xref: /dragonfly/contrib/mpfr/src/sin_cos.c (revision ab6d115f)

/* mpfr_sin_cos -- sine and cosine of a floating-point number

Copyright 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"

/* (y, z) <- (sin(x), cos(x)), return value is 0 iff both results are exact
   ie, iff x = 0 */
int
mpfr_sin_cos (mpfr_ptr y, mpfr_ptr z, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
  mpfr_prec_t prec, m;
  int neg, reduce;
  mpfr_t c, xr;
  mpfr_srcptr xx;
  mpfr_exp_t err, expx;
  int inexy, inexz;
  MPFR_ZIV_DECL (loop);
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_ASSERTN (y != z);

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN(x) || MPFR_IS_INF(x))
        {
          MPFR_SET_NAN (y);
          MPFR_SET_NAN (z);
          MPFR_RET_NAN;
        }
      else /* x is zero */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (x));
          MPFR_SET_ZERO (y);
          MPFR_SET_SAME_SIGN (y, x);
          /* y = 0, thus exact, but z is inexact in case of underflow
             or overflow */
          inexy = 0; /* y is exact */
          inexz = mpfr_set_ui (z, 1, rnd_mode);
          return INEX(inexy,inexz);
        }
    }

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

  MPFR_SAVE_EXPO_MARK (expo);

  prec = MAX (MPFR_PREC (y), MPFR_PREC (z));
  m = prec + MPFR_INT_CEIL_LOG2 (prec) + 13;
  expx = MPFR_GET_EXP (x);

  /* When x is close to 0, say 2^(-k), then there is a cancellation of about
     2k bits in 1-cos(x)^2. FIXME: in that case, it would be more efficient
     to compute sin(x) directly. VL: This is partly done by using
     MPFR_FAST_COMPUTE_IF_SMALL_INPUT from the mpfr_sin and mpfr_cos
     functions. Moreover, any overflow on m is avoided. */
  if (expx < 0)
    {
      /* Warning: in case y = x, and the first call to
         MPFR_FAST_COMPUTE_IF_SMALL_INPUT succeeds but the second fails,
         we will have clobbered the original value of x.
         The workaround is to first compute z = cos(x) in that case, since
         y and z are different. */
      if (y != x)
        /* y and x differ, thus we can safely try to compute y first */
        {
          MPFR_FAST_COMPUTE_IF_SMALL_INPUT (
            y, x, -2 * expx, 2, 0, rnd_mode,
            { inexy = _inexact;
              goto small_input; });
          if (0)
            {
            small_input:
              /* we can go here only if we can round sin(x) */
              MPFR_FAST_COMPUTE_IF_SMALL_INPUT (
                z, __gmpfr_one, -2 * expx, 1, 0, rnd_mode,
                { inexz = _inexact;
                  MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
                  goto end; });
            }

          /* if we go here, one of the two MPFR_FAST_COMPUTE_IF_SMALL_INPUT
             calls failed */
        }
      else /* y and x are the same variable: try to compute z first, which
              necessarily differs */
        {
          MPFR_FAST_COMPUTE_IF_SMALL_INPUT (
            z, __gmpfr_one, -2 * expx, 1, 0, rnd_mode,
            { inexz = _inexact;
              goto small_input2; });
          if (0)
            {
            small_input2:
              /* we can go here only if we can round cos(x) */
              MPFR_FAST_COMPUTE_IF_SMALL_INPUT (
                y, x, -2 * expx, 2, 0, rnd_mode,
                { inexy = _inexact;
                  MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
                  goto end; });
            }
        }
      m += 2 * (-expx);
    }

  if (prec >= MPFR_SINCOS_THRESHOLD)
    {
      MPFR_SAVE_EXPO_FREE (expo);
      return mpfr_sincos_fast (y, z, x, rnd_mode);
    }

  mpfr_init (c);
  mpfr_init (xr);

