--- /dev/null
+/* mpfr_lngamma -- lngamma function
+
+Copyright 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"
+
+/* assuming b[0]...b[2(n-1)] are computed, computes and stores B[2n]*(2n+1)!
+
+ t/(exp(t)-1) = sum(B[j]*t^j/j!, j=0..infinity)
+ thus t = (exp(t)-1) * sum(B[j]*t^j/j!, n=0..infinity).
+ Taking the coefficient of degree n+1 > 1, we get:
+ 0 = sum(1/(n+1-k)!*B[k]/k!, k=0..n)
+ which gives:
+ B[n] = -sum(binomial(n+1,k)*B[k], k=0..n-1)/(n+1).
+
+ Let C[n] = B[n]*(n+1)!.
+ Then C[n] = -sum(binomial(n+1,k)*C[k]*n!/(k+1)!, k=0..n-1),
+ which proves that the C[n] are integers.
+*/
+static mpz_t*
+bernoulli (mpz_t *b, unsigned long n)
+{
+ if (n == 0)
+ {
+ b = (mpz_t *) (*__gmp_allocate_func) (sizeof (mpz_t));
+ mpz_init_set_ui (b[0], 1);
+ }
+ else
+ {
+ mpz_t t;
+ unsigned long k;
+
+ b = (mpz_t *) (*__gmp_reallocate_func)
+ (b, n * sizeof (mpz_t), (n + 1) * sizeof (mpz_t));
+ mpz_init (b[n]);
+ /* b[n] = -sum(binomial(2n+1,2k)*C[k]*(2n)!/(2k+1)!, k=0..n-1) */
+ mpz_init_set_ui (t, 2 * n + 1);
+ mpz_mul_ui (t, t, 2 * n - 1);
+ mpz_mul_ui (t, t, 2 * n);
+ mpz_mul_ui (t, t, n);
+ mpz_div_ui (t, t, 3); /* exact: t=binomial(2*n+1,2*k)*(2*n)!/(2*k+1)!
+ for k=n-1 */
+ mpz_mul (b[n], t, b[n-1]);
+ for (k = n - 1; k-- > 0;)
+ {
+ mpz_mul_ui (t, t, 2 * k + 1);
+ mpz_mul_ui (t, t, 2 * k + 2);
+ mpz_mul_ui (t, t, 2 * k + 2);
+ mpz_mul_ui (t, t, 2 * k + 3);
+ mpz_div_ui (t, t, 2 * (n - k) + 1);
+ mpz_div_ui (t, t, 2 * (n - k));
+ mpz_addmul (b[n], t, b[k]);
+ }
+ /* take into account C[1] */
+ mpz_mul_ui (t, t, 2 * n + 1);
+ mpz_div_2exp (t, t, 1);
+ mpz_sub (b[n], b[n], t);
+ mpz_neg (b[n], b[n]);
+ mpz_clear (t);
+ }
+ return b;
+}
+
+/* given a precision p, return alpha, such that the argument reduction
+ will use k = alpha*p*log(2).
+
+ Warning: we should always have alpha >= log(2)/(2Pi) ~ 0.11,
+ and the smallest value of alpha multiplied by the smallest working
+ precision should be >= 4.
+*/
+static double
+mpfr_gamma_alpha (mp_prec_t p)
+{
+ if (p <= 100)
+ return 0.6;
+ else if (p <= 200)
+ return 0.8;
+ else if (p <= 500)
+ return 0.8;
+ else if (p <= 1000)
+ return 1.3;
+ else if (p <= 2000)
+ return 1.7;
+ else if (p <= 5000)
+ return 2.2;
+ else if (p <= 10000)
+ return 3.4;
+ else /* heuristic fit from above */
+ return 0.26 * (double) MPFR_INT_CEIL_LOG2 ((unsigned long) p);
+}
+
+#ifndef IS_GAMMA
+static int
+unit_bit (mpfr_srcptr (x))
+{
+ mp_exp_t expo;
+ mp_prec_t prec;
+ mp_limb_t x0;
+
+ expo = MPFR_GET_EXP (x);
+ if (expo <= 0)
+ return 0; /* |x| < 1 */
+
+ prec = MPFR_PREC (x);
+ if (expo > prec)
+ return 0; /* y is a multiple of 2^(expo-prec), thus an even integer */
+
+ /* Now, the unit bit is represented. */
+
+ prec = ((prec - 1) / BITS_PER_MP_LIMB + 1) * BITS_PER_MP_LIMB - expo;
+ /* number of represented fractional bits (including the trailing 0's) */
+
+ x0 = *(MPFR_MANT (x) + prec / BITS_PER_MP_LIMB);
+ /* limb containing the unit bit */
+
+ return (x0 >> (prec % BITS_PER_MP_LIMB)) & 1;
+}
+#endif
+
+/* lngamma(x) = log(gamma(x)).
