summaryrefslogtreecommitdiff
path: root/Build/source/libs/mpfr/mpfr-src/src/zeta.c
diff options
context:
space:
mode:
Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/zeta.c')
-rw-r--r--Build/source/libs/mpfr/mpfr-src/src/zeta.c689
1 files changed, 0 insertions, 689 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/zeta.c b/Build/source/libs/mpfr/mpfr-src/src/zeta.c
deleted file mode 100644
index ec2d88de7ec..00000000000
--- a/Build/source/libs/mpfr/mpfr-src/src/zeta.c
+++ /dev/null
@@ -1,689 +0,0 @@
-/* mpfr_zeta -- compute the Riemann Zeta function
-
-Copyright 2003-2020 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. */
-
-#include <float.h> /* for DBL_MAX */
-
-#define MPFR_NEED_LONGLONG_H
-#include "mpfr-impl.h"
-
-/*
- Parameters:
- s - the input floating-point number
- n, p - parameters from the algorithm
- tc - an array of p floating-point numbers tc[1]..tc[p]
- Output:
- b is the result, i.e.
- sum(tc[i]*product((s+2j)*(s+2j-1)/n^2,j=1..i-1), i=1..p)*s*n^(-s-1)
-*/
-static void
-mpfr_zeta_part_b (mpfr_t b, mpfr_srcptr s, int n, int p, mpfr_t *tc)
-{
- mpfr_t s1, d, u;
- unsigned long n2;
- int l, t;
- MPFR_GROUP_DECL (group);
-
- if (p == 0)
- {
- MPFR_SET_ZERO (b);
- MPFR_SET_POS (b);
- return;
- }
-
- n2 = n * n;
- MPFR_GROUP_INIT_3 (group, MPFR_PREC (b), s1, d, u);
-
- /* t equals 2p-2, 2p-3, ... ; s1 equals s+t */
- t = 2 * p - 2;
- mpfr_set (d, tc[p], MPFR_RNDN);
- for (l = 1; l < p; l++)
- {
- mpfr_add_ui (s1, s, t, MPFR_RNDN); /* s + (2p-2l) */
- mpfr_mul (d, d, s1, MPFR_RNDN);
- t = t - 1;
- mpfr_add_ui (s1, s, t, MPFR_RNDN); /* s + (2p-2l-1) */
- mpfr_mul (d, d, s1, MPFR_RNDN);
- t = t - 1;
- mpfr_div_ui (d, d, n2, MPFR_RNDN);
- mpfr_add (d, d, tc[p-l], MPFR_RNDN);
- /* since s is positive and the tc[i] have alternate signs,
- the following is unlikely */
- if (MPFR_UNLIKELY (mpfr_cmpabs (d, tc[p-l]) > 0))
- mpfr_set (d, tc[p-l], MPFR_RNDN);
- }
- mpfr_mul (d, d, s, MPFR_RNDN);
- mpfr_add (s1, s, __gmpfr_one, MPFR_RNDN);
- mpfr_neg (s1, s1, MPFR_RNDN);
- mpfr_ui_pow (u, n, s1, MPFR_RNDN);
- mpfr_mul (b, d, u, MPFR_RNDN);
-
- MPFR_GROUP_CLEAR (group);
-}
-
-/* Input: p - an integer
- Output: fills tc[1..p], tc[i] = bernoulli(2i)/(2i)!
- tc[1]=1/12, tc[2]=-1/720, tc[3]=1/30240, ...
- Assumes all the tc[i] have the same precision.
-
- Uses the recurrence (4.60) from the book "Modern Computer Arithmetic"
- by Brent and Zimmermann for C_k = bernoulli(2k)/(2k)!:
- sum(C_k/(2k+1-2j)!/4^(k-j), j=0..k) = 1/(2k)!/4^k
- If we put together the terms involving C_0 and C_1 we get:
- sum(D_k/(2k+1-2j)!/4^(k-j), j=1..k) = 0
- with D_1 = C_0/4/(2k+1)/(2k)+C_1-1/(2k)/4=(k-1)/(12k+6),
- and D_k = C_k for k >= 2.
