path: root/atan.c

/* mpfr_atan -- arc-tangent of a floating-point number

Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.

This file is part of the GNU MPFR Library, and was contributed by Mathieu Dutour.

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"

/* If x = p/2^r, put in y an approximation of atan(x)/x using 2^m terms
   for the series expansion, with an error of at most 1 ulp.
   Assumes |x| < 1.

   If X=x^2, we want 1 - X/3 + X^2/5 - ... + (-1)^k*X^k/(2k+1) + ...
*/
static void
mpfr_atan_aux (mpfr_ptr y, mpz_ptr p, long r, int m, mpz_t *tab)
{
  mpz_t *S, *Q, *ptoj;
  unsigned long n, i, k, j, l;
  mp_exp_t diff, expo;
  int im, done;
  mp_prec_t mult, *accu, *log2_nb_terms;
  mp_prec_t precy = MPFR_PREC(y);

  if (mpz_cmp_ui (p, 0) == 0)
    {
      mpfr_set_ui (y, 1, GMP_RNDN); /* limit(atan(x)/x, x=0) */
      return;
    }

  accu = (mp_prec_t*) (*__gmp_allocate_func) ((2 * m + 2) * sizeof (mp_prec_t));
  log2_nb_terms = accu + m + 1;

  /* Set Tables */
  S    = tab;           /* S */
  ptoj = S + 1*(m+1);   /* p^2^j Precomputed table */
  Q    = S + 2*(m+1);   /* Product of Odd integer  table  */

  /* From p to p^2, and r to 2r */
  mpz_mul (p, p, p);
  MPFR_ASSERTD (2 * r > r);
  r = 2 * r;

  /* Normalize p */
  n = mpz_scan1 (p, 0);
  mpz_tdiv_q_2exp (p, p, n); /* exact */
  MPFR_ASSERTD (r > n);
  r -= n;
  /* since |p/2^r| < 1, and p is a non-zero integer, necessarily r > 0 */

  MPFR_ASSERTD (mpz_sgn (p) > 0);
  MPFR_ASSERTD (m > 0);