+ We use formula [6.1.40] from Abramowitz&Stegun:
+ lngamma(z) = (z-1/2)*log(z) - z + 1/2*log(2*Pi)
+ + sum (Bernoulli[2n]/(2m)/(2m-1)/z^(2m-1),m=1..infinity)
+ According to [6.1.42], if the sum is truncated after m=n, the error
+ R_n(z) is bounded by |B[2n+2]|*K(z)/(2n+1)/(2n+2)/|z|^(2n+1)
+ where K(z) = max (z^2/(u^2+z^2)) for u >= 0.
+ For z real, |K(z)| <= 1 thus R_n(z) is bounded by the first neglected term.
+ */
+#ifdef IS_GAMMA
+#define GAMMA_FUNC mpfr_gamma_aux
+#else
+#define GAMMA_FUNC mpfr_lngamma_aux
+#endif
+
+static int
+GAMMA_FUNC (mpfr_ptr y, mpfr_srcptr z0, mp_rnd_t rnd)
+{
+ mp_prec_t precy, w; /* working precision */
+ mpfr_t s, t, u, v, z;
+ unsigned long m, k, maxm;
+ mpz_t *INITIALIZED(B); /* variable B declared as initialized */
+ int inexact, compared;
+ mp_exp_t err_s, err_t;
+ unsigned long Bm = 0; /* number of allocated B[] */
+ unsigned long oldBm;
+ double d;
+ MPFR_SAVE_EXPO_DECL (expo);
+
+ compared = mpfr_cmp_ui (z0, 1);
+
+ MPFR_SAVE_EXPO_MARK (expo);
+
+#ifndef IS_GAMMA /* lngamma or lgamma */
+ if (compared == 0 || (compared > 0 && mpfr_cmp_ui (z0, 2) == 0))
+ {
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_set_ui (y, 0, GMP_RNDN); /* lngamma(1 or 2) = +0 */
+ }
+
+ /* Deal here with tiny inputs. We have for -0.3 <= x <= 0.3:
+ - log|x| - gamma*x <= log|gamma(x)| <= - log|x| - gamma*x + x^2 */
+ if (MPFR_EXP(z0) <= - (mp_exp_t) MPFR_PREC(y))
+ {
+ mpfr_t l, h, g;
+ int ok, inex2;
+ mp_prec_t prec = MPFR_PREC(y) + 14;
+ MPFR_ZIV_DECL (loop);
+
+ MPFR_ZIV_INIT (loop, prec);
+ do
+ {
+ mpfr_init2 (l, prec);
+ if (MPFR_IS_POS(z0))
+ {
+ mpfr_log (l, z0, GMP_RNDU); /* upper bound for log(z0) */
+ mpfr_init2 (h, MPFR_PREC(l));
+ }
+ else
+ {
+ mpfr_init2 (h, MPFR_PREC(z0));
+ mpfr_neg (h, z0, GMP_RNDN); /* exact */
+ mpfr_log (l, h, GMP_RNDU); /* upper bound for log(-z0) */
+ mpfr_set_prec (h, MPFR_PREC(l));
+ }
+ mpfr_neg (l, l, GMP_RNDD); /* lower bound for -log(|z0|) */
+ mpfr_set (h, l, GMP_RNDD); /* exact */
+ mpfr_nextabove (h); /* upper bound for -log(|z0|), avoids two calls
+ to mpfr_log */
+ mpfr_init2 (g, MPFR_PREC(l));
+ /* if z0>0, we need an upper approximation of Euler's constant
+ for the left bound */
+ mpfr_const_euler (g, MPFR_IS_POS(z0) ? GMP_RNDU : GMP_RNDD);
+ mpfr_mul (g, g, z0, GMP_RNDD);
+ mpfr_sub (l, l, g, GMP_RNDD);
+ mpfr_const_euler (g, MPFR_IS_POS(z0) ? GMP_RNDD : GMP_RNDU); /* cached */
+ mpfr_mul (g, g, z0, GMP_RNDU);
+ mpfr_sub (h, h, g, GMP_RNDD);
+ mpfr_mul (g, z0, z0, GMP_RNDU);
+ mpfr_add (h, h, g, GMP_RNDU);
+ inexact = mpfr_prec_round (l, MPFR_PREC(y), rnd);
+ inex2 = mpfr_prec_round (h, MPFR_PREC(y), rnd);
+ /* Caution: we not only need l = h, but both inexact flags should
+ agree. Indeed, one of the inexact flags might be zero. In that
