
if (r >= UV_MAX/nr) return 0; /* Unavoidable overflow */
r *= nr;
r /= dr;
n--;
} else {
r *= n--;
r /= d;
}
}
return r;
}
UV totient(UV n) {
UV i, nfacs, totient, lastf, facs[MPU_MAX_FACTORS+1];
if (n <= 1) return n;
totient = 1;
/* phi(2m) = 2phi(m) if m even, phi(m) if m odd */
while ((n & 0x3) == 0) { n >>= 1; totient <<= 1; }
if ((n & 0x1) == 0) { n >>= 1; }
/* factor and calculate totient */
nfacs = factor(n, facs);
lastf = 0;
for (i = 0; i < nfacs; i++) {
UV f = facs[i];
if (f == lastf) { totient *= f; }
else { totient *= f-1; lastf = f; }
}
return totient;
}
static const UV jordan_overflow[5] =
#if BITS_PER_WORD == 64
{UVCONST(4294967311), 2642249, 65537, 7133, 1627};
#else
{UVCONST( 65537), 1627, 257, 85, 41};
#endif
UV jordan_totient(UV k, UV n) {
UV factors[MPU_MAX_FACTORS+1];
int nfac, i;
UV j, totient;
if (k == 0 || n <= 1) return (n == 1);
if (k > 6 || (k > 1 && n >= jordan_overflow[k-2])) return 0;
totient = 1;
/* Similar to Euler totient, shortcut even inputs */
while ((n & 0x3) == 0) { n >>= 1; totient *= (1<<k); }
if ((n & 0x1) == 0) { n >>= 1; totient *= ((1<<k)-1); }
nfac = factor(n,factors);
for (i = 0; i < nfac; i++) {
UV p = factors[i];
UV pk = p;
for (j = 1; j < k; j++) pk *= p;
totient *= (pk-1);
while (i+1 < nfac && p == factors[i+1]) {
i++;
totient *= pk;
}
}
return totient;
}
UV carmichael_lambda(UV n) {
UV fac[MPU_MAX_FACTORS+1];
int i, nfactors;
UV lambda = 1;
if (n < 8) return totient(n);
if ((n & (n-1)) == 0) return n >> 2;
i = ctz(n);
if (i > 0) {
n >>= i;
lambda <<= (i>2) ? i-2 : i-1;
}
nfactors = factor(n, fac);
for (i = 0; i < nfactors; i++) {
UV p = fac[i], pk = p-1;
while (i+1 < nfactors && p == fac[i+1]) {
i++;
pk *= p;
}
lambda = lcm_ui(lambda, pk);
}
return lambda;
}
int moebius(UV n) {
UV factors[MPU_MAX_FACTORS+1];
UV i, nfactors;
if (n <= 1) return (int)n;
if ( n >= 49 && (!(n% 4) || !(n% 9) || !(n%25) || !(n%49)) )
return 0;
nfactors = factor(n, factors);
for (i = 1; i < nfactors; i++)
if (factors[i] == factors[i-1])
return 0;
return (nfactors % 2) ? -1 : 1;
}
UV exp_mangoldt(UV n) {
if (n <= 1) return 1;
else if ((n & (n-1)) == 0) return 2; /* Power of 2 */
else if ((n & 1) == 0) return 1; /* Even number (not 2) */
else if (is_prob_prime(n)) return n;
else {
int k = powerof(n);
if (k >= 2) {
n = (k==2) ? isqrt(n) : (UV)(pow(n,1.0/k)+0.0000001);
if (is_prob_prime(n)) return n;
}
return 1;
}
}
UV znorder(UV a, UV n) {
UV fac[MPU_MAX_FACTORS+1];
UV exp[MPU_MAX_FACTORS+1];
int i, nfactors;
UV j, phi, k = 1;
