modified: SpatialOmicsCoord.py

[GalaxyCodeBases.git] / c_cpp / etc / calc / zprime.c

/*
 * zprime - rapid small prime routines
 *
 * Copyright (C) 1999-2007  Landon Curt Noll
 *
 * Calc is open software; you can redistribute it and/or modify it under
 * the terms of the version 2.1 of the GNU Lesser General Public License
 * as published by the Free Software Foundation.
 *
 * Calc is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
 * or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General
 * Public License for more details.
 *
 * A copy of version 2.1 of the GNU Lesser General Public License is
 * distributed with calc under the filename COPYING-LGPL.  You should have
 * received a copy with calc; if not, write to Free Software Foundation, Inc.
 * 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
 *
 * @(#) $Revision: 30.2 $
 * @(#) $Id: zprime.c,v 30.2 2013/08/11 08:41:38 chongo Exp $
 * @(#) $Source: /usr/local/src/bin/calc/RCS/zprime.c,v $
 *
 * Under source code control:   1994/05/29 04:34:36
 * File existed as early as:    1994
 *
 * chongo <was here> /\oo/\     http://www.isthe.com/chongo/
 * Share and enjoy!  :-)        http://www.isthe.com/chongo/tech/comp/calc/
 */


#include "zmath.h"
#include "prime.h"
#include "jump.h"
#include "config.h"
#include "zrand.h"
#include "have_const.h"


/*
 * When performing a probabilistic primality test, check to see
 * if the number has a factor <= PTEST_PRECHECK.
 *
 * XXX - what should this value be?  Perhaps this should be a function
 *       of the size of the text value and the number of tests?
 */
#define PTEST_PRECHECK ((FULL)101)

/*
 * product of primes that fit into a long
 */
STATIC CONST FULL pfact_tbl[MAX_PFACT_VAL+1] = {
    1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310, 30030, 30030,
    30030, 30030, 510510, 510510, 9699690, 9699690, 9699690, 9699690,
    223092870, 223092870, 223092870, 223092870, 223092870, 223092870
#if FULL_BITS == 64
    , U(6469693230), U(6469693230), U(200560490130), U(200560490130),
    U(200560490130), U(200560490130), U(200560490130), U(200560490130),
    U(7420738134810), U(7420738134810), U(7420738134810), U(7420738134810),
    U(304250263527210), U(304250263527210), U(13082761331670030),
    U(13082761331670030), U(13082761331670030), U(13082761331670030),
    U(614889782588491410), U(614889782588491410), U(614889782588491410),
    U(614889782588491410), U(614889782588491410), U(614889782588491410)
#endif
};

/*
 * determine the top 1 bit of a 8 bit value:
 *
 *      topbit[0] == 0 by convention
 *      topbit[x] gives the highest 1 bit of x
 */
STATIC CONST unsigned char topbit[256] = {
    0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3,
    4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
    5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
    5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
    6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
    6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
    6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
    6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
    7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
    7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
    7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
    7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
    7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
    7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
    7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
    7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
};

/*
 * integer square roots of powers of 2
 *
 * isqrt_pow2[x] == (int)(sqrt(2 to the x power))             (for 0 <= x < 64)
 *
 * We have enough table entries for a FULL that is 64 bits long.
 */
STATIC CONST FULL isqrt_pow2[64] = {
    1, 1, 2, 2, 4, 5, 8, 11,                                    /*  0 ..  7 */
    16, 22, 32, 45, 64, 90, 128, 181,                           /*  8 .. 15 */
    256, 362, 512, 724, 1024, 1448, 2048, 2896,                 /* 16 .. 23 */
    4096, 5792, 8192, 11585, 16384, 23170, 32768, 46340,        /* 24 .. 31 */
    65536, 92681, 131072, 185363,                               /* 32 .. 35 */
    262144, 370727, 524288, 741455,                             /* 36 .. 39 */
    1048576, 1482910, 2097152, 2965820,                         /* 40 .. 43 */
    4194304, 5931641, 8388608, 11863283,                        /* 44 .. 47 */
    16777216, 23726566, 33554432, 47453132,                     /* 48 .. 51 */
    67108864, 94906265, 134217728, 189812531,                   /* 52 .. 55 */
    268435456, 379625062, 536870912, 759250124,                 /* 56 .. 59 */
    1073741824, 1518500249, 0x80000000, 0xb504f333              /* 60 .. 63 */
};

/*
 * static functions
 */
S_FUNC FULL fsqrt(FULL v);              /* quick square root of v */
S_FUNC long pix(FULL x);                /* pi of x */
S_FUNC FULL small_factor(ZVALUE n, FULL limit);   /* factor or 0 */


/*
 * Determine if a value is a small (32 bit) prime
 *
 * Returns:
 *      1       z is a prime <= MAX_SM_VAL
 *      0       z is not a prime <= MAX_SM_VAL
 *      -1      z > MAX_SM_VAL
 */
FLAG
zisprime(ZVALUE z)
{
        FULL n;                         /* number to test */
        FULL isqr;                      /* factor limit */
        CONST unsigned short *tp;       /* pointer to a prime factor */

        z.sign = 0;
        if (zisleone(z)) {
                return 0;
        }

        /* even numbers > 2 are not prime */
        if (ziseven(z)) {
                /*
                 * "2 is the greatest odd prime because it is the least even!"
                 *                                      - Dr. Dan Jurca  1978
                 */
                return zisabstwo(z);
        }

        /* ignore non-small values */
        if (zge32b(z)) {
                return -1;
        }

        /* we now know that we are dealing with a value 0 <= n < 2^32 */
        n = ztofull(z);

        /* lookup small cases in pr_map */
        if (n <= MAX_MAP_VAL) {
                return (pr_map_bit(n) ? 1 : 0);
        }