  /* check if p=1 (special case) */
  l = 0;
  /*
    We compute by binary splitting, with X = x^2 = p/2^r:
    P(a,b) = p if a+1=b, P(a,c)*P(c,b) otherwise
    Q(a,b) = (2a+1)*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise
    S(a,b) = p*(2a+1) if a+1=b, Q(c,b)*S(a,c)+Q(a,c)*P(a,c)*S(c,b) otherwise
    Then atan(x)/x ~ S(0,i)/Q(0,i) for i so that (p/2^r)^i/i is small enough.
    The factor 2^(r*(b-a)) in Q(a,b) is implicit, thus we have to take it
    into account when we compute with Q.
  */
  accu[0] = 0; /* accu[k] = Mult[0] + ... + Mult[k], where Mult[j] is the
                  number of bits of the corresponding term S[j]/Q[j] */
  if (mpz_cmp_ui (p, 1) != 0)
    {
      /* p <> 1: precompute ptoj table */
      mpz_set (ptoj[0], p);
      for (im = 1 ; im <= m ; im ++)
        mpz_mul (ptoj[im], ptoj[im - 1], ptoj[im - 1]);
      /* main loop */
      n = 1UL << m;
      /* the ith term being X^i/(2i+1) with X=p/2^r, we can stop when
         p^i/2^(r*i) < 2^(-precy), i.e. r*i > precy + log2(p^i) */
      for (i = k = done = 0; (i < n) && (done == 0); i += 2, k ++)
        {
          /* initialize both S[k],Q[k] and S[k+1],Q[k+1] */
          mpz_set_ui (Q[k+1], 2 * i + 3); /* Q(i+1,i+2) */
          mpz_mul_ui (S[k+1], p, 2 * i + 1); /* S(i+1,i+2) */
          mpz_mul_2exp (S[k], Q[k+1], r);
          mpz_sub (S[k], S[k], S[k+1]); /* S(i,i+2) */
          mpz_mul_ui (Q[k], Q[k+1], 2 * i + 1); /* Q(i,i+2) */
          log2_nb_terms[k] = 1; /* S[k]/Q[k] corresponds to 2 terms */
          for (j = (i + 2) >> 1, l = 1; (j & 1) == 0; l ++, j >>= 1, k --)
            {
              /* invariant: S[k-1]/Q[k-1] and S[k]/Q[k] correspond
                 to 2^l terms each. We combine them into S[k-1]/Q[k-1] */
              MPFR_ASSERTD (k > 0);
              mpz_mul (S[k], S[k], Q[k-1]);
              mpz_mul (S[k], S[k], ptoj[l]);
              mpz_mul (S[k-1], S[k-1], Q[k]);
              mpz_mul_2exp (S[k-1], S[k-1], r << l);
              mpz_add (S[k-1], S[k-1], S[k]);
              mpz_mul (Q[k-1], Q[k-1], Q[k]);
              log2_nb_terms[k-1] = l + 1;
              /* now S[k-1]/Q[k-1] corresponds to 2^(l+1) terms */
              MPFR_MPZ_SIZEINBASE2(mult, ptoj[l+1]);
              /* FIXME: precompute bits(ptoj[l+1]) outside the loop? */
              mult = (r << (l + 1)) - mult - 1;
              accu[k-1] = (k == 1) ? mult : accu[k-2] + mult;
              if (accu[k-1] > precy)
                done = 1;
            }
        }
    }
  else /* special case p=1: the ith term being X^i/(2i+1) with X=1/2^r,
          we can stop when r*i > precy i.e. i > precy/r */
    {
      n = 1UL << m;
      for (i = k = 0; (i < n) && (i <= precy / r); i += 2, k ++)
        {
          mpz_set_ui (Q[k + 1], 2 * i + 3);
          mpz_mul_2exp (S[k], Q[k+1], r);
          mpz_sub_ui (S[k], S[k], 1 + 2 * i);
          mpz_mul_ui (Q[k], Q[k + 1], 1 + 2 * i);
          log2_nb_terms[k] = 1; /* S[k]/Q[k] corresponds to 2 terms */
          for (j = (i + 2) >> 1, l = 1; (j & 1) == 0; l++, j >>= 1, k --)
            {
              MPFR_ASSERTD (k > 0);
              mpz_mul (S[k], S[k], Q[k-1]);
              mpz_mul (S[k-1], S[k-1], Q[k]);
              mpz_mul_2exp (S[k-1], S[k-1], r << l);
              mpz_add (S[k-1], S[k-1], S[k]);
              mpz_mul (Q[k-1], Q[k-1], Q[k]);
              log2_nb_terms[k-1] = l + 1;
            }
        }
    }

  /* we need to combine S[0]/Q[0]...S[k-1]/Q[k-1] */
  l = 0; /* number of terms accumulated in S[k]/Q[k] */
  while (k > 1)
    {
      k --;
      /* combine S[k-1]/Q[k-1] and S[k]/Q[k] */
      j = log2_nb_terms[k-1];
      mpz_mul (S[k], S[k], Q[k-1]);
      if (mpz_cmp_ui (p, 1) != 0)
        mpz_mul (S[k], S[k], ptoj[j]);
      mpz_mul (S[k-1], S[k-1], Q[k]);
      l += 1 << log2_nb_terms[k];
      mpz_mul_2exp (S[k-1], S[k-1], r * l);
      mpz_add (S[k-1], S[k-1], S[k]);
      mpz_mul (Q[k-1], Q[k-1], Q[k]);
    }
  (*__gmp_free_func) (accu, (2 * m + 2) * sizeof (mp_prec_t));

  MPFR_MPZ_SIZEINBASE2 (diff, S[0]);
  diff -= 2 * precy;
  expo = diff;
  if (diff >= 0)
    mpz_tdiv_q_2exp (S[0], S[0], diff);
  else
    mpz_mul_2exp (S[0], S[0], -diff);

  MPFR_MPZ_SIZEINBASE2 (diff, Q[0]);
  diff -= precy;
  expo -= diff;
  if (diff >= 0)
    mpz_tdiv_q_2exp (Q[0], Q[0], diff);
  else
    mpz_mul_2exp (Q[0], Q[0], -diff);

  mpz_tdiv_q (S[0], S[0], Q[0]);
  mpfr_set_z (y, S[0], GMP_RNDD);
  MPFR_SET_EXP (y, MPFR_EXP(y) + expo - r * (i - 1));
}

int
mpfr_atan (mpfr_ptr atan, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
  mpfr_t xp, arctgt, sk, tmp, tmp2;
  mpz_t  ukz;
  mpz_t *tabz;
  mp_exp_t exptol;
  mp_prec_t prec, realprec;
  unsigned long twopoweri;
  int comparaison, inexact, inexact2;
  int i, n0, oldn0;
  MPFR_GROUP_DECL (group);
  MPFR_SAVE_EXPO_DECL (expo);
  MPFR_ZIV_DECL (loop);

