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/* mpfr_lngamma -- lngamma function
Copyright 2005-2020 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"
/* 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 void
mpfr_gamma_alpha (mpfr_t s, mpfr_prec_t p)
{
MPFR_LOG_FUNC
(("p=%Pu", p),
("s[%Pu]=%.*Rg", mpfr_get_prec (s), mpfr_log_prec, s));
if (p <= 100)
mpfr_set_ui_2exp (s, 614, -10, MPFR_RNDN); /* about 0.6 */
else if (p <= 500)
mpfr_set_ui_2exp (s, 819, -10, MPFR_RNDN); /* about 0.8 */
else if (p <= 1000)
mpfr_set_ui_2exp (s, 1331, -10, MPFR_RNDN); /* about 1.3 */
else if (p <= 2000)
mpfr_set_ui_2exp (s, 1741, -10, MPFR_RNDN); /* about 1.7 */
else if (p <= 5000)
mpfr_set_ui_2exp (s, 2253, -10, MPFR_RNDN); /* about 2.2 */
else if (p <= 10000)
mpfr_set_ui_2exp (s, 3482, -10, MPFR_RNDN); /* about 3.4 */
else
mpfr_set_ui_2exp (s, 9, -1, MPFR_RNDN); /* 4.5 */
}
#ifdef IS_GAMMA
/* This function is called in case of intermediate overflow/underflow.
The s1 and s2 arguments are temporary MPFR numbers, having the
working precision. If the result could be determined, then the
flags are updated via pexpo, y is set to the result, and the
(non-zero) ternary value is returned. Otherwise 0 is returned
in order to perform the next Ziv iteration. */
static int
mpfr_explgamma (mpfr_ptr y, mpfr_srcptr x, mpfr_save_expo_t *pexpo,
mpfr_ptr s1, mpfr_ptr s2, mpfr_rnd_t rnd)
{
mpfr_t t1, t2;
int inex1, inex2, sign;
MPFR_BLOCK_DECL (flags1);
MPFR_BLOCK_DECL (flags2);
MPFR_GROUP_DECL (group);
MPFR_BLOCK (flags1, inex1 = mpfr_lgamma (s1, &sign, x, MPFR_RNDD));
MPFR_ASSERTN (inex1 != 0);
/* s1 = RNDD(lngamma(x)), inexact */
if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags1)))
{
if (MPFR_IS_POS (s1))
{
MPFR_SAVE_EXPO_UPDATE_FLAGS (*pexpo, MPFR_FLAGS_OVERFLOW);
return mpfr_overflow (y, rnd, sign);
}
else
{
MPFR_SAVE_EXPO_UPDATE_FLAGS (*pexpo, MPFR_FLAGS_UNDERFLOW);
return mpfr_underflow (y, rnd == MPFR_RNDN ? MPFR_RNDZ : rnd, sign);
}
}
mpfr_set (s2, s1, MPFR_RNDN); /* exact */
mpfr_nextabove (s2); /* v = RNDU(lngamma(z0)) */
if (sign < 0)
rnd = MPFR_INVERT_RND (rnd); /* since the result with be negated */
MPFR_GROUP_INIT_2 (group, MPFR_PREC (y), t1, t2);
MPFR_BLOCK (flags1, inex1 = mpfr_exp (t1, s1, rnd));
MPFR_BLOCK (flags2, inex2 = mpfr_exp (t2, s2, rnd));
/* t1 is the rounding with mode 'rnd' of a lower bound on |Gamma(x)|,
t2 is the rounding with mode 'rnd' of an upper bound, thus if both
are equal, so is the wanted result. If t1 and t2 differ or the flags
differ, at some point of Ziv's loop they should agree. */
if (mpfr_equal_p (t1, t2) && flags1 == flags2)
{
MPFR_ASSERTN ((inex1 > 0 && inex2 > 0) || (inex1 < 0 && inex2 < 0));
mpfr_set4 (y, t1, MPFR_RNDN, sign); /* exact */
if (sign < 0)
inex1 = - inex1;
MPFR_SAVE_EXPO_UPDATE_FLAGS (*pexpo, flags1);
}
else
inex1 = 0; /* couldn't determine the result */
MPFR_GROUP_CLEAR (group);
return inex1;
}
#else
static int
unit_bit (mpfr_srcptr x)
{
mpfr_exp_t expo;
mpfr_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 = MPFR_PREC2LIMBS (prec) * GMP_NUMB_BITS - expo;
/* number of represented fractional bits (including the trailing 0's) */
x0 = *(MPFR_MANT (x) + prec / GMP_NUMB_BITS);
/* limb containing the unit bit */
return (x0 >> (prec % GMP_NUMB_BITS)) & 1;
}
#endif
/* FIXME: There is an internal overflow when z is very large.
