/* mpfr_const_euler -- Euler's constant
Copyright 2001-2017 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
http://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"
/* 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);
}
static void mpfr_const_euler_S2 (mpfr_ptr, unsigned long);
static void mpfr_const_euler_R (mpfr_ptr, unsigned long);
int
mpfr_const_euler_internal (mpfr_t x, mpfr_rnd_t rnd)
{
mpfr_prec_t prec = MPFR_PREC(x), m, log2m;
mpfr_t y, z;
unsigned long n;
int inexact;
MPFR_ZIV_DECL (loop);
log2m = MPFR_INT_CEIL_LOG2 (prec);
m = prec + 2 * log2m + 23;
mpfr_init2 (y, m);
mpfr_init2 (z, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_exp_t exp_S, err;
/* since prec >= 1, we have m >= 24 here, which ensures n >= 9 below */
n = 1 + (unsigned long) ((double) m * LOG2 / 2.0);
MPFR_ASSERTD (n >= 9);
mpfr_const_euler_S2 (y, n); /* error <= 3 ulps */
exp_S = MPFR_EXP(y);
mpfr_set_ui (z, n, MPFR_RNDN);
mpfr_log (z, z, MPFR_RNDD); /* error <= 1 ulp */
mpfr_sub (y, y, z, MPFR_RNDN); /* S'(n) - log(n) */
/* the error is less than 1/2 + 3*2^(exp_S-EXP(y)) + 2^(EXP(z)-EXP(y))
<= 1/2 + 2^(exp_S+2-EXP(y)) + 2^(EXP(z)-EXP(y))
<= 1/2 + 2^(1+MAX(exp_S+2,EXP(z))-EXP(y)) */
err = 1 + MAX(exp_S + 2, MPFR_EXP(z)) - MPFR_EXP(y);
err = (err >= -1) ? err + 1 : 0; /* error <= 2^err ulp(y) */
exp_S = MPFR_EXP(y);
mpfr_const_euler_R (z, n); /* err <= ulp(1/2) = 2^(-m) */
mpfr_sub (y, y, z, MPFR_RNDN);
/* err <= 1/2 ulp(y) + 2^(-m) + 2^(err + exp_S - EXP(y)) ulp(y).
Since the result is between 0.5 and 1, ulp(y) = 2^(-m).
So we get 3/2*ulp(y) + 2^(err + exp_S - EXP(y)) ulp(y).
3/2 + 2^e <= 2^(e+1) for e>=1, and <= 2^2 otherwise */
err = err + exp_S - MPFR_EXP(y);
err = (err >= 1) ? err + 1 : 2;
if (MPFR_LIKELY (MPFR_CAN_ROUND (y, m - err, prec, rnd)))
break;
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (y, m);
mpfr_set_prec (z, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (x, y, rnd);
mpfr_clear (y);
mpfr_clear (z);
return inexact; /* always inexact */
}
static void
mpfr_const_euler_S2_aux (mpz_t P, mpz_t Q, mpz_t T, unsigned long n,
unsigned long a, unsigned long b, int need_P)
{
if (a + 1 == b)
{
mpz_set_ui (P, n);
if (a > 1)
mpz_mul_si (P, P, 1 - (long) a);
mpz_set (T, P);
mpz_set_ui (Q, a);
mpz_mul_ui (Q, Q, a);
}
else
{
unsigned long c = (a + b) / 2;
mpz_t P2, Q2, T2;
mpfr_const_euler_S2_aux (P, Q, T, n, a, c, 1);
mpz_init (P2);
mpz_init (Q2);
mpz_init (T2);
mpfr_const_euler_S2_aux (P2, Q2, T2, n, c, b, 1);
mpz_mul (T, T, Q2);
mpz_mul (T2, T2, P);
mpz_add (T, T, T2);
if (need_P)
mpz_mul (P, P, P2);
mpz_mul (Q, Q, Q2);
mpz_clear (P2);
mpz_clear (Q2);
mpz_clear (T2);
/* divide by 2 if possible */
{
unsigned long v2;
v2 = mpz_scan1 (P, 0);
c = mpz_scan1 (Q, 0);
if (c < v2)
v2 = c;
c = mpz_scan1 (T, 0);
if (c < v2)
v2 = c;
if (v2)
{
mpz_tdiv_q_2exp (P, P, v2);
mpz_tdiv_q_2exp (Q, Q, v2);
mpz_tdiv_q_2exp (T, T, v2);
}
}
}
}
/* computes S(n) = sum(n^k*(-1)^(k-1)/k!/k, k=1..ceil(4.319136566 * n))
using binary splitting.
We have S(n) = sum(f(k), k=1..N) with N=ceil(4.319136566 * n)
and f(k) = n^k*(-1)*(k-1)/k!/k,
thus f(k)/f(k-1) = -n*(k-1)/k^2
*/
static void
mpfr_const_euler_S2 (mpfr_t x, unsigned long n)
{
mpz_t P, Q, T;
unsigned long N = (unsigned long) (ALPHA * (double) n + 1.0);
mpz_init (P);
mpz_init (Q);
mpz_init (T);
mpfr_const_euler_S2_aux (P, Q, T, n, 1, N + 1, 0);
mpfr_set_z (x, T, MPFR_RNDN);
mpfr_div_z (x, x, Q, MPFR_RNDN);
mpz_clear (P);
mpz_clear (Q);
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).
*/
static 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_fdiv_q_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_fdiv_q_ui (s, s, n); /* err <= 1 + (n+1)/2 */
MPFR_ASSERTN (MPFR_PREC(x) >= mpz_sizeinbase(s, 2));
mpfr_set_z (x, s, MPFR_RNDD); /* exact */
mpfr_div_2ui (x, x, m, MPFR_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, -(long)n, MPFR_RNDD); /* assumed exact */
mpfr_exp (y, y, MPFR_RNDD); /* err <= ulp(y) <= exp(-n)*2^(1-m) */
mpfr_mul (x, x, y, MPFR_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);
}