if (n <= 1) return n; /* znorder(x,0) = 0, znorder(x,1) = 1 */
if (a <= 1) return a; /* znorder(0,x) = 0, znorder(1,x) = 1 (x > 1) */
if (gcd_ui(a,n) > 1) return 0;
/* Abhijit Das, algorithm 1.7, applied to Carmichael Lambda */
phi = carmichael_lambda(n);
nfactors = factor_exp(phi, fac, exp);
for (i = 0; i < nfactors; i++) {
UV b, ek, pi = fac[i], ei = exp[i];
UV phidiv = phi / pi;
for (j = 1; j < ei; j++)
phidiv /= pi;
b = powmod(a, phidiv, n);
for (ek = 0; b != 1; b = powmod(b, pi, n)) {
if (ek++ >= ei) return 0;
k *= pi;
}
}
return k;
}
UV znprimroot(UV n) {
UV fac[MPU_MAX_FACTORS+1];
UV exp[MPU_MAX_FACTORS+1];
UV a, phi;
int i, nfactors;
if (n <= 4) return (n == 0) ? 0 : n-1;
if (n % 4 == 0) return 0;
if (is_prob_prime(n)) {
phi = n-1;
} else { /* prim root exists if n is 2, 4, p^a, or 2(p^a) for odd prime p */
if (n & 0x3) nfactors = factor_exp( (n&1) ? n : n>>1, fac, exp);
else nfactors = 0;
if (nfactors != 1) return 0;
phi = fac[0]-1; /* n = p^a for odd prime p. Calculate totient. */
for (i = 1; i < (int)exp[0]; i++)
phi *= fac[0];
}
nfactors = factor_exp(phi, fac, exp);
for (i = 0; i < nfactors; i++)
exp[i] = phi / fac[i]; /* exp[i] = phi(n) / i-th-factor-of-phi(n) */
for (a = 2; a < n; a++) {
if (kronecker_uu(a, n) == 0) continue;
for (i = 0; i < nfactors; i++)
if (powmod(a, exp[i], n) == 1)
break;
if (i == nfactors) return a;
}
return 0;
}
IV gcdext(IV a, IV b, IV* u, IV* v, IV* cs, IV* ct) {
IV s = 0; IV os = 1;
IV t = 1; IV ot = 0;
IV r = b; IV or = a;
while (r != 0) {
IV quot = or / r;
{ IV tmp = r; r = or - quot * r; or = tmp; }
{ IV tmp = s; s = os - quot * s; os = tmp; }
{ IV tmp = t; t = ot - quot * t; ot = tmp; }
}
if (or < 0) /* correct sign */
{ or = -or; os = -os; ot = -ot; }
if (u != 0) *u = os;
if (v != 0) *v = ot;
if (cs != 0) *cs = s;
if (ct != 0) *ct = t;
return or;
}
/* Calculate 1/a mod n. */
UV modinverse(UV a, UV n) {
IV t = 0; UV nt = 1;
UV r = n; UV nr = a;
while (nr != 0) {
UV quot = r / nr;
{ UV tmp = nt; nt = t - quot*nt; t = tmp; }
{ UV tmp = nr; nr = r - quot*nr; r = tmp; }
}
if (r > 1) return 0; /* No inverse */
if (t < 0) t += n;
return t;
}
UV divmod(UV a, UV b, UV n) { /* a / b mod n */
UV binv = modinverse(b, n);
if (binv == 0) return 0;
return mulmod(a, binv, n);
}
/* status: 1 ok, -1 no inverse, 0 overflow */
UV chinese(UV* a, UV* n, UV num, int* status) {
UV p, gcd, i, j, lcm, sum;
*status = 1;
if (num == 0) return 0;
/* Sort modulii, largest first */
for (i = 1; i < num; i++)
for (j = i; j > 0 && n[j-1] < n[j]; j--)
