/* mpz_fac_ui(result, n) -- Set RESULT to N!. Copyright 1991, 1993, 1994, 1995, 2000, 2001, 2002, 2003 Free Software Foundation, Inc. Copyright 2009 Robert Gerbicz This file is part of the MPIR Library. The MPIR Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. The MPIR Library 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. You should have received a copy of the GNU Lesser General Public License along with the MPIR Library; see the file COPYING.LIB. If not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. */ #include "mpir.h" #include "gmp-impl.h" #include "longlong.h" /* These constants are generated by gen-fac_ui.c */ #if GMP_NUMB_BITS == 32 /* This table is 0!,1!,2!,3!,...,n! where n! has <= GMP_NUMB_BITS bits */ #define ONE_LIMB_FACTORIAL_TABLE CNST_LIMB(0x1),CNST_LIMB(0x1),CNST_LIMB(0x2),CNST_LIMB(0x6),CNST_LIMB(0x18),CNST_LIMB(0x78),CNST_LIMB(0x2d0),CNST_LIMB(0x13b0),CNST_LIMB(0x9d80),CNST_LIMB(0x58980),CNST_LIMB(0x375f00),CNST_LIMB(0x2611500),CNST_LIMB(0x1c8cfc00) /* is 2^(GMP_LIMB_BITS+1)/exp(1) */ #define FAC2OVERE CNST_LIMB(0xbc5c254b) /* FACMULn is largest odd x such that x*(x+2)*...*(x+2(n-1))<=2^GMP_NUMB_BITS-1 */ #define FACMUL2 CNST_LIMB(0xffff) #define FACMUL3 CNST_LIMB(0x657) #define FACMUL4 CNST_LIMB(0xfd) #endif /* 32 bits */ #if GMP_NUMB_BITS == 64 /* This table is 0!,1!,2!,3!,...,n! where n! has <= GMP_NUMB_BITS bits */ #define ONE_LIMB_FACTORIAL_TABLE CNST_LIMB(0x1),CNST_LIMB(0x1),CNST_LIMB(0x2),CNST_LIMB(0x6),CNST_LIMB(0x18),CNST_LIMB(0x78),CNST_LIMB(0x2d0),CNST_LIMB(0x13b0),CNST_LIMB(0x9d80),CNST_LIMB(0x58980),CNST_LIMB(0x375f00),CNST_LIMB(0x2611500),CNST_LIMB(0x1c8cfc00),CNST_LIMB(0x17328cc00),CNST_LIMB(0x144c3b2800),CNST_LIMB(0x13077775800),CNST_LIMB(0x130777758000),CNST_LIMB(0x1437eeecd8000),CNST_LIMB(0x16beecca730000),CNST_LIMB(0x1b02b9306890000),CNST_LIMB(0x21c3677c82b40000) /* is 2^(GMP_LIMB_BITS+1)/exp(1) */ #define FAC2OVERE CNST_LIMB(0xbc5c254b96be9524) /* FACMULn is largest odd x such that x*(x+2)*...*(x+2(n-1))<=2^GMP_NUMB_BITS-1 */ #define FACMUL2 CNST_LIMB(0xffffffff) #define FACMUL3 CNST_LIMB(0x285143) #define FACMUL4 CNST_LIMB(0xfffd) #endif /* 64 bits */ static void odd_product _PROTO ((unsigned long low, unsigned long high, mpz_t * st)); static void ap_product_small _PROTO ((mpz_t ret, mp_limb_t start, mp_limb_t step, unsigned long count, unsigned long nm)); static void binary_splitting _PROTO ((mpz_ptr result,unsigned long int *a,unsigned long int L)); static void large_mpz_fac_ui _PROTO ((mpz_ptr result, unsigned long int n)); static unsigned long int ulsqrt _PROTO ((unsigned long int n)); /* must be >=2 */ #define APCONST 5 /* for single non-zero limb */ #define MPZ_SET_1_NZ(z,n) \ do { \ mpz_ptr __z = (z); \ ASSERT ((n) != 0); \ PTR(__z)[0] = (n); \ SIZ(__z) = 1; \ } while (0) /* for src>0 and n>0 */ #define MPZ_MUL_1_POS(dst,src,n) \ do { \ mpz_ptr __dst = (dst); \ mpz_srcptr __src = (src); \ mp_size_t __size = SIZ(__src); \ mp_ptr __dst_p; \ mp_limb_t __c; \ \ ASSERT (__size > 0); \ ASSERT ((n) != 0); \ \ MPZ_REALLOC (__dst, __size+1); \ __dst_p = PTR(__dst); \ \ __c = mpn_mul_1 (__dst_p, PTR(__src), __size, n); \ __dst_p[__size] = __c; \ SIZ(__dst) = __size + (__c != 0); \ } while (0) #if BITS_PER_ULONG == BITS_PER_MP_LIMB #define BSWAP_ULONG(x,y) BSWAP_LIMB(x,y) #endif /* We used to have a case here for limb==2*long, doing a BSWAP_LIMB followed by a shift down to get the high part. But it provoked incorrect code from "HP aC++/ANSI C B3910B A.05.52 [Sep 05 2003]" in ILP32 mode. This case would have been nice for gcc ia64 where BSWAP_LIMB is a mux1, but we can get that directly muxing a 4-byte ulong if it matters enough. */ #if ! defined (BSWAP_ULONG) #define BSWAP_ULONG(dst, src) \ do { \ unsigned long __bswapl_src = (src); \ unsigned long __bswapl_dst = 0; \ int __i; \ for (__i = 0; __i < sizeof(unsigned long); __i++) \ { \ __bswapl_dst = (__bswapl_dst << 8) | (__bswapl_src & 0xFF); \ __bswapl_src >>= 8; \ } \ (dst) = __bswapl_dst; \ } while (0) #endif /* x is bit reverse of y */ /* Note the divides below are all exact */ #define BITREV_ULONG(x,y) \ do { \ unsigned long __dst; \ BSWAP_ULONG(__dst,y); \ __dst = ((__dst>>4)&(ULONG_MAX/17)) | ((__dst<<4)&((ULONG_MAX/17)*16)); \ __dst = ((__dst>>2)&(ULONG_MAX/5) ) | ((__dst<<2)&((ULONG_MAX/5)*4) ); \ __dst = ((__dst>>1)&(ULONG_MAX/3) ) | ((__dst<<1)&((ULONG_MAX/3)*2) ); \ (x) = __dst; \ } while(0) /* above could be improved if cpu has a nibble/bit swap/muxing instruction */ /* above code is serialized, possible to write as a big parallel expression */ void mpz_fac_ui (mpz_ptr x, unsigned long n) { unsigned long z, stt; int i, j; mpz_t t1, st[8 * sizeof (unsigned long) + 1 - APCONST]; mp_limb_t d[4]; static const mp_limb_t table[] = { ONE_LIMB_FACTORIAL_TABLE }; if (n < numberof (table)) { MPZ_SET_1_NZ (x, table[n]); return; } /* NOTE : MUST have n>=3 here */ ASSERT (n >= 3); /* for estimating the alloc sizes the calculation of these formula's is not exact and also the formulas are only approximations, also we ignore the few "side" calculations, correct allocation seems to speed up the small sizes better, having very little effect on the large sizes */ /* estimate space for stack entries see below number of bits for n! is (1+log_2(2*pi)/2)-n*log_2(exp(1))+(n+1/2)*log_2(n)= 2.325748065-n*1.442695041+(n+0.5)*log_2(n) */ umul_ppmm (d[1], d[0], (mp_limb_t) n, (mp_limb_t) FAC2OVERE); /* d[1] is 2n/e, d[0] ignored */ count_leading_zeros (z, d[1]); z = GMP_LIMB_BITS - z - 1; /* z=floor(log_2(2n/e)) */ umul_ppmm (d[1], d[0], (mp_limb_t) n, (mp_limb_t) z); /* d=n*floor(log_2(2n/e)) */ d[0] = (d[0] >> 2) | (d[1] << (GMP_LIMB_BITS - 2)); d[1] >>= 2; /* d=n*floor(log_2(2n/e))/4 */ z = d[0] + 1; /* have to ignore any overflow */ /* so z is the number of bits wanted for st[0] */ if (n <= ((unsigned long) 1) << (APCONST)) { mpz_realloc2 (x, 4 * z); ap_product_small (x, CNST_LIMB(2), CNST_LIMB(1), n - 1, 4L); return; } if (n <= ((unsigned long) 1) << (APCONST + 1)) { /* use n!