+ case if we assume lngamma(z0) cannot be exact, the other flag
+ should be correct. We are conservative here and request that both
+ inexact flags agree. */
+ ok = SAME_SIGN (inexact, inex2) && mpfr_cmp (l, h) == 0;
+ if (ok)
+ mpfr_set (y, h, rnd); /* exact */
+ mpfr_clear (l);
+ mpfr_clear (h);
+ mpfr_clear (g);
+ if (ok)
+ {
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd);
+ }
+ /* since we have log|gamma(x)| = - log|x| - gamma*x + O(x^2),
+ if x ~ 2^(-n), then we have a n-bit approximation, thus
+ we can try again with a working precision of n bits,
+ especially when n >> PREC(y).
+ Otherwise we would use the reflection formula evaluating x-1,
+ which would need precision n. */
+ MPFR_ZIV_NEXT (loop, prec);
+ }
+ while (prec <= -MPFR_EXP(z0));
+ MPFR_ZIV_FREE (loop);
+ }
+#endif
+
+ precy = MPFR_PREC(y);
+
+ mpfr_init2 (s, MPFR_PREC_MIN);
+ mpfr_init2 (t, MPFR_PREC_MIN);
+ mpfr_init2 (u, MPFR_PREC_MIN);
+ mpfr_init2 (v, MPFR_PREC_MIN);
+ mpfr_init2 (z, MPFR_PREC_MIN);
+
+ if (compared < 0)
+ {
+ mp_exp_t err_u;
+
+ /* use reflection formula:
+ gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x)
+ thus lngamma(x) = log(Pi*(x-1)/sin(Pi*(2-x))) - lngamma(2-x) */
+
+ w = precy + MPFR_INT_CEIL_LOG2 (precy);
+ while (1)
+ {
+ w += MPFR_INT_CEIL_LOG2 (w) + 14;
+ MPFR_ASSERTD(w >= 3);
+ mpfr_set_prec (s, w);
+ mpfr_set_prec (t, w);
+ mpfr_set_prec (u, w);
+ mpfr_set_prec (v, w);
+ /* In the following, we write r for a real of absolute value
+ at most 2^(-w). Different instances of r may represent different
+ values. */
+ mpfr_ui_sub (s, 2, z0, GMP_RNDD); /* s = (2-z0) * (1+2r) >= 1 */
+ mpfr_const_pi (t, GMP_RNDN); /* t = Pi * (1+r) */
+ mpfr_lngamma (u, s, GMP_RNDN); /* lngamma(2-x) */
+ /* Let s = (2-z0) + h. By construction, -(2-z0)*2^(1-w) <= h <= 0.
+ We have lngamma(s) = lngamma(2-z0) + h*Psi(z), z in [2-z0+h,2-z0].
+ Since 2-z0+h = s >= 1 and |Psi(x)| <= max(1,log(x)) for x >= 1,
+ the error on u is bounded by
+ ulp(u)/2 + (2-z0)*max(1,log(2-z0))*2^(1-w)
+ = (1/2 + (2-z0)*max(1,log(2-z0))*2^(1-E(u))) ulp(u) */
+ d = (double) MPFR_GET_EXP(s) * 0.694; /* upper bound for log(2-z0) */
+ err_u = MPFR_GET_EXP(s) + __gmpfr_ceil_log2 (d) + 1 - MPFR_GET_EXP(u);
+ err_u = (err_u >= 0) ? err_u + 1 : 0;
+ /* now the error on u is bounded by 2^err_u ulps */
+
+ mpfr_mul (s, s, t, GMP_RNDN); /* Pi*(2-x) * (1+r)^4 */
+ err_s = MPFR_GET_EXP(s); /* 2-x <= 2^err_s */
+ mpfr_sin (s, s, GMP_RNDN); /* sin(Pi*(2-x)) */
+ /* the error on s is bounded by 1/2*ulp(s) + [(1+2^(-w))^4-1]*(2-x)
+ <= 1/2*ulp(s) + 5*2^(-w)*(2-x) for w >= 3
+ <= (1/2 + 5 * 2^(-E(s)) * (2-x)) ulp(s) */
+ err_s += 3 - MPFR_GET_EXP(s);
+ err_s = (err_s >= 0) ? err_s + 1 : 0;
+ /* the error on s is bounded by 2^err_s ulp(s), thus by
+ 2^(err_s+1)*2^(-w)*|s| since ulp(s) <= 2^(1-w)*|s|.