{ p=n[j-1]; n[j-1]=n[j]; n[j]=p; p=a[j-1]; a[j-1]=a[j]; a[j]=p; }
if (n[0] > IV_MAX) { *status = 0; return 0; }
lcm = n[0]; sum = a[0] % n[0];
for (i = 1; i < num; i++) {
IV u, v, t, s;
UV vs, ut;
gcd = gcdext(lcm, n[i], &u, &v, &s, &t);
if (gcd != 1 && ((sum % gcd) != (a[i] % gcd))) { *status = -1; return 0; }
if (s < 0) s = -s;
if (t < 0) t = -t;
if (s > (IV)(IV_MAX/lcm)) { *status = 0; return 0; }
lcm *= s;
if (u < 0) u += lcm;
if (v < 0) v += lcm;
vs = mulmod((UV)v, (UV)s, lcm);
ut = mulmod((UV)u, (UV)t, lcm);
sum = addmod( mulmod(vs, sum, lcm), mulmod(ut, a[i], lcm), lcm );
}
return sum;
}
long double chebyshev_function(UV n, int which)
{
long double logp, logn = logl(n);
UV sqrtn = which ? isqrt(n) : 0; /* for theta, p <= sqrtn always false */
KAHAN_INIT(sum);
if (n < primes_small[NPRIMES_SMALL-1]) {
UV p, pi;
for (pi = 1; (p = primes_small[pi]) <= n; pi++) {
logp = logl(p);
if (p <= sqrtn) logp *= floorl(logn/logp+1e-15);
KAHAN_SUM(sum, logp);
}
} else {
UV seg_base, seg_low, seg_high;
unsigned char* segment;
void* ctx;
long double logl2 = logl(2);
long double logl3 = logl(3);
long double logl5 = logl(5);
if (!which) {
KAHAN_SUM(sum,logl2); KAHAN_SUM(sum,logl3); KAHAN_SUM(sum,logl5);
} else {
KAHAN_SUM(sum, logl2 * floorl(logn/logl2 + 1e-15));
KAHAN_SUM(sum, logl3 * floorl(logn/logl3 + 1e-15));
KAHAN_SUM(sum, logl5 * floorl(logn/logl5 + 1e-15));
}
ctx = start_segment_primes(7, n, &segment);
while (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
START_DO_FOR_EACH_SIEVE_PRIME( segment, seg_low - seg_base, seg_high - seg_base ) {
p += seg_base;
logp = logl(p);
if (p <= sqrtn) logp *= floorl(logn/logp+1e-15);
KAHAN_SUM(sum, logp);
} END_DO_FOR_EACH_SIEVE_PRIME
}
end_segment_primes(ctx);
}
return sum;
}
/*
* See:
* "Multiple-Precision Exponential Integral and Related Functions"
* by David M. Smith
* "On the Evaluation of the Complex-Valued Exponential Integral"
* by Vincent Pegoraro and Philipp Slusallek
* "Numerical Recipes" 3rd edition
* by William H. Press et al.
* "Rational Chevyshev Approximations for the Exponential Integral E_1(x)"
* by W. J. Cody and Henry C. Thacher, Jr.
*
* Any mistakes here are completely my fault. This code has not been
* verified for anything serious. For better results, see:
* http://www.trnicely.net/pi/pix_0000.htm
* which although the author claims are demonstration programs, will
* undoubtedly produce more reliable results than this code does (I don't
* know of any obvious issues with this code, but it just hasn't been used
* by many people).