=odd(1,n)*(n/2)!*2^(n/2) */ mpz_init2 (t1, 2 * z); mpz_realloc2 (x, 4 * z); ap_product_small (x, CNST_LIMB(2), CNST_LIMB(1), n / 2 - 1, 4L); ap_product_small (t1, CNST_LIMB(3), CNST_LIMB(2), (n - 1) / 2, 4L); mpz_mul (x, x, t1); mpz_clear (t1); mpz_mul_2exp (x, x, n / 2); return; } if (n <= ((unsigned long) 1) << (APCONST + 2)) { /* use n!=C_2(1,n/2)^2*C_2(n/2,n)*(n/4)!*2^(n/2+n/4) all int divs so need (BITS_IN_N-APCONST+1)=(APCONST+3-APCONST+1)=4 stack entries */ mpz_init2 (t1, 2 * z); mpz_realloc2 (x, 4 * z); for (i = 0; i < 4; i++) { mpz_init2 (st[i], z); z >>= 1; } odd_product (1, n / 2, st); mpz_set (x, st[0]); odd_product (n / 2, n, st); mpz_mul (x, x, x); ASSERT (n / 4 <= FACMUL4 + 6); ap_product_small (t1, CNST_LIMB(2), CNST_LIMB(1), n / 4 - 1, 4L); /* must have 2^APCONST odd numbers max */ mpz_mul (t1, t1, st[0]); for (i = 0; i < 4; i++) mpz_clear (st[i]); mpz_mul (x, x, t1); mpz_clear (t1); mpz_mul_2exp (x, x, n / 2 + n / 4); return; } if (ABOVE_THRESHOLD(n,FAC_UI_THRESHOLD)){large_mpz_fac_ui(x,n);return;} count_leading_zeros (stt, (mp_limb_t) n); stt = GMP_LIMB_BITS - stt + 1 - APCONST; for (i = 0; i < (signed long) stt; i++) { mpz_init2 (st[i], z); z >>= 1; } count_leading_zeros (z, (mp_limb_t) (n / 3)); /* find z st 2^z>n/3 range for z is 1 <= z <= 8 * sizeof(unsigned long)-1 */ z = GMP_LIMB_BITS - z; /* n! = 2^e * PRODUCT_{i=0}^{i=z-1} C_2( n/2^{i+1}, n/2^i )^{i+1} where 2^e || n! 3.2^z>n C_2(a,b)=PRODUCT of odd z such that a>= 1) { MPZ_SET_1_NZ (x, 1); for (i = 8 * sizeof (unsigned long) - j; i >= j; i -= 2 * j) if ((signed long) z >= i) { odd_product (n >> i, n >> (i - 1), st); /* largest odd product when j=i=1 then we have odd_product(n/2,n,st) which is approx (2n/e)^(n/4) so log_base2(largest oddproduct)=n*log_base2(2n/e)/4 number of bits is n*log_base2(2n/e)/4+1 */ if (i != j) mpz_pow_ui (st[0], st[0], i / j); mpz_mul (x, x, st[0]); } if ((signed long) z >= j && j != 1) { mpz_mul (t1, t1, x); mpz_mul (t1, t1, t1); } } for (i = 0; i < (signed long) stt; i++) mpz_clear (st[i]); mpz_mul (x, x, t1); mpz_clear (t1); popc_limb (i, (mp_limb_t) n); mpz_mul_2exp (x, x, n - i); return; } /* start,step are mp_limb_t although they will fit in unsigned long */ static void ap_product_small (mpz_t ret, mp_limb_t start, mp_limb_t step, unsigned long count, unsigned long nm) { unsigned long a; mp_limb_t b; ASSERT (count <= (((unsigned long) 1) << APCONST)); /* count can never be zero ? check this and remove test