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

  /* Singular cases */
  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x))
        {
          MPFR_SET_NAN (atan);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (x))
        {
          if (MPFR_IS_POS (x))  /* arctan(+inf) = Pi/2 */
            inexact = mpfr_const_pi (atan, rnd_mode);
          else /* arctan(-inf) = -Pi/2 */
            {
              inexact = -mpfr_const_pi (atan,
                                        MPFR_INVERT_RND (rnd_mode));
              MPFR_CHANGE_SIGN (atan);
            }
          inexact2 = mpfr_div_2ui (atan, atan, 1, rnd_mode);
          if (MPFR_UNLIKELY (inexact2))
            inexact = inexact2; /* An underflow occurs */
          MPFR_RET (inexact);
        }
      else /* x is necessarily 0 */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (x));
          MPFR_SET_ZERO (atan);
          MPFR_SET_SAME_SIGN (atan, x);
          MPFR_RET (0);
        }
    }

  /* atan(x) = x - x^3/3 + x^5/5...
     so the error is < 2^(3*EXP(x)-1)
     so `EXP(x)-(3*EXP(x)-1)` = -2*EXP(x)+1 */
  MPFR_FAST_COMPUTE_IF_SMALL_INPUT (atan, x, -2 * MPFR_GET_EXP (x), 1, 0,
                                    rnd_mode, {});

  /* Set x_p=|x| */
  MPFR_TMP_INIT_ABS (xp, x);

  /* Other simple case arctan(-+1)=-+pi/4 */
  comparaison = mpfr_cmp_ui (xp, 1);
  if (MPFR_UNLIKELY (comparaison == 0))
    {
      int neg = MPFR_IS_NEG (x);
      inexact = mpfr_const_pi (atan, MPFR_IS_POS (x) ? rnd_mode
                               : MPFR_INVERT_RND (rnd_mode));
      if (neg)
        {
          inexact = -inexact;
          MPFR_CHANGE_SIGN (atan);
        }
      inexact2 = mpfr_div_2ui (atan, atan, 2, rnd_mode);
      if (MPFR_UNLIKELY (inexact2))
        inexact = inexact2; /* an underflow occurs */
      return inexact;
    }

  realprec = MPFR_PREC (atan) + MPFR_INT_CEIL_LOG2 (MPFR_PREC (atan)) + 4;
  prec = realprec + BITS_PER_MP_LIMB;

  MPFR_SAVE_EXPO_MARK (expo);

  /* Initialisation    */
  mpz_init (ukz);
  MPFR_GROUP_INIT_4 (group, prec, sk, tmp, tmp2, arctgt);
  oldn0 = 0;
  tabz = (mpz_t *) 0;

  MPFR_ZIV_INIT (loop, prec);
  for (;;)
    {
      /* First, if |x| < 1, we need to have more prec to be able to round (sup)
         n0 = ceil(log(prec_requested + 2 + 1+ln(2.4)/ln(2))/log(2)) */
      mp_prec_t sup;
#if 0
      sup = 1;
      if (MPFR_GET_EXP (xp) < 0
          && (mpfr_uexp_t) (2-MPFR_GET_EXP (xp)) > realprec)
        sup = (mpfr_uexp_t) (2-MPFR_GET_EXP (xp)) - realprec;
#else
      sup = MPFR_GET_EXP (xp) < 0 ? 2-MPFR_GET_EXP (xp) : 1;
#endif
      n0 = MPFR_INT_CEIL_LOG2 ((realprec + sup) + 3);
      MPFR_ASSERTD (3*n0 > 2);
      prec = (realprec + sup) + 1 + MPFR_INT_CEIL_LOG2 (3*n0-2);

