--- /dev/null
+/* mpfr_const_euler -- Euler's constant
+
+Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
+Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, 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, mp_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, mp_rnd_t rnd)
+{
+ mp_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 (;;)
+ {
+ mp_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, GMP_RNDN);
+ mpfr_log (z, z, GMP_RNDD); /* error <= 1 ulp */
+ mpfr_sub (y, y, z, GMP_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, GMP_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, GMP_RNDN);
+ mpfr_div_z (x, x, Q, GMP_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_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 */
+ MPFR_ASSERTN (MPFR_PREC(x) >= mpz_sizeinbase(s, 2));
+ 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, -(long)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);
+}
projects / msp430-gcc.git / blobdiff