  MPFR_ZIV_INIT (loop, m);
  for (;;)
    {
      /* the following is copied from sin.c */
      if (expx >= 2) /* reduce the argument */
        {
          reduce = 1;
          mpfr_set_prec (c, expx + m - 1);
          mpfr_set_prec (xr, m);
          mpfr_const_pi (c, MPFR_RNDN);
          mpfr_mul_2ui (c, c, 1, MPFR_RNDN);
          mpfr_remainder (xr, x, c, MPFR_RNDN);
          mpfr_div_2ui (c, c, 1, MPFR_RNDN);
          if (MPFR_SIGN (xr) > 0)
            mpfr_sub (c, c, xr, MPFR_RNDZ);
          else
            mpfr_add (c, c, xr, MPFR_RNDZ);
          if (MPFR_IS_ZERO(xr)
              || MPFR_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m
              || MPFR_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m)
            goto next_step;
          xx = xr;
        }
      else /* the input argument is already reduced */
        {
          reduce = 0;
          xx = x;
        }

      neg = MPFR_IS_NEG (xx); /* gives sign of sin(x) */
      mpfr_set_prec (c, m);
      mpfr_cos (c, xx, MPFR_RNDZ);
      /* If no argument reduction was performed, the error is at most ulp(c),
         otherwise it is at most ulp(c) + 2^(2-m). Since |c| < 1, we have
         ulp(c) <= 2^(-m), thus the error is bounded by 2^(3-m) in that later
         case. */
      if (reduce == 0)
        err = m;
      else
        err = MPFR_GET_EXP (c) + (mpfr_exp_t) (m - 3);
      if (!mpfr_can_round (c, err, MPFR_RNDN, MPFR_RNDZ,
                           MPFR_PREC (z) + (rnd_mode == MPFR_RNDN)))
        goto next_step;

      /* we can't set z now, because in case z = x, and the mpfr_can_round()
         call below fails, we will have clobbered the input */
      mpfr_set_prec (xr, MPFR_PREC(c));
      mpfr_swap (xr, c); /* save the approximation of the cosine in xr */
      mpfr_sqr (c, xr, MPFR_RNDU); /* the absolute error is bounded by
                                      2^(5-m) if reduce=1, and by 2^(2-m)
                                      otherwise */
      mpfr_ui_sub (c, 1, c, MPFR_RNDN); /* error bounded by 2^(6-m) if reduce
                                           is 1, and 2^(3-m) otherwise */
      mpfr_sqrt (c, c, MPFR_RNDN); /* the absolute error is bounded by
                                      2^(6-m-Exp(c)) if reduce=1, and
                                      2^(3-m-Exp(c)) otherwise */
      err = 3 + 3 * reduce - MPFR_GET_EXP (c);
      if (neg)
        MPFR_CHANGE_SIGN (c);

      /* the absolute error on c is at most 2^(err-m), which we must put
         in the form 2^(EXP(c)-err). */
      err = MPFR_GET_EXP (c) + (mpfr_exp_t) m - err;
      if (mpfr_can_round (c, err, MPFR_RNDN, MPFR_RNDZ,
                          MPFR_PREC (y) + (rnd_mode == MPFR_RNDN)))
        break;
      /* check for huge cancellation */
      if (err < (mpfr_exp_t) MPFR_PREC (y))
        m += MPFR_PREC (y) - err;
      /* Check if near 1 */
      if (MPFR_GET_EXP (c) == 1
          && MPFR_MANT (c)[MPFR_LIMB_SIZE (c)-1] == MPFR_LIMB_HIGHBIT)
        m += m;

    next_step:
      MPFR_ZIV_NEXT (loop, m);
      mpfr_set_prec (c, m);
    }
  MPFR_ZIV_FREE (loop);

  inexy = mpfr_set (y, c, rnd_mode);
  inexz = mpfr_set (z, xr, rnd_mode);

  mpfr_clear (c);
  mpfr_clear (xr);

 end:
  MPFR_SAVE_EXPO_FREE (expo);
  /* FIXME: add a test for bug before revision 7355 */
  inexy = mpfr_check_range (y, inexy, rnd_mode);
  inexz = mpfr_check_range (z, inexz, rnd_mode);
  MPFR_RET (INEX(inexy,inexz));
}