Simple overflow detection with possible false negatives?
For the particular cases near the overflow boundary,
scaling by a power of two?
*/
/* 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[2m]/(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, mpfr_rnd_t rnd)
{
mpfr_prec_t precy, w; /* working precision */
mpfr_t s, t, u, v, z;
unsigned long m, k, maxm, l;
int compared, inexact;
mpfr_exp_t err_s, err_t;
double d;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (z0), mpfr_log_prec, z0, rnd),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, inexact));
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, MPFR_RNDN); /* lngamma(1 or 2) = +0 */
}
/* Deal with very large inputs: according to [6.1.42], if we denote
R_n(z) = lngamma(z) - (z-1/2)*log(z) + z - 1/2*log(2*Pi), we have
|R_n(z)| <= B_2/2/z, thus for z >= 2 we have
|lngamma(z) - [z*(log(z) - 1)]| < 1/2*log(z) + 1. */
if (compared > 0 && MPFR_GET_EXP (z0) >= (mpfr_exp_t) MPFR_PREC(y) + 2)
{
/* Since PREC(y) >= 2, this ensures EXP(z0) >= 4, thus |z0| >= 8,
thus 1/2*log(z0) + 1 < log(z0).
Since the largest possible z is < 2^(2^62) on a 64-bit machine,
the largest value of log(z) is 2^62*log(2.) < 3.2e18 < 2^62,
thus if we use at least 62 bits of precision, then log(t)-1 will
be exact */
mpfr_init2 (t, MPFR_PREC(y) >= 52 ? MPFR_PREC(y) + 10 : 62);
mpfr_log (t, z0, MPFR_RNDU); /* error < 1 ulp */
inexact = mpfr_sub_ui (t, t, 1, MPFR_RNDU); /* err < 2 ulps, since the
exponent of t might have
decreased */
MPFR_ASSERTD(inexact == 0);
mpfr_mul (t, z0, t, MPFR_RNDU); /* err < 1+2*2=5 ulps according to
"Generic error on multiplication"
in algorithms.tex */
if (MPFR_IS_INF(t))
{
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
inexact = mpfr_overflow (y, rnd, 1);
return inexact;
}
if (MPFR_GET_EXP(t) - MPFR_PREC(t) >= 62)
{
/* then ulp(t) >= 2^62 > log(z0) thus the total error is bounded
by 6 ulp(t) */
if (MPFR_CAN_ROUND (t, MPFR_PREC(t) - 3, MPFR_PREC(y), rnd))
{
inexact = mpfr_set (y, t, rnd);
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd);
}
}
mpfr_clear (t);
}
/* 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_GET_EXP (z0) <= - (mpfr_exp_t) MPFR_PREC(y))
{
mpfr_t l, h, g;
int ok, inex1, inex2;
mpfr_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, MPFR_RNDU); /* upper bound for log(z0) */
mpfr_init2 (h, MPFR_PREC(l));
}
else
{
mpfr_init2 (h, MPFR_PREC(z0));
mpfr_neg (h, z0, MPFR_RNDN); /* exact */
mpfr_log (l, h, MPFR_RNDU); /* upper bound for log(-z0) */
mpfr_set_prec (h, MPFR_PREC(l));
}
mpfr_neg (l, l, MPFR_RNDD); /* lower bound for -log(|z0|) */
mpfr_set (h, l, MPFR_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) ? MPFR_RNDU : MPFR_RNDD);
mpfr_mul (g, g, z0, MPFR_RNDD);
mpfr_sub (l, l, g, MPFR_RNDD);
mpfr_const_euler (g, MPFR_IS_POS(z0) ? MPFR_RNDD : MPFR_RNDU); /* cached */
mpfr_mul (g, g, z0, MPFR_RNDU);
mpfr_sub (h, h, g, MPFR_RNDD);
mpfr_sqr (g, z0, MPFR_RNDU);
mpfr_add (h, h, g, MPFR_RNDU);
