예제 #1
0
int
mpfr_sinh_cosh (mpfr_ptr sh, mpfr_ptr ch, mpfr_srcptr xt, mpfr_rnd_t rnd_mode)
{
    mpfr_t x;
    int inexact_sh, inexact_ch;

    MPFR_ASSERTN (sh != ch);

    MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg rnd=%d",
      mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
     ("sh[%Pu]=%.*Rg ch[%Pu]=%.*Rg",
      mpfr_get_prec (sh), mpfr_log_prec, sh,
      mpfr_get_prec (ch), mpfr_log_prec, ch));

    if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
    {
        if (MPFR_IS_NAN (xt))
        {
            MPFR_SET_NAN (ch);
            MPFR_SET_NAN (sh);
            MPFR_RET_NAN;
        }
        else if (MPFR_IS_INF (xt))
        {
            MPFR_SET_INF (sh);
            MPFR_SET_SAME_SIGN (sh, xt);
            MPFR_SET_INF (ch);
            MPFR_SET_POS (ch);
            MPFR_RET (0);
        }
        else /* xt is zero */
        {
            MPFR_ASSERTD (MPFR_IS_ZERO (xt));
            MPFR_SET_ZERO (sh);                   /* sinh(0) = 0 */
            MPFR_SET_SAME_SIGN (sh, xt);
            inexact_sh = 0;
            inexact_ch = mpfr_set_ui (ch, 1, rnd_mode); /* cosh(0) = 1 */
            return INEX(inexact_sh,inexact_ch);
        }
    }

    /* Warning: if we use MPFR_FAST_COMPUTE_IF_SMALL_INPUT here, make sure
       that the code also works in case of overlap (see sin_cos.c) */

    MPFR_TMP_INIT_ABS (x, xt);

    {
        mpfr_t s, c, ti;
        mpfr_exp_t d;
        mpfr_prec_t N;    /* Precision of the intermediary variables */
        long int err;    /* Precision of error */
        MPFR_ZIV_DECL (loop);
        MPFR_SAVE_EXPO_DECL (expo);
        MPFR_GROUP_DECL (group);

        MPFR_SAVE_EXPO_MARK (expo);

        /* compute the precision of intermediary variable */
        N = MPFR_PREC (ch);
        N = MAX (N, MPFR_PREC (sh));
        /* the optimal number of bits : see algorithms.ps */
        N = N + MPFR_INT_CEIL_LOG2 (N) + 4;

        /* initialise of intermediary variables */
        MPFR_GROUP_INIT_3 (group, N, s, c, ti);

        /* First computation of sinh_cosh */
        MPFR_ZIV_INIT (loop, N);
        for (;;)
        {
            MPFR_BLOCK_DECL (flags);

            /* compute sinh_cosh */
            MPFR_BLOCK (flags, mpfr_exp (s, x, MPFR_RNDD));
            if (MPFR_OVERFLOW (flags))
                /* exp(x) does overflow */
            {
                /* since cosh(x) >= exp(x), cosh(x) overflows too */
                inexact_ch = mpfr_overflow (ch, rnd_mode, MPFR_SIGN_POS);
                /* sinh(x) may be representable */
                inexact_sh = mpfr_sinh (sh, xt, rnd_mode);
                MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
                break;
            }
            d = MPFR_GET_EXP (s);
            mpfr_ui_div (ti, 1, s, MPFR_RNDU);  /* 1/exp(x) */
            mpfr_add (c, s, ti, MPFR_RNDU);     /* exp(x) + 1/exp(x) */
            mpfr_sub (s, s, ti, MPFR_RNDN);     /* exp(x) - 1/exp(x) */
            mpfr_div_2ui (c, c, 1, MPFR_RNDN);  /* 1/2(exp(x) + 1/exp(x)) */
            mpfr_div_2ui (s, s, 1, MPFR_RNDN);  /* 1/2(exp(x) - 1/exp(x)) */