/*************** asymptotically fast implementation below ********************/

/* truncate Q from R to at most prec bits.
   Return the number of truncated bits.
 */
static mpfr_prec_t
reduce (mpz_t Q, mpz_srcptr R, mpfr_prec_t prec)
{
  mpfr_prec_t l = mpz_sizeinbase (R, 2);

  l = (l > prec) ? l - prec : 0;
  mpz_fdiv_q_2exp (Q, R, l);
  return l;
}

/* truncate S and C so that the smaller has prec bits.
   Return the number of truncated bits.
 */
static unsigned long
reduce2 (mpz_t S, mpz_t C, mpfr_prec_t prec)
{
  unsigned long ls = mpz_sizeinbase (S, 2);
  unsigned long lc = mpz_sizeinbase (C, 2);
  unsigned long l;

  l = (ls < lc) ? ls : lc; /* smaller length */
  l = (l > prec) ? l - prec : 0;
  mpz_fdiv_q_2exp (S, S, l);
  mpz_fdiv_q_2exp (C, C, l);
  return l;
}

/* return in S0/Q0 a rational approximation of sin(X) with absolute error
                     bounded by 9*2^(-prec), where 0 <= X=p/2^r <= 1/2,
   and in    C0/Q0 a rational approximation of cos(X), with relative error
                     bounded by 9*2^(-prec) (and also absolute error, since
                     |cos(X)| <= 1).
   We have sin(X)/X = sum((-1)^i*(p/2^r)^i/(2i+1)!, i=0..infinity).
   We use the following binary splitting formula:
   P(a,b) = (-p)^(b-a)
   Q(a,b) = (2a)*(2a+1)*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise
   T(a,b) = 1 if a+1=b, Q(c,b)*T(a,c)+P(a,c)*T(c,b) otherwise.

   Since we use P(a,b) for b-a=2^k only, we compute only p^(2^k).
   We do not store the factor 2^r in Q().

   Then sin(X)/X ~ T(0,i)/Q(0,i) for i so that (p/2^r)^i/i! is small enough.

   Return l such that Q0 has to be multiplied by 2^l.

   Assumes prec >= 10.
*/
static unsigned long
sin_bs_aux (mpz_t Q0, mpz_t S0, mpz_t C0, mpz_srcptr p, mpfr_prec_t r,
            mpfr_prec_t prec)
{
  mpz_t T[GMP_NUMB_BITS], Q[GMP_NUMB_BITS], ptoj[GMP_NUMB_BITS], pp;
  mpfr_prec_t log2_nb_terms[GMP_NUMB_BITS], mult[GMP_NUMB_BITS];
  mpfr_prec_t accu[GMP_NUMB_BITS], size_ptoj[GMP_NUMB_BITS];
  mpfr_prec_t prec_i_have, r0 = r;
  unsigned long alloc, i, j, k;
  mpfr_prec_t l;

  if (MPFR_UNLIKELY(mpz_cmp_ui (p, 0) == 0)) /* sin(x)/x -> 1 */
    {
      mpz_set_ui (Q0, 1);
      mpz_set_ui (S0, 1);
      mpz_set_ui (C0, 1);
      return 0;
    }

  /* check that X=p/2^r <= 1/2 */
  MPFR_ASSERTN(mpz_sizeinbase (p, 2) - (mpfr_exp_t) r <= -1);

  mpz_init (pp);

  /* normalize p (non-zero here) */
  l = mpz_scan1 (p, 0);
  mpz_fdiv_q_2exp (pp, p, l); /* p = pp * 2^l */
  mpz_mul (pp, pp, pp);
  r = 2 * (r - l);            /* x^2 = (p/2^r0)^2 = pp / 2^r */

  /* now p is odd */
  alloc = 2;
  mpz_init_set_ui (T[0], 6);
  mpz_init_set_ui (Q[0], 6);
  mpz_init_set (ptoj[0], pp); /* ptoj[i] = pp^(2^i) */
  mpz_init (T[1]);
  mpz_init (Q[1]);
  mpz_init (ptoj[1]);
  mpz_mul (ptoj[1], pp, pp);  /* ptoj[1] = pp^2 */
  size_ptoj[1] = mpz_sizeinbase (ptoj[1], 2);

  mpz_mul_2exp (T[0], T[0], r);
  mpz_sub (T[0], T[0], pp);      /* 6*2^r - pp = 6*2^r*(1 - x^2/6) */
  log2_nb_terms[0] = 1;