*/
static long double const euler_mascheroni = 0.57721566490153286060651209008240243104215933593992L;
static long double const li2 = 1.045163780117492784844588889194613136522615578151L;
long double _XS_ExponentialIntegral(long double x) {
long double val, term;
unsigned int n;
KAHAN_INIT(sum);
if (x == 0) croak("Invalid input to ExponentialIntegral: x must be != 0");
/* Protect against messed up rounding modes */
if (x >= 12000) return INFINITY;
if (x <= -12000) return 0;
if (x < -1) {
/* Continued fraction, good for x < -1 */
long double lc = 0;
long double ld = 1.0L / (1.0L - (long double)x);
val = ld * (-expl(x));
for (n = 1; n <= 100000; n++) {
long double old, t, n2;
t = (long double)(2*n + 1) - (long double) x;
n2 = n * n;
lc = 1.0L / (t - n2 * lc);
ld = 1.0L / (t - n2 * ld);
old = val;
val *= ld/lc;
if ( fabsl(val-old) <= LDBL_EPSILON*fabsl(val) )
break;
}
} else if (x < 0) {
/* Rational Chebyshev approximation (Cody, Thacher), good for -1 < x < 0 */
static const long double C6p[7] = { -148151.02102575750838086L,
150260.59476436982420737L,
89904.972007457256553251L,
15924.175980637303639884L,
2150.0672908092918123209L,
116.69552669734461083368L,
5.0196785185439843791020L };
static const long double C6q[7] = { 256664.93484897117319268L,
184340.70063353677359298L,
52440.529172056355429883L,
8125.8035174768735759866L,
750.43163907103936624165L,
40.205465640027706061433L,
1.0000000000000000000000L };
long double sumn = C6p[0]-x*(C6p[1]-x*(C6p[2]-x*(C6p[3]-x*(C6p[4]-x*(C6p[5]-x*C6p[6])))));
long double sumd = C6q[0]-x*(C6q[1]-x*(C6q[2]-x*(C6q[3]-x*(C6q[4]-x*(C6q[5]-x*C6q[6])))));
val = logl(-x) - sumn/sumd;
} else if (x < -logl(LDBL_EPSILON)) {
/* Convergent series */
long double fact_n = x;
for (n = 2; n <= 200; n++) {
long double invn = 1.0L / n;
fact_n *= (long double)x * invn;
term = fact_n * invn;
KAHAN_SUM(sum, term);
/* printf("C after adding %.20Lf, val = %.20Lf\n", term, sum); */
if ( term < LDBL_EPSILON*sum) break;
}
KAHAN_SUM(sum, euler_mascheroni);
KAHAN_SUM(sum, logl(x));
KAHAN_SUM(sum, x);
val = sum;
} else {
/* Asymptotic divergent series */
long double invx = 1.0L / x;
term = 1.0;
for (n = 1; n <= 200; n++) {
long double last_term = term;
term = term * ( (long double)n * invx );
if (term < LDBL_EPSILON*sum) break;
if (term < last_term) {
KAHAN_SUM(sum, term);
/* printf("A after adding %.20llf, sum = %.20llf\n", term, sum); */
} else {
KAHAN_SUM(sum, (-last_term/3) );
/* printf("A after adding %.20llf, sum = %.20llf\n", -last_term/3, sum); */
break;
}
}
term = expl(x) * invx;
val = term * sum + term;
}
return val;
}
long double _XS_LogarithmicIntegral(long double x) {
if (x == 0) return 0;
if (x == 1) return -INFINITY;
if (x == 2) return li2;
if (x < 0) croak("Invalid input to LogarithmicIntegral: x must be >= 0");
if (x >= LDBL_MAX) return INFINITY;
return _XS_ExponentialIntegral(logl(x));
}
/* Thanks to Kim Walisch for this idea */
UV _XS_Inverse_Li(UV x) {
double nlogn = (double)x * log((double)x);
UV lo = (UV) (nlogn);
UV hi = (UV) (nlogn * 2 + 2);
if (x == 0) return 0;
if (hi <= lo) hi = UV_MAX;
while (lo < hi) {
UV mid = lo + (hi-lo)/2;
if (_XS_LogarithmicIntegral(mid) < x) lo = mid+1;
else hi = mid;
}
return lo;
}
/*
* Storing the first 10-20 Zeta values makes sense. Past that it is purely
* to avoid making the call to generate them ourselves. We could cache the
