niceSOFT/mpfr | Gitverse

src/cos.c
299 строк9 KB
Vincent Lefèvre
03 янв 2018, 13:53
03 янв 2018, 13:532fce388
/* mpfr_cos -- cosine of a floating-point number

Copyright 2001-2018 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"

static int
mpfr_cos_fast (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
  int inex;

  inex = mpfr_sincos_fast (NULL, y, x, rnd_mode);
  inex = inex >> 2; /* 0: exact, 1: rounded up, 2: rounded down */
  return (inex == 2) ? -1 : inex;
}

/* f <- 1 - r/2! + r^2/4! + ... + (-1)^l r^l/(2l)! + ...
   Assumes |r| < 1/2, and f, r have the same precision.
   Returns e such that the error on f is bounded by 2^e ulps.
*/
static int
mpfr_cos2_aux (mpfr_ptr f, mpfr_srcptr r)
{
  mpz_t x, t, s;
  mpfr_exp_t ex, l, m;
  mpfr_prec_t p, q;
  unsigned long i, maxi, imax;

  MPFR_ASSERTD(mpfr_get_exp (r) <= -1);

  /* compute minimal i such that i*(i+1) does not fit in an unsigned long,
     assuming that there are no padding bits. */
  maxi = 1UL << (sizeof(unsigned long) * CHAR_BIT / 2);
  if (maxi * (maxi / 2) == 0) /* test checked at compile time */
    {
      /* can occur only when there are padding bits. */
      /* maxi * (maxi-1) is representable iff maxi * (maxi / 2) != 0 */
      do
        maxi /= 2;
      while (maxi * (maxi / 2) == 0);
    }

  mpz_init (x);
  mpz_init (s);
  mpz_init (t);
  ex = mpfr_get_z_2exp (x, r); /* r = x*2^ex */

  /* remove trailing zeroes */
  l = mpz_scan1 (x, 0);
  ex += l;
  mpz_fdiv_q_2exp (x, x, l);

  /* since |r| < 1, r = x*2^ex, and x is an integer, necessarily ex < 0 */

  p = mpfr_get_prec (f); /* same than r */
  /* bound for number of iterations */
  imax = p / (-mpfr_get_exp (r));
  imax += (imax == 0);
  q = 2 * MPFR_INT_CEIL_LOG2(imax) + 4; /* bound for (3l)^2 */

  mpz_set_ui (s, 1); /* initialize sum with 1 */
  mpz_mul_2exp (s, s, p + q); /* scale all values by 2^(p+q) */
  mpz_set (t, s); /* invariant: t is previous term */
  for (i = 1; (m = mpz_sizeinbase (t, 2)) >= q; i += 2)
    {
      /* adjust precision of x to that of t */
      l = mpz_sizeinbase (x, 2);
      if (l > m)
        {
          l -= m;
          mpz_fdiv_q_2exp (x, x, l);
          ex += l;
        }
      /* multiply t by r */
      mpz_mul (t, t, x);
      mpz_fdiv_q_2exp (t, t, -ex);
      /* divide t by i*(i+1) */
      if (i < maxi)
        mpz_fdiv_q_ui (t, t, i * (i + 1));
      else
        {
          mpz_fdiv_q_ui (t, t, i);
          mpz_fdiv_q_ui (t, t, i + 1);
        }
      /* if m is the (current) number of bits of t, we can consider that
         all operations on t so far had precision >= m, so we can prove
         by induction that the relative error on t is of the form
         (1+u)^(3l)-1, where |u| <= 2^(-m), and l=(i+1)/2 is the # of loops.
         Since |(1+x^2)^(1/x) - 1| <= 4x/3 for |x| <= 1/2,
         for |u| <= 1/(3l)^2, the absolute error is bounded by
         4/3*(3l)*2^(-m)*t <= 4*l since |t| < 2^m.
         Therefore the error on s is bounded by 2*l*(l+1). */
      /* add or subtract to s */
      if (i % 4 == 1)
        mpz_sub (s, s, t);
      else
        mpz_add (s, s, t);
    }

  mpfr_set_z (f, s, MPFR_RNDN);
  mpfr_div_2ui (f, f, p + q, MPFR_RNDN);

  mpz_clear (x);
  mpz_clear (s);
  mpz_clear (t);

  l = (i - 1) / 2; /* number of iterations */
  return 2 * MPFR_INT_CEIL_LOG2 (l + 1) + 1; /* bound is 2l(l+1) */
}

int
mpfr_cos (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
  mpfr_prec_t K0, K, precy, m, k, l;
  int inexact, reduce = 0;
  mpfr_t r, s, xr, c;
  mpfr_exp_t exps, cancel = 0, expx;
  MPFR_ZIV_DECL (loop);
  MPFR_SAVE_EXPO_DECL (expo);
  MPFR_GROUP_DECL (group);

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

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (x));
          return mpfr_set_ui (y, 1, rnd_mode);
        }
    }

  MPFR_SAVE_EXPO_MARK (expo);

  /* cos(x) = 1-x^2/2 + ..., so error < 2^(2*EXP(x)-1) */
  expx = MPFR_GET_EXP (x);
  MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_one, -2 * expx,
                                    1, 0, rnd_mode, expo, {});

  /* Compute initial precision */
  precy = MPFR_PREC (y);

  if (precy >= MPFR_SINCOS_THRESHOLD)
    {
      inexact = mpfr_cos_fast (y, x, rnd_mode);
      goto end;
    }