+ Now n*2^(-w) can always be written |(1+r)^n-1| for some
+ |r|<=2^(-w), thus taking n=2^(err_s+1) we see that
+ |S - s| <= |(1+r)^(2^(err_s+1))-1| * |s|, where S is the
+ true value.
+ In fact if ulp(s) <= ulp(S) the same inequality holds for
+ |S| instead of |s| in the right hand side, i.e., we can
+ write s = (1+r)^(2^(err_s+1)) * S.
+ But if ulp(S) < ulp(s), we need to add one ``bit'' to the error,
+ to get s = (1+r)^(2^(err_s+2)) * S. This is true since with
+ E = n*2^(-w) we have |s - S| <= E * |s|, thus
+ |s - S| <= E/(1-E) * |S|.
+ Now E/(1-E) is bounded by 2E as long as E<=1/2,
+ and 2E can be written (1+r)^(2n)-1 as above.
+ */
+ err_s += 2; /* exponent of relative error */
+
+ mpfr_sub_ui (v, z0, 1, GMP_RNDN); /* v = (x-1) * (1+r) */
+ mpfr_mul (v, v, t, GMP_RNDN); /* v = Pi*(x-1) * (1+r)^3 */
+ mpfr_div (v, v, s, GMP_RNDN); /* Pi*(x-1)/sin(Pi*(2-x)) */
+ mpfr_abs (v, v, GMP_RNDN);
+ /* (1+r)^(3+2^err_s+1) */
+ err_s = (err_s <= 1) ? 3 : err_s + 1;
+ /* now (1+r)^M with M <= 2^err_s */
+ mpfr_log (v, v, GMP_RNDN);
+ /* log(v*(1+e)) = log(v)+log(1+e) where |e| <= 2^(err_s-w).
+ Since |log(1+e)| <= 2*e for |e| <= 1/4, the error on v is
+ bounded by ulp(v)/2 + 2^(err_s+1-w). */
+ if (err_s + 2 > w)
+ {
+ w += err_s + 2;
+ }
+ else
+ {
+ err_s += 1 - MPFR_GET_EXP(v);
+ err_s = (err_s >= 0) ? err_s + 1 : 0;
+ /* the error on v is bounded by 2^err_s ulps */
+ err_u += MPFR_GET_EXP(u); /* absolute error on u */
+ err_s += MPFR_GET_EXP(v); /* absolute error on v */
+ mpfr_sub (s, v, u, GMP_RNDN);
+ /* the total error on s is bounded by ulp(s)/2 + 2^(err_u-w)
+ + 2^(err_s-w) <= ulp(s)/2 + 2^(max(err_u,err_s)+1-w) */
+ err_s = (err_s >= err_u) ? err_s : err_u;
+ err_s += 1 - MPFR_GET_EXP(s); /* error is 2^err_s ulp(s) */
+ err_s = (err_s >= 0) ? err_s + 1 : 0;
+ if (mpfr_can_round (s, w - err_s, GMP_RNDN, GMP_RNDZ, precy
+ + (rnd == GMP_RNDN)))
+ goto end;
+ }
+ }
+ }
+
+ /* now z0 > 1 */
+
+ MPFR_ASSERTD (compared > 0);
+
+ /* since k is O(w), the value of log(z0*...*(z0+k-1)) is about w*log(w),
+ so there is a cancellation of ~log(w) in the argument reconstruction */
+ w = precy + MPFR_INT_CEIL_LOG2 (precy);
+
+ do
+ {
+ w += MPFR_INT_CEIL_LOG2 (w) + 13;
+ MPFR_ASSERTD (w >= 3);
+
+ mpfr_set_prec (s, 53);