        /* ignore Saber-C warning #530 about empty for statement */
        /*        ok to ignore in proc zisprime */
        /* a number >=2^16 and < 2^32 */
        for (isqr=fsqrt(n), tp=prime; (*tp <= isqr) && (n % *tp); ++tp) {
        }
        return ((*tp <= isqr && *tp != 1) ? 0 : 1);
}


/*
 * Determine the next small (32 bit) prime > a 32 bit value.
 *
 * given:
 *      z               search point
 *
 * Returns:
 *      0       next prime is 2^32+15
 *      1       abs(z) >= 2^32
 *      smallest prime > abs(z) otherwise
 */
FULL
znprime(ZVALUE z)
{
        FULL n;                 /* search point */

        z.sign = 0;

        /* ignore large values */
        if (zge32b(z)) {
                return (FULL)1;
        }

        /* deal a search point of 0 or 1 */
        if (zisabsleone(z)) {
                return (FULL)2;
        }

        /* deal with returning a value that is beyond our reach */
        n = ztofull(z);
        if (n >= MAX_SM_PRIME) {
                return (FULL)0;
        }

        /* return the next prime */
        return next_prime(n);
}


/*
 * Compute the next prime beyond a small (32 bit) value.
 *
 * This function assumes that 2 <= n < 2^32-5.
 *
 * given:
 *      n               search point
 */
FULL
next_prime(FULL n)
{
        CONST unsigned short *tp;       /* pointer to a prime factor */
        CONST unsigned char *j;         /* current jump increment */
        int tmp;

        /* find our search point */
        n = ((n & 0x1) ? n+2 : n+1);

        /* if we can just search the bit map, then search it */
        if (n <= MAX_MAP_PRIME) {

                /* search until we find a 1 bit */
                while (pr_map_bit(n) == 0) {
                        n += (FULL)2;
                }

        /* too large for our table, find the next prime the hard way */
        } else {
                FULL isqr;              /* factor limit */

                /*
                 * Our search for a prime may cause us to increment n over
                 * a perfect square, but never two perfect squares.  The largest
                 * prime gap <= 2614941711251 is 651.  Shanks conjectures that
                 * the largest gap below P is about ln(P)^2.
                 *
                 * The value fsqrt(n)^2 will always be the perfect square
                 * that is <= n.  Given the smallness of prime gaps we will
                 * deal with, we know that n could carry us across the next
                 * perfect square (fsqrt(n)+1)^2 but not the following
                 * perfect square (fsqrt(n)+2)^2.
                 *
                 * Now the factor search limit for values < (fsqrt(n)+2)^2
                 * is the same limit for (fsqrt(n)+1)^2; namely fsqrt(n)+1.
                 * Therefore setting our limit at fsqrt(n)+1 and never
                 * bothering with it after that is safe.
                 */
                isqr = fsqrt(n)+1;

                /*
                 * If our factor limit is even, then we can reduce it to
                 * the next lowest odd value.  We already tested if n
                 * was even and all of our remaining potential factors
                 * are odd.
                 */
                if ((isqr & 0x1) == 0) {
                        --isqr;
                }

                /*
                 * Skip to next value not divisible by a trivial prime.
                 */
                n = firstjmp(n, tmp);
                j = jmp + jmpptr(n);

                /*
                 * Look for tiny prime factors of increasing n until we
                 * find a prime.
                 */
                do {
                        /* ignore Saber-C warning #530 - empty for statement */
                        /*        ok to ignore in proc next_prime */
                        /* XXX - speed up test for large n by using gcds */
                        /* find a factor, or give up if not found */
                        for (tp=JPRIME; (*tp <= isqr) && (n % *tp); ++tp) {
                        }
                } while (*tp <= isqr && *tp != 1 && (n += nxtjmp(j)));
        }

        /* return the prime that we found */
        return n;
}


/*
 * Determine the previous small (32 bit) prime < a 32 bit value
 *
 * given:
 *      z               search point
 *
 * Returns:
 *      1       abs(z) >= 2^32
 *      0       abs(z) <= 2
 *      greatest prime < abs(z) otherwise
 */
FULL
zpprime(ZVALUE z)
{
        CONST unsigned short *tp;       /* pointer to a prime factor */
        FULL isqr;                      /* isqrt(z) */
        FULL n;                         /* search point */
        CONST unsigned char *j;         /* current jump increment */
        int tmp;

        z.sign  = 0;

        /* ignore large values */
        if (zge32b(z)) {
                return (FULL)1;
        }

        /* deal with special case small values */
        n = ztofull(z);
        switch ((int)n) {
        case 0:
        case 1:
        case 2:
                /* ignore values <= 2 */
                return (FULL)0;
        case 3:
                /* 3 returns the only even prime */
                return (FULL)2;
        }

        /* deal with values above the bit map */
        if (n > NXT_MAP_PRIME) {

                /* find our search point */
                n = ((n & 0x1) ? n-2 : n-1);

                /* our factor limit - see next_prime for why this works */
                isqr = fsqrt(n)+1;
                if ((isqr & 0x1) == 0) {
                        --isqr;
                }

                /*
                 * Skip to previous value not divisible by a trivial prime.
                 */
                tmp = jmpindxval(n);
                if (tmp >= 0) {

                        /* find next value not divisible by a trivial prime */
                        n += tmp;

                        /* find the previous jump index */
                        j = jmp + jmpptr(n);

                        /* jump back */
                        n -= prevjmp(j);

                /* already not divisible by a trivial prime */
                } else {
                        /* find the current jump index */
                        j = jmp + jmpptr(n);
                }