below */ if (count == 0) { MPZ_SET_1_NZ (ret, 1); return; } if (count == 1) { MPZ_SET_1_NZ (ret, start); return; } switch (nm) { case 1: MPZ_SET_1_NZ (ret, start); b = start + step; for (a = 0; a < count - 1; b += step, a++) MPZ_MUL_1_POS (ret, ret, b); return; case 2: MPZ_SET_1_NZ (ret, start * (start + step)); if (count == 2) return; for (b = start + 2 * step, a = count / 2 - 1; a != 0; a--, b += 2 * step) MPZ_MUL_1_POS (ret, ret, b * (b + step)); if (count % 2 == 1) MPZ_MUL_1_POS (ret, ret, b); return; case 3: if (count == 2) { MPZ_SET_1_NZ (ret, start * (start + step)); return; } MPZ_SET_1_NZ (ret, start * (start + step) * (start + 2 * step)); if (count == 3) return; for (b = start + 3 * step, a = count / 3 - 1; a != 0; a--, b += 3 * step) MPZ_MUL_1_POS (ret, ret, b * (b + step) * (b + 2 * step)); if (count % 3 == 2) b = b * (b + step); if (count % 3 != 0) MPZ_MUL_1_POS (ret, ret, b); return; default: /* ie nm=4 */ if (count == 2) { MPZ_SET_1_NZ (ret, start * (start + step)); return; } if (count == 3) { MPZ_SET_1_NZ (ret, start * (start + step) * (start + 2 * step)); return; } MPZ_SET_1_NZ (ret, start * (start + step) * (start + 2 * step) * (start + 3 * step)); if (count == 4) return; for (b = start + 4 * step, a = count / 4 - 1; a != 0; a--, b += 4 * step) MPZ_MUL_1_POS (ret, ret, b * (b + step) * (b + 2 * step) * (b + 3 * step)); if (count % 4 == 2) b = b * (b + step); if (count % 4 == 3) b = b * (b + step) * (b + 2 * step); if (count % 4 != 0) MPZ_MUL_1_POS (ret, ret, b); return; } } /* return value in st[0] odd_product(l,h)=sqrt((h/e)^h/(l/e)^l) using Stirling approx and e=exp(1) so st[0] needs enough bits for above, st[1] needs half these bits and st[2] needs 1/4 of these bits etc */ static void odd_product (unsigned long low, unsigned long high, mpz_t * st) { unsigned long stc = 1, stn = 0, n, y, mask, a, nm = 1; signed long z; low++; if (low % 2 == 0) low++; if (high == 0) high = 1; if (high % 2 == 0) high--; /* must have high>=low ? check this and remove test below */ if (high < low) { MPZ_SET_1_NZ (st[0], 1); return; } if (high == low) { MPZ_SET_1_NZ (st[0], low); return; } if (high <= FACMUL2 + 2) { nm = 2; if (high <= FACMUL3 + 4) { nm = 3; if (high <= FACMUL4 + 6) nm = 4; } } high = (high - low) / 2 + 1; /* high is now count,high<=2^(BITS_PER_ULONG-1) */ if (high <= (((unsigned long) 1) << APCONST)) { ap_product_small (st[0], (mp_limb_t) low, CNST_LIMB(2), high, nm); return; } count_leading_zeros (n, (mp_limb_t) high); /* assumes clz above is LIMB based not NUMB based */ n = GMP_LIMB_BITS - n - APCONST; mask = (((unsigned long) 1) << n); a = mask << 1; mask--; /* have 2^(BITS_IN_N-APCONST) iterations so need (BITS_IN_N-APCONST+1) stack entries */ for (z = mask; z >= 0; z--) { BITREV_ULONG (y, z); y >>= (BITS_PER_ULONG - n); ap_product_small (st[stn], (mp_limb_t) (low + 2 * ((~y) & mask)), (mp_limb_t) a, (high + y) >> n, nm); ASSERT (((high + y) >> n) <= (((unsigned long) 1) << APCONST)); stn++; y = stc++; while ((y & 1) == 0) { mpz_mul (st[stn - 2], st[stn - 2], st[stn - 1]); stn--; y >>= 1; } } ASSERT (stn == 1); return; } // Computation of n factorial by computing the prime factoriation of n!, // using iterated squaring and multiplication by a "small" number idea and binary splitting // written by Robert Gerbicz the algorithm is due to Schoenhage. See Peter // Luschny's website for many factorial implementations, including one of // this algorithm. See page 226 in: "Fast Algorithms, A Multitape Turing // Machine Implementation" by A. Schonhage, A. F. W. Grotefeld and E. Vetter, // BI Wissenschafts-Verlag, Mannheim, 1994. static void binary_splitting (mpz_ptr result,unsigned long int *a,unsigned long int L) { // mulptiplication by binary splitting unsigned long int i,L0,L1; mpz_t temp; if(L==0) { mpz_set_ui (result, 1); return; } if(L<=3) { mpz_set_ui(result,a[0]); for(i=1;i>1,n64=(n>>6)+1,memalloc; int h,expo; mpz_t temp; mpz_init(temp); isprime=__GMP_ALLOCATE_FUNC_TYPE(n64,unsigned long int); ASSERT(n>=2);// cant handle n<2 Bit[0]=1; for(i=1;i<32;i++) Bit[i]=Bit[i-1]<<1; // Bit[i]=2^i // determine all odd primes up to n by sieve for(i=0;i>6]&Bit[(p>>1)&31]) { for(i=(p*p-1)>>1;i<=n2;i+=p) isprime[i>>5]&=~Bit[i&31]; } } primepi=0; for(i=0;i<=n2;i++) primepi+=((isprime[i>>5]&Bit[i&31])>0); memalloc=primepi; primes=__GMP_ALLOCATE_FUNC_TYPE(memalloc,unsigned long int); S=__GMP_ALLOCATE_FUNC_TYPE(memalloc,unsigned long int); exponent=__GMP_ALLOCATE_FUNC_TYPE(memalloc,unsigned long int); primepi=0; // 1 is not prime and hasn't cancelled, so start from 1*2+1=3 for(i=1;i<=n2;i++) if((isprime[i>>5]&Bit[i&31])>0) { p=2*i+1; N=n; e=0; while(N) N/=p,e+=N; primes[primepi]=p; // store prime exponent[primepi]=e; // exponent of p in the factorization of n! primepi++; } __GMP_FREE_FUNC_TYPE(isprime,n64,unsigned long int); mpz_set_ui(result,1); expo=0,p2=1; N=n; while(N) N>>=1,p2<<=1,expo++; for(h=expo;h>=0;h--) { // collect all primes for which in the factorization of n! the primes[g]'s exponent's h-th bit is 1 count=0; // note that exponent[] is a decreasing array for(g=0;(g=p2);g++) if(((exponent[g]>>h)&1)==1) { S[count]=primes[g]; count++; } binary_splitting(temp,S,count); // build the product by binary splitting mpz_pow_ui(result,result,2); // squaring mpz_mul(result,result,temp); // multiplcation by a not so large number p2>>=1; } N=n; e=0; while(N) N>>=1,e+=N; mpz_mul_2exp(result,result,e); // shift the number to finally get n! __GMP_FREE_FUNC_TYPE(S,memalloc,unsigned long int); __GMP_FREE_FUNC_TYPE(primes,memalloc,unsigned long int); __GMP_FREE_FUNC_TYPE(exponent,memalloc,unsigned long int); mpz_clear(temp); return; }