  /* already take into account the factor x=p/2^r in sin(x) = x * (...) */
  mult[0] = r  - mpz_sizeinbase (pp, 2) + r0 - mpz_sizeinbase (p, 2);
  /* we have x^3 < 1/2^mult[0] */

  for (i = 2, k = 0, prec_i_have = mult[0]; prec_i_have < prec; i += 2)
    {
      /* i is even here */
      /* invariant: Q[0]*Q[1]*...*Q[k] equals (2i-1)!,
         we have already summed terms of index < i
         in S[0]/Q[0], ..., S[k]/Q[k] */
      k ++;
      if (k + 1 >= alloc) /* necessarily k + 1 = alloc */
        {
          alloc ++;
          mpz_init (T[k+1]);
          mpz_init (Q[k+1]);
          mpz_init (ptoj[k+1]);
          mpz_mul (ptoj[k+1], ptoj[k], ptoj[k]); /* pp^(2^(k+1)) */
          size_ptoj[k+1] = mpz_sizeinbase (ptoj[k+1], 2);
        }
      /* for i even, we have Q[k] = (2*i)*(2*i+1), T[k] = 1,
         then                Q[k+1] = (2*i+2)*(2*i+3), T[k+1] = 1,
         which reduces to T[k] = (2*i+2)*(2*i+3)*2^r-pp,
         Q[k] = (2*i)*(2*i+1)*(2*i+2)*(2*i+3). */
      log2_nb_terms[k] = 1;
      mpz_set_ui (Q[k], (2 * i + 2) * (2 * i + 3));
      mpz_mul_2exp (T[k], Q[k], r);
      mpz_sub (T[k], T[k], pp);
      mpz_mul_ui (Q[k], Q[k], (2 * i) * (2 * i + 1));
      /* the next term of the series is divided by Q[k] and multiplied
         by pp^2/2^(2r), thus the mult. factor < 1/2^mult[k] */
      mult[k] = mpz_sizeinbase (Q[k], 2) + 2 * r - size_ptoj[1] - 1;
      /* the absolute contribution of the next term is 1/2^accu[k] */
      accu[k] = (k == 0) ? mult[k] : mult[k] + accu[k-1];
      prec_i_have = accu[k]; /* the current term is < 1/2^accu[k] */
      j = (i + 2) / 2;
      l = 1;
      while ((j & 1) == 0) /* combine and reduce */
        {
          mpz_mul (T[k], T[k], ptoj[l]);
          mpz_mul (T[k-1], T[k-1], Q[k]);
          mpz_mul_2exp (T[k-1], T[k-1], r << l);
          mpz_add (T[k-1], T[k-1], T[k]);
          mpz_mul (Q[k-1], Q[k-1], Q[k]);
          log2_nb_terms[k-1] ++; /* number of terms in S[k-1]
                                    is a power of 2 by construction */
          prec_i_have = mpz_sizeinbase (Q[k], 2);
          mult[k-1] += prec_i_have + (r << l) - size_ptoj[l] - 1;
          accu[k-1] = (k == 1) ? mult[k-1] : mult[k-1] + accu[k-2];
          prec_i_have = accu[k-1];
          l ++;
          j >>= 1;
          k --;
        }
    }

  /* accumulate all products in T[0] and Q[0]. Warning: contrary to above,
     here we do not have log2_nb_terms[k-1] = log2_nb_terms[k]+1. */
  l = 0; /* number of accumulated terms in the right part T[k]/Q[k] */
  while (k > 0)
    {
      j = log2_nb_terms[k-1];
      mpz_mul (T[k], T[k], ptoj[j]);
      mpz_mul (T[k-1], T[k-1], Q[k]);
      l += 1 << log2_nb_terms[k];
      mpz_mul_2exp (T[k-1], T[k-1], r * l);
      mpz_add (T[k-1], T[k-1], T[k]);
      mpz_mul (Q[k-1], Q[k-1], Q[k]);
      k--;
    }

  l = r0 + r * (i - 1); /* implicit multiplier 2^r for Q0 */
  /* at this point T[0]/(2^l*Q[0]) is an approximation of sin(x) where the 1st
     neglected term has contribution < 1/2^prec, thus since the series has
     alternate signs, the error is < 1/2^prec */