-
- FIXME: we have C_k = (-1)^(k-1) 2/(2pi)^(2k) * zeta(2k),
- see for example formula (4.65) from the above book,
- thus since |zeta(2k)-1| < 2^(1-2k) for k >= 2, we have:
- |C_k - E_k| < E_k * 2^(1-2k) for k >= 2 and E_k := (-1)^(k-1) 2/(2pi)^(2k).
- Then if 2k-1 >= prec we can evaluate E_k instead, which only requires one
- multiplication per term, instead of O(k) small divisions.
-*/
-static void
-mpfr_zeta_c (int p, mpfr_t *tc)
-{
- if (p > 0)
- {
- mpfr_t d;
- int k, l;
- mpfr_prec_t prec = MPFR_PREC (tc[1]);
-
- mpfr_init2 (d, prec);
- mpfr_div_ui (tc[1], __gmpfr_one, 12, MPFR_RNDN);
- for (k = 2; k <= p; k++)
- {
- mpfr_set_ui (d, k-1, MPFR_RNDN);
- mpfr_div_ui (d, d, 12*k+6, MPFR_RNDN);
- for (l=2; l < k; l++)
- {
- mpfr_div_ui (d, d, 4*(2*k-2*l+3)*(2*k-2*l+2), MPFR_RNDN);
- mpfr_add (d, d, tc[l], MPFR_RNDN);
- }
- mpfr_div_ui (tc[k], d, 24, MPFR_RNDN);
- MPFR_CHANGE_SIGN (tc[k]);
- }
- mpfr_clear (d);
- }
-}
-
-/* Input: s - a floating-point number
- n - an integer
- Output: sum - a floating-point number approximating sum(1/i^s, i=1..n-1) */
-static void
-mpfr_zeta_part_a (mpfr_t sum, mpfr_srcptr s, int n)
-{
- mpfr_t u, s1;
- int i;
- MPFR_GROUP_DECL (group);
-
- MPFR_GROUP_INIT_2 (group, MPFR_PREC (sum), u, s1);
-
- mpfr_neg (s1, s, MPFR_RNDN);
- mpfr_ui_pow (u, n, s1, MPFR_RNDN);
- mpfr_div_2ui (u, u, 1, MPFR_RNDN);
- mpfr_set (sum, u, MPFR_RNDN);
- for (i=n-1; i>1; i--)
- {
- mpfr_ui_pow (u, i, s1, MPFR_RNDN);
- mpfr_add (sum, sum, u, MPFR_RNDN);
- }
- mpfr_add (sum, sum, __gmpfr_one, MPFR_RNDN);
-
- MPFR_GROUP_CLEAR (group);
-}
-
-/* Input: s - a floating-point number >= 1/2.
- rnd_mode - a rounding mode.
- Assumes s is neither NaN nor Infinite.
- Output: z - Zeta(s) rounded to the precision of z with direction rnd_mode
-*/
-static int
-mpfr_zeta_pos (mpfr_t z, mpfr_srcptr s, mpfr_rnd_t rnd_mode)
-{
- mpfr_t b, c, z_pre, f, s1;
- double beta, sd, dnep;
- mpfr_t *tc1;
- mpfr_prec_t precz, precs, d, dint;
- int p, n, l, add;
- int inex;
- MPFR_GROUP_DECL (group);
- MPFR_ZIV_DECL (loop);
-
- MPFR_ASSERTD (MPFR_IS_POS (s) && MPFR_GET_EXP (s) >= 0);
-
- precz = MPFR_PREC (z);
- precs = MPFR_PREC (s);
-
- /* Zeta(x) = 1+1/2^x+1/3^x+1/4^x+1/5^x+O(1/6^x)
- so with 2^(EXP(x)-1) <= x < 2^EXP(x)
- So for x > 2^3, k^x > k^8, so 2/k^x < 2/k^8
- Zeta(x) = 1 + 1/2^x*(1+(2/3)^x+(2/4)^x+...)
