/* mpc_sin_cos -- combined sine and cosine of a complex number.

Copyright (C) 2010, 2011, 2012 INRIA

This file is part of GNU MPC.

GNU MPC 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.

GNU MPC 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 this program. If not, see http://www.gnu.org/licenses/ .

*/

#include <stdio.h>

#include "mpc-impl.h"

static int

mpc_sin_cos_nonfinite (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op,

   mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos)

   /* assumes that op (that is, its real or imaginary part) is not finite */

{

   int overlap;

   mpc_t op_loc;

   overlap = (rop_sin == op || rop_cos == op);

   if (overlap) {

      mpc_init3 (op_loc, MPC_PREC_RE (op), MPC_PREC_IM (op));

      mpc_set (op_loc, op, MPC_RNDNN);

   }

   else

      op_loc [0] = op [0];

   if (rop_sin != NULL) {

      if (mpfr_nan_p (mpc_realref (op_loc)) || mpfr_nan_p (mpc_imagref (op_loc))) {

         mpc_set (rop_sin, op_loc, rnd_sin);

         if (mpfr_nan_p (mpc_imagref (op_loc))) {

            /* sin(x +i*NaN) = NaN +i*NaN, except for x=0 */

            /* sin(-0 +i*NaN) = -0 +i*NaN */

            /* sin(+0 +i*NaN) = +0 +i*NaN */

            if (!mpfr_zero_p (mpc_realref (op_loc)))

               mpfr_set_nan (mpc_realref (rop_sin));

         }

         else /* op = NaN + i*y */

            if (!mpfr_inf_p (mpc_imagref (op_loc)) && !mpfr_zero_p (mpc_imagref (op_loc)))

            /* sin(NaN -i*Inf) = NaN -i*Inf */

            /* sin(NaN -i*0) = NaN -i*0 */

            /* sin(NaN +i*0) = NaN +i*0 */

            /* sin(NaN +i*Inf) = NaN +i*Inf */

            /* sin(NaN +i*y) = NaN +i*NaN, when 0<|y|

            mpfr_set_nan (mpc_imagref (rop_sin));

      }

      else if (mpfr_inf_p (mpc_realref (op_loc))) {

         mpfr_set_nan (mpc_realref (rop_sin));

         if (!mpfr_inf_p (mpc_imagref (op_loc)) && !mpfr_zero_p (mpc_imagref (op_loc)))

            /* sin(+/-Inf +i*y) = NaN +i*NaN, when 0<|y|

            mpfr_set_nan (mpc_imagref (rop_sin));

         else

            /* sin(+/-Inf -i*Inf) = NaN -i*Inf */

            /* sin(+/-Inf +i*Inf) = NaN +i*Inf */

            /* sin(+/-Inf -i*0) = NaN -i*0 */

            /* sin(+/-Inf +i*0) = NaN +i*0 */

            mpfr_set (mpc_imagref (rop_sin), mpc_imagref (op_loc), MPC_RND_IM (rnd_sin));

      }

      else if (mpfr_zero_p (mpc_realref (op_loc))) {

         /* sin(-0 -i*Inf) = -0 -i*Inf */

         /* sin(+0 -i*Inf) = +0 -i*Inf */

         /* sin(-0 +i*Inf) = -0 +i*Inf */

         /* sin(+0 +i*Inf) = +0 +i*Inf */

         mpc_set (rop_sin, op_loc, rnd_sin);

      }

      else {

         /* sin(x -i*Inf) = +Inf*(sin(x) -i*cos(x)) */

         /* sin(x +i*Inf) = +Inf*(sin(x) +i*cos(x)) */

         mpfr_t s, c;

         mpfr_init2 (s, 2);

         mpfr_init2 (c, 2);

         mpfr_sin_cos (s, c, mpc_realref (op_loc), MPFR_RNDZ);

         mpfr_set_inf (mpc_realref (rop_sin), MPFR_SIGN (s));

         mpfr_set_inf (mpc_imagref (rop_sin), MPFR_SIGN (c)*MPFR_SIGN (mpc_imagref (op_loc)));

         mpfr_clear (s);

         mpfr_clear (c);