  /* we truncate Q0 to prec bits: the relative error is at most 2^(1-prec),
     which means that Q0 = Q[0] * (1+theta) with |theta| <= 2^(1-prec)
     [up to a power of two] */
  l += reduce (Q0, Q[0], prec);
  l -= reduce (T[0], T[0], prec);
  /* multiply by x = p/2^l */
  mpz_mul (S0, T[0], p);
  l -= reduce (S0, S0, prec); /* S0 = T[0] * (1 + theta)^2 up to power of 2 */
  /* sin(X) ~ S0/Q0*(1 + theta)^3 + err with |theta| <= 2^(1-prec) and
              |err| <= 2^(-prec), thus since |S0/Q0| <= 1:
     |sin(X) - S0/Q0| <= 4*|theta*S0/Q0| + |err| <= 9*2^(-prec) */

  mpz_clear (pp);
  for (j = 0; j < alloc; j ++)
    {
      mpz_clear (T[j]);
      mpz_clear (Q[j]);
      mpz_clear (ptoj[j]);
    }

  /* compute cos(X) from sin(X): sqrt(1-(S/Q)^2) = sqrt(Q^2-S^2)/Q
     = sqrt(Q0^2*2^(2l)-S0^2)/Q0.
     Write S/Q = sin(X) + eps with |eps| <= 9*2^(-prec),
     then sqrt(Q^2-S^2) = sqrt(Q^2-Q^2*(sin(X)+eps)^2)
                        = sqrt(Q^2*cos(X)^2-Q^2*(2*sin(X)*eps+eps^2))
                        = sqrt(Q^2*cos(X)^2-Q^2*eps1) with |eps1|<=9*2^(-prec)
                          [using X<=1/2 and eps<=9*2^(-prec) and prec>=10]

                        Since we truncate the square root, we get:
                          sqrt(Q^2*cos(X)^2-Q^2*eps1)+eps2 with |eps2|<1
                        = Q*sqrt(cos(X)^2-eps1)+eps2
                        = Q*cos(X)*(1+eps3)+eps2 with |eps3| <= 6*2^(-prec)
                        = Q*cos(X)*(1+eps3+eps2/(Q*cos(X)))
                        = Q*cos(X)*(1+eps4) with |eps4| <= 9*2^(-prec)
                          since |Q| >= 2^(prec-1) */
  /* we assume that Q0*2^l >= 2^(prec-1) */
  MPFR_ASSERTN(l + mpz_sizeinbase (Q0, 2) >= prec);
  mpz_mul (C0, Q0, Q0);
  mpz_mul_2exp (C0, C0, 2 * l);
  mpz_submul (C0, S0, S0);
  mpz_sqrt (C0, C0);

  return l;
}

/* Put in s and c approximations of sin(x) and cos(x) respectively.
   Assumes 0 < x < Pi/4 and PREC(s) = PREC(c) >= 10.
   Return err such that the relative error is bounded by 2^err ulps.
*/
static int
sincos_aux (mpfr_t s, mpfr_t c, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
  mpfr_prec_t prec_s, sh;
  mpz_t Q, S, C, Q2, S2, C2, y;
  mpfr_t x2;
  unsigned long l, l2, j, err;

  MPFR_ASSERTD(MPFR_PREC(s) == MPFR_PREC(c));

  prec_s = MPFR_PREC(s);

  mpfr_init2 (x2, MPFR_PREC(x));
  mpz_init (Q);
  mpz_init (S);
  mpz_init (C);
  mpz_init (Q2);
  mpz_init (S2);
  mpz_init (C2);
  mpz_init (y);

  mpfr_set (x2, x, MPFR_RNDN); /* exact */
  mpz_set_ui (Q, 1);
  l = 0;
  mpz_set_ui (S, 0); /* sin(0) = S/(2^l*Q), exact */
  mpz_set_ui (C, 1); /* cos(0) = C/(2^l*Q), exact */