                /* factor values until we find a prime */
                do {
                        /* ignore Saber-C warning #530 - empty for statement */
                        /*        ok to ignore in proc zpprime */
                        /* XXX - speed up test for large n by using gcds */
                        /* find a factor, or give up if not found */
                        for (tp=prime; (*tp <= isqr) && (n % *tp); ++tp) {
                        }
                } while (*tp <= isqr && *tp != 1 && (n -= prevjmp(j)));

        /* deal with values within the bit map */
        } else if (n <= MAX_MAP_PRIME) {

                /* find our search point */
                n = ((n & 0x1) ? n-2 : n-1);

                /* search until we find a 1 bit */
                while (pr_map_bit(n) == 0) {
                        n -= (FULL)2;
                }

        /* deal with values that could cross into the bit map */
        } else {
                /* MAX_MAP_PRIME < n <= NXT_MAP_PRIME returns MAX_MAP_PRIME */
                return MAX_MAP_PRIME;
        }

        /* return what we found */
        return n;
}


/*
 * Compute the number of primes <= a ZVALUE that can fit into a FULL
 *
 * given:
 *      z               compute primes <= z
 *
 * Returns:
 *      -1      error
 *      >=0     number of primes <= x
 */
long
zpix(ZVALUE z)
{
        /* pi(<0) is always 0 */
        if (zisneg(z)) {
                return (long)0;
        }

        /* firewall */
        if (zge32b(z)) {
                return (long)-1;
        }
        return pix(ztofull(z));
}


/*
 * Compute the number of primes <= a ZVALUE
 *
 * given:
 *      x               value of z
 *
 * Returns:
 *      -1      error
 *      >=0     number of primes <= x
 */
S_FUNC long
pix(FULL x)
{
        long count;                     /* pi(x) */
        FULL top;                       /* top of the range to test */
        CONST unsigned short *tp;       /* pointer to a tiny prime */
        FULL i;

        /* compute pi(x) using the 2^8 step table */
        if (x <= MAX_PI10B) {

                /* x within the prime table, so use it */
                if (x < MAX_MAP_PRIME) {
                        /* firewall - pix(x) ==0 for x < 2 */
                        if (x < 2) {
                                count = 0;

                        } else {
                                /* determine how and where we will count */
                                if (x < 1024) {
                                        count = 1;
                                        tp = prime;
                                } else {
                                        count = pi10b[x>>10];
                                        tp = prime+count-1;
                                }
                                /* count primes in the table */
                                while (*tp++ <= x) {
                                        ++count;
                                }
                        }

                /* x is larger than the prime table, so count the hard way */
                } else {

                        /* case: count down from pi18b entry to x */
                        if (x & 0x200) {
                                top = (x | 0x3ff);
                                count = pi10b[(top+1)>>10];
                                for (i=next_prime(x); i <= top;
                                     i=next_prime(i)) {
                                        --count;
                                }

                        /* case: count up from pi10b entry to x */
                        } else {
                                count = pi10b[x>>10];
                                for (i=next_prime(x&(~0x3ff));
                                     i <= x; i = next_prime(i)) {
                                        ++count;
                                }
                        }
                }

        /* compute pi(x) using the 2^18 interval table */
        } else {

                /* compute sum of intervals up to our interval */
                for (count=0, i=0; i < (x>>18); ++i) {
                        count += pi18b[i];
                }

                /* case: count down from pi18b entry to x */
                if (x & 0x20000) {
                        top = (x | 0x3ffff);
                        count += pi18b[i];
                        if (top > MAX_SM_PRIME) {
                                if (x < MAX_SM_PRIME) {
                                        for (i=next_prime(x); i < MAX_SM_PRIME;
                                             i=next_prime(i)) {
                                                --count;
                                        }
                                        --count;
                                }
                        } else {
                                for (i=next_prime(x); i<=top; i=next_prime(i)) {
                                        --count;
                                }
                        }

                /* case: count up from pi18b entry to x */
                } else {
                        for (i=next_prime(x&(~0x3ffff));
                             i <= x; i = next_prime(i)) {
                                ++count;
                        }
                }
        }
        return count;
}


/*
 * Compute the smallest prime factor < limit
 *
 * given:
 *      n               number to factor
 *      zlimit          ending search point
 *      res             factor, if found, or NULL
 *
 * Returns:
 *      -1      error, limit >= 2^32
 *      0       no factor found, res is not changed
 *      1       factor found, res (if non-NULL) is smallest prime factor
 *
 * NOTE: This routine will not return a factor == the test value
 *       except when the test value is 1 or -1.
 */
FLAG
zfactor(ZVALUE n, ZVALUE zlimit, ZVALUE *res)
{
        FULL f;                 /* factor found, or 0 */

        /*
         * determine the limit
         */
        if (zge32b(zlimit)) {
                /* limit is too large to be reasonable */
                return -1;
        }
        n.sign = 0;             /* ignore sign of n */

        /*
         * find the smallest factor <= limit, if possible
         */
        f = small_factor(n, ztofull(zlimit));

        /*
         * report the results
         */
        if (f > 0) {
                /* return factor if requested */
                if (res) {
                        utoz(f, res);
                }
                /* report a factor was found */
                return 1;
        }
        /* no factor was found */
        return 0;
}


/*
 * Find a smallest prime factor <= some small (32 bit) limit of a value
 *
 * given:
 *      z               number to factor
 *      limit           largest factor we will test
 *
 * Returns:
 *      0       no prime <= the limit was found
 *      != 0    the smallest prime factor
 */
S_FUNC FULL
small_factor(ZVALUE z, FULL limit)
{
        FULL top;                       /* current max factor level */
        CONST unsigned short *tp;       /* pointer to a tiny prime */
        FULL factlim;                   /* highest factor to test */
        CONST unsigned short *p;        /* test factor */
        FULL factor;                    /* test factor */
        HALF tlim;                      /* limit on prime table use */
        HALF divval[2];                 /* divisor value */
        ZVALUE div;                     /* test factor/divisor */
        ZVALUE tmp;
        CONST unsigned char *j;