- = 1 + 1/2^x*(1+sum((2/k)^x,k=3..infinity))
- <= 1 + 1/2^x*(1+sum((2/k)^8,k=3..infinity))
- And sum((2/k)^8,k=3..infinity) = -257+128*Pi^8/4725 ~= 0.0438035
- So Zeta(x) <= 1 + 1/2^x*2 for x >= 8
- The error is < 2^(-x+1) <= 2^(-2^(EXP(x)-1)+1) */
- if (MPFR_GET_EXP (s) > 3)
- {
- mpfr_exp_t err;
- err = MPFR_GET_EXP (s) - 1;
- if (err > (mpfr_exp_t) (sizeof (mpfr_exp_t)*CHAR_BIT-2))
- err = MPFR_EMAX_MAX;
- else
- err = ((mpfr_exp_t)1) << err;
- err = 1 - (-err+1); /* GET_EXP(one) - (-err+1) = err :) */
- MPFR_FAST_COMPUTE_IF_SMALL_INPUT (z, __gmpfr_one, err, 0, 1,
- rnd_mode, {});
- }
-
- d = precz + MPFR_INT_CEIL_LOG2(precz) + 10;
-
- /* we want that s1 = s-1 is exact, i.e. we should have PREC(s1) >= EXP(s) */
- dint = (mpfr_uexp_t) MPFR_GET_EXP (s);
- mpfr_init2 (s1, MAX (precs, dint));
- inex = mpfr_sub (s1, s, __gmpfr_one, MPFR_RNDN);
- MPFR_ASSERTD (inex == 0);
-
- /* case s=1 should have already been handled */
- MPFR_ASSERTD (!MPFR_IS_ZERO (s1));
-
- MPFR_GROUP_INIT_4 (group, MPFR_PREC_MIN, b, c, z_pre, f);
-
- MPFR_ZIV_INIT (loop, d);
- for (;;)
- {
- /* Principal loop: we compute, in z_pre,
- an approximation of Zeta(s), that we send to can_round */
- if (MPFR_GET_EXP (s1) <= -(mpfr_exp_t) ((mpfr_prec_t) (d-3)/2))
- /* Branch 1: when s-1 is very small, one
- uses the approximation Zeta(s)=1/(s-1)+gamma,
- where gamma is Euler's constant */
- {
- dint = MAX (d + 3, precs);
- /* branch 1, with internal precision dint */
- MPFR_GROUP_REPREC_4 (group, dint, b, c, z_pre, f);
- mpfr_div (z_pre, __gmpfr_one, s1, MPFR_RNDN);
- mpfr_const_euler (f, MPFR_RNDN);
- mpfr_add (z_pre, z_pre, f, MPFR_RNDN);
- }
- else /* Branch 2 */
- {
- size_t size;
-
- /* branch 2 */
- /* Computation of parameters n, p and working precision */
- dnep = (double) d * LOG2;
- sd = mpfr_get_d (s, MPFR_RNDN);
- /* beta = dnep + 0.61 + sd * log (6.2832 / sd);
- but a larger value is OK */
-#define LOG6dot2832 1.83787940484160805532
- beta = dnep + 0.61 + sd * (LOG6dot2832 - LOG2 *
- __gmpfr_floor_log2 (sd));
- if (beta <= 0.0)
- {
- p = 0;
- /* n = 1 + (int) (exp ((dnep - LOG2) / sd)); */
- n = 1 + (int) __gmpfr_ceil_exp2 ((d - 1.0) / sd);
- }
- else
- {
- p = 1 + (int) beta / 2;
- n = 1 + (int) ((sd + 2.0 * (double) p - 1.0) / 6.2832);
- }
- /* add = 4 + floor(1.5 * log(d) / log (2)).
