| /* mpfr_gamma -- gamma function |
| |
| Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. |
| Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, 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 mp_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 mp_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, GMP_RNDZ); |
| mpfr_set_str_binary (y, "10.101101111110000101010001011000101001"); /* upper bound of e */ |
| mpfr_div (x, x, y, GMP_RNDZ); |
| mpfr_pow_ui (x, x, n, GMP_RNDZ); |
| mpfr_const_pi (y, GMP_RNDZ); |
| mpfr_mul_ui (y, y, 2 * n, GMP_RNDZ); |
| mpfr_sqrt (y, y, GMP_RNDZ); |
| mpfr_mul (x, x, y, GMP_RNDZ); |
| mpfr_log2 (x, x, GMP_RNDZ); |
| r = mpfr_get_ui (x, GMP_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, mp_rnd_t rnd_mode) |
| { |
| mpfr_t xp, GammaTrial, tmp, tmp2; |
| mpz_t fact; |
| mp_prec_t realprec; |
| int compared, inex, is_integer; |
| MPFR_GROUP_DECL (group); |
| MPFR_SAVE_EXPO_DECL (expo); |
| MPFR_ZIV_DECL (loop); |
| |
| MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode), |
| ("gamma[%#R]=%R inexact=%d", gamma, 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_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_EXP(x) + 2 <= -2 * (mp_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(gamma))) |
| { |
| int positive = MPFR_IS_POS (x); |
| inex = mpfr_ui_div (gamma, 1, x, rnd_mode); |
| if (inex == 0) /* x is a power of two */ |
| { |
| if (positive) |
| { |
| if (rnd_mode == GMP_RNDU || rnd_mode == GMP_RNDN) |
| inex = 1; |
| else /* round to zero or to -Inf */ |
| { |
| mpfr_nextbelow (gamma); /* 2^k - epsilon */ |
| inex = -1; |
| } |
| } |
| else /* negative */ |
| { |
| if (rnd_mode == GMP_RNDU || rnd_mode == GMP_RNDZ) |
| { |
| mpfr_nextabove (gamma); /* -2^k + epsilon */ |
| inex = 1; |
| } |
| else /* round to nearest and to -Inf */ |
| inex = -1; |
| } |
| } |
| return inex; |
| } |
| |
| 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, GMP_RNDN)) |
| { |
| unsigned long int u; |
| mp_prec_t p = MPFR_PREC(gamma); |
| u = mpfr_get_ui (x, GMP_RNDN); |
| if (u < 44787929UL && bits_fac (u - 1) <= p + (rnd_mode == GMP_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. */ |
| } |
| |
| /* 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_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, GMP_RNDZ); |
| mpfr_sub_ui (yp, x, 2, GMP_RNDZ); |
| mpfr_pow (xp, xp, yp, GMP_RNDZ); /* (x/e)^(x-2) */ |
| mpfr_set_str_binary (yp, EXPM1_STR); |
| mpfr_mul (xp, xp, yp, GMP_RNDZ); /* x^(x-2) / e^(x-1) */ |
| mpfr_mul (xp, xp, yp, GMP_RNDZ); /* x^(x-2) / e^x */ |
| mpfr_mul (xp, xp, x, GMP_RNDZ); /* lower bound on x^(x-1) / e^x */ |
| MPFR_BLOCK (flags, mpfr_mul_2ui (xp, xp, 1, GMP_RNDZ)); |
| mpfr_clear (xp); |
| mpfr_clear (yp); |
| return MPFR_OVERFLOW (flags) ? mpfr_overflow (gamma, rnd_mode, 1) |
| : mpfr_gamma_aux (gamma, x, rnd_mode); |
| } |
| |
| /* now compared < 0 */ |
| |
| MPFR_SAVE_EXPO_MARK (expo); |
| |
| /* 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; |
| mp_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, GMP_RNDD); |
| mpfr_log (xp, xp, GMP_RNDD); |
| mpfr_sub_ui (xp, xp, 1, GMP_RNDD); |
| mpfr_mul (xp, xp, x, GMP_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, GMP_RNDN); |
| MPFR_ASSERTD (ck == 0); |
| mpfr_const_pi (tmp2, GMP_RNDN); |
| mpfr_mul (tmp2, tmp2, tmp, GMP_RNDN); /* Pi*(2-x) */ |
| mpfr_sin (tmp, tmp2, GMP_RNDN); /* sin(Pi*(2-x)) */ |
| sgn = mpfr_sgn (tmp); |
| mpfr_abs (tmp, tmp, GMP_RNDN); |
| mpfr_mul_ui (tmp2, tmp2, 3, GMP_RNDU); /* 3Pi(2-x) */ |
| mpfr_add_ui (tmp2, tmp2, 1, GMP_RNDU); /* 3Pi(2-x)+1 */ |
| mpfr_div_2ui (tmp2, tmp2, mpfr_get_prec (tmp), GMP_RNDU); |
| /* if tmp2<|tmp|, we get a lower bound */ |
| if (mpfr_cmp (tmp2, tmp) < 0) |
| { |
| mpfr_sub (tmp, tmp, tmp2, GMP_RNDZ); /* low bnd on |sin(Pi*(2-x))| */ |
| mpfr_ui_div (tmp, 12, tmp, GMP_RNDU); /* upper bound */ |
| mpfr_log (tmp, tmp, GMP_RNDU); |
| mpfr_add (tmp, tmp, xp, GMP_RNDU); |
| 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 == GMP_RNDN) ? GMP_RNDZ : rnd_mode, -sgn); |
| } |
| } |
| |
| realprec = MPFR_PREC (gamma); |
| /* we want both 1-x and 2-x to be exact */ |
| { |
| mp_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 (;;) |
| { |
| mp_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, GMP_RNDN); |
| MPFR_ASSERTD(ck == 0); /* 2-x, exact */ |
| mpfr_gamma (tmp, xp, GMP_RNDN); /* gamma(2-x), error (1+u) */ |
| mpfr_const_pi (tmp2, GMP_RNDN); /* Pi, error (1+u) */ |
| mpfr_mul (GammaTrial, tmp2, xp, GMP_RNDN); /* Pi*(2-x), error (1+u)^2 */ |
| err_g = MPFR_GET_EXP(GammaTrial); |
| mpfr_sin (GammaTrial, GammaTrial, GMP_RNDN); /* sin(Pi*(2-x)) */ |
| 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, GMP_RNDN); |
| MPFR_ASSERTD(ck == 0); /* x-1, exact */ |
| mpfr_mul (xp, tmp2, xp, GMP_RNDN); /* Pi*(x-1), error (1+u)^2 */ |
| mpfr_mul (GammaTrial, GammaTrial, tmp, GMP_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, GMP_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; |
| MPFR_ZIV_NEXT (loop, realprec); |
| } |
| MPFR_ZIV_FREE (loop); |
| |
| 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); |
| } |