        /*
         * catch impossible ranges
         */
        if (limit < 2) {
                /* range is too small */
                return 0;
        }

        /*
         * perform the even test
         */
        if (ziseven(z)) {
                if (zistwo(z)) {
                        /* z is 2, so don't return 2 as a factor */
                        return 0;
                }
                return 2;

        /*
         * value is odd
         */
        } else if (limit == 2) {
                /* limit is 2, value is odd, no factors will ever be found */
                return 0;
        }

        /*
         * force the factor limit to be odd
         */
        if ((limit & 0x1) == 0) {
                --limit;
        }

        /*
         * case: number to factor fits into a FULL
         */
        if (!zgtmaxufull(z)) {
                FULL val = ztofull(z);  /* find the smallest factor of val */
                FULL isqr;              /* sqrt of val */

                /*
                 * special case: val is a prime <= MAX_MAP_PRIME
                 */
                if (val <= MAX_MAP_PRIME && pr_map_bit(val)) {
                        /* z is prime, so no factors will be found */
                        return 0;
                }

                /*
                 * we need not search above the sqrt of val
                 */
                isqr = fsqrt(val);
                if (limit > isqr) {
                        /* limit is largest odd value <= sqrt of val */
                        limit = ((isqr & 0x1) ? isqr : isqr-1);
                }

                /*
                 * search for a small prime factor
                 */
                top = ((limit < MAX_MAP_VAL) ? limit : MAX_MAP_VAL);
                for (tp = prime; *tp <= top && *tp != 1; ++tp) {
                        if (val%(*tp) == 0) {
                                return ((FULL)*tp);
                        }
                }

#if FULL_BITS == 64
                /*
                 * Our search will carry us beyond the prime table.  We will
                 * continue to values until we reach our limit or until a
                 * factor is found.
                 *
                 * It is faster to simply test odd values and ignore non-prime
                 * factors because the work needed to find the next prime is
                 * more than the work one saves in not factor with non-prime
                 * values.
                 *
                 * We can improve on this method by skipping odd values that
                 * are a multiple of 3, 5, 7 and 11.  We use a table of
                 * bytes that indicate the offsets between odd values that
                 * are not a multiple of 3,4,5,7 & 11.
                 */
                /* XXX - speed up test for large z by using gcds */
                j = jmp + jmpptr(NXT_MAP_PRIME);
                for (top=NXT_MAP_PRIME; top <= limit; top += nxtjmp(j)) {
                        if ((val % top) == 0) {
                                return top;
                        }
                }
#endif /* FULL_BITS == 64 */

                /* no prime factors found */
                return 0;
        }

        /*
         * Find a factor of a value that is too large to fit into a FULL.
         *
         * determine if/what our sqrt factor limit will be
         */
        if (zge64b(z)) {
                /* we have no factor limit, avoid highest factor */
                factlim = MAX_SM_PRIME-1;
        } else if (zge32b(z)) {
                /* determine if limit is too small to matter */
                if (limit < BASE) {
                        factlim = limit;
                } else {
                        /* find the isqrt(z) */
                        if (!zsqrt(z, &tmp, 0)) {
                                /* sqrt is exact */
                                factlim = ztofull(tmp);
                        } else {
                                /* sqrt is inexact */
                                factlim = ztofull(tmp)+1;
                        }
                        zfree(tmp);

                        /* avoid highest factor */
                        if (factlim >= MAX_SM_PRIME) {
                                factlim = MAX_SM_PRIME-1;
                        }
                }
        } else {
                /* determine our factor limit */
                factlim = fsqrt(ztofull(z));
                if (factlim >= MAX_SM_PRIME) {
                        factlim = MAX_SM_PRIME-1;
                }
        }
        if (factlim > limit) {
                factlim = limit;
        }

        /*
         * walk the prime table looking for factors
         *
         * XXX - consider using gcd of products of primes to speed this
         *       section up
         */
        tlim = (HALF)((factlim >= MAX_MAP_PRIME) ? MAX_MAP_PRIME-1 : factlim);
        div.sign = 0;
        div.v = divval;
        div.len = 1;
        for (p=prime; (HALF)*p <= tlim; ++p) {

                /* setup factor */
                div.v[0] = (HALF)(*p);

                if (zdivides(z, div))
                        return (FULL)(*p);
        }
        if ((FULL)*p > factlim) {
                /* no factor found */
                return (FULL)0;
        }

        /*
         * test the highest factor possible
         */
        div.v[0] = MAX_MAP_PRIME;

        if (zdivides(z, div))
                return (FULL)MAX_MAP_PRIME;

        /*
         * generate higher test factors as needed
         *
         * XXX - consider using gcd of products of primes to speed this
         *       section up
         */
#if BASEB == 16
        div.len = 2;
#endif
        factor = NXT_MAP_PRIME;
        j = jmp + jmpptr(factor);
        for(; factor <= factlim; factor += nxtjmp(j)) {

                /* setup factor */
#if BASEB == 32
                div.v[0] = (HALF)factor;
#else
                div.v[0] = (HALF)(factor & BASE1);
                div.v[1] = (HALF)(factor >> BASEB);
#endif

                if (zdivides(z, div))
                        return (FULL)(factor);
        }
        if (factor >= factlim) {
                /* no factor found */
                return (FULL)0;
        }

