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/* mpfr_const_euler -- Euler's constant
Copyright 2001-2020 Free Software Foundation, Inc.
Contributed by Fredrik Johansson.
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. */
/* The approximation error bound uses Theorem 1 and Remark 2 in
http://arxiv.org/pdf/1312.0039v1.pdf */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* Declare the cache */
MPFR_DECL_INIT_CACHE (__gmpfr_cache_const_euler, mpfr_const_euler_internal)
/* Set User Interface */
#undef mpfr_const_euler
int
mpfr_const_euler (mpfr_ptr x, mpfr_rnd_t rnd_mode) {
return mpfr_cache (x, __gmpfr_cache_const_euler, rnd_mode);
}
typedef struct
{
mpz_t P;
mpz_t Q;
mpz_t T;
mpz_t C;
mpz_t D;
mpz_t V;
} mpfr_const_euler_bs_struct;
typedef mpfr_const_euler_bs_struct mpfr_const_euler_bs_t[1];
static void
mpfr_const_euler_bs_init (mpfr_const_euler_bs_t s)
{
mpz_init (s->P);
mpz_init (s->Q);
mpz_init (s->T);
mpz_init (s->C);
mpz_init (s->D);
mpz_init (s->V);
}
static void
mpfr_const_euler_bs_clear (mpfr_const_euler_bs_t s)
{
mpz_clear (s->P);
mpz_clear (s->Q);
mpz_clear (s->T);
mpz_clear (s->C);
mpz_clear (s->D);
mpz_clear (s->V);
}
static void
mpfr_const_euler_bs_1 (mpfr_const_euler_bs_t s,
unsigned long n1, unsigned long n2, unsigned long N,
int cont)
{
if (n2 - n1 == 1)
{
mpz_set_ui (s->P, N);
mpz_mul (s->P, s->P, s->P);
mpz_set_ui (s->Q, n1 + 1);
mpz_mul (s->Q, s->Q, s->Q);
mpz_set_ui (s->C, 1);
mpz_set_ui (s->D, n1 + 1);
mpz_set (s->T, s->P);
mpz_set (s->V, s->P);
}
else
{
mpfr_const_euler_bs_t L, R;
mpz_t t, u, v;
unsigned long m = (n1 + n2) / 2;
mpfr_const_euler_bs_init (L);
mpfr_const_euler_bs_init (R);
mpfr_const_euler_bs_1 (L, n1, m, N, 1);
mpfr_const_euler_bs_1 (R, m, n2, N, 1);
mpz_init (t);
mpz_init (u);
mpz_init (v);
if (cont)
mpz_mul (s->P, L->P, R->P);
mpz_mul (s->Q, L->Q, R->Q);
mpz_mul (s->D, L->D, R->D);
/* T = LP RT + RQ LT*/
mpz_mul (t, L->P, R->T);
mpz_mul (v, R->Q, L->T);
mpz_add (s->T, t, v);
/* C = LC RD + RC LD */
if (cont)
{
mpz_mul (s->C, L->C, R->D);
mpz_addmul (s->C, R->C, L->D);
}
/* V = RD (RQ LV + LC LP RT) + LD LP RV */
mpz_mul (u, L->P, R->V);
mpz_mul (u, u, L->D);
mpz_mul (v, R->Q, L->V);
mpz_addmul (v, t, L->C);
mpz_mul (v, v, R->D);
mpz_add (s->V, u, v);
mpfr_const_euler_bs_clear (L);
mpfr_const_euler_bs_clear (R);
mpz_clear (t);
mpz_clear (u);
mpz_clear (v);
}
}
static void
mpfr_const_euler_bs_2 (mpz_t P, mpz_t Q, mpz_t T,
unsigned long n1, unsigned long n2, unsigned long N,
int cont)
{
if (n2 - n1 == 1)
{
if (n1 == 0)
{
mpz_set_ui (P, 1);
mpz_set_ui (Q, 4 * N);
}
else
{
mpz_set_ui (P, 2 * n1 - 1);
mpz_pow_ui (P, P, 3);
mpz_set_ui (Q, 32 * n1);
mpz_mul_ui (Q, Q, N);
mpz_mul_ui (Q, Q, N);
}
mpz_set (T, P);
}
else
{
mpz_t P2, Q2, T2;
