/* mpfr_lngamma -- lngamma function
Copyright 2005, 2006, 2007 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.
This file is part of the MPFR Library.
The 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 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 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 *B;
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))
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;
mpfr_init2 (l, MPFR_PREC(y) + 14);
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);
}
}
#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
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