* calculated values. These all have 1 subtracted from them. */
static const long double riemann_zeta_table[] = {
0.6449340668482264364724151666460251892L, /* zeta(2) */
0.2020569031595942853997381615114499908L,
0.0823232337111381915160036965411679028L,
0.0369277551433699263313654864570341681L,
0.0173430619844491397145179297909205279L,
0.0083492773819228268397975498497967596L,
0.0040773561979443393786852385086524653L,
0.0020083928260822144178527692324120605L,
0.0009945751278180853371459589003190170L,
0.0004941886041194645587022825264699365L,
0.0002460865533080482986379980477396710L,
0.0001227133475784891467518365263573957L,
0.0000612481350587048292585451051353337L,
0.0000305882363070204935517285106450626L,
0.0000152822594086518717325714876367220L,
0.0000076371976378997622736002935630292L, /* zeta(17) Past here all we're */
0.0000038172932649998398564616446219397L, /* zeta(18) getting is speed. */
0.0000019082127165539389256569577951013L,
0.0000009539620338727961131520386834493L,
0.0000004769329867878064631167196043730L,
0.0000002384505027277329900036481867530L,
0.0000001192199259653110730677887188823L,
0.0000000596081890512594796124402079358L,
0.0000000298035035146522801860637050694L,
0.0000000149015548283650412346585066307L,
0.0000000074507117898354294919810041706L,
0.0000000037253340247884570548192040184L,
0.0000000018626597235130490064039099454L,
0.0000000009313274324196681828717647350L,
0.0000000004656629065033784072989233251L,
0.0000000002328311833676505492001455976L,
0.0000000001164155017270051977592973835L,
0.0000000000582077208790270088924368599L,
0.0000000000291038504449709968692942523L,
0.0000000000145519218910419842359296322L,
0.0000000000072759598350574810145208690L,
0.0000000000036379795473786511902372363L,
0.0000000000018189896503070659475848321L,
0.0000000000009094947840263889282533118L,
0.0000000000004547473783042154026799112L,
0.0000000000002273736845824652515226821L,
0.0000000000001136868407680227849349105L,
0.0000000000000568434198762758560927718L,
0.0000000000000284217097688930185545507L,
0.0000000000000142108548280316067698343L,
0.00000000000000710542739521085271287735L,
0.00000000000000355271369133711367329847L,
0.00000000000000177635684357912032747335L,
0.000000000000000888178421093081590309609L,
0.000000000000000444089210314381336419777L,
0.000000000000000222044605079804198399932L,
0.000000000000000111022302514106613372055L,
0.0000000000000000555111512484548124372374L,
0.0000000000000000277555756213612417258163L,
0.0000000000000000138777878097252327628391L,
};
#define NPRECALC_ZETA (sizeof(riemann_zeta_table)/sizeof(riemann_zeta_table[0]))
/* Riemann Zeta on the real line, with 1 subtracted.
* Compare to Math::Cephes zetac. Also zeta with q=1 and subtracting 1.
*
* The Cephes zeta function uses a series (2k)!/B_2k which converges rapidly
* and has a very wide range of values. We use it here for some values.
*
* Note: Calculations here are done on long doubles and we try to generate as
* much accuracy as possible. They will get returned to Perl as an NV,
* which is typically a 64-bit double with 15 digits.
*
* For values 0.5 to 5, this code uses the rational Chebyshev approximation
* from Cody and Thacher. This method is extraordinarily fast and very
* accurate over its range (slightly better than Cephes for most values). If
* we had quad floats, we could use the 9-term polynomial.