      /* Initialisation */
      MPFR_GROUP_REPREC_4 (group, prec, sk, tmp, tmp2, arctgt);
      if (MPFR_LIKELY (oldn0 == 0))
        {
          oldn0 = 3*(n0+1);
          tabz = (mpz_t *) (*__gmp_allocate_func) (oldn0*sizeof (mpz_t));
          for (i = 0; i < oldn0; i++)
            mpz_init (tabz[i]);
        }
      else if (MPFR_UNLIKELY (oldn0 < 3*n0+1))
        {
          tabz = (mpz_t *) (*__gmp_reallocate_func)
            (tabz, oldn0*sizeof (mpz_t), 3*(n0+1)*sizeof (mpz_t));
          for (i = oldn0; i < 3*(n0+1); i++)
            mpz_init (tabz[i]);
          oldn0 = 3*(n0+1);
        }

      if (comparaison > 0)
        mpfr_ui_div (sk, 1, xp, GMP_RNDN);
      else
        mpfr_set (sk, xp, GMP_RNDN);

      /* sk is 1/|x| if |x| > 1, and |x| otherwise, i.e. min(|x|, 1/|x|) */

      /* Assignation  */
      MPFR_SET_ZERO (arctgt);
      twopoweri = 1<<0;
      MPFR_ASSERTD (n0 >= 4);
      /* FIXME: further reduce the argument so that it is less than
         1/n where n is the output precision. In such a way, the
         first calls to mpfr_atan_aux will not be too expensive,
         since the number of needed terms will be n/log(n), so the
         factorial contribution will be O(n). */
      for (i = 0 ; i < n0; i++)
        {
          if (MPFR_UNLIKELY (MPFR_IS_ZERO (sk)))
            break;
          /* Calculation of trunc(tmp) --> mpz */
          mpfr_mul_2ui (tmp, sk, twopoweri, GMP_RNDN);
          mpfr_trunc (tmp, tmp);
          if (!MPFR_IS_ZERO (tmp))
            {
              exptol = mpfr_get_z_exp (ukz, tmp);
              /* since the s_k are decreasing (see algorithms.tex),
                 and s_0 = min(|x|, 1/|x|) < 1, we have sk < 1,
                 thus exptol < 0 */
              MPFR_ASSERTD (exptol < 0);
              mpz_tdiv_q_2exp (ukz, ukz, (unsigned long int) (-exptol));
              /* Calculation of arctan(Ak) */
              mpfr_set_z (tmp, ukz, GMP_RNDN);
              mpfr_div_2ui (tmp, tmp, twopoweri, GMP_RNDN);
              mpfr_atan_aux (tmp2, ukz, twopoweri, n0 - i, tabz);
              mpfr_mul (tmp2, tmp2, tmp, GMP_RNDN);
              /* Addition */
              mpfr_add (arctgt, arctgt, tmp2, GMP_RNDN);
              /* Next iteration */
              mpfr_sub (tmp2, sk, tmp, GMP_RNDN);
              mpfr_mul (sk, sk, tmp, GMP_RNDN);
              mpfr_add_ui (sk, sk, 1, GMP_RNDN);
              mpfr_div (sk, tmp2, sk, GMP_RNDN);
            }
          twopoweri <<= 1;
        }
      /* Add last step (Arctan(sk) ~= sk */
      mpfr_add (arctgt, arctgt, sk, GMP_RNDN);
      if (comparaison > 0)
        {
          mpfr_const_pi (tmp, GMP_RNDN);
          mpfr_div_2ui (tmp, tmp, 1, GMP_RNDN);
          mpfr_sub (arctgt, tmp, arctgt, GMP_RNDN);
        }
      MPFR_SET_POS (arctgt);

      if (MPFR_LIKELY (MPFR_CAN_ROUND (arctgt, realprec, MPFR_PREC (atan),
                                       rnd_mode)))
        break;
      MPFR_ZIV_NEXT (loop, realprec);
    }
  MPFR_ZIV_FREE (loop);

  inexact = mpfr_set4 (atan, arctgt, rnd_mode, MPFR_SIGN (x));

  for (i = 0 ; i < oldn0 ; i++)
    mpz_clear (tabz[i]);
  mpz_clear (ukz);
  (*__gmp_free_func) (tabz, oldn0*sizeof (mpz_t));
  MPFR_GROUP_CLEAR (group);

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