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/* mpfr_const_euler -- Euler's constant
Copyright 2001 Free Software Foundation.
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. If not, write to
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include <stdio.h>
#include <stdlib.h>
#include "gmp.h"
#include "gmp-impl.h"
#include "longlong.h"
#include "mpfr.h"
#include "mpfr-impl.h"
static void mpfr_const_euler_S _PROTO ((mpfr_ptr, unsigned long));
static void mpfr_const_euler_R _PROTO ((mpfr_ptr, unsigned long));
int
mpfr_const_euler (mpfr_t x, mp_rnd_t rnd)
{
mp_prec_t prec = MPFR_PREC(x), m, log2m;
mpfr_t y, z;
unsigned long n;
log2m = _mpfr_ceil_log2 ((double) prec);
m = prec + log2m;
mpfr_init (y);
mpfr_init (z);
do
{
m += BITS_PER_MP_LIMB;
n = 1 + (unsigned long)((double) m * LOG2 / 2.0);
if (n < 9)
n = 9;
MPFR_ASSERTD (n >= 9);
mpfr_set_prec (y, m + log2m);
mpfr_set_prec (z, m + log2m);
mpfr_const_euler_S (y, n);
mpfr_set_ui (z, n, GMP_RNDN);
mpfr_log (z, z, GMP_RNDD);
mpfr_sub (y, y, z, GMP_RNDN); /* S'(n) - log(n) */
mpfr_set_prec (z, m);
mpfr_const_euler_R (z, n);
mpfr_sub (y, y, z, GMP_RNDN);
}
while (!mpfr_can_round (y, m - 3, GMP_RNDN, rnd, prec));
mpfr_set (x, y, rnd);
mpfr_clear (y);
mpfr_clear (z);
return 1; /* always inexact */
}
/* computes S(n) = sum(n^k*(-1)^(k-1)/k!/k, k=1..ceil(4.319136566 * n))
with an error of at most ulp(x).
[S(n) >= 2 for n >= 5]
*/
void
mpfr_const_euler_S (mpfr_t x, unsigned long n)
{
unsigned long N, k, m;
mpz_t a, s, t;
N = (long) (ALPHA * (double) n + 1.0); /* ceil(alpha * n) */
m = MPFR_PREC(x) + (unsigned long) ((double) n / LOG2)
+ _mpfr_ceil_log2 ((double) N) + 1;
mpz_init_set_ui (a, 1);
mpz_mul_2exp (a, a, m); /* a=-2^m where m is the precision of x */
mpz_init_set_ui (s, 0);
mpz_init (t);
/* here, a and s are exact */
for (k = 1; k <= N; k++)
{
mpz_mul_ui (a, a, n);
mpz_div_ui (a, a, k);
mpz_div_ui (t, a, k);
if (k % 2)
mpz_add (s, s, t);
else
mpz_sub (s, s, t);
}
/* the error on s is at most N (e^n + 1),
thus that the error on x is at most one ulp */
mpfr_set_z (x, s, GMP_RNDD);
mpfr_div_2ui (x, x, m, GMP_RNDD);
mpz_clear (a);
mpz_clear (s);
mpz_clear (t);
}
/* computes R(n) = exp(-n)/n * sum(k!/(-n)^k, k=0..n-2)
with error at most 4*ulp(x). Assumes n>=2.
Since x <= exp(-n)/n <= 1/8, then 4*ulp(x) <= ulp(1).
*/
void
mpfr_const_euler_R (mpfr_t x, unsigned long n)
{
unsigned long k, m;
mpz_t a, s;
mpfr_t y;
MPFR_ASSERTN (n >= 2); /* ensures sum(k!/(-n)^k, k=0..n-2) >= 2/3 */
/* as we multiply the sum by exp(-n), we need only PREC(x) - n/LOG2 bits */
m = MPFR_PREC(x) - (unsigned long) ((double) n / LOG2);
mpz_init_set_ui (a, 1);
mpz_mul_2exp (a, a, m);
mpz_init_set (s, a);
for (k = 1; k <= n; k++)
{
mpz_mul_ui (a, a, k);
mpz_div_ui (a, a, n);
/* the error e(k) on a is e(k) <= 1 + k/n*e(k-1) with e(0)=0,
i.e. e(k) <= k */
if (k % 2)
mpz_sub (s, s, a);
else
mpz_add (s, s, a);
}
/* the error on s is at most 1+2+...+n = n*(n+1)/2 */
mpz_div_ui (s, s, n); /* err <= 1 + (n+1)/2 */
if (MPFR_PREC(x) < mpz_sizeinbase(s, 2))
{
fprintf (stderr, "prec(x) is too small in mpfr_const_euler_R\n");
exit (1);
}
mpfr_set_z (x, s, GMP_RNDD); /* exact */
mpfr_div_2ui (x, x, m, GMP_RNDD);
/* now x = 1/n * sum(k!/(-n)^k, k=0..n-2) <= 1/n */
/* err(x) <= (n+1)/2^m <= (n+1)*exp(n)/2^PREC(x) */
mpfr_init2 (y, m);
mpfr_set_si (y, -n, GMP_RNDD); /* assumed exact */
mpfr_exp (y, y, GMP_RNDD); /* err <= ulp(y) <= exp(-n)*2^(1-m) */
mpfr_mul (x, x, y, GMP_RNDD);
/* err <= ulp(x) + (n + 1 + 2/n) / 2^prec(x)
<= ulp(x) + (n + 1 + 2/n) ulp(x)/x since x*2^(-prec(x)) < ulp(x)
<= ulp(x) + (n + 1 + 2/n) 3/(2n) ulp(x) since x >= 2/3*n for n >= 2
<= 4 * ulp(x) for n >= 2 */
mpfr_clear (y);
mpz_clear (a);
mpz_clear (s);
}
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