*/
long double ld_riemann_zeta(long double x) {
int i;
if (x < 0) croak("Invalid input to RiemannZeta: x must be >= 0");
if (x == 1) return INFINITY;
if (x == (unsigned int)x) {
int k = x - 2;
if ((k >= 0) && (k < (int)NPRECALC_ZETA))
return riemann_zeta_table[k];
}
/* Cody / Thacher rational Chebyshev approximation for small values */
if (x >= 0.5 && x <= 5.0) {
static const long double C8p[9] = { 1.287168121482446392809e10L,
1.375396932037025111825e10L,
5.106655918364406103683e09L,
8.561471002433314862469e08L,
7.483618124380232984824e07L,
4.860106585461882511535e06L,
2.739574990221406087728e05L,
4.631710843183427123061e03L,
5.787581004096660659109e01L };
static const long double C8q[9] = { 2.574336242964846244667e10L,
5.938165648679590160003e09L,
9.006330373261233439089e08L,
8.042536634283289888587e07L,
5.609711759541920062814e06L,
2.247431202899137523543e05L,
7.574578909341537560115e03L,
-2.373835781373772623086e01L,
1.000000000000000000000L };
long double sumn = C8p[0]+x*(C8p[1]+x*(C8p[2]+x*(C8p[3]+x*(C8p[4]+x*(C8p[5]+x*(C8p[6]+x*(C8p[7]+x*C8p[8])))))));
long double sumd = C8q[0]+x*(C8q[1]+x*(C8q[2]+x*(C8q[3]+x*(C8q[4]+x*(C8q[5]+x*(C8q[6]+x*(C8q[7]+x*C8q[8])))))));
long double sum = (sumn - (x-1)*sumd) / ((x-1)*sumd);
return sum;
}
if (x > 17000.0)
return 0.0;
#if 0
{
KAHAN_INIT(sum);
/* Simple defining series, works well. */
for (i = 5; i <= 1000000; i++) {
long double term = powl(i, -x);
KAHAN_SUM(sum, term);
if (term < LDBL_EPSILON*sum) break;
}
KAHAN_SUM(sum, powl(4, -x) );
KAHAN_SUM(sum, powl(3, -x) );
KAHAN_SUM(sum, powl(2, -x) );
return sum;
}
#endif
/* The 2n!/B_2k series used by the Cephes library. */
{
/* gp/pari: factorial(2n)/bernfrac(2n) */
static const long double A[] = {
12.0L,
-720.0L,
30240.0L,
-1209600.0L,
47900160.0L,
-1892437580.3183791606367583212735166426L,
74724249600.0L,
-2950130727918.1642244954382084600497650L,
116467828143500.67248729113000661089202L,
-4597978722407472.6105457273596737891657L,
181521054019435467.73425331153534235290L,
-7166165256175667011.3346447367083352776L,
282908877253042996618.18640556532523927L,
};
long double a, b, s, t;
const long double w = 10.0;
s = 0.0;
b = 0.0;
for (i = 2; i < 11; i++) {
b = powl( i, -x );
s += b;
if (fabsl(b) < fabsl(LDBL_EPSILON * s))
return s;
}
s = s + b*w/(x-1.0) - 0.5 * b;
a = 1.0;
for (i = 0; i < 13; i++) {
long double k = 2*i;
a *= x + k;
b /= w;
t = a*b/A[i];
s = s + t;
if (fabsl(t) < fabsl(LDBL_EPSILON * s))
break;
a *= x + k + 1.0;
b /= w;
}
return s;
}
}
long double _XS_RiemannR(long double x) {
long double part_term, term, flogx, ki;
unsigned int k;
KAHAN_INIT(sum);
if (x <= 0) croak("Invalid input to ReimannR: x must be > 0");
if (x > 1e19) {
const signed char* amob = _moebius_range(0, 100);
KAHAN_SUM(sum, _XS_ExponentialIntegral(logl(x)));
for (k = 2; k <= 100; k++) {
if (amob[k] == 0) continue;
ki = 1.0L / (long double) k;
part_term = powl(x,ki);
if (part_term > LDBL_MAX) return INFINITY;
term = amob[k] * ki * _XS_ExponentialIntegral(logl(part_term));
KAHAN_SUM(sum, term);
if (fabsl(term) < fabsl(LDBL_EPSILON*sum)) break;
}
Safefree(amob);
return sum;
}
KAHAN_SUM(sum, 1.0);
flogx = logl(x);
part_term = 1;
for (k = 1; k <= 10000; k++) {
part_term *= flogx / k;
if (k-1 < NPRECALC_ZETA) term = part_term / (k+k*riemann_zeta_table[k-1]);
else term = part_term / (k+k*ld_riemann_zeta(k+1));
KAHAN_SUM(sum, term);
/* printf("R %5d after adding %.18Lg, sum = %.19Lg\n", k, term, sum); */
if (fabsl(term) < fabsl(LDBL_EPSILON*sum)) break;
}
return sum;
}