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Diffstat (limited to 'src/gamma.c')
| -rw-r--r-- | src/gamma.c | 439 |
1 files changed, 439 insertions, 0 deletions
diff --git a/src/gamma.c b/src/gamma.c new file mode 100644 index 0000000..dbfacec --- /dev/null +++ b/src/gamma.c @@ -0,0 +1,439 @@ +/* mpfr_gamma -- gamma function + +Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel projects, INRIA. + +This file is part of the GNU MPFR Library. + +The GNU MPFR 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 3 of the License, or (at your +option) any later version. + +The GNU MPFR 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 GNU MPFR Library; see the file COPYING.LESSER. If not, see +http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., +51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ + +#define MPFR_NEED_LONGLONG_H +#include "mpfr-impl.h" + +#define IS_GAMMA +#include "lngamma.c" +#undef IS_GAMMA + +/* return a sufficient precision such that 2-x is exact, assuming x < 0 */ +static mpfr_prec_t +mpfr_gamma_2_minus_x_exact (mpfr_srcptr x) +{ + /* Since x < 0, 2-x = 2+y with y := -x. + If y < 2, a precision w >= PREC(y) + EXP(2)-EXP(y) = PREC(y) + 2 - EXP(y) + is enough, since no overlap occurs in 2+y, so no carry happens. + If y >= 2, either ULP(y) <= 2, and we need w >= PREC(y)+1 since a + carry can occur, or ULP(y) > 2, and we need w >= EXP(y)-1: + (a) if EXP(y) <= 1, w = PREC(y) + 2 - EXP(y) + (b) if EXP(y) > 1 and EXP(y)-PREC(y) <= 1, w = PREC(y) + 1 + (c) if EXP(y) > 1 and EXP(y)-PREC(y) > 1, w = EXP(y) - 1 */ + return (MPFR_GET_EXP(x) <= 1) ? MPFR_PREC(x) + 2 - MPFR_GET_EXP(x) + : ((MPFR_GET_EXP(x) <= MPFR_PREC(x) + 1) ? MPFR_PREC(x) + 1 + : MPFR_GET_EXP(x) - 1); +} + +/* return a sufficient precision such that 1-x is exact, assuming x < 1 */ +static mpfr_prec_t +mpfr_gamma_1_minus_x_exact (mpfr_srcptr x) +{ + if (MPFR_IS_POS(x)) + return MPFR_PREC(x) - MPFR_GET_EXP(x); + else if (MPFR_GET_EXP(x) <= 0) + return MPFR_PREC(x) + 1 - MPFR_GET_EXP(x); + else if (MPFR_PREC(x) >= MPFR_GET_EXP(x)) + return MPFR_PREC(x) + 1; + else + return MPFR_GET_EXP(x); +} + +/* returns a lower bound of the number of significant bits of n! + (not counting the low zero bits). + We know n! >= (n/e)^n*sqrt(2*Pi*n) for n >= 1, and the number of zero bits + is floor(n/2) + floor(n/4) + floor(n/8) + ... + This approximation is exact for n <= 500000, except for n = 219536, 235928, + 298981, 355854, 464848, 493725, 498992 where it returns a value 1 too small. +*/ +static unsigned long +bits_fac (unsigned long n) +{ + mpfr_t x, y; + unsigned long r, k; + mpfr_init2 (x, 38); + mpfr_init2 (y, 38); + mpfr_set_ui (x, n, MPFR_RNDZ); + mpfr_set_str_binary (y, "10.101101111110000101010001011000101001"); /* upper bound of e */ + mpfr_div (x, x, y, MPFR_RNDZ); + mpfr_pow_ui (x, x, n, MPFR_RNDZ); + mpfr_const_pi (y, MPFR_RNDZ); + mpfr_mul_ui (y, y, 2 * n, MPFR_RNDZ); + mpfr_sqrt (y, y, MPFR_RNDZ); + mpfr_mul (x, x, y, MPFR_RNDZ); + mpfr_log2 (x, x, MPFR_RNDZ); + r = mpfr_get_ui (x, MPFR_RNDU); + for (k = 2; k <= n; k *= 2) + r -= n / k; + mpfr_clear (x); + mpfr_clear (y); + return r; +} + +/* We use the reflection formula + Gamma(1+t) Gamma(1-t) = - Pi t / sin(Pi (1 + t)) + in order to treat the case x <= 1, + i.e. with x = 1-t, then Gamma(x) = -Pi*(1-x)/sin(Pi*(2-x))/GAMMA(2-x) +*/ +int +mpfr_gamma (mpfr_ptr gamma, mpfr_srcptr x, mpfr_rnd_t rnd_mode) +{ + mpfr_t xp, GammaTrial, tmp, tmp2; + mpz_t fact; + mpfr_prec_t realprec; + int compared, is_integer; + int inex = 0; /* 0 means: result gamma not set yet */ + MPFR_GROUP_DECL (group); + MPFR_SAVE_EXPO_DECL (expo); + MPFR_ZIV_DECL (loop); + + MPFR_LOG_FUNC + (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), + ("gamma[%Pu]=%.*Rg inexact=%d", + mpfr_get_prec (gamma), mpfr_log_prec, gamma, inex)); + + /* Trivial cases */ + if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) + { + if (MPFR_IS_NAN (x)) + { + MPFR_SET_NAN (gamma); + MPFR_RET_NAN; + } + else if (MPFR_IS_INF (x)) + { + if (MPFR_IS_NEG (x)) + { + MPFR_SET_NAN (gamma); + MPFR_RET_NAN; + } + else + { + MPFR_SET_INF (gamma); + MPFR_SET_POS (gamma); + MPFR_RET (0); /* exact */ + } + } + else /* x is zero */ + { + MPFR_ASSERTD(MPFR_IS_ZERO(x)); + MPFR_SET_INF(gamma); + MPFR_SET_SAME_SIGN(gamma, x); + mpfr_set_divby0 (); + MPFR_RET (0); /* exact */ + } + } + + /* Check for tiny arguments, where gamma(x) ~ 1/x - euler + .... + We know from "Bound on Runs of Zeros and Ones for Algebraic Functions", + Proceedings of Arith15, T. Lang and J.-M. Muller, 2001, that the maximal + number of consecutive zeroes or ones after the round bit is n-1 for an + input of n bits. But we need a more precise lower bound. Assume x has + n bits, and 1/x is near a floating-point number y of n+1 bits. We can + write x = X*2^e, y = Y/2^f with X, Y integers of n and n+1 bits. + Thus X*Y^2^(e-f) is near from 1, i.e., X*Y is near from 2^(f-e). + Two cases can happen: + (i) either X*Y is exactly 2^(f-e), but this can happen only if X and Y + are themselves powers of two, i.e., x is a power of two; + (ii) or X*Y is at distance at least one from 2^(f-e), thus + |xy-1| >= 2^(e-f), or |y-1/x| >= 2^(e-f)/x = 2^(-f)/X >= 2^(-f-n). + Since ufp(y) = 2^(n-f) [ufp = unit in first place], this means + that the distance |y-1/x| >= 2^(-2n) ufp(y). + Now assuming |gamma(x)-1/x| <= 1, which is true for x <= 1, + if 2^(-2n) ufp(y) >= 2, the error is at most 2^(-2n-1) ufp(y), + and round(1/x) with precision >= 2n+2 gives the correct result. + If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). + A sufficient condition is thus EXP(x) + 2 <= -2 MAX(PREC(x),PREC(Y)). + */ + if (MPFR_GET_EXP (x) + 2 + <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(gamma))) + { + int sign = MPFR_SIGN (x); /* retrieve sign before possible override */ + int special; + MPFR_BLOCK_DECL (flags); + + MPFR_SAVE_EXPO_MARK (expo); + + /* for overflow cases, see below; this needs to be done + before x possibly gets overridden. */ + special = + MPFR_GET_EXP (x) == 1 - MPFR_EMAX_MAX && + MPFR_IS_POS_SIGN (sign) && + MPFR_IS_LIKE_RNDD (rnd_mode, sign) && + mpfr_powerof2_raw (x); + + MPFR_BLOCK (flags, inex = mpfr_ui_div (gamma, 