+ /* we need z >= w*log(2)/(2*Pi) to get an absolute error less than 2^(-w)
+ but the optimal value is about 0.155665*w */
+ /* FIXME: replace double by mpfr_t types. */
+ mpfr_set_d (s, mpfr_gamma_alpha (precy) * (double) w, GMP_RNDU);
+ if (mpfr_cmp (z0, s) < 0)
+ {
+ mpfr_sub (s, s, z0, GMP_RNDU);
+ k = mpfr_get_ui (s, GMP_RNDU);
+ if (k < 3)
+ k = 3;
+ }
+ else
+ k = 3;
+
+ mpfr_set_prec (s, w);
+ mpfr_set_prec (t, w);
+ mpfr_set_prec (u, w);
+ mpfr_set_prec (v, w);
+ mpfr_set_prec (z, w);
+
+ mpfr_add_ui (z, z0, k, GMP_RNDN);
+ /* z = (z0+k)*(1+t1) with |t1| <= 2^(-w) */
+
+ /* z >= 4 ensures the relative error on log(z) is small,
+ and also (z-1/2)*log(z)-z >= 0 */
+ MPFR_ASSERTD (mpfr_cmp_ui (z, 4) >= 0);
+
+ mpfr_log (s, z, GMP_RNDN); /* log(z) */
+ /* we have s = log((z0+k)*(1+t1))*(1+t2) with |t1|, |t2| <= 2^(-w).
+ Since w >= 2 and z0+k >= 4, we can write log((z0+k)*(1+t1))
+ = log(z0+k) * (1+t3) with |t3| <= 2^(-w), thus we have
+ s = log(z0+k) * (1+t4)^2 with |t4| <= 2^(-w) */
+ mpfr_mul_2ui (t, z, 1, GMP_RNDN); /* t = 2z * (1+t5) */
+ mpfr_sub_ui (t, t, 1, GMP_RNDN); /* t = 2z-1 * (1+t6)^3 */
+ /* since we can write 2z*(1+t5) = (2z-1)*(1+t5') with
+ t5' = 2z/(2z-1) * t5, thus |t5'| <= 8/7 * t5 */
+ mpfr_mul (s, s, t, GMP_RNDN); /* (2z-1)*log(z) * (1+t7)^6 */
+ mpfr_div_2ui (s, s, 1, GMP_RNDN); /* (z-1/2)*log(z) * (1+t7)^6 */
+ mpfr_sub (s, s, z, GMP_RNDN); /* (z-1/2)*log(z)-z */
+ /* s = [(z-1/2)*log(z)-z]*(1+u)^14, s >= 1/2 */
+
+ mpfr_ui_div (u, 1, z, GMP_RNDN); /* 1/z * (1+u), u <= 1/4 since z >= 4 */
+
+ /* the first term is B[2]/2/z = 1/12/z: t=1/12/z, C[2]=1 */
+ mpfr_div_ui (t, u, 12, GMP_RNDN); /* 1/(12z) * (1+u)^2, t <= 3/128 */
+ mpfr_set (v, t, GMP_RNDN); /* (1+u)^2, v < 2^(-5) */
+ mpfr_add (s, s, v, GMP_RNDN); /* (1+u)^15 */
+
+ mpfr_mul (u, u, u, GMP_RNDN); /* 1/z^2 * (1+u)^3 */
+
+ if (Bm == 0)
+ {
+ B = bernoulli ((mpz_t *) 0, 0);
+ B = bernoulli (B, 1);
+ Bm = 2;
+ }
+
+ /* m <= maxm ensures that 2*m*(2*m+1) <= ULONG_MAX */
+ maxm = 1UL << (BITS_PER_MP_LIMB / 2 - 1);
+
+ /* s:(1+u)^15, t:(1+u)^2, t <= 3/128 */
+
+ for (m = 2; MPFR_GET_EXP(v) + (mp_exp_t) w >= MPFR_GET_EXP(s); m++)
+ {
+ mpfr_mul (t, t, u, GMP_RNDN); /* (1+u)^(10m-14) */
+ if (m <= maxm)
+ {
+ mpfr_mul_ui (t, t, 2*(m-1)*(2*m-3), GMP_RNDN);
+ mpfr_div_ui (t, t, 2*m*(2*m-1), GMP_RNDN);