            /* it may be that s is zero (in fact, it can only occur when exp(x)=1,
               and thus ti=1 too) */
            if (MPFR_IS_ZERO (s))
                err = N; /* double the precision */
            else
            {
                /* calculation of the error */
                d = d - MPFR_GET_EXP (s) + 2;
                /* error estimate: err = N-(__gmpfr_ceil_log2(1+pow(2,d)));*/
                err = N - (MAX (d, 0) + 1);
                if (MPFR_LIKELY (MPFR_CAN_ROUND (s, err, MPFR_PREC (sh),
                                                 rnd_mode) &&               \
                                 MPFR_CAN_ROUND (c, err, MPFR_PREC (ch),
                                                 rnd_mode)))
                {
                    inexact_sh = mpfr_set4 (sh, s, rnd_mode, MPFR_SIGN (xt));
                    inexact_ch = mpfr_set (ch, c, rnd_mode);
                    break;
                }
            }
            /* actualisation of the precision */
            N += err;
            MPFR_ZIV_NEXT (loop, N);
            MPFR_GROUP_REPREC_3 (group, N, s, c, ti);
        }
        MPFR_ZIV_FREE (loop);
        MPFR_GROUP_CLEAR (group);
        MPFR_SAVE_EXPO_FREE (expo);
    }

    /* now, let's raise the flags if needed */
    inexact_sh = mpfr_check_range (sh, inexact_sh, rnd_mode);
    inexact_ch = mpfr_check_range (ch, inexact_ch, rnd_mode);

    return INEX(inexact_sh,inexact_ch);
}
/* Don't need to save/restore exponent range: the cache does it.
   Catalan's constant is G = sum((-1)^k/(2*k+1)^2, k=0..infinity).
   We compute it using formula (31) of Victor Adamchik's page
   "33 representations for Catalan's constant"
   http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm

   G = Pi/8*log(2+sqrt(3)) + 3/8*sum(k!^2/(2k)!/(2k+1)^2,k=0..infinity)
*/
int
mpfr_const_catalan_internal (mpfr_ptr g, mpfr_rnd_t rnd_mode)
{
  mpfr_t x, y, z;
  mpz_t T, P, Q;
  mpfr_prec_t pg, p;
  int inex;
  MPFR_ZIV_DECL (loop);
  MPFR_GROUP_DECL (group);

  MPFR_LOG_FUNC (("rnd_mode=%d", rnd_mode), ("g[%#R]=%R inex=%d", g, g, inex));

  /* Here are the WC (max prec = 100.000.000)
     Once we have found a chain of 11, we only look for bigger chain.
     Found 3 '1' at 0
     Found 5 '1' at 9
     Found 6 '0' at 34
     Found 9 '1' at 176
     Found 11 '1' at 705
     Found 12 '0' at 913
     Found 14 '1' at 12762
     Found 15 '1' at 152561
     Found 16 '0' at 171725
     Found 18 '0' at 525355
     Found 20 '0' at 529245
     Found 21 '1' at 6390133
     Found 22 '0' at 7806417
     Found 25 '1' at 11936239
     Found 27 '1' at 51752950
  */
  pg = MPFR_PREC (g);
  p = pg + MPFR_INT_CEIL_LOG2 (pg) + 7;

  MPFR_GROUP_INIT_3 (group, p, x, y, z);
  mpz_init (T);
  mpz_init (P);
  mpz_init (Q);

  MPFR_ZIV_INIT (loop, p);
  for (;;) {
    mpfr_sqrt_ui (x, 3, MPFR_RNDU);
    mpfr_add_ui (x, x, 2, MPFR_RNDU);
    mpfr_log (x, x, MPFR_RNDU);
    mpfr_const_pi (y, MPFR_RNDU);
    mpfr_mul (x, x, y, MPFR_RNDN);
    S (T, P, Q, 0, (p - 1) / 2);
    mpz_mul_ui (T, T, 3);
    mpfr_set_z (y, T, MPFR_RNDU);
    mpfr_set_z (z, Q, MPFR_RNDD);
    mpfr_div (y, y, z, MPFR_RNDN);
    mpfr_add (x, x, y, MPFR_RNDN);
    mpfr_div_2ui (x, x, 3, MPFR_RNDN);

    if (MPFR_LIKELY (MPFR_CAN_ROUND (x, p - 5, pg, rnd_mode)))
      break;

    MPFR_ZIV_NEXT (loop, p);
    MPFR_GROUP_REPREC_3 (group, p, x, y, z);
  }
  MPFR_ZIV_FREE (loop);
  inex = mpfr_set (g, x, rnd_mode);

  MPFR_GROUP_CLEAR (group);
  mpz_clear (T);
  mpz_clear (P);
  mpz_clear (Q);

  return inex;
}