--- /dev/null
+/* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument.
+
+Copyright 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"
+
+int
+mpfr_zeta_ui (mpfr_ptr z, unsigned long m, mp_rnd_t r)
+{
+ MPFR_ZIV_DECL (loop);
+
+ if (m == 0)
+ {
+ mpfr_set_ui (z, 1, r);
+ mpfr_div_2ui (z, z, 1, r);
+ MPFR_CHANGE_SIGN (z);
+ MPFR_RET (0);
+ }
+ else if (m == 1)
+ {
+ MPFR_SET_INF (z);
+ MPFR_SET_POS (z);
+ return 0;
+ }
+ else /* m >= 2 */
+ {
+ mp_prec_t p = MPFR_PREC(z);
+ unsigned long n, k, err, kbits;
+ mpz_t d, t, s, q;
+ mpfr_t y;
+ int inex;
+
+ if (m >= p) /* 2^(-m) < ulp(1) = 2^(1-p). This means that
+ 2^(-m) <= 1/2*ulp(1). We have 3^(-m)+4^(-m)+... < 2^(-m)
+ i.e. zeta(m) < 1+2*2^(-m) for m >= 3 */
+
+ {
+ if (m == 2) /* necessarily p=2 */
+ return mpfr_set_ui_2exp (z, 13, -3, r);
+ else if (r == GMP_RNDZ || r == GMP_RNDD || (r == GMP_RNDN && m > p))
+ {
+ mpfr_set_ui (z, 1, r);
+ return -1;
+ }
+ else
+ {
+ mpfr_set_ui (z, 1, r);
+ mpfr_nextabove (z);
+ return 1;
+ }
+ }
+
+ /* now treat also the case where zeta(m) - (1+1/2^m) < 1/2*ulp(1),
+ and the result is either 1+2^(-m) or 1+2^(-m)+2^(1-p). */
+ mpfr_init2 (y, 31);
+
+ if (m >= p / 2) /* otherwise 4^(-m) > 2^(-p) */
+ {
+ /* the following is a lower bound for log(3)/log(2) */
+ mpfr_set_str_binary (y, "1.100101011100000000011010001110");
+ mpfr_mul_ui (y, y, m, GMP_RNDZ); /* lower bound for log2(3^m) */
+ if (mpfr_cmp_ui (y, p + 2) >= 0)
+ {
+ mpfr_clear (y);
+ mpfr_set_ui (z, 1, GMP_RNDZ);
+ mpfr_div_2ui (z, z, m, GMP_RNDZ);
+ mpfr_add_ui (z, z, 1, GMP_RNDZ);
+ if (r != GMP_RNDU)
+ return -1;
+ mpfr_nextabove (z);
+ return 1;
+ }
+ }
+
+ mpz_init (s);
+ mpz_init (d);
+ mpz_init (t);
+ mpz_init (q);
+
+ p += MPFR_INT_CEIL_LOG2(p); /* account of the n term in the error */
+
+ p += MPFR_INT_CEIL_LOG2(p) + 15; /* initial value */
+
+ MPFR_ZIV_INIT (loop, p);
+ for(;;)
+ {
+ /* 0.39321985067869744 = log(2)/log(3+sqrt(8)) */
+ n = 1 + (unsigned long) (0.39321985067869744 * (double) p);
+ err = n + 4;
+
+ mpfr_set_prec (y, p);
+
+ /* computation of the d[k] */
+ mpz_set_ui (s, 0);
+ mpz_set_ui (t, 1);
+ mpz_mul_2exp (t, t, 2 * n - 1); /* t[n] */
+ mpz_set (d, t);
+ for (k = n; k > 0; k--)
+ {
+ count_leading_zeros (kbits, k);
+ kbits = BITS_PER_MP_LIMB - kbits;
+ /* if k^m is too large, use mpz_tdiv_q */
+ if (m * kbits > 2 * BITS_PER_MP_LIMB)
+ {
+ /* if we know in advance that k^m > d, then floor(d/k^m) will
+ be zero below, so there is no need to compute k^m */
+ kbits = (kbits - 1) * m + 1;
+ /* k^m has at least kbits bits */
+ if (kbits > mpz_sizeinbase (d, 2))
+ mpz_set_ui (q, 0);
+ else
+ {
+ mpz_ui_pow_ui (q, k, m);
+ mpz_tdiv_q (q, d, q);
+ }
+ }
+ else /* use several mpz_tdiv_q_ui calls */
+ {
+ unsigned long km = k, mm = m - 1;
+ while (mm > 0 && km < ULONG_MAX / k)
+ {
+ km *= k;
+ mm --;
+ }
+ mpz_tdiv_q_ui (q, d, km);
+ while (mm > 0)
+ {
+ km = k;
+ mm --;
+ while (mm > 0 && km < ULONG_MAX / k)
+ {
+ km *= k;
+ mm --;
+ }
+ mpz_tdiv_q_ui (q, q, km);
+ }
+ }
+ if (k % 2)
+ mpz_add (s, s, q);
+ else
+ mpz_sub (s, s, q);
+
+ /* we have d[k] = sum(t[i], i=k+1..n)
+ with t[i] = n*(n+i-1)!*4^i/(n-i)!/(2i)!
+ t[k-1]/t[k] = k*(2k-1)/(n-k+1)/(n+k-1)/2 */
+#if (BITS_PER_MP_LIMB == 32)
+#define KMAX 46341 /* max k such that k*(2k-1) < 2^32 */
+#elif (BITS_PER_MP_LIMB == 64)
+#define KMAX 3037000500
+#endif
+#ifdef KMAX
+ if (k <= KMAX)
+ mpz_mul_ui (t, t, k * (2 * k - 1));
+ else
+#endif
+ {
+ mpz_mul_ui (t, t, k);
+ mpz_mul_ui (t, t, 2 * k - 1);
+ }
+ mpz_div_2exp (t, t, 1);
+ if (n < 1UL << (BITS_PER_MP_LIMB / 2))
+ /* (n - k + 1) * (n + k - 1) < n^2 */
+ mpz_divexact_ui (t, t, (n - k + 1) * (n + k - 1));
+ else
+ {
+ mpz_divexact_ui (t, t, n - k + 1);
+ mpz_divexact_ui (t, t, n + k - 1);
+ }
+ mpz_add (d, d, t);
+ }
+
+ /* multiply by 1/(1-2^(1-m)) = 1 + 2^(1-m) + 2^(2-m) + ... */
+ mpz_div_2exp (t, s, m - 1);
+ do
+ {
+ err ++;
+ mpz_add (s, s, t);
+ mpz_div_2exp (t, t, m - 1);
+ }
+ while (mpz_cmp_ui (t, 0) > 0);
+
+ /* divide by d[n] */
+ mpz_mul_2exp (s, s, p);
+ mpz_tdiv_q (s, s, d);
+ mpfr_set_z (y, s, GMP_RNDN);
+ mpfr_div_2ui (y, y, p, GMP_RNDN);
+
+ err = MPFR_INT_CEIL_LOG2 (err);
+
+ if (MPFR_LIKELY(MPFR_CAN_ROUND (y, p - err, MPFR_PREC(z), r)))
+ break;
+
+ MPFR_ZIV_NEXT (loop, p);
+ }
+ MPFR_ZIV_FREE (loop);
+
+ mpz_clear (d);
+ mpz_clear (t);
+ mpz_clear (q);
+ mpz_clear (s);
+ inex = mpfr_set (z, y, r);
+ mpfr_clear (y);
+ return inex;
+ }
+}
projects / msp430-gcc.git / blobdiff