+ mpfr_div_ui (t, t, 2*m*(2*m+1), GMP_RNDN);
+ }
+ else
+ {
+ mpfr_mul_ui (t, t, 2*(m-1), GMP_RNDN);
+ mpfr_mul_ui (t, t, 2*m-3, GMP_RNDN);
+ mpfr_div_ui (t, t, 2*m, GMP_RNDN);
+ mpfr_div_ui (t, t, 2*m-1, GMP_RNDN);
+ mpfr_div_ui (t, t, 2*m, GMP_RNDN);
+ mpfr_div_ui (t, t, 2*m+1, GMP_RNDN);
+ }
+ /* (1+u)^(10m-8) */
+ /* invariant: t=1/(2m)/(2m-1)/z^(2m-1)/(2m+1)! */
+ if (Bm <= m)
+ {
+ B = bernoulli (B, m); /* B[2m]*(2m+1)!, exact */
+ Bm ++;
+ }
+ mpfr_mul_z (v, t, B[m], GMP_RNDN); /* (1+u)^(10m-7) */
+ MPFR_ASSERTD(MPFR_GET_EXP(v) <= - (2 * m + 3));
+ mpfr_add (s, s, v, GMP_RNDN);
+ }
+ /* m <= 1/2*Pi*e*z ensures that |v[m]| < 1/2^(2m+3) */
+ MPFR_ASSERTD ((double) m <= 4.26 * mpfr_get_d (z, GMP_RNDZ));
+
+ /* We have sum([(1+u)^(10m-7)-1]*1/2^(2m+3), m=2..infinity)
+ <= 1.46*u for u <= 2^(-3).
+ We have 0 < lngamma(z) - [(z - 1/2) ln(z) - z + 1/2 ln(2 Pi)] < 0.021
+ for z >= 4, thus since the initial s >= 0.85, the different values of
+ s differ by at most one binade, and the total rounding error on s
+ in the for-loop is bounded by 2*(m-1)*ulp(final_s).
+ The error coming from the v's is bounded by
+ 1.46*2^(-w) <= 2*ulp(final_s).
+ Thus the total error so far is bounded by [(1+u)^15-1]*s+2m*ulp(s)
+ <= (2m+47)*ulp(s).
+ Taking into account the truncation error (which is bounded by the last
+ term v[] according to 6.1.42 in A&S), the bound is (2m+48)*ulp(s).
+ */
+
+ /* add 1/2*log(2*Pi) and subtract log(z0*(z0+1)*...*(z0+k-1)) */
+ mpfr_const_pi (v, GMP_RNDN); /* v = Pi*(1+u) */
+ mpfr_mul_2ui (v, v, 1, GMP_RNDN); /* v = 2*Pi * (1+u) */
+ if (k)
+ {
+ unsigned long l;
+ mpfr_set (t, z0, GMP_RNDN); /* t = z0*(1+u) */
+ for (l = 1; l < k; l++)
+ {
+ mpfr_add_ui (u, z0, l, GMP_RNDN); /* u = (z0+l)*(1+u) */
+ mpfr_mul (t, t, u, GMP_RNDN); /* (1+u)^(2l+1) */
+ }
+ /* now t: (1+u)^(2k-1) */
+ /* instead of computing log(sqrt(2*Pi)/t), we compute
+ 1/2*log(2*Pi/t^2), which trades a square root for a square */
+ mpfr_mul (t, t, t, GMP_RNDN); /* (z0*...*(z0+k-1))^2, (1+u)^(4k-1) */
+ mpfr_div (v, v, t, GMP_RNDN);
+ /* 2*Pi/(z0*...*(z0+k-1))^2 (1+u)^(4k+1) */
+ }
+#ifdef IS_GAMMA
+ err_s = MPFR_GET_EXP(s);
+ mpfr_exp (s, s, GMP_RNDN);
+ /* before the exponential, we have s = s0 + h where
+ |h| <= (2m+48)*ulp(s), thus exp(s0) = exp(s) * exp(-h).