- We should have add >= 10, which is always fulfilled since
- d = precz + 11 >= 12, thus ceil(log2(d)) >= 4 */
- add = 4 + (3 * MPFR_INT_CEIL_LOG2 (d)) / 2;
- MPFR_ASSERTD(add >= 10);
- dint = d + add;
- if (dint < precs)
- dint = precs;
-
- /* internal precision is dint */
-
- size = (p + 1) * sizeof(mpfr_t);
- tc1 = (mpfr_t*) mpfr_allocate_func (size);
- for (l=1; l<=p; l++)
- mpfr_init2 (tc1[l], dint);
- MPFR_GROUP_REPREC_4 (group, dint, b, c, z_pre, f);
-
- /* precision of z is precz */
-
- /* Computation of the coefficients c_k */
- mpfr_zeta_c (p, tc1);
- /* Computation of the 3 parts of the function Zeta. */
- mpfr_zeta_part_a (z_pre, s, n);
- mpfr_zeta_part_b (b, s, n, p, tc1);
- /* s1 = s-1 is already computed above */
- mpfr_div (c, __gmpfr_one, s1, MPFR_RNDN);
- mpfr_ui_pow (f, n, s1, MPFR_RNDN);
- mpfr_div (c, c, f, MPFR_RNDN);
- mpfr_add (z_pre, z_pre, c, MPFR_RNDN);
- mpfr_add (z_pre, z_pre, b, MPFR_RNDN);
- for (l=1; l<=p; l++)
- mpfr_clear (tc1[l]);
- mpfr_free_func (tc1, size);
- /* End branch 2 */
- }
-
- if (MPFR_LIKELY (MPFR_CAN_ROUND (z_pre, d-3, precz, rnd_mode)))
- break;
- MPFR_ZIV_NEXT (loop, d);
- }
- MPFR_ZIV_FREE (loop);
-
- inex = mpfr_set (z, z_pre, rnd_mode);
-
- MPFR_GROUP_CLEAR (group);
- mpfr_clear (s1);
-
- return inex;
-}
-
-/* return add = 1 + floor(log(c^3*(13+m1))/log(2))
- where c = (1+eps)*(1+eps*max(8,m1)),
- m1 = 1 + max(1/eps,2*sd)*(1+eps),
- eps = 2^(-precz-14)
- sd = abs(s-1)
- */
-static long
-compute_add (mpfr_srcptr s, mpfr_prec_t precz)
-{
- mpfr_t t, u, m1;
- long add;
-
- mpfr_inits2 (64, t, u, m1, (mpfr_ptr) 0);
- if (mpfr_cmp_ui (s, 1) >= 0)
- mpfr_sub_ui (t, s, 1, MPFR_RNDU);
- else
- mpfr_ui_sub (t, 1, s, MPFR_RNDU);
- /* now t = sd = abs(s-1), rounded up */
- mpfr_set_ui_2exp (u, 1, - precz - 14, MPFR_RNDU);
- /* u = eps */
- /* since 1/eps = 2^(precz+14), if EXP(sd) >= precz+14, then
- sd >= 1/2*2^(precz+14) thus 2*sd >= 2^(precz+14) >= 1/eps */
- if (mpfr_get_exp (t) >= precz + 14)
- mpfr_mul_2ui (t, t, 1, MPFR_RNDU);
- else
- mpfr_set_ui_2exp (t, 1, precz + 14, MPFR_RNDU);
- /* now t = max(1/eps,2*sd) */
- mpfr_add_ui (u, u, 1, MPFR_RNDU); /* u = 1+eps, rounded up */
- mpfr_mul (t, t, u, MPFR_RNDU); /* t = max(1/eps,2*sd)*(1+eps) */
- mpfr_add_ui (m1, t, 1, MPFR_RNDU);
- if (mpfr_get_exp (m1) <= 3)
- mpfr_set_ui (t, 8, MPFR_RNDU);
- else
- mpfr_set (t, m1, MPFR_RNDU);
- /* now t = max(8,m1) */
- mpfr_div_2ui (t, t, precz + 14, MPFR_RNDU); /* eps*max(8,m1) */
- mpfr_add_ui (t, t, 1, MPFR_RNDU); /* 1+eps*max(8,m1) */
- mpfr_mul (t, t, u, MPFR_RNDU); /* t = c */
- mpfr_add_ui (u, m1, 13, MPFR_RNDU); /* 13+m1 */
- mpfr_mul (u, u, t, MPFR_RNDU); /* c*(13+m1) */
- mpfr_sqr (t, t, MPFR_RNDU); /* c^2 */
- mpfr_mul (u, u, t, MPFR_RNDU); /* c^3*(13+m1) */
- add = mpfr_get_exp (u);
- mpfr_clears (t, u, m1, (mpfr_ptr) 0);
- return add;
-}
-
-/* return in z a lower bound (for rnd = RNDD) or upper bound (for rnd = RNDU)
- of |zeta(s)|/2, using:
- log(|zeta(s)|/2) = (s-1)*log(2*Pi) + lngamma(1-s)
- + log(|sin(Pi*s/2)| * zeta(1-s)).
- Assumes s < 1/2 and s1 = 1-s exactly, thus s1 > 1/2.
