path: root/src/cos.c

/* mpfr_cos -- cosine of a floating-point number

Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
Contributed by the AriC and Caramel 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 << (CHAR_BIT * sizeof(unsigned long) / 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)
    {
      MPFR_SAVE_EXPO_FREE (expo);
      return mpfr_cos_fast (y, x, rnd_mode);
    }

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

  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);
    }

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