/* mpfr_gamma -- gamma function Copyright 2001-2019 Free Software Foundation, Inc. Contributed by the AriC and Caramba 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 https://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 and x is not an integer */ 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. Note: case (c) cannot happen in practice since this would imply that y is integer, thus x is negative integer */ return (MPFR_GET_EXP(x) <= 1) ? MPFR_PREC(x) + 2 - MPFR_GET_EXP(x) : MPFR_PREC(x) + 1; } /* return a sufficient precision such that 1-x is exact, assuming x < 1 and x is not an integer */ 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 /* necessarily MPFR_PREC(x) > MPFR_GET_EXP(x) since otherwise x would be an integer */ return MPFR_PREC(x) + 1; } /* 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_SAVE_EXPO_DECL (expo); MPFR_ASSERTD (n >= 1); MPFR_SAVE_EXPO_MARK (expo); 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); /* lower bound on ceil(x) */ for (k = 2; k <= n; k *= 2) { /* Note: the approximation is accurate enough so that the subtractions do not wrap. */ MPFR_ASSERTD (r >= n / k); r -= n / k; } mpfr_clear (x); mpfr_clear (y); MPFR_SAVE_EXPO_FREE (expo); 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 + ... can be approximated by 1/x, with some error term ~= - euler. We need to make sure that there are no breakpoints (discontinuity points of the rounding function) between gamma(x) and 1/x (included), where the possible breakpoints (for all rounding modes) are the numbers that fit on PREC(gamma)+1 bits. There will be a special case when |x| is a power of two, since such values are breakpoints. We will choose n minimum such that x fits on n bits and the breakpoints fit on n+1 bits, thus n = MAX(MPFR_PREC(x), MPFR_PREC(gamma)). 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 for 1/x is n-1 for an input x of n bits [this is an actually much older result!]. But we need a more precise lower bound. Assume that 1/x is near a breakpoint y. From the definition of n, the input x fits on n bits and the breakpoint y fits on 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 respectively. Thus X*Y^2^(e-f) is near 1, i.e., X*Y is near the integer 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 0 < x <= 1, if 2^(-2n) ufp(y) >= 1, then gamma(x) and 1/x round in the same way, so that rounding 1/x gives the correct result and correct (nonzero) ternary value. If x < 2^E, then y >= 2^(-E), thus ufp(y) >= 2^(-E). A sufficient condition is thus EXP(x) <= -2n, where n = MAX(MPFR_PREC(x), MPFR_PREC(gamma)). */ /* TODO: The above proof uses the same precision for input and output. Without this assumption, one might obtain a bound like PREC(x) + PREC(y) instead of 2 MAX(PREC(x),PREC(y)). */ /* TODO: Handle the very small arguments that do not satisfy the condition, by using the approximation 1/x - euler and a Ziv loop. Otherwise, after some tests, even Gamma(1+x)/x would be faster than the generic code. */ if (MPFR_GET_EXP (x) <= -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); MPFR_ASSERTD (u >= 2); 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, zp; mpfr_exp_t expxp; MPFR_BLOCK_DECL (flags); MPFR_GROUP_DECL (group); /* quick test for the default exponent range */ if (mpfr_get_emax () >= 1073741823UL && MPFR_GET_EXP(x) <= 25) { MPFR_SAVE_EXPO_FREE (expo); return mpfr_gamma_aux (gamma, x, rnd_mode); } MPFR_GROUP_INIT_3 (group, 53, xp, yp, zp); /* 1/e rounded down to 53 bits */ mpfr_set_str_binary (zp, "0.010111100010110101011000110110001011001110111100111"); mpfr_mul (xp, x, zp, MPFR_RNDZ); mpfr_sub_ui (yp, x, 2, MPFR_RNDZ); mpfr_pow (xp, xp, yp, MPFR_RNDZ); /* (x/e)^(x-2) */ mpfr_mul (xp, xp, zp, MPFR_RNDZ); /* x^(x-2) / e^(x-1) */ mpfr_mul (xp, xp, zp, 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_GROUP_CLEAR (group); 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); MPFR_DBGRES (ck = mpfr_ui_sub (tmp, 2, x, MPFR_RNDN)); MPFR_ASSERTD (ck == 0); /* tmp = 2-x exactly */ 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); }