--- /dev/null
+/* 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);
+}
projects / msp430-gcc.git / blobdiff