        /*
         * test the highest factor possible
         */
#if BASEB == 32
        div.v[0] = MAX_SM_PRIME;
#else
        div.v[0] = (MAX_SM_PRIME & BASE1);
        div.v[1] = (MAX_SM_PRIME >> BASEB);
#endif
        if (zdivides(z, div))
                return (FULL)MAX_SM_PRIME;

        /*
         * no factor found
         */
        return (FULL)0;
}


/*
 * Compute the product of the primes up to the specified number.
 */
void
zpfact(ZVALUE z, ZVALUE *dest)
{
        long n;                         /* limiting number to multiply by */
        long p;                         /* current prime */
        CONST unsigned short *tp;       /* pointer to a tiny prime */
        CONST unsigned char *j;         /* current jump increment */
        ZVALUE res, temp;

        /* firewall */
        if (zisneg(z)) {
                math_error("Negative argument for factorial");
                /*NOTREACHED*/
        }
        if (zge24b(z)) {
                math_error("Very large factorial");
                /*NOTREACHED*/
        }
        n = ztolong(z);

        /*
         * Deal with table lookup pfact values
         */
        if (n <= MAX_PFACT_VAL) {
                utoz(pfact_tbl[n], dest);
                return;
        }

        /*
         * Multiply by the primes in the static table
         */
        utoz(pfact_tbl[MAX_PFACT_VAL], &res);
        for (tp=(&prime[NXT_PFACT_VAL]); *tp != 1 && (long)(*tp) <= n; ++tp) {
                zmuli(res, *tp, &temp);
                zfree(res);
                res = temp;
        }

        /*
         * if needed, multiply by primes beyond the static table
         */
        j = jmp + jmpptr(NXT_MAP_PRIME);
        for (p = NXT_MAP_PRIME; p <= n; p += nxtjmp(j)) {
                FULL isqr;      /* isqrt(p) */

                /* our factor limit - see next_prime for why this works */
                isqr = fsqrt(p)+1;
                if ((isqr & 0x1) == 0) {
                        --isqr;
                }

                /* ignore Saber-C warning #530 about empty for statement */
                /*        ok to ignore in proc zpfact */
                /* find the next prime */
                for (tp=prime; (*tp <= isqr) && (p % (long)(*tp)); ++tp) {
                }
                if (*tp <= isqr && *tp != 1) {
                        continue;
                }

                /* multiply by the next prime */
                zmuli(res, p, &temp);
                zfree(res);
                res = temp;
        }
        *dest = res;
}


/*
 * Perform a probabilistic primality test (algorithm P in Knuth vol2, 4.5.4).
 * Returns FALSE if definitely not prime, or TRUE if probably prime.
 * Count determines how many times to check for primality.
 * The chance of a non-prime passing this test is less than (1/4)^count.
 * For example, a count of 100 fails for only 1 in 10^60 numbers.
 *
 * It is interesting to note that ptest(a,1,x) (for any x >= 0) of this
 * test will always return TRUE for a prime, and rarely return TRUE for
 * a non-prime.  The 1/4 is appears in practice to be a poor upper
 * bound.  Even so the only result that is EXACT and TRUE is when
 * this test returns FALSE for a non-prime.  When ptest returns TRUE,
 * one cannot determine if the value in question is prime, or the value
 * is one of those rare non-primes that produces a false positive.
 *
 * The absolute value of count determines how many times to check
 * for primality.  If count < 0, then the trivial factor check is
 * omitted.
 * skip = 0 uses random bases
 * skip = 1 uses prime bases 2, 3, 5, ...
 * skip > 1 or < 0 uses bases skip, skip + 1, ...
 */
BOOL
zprimetest(ZVALUE z, long count, ZVALUE skip)
{
        long limit = 0;         /* test odd values from skip up to limit */
        ZVALUE zbase;           /* base as a ZVALUE */
        long i, ij, ik;
        ZVALUE zm1, z1, z2, z3;
        int type;               /* random, prime or consecutive integers */
        CONST unsigned short *pr;       /* pointer to small prime */

        /*
         * firewall - ignore sign of z, values 0 and 1 are not prime
         */
        z.sign = 0;
        if (zisleone(z)) {
                return 0;
        }

        /*
         * firewall - All even values, except 2, are not prime
         */
        if (ziseven(z))
                return zistwo(z);

        if (z.len == 1 && *z.v == 3)
                return 1;               /* 3 is prime */

        /*
         * we know that z is an odd value > 1
         */

        /*
         * Perform trivial checks if count is not negative
         */
        if (count >= 0) {

                /*
                 * If the number is a small (32 bit) value, do a direct test
                 */
                if (!zge32b(z)) {
                        return zisprime(z);
                }

                /*
                 * See if the number has a tiny factor.
                 */
                if (small_factor(z, PTEST_PRECHECK) != 0) {
                        /* a tiny factor was found */
                        return FALSE;
                }

                /*
                 * If our count is zero, do nothing more
                 */
                if (count == 0) {
                        /* no test was done, so no test failed! */
                        return TRUE;
                }