1, x, rnd_mode)); + if (inex == 0) /* x is a power of two */ + { + /* return RND(1/x - euler) = RND(+/- 2^k - eps) with eps > 0 */ + if (rnd_mode == MPFR_RNDN || MPFR_IS_LIKE_RNDU (rnd_mode, sign)) + inex = 1; + else + { + mpfr_nextbelow (gamma); + inex = -1; + } + } + else if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags))) + { + /* Overflow in the division 1/x. This is a real overflow, except + in RNDZ or RNDD when 1/x = 2^emax, i.e. x = 2^(-emax): due to + the "- euler", the rounded value in unbounded exponent range + is 0.111...11 * 2^emax (not an overflow). */ + if (!special) + MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, flags); + } + MPFR_SAVE_EXPO_FREE (expo); + /* Note: an overflow is possible with an infinite result; + in this case, the overflow flag will automatically be + restored by mpfr_check_range. */ + return mpfr_check_range (gamma, inex, rnd_mode); + } + + is_integer = mpfr_integer_p (x); + /* gamma(x) for x a negative integer gives NaN */ + if (is_integer && MPFR_IS_NEG(x)) + { + MPFR_SET_NAN (gamma); + MPFR_RET_NAN; + } + + compared = mpfr_cmp_ui (x, 1); + if (compared == 0) + return mpfr_set_ui (gamma, 1, rnd_mode); + + /* if x is an integer that fits into an unsigned long, use mpfr_fac_ui + if argument is not too large. + If precision is p, fac_ui costs O(u*p), whereas gamma costs O(p*M(p)), + so for u <= M(p), fac_ui should be faster. + We approximate here M(p) by p*log(p)^2, which is not a bad guess. + Warning: since the generic code does not handle exact cases, + we want all cases where gamma(x) is exact to be treated here. + */ + if (is_integer && mpfr_fits_ulong_p (x, MPFR_RNDN)) + { + unsigned long int u; + mpfr_prec_t p = MPFR_PREC(gamma); + u = mpfr_get_ui (x, MPFR_RNDN); + if (u < 44787929UL && bits_fac (u - 1) <= p + (rnd_mode == MPFR_RNDN)) + /* bits_fac: lower bound on the number of bits of m, + where gamma(x) = (u-1)! = m*2^e with m odd. */ + return mpfr_fac_ui (gamma, u - 1, rnd_mode); + /* if bits_fac(...) > p (resp. p+1 for rounding to nearest), + then gamma(x) cannot be exact in precision p (resp. p+1). + FIXME: remove the test u < 44787929UL after changing bits_fac + to return a mpz_t or mpfr_t. */ + } + + MPFR_SAVE_EXPO_MARK (expo); + + /* check for overflow: according to (6.1.37) in Abramowitz & Stegun, + gamma(x) >= exp(-x) * x^(x-1/2) * sqrt(2*Pi) + >= 2 * (x/e)^x / x for x >= 1 */ + if (compared > 0) + { + mpfr_t yp; + mpfr_exp_t expxp; + MPFR_BLOCK_DECL (flags); + + /* 1/e rounded down to 53 bits */ +#define EXPM1_STR "0.010111100010110101011000110110001011001110111100111" + mpfr_init2 (xp, 53); + mpfr_init2 (yp, 53); + mpfr_set_str_binary (xp, EXPM1_STR); + mpfr_mul (xp, x, xp, MPFR_RNDZ); + mpfr_sub_ui (yp, x, 2, MPFR_RNDZ); + mpfr_pow (xp, xp, yp, MPFR_RNDZ); /* (x/e)^(x-2) */ + mpfr_set_str_binary (yp, EXPM1_STR); + mpfr_mul (xp, xp, yp, MPFR_RNDZ); /* x^(x-2) / e^(x-1) */ + mpfr_mul (xp, xp, yp, MPFR_RNDZ); /* x^(x-2) / e^x */ + mpfr_mul (xp, xp, x, MPFR_RNDZ); /* lower bound on x^(x-1) / e^x */ + MPFR_BLOCK (flags, mpfr_mul_2ui (xp, xp, 1, MPFR_RNDZ)); + expxp = MPFR_GET_EXP (xp); + mpfr_clear (xp); + mpfr_clear (yp); + MPFR_SAVE_EXPO_FREE (expo); + return MPFR_OVERFLOW (flags) || expxp > __gmpfr_emax ? + mpfr_overflow (gamma, rnd_mode, 1) : + mpfr_gamma_aux (gamma, x, rnd_mode); + } + + /* now compared < 0 */ + + /* check for underflow: for x < 1, + gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x). + Since gamma(2-x) >= 2 * ((2-x)/e)^(2-x) / (2-x), we have + |gamma(x)| <= Pi*(1-x)*(2-x)/2/((2-x)/e)^(2-x) / |sin(Pi*(2-x))| + <= 12 * ((2-x)/e)^x / |sin(Pi*(2-x))|. + To avoid an underflow in ((2-x)/e)^x, we compute the logarithm. + */ + if (MPFR_IS_NEG(x)) + { + int underflow = 0, sgn, ck; + mpfr_prec_t w; + + mpfr_init2 (xp, 53); + mpfr_init2 (tmp, 53); + mpfr_init2 (tmp2, 53); + /* we want an upper bound for x * [log(2-x)-1]. + since x < 0, we need a lower bound on log(2-x) */ + mpfr_ui_sub (xp, 2, x, MPFR_RNDD); + mpfr_log (xp, xp, MPFR_RNDD); + mpfr_sub_ui (xp, xp, 1, MPFR_RNDD); + mpfr_mul (xp, xp, x, MPFR_RNDU); + + /* we need an upper bound on 1/|sin(Pi*(2-x))|, + thus a lower bound on |sin(Pi*(2-x))|. + If 2-x is exact, then the error of Pi*(2-x) is (1+u)^2 with u = 2^(-p) + thus the error on sin(Pi*(2-x)) is less than 1/2ulp + 3Pi(2-x)u, + assuming u <= 1, thus <= u + 3Pi(2-x)u */ + + w = mpfr_gamma_2_minus_x_exact (x); /* 2-x is exact for prec >= w */ + w += 17; /* to get tmp2 small enough */ + mpfr_set_prec (tmp, w); + mpfr_set_prec (tmp2, w); + ck = mpfr_ui_sub (tmp, 2, x, MPFR_RNDN); + MPFR_ASSERTD (ck == 0); (void) ck; /* use ck to avoid a warning */ + mpfr_const_pi (tmp2, MPFR_RNDN); + mpfr_mul (tmp2, tmp2, tmp, MPFR_RNDN); /* Pi*(2-x) */ + mpfr_sin (tmp, tmp2, MPFR_RNDN); /* sin(Pi*(2-x)) */ + sgn = mpfr_sgn (tmp); + mpfr_abs (tmp, tmp, MPFR_RNDN); + mpfr_mul_ui (tmp2, tmp2, 3, MPFR_RNDU); /* 3Pi(2-x) */ + mpfr_add_ui (tmp2, tmp2, 1, MPFR_RNDU); /* 3Pi(2-x)+1 */ + mpfr_div_2ui (tmp2, tmp2, mpfr_get_prec (tmp), MPFR_RNDU); + /* if tmp2<|tmp|, we get a lower bound */ + if (mpfr_cmp (tmp2, tmp) < 0) + { + mpfr_sub (tmp, tmp, tmp2, MPFR_RNDZ); /* low bnd on |sin(Pi*(2-x))| */ + mpfr_ui_div (tmp, 12, tmp, MPFR_RNDU); /* upper bound */ + mpfr_log2 (tmp, tmp, MPFR_RNDU); + mpfr_add (xp, tmp, xp, MPFR_RNDU); + /* The assert below checks that expo.saved_emin - 2 always + fits in a long. FIXME if we want to allow mpfr_exp_t to + be a long long, for instance. */ + MPFR_ASSERTN (MPFR_EMIN_MIN - 2 >= LONG_MIN); + underflow = mpfr_cmp_si (xp, expo.saved_emin - 2) <= 0; + } + + mpfr_clear (xp); + mpfr_clear (tmp); + mpfr_clear (tmp2); + if (underflow) /* the sign is the opposite of that of sin(Pi*(2-x)) */ + { + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_underflow (gamma, (rnd_mode == MPFR_RNDN) ? MPFR_RNDZ : rnd_mode, -sgn); + } + } + + realprec = MPFR_PREC (gamma); + /* we want both 1-x and 2-x to be exact */ + { + mpfr_prec_t w; + w = mpfr_gamma_1_minus_x_exact (x); + if (realprec < w) + realprec = w; + w = mpfr_gamma_2_minus_x_exact (x); + if (realprec < w) + realprec = w; + } + realprec = realprec + MPFR_INT_CEIL_LOG2 (realprec) + 20; + MPFR_ASSERTD(realprec >= 5); + + MPFR_GROUP_INIT_4 (group, realprec + MPFR_INT_CEIL_LOG2 (realprec) + 20, + xp, tmp, tmp2, GammaTrial); + mpz_init (fact); + MPFR_ZIV_INIT (loop, realprec); + for (;;) + { + mpfr_exp_t err_g; + int ck; + MPFR_GROUP_REPREC_4 (group, realprec, xp, tmp, tmp2, GammaTrial); + + /* reflection formula: gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x) */ + + ck = mpfr_ui_sub (xp, 2, x, MPFR_RNDN); /* 2-x, exact */ + MPFR_ASSERTD(ck == 0); (void) ck; /* use ck to avoid a warning */ + mpfr_gamma (tmp, xp, MPFR_RNDN); /* gamma(2-x), error (1+u) */ + mpfr_const_pi (tmp2, MPFR_RNDN); /* Pi, error (1+u) */ + mpfr_mul (GammaTrial, tmp2, xp, MPFR_RNDN); /* Pi*(2-x), error (1+u)^2 */ + err_g = MPFR_GET_EXP(GammaTrial); + mpfr_sin (GammaTrial, GammaTrial, MPFR_RNDN); /* sin(Pi*(2-x)) */ + /* If tmp is +Inf, we compute exp(lngamma(x)). */ + if (mpfr_inf_p (tmp)) + { + inex = mpfr_explgamma (gamma, x, &expo, tmp, tmp2, rnd_mode); + if (inex) + goto end; + else + goto ziv_next; + } + err_g = err_g + 1 - MPFR_GET_EXP(GammaTrial); + /* let g0 the true value of Pi*(2-x), g the computed value. + We have g = g0 + h with |h| <= |(1+u^2)-1|*g. + Thus sin(g) = sin(g0) + h' with |h'| <= |(1+u^2)-1|*g. + The relative error is thus bounded by |(1+u^2)-1|*g/sin(g) + <= |(1+u^2)-1|*2^err_g. <= 2.25*u*2^err_g for |u|<=1/4. + With the rounding error, this gives (0.5 + 2.25*2^err_g)*u. */ + ck = mpfr_sub_ui (xp, x, 1, MPFR_RNDN); /* x-1, exact */ + MPFR_ASSERTD(ck == 0); (void) ck; /* use ck to avoid a warning */ + mpfr_mul (xp, tmp2, xp, MPFR_RNDN); /* Pi*(x-1), error (1+u)^2 */ + mpfr_mul (GammaTrial, GammaTrial, tmp, MPFR_RNDN); + /* [1 + (0.5 + 2.25*2^err_g)*u]*(1+u)^2 = 1 + (2.5 + 2.25*2^err_g)*u + + (0.5 + 2.25*2^err_g)*u*(2u+u^2) + u^2. + For err_g <= realprec-2, we have (0.5 + 2.25*2^err_g)*u <= + 0.5*u + 2.25/4 <= 0.6875 and u^2 <= u/4, thus + (0.5 + 2.25*2^err_g)*u*(2u+u^2) + u^2 <= 0.6875*(2u+u/4) + u/4 + <= 1.8*u, thus the rel. error is bounded by (4.5 + 2.25*2^err_g)*u. */ + mpfr_div (GammaTrial, xp, GammaTrial, MPFR_RNDN); + /* the error is of the form (1+u)^3/[1 + (4.5 + 2.25*2^err_g)*u]. + For realprec >= 5 and err_g <= realprec-2, [(4.5 + 2.25*2^err_g)*u]^2 + <= 0.71, and for |y|<=0.71, 1/(1-y) can be written 1+a*y with a<=4. + (1+u)^3 * (1+4*(4.5 + 2.25*2^err_g)*u) + = 1 + (21 + 9*2^err_g)*u + (57+27*2^err_g)*u^2 + (55+27*2^err_g)*u^3 + + (18+9*2^err_g)*u^4 + <= 1 + (21 + 9*2^err_g)*u + (57+27*2^err_g)*u^2 + (56+28*2^err_g)*u^3 + <= 1 + (21 + 9*2^err_g)*u + (59+28*2^err_g)*u^2 + <= 1 + (23 + 10*2^err_g)*u. + The final error is thus bounded by (23 + 10*2^err_g) ulps, + which is <= 2^6 for err_g<=2, and <= 2^(err_g+4) for err_g >= 2. */ + err_g = (err_g <= 2) ? 6 : err_g + 4; + + if (MPFR_LIKELY (MPFR_CAN_ROUND (GammaTrial, realprec - err_g, + MPFR_PREC(gamma), rnd_mode))) + break; + + ziv_next: + MPFR_ZIV_NEXT (loop, realprec); + } + + end: + MPFR_ZIV_FREE (loop); + + if (inex == 0) + inex = mpfr_set (gamma, GammaTrial, rnd_mode); + MPFR_GROUP_CLEAR (group); + mpz_clear (fact); + + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (gamma, inex, rnd_mode); +} |