- y and p are temporary variables.
- At input, p is Pi rounded down.
- The comments in the code are for rnd = RNDD. */
-static void
-mpfr_reflection_overflow (mpfr_t z, mpfr_t s1, const mpfr_t s, mpfr_t y,
- mpfr_t p, mpfr_rnd_t rnd)
-{
- mpz_t sint;
-
- MPFR_ASSERTD (rnd == MPFR_RNDD || rnd == MPFR_RNDU);
-
- /* Since log is increasing, we want lower bounds on |sin(Pi*s/2)| and
- zeta(1-s). */
- mpz_init (sint);
- mpfr_get_z (sint, s, MPFR_RNDD); /* sint = floor(s) */
- /* We first compute a lower bound of |sin(Pi*s/2)|, which is a periodic
- function of period 2. Thus:
- if 2k < s < 2k+1, then |sin(Pi*s/2)| is increasing;
- if 2k-1 < s < 2k, then |sin(Pi*s/2)| is decreasing.
- These cases are distinguished by testing bit 0 of floor(s) as if
- represented in two's complement (or equivalently, as an unsigned
- integer mod 2):
- 0: sint = 0 mod 2, thus 2k < s < 2k+1 and |sin(Pi*s/2)| is increasing;
- 1: sint = 1 mod 2, thus 2k-1 < s < 2k and |sin(Pi*s/2)| is decreasing.
- Let's recall that the comments are for rnd = RNDD. */
- if (mpz_tstbit (sint, 0) == 0) /* |sin(Pi*s/2)| is increasing: round down
- Pi*s to get a lower bound. */
- {
- mpfr_mul (y, p, s, rnd);
- if (rnd == MPFR_RNDD)
- mpfr_nextabove (p); /* we will need p rounded above afterwards */
- }
- else /* |sin(Pi*s/2)| is decreasing: round up Pi*s to get a lower bound. */
- {
- if (rnd == MPFR_RNDD)
- mpfr_nextabove (p);
- mpfr_mul (y, p, s, MPFR_INVERT_RND(rnd));
- }
- mpfr_div_2ui (y, y, 1, MPFR_RNDN); /* exact, rounding mode doesn't matter */
- /* The rounding direction of sin depends on its sign. We have:
- if -4k-2 < s < -4k, then -2k-1 < s/2 < -2k, thus sin(Pi*s/2) < 0;
- if -4k < s < -4k+2, then -2k < s/2 < -2k+1, thus sin(Pi*s/2) > 0.
- These cases are distinguished by testing bit 1 of floor(s) as if
- represented in two's complement (or equivalently, as an unsigned
- integer mod 4):
- 0: sint = {0,1} mod 4, thus -2k < s/2 < -2k+1 and sin(Pi*s/2) > 0;
- 1: sint = {2,3} mod 4, thus -2k-1 < s/2 < -2k and sin(Pi*s/2) < 0.
- Let's recall that the comments are for rnd = RNDD. */
- if (mpz_tstbit (sint, 1) == 0) /* -2k < s/2 < -2k+1; sin(Pi*s/2) > 0 */
- {
- /* Round sin down to get a lower bound of |sin(Pi*s/2)|. */
- mpfr_sin (y, y, rnd);
- }
- else /* -2k-1 < s/2 < -2k; sin(Pi*s/2) < 0 */
- {
- /* Round sin up to get a lower bound of |sin(Pi*s/2)|. */
- mpfr_sin (y, y, MPFR_INVERT_RND(rnd));
- mpfr_abs (y, y, MPFR_RNDN); /* exact, rounding mode doesn't matter */
- }
- mpz_clear (sint);
- /* now y <= |sin(Pi*s/2)| when rnd=RNDD, y >= |sin(Pi*s/2)| when rnd=RNDU */
- mpfr_zeta_pos (z, s1, rnd); /* zeta(1-s) */