        } else {
                /* use the absolute value of count */
                count = -count;
        }
        if (z.len < conf->redc2) {
                return zredcprimetest(z, count, skip);
        }

        if (ziszero(skip)) {
                type = 0;
                zbase = _zero_;
        } else if (zisone(skip)) {
                type = 1;
                itoz(2, &zbase);
                limit = 1 << 16;
                if (!zge16b(z))
                        limit = ztolong(z);
        } else {
                type = 2;
                if (zrel(skip, z) >= 0 || zisneg(skip))
                        zmod(skip, z, &zbase, 0);
                else
                        zcopy(skip, &zbase);
        }
        /*
         * Loop over various bases, testing each one.
         */
        zsub(z, _one_, &zm1);
        ik = zlowbit(zm1);
        zshift(zm1, -ik, &z1);
        pr = prime;
        for (i = 0; i < count; i++) {
                switch (type) {
                case 0:
                        zfree(zbase);
                        zrandrange(_two_, zm1, &zbase);
                        break;
                case 1:
                        if (i == 0)
                                break;
                        zfree(zbase);
                        if (*pr == 1 || (long)*pr >= limit) {
                                zfree(z1);
                                zfree(zm1);
                                return TRUE;
                        }
                        itoz((long) *pr++, &zbase);
                        break;
                default:
                        if (i == 0)
                                break;
                        zadd(zbase, _one_, &z3);
                        zfree(zbase);
                        zbase = z3;
                }

                ij = 0;
                zpowermod(zbase, z1, z, &z3);
                for (;;) {
                        if (zisone(z3)) {
                                if (ij) {
                                /* number is definitely not prime */
                                        zfree(z3);
                                        zfree(zm1);
                                        zfree(z1);
                                        zfree(zbase);
                                        return FALSE;
                                }
                                break;
                        }
                        if (!zcmp(z3, zm1))
                                break;
                        if (++ij >= ik) {
                                /* number is definitely not prime */
                                zfree(z3);
                                zfree(zm1);
                                zfree(z1);
                                zfree(zbase);
                                return FALSE;
                        }
                        zsquare(z3, &z2);
                        zfree(z3);
                        zmod(z2, z, &z3, 0);
                        zfree(z2);
                }
                zfree(z3);
        }
        zfree(zm1);
        zfree(z1);
        zfree(zbase);

        /* number might be prime */
        return TRUE;
}


/*
 * Called by zprimetest when simple cases have been eliminated
 * and z.len < conf->redc2.  Here count > 0, z is odd and > 3.
 */
BOOL
zredcprimetest(ZVALUE z, long count, ZVALUE skip)
{
        long limit = 0;         /* test odd values from skip up to limit */
        ZVALUE zbase;           /* base as a ZVALUE */
        REDC *rp;
        long i, ij, ik;
        ZVALUE zm1, z1, z2, z3;
        ZVALUE zredcm1;
        int type;               /* random, prime or consecutive integers */
        CONST unsigned short *pr;       /* pointer to small prime */


        rp = zredcalloc(z);
        zsub(z, rp->one, &zredcm1);
        if (ziszero(skip)) {
                zbase = _zero_;
                type = 0;
        } else if (zisone(skip)) {
                itoz(2, &zbase);
                type = 1;
                limit = 1 << 16;
                if (!zge16b(z))
                        limit = ztolong(z);
        } else {
                zredcencode(rp, skip, &zbase);
                type = 2;
        }
        /*
         * Loop over various "random" numbers, testing each one.
         */
        zsub(z, _one_, &zm1);
        ik = zlowbit(zm1);
        zshift(zm1, -ik, &z1);
        pr = prime;

        for (i = 0; i < count; i++) {
                switch (type) {
                case 0:
                        do {
                                zfree(zbase);
                                zrandrange(_one_, z, &zbase);
                        }
                        while (!zcmp(zbase, rp->one) ||
                                        !zcmp(zbase, zredcm1));
                        break;
                case 1:
                        if (i == 0) {
                                break;
                        }
                        zfree(zbase);
                        if (*pr == 1 || (long)*pr >= limit) {
                                zfree(z1);
                                zfree(zm1);
                                if (z.len < conf->redc2) {
                                        zredcfree(rp);
                                        zfree(zredcm1);
                                }
                                return TRUE;
                        }
                        itoz((long) *pr++, &z3);
                        zredcencode(rp, z3, &zbase);
                        zfree(z3);
                        break;
                default:
                        if (i == 0)
                                break;
                        zadd(zbase, rp->one, &z3);
                        zfree(zbase);
                        zbase = z3;
                        if (zrel(zbase, z) >= 0) {
                                zsub(zbase, z, &z3);
                                zfree(zbase);
                                zbase = z3;
                        }
                }

                ij = 0;
                zredcpower(rp, zbase, z1, &z3);
                for (;;) {
                        if (!zcmp(z3, rp->one)) {
                                if (ij) {
                                /* number is definitely not prime */
                                        zfree(z3);
                                        zfree(zm1);
                                        zfree(z1);
                                        zfree(zbase);
                                        zredcfree(rp);
                                        zfree(zredcm1);
                                        return FALSE;
                                }
                                break;
                        }
                        if (!zcmp(z3, zredcm1))
                                break;
                        if (++ij >= ik) {
                                /* number is definitely not prime */
                                zfree(z3);
                                zfree(zm1);
                                zfree(z1);
                                zfree(zbase);
                                zredcfree(rp);
                                zfree(zredcm1);
                                return FALSE;
                        }
                        zredcsquare(rp, z3, &z2);
                        zfree(z3);
                        z3 = z2;
                }
                zfree(z3);
        }
        zfree(zbase);
        zredcfree(rp);
        zfree(zredcm1);
        zfree(zm1);
        zfree(z1);

        /* number might be prime */
        return TRUE;
}


/*
 * znextcand - find the next integer that passes ptest().
 * The signs of z and mod are ignored.  Result is the least integer
 * greater than abs(z) congruent to res modulo abs(mod), or if there
 * is no such integer, zero.
 *
 * given:
 *      z               search point > 2
 *      count           ptests to perform per candidate
 *      skip            ptests to skip
 *      res             return congruent to res modulo abs(mod)
 *      mod             congruent to res modulo abs(mod)
 *      cand            candidate found
 */
BOOL
znextcand(ZVALUE z, long count, ZVALUE skip, ZVALUE res, ZVALUE mod,
          ZVALUE *cand)
{
        ZVALUE tmp1;
        ZVALUE tmp2;

        z.sign  = 0;
        mod.sign = 0;
        if (ziszero(mod)) {
                if (zrel(res, z) > 0 && zprimetest(res, count, skip)) {
                        zcopy(res, cand);
                        return TRUE;
                }
                return FALSE;
        }
        if (ziszero(z) && zisone(mod)) {
                zcopy(_two_, cand);
                return TRUE;
        }
        zsub(res, z, &tmp1);
        if (zmod(tmp1, mod, &tmp2, 0))
                zadd(z, tmp2, cand);
        else
                zadd(z, mod, cand);