+ For |h| <= 1/4, we have |exp(h)-1| <= 1.2*|h| thus
+ |exp(s) - exp(s0)| <= 1.2 * exp(s) * (2m+48)* 2^(EXP(s)-w). */
+ d = 1.2 * (2.0 * (double) m + 48.0);
+ /* the error on s is bounded by d*2^err_s * 2^(-w) */
+ mpfr_sqrt (t, v, GMP_RNDN);
+ /* let v0 be the exact value of v. We have v = v0*(1+u)^(4k+1),
+ thus t = sqrt(v0)*(1+u)^(2k+3/2). */
+ mpfr_mul (s, s, t, GMP_RNDN);
+ /* the error on input s is bounded by (1+u)^(d*2^err_s),
+ and that on t is (1+u)^(2k+3/2), thus the
+ total error is (1+u)^(d*2^err_s+2k+5/2) */
+ err_s += __gmpfr_ceil_log2 (d);
+ err_t = __gmpfr_ceil_log2 (2.0 * (double) k + 2.5);
+ err_s = (err_s >= err_t) ? err_s + 1 : err_t + 1;
+#else
+ mpfr_log (t, v, GMP_RNDN);
+ /* let v0 be the exact value of v. We have v = v0*(1+u)^(4k+1),
+ thus log(v) = log(v0) + (4k+1)*log(1+u). Since |log(1+u)/u| <= 1.07
+ for |u| <= 2^(-3), the absolute error on log(v) is bounded by
+ 1.07*(4k+1)*u, and the rounding error by ulp(t). */
+ mpfr_div_2ui (t, t, 1, GMP_RNDN);
+ /* the error on t is now bounded by ulp(t) + 0.54*(4k+1)*2^(-w).
+ We have sqrt(2*Pi)/(z0*(z0+1)*...*(z0+k-1)) <= sqrt(2*Pi)/k! <= 0.5
+ since k>=3, thus t <= -0.5 and ulp(t) >= 2^(-w).
+ Thus the error on t is bounded by (2.16*k+1.54)*ulp(t). */
+ err_t = MPFR_GET_EXP(t) + (mp_exp_t)
+ __gmpfr_ceil_log2 (2.2 * (double) k + 1.6);
+ err_s = MPFR_GET_EXP(s) + (mp_exp_t)
+ __gmpfr_ceil_log2 (2.0 * (double) m + 48.0);
+ mpfr_add (s, s, t, GMP_RNDN); /* this is a subtraction in fact */
+ /* the final error in ulp(s) is
+ <= 1 + 2^(err_t-EXP(s)) + 2^(err_s-EXP(s))
+ <= 2^(1+max(err_t,err_s)-EXP(s)) if err_t <> err_s
+ <= 2^(2+max(err_t,err_s)-EXP(s)) if err_t = err_s */
+ err_s = (err_t == err_s) ? 1 + err_s : ((err_t > err_s) ? err_t : err_s);
+ err_s += 1 - MPFR_GET_EXP(s);
+#endif
+ }
+ while (MPFR_UNLIKELY (!MPFR_CAN_ROUND (s, w - err_s, precy, rnd)));
+
+ oldBm = Bm;
+ while (Bm--)
+ mpz_clear (B[Bm]);
+ (*__gmp_free_func) (B, oldBm * sizeof (mpz_t));
+
+ end:
+ inexact = mpfr_set (y, s, rnd);
+
+ mpfr_clear (s);
+ mpfr_clear (t);
+ mpfr_clear (u);
+ mpfr_clear (v);
+ mpfr_clear (z);
+
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd);
+}
+
+#ifndef IS_GAMMA
+
+int
+mpfr_lngamma (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd)
+{
+ int inex;
+
+ MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd),
+ ("lngamma[%#R]=%R inexact=%d", y, y, inex));
+
+ /* special cases */
+ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
+ {
+ if (MPFR_IS_NAN (x) || MPFR_IS_NEG (x))
+ {
+ MPFR_SET_NAN (y);
+ MPFR_RET_NAN;
+ }
+ else /* lngamma(+Inf) = lngamma(+0) = +Inf */
+ {
+ MPFR_SET_INF (y);
+ MPFR_SET_POS (y);
+ MPFR_RET (0); /* exact */
+ }
+ }
+
+ /* if x < 0 and -2k-1 <= x <= -2k, then lngamma(x) = NaN */
+ if (MPFR_IS_NEG (x) && (unit_bit (x) == 0 || mpfr_integer_p (x)))
+ {
+ MPFR_SET_NAN (y);
+ MPFR_RET_NAN;
+ }
+
+ inex = mpfr_lngamma_aux (y, x, rnd);