- mpfr_mul (z, z, y, rnd);
- /* now z <= |sin(Pi*s/2)|*zeta(1-s) */
- mpfr_log (z, z, rnd);
- /* now z <= log(|sin(Pi*s/2)|*zeta(1-s)) */
- mpfr_lngamma (y, s1, rnd);
- mpfr_add (z, z, y, rnd);
- /* z <= lngamma(1-s) + log(|sin(Pi*s/2)|*zeta(1-s)) */
- /* since s-1 < 0, we want to round log(2*pi) upwards */
- mpfr_mul_2ui (y, p, 1, MPFR_INVERT_RND(rnd));
- mpfr_log (y, y, MPFR_INVERT_RND(rnd));
- mpfr_mul (y, y, s1, MPFR_INVERT_RND(rnd));
- mpfr_sub (z, z, y, rnd);
- mpfr_exp (z, z, rnd);
- if (rnd == MPFR_RNDD)
- mpfr_nextbelow (p); /* restore original p */
-}
-
-int
-mpfr_zeta (mpfr_t z, mpfr_srcptr s, mpfr_rnd_t rnd_mode)
-{
- mpfr_t z_pre, s1, y, p;
- long add;
- mpfr_prec_t precz, prec1, precs, precs1;
- int inex;
- MPFR_GROUP_DECL (group);
- MPFR_ZIV_DECL (loop);
- MPFR_SAVE_EXPO_DECL (expo);
-
- MPFR_LOG_FUNC (
- ("s[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (s), mpfr_log_prec, s, rnd_mode),
- ("z[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (z), mpfr_log_prec, z, inex));
-
- /* Zero, Nan or Inf ? */
- if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (s)))
- {
- if (MPFR_IS_NAN (s))
- {
- MPFR_SET_NAN (z);
- MPFR_RET_NAN;
- }
- else if (MPFR_IS_INF (s))
- {
- if (MPFR_IS_POS (s))
- return mpfr_set_ui (z, 1, MPFR_RNDN); /* Zeta(+Inf) = 1 */
- MPFR_SET_NAN (z); /* Zeta(-Inf) = NaN */
- MPFR_RET_NAN;
- }
- else /* s iz zero */
- {
- MPFR_ASSERTD (MPFR_IS_ZERO (s));
- return mpfr_set_si_2exp (z, -1, -1, rnd_mode);
- }
- }
-
- /* s is neither Nan, nor Inf, nor Zero */
-
- /* check tiny s: we have zeta(s) = -1/2 - 1/2 log(2 Pi) s + ... around s=0,
- and for |s| <= 2^(-4), we have |zeta(s) + 1/2| <= |s|.
- EXP(s) + 1 < -PREC(z) is a sufficient condition to be able to round
- correctly, for any PREC(z) >= 1 (see algorithms.tex for details). */
- if (MPFR_GET_EXP (s) + 1 < - (mpfr_exp_t) MPFR_PREC(z))
- {
- int signs = MPFR_SIGN(s);
-
- MPFR_SAVE_EXPO_MARK (expo);
- mpfr_set_si_2exp (z, -1, -1, rnd_mode); /* -1/2 */
- if (rnd_mode == MPFR_RNDA)
- rnd_mode = MPFR_RNDD; /* the result is around -1/2, thus negative */
- if ((rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDZ) && signs < 0)
- {
- mpfr_nextabove (z); /* z = -1/2 + epsilon */
- inex = 1;
- }
- else if (rnd_mode == MPFR_RNDD && signs > 0)
- {
- mpfr_nextbelow (z); /* z = -1/2 - epsilon */
- inex = -1;
- }
- else
- {
- if (rnd_mode == MPFR_RNDU) /* s > 0: z = -1/2 */
- inex = 1;
- else if (rnd_mode == MPFR_RNDD)
- inex = -1; /* s < 0: z = -1/2 */
- else /* (MPFR_RNDZ and s > 0) or MPFR_RNDN: z = -1/2 */
- inex = (signs > 0) ? 1 : -1;
- }
- MPFR_SAVE_EXPO_FREE (expo);
- return mpfr_check_range (z, inex, rnd_mode);