        /*
         * Now *cand is least integer greater than abs(z) and congruent
         * to res modulo mod.
         */
        zfree(tmp1);
        zfree(tmp2);
        if (zprimetest(*cand, count, skip))
                return TRUE;
        zgcd(*cand, mod, &tmp1);
        if (!zisone(tmp1)) {
                zfree(tmp1);
                zfree(*cand);
                return FALSE;
        }
        zfree(tmp1);
        if (ziseven(*cand)) {
                zadd(*cand, mod, &tmp1);
                zfree(*cand);
                *cand = tmp1;
                if (zprimetest(*cand, count, skip))
                        return TRUE;
        }
        /*
         * *cand is now least odd integer > abs(z) and congruent to
         * res modulo mod.
         */
        if (zisodd(mod))
                zshift(mod, 1, &tmp1);
        else
                zcopy(mod, &tmp1);
        do {
                zadd(*cand, tmp1, &tmp2);
                zfree(*cand);
                *cand = tmp2;
        } while (!zprimetest(*cand, count, skip));
        zfree(tmp1);
        return TRUE;
}


/*
 * zprevcand - find the nearest previous integer that passes ptest().
 * The signs of z and mod are ignored.  Result is greatest positive integer
 * less than abs(z) congruent to res modulo abs(mod), or if there
 * is no such integer, zero.
 *
 * given:
 *      z               search point > 2
 *      count           ptests to perform per candidate
 *      skip            ptests to skip
 *      res             return congruent to res modulo abs(mod)
 *      mod             congruent to res modulo abs(mod)
 *      cand            candidate found
 */
BOOL
zprevcand(ZVALUE z, long count, ZVALUE skip, ZVALUE res, ZVALUE mod,
          ZVALUE *cand)
{
        ZVALUE tmp1;
        ZVALUE tmp2;

        z.sign  = 0;
        mod.sign = 0;
        if (ziszero(mod)) {
                if (zispos(res)&&zrel(res, z)<0 && zprimetest(res,count,skip)) {
                        zcopy(res, cand);
                        return TRUE;
                }
                return FALSE;
        }
        zsub(z, res, &tmp1);
        if (zmod(tmp1, mod, &tmp2, 0))
                zsub(z, tmp2, cand);
        else
                zsub(z, mod, cand);
        /*
         * *cand is now the greatest integer < z that is congruent to res
         * modulo mod.
         */
        zfree(tmp1);
        zfree(tmp2);
        if (zisneg(*cand)) {
                zfree(*cand);
                return FALSE;
        }
        if (zprimetest(*cand, count, skip))
                return TRUE;
        zgcd(*cand, mod, &tmp1);
        if (!zisone(tmp1)) {
                zfree(tmp1);
                zmod(*cand, mod, &tmp1, 0);
                zfree(*cand);
                if (zprimetest(tmp1, count, skip)) {
                        *cand = tmp1;
                        return TRUE;
                }
                if (ziszero(tmp1)) {
                        zfree(tmp1);
                        if (zprimetest(mod, count, skip)) {
                                zcopy(mod, cand);
                                return TRUE;
                        }
                        return FALSE;
                }
                zfree(tmp1);
                return FALSE;
        }
        zfree(tmp1);
        if (ziseven(*cand)) {
                zsub(*cand, mod, &tmp1);
                zfree(*cand);
                if (zisneg(tmp1)) {
                        zfree(tmp1);
                        return FALSE;
                }
                *cand = tmp1;
                if (zprimetest(*cand, count, skip))
                        return TRUE;
        }
        /*
         * *cand is now the greatest odd integer < z that is congruent to
         * res modulo mod.
         */
        if (zisodd(mod))
                zshift(mod, 1, &tmp1);
        else
                zcopy(mod, &tmp1);

        do {
                zsub(*cand, tmp1, &tmp2);
                zfree(*cand);
                *cand = tmp2;
        } while (!zprimetest(*cand, count, skip) && !zisneg(*cand));
        zfree(tmp1);
        if (zisneg(*cand)) {
                        zadd(*cand, mod, &tmp1);
                        zfree(*cand);
                        *cand = tmp1;
                        if (zistwo(*cand))
                                return TRUE;
                        zfree(*cand);
                        return FALSE;
        }
        return TRUE;
}


/*
 * Find the lowest prime factor of a number if one can be found.
 * Search is conducted for the first count primes.
 *
 * Returns:
 *      1       no factor found or z < 3
 *      >1      factor found
 */
FULL
zlowfactor(ZVALUE z, long count)
{
        FULL factlim;                   /* highest factor to test */
        CONST unsigned short *p;        /* test factor */
        FULL factor;                    /* test factor */
        HALF tlim;                      /* limit on prime table use */
        HALF divval[2];                 /* divisor value */
        ZVALUE div;                     /* test factor/divisor */
        ZVALUE tmp;

        z.sign = 0;

        /*
         * firewall
         */
        if (count <= 0 || zisleone(z) || zistwo(z)) {
                /* number is < 3 or count is <= 0 */
                return (FULL)1;
        }