  K0 = __gmpfr_isqrt (precy / 3);
  m = precy + 2 * MPFR_INT_CEIL_LOG2 (precy) + 2 * K0 + 4;

  if (expx >= 3)
    {
      reduce = 1;
      /* As expx + m - 1 will silently be converted into mpfr_prec_t
         in the mpfr_init2 call, the assert below may be useful to
         avoid undefined behavior. */
      MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX);
      mpfr_init2 (c, expx + m - 1);
      mpfr_init2 (xr, m);
    }

  MPFR_GROUP_INIT_2 (group, m, r, s);
  MPFR_ZIV_INIT (loop, m);
  for (;;)
    {
      /* If |x| >= 4, first reduce x cmod (2*Pi) into xr, using mpfr_remainder:
         let e = EXP(x) >= 3, and m the target precision:
         (1) c <- 2*Pi              [precision e+m-1, nearest]
         (2) xr <- remainder (x, c) [precision m, nearest]
         We have |c - 2*Pi| <= 1/2ulp(c) = 2^(3-e-m)
                 |xr - x - k c| <= 1/2ulp(xr) <= 2^(1-m)
                 |k| <= |x|/(2*Pi) <= 2^(e-2)
         Thus |xr - x - 2kPi| <= |k| |c - 2Pi| + 2^(1-m) <= 2^(2-m).
         It follows |cos(xr) - cos(x)| <= 2^(2-m). */
      if (reduce)
        {
          mpfr_const_pi (c, MPFR_RNDN);
          mpfr_mul_2ui (c, c, 1, MPFR_RNDN); /* 2Pi */
          mpfr_remainder (xr, x, c, MPFR_RNDN);
          if (MPFR_IS_ZERO(xr))
            goto ziv_next;
          /* now |xr| <= 4, thus r <= 16 below */
          mpfr_mul (r, xr, xr, MPFR_RNDU); /* err <= 1 ulp */
        }
      else
        mpfr_mul (r, x, x, MPFR_RNDU); /* err <= 1 ulp */

      /* now |x| < 4 (or xr if reduce = 1), thus |r| <= 16 */

      /* we need |r| < 1/2 for mpfr_cos2_aux, i.e., EXP(r) - 2K <= -1 */
      K = K0 + 1 + MAX(0, MPFR_GET_EXP(r)) / 2;
      /* since K0 >= 0, if EXP(r) < 0, then K >= 1, thus EXP(r) - 2K <= -3;
         otherwise if EXP(r) >= 0, then K >= 1/2 + EXP(r)/2, thus
         EXP(r) - 2K <= -1 */

      MPFR_SET_EXP (r, MPFR_GET_EXP (r) - 2 * K); /* Can't overflow! */

      /* s <- 1 - r/2! + ... + (-1)^l r^l/(2l)! */
      l = mpfr_cos2_aux (s, r);
      /* l is the error bound in ulps on s */
      MPFR_SET_ONE (r);
      for (k = 0; k < K; k++)
        {
          mpfr_sqr (s, s, MPFR_RNDU);            /* err <= 2*olderr */
          MPFR_SET_EXP (s, MPFR_GET_EXP (s) + 1); /* Can't overflow */
          mpfr_sub (s, s, r, MPFR_RNDN);         /* err <= 4*olderr */
          if (MPFR_IS_ZERO(s))
            goto ziv_next;
          MPFR_ASSERTD (MPFR_GET_EXP (s) <= 1);
        }

      /* The absolute error on s is bounded by (2l+1/3)*2^(2K-m)
         2l+1/3 <= 2l+1.
         If |x| >= 4, we need to add 2^(2-m) for the argument reduction
         by 2Pi: if K = 0, this amounts to add 4 to 2l+1/3, i.e., to add
         2 to l; if K >= 1, this amounts to add 1 to 2*l+1/3. */
      l = 2 * l + 1;
      if (reduce)
        l += (K == 0) ? 4 : 1;
      k = MPFR_INT_CEIL_LOG2 (l) + 2 * K;
      /* now the error is bounded by 2^(k-m) = 2^(EXP(s)-err) */

      exps = MPFR_GET_EXP (s);
      if (MPFR_LIKELY (MPFR_CAN_ROUND (s, exps + m - k, precy, rnd_mode)))
        break;

      if (MPFR_UNLIKELY (exps == 1))
        /* s = 1 or -1, and except x=0 which was already checked above,
           cos(x) cannot be 1 or -1, so we can round if the error is less
           than 2^(-precy) for directed rounding, or 2^(-precy-1) for rounding
           to nearest. */
        {
          if (m > k && (m - k >= precy + (rnd_mode == MPFR_RNDN)))
            {
              /* If round to nearest or away, result is s = 1 or -1,
                 otherwise it is round(nexttoward (s, 0)). However in order to
                 have the inexact flag correctly set below, we set |s| to
                 1 - 2^(-m) in all cases. */
              mpfr_nexttozero (s);
              break;
            }
        }

      if (exps < cancel)
        {
          m += cancel - exps;
          cancel = exps;
        }

    ziv_next:
      MPFR_ZIV_NEXT (loop, m);
      MPFR_GROUP_REPREC_2 (group, m, r, s);
      if (reduce)
        {
          mpfr_set_prec (xr, m);
          mpfr_set_prec (c, expx + m - 1);
        }
    }
  MPFR_ZIV_FREE (loop);
  inexact = mpfr_set (y, s, rnd_mode);
  MPFR_GROUP_CLEAR (group);
  if (reduce)
    {
      mpfr_clear (xr);
      mpfr_clear (c);
    }

 end:
  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, inexact, rnd_mode);
}