+ return inex;
+}
+
+int
+mpfr_lgamma (mpfr_ptr y, int *signp, mpfr_srcptr x, mp_rnd_t rnd)
+{
+ int inex;
+
+ MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd),
+ ("lgamma[%#R]=%R inexact=%d", y, y, inex));
+
+ *signp = 1; /* most common case */
+
+ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
+ {
+ if (MPFR_IS_NAN (x))
+ {
+ MPFR_SET_NAN (y);
+ MPFR_RET_NAN;
+ }
+ else
+ {
+ *signp = MPFR_INT_SIGN (x);
+ MPFR_SET_INF (y);
+ MPFR_SET_POS (y);
+ MPFR_RET (0);
+ }
+ }
+
+ if (MPFR_IS_NEG (x))
+ {
+ if (mpfr_integer_p (x))
+ {
+ MPFR_SET_INF (y);
+ MPFR_SET_POS (y);
+ MPFR_RET (0);
+ }
+
+ if (unit_bit (x) == 0)
+ *signp = -1;
+
+ /* For tiny negative x, we have gamma(x) = 1/x - euler + O(x),
+ thus |gamma(x)| = -1/x + euler + O(x), and
+ log |gamma(x)| = -log(-x) - euler*x + O(x^2).
+ More precisely we have for -0.4 <= x < 0:
+ -log(-x) <= log |gamma(x)| <= -log(-x) - x.
+ Since log(x) is not representable, we may have an instance of the
+ Table Maker Dilemma. The only way to ensure correct rounding is to
+ compute an interval [l,h] such that l <= -log(-x) and
+ -log(-x) - x <= h, and check whether l and h round to the same number
+ for the target precision and rounding modes. */
+ if (MPFR_EXP(x) + 1 <= - (mp_exp_t) MPFR_PREC(y))
+ /* since PREC(y) >= 1, this ensures EXP(x) <= -2,
+ thus |x| <= 0.25 < 0.4 */
+ {
+ mpfr_t l, h;
+ int ok, inex2;
+ mp_prec_t w = MPFR_PREC (y) + 14;
+
+ while (1)
+ {
+ mpfr_init2 (l, w);
+ mpfr_init2 (h, w);
+ /* we want a lower bound on -log(-x), thus an upper bound
+ on log(-x), thus an upper bound on -x. */
+ mpfr_neg (l, x, GMP_RNDU); /* upper bound on -x */
+ mpfr_log (l, l, GMP_RNDU); /* upper bound for log(-x) */
+ mpfr_neg (l, l, GMP_RNDD); /* lower bound for -log(-x) */
+ mpfr_neg (h, x, GMP_RNDD); /* lower bound on -x */
+ mpfr_log (h, h, GMP_RNDD); /* lower bound on log(-x) */
+ mpfr_neg (h, h, GMP_RNDU); /* upper bound for -log(-x) */
+ mpfr_sub (h, h, x, GMP_RNDU); /* upper bound for -log(-x) - x */
+ inex = mpfr_prec_round (l, MPFR_PREC (y), rnd);
+ inex2 = mpfr_prec_round (h, MPFR_PREC (y), rnd);
+ /* Caution: we not only need l = h, but both inexact flags
+ should agree. Indeed, one of the inexact flags might be
+ zero. In that case if we assume ln|gamma(x)| cannot be
+ exact, the other flag should be correct. We are conservative
+ here and request that both inexact flags agree. */
+ ok = SAME_SIGN (inex, inex2) && mpfr_equal_p (l, h);
+ if (ok)
+ mpfr_set (y, h, rnd); /* exact */
+ mpfr_clear (l);
+ mpfr_clear (h);
+ if (ok)
+ return inex;
+ /* if ulp(log(-x)) <= |x| there is no reason to loop,
+ since the width of [l, h] will be at least |x| */
+ if (MPFR_EXP(l) < MPFR_EXP(x) + (mp_exp_t) w)
+ break;
+ w += MPFR_INT_CEIL_LOG2(w) + 3;
+ }
+ }
+ }
+
+ inex = mpfr_lngamma_aux (y, x, rnd);
+ return inex;
+}
+
+#endif
projects / msp430-gcc.git / blobdiff