unsigned long m = (n1 + n2) / 2;
mpz_init (P2);
mpz_init (Q2);
mpz_init (T2);
mpfr_const_euler_bs_2 (P, Q, T, n1, m, N, 1);
mpfr_const_euler_bs_2 (P2, Q2, T2, m, n2, N, 1);
mpz_mul (T, T, Q2);
mpz_mul (T2, T2, P);
mpz_add (T, T, T2);
if (cont)
mpz_mul (P, P, P2);
mpz_mul (Q, Q, Q2);
mpz_clear (P2);
mpz_clear (Q2);
mpz_clear (T2);
}
}
int
mpfr_const_euler_internal (mpfr_t x, mpfr_rnd_t rnd)
{
mpfr_const_euler_bs_t sum;
mpz_t t, u, v;
unsigned long n, N;
mpfr_prec_t prec, wp, magn;
mpfr_t y;
int inexact;
MPFR_ZIV_DECL (loop);
prec = mpfr_get_prec (x);
wp = prec + MPFR_INT_CEIL_LOG2 (prec) + 5;
mpfr_init2 (y, wp);
mpfr_const_euler_bs_init (sum);
mpz_init (t);
mpz_init (u);
mpz_init (v);
MPFR_ZIV_INIT (loop, wp);
for (;;)
{
/* The approximation error is bounded by 24 exp(-8n) when
n > 1, which is smaller than 2^-wp if
n > (wp + log_2(24)) * (log(2)/8).
Note log2(24) < 5 and log(2)/8 < 866434 / 10000000. */
mpz_set_ui (t, wp + 5);
mpz_mul_ui (t, t, 866434);
mpz_cdiv_q_ui (t, t, 10000000);
n = mpz_get_ui (t);
/* It is sufficient to take N >= alpha*n + 1
where alpha = 3/LambertW(3/e) = 4.970625759544... */
mpz_set_ui (t, n);
mpz_mul_ui (t, t, 4970626);
mpz_cdiv_q_ui (t, t, 1000000);
mpz_add_ui (t, t, 1);
N = mpz_get_ui (t);
/* V / ((T + Q) * D) = S / I
where S = sum_{k=0}^{N-1} H_k n^(2k) / (k!)^2,
I = sum_{k=0}^{N-1} n^(2k) / (k!)^2 */
mpfr_const_euler_bs_1 (sum, 0, N, n, 0);
mpz_add (sum->T, sum->T, sum->Q);
mpz_mul (t, sum->T, sum->D);
mpz_mul_2exp (u, sum->V, wp);
mpz_tdiv_q (v, u, t);
/* v * 2^-wp = S/I with error < 1 */
/* C / (D * V) = U where
U = (1/(4n)) sum_{k=0}^{2n-1} [(2k)!]^3 / ((k!)^4 8^(2k) (2n)^(2k)) */
mpfr_const_euler_bs_2 (sum->C, sum->D, sum->V, 0, 2*n, n, 0);
mpz_mul (t, sum->Q, sum->Q);
mpz_mul (t, t, sum->V);
mpz_mul (u, sum->T, sum->T);
mpz_mul (u, u, sum->D);
mpz_mul_2exp (t, t, wp);
mpz_tdiv_q (t, t, u);
/* t * 2^-wp = U/I^2 with error < 1 */
/* gamma = S/I - U/I^2 - log(n) with error at most 2^-wp */
mpz_sub (v, v, t);
/* v * 2^-wp now equals gamma + log(n) with error at most 3*2^-wp */
/* log(n) < 2^ceil(log2(n)) */
magn = MPFR_INT_CEIL_LOG2(n);
mpfr_set_prec (y, wp + magn);
mpfr_set_ui (y, n, MPFR_RNDZ); /* exact */
mpfr_log (y, y, MPFR_RNDZ); /* error < 2^-wp */
mpfr_mul_2ui (y, y, wp, MPFR_RNDZ);
mpfr_z_sub (y, v, y, MPFR_RNDZ);
mpfr_div_2ui (y, y, wp, MPFR_RNDZ);
/* rounding error from the last subtraction < 2^-wp */
/* so y = gamma with error < 5*2^-wp */
if (MPFR_LIKELY (MPFR_CAN_ROUND (y, wp - 3, prec, rnd)))
break;
MPFR_ZIV_NEXT (loop, wp);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (x, y, rnd);
mpfr_clear (y);
mpz_clear (t);
mpz_clear (u);
mpz_clear (v);
mpfr_const_euler_bs_clear (sum);
return inexact; /* always inexact */
}
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