path: root/cos.c

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

Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.

This file is part of the MPFR Library.

The 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 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 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. */

#include <limits.h>

#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"

#if 1 /* new code, using mpz */

/* 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;
  mp_exp_t ex, l, m;
  mp_prec_t p, q;
  unsigned long i, maxi, imax;

  /* 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_exp (x, r); /* r = x*2^ex */

  /* remove trailing zeroes */
  l = mpz_scan1 (x, 0);
  ex += l;
  mpz_div_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));
  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_div_2exp (x, x, l);
	  ex += l;
	}
      /* multiply t by r */
      mpz_mul (t, t, x);
      mpz_div_2exp (t, t, -ex);
      /* divide t by i*(i+1) */
      if (i < maxi)
        mpz_div_ui (t, t, i * (i + 1));
      else
        {
          mpz_div_ui (t, t, i);
          mpz_div_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, GMP_RNDN);
  mpfr_div_2ui (f, f, p + q, GMP_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) */
}

#else /* previous code, using mpf */

/* s <- 1 - r/2! + r^2/4! + ... + (-1)^l r^l/(2l)! + ...
   Assumes |r| < 1.
   Returns e such that the error is bounded by 2^e ulps.
   (Let the index l0 of the last term (-1)^l r^l/(2l)!,
    the absolute error on s is at most 2 * l0 * 2^(-m).)
*/
static int
mpfr_cos2_aux (mpfr_ptr s, mpfr_srcptr r)
{
  unsigned int l, b = 2;
  mp_exp_t prec, m = MPFR_PREC (s);
  mpfr_t t;

  MPFR_ASSERTD (MPFR_GET_EXP (r) <= 0);

  mpfr_init2 (t, m);

  /* First step for l==1 can be simplified,
     futhermore multiply by 1 is not efficient since it is an exact
     multiplication (mulhigh failed and we must do a complete mul) */
  mpfr_div_2ui (t, r, 1, GMP_RNDN); /* exact */
  mpfr_sub (s, __gmpfr_one, t, GMP_RNDD);
  MPFR_ASSERTD (MPFR_GET_EXP (s) == 0);        /* check 1/2 <= s < 1 */

  for (l = 2; MPFR_GET_EXP (t) + m >= 0; l++)
    {
      mpfr_mul (t, t, r, GMP_RNDU);                /* err <= (3l-1) ulp */
      mpfr_div_ui (t, t, (unsigned long) (2*l-1)*(2*l), GMP_RNDU);
                                                   /* err <= 3l ulp */
      MPFR_ASSERTD (MPFR_IS_POS (t));
      MPFR_ASSERTD (MPFR_IS_POS (s));
      if (l % 2 == 0)
        mpfr_add (s, s, t, GMP_RNDD);
      else
        mpfr_sub (s, s, t, GMP_RNDD);
      MPFR_ASSERTD (MPFR_GET_EXP (s) == 0);        /* check 1/2 <= s < 1 */
      /* err(s) <= l * 2^(-m) */
      if (MPFR_UNLIKELY (3 * l > (1U << b)))
        b++;
      /* now 3l <= 2^b, we want 3l*ulp(t) <= 2^(-m)
         i.e. b+EXP(t)-PREC(t) <= -m */
      prec = m + MPFR_GET_EXP (t) + b;
      if (MPFR_LIKELY (prec >= MPFR_PREC_MIN))
	mpfr_prec_round (t, prec, GMP_RNDN);
    }
  mpfr_clear (t);

  return 1 + MPFR_INT_CEIL_LOG2 (l); /* bound is 2l ulps */
}

#endif

int
mpfr_cos (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
  mp_prec_t K0, K, precy, m, k, l, precx;
  int inexact;
  mpfr_t r, s;
  mp_exp_t exps, cancel = 0;
  MPFR_ZIV_DECL (loop);
  MPFR_SAVE_EXPO_DECL (expo);
  MPFR_GROUP_DECL (group);

  MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
                 ("y[%#R]=%R inexact=%d", y, 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, GMP_RNDN);
        }
    }

  MPFR_SAVE_EXPO_MARK (expo);

  /* cos(x) = 1-x^2/2 + ..., so error < 2^(2*EXP(x)-1) */
  MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, __gmpfr_one, -2 * MPFR_GET_EXP (x), 1,
                                    0, rnd_mode, inexact = _inexact; goto end);

  /* Compute initial precision */
  precy = MPFR_PREC (y);
  /* We can choose everything we want for K0.
     This formula has been created by trying many things...
     and is far from perfect */
  K0 = (MPFR_GET_EXP (x) > 0) ? (MPFR_GET_EXP (x)) : 0 ;
  precx = MPFR_PREC (x);
  if (precx > precy)
    precx = precy;
  precx = __gmpfr_isqrt (precx) * __gmpfr_isqrt (precy);
  K0 = __gmpfr_isqrt (precx / (1 + K0 + MPFR_INT_CEIL_LOG2 (precy) / 8) );
  m = precy + 3 * K0 + 4;
  if (MPFR_GET_EXP (x) >= 0)
    m += 5 * MPFR_GET_EXP (x);
  else
    m += -MPFR_GET_EXP (x);

  MPFR_GROUP_INIT_2 (group, m, r, s);
  MPFR_ZIV_INIT (loop, m);
  for (;;)
    {
      mpfr_mul (r, x, x, GMP_RNDU); /* err <= 1 ulp */

      /* we need that |r| < 1 for mpfr_cos2_aux, i.e. up(x^2)/2^(2K) < 1 */
      K = K0 + MAX (MPFR_GET_EXP (r), 0);

      /*mpfr_div_2ui (r, r, 2 * K, GMP_RNDN); r = (x/2^K)^2, err <= 1 ulp */
      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, GMP_RNDU);            /* err <= 2*olderr */
          MPFR_SET_EXP (s, MPFR_GET_EXP (s)+1); /* Can't overflow */
          mpfr_sub (s, s, r, GMP_RNDN);         /* err <= 4*olderr */
          MPFR_ASSERTD (MPFR_GET_EXP (s) <= 1);
        }

      /* absolute error on s is bounded by (2l+1/3)*2^(2K-m)
         2l+1/3 <= 2l+1 */
      k = MPFR_INT_CEIL_LOG2 (2*l+1) + 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 (exps + m - k > precy
              /* if round to nearest or away, result is s,
                 otherwise it is round(nexttoward (s, 0)) */
              && MPFR_IS_LIKE_RNDZ (rnd_mode, MPFR_IS_NEG (s)))
            mpfr_nexttozero (s);
          break;
        }

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

      MPFR_ZIV_NEXT (loop, m);
      MPFR_GROUP_REPREC_2 (group, m, r, s);
    }
  MPFR_ZIV_FREE (loop);
  inexact = mpfr_set (y, s, rnd_mode);
  MPFR_GROUP_CLEAR (group);

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