        /*
         * test for the first factor
         */
        if (ziseven(z)) {
                return (FULL)2;
        }
        if (count <= 1) {
                /* count was 1, tested the one and only factor */
                return (FULL)1;
        }

        /*
         * determine if/what our sqrt factor limit will be
         */
        if (zge64b(z)) {
                /* we have no factor limit, avoid highest factor */
                factlim = MAX_SM_PRIME-1;
        } else if (zge32b(z)) {
                /* find the isqrt(z) */
                if (!zsqrt(z, &tmp, 0)) {
                        /* sqrt is exact */
                        factlim = ztofull(tmp);
                } else {
                        /* sqrt is inexact */
                        factlim = ztofull(tmp)+1;
                }
                zfree(tmp);

                /* avoid highest factor */
                if (factlim >= MAX_SM_PRIME) {
                        factlim = MAX_SM_PRIME-1;
                }
        } else {
                /* determine our factor limit */
                factlim = fsqrt(ztofull(z));
        }
        if (factlim >= MAX_SM_PRIME) {
                factlim = MAX_SM_PRIME-1;
        }

        /*
         * walk the prime table looking for factors
         */
        tlim = (HALF)((factlim >= MAX_MAP_PRIME) ? MAX_MAP_PRIME-1 : factlim);
        div.sign = 0;
        div.v = divval;
        div.len = 1;
        for (p=prime, --count; count > 0 && (HALF)*p <= tlim; ++p, --count) {

                /* setup factor */
                div.v[0] = (HALF)(*p);

                if (zdivides(z, div))
                        return (FULL)(*p);
        }
        if (count <= 0 || (FULL)*p > factlim) {
                /* no factor found */
                return (FULL)1;
        }

        /*
         * test the highest factor possible
         */
        div.v[0] = MAX_MAP_PRIME;
        if (zdivides(z, div))
                return (FULL)MAX_MAP_PRIME;

        /*
         * generate higher test factors as needed
         */
#if BASEB == 16
        div.len = 2;
#endif
        for(factor = NXT_MAP_PRIME;
            count > 0 && factor <= factlim;
            factor = next_prime(factor), --count) {

                /* setup factor */
#if BASEB == 32
                div.v[0] = (HALF)factor;
#else
                div.v[0] = (HALF)(factor & BASE1);
                div.v[1] = (HALF)(factor >> BASEB);
#endif

                if (zdivides(z, div))
                        return (FULL)(factor);
        }
        if (count <= 0 || factor >= factlim) {
                /* no factor found */
                return (FULL)1;
        }

        /*
         * test the highest factor possible
         */
#if BASEB == 32
        div.v[0] = MAX_SM_PRIME;
#else
        div.v[0] = (MAX_SM_PRIME & BASE1);
        div.v[1] = (MAX_SM_PRIME >> BASEB);
#endif
        if (zdivides(z, div))
                return (FULL)MAX_SM_PRIME;

        /*
         * no factor found
         */
        return (FULL)1;
}


/*
 * Compute the least common multiple of all the numbers up to the
 * specified number.
 */
void
zlcmfact(ZVALUE z, ZVALUE *dest)
{
        long n;                         /* limiting number to multiply by */
        long p;                         /* current prime */
        long pp = 0;                    /* power of prime */
        long i;                         /* test value */
        CONST unsigned short *pr;       /* pointer to a small prime */
        ZVALUE res, temp;

        if (zisneg(z) || ziszero(z)) {
                math_error("Non-positive argument for lcmfact");
                /*NOTREACHED*/
        }
        if (zge24b(z)) {
                math_error("Very large lcmfact");
                /*NOTREACHED*/
        }
        n = ztolong(z);
        /*
         * Multiply by powers of the necessary odd primes in order.
         * The power for each prime is the highest one which is not
         * more than the specified number.
         */
        res = _one_;
        for (pr=prime; (long)(*pr) <= n && *pr > 1; ++pr) {
                i = p = *pr;
                while (i <= n) {
                        pp = i;
                        i *= p;
                }
                zmuli(res, pp, &temp);
                zfree(res);
                res = temp;
        }
        for (p = NXT_MAP_PRIME; p <= n; p = (long)next_prime(p)) {
                i = p;
                while (i <= n) {
                        pp = i;
                        i *= p;
                }
                zmuli(res, pp, &temp);
                zfree(res);
                res = temp;
        }
        /*
         * Finish by scaling by the necessary power of two.
         */
        zshift(res, zhighbit(z), dest);
        zfree(res);
}


/*
 * fsqrt - fast square root of a FULL value
 *
 * We will determine the square root of a given value.
 * Starting with the integer square root of the largest power of
 * two <= the value, we will perform 3 Newton interations to
 * arive at our guess.
 *
 * We have verified that fsqrt(x) == (FULL)sqrt((double)x), or
 * fsqrt(x)-1 == (FULL)sqrt((double)x) for all 0 <= x < 2^32.
 *
 * given:
 *      x               compute the integer square root of x
 */
S_FUNC FULL
fsqrt(FULL x)
{
        FULL y;         /* (FULL)temporary value */
        int i;

        /* firewall - deal with 0 */
        if (x == 0) {
            return 0;
        }

        /* ignore Saber-C warning #530 about empty for statement */
        /*        ok to ignore in proc fsqrt */
        /* determine our initial guess */
        for (i=0, y=x; y >= (FULL)256; i+=8, y>>=8) {
        }
        y = isqrt_pow2[i + topbit[y]];

        /* perform 3 Newton interations */
        y = (y+x/y)>>1;
        y = (y+x/y)>>1;
        y = (y+x/y)>>1;
#if FULL_BITS == 64
        y = (y+x/y)>>1;
#endif

        /* return the result */
        return y;
}