      }

   }

   if (rop_cos != NULL) {

      if (mpfr_nan_p (mpc_realref (op_loc))) {

         /* cos(NaN + i * NaN) = NaN + i * NaN */

         /* cos(NaN - i * Inf) = +Inf + i * NaN */

         /* cos(NaN + i * Inf) = +Inf + i * NaN */

         /* cos(NaN - i * 0) = NaN - i * 0 */

         /* cos(NaN + i * 0) = NaN + i * 0 */

         /* cos(NaN + i * y) = NaN + i * NaN, when y != 0 */

         if (mpfr_inf_p (mpc_imagref (op_loc)))

            mpfr_set_inf (mpc_realref (rop_cos), +1);

         else

            mpfr_set_nan (mpc_realref (rop_cos));

         if (mpfr_zero_p (mpc_imagref (op_loc)))

            mpfr_set (mpc_imagref (rop_cos), mpc_imagref (op_loc), MPC_RND_IM (rnd_cos));

         else

            mpfr_set_nan (mpc_imagref (rop_cos));

      }

      else if (mpfr_nan_p (mpc_imagref (op_loc))) {

          /* cos(-Inf + i * NaN) = NaN + i * NaN */

          /* cos(+Inf + i * NaN) = NaN + i * NaN */

          /* cos(-0 + i * NaN) = NaN - i * 0 */

          /* cos(+0 + i * NaN) = NaN + i * 0 */

          /* cos(x + i * NaN) = NaN + i * NaN, when x != 0 */

         if (mpfr_zero_p (mpc_realref (op_loc)))

            mpfr_set (mpc_imagref (rop_cos), mpc_realref (op_loc), MPC_RND_IM (rnd_cos));

         else

            mpfr_set_nan (mpc_imagref (rop_cos));

         mpfr_set_nan (mpc_realref (rop_cos));

      }

      else if (mpfr_inf_p (mpc_realref (op_loc))) {

         /* cos(-Inf -i*Inf) = cos(+Inf +i*Inf) = -Inf +i*NaN */

         /* cos(-Inf +i*Inf) = cos(+Inf -i*Inf) = +Inf +i*NaN */

         /* cos(-Inf -i*0) = cos(+Inf +i*0) = NaN -i*0 */

         /* cos(-Inf +i*0) = cos(+Inf -i*0) = NaN +i*0 */

         /* cos(-Inf +i*y) = cos(+Inf +i*y) = NaN +i*NaN, when y != 0 */

         const int same_sign =

            mpfr_signbit (mpc_realref (op_loc)) == mpfr_signbit (mpc_imagref (op_loc));

         if (mpfr_inf_p (mpc_imagref (op_loc)))

            mpfr_set_inf (mpc_realref (rop_cos), (same_sign ? -1 : +1));

         else

            mpfr_set_nan (mpc_realref (rop_cos));

         if (mpfr_zero_p (mpc_imagref (op_loc)))

            mpfr_setsign (mpc_imagref (rop_cos), mpc_imagref (op_loc), same_sign,

                          MPC_RND_IM(rnd_cos));

         else

            mpfr_set_nan (mpc_imagref (rop_cos));

      }

      else if (mpfr_zero_p (mpc_realref (op_loc))) {

         /* cos(-0 -i*Inf) = cos(+0 +i*Inf) = +Inf -i*0 */

         /* cos(-0 +i*Inf) = cos(+0 -i*Inf) = +Inf +i*0 */

         const int same_sign =

            mpfr_signbit (mpc_realref (op_loc)) == mpfr_signbit (mpc_imagref (op_loc));

         mpfr_setsign (mpc_imagref (rop_cos), mpc_realref (op_loc), same_sign,

                       MPC_RND_IM (rnd_cos));

         mpfr_set_inf (mpc_realref (rop_cos), +1);

      }

      else {

         /* cos(x -i*Inf) = +Inf*cos(x) +i*Inf*sin(x), when x != 0 */

         /* cos(x +i*Inf) = +Inf*cos(x) -i*Inf*sin(x), when x != 0 */

         mpfr_t s, c;

         mpfr_init2 (c, 2);

         mpfr_init2 (s, 2);

         mpfr_sin_cos (s, c, mpc_realref (op_loc), MPFR_RNDN);

         mpfr_set_inf (mpc_realref (rop_cos), mpfr_sgn (c));

         mpfr_set_inf (mpc_imagref (rop_cos),

            (mpfr_sgn (mpc_imagref (op_loc)) == mpfr_sgn (s) ? -1 : +1));

         mpfr_clear (s);

         mpfr_clear (c);