- }
-
- /* Check for case s= -2n */
- if (MPFR_IS_NEG (s))
- {
- mpfr_t tmp;
- tmp[0] = *s;
- MPFR_EXP (tmp) = MPFR_GET_EXP (s) - 1;
- if (mpfr_integer_p (tmp))
- {
- MPFR_SET_ZERO (z);
- MPFR_SET_POS (z);
- MPFR_RET (0);
- }
- }
-
- /* Check for case s=1 before changing the exponent range */
- if (mpfr_cmp (s, __gmpfr_one) == 0)
- {
- MPFR_SET_INF (z);
- MPFR_SET_POS (z);
- MPFR_SET_DIVBY0 ();
- MPFR_RET (0);
- }
-
- MPFR_SAVE_EXPO_MARK (expo);
-
- /* Compute Zeta */
- if (MPFR_IS_POS (s) && MPFR_GET_EXP (s) >= 0) /* Case s >= 1/2 */
- inex = mpfr_zeta_pos (z, s, rnd_mode);
- else /* use reflection formula
- zeta(s) = 2^s*Pi^(s-1)*sin(Pi*s/2)*gamma(1-s)*zeta(1-s) */
- {
- int overflow = 0;
-
- precz = MPFR_PREC (z);
- precs = MPFR_PREC (s);
-
- /* Precision precs1 needed to represent 1 - s, and s + 2,
- without any truncation */
- precs1 = precs + 2 + MAX (0, - MPFR_GET_EXP (s));
- /* Precision prec1 is the precision on elementary computations;
- it ensures a final precision prec1 - add for zeta(s) */
- add = compute_add (s, precz);
- prec1 = precz + add;
- /* FIXME: To avoid that the working precision (prec1) depends on the
- input precision, one would need to take into account the error made
- when s1 is not exactly 1-s when computing zeta(s1) and gamma(s1)
- below, and also in the case y=Inf (i.e. when gamma(s1) overflows).
- Make sure that underflows do not occur in intermediate computations.
- Due to the limited precision, they are probably not possible
- in practice; add some MPFR_ASSERTN's to be sure that problems
- do not remain undetected? */
- prec1 = MAX (prec1, precs1) + 10;
-
- MPFR_GROUP_INIT_4 (group, prec1, z_pre, s1, y, p);
- MPFR_ZIV_INIT (loop, prec1);
- for (;;)
- {
- mpfr_exp_t ey;
- mpfr_t z_up;
-
- mpfr_const_pi (p, MPFR_RNDD); /* p is Pi */
-
- mpfr_sub (s1, __gmpfr_one, s, MPFR_RNDN); /* s1 = 1-s */
- mpfr_gamma (y, s1, MPFR_RNDN); /* gamma(1-s) */
- if (MPFR_IS_INF (y)) /* zeta(s) < 0 for -4k-2 < s < -4k,
- zeta(s) > 0 for -4k < s < -4k+2 */
- {
- /* FIXME: An overflow in gamma(s1) does not imply that
- zeta(s) will overflow. A solution:
- 1. Compute
- log(|zeta(s)|/2) = (s-1)*log(2*pi) + lngamma(1-s)
- + log(abs(sin(Pi*s/2)) * zeta(1-s))
- (possibly sharing computations with the normal case)
- with a rather good accuracy (see (2)).
- Memorize the sign of sin(...) for the final sign.
- 2. Take the exponential, ~= |zeta(s)|/2. If there is an
- overflow, then this means an overflow on the final result
- (due to the multiplication by 2, which has not been done
- yet).
- 3. Ziv test.
- 4. Correct the sign from the sign of sin(...).