  /* Invariant: x = X + x2/2^(sh-1), where the part X was already treated,
     S/(2^l*Q) ~ sin(X), C/(2^l*Q) ~ cos(X), and x2/2^(sh-1) < Pi/4.
     'sh-1' is the number of already shifted bits in x2.
  */

  for (sh = 1, j = 0; mpfr_cmp_ui (x2, 0) != 0 && sh <= prec_s; sh <<= 1, j++)
    {
      if (sh > prec_s / 2) /* sin(x) = x + O(x^3), cos(x) = 1 + O(x^2) */
        {
          l2 = -mpfr_get_z_2exp (S2, x2); /* S2/2^l2 = x2 */
          l2 += sh - 1;
          mpz_set_ui (Q2, 1);
          mpz_set_ui (C2, 1);
          mpz_mul_2exp (C2, C2, l2);
          mpfr_set_ui (x2, 0, MPFR_RNDN);
        }
      else
        {
          /* y <- trunc(x2 * 2^sh) = trunc(x * 2^(2*sh-1)) */
          mpfr_mul_2exp (x2, x2, sh, MPFR_RNDN); /* exact */
          mpfr_get_z (y, x2, MPFR_RNDZ); /* round towards zero: now
                                           0 <= x2 < 2^sh, thus
                                           0 <= x2/2^(sh-1) < 2^(1-sh) */
          if (mpz_cmp_ui (y, 0) == 0)
            continue;
          mpfr_sub_z (x2, x2, y, MPFR_RNDN); /* should be exact */
          l2 = sin_bs_aux (Q2, S2, C2, y, 2 * sh - 1, prec_s);
          /* we now have |S2/Q2/2^l2 - sin(X)| <= 9*2^(prec_s)
             and |C2/Q2/2^l2 - cos(X)| <= 6*2^(prec_s), with X=y/2^(2sh-1) */
        }
      if (sh == 1) /* S=0, C=1 */
        {
          l = l2;
          mpz_swap (Q, Q2);
          mpz_swap (S, S2);
          mpz_swap (C, C2);
        }
      else
        {
          /* s <- s*c2+c*s2, c <- c*c2-s*s2, using Karatsuba:
             a = s+c, b = s2+c2, t = a*b, d = s*s2, e = c*c2,
             s <- t - d - e, c <- e - d */
          mpz_add (y, S, C); /* a */
          mpz_mul (C, C, C2); /* e */
          mpz_add (C2, C2, S2); /* b */
          mpz_mul (S2, S, S2); /* d */
          mpz_mul (y, y, C2); /* a*b */
          mpz_sub (S, y, S2); /* t - d */
          mpz_sub (S, S, C); /* t - d - e */
          mpz_sub (C, C, S2); /* e - d */
          mpz_mul (Q, Q, Q2);
          /* after j loops, the error is <= (11j-2)*2^(prec_s) */
          l += l2;
          /* reduce Q to prec_s bits */
          l += reduce (Q, Q, prec_s);
          /* reduce S,C to prec_s bits, error <= 11*j*2^(prec_s) */
          l -= reduce2 (S, C, prec_s);
        }
    }

  j = 11 * j;
  for (err = 0; j > 1; j = (j + 1) / 2, err ++);

  mpfr_set_z (s, S, MPFR_RNDN);
  mpfr_div_z (s, s, Q, MPFR_RNDN);
  mpfr_div_2exp (s, s, l, MPFR_RNDN);

  mpfr_set_z (c, C, MPFR_RNDN);
  mpfr_div_z (c, c, Q, MPFR_RNDN);
  mpfr_div_2exp (c, c, l, MPFR_RNDN);

  mpz_clear (Q);
  mpz_clear (S);
  mpz_clear (C);
  mpz_clear (Q2);
  mpz_clear (S2);
  mpz_clear (C2);
  mpz_clear (y);
  mpfr_clear (x2);
  return err;
}

/* Assumes x is neither NaN, +/-Inf, nor +/- 0.
   One of s and c might be NULL, in which case the corresponding value is
   not computed.
   Assumes s differs from c.
 */
int
mpfr_sincos_fast (mpfr_t s, mpfr_t c, mpfr_srcptr x, mpfr_rnd_t rnd)
{
  int inexs, inexc;
  mpfr_t x_red, ts, tc;
  mpfr_prec_t w;
  mpfr_exp_t err, errs, errc;
  MPFR_ZIV_DECL (loop);