inex1 = 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 (inex1, 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_ZIV_FREE (loop);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex1, 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_GET_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);
inexact = 0; /* 0 means: result y not set yet */
if (compared < 0)
{
mpfr_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);
w += MPFR_INT_CEIL_LOG2 (w) + 14;
MPFR_ZIV_INIT (loop, w);
while (1)
{
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, MPFR_RNDD); /* s = (2-z0) * (1+2r) >= 1 */
mpfr_const_pi (t, MPFR_RNDN); /* t = Pi * (1+r) */
mpfr_lngamma (u, s, MPFR_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) */
if (MPFR_IS_ZERO(u)) /* in that case the error on u is zero */
err_u = 0;
else
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, MPFR_RNDN); /* Pi*(2-x) * (1+r)^4 */
err_s = MPFR_GET_EXP(s); /* 2-x <= 2^err_s */
mpfr_sin (s, s, MPFR_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, MPFR_RNDN); /* v = (x-1) * (1+r) */
mpfr_mul (v, v, t, MPFR_RNDN); /* v = Pi*(x-1) * (1+r)^3 */
mpfr_div (v, v, s, MPFR_RNDN); /* Pi*(x-1)/sin(Pi*(2-x)) */
mpfr_abs (v, v, MPFR_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, MPFR_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
{
/* if v = 0 here, it was 1 before the call to mpfr_log,
thus the error on v was zero */
if (!MPFR_IS_ZERO(v))
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 */
if (!MPFR_IS_ZERO(v)) /* same as above */
err_s += MPFR_GET_EXP(v); /* absolute error on v */
mpfr_sub (s, v, u, MPFR_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, precy, rnd))
goto end;
}
MPFR_ZIV_NEXT (loop, w);
}
MPFR_ZIV_FREE (loop);
}
/* 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);
w += MPFR_INT_CEIL_LOG2 (w) + 13;
MPFR_ZIV_INIT (loop, w);
while (1)
{
MPFR_ASSERTD (w >= 3);
/* argument reduction: we compute gamma(z0 + k), where the series
has error term B_{2n}/(z0+k)^(2n) ~ (n/(Pi*e*(z0+k)))^(2n)
and we need k steps of argument reconstruction. Assuming k is large
with respect to z0, and k = n, we get 1/(Pi*e)^(2n) ~ 2^(-w), i.e.,
k ~ w*log(2)/2/log(Pi*e) ~ 0.1616 * w.
However, since the series is slightly more expensive to compute,
the optimal value seems to be k ~ 0.25 * w experimentally (with
caching of Bernoulli numbers).
For only one computation of gamma with large precision, it is better
to set k to a larger value, say k ~ w. */
mpfr_set_prec (s, 53);
mpfr_gamma_alpha (s, w);
mpfr_set_ui_2exp (s, 4, -4, MPFR_RNDU);
mpfr_mul_ui (s, s, w, MPFR_RNDU);
if (mpfr_cmp (z0, s) < 0)
{
mpfr_sub (s, s, z0, MPFR_RNDU);
k = mpfr_get_ui (s, MPFR_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, MPFR_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, MPFR_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, MPFR_RNDN); /* t = 2z * (1+t5) */
mpfr_sub_ui (t, t, 1, MPFR_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, MPFR_RNDN); /* (2z-1)*log(z) * (1+t7)^6 */
mpfr_div_2ui (s, s, 1, MPFR_RNDN); /* (z-1/2)*log(z) * (1+t7)^6 */