- 5. Round then multiply by 2. Here, an overflow in either
- operation means a real overflow. */
- mpfr_reflection_overflow (z_pre, s1, s, y, p, MPFR_RNDD);
- /* z_pre is a lower bound of |zeta(s)|/2, thus if it overflows,
- or has exponent emax, then |zeta(s)| overflows too. */
- if (MPFR_IS_INF (z_pre) || MPFR_GET_EXP(z_pre) == __gmpfr_emax)
- { /* determine the sign of overflow */
- mpfr_div_2ui (s1, s, 2, MPFR_RNDN); /* s/4, exact */
- mpfr_frac (s1, s1, MPFR_RNDN); /* exact, -1 < s1 < 0 */
- overflow = (mpfr_cmp_si_2exp (s1, -1, -1) > 0) ? -1 : 1;
- break;
- }
- else /* EXP(z_pre) < __gmpfr_emax */
- {
- int ok = 0;
- mpfr_t z_down;
- mpfr_init2 (z_up, mpfr_get_prec (z_pre));
- mpfr_reflection_overflow (z_up, s1, s, y, p, MPFR_RNDU);
- /* if the lower approximation z_pre does not overflow, but
- z_up does, we need more precision */
- if (MPFR_IS_INF (z_up) || MPFR_GET_EXP(z_up) == __gmpfr_emax)
- goto next_loop;
- /* check if z_pre and z_up round to the same number */
- mpfr_init2 (z_down, precz);
- mpfr_set (z_down, z_pre, rnd_mode);
- /* Note: it might be that EXP(z_down) = emax here, in that
- case we will have overflow below when we multiply by 2 */
- mpfr_prec_round (z_up, precz, rnd_mode);
- ok = mpfr_cmp (z_down, z_up) == 0;
- mpfr_clear (z_up);
- mpfr_clear (z_down);
- if (ok)
- {
- /* get correct sign and multiply by 2 */
- mpfr_div_2ui (s1, s, 2, MPFR_RNDN); /* s/4, exact */
- mpfr_frac (s1, s1, MPFR_RNDN); /* exact, -1 < s1 < 0 */
- if (mpfr_cmp_si_2exp (s1, -1, -1) > 0)
- mpfr_neg (z_pre, z_pre, rnd_mode);
- mpfr_mul_2ui (z_pre, z_pre, 1, rnd_mode);
- break;
- }
- else
- goto next_loop;
- }
- }
- mpfr_zeta_pos (z_pre, s1, MPFR_RNDN); /* zeta(1-s) */
- mpfr_mul (z_pre, z_pre, y, MPFR_RNDN); /* gamma(1-s)*zeta(1-s) */
-
- /* multiply z_pre by 2^s*Pi^(s-1) where p=Pi, s1=1-s */
- mpfr_mul_2ui (y, p, 1, MPFR_RNDN); /* 2*Pi */
- mpfr_neg (s1, s1, MPFR_RNDN); /* s-1 */
- mpfr_pow (y, y, s1, MPFR_RNDN); /* (2*Pi)^(s-1) */
- mpfr_mul (z_pre, z_pre, y, MPFR_RNDN);
- mpfr_mul_2ui (z_pre, z_pre, 1, MPFR_RNDN);
-
- /* multiply z_pre by sin(Pi*s/2) */
- mpfr_mul (y, s, p, MPFR_RNDN);
- mpfr_div_2ui (p, y, 1, MPFR_RNDN); /* p = s*Pi/2 */
- /* FIXME: sinpi will be available, we should replace the mpfr_sin
- call below by mpfr_sinpi(s/2), where s/2 will be exact.
- Can mpfr_sin underflow? Moreover, the code below should be
- improved so that the "if" condition becomes unlikely, e.g.
- by taking a slightly larger working precision. */
- mpfr_sin (y, p, MPFR_RNDN); /* y = sin(Pi*s/2) */
- ey = MPFR_GET_EXP (y);
- if (ey < 0) /* take account of cancellation in sin(p) */
- {
- mpfr_t t;
-
- MPFR_ASSERTN (- ey < MPFR_PREC_MAX - prec1);
- mpfr_init2 (t, prec1 - ey);
- mpfr_const_pi (t, MPFR_RNDD);
- mpfr_mul (t, s, t, MPFR_RNDN);
- mpfr_div_2ui (t, t, 1, MPFR_RNDN);
- mpfr_sin (y, t, MPFR_RNDN);
- mpfr_clear (t);
- }
- mpfr_mul (z_pre, z_pre, y, MPFR_RNDN);
-
- if (MPFR_LIKELY (MPFR_CAN_ROUND (z_pre, prec1 - add, precz,
- rnd_mode)))
- break;
-
- next_loop:
- MPFR_ZIV_NEXT (loop, prec1);
- MPFR_GROUP_REPREC_4 (group, prec1, z_pre, s1, y, p);
- }
- MPFR_ZIV_FREE (loop);
- if (overflow != 0)
- {
- inex = mpfr_overflow (z, rnd_mode, overflow);
- MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
- }
- else
- inex = mpfr_set (z, z_pre, rnd_mode);
- MPFR_GROUP_CLEAR (group);
- }
-
- MPFR_SAVE_EXPO_FREE (expo);
- return mpfr_check_range (z, inex, rnd_mode);
-}