  MPFR_ASSERTN(s != c);
  if (s == NULL)
    w = MPFR_PREC(c);
  else if (c == NULL)
    w = MPFR_PREC(s);
  else
    w = MPFR_PREC(s) >= MPFR_PREC(c) ? MPFR_PREC(s) : MPFR_PREC(c);
  w += MPFR_INT_CEIL_LOG2(w) + 9; /* ensures w >= 10 (needed by sincos_aux) */
  mpfr_init2 (ts, w);
  mpfr_init2 (tc, w);

  MPFR_ZIV_INIT (loop, w);
  for (;;)
    {
      /* if 0 < x <= Pi/4, we can call sincos_aux directly */
      if (MPFR_IS_POS(x) && mpfr_cmp_ui_2exp (x, 1686629713, -31) <= 0)
        {
          err = sincos_aux (ts, tc, x, MPFR_RNDN);
        }
      /* if -Pi/4 <= x < 0, use sin(-x)=-sin(x) */
      else if (MPFR_IS_NEG(x) && mpfr_cmp_si_2exp (x, -1686629713, -31) >= 0)
        {
          mpfr_init2 (x_red, MPFR_PREC(x));
          mpfr_neg (x_red, x, rnd); /* exact */
          err = sincos_aux (ts, tc, x_red, MPFR_RNDN);
          mpfr_neg (ts, ts, MPFR_RNDN);
          mpfr_clear (x_red);
        }
      else /* argument reduction is needed */
        {
          long q;
          mpfr_t pi;
          int neg = 0;

          mpfr_init2 (x_red, w);
          mpfr_init2 (pi, (MPFR_EXP(x) > 0) ? w + MPFR_EXP(x) : w);
          mpfr_const_pi (pi, MPFR_RNDN);
          mpfr_div_2exp (pi, pi, 1, MPFR_RNDN); /* Pi/2 */
          mpfr_remquo (x_red, &q, x, pi, MPFR_RNDN);
          /* x = q * (Pi/2 + eps1) + x_red + eps2,
             where |eps1| <= 1/2*ulp(Pi/2) = 2^(-w-MAX(0,EXP(x))),
             and eps2 <= 1/2*ulp(x_red) <= 1/2*ulp(Pi/2) = 2^(-w)
             Since |q| <= x/(Pi/2) <= |x|, we have
             q*|eps1| <= 2^(-w), thus
             |x - q * Pi/2 - x_red| <= 2^(1-w) */
          /* now -Pi/4 <= x_red <= Pi/4: if x_red < 0, consider -x_red */
          if (MPFR_IS_NEG(x_red))
            {
              mpfr_neg (x_red, x_red, MPFR_RNDN);
              neg = 1;
            }
          err = sincos_aux (ts, tc, x_red, MPFR_RNDN);
          err ++; /* to take into account the argument reduction */
          if (neg) /* sin(-x) = -sin(x), cos(-x) = cos(x) */
            mpfr_neg (ts, ts, MPFR_RNDN);
          if (q & 2) /* sin(x+Pi) = -sin(x), cos(x+Pi) = -cos(x) */
            {
              mpfr_neg (ts, ts, MPFR_RNDN);
              mpfr_neg (tc, tc, MPFR_RNDN);
            }
          if (q & 1) /* sin(x+Pi/2) = cos(x), cos(x+Pi/2) = -sin(x) */
            {
              mpfr_neg (ts, ts, MPFR_RNDN);
              mpfr_swap (ts, tc);
            }
          mpfr_clear (x_red);
          mpfr_clear (pi);
        }
      /* adjust errors with respect to absolute values */
      errs = err - MPFR_EXP(ts);
      errc = err - MPFR_EXP(tc);
      if ((s == NULL || MPFR_CAN_ROUND (ts, w - errs, MPFR_PREC(s), rnd)) &&
          (c == NULL || MPFR_CAN_ROUND (tc, w - errc, MPFR_PREC(c), rnd)))
        break;
      MPFR_ZIV_NEXT (loop, w);
      mpfr_set_prec (ts, w);
      mpfr_set_prec (tc, w);
    }
  MPFR_ZIV_FREE (loop);

  inexs = (s == NULL) ? 0 : mpfr_set (s, ts, rnd);
  inexc = (c == NULL) ? 0 : mpfr_set (c, tc, rnd);

  mpfr_clear (ts);
  mpfr_clear (tc);
  return INEX(inexs,inexc);
}