      }

   }

   if (overlap)

      mpc_clear (op_loc);

   return MPC_INEX12 (MPC_INEX (0,0), MPC_INEX (0,0));

      /* everything is exact */

}

static int

mpc_sin_cos_real (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op,

   mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos)

   /* assumes that op is real */

{

   int inex_sin_re = 0, inex_cos_re = 0;

      /* Until further notice, assume computations exact; in particular,

         by definition, for not computed values.                         */

   mpfr_t s, c;

   int inex_s, inex_c;

   int sign_im = mpfr_signbit (mpc_imagref (op));

   /* sin(x +-0*i) = sin(x) +-0*i*sign(cos(x)) */

   /* cos(x +-i*0) = cos(x) -+i*0*sign(sin(x)) */

   if (rop_sin != 0)

      mpfr_init2 (s, MPC_PREC_RE (rop_sin));

   else

      mpfr_init2 (s, 2); /* We need only the sign. */

   if (rop_cos != NULL)

      mpfr_init2 (c, MPC_PREC_RE (rop_cos));

   else

      mpfr_init2 (c, 2);

   inex_s = mpfr_sin (s, mpc_realref (op), MPC_RND_RE (rnd_sin));

   inex_c = mpfr_cos (c, mpc_realref (op), MPC_RND_RE (rnd_cos));

      /* We cannot use mpfr_sin_cos since we may need two distinct rounding

         modes and the exact return values. If we need only the sign, an

         arbitrary rounding mode will work.                                 */

   if (rop_sin != NULL) {

      mpfr_set (mpc_realref (rop_sin), s, MPFR_RNDN); /* exact */

      inex_sin_re = inex_s;

      mpfr_set_zero (mpc_imagref (rop_sin),

         (     ( sign_im && !mpfr_signbit(c))

            || (!sign_im &&  mpfr_signbit(c)) ? -1 : 1));

   }

   if (rop_cos != NULL) {

      mpfr_set (mpc_realref (rop_cos), c, MPFR_RNDN); /* exact */

      inex_cos_re = inex_c;

      mpfr_set_zero (mpc_imagref (rop_cos),

         (     ( sign_im &&  mpfr_signbit(s))

            || (!sign_im && !mpfr_signbit(s)) ? -1 : 1));

   }

   mpfr_clear (s);

   mpfr_clear (c);

   return MPC_INEX12 (MPC_INEX (inex_sin_re, 0), MPC_INEX (inex_cos_re, 0));

}

static int

mpc_sin_cos_imag (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op,

   mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos)

   /* assumes that op is purely imaginary, but not zero */

{

   int inex_sin_im = 0, inex_cos_re = 0;

      /* assume exact if not computed */

   int overlap;

   mpc_t op_loc;

   overlap = (rop_sin == op || rop_cos == op);

   if (overlap) {

      mpc_init3 (op_loc, MPC_PREC_RE (op), MPC_PREC_IM (op));

      mpc_set (op_loc, op, MPC_RNDNN);

   }

   else

      op_loc [0] = op [0];

   if (rop_sin != NULL) {

      /* sin(+-O +i*y) = +-0 +i*sinh(y) */

      mpfr_set (mpc_realref(rop_sin), mpc_realref(op_loc), MPFR_RNDN);

      inex_sin_im = mpfr_sinh (mpc_imagref(rop_sin), mpc_imagref(op_loc), MPC_RND_IM(rnd_sin));

   }

   if (rop_cos != NULL) {

      /* cos(-0 - i * y) = cos(+0 + i * y) = cosh(y) - i * 0,

         cos(-0 + i * y) = cos(+0 - i * y) = cosh(y) + i * 0,

         where y > 0 */

      inex_cos_re = mpfr_cosh (mpc_realref (rop_cos), mpc_imagref (op_loc), MPC_RND_RE (rnd_cos));

      mpfr_set_ui (mpc_imagref (rop_cos), 0ul, MPC_RND_IM (rnd_cos));

      if (mpfr_signbit (mpc_realref (op_loc)) ==  mpfr_signbit (mpc_imagref (op_loc)))

         MPFR_CHANGE_SIGN (mpc_imagref (rop_cos));

   }

   if (overlap)

      mpc_clear (op_loc);

   return MPC_INEX12 (MPC_INEX (0, inex