mpfr_sub (s, s, z, MPFR_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, MPFR_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, MPFR_RNDN); /* 1/(12z) * (1+u)^2, t <= 3/128 */
mpfr_set (v, t, MPFR_RNDN); /* (1+u)^2, v < 2^(-5) */
mpfr_add (s, s, v, MPFR_RNDN); /* (1+u)^15 */
mpfr_sqr (u, u, MPFR_RNDN); /* 1/z^2 * (1+u)^3 */
/* m <= maxm ensures that 2*m*(2*m+1) <= ULONG_MAX */
maxm = 1UL << (sizeof(unsigned long) * CHAR_BIT / 2 - 1);
/* s:(1+u)^15, t:(1+u)^2, t <= 3/128 */
for (m = 2; MPFR_GET_EXP(v) + (mpfr_exp_t) w >= MPFR_GET_EXP(s); m++)
{
mpfr_mul (t, t, u, MPFR_RNDN); /* (1+u)^(10m-14) */
if (m <= maxm)
{
mpfr_mul_ui (t, t, 2*(m-1)*(2*m-3), MPFR_RNDN);
mpfr_div_ui (t, t, 2*m*(2*m-1), MPFR_RNDN);
mpfr_div_ui (t, t, 2*m*(2*m+1), MPFR_RNDN);
}
else
{
mpfr_mul_ui (t, t, 2*(m-1), MPFR_RNDN);
mpfr_mul_ui (t, t, 2*m-3, MPFR_RNDN);
mpfr_div_ui (t, t, 2*m, MPFR_RNDN);
mpfr_div_ui (t, t, 2*m-1, MPFR_RNDN);
mpfr_div_ui (t, t, 2*m, MPFR_RNDN);
mpfr_div_ui (t, t, 2*m+1, MPFR_RNDN);
}
/* (1+u)^(10m-8) */
/* invariant: t=1/(2m)/(2m-1)/z^(2m-1)/(2m+1)! */
mpfr_mul_z (v, t, mpfr_bernoulli_cache(m), MPFR_RNDN); /* (1+u)^(10m-7) */
MPFR_ASSERTD(MPFR_GET_EXP(v) <= - (2 * m + 3));
mpfr_add (s, s, v, MPFR_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, MPFR_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, MPFR_RNDN); /* v = Pi*(1+u) */
mpfr_mul_2ui (v, v, 1, MPFR_RNDN); /* v = 2*Pi * (1+u) */
/* k >= 3 */
mpfr_set (t, z0, MPFR_RNDN); /* t = z0*(1+u) */
l = 1;
/* replace #if 1 by #if 0 for the naive argument reconstruction */
#if 1
/* We multiply by (z0+1)*(z0+2)*...*(z0+k-1) by blocks of j consecutive
terms where j ~ sqrt(k).
If we multiply naively by z0+1, then by z0+2, ..., then by z0+j,
the multiplicative term for the rounding error is (1+u)^(2j).
The multiplicative term is not larger when we multiply by
Z[j] + c[j-1]*Z[j-1] + ... + c[2]*Z[2] + c[1]*z0 + c[0]
with c[p] integers, and Z[p] = z0^p * (1+u)^(p-1).
Note that all terms are positive.
Indeed, since c[1] is exact, c[1]*z0 corresponds to (1+u),
then c[1]*z0 + c[0] corresponds to (1+u)^2,
c[2]*Z[2] + c[1]*z0 + c[0] to (1+u)^3, ...,
c[j-1]*Z[j-1] + ... + c[0] to (1+u)^j,
and Z[j] + c[j-1]*Z[j-1] + ... + c[1]*z0 + c[0] to (1+u)^(j+1).
With the accumulation in t, we get (1+u)^(j+2) and j+2 <= 2j. */
{
unsigned long j, i, p;
mpfr_t *Z;
mpz_t *c;
for (j = 2; (j + 1) * (j + 1) < k; j++);
/* Z[i] stores z0^i for i <= j */
Z = (mpfr_t *) mpfr_allocate_func ((j + 1) * sizeof (mpfr_t));
for (i = 2; i <= j; i++)
mpfr_init2 (Z[i], w);
mpfr_sqr (Z[2], z0, MPFR_RNDN);
for (i = 3; i <= j; i++)
if ((i & 1) == 0)
mpfr_sqr (Z[i], Z[i >> 1], MPFR_RNDN);
else
mpfr_mul (Z[i], Z[i-1], z0, MPFR_RNDN);
c = (mpz_t *) mpfr_allocate_func ((j + 1) * sizeof (mpz_t));
for (i = 0; i <= j; i++)
mpz_init (c[i]);
for (; l + j <= k; l += j)
{
/* c[i] is the coefficient of x^i in (x+l)*...*(x+l+j-1) */
mpz_set_ui (c[0], 1);
for (i = 0; i < j; i++)
/* multiply (x+l)*(x+l+1)*...*(x+l+i-1) by x+l+i:
(b[i]*x^i + b[i-1]*x^(i-1) + ... + b[0])*(x+l+i) =
b[i]*x^(i+1) + (b[i-1]+(l+i)*b[i])*x^i + ...
+ (b[0]+(l+i)*b[1])*x + i*b[0] */
{
mpz_set (c[i+1], c[i]); /* b[i]*x^(i+1) */
for (p = i; p > 0; p--)
{
mpz_mul_ui (c[p], c[p], l + i);
mpz_add (c[p], c[p], c[p-1]); /* b[p-1]+(l+i)*b[p] */
}
mpz_mul_ui (c[0], c[0], l+i); /* i*b[0] */
}
/* now compute z0^j + c[j-1]*z0^(j-1) + ... + c[1]*z0 + c[0] */
mpfr_set_z (u, c[0], MPFR_RNDN);
for (i = 0; i < j; i++)
{
mpfr_mul_z (z, (i == 0) ? z0 : Z[i+1], c[i+1], MPFR_RNDN);
mpfr_add (u, u, z, MPFR_RNDN);
}
mpfr_mul (t, t, u, MPFR_RNDN);
}
for (i = 0; i <= j; i++)
mpz_clear (c[i]);
mpfr_free_func (c, (j + 1) * sizeof (mpz_t));
for (i = 2; i <= j; i++)
mpfr_clear (Z[i]);
mpfr_free_func (Z, (j + 1) * sizeof (mpfr_t));
}
#endif /* end of fast argument reconstruction */
for (; l < k; l++)
{
mpfr_add_ui (u, z0, l, MPFR_RNDN); /* u = (z0+l)*(1+u) */
mpfr_mul (t, t, u, MPFR_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_sqr (t, t, MPFR_RNDN); /* (z0*...*(z0+k-1))^2, (1+u)^(4k-1) */
mpfr_div (v, v, t, MPFR_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, MPFR_RNDN);
/* If s is +Inf, we compute exp(lngamma(z0)). */
if (mpfr_inf_p (s))
{
inexact = mpfr_explgamma (y, z0, &expo, s, t, rnd);
if (inexact)
goto end0;
else
goto ziv_next;
}
/* 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, MPFR_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, MPFR_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); */
/* since d = 1.2 * (2m+48), ceil(log2(d)) = 2 + ceil(log2(0.6*m+14.4))
<= 2 + ceil(log2(0.6*m+15)) */
{
unsigned long mm = (1 + m / 5) * 3; /* 0.6*m <= mm */
err_s += 2 + __gmpfr_int_ceil_log2 (mm + 15);
}
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, MPFR_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, MPFR_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) + (mpfr_exp_t)
__gmpfr_ceil_log2 (2.2 * (double) k + 1.6);
err_s = MPFR_GET_EXP(s) + (mpfr_exp_t)
__gmpfr_ceil_log2 (2.0 * (double) m + 48.0);
mpfr_add (s, s, t, MPFR_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
if (MPFR_LIKELY (MPFR_CAN_ROUND (s, w - err_s, precy, rnd)))
break;
#ifdef IS_GAMMA
ziv_next:
#endif
MPFR_ZIV_NEXT (loop, w);
}
#ifdef IS_GAMMA
end0:
#endif
end:
if (inexact == 0)
inexact = mpfr_set (y, s, rnd);
MPFR_ZIV_FREE (loop);
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, mpfr_rnd_t rnd)
{
int inex;
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, inex));
/* special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x) ||
(MPFR_IS_NEG (x) && mpfr_integer_p (x))))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* lngamma(+/-Inf) = lngamma(nonpositive integer) = +Inf */
{
if (!MPFR_IS_INF (x))
MPFR_SET_DIVBY0 ();
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0); /* exact */
}
}
/* if -2k-1 < x < -2k <= 0, then lngamma(x) = NaN */
if (MPFR_IS_NEG (x) && unit_bit (x) == 0)
{
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, mpfr_rnd_t rnd)
{
int inex;
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd),
("y[%Pu]=%.*Rg signp=%d inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, *signp, 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
{
if (MPFR_IS_ZERO (x))
MPFR_SET_DIVBY0 ();
*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_SET_DIVBY0 ();
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 <= - (mpfr_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;
mpfr_prec_t w = MPFR_PREC (y) + 14;
mpfr_exp_t expl;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_SAVE_EXPO_MARK (expo);
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, MPFR_RNDU); /* upper bound on -x */
mpfr_log (l, l, MPFR_RNDU); /* upper bound for log(-x) */
mpfr_neg (l, l, MPFR_RNDD); /* lower bound for -log(-x) */
mpfr_neg (h, x, MPFR_RNDD); /* lower bound on -x */
mpfr_log (h, h, MPFR_RNDD); /* lower bound on log(-x) */
mpfr_neg (h, h, MPFR_RNDU); /* upper bound for -log(-x) */
mpfr_sub (h, h, x, MPFR_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 */
else
expl = MPFR_EXP (l);
mpfr_clear (l);
mpfr_clear (h);
if (ok)
{
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd);
}
/* if ulp(log(-x)) <= |x| there is no reason to loop,
since the width of [l, h] will be at least |x| */
if (expl < MPFR_EXP (x) + w)
break;
w += MPFR_INT_CEIL_LOG2(w) + 3;
}
MPFR_SAVE_EXPO_FREE (expo);
}
}
inex = mpfr_lngamma_aux (y, x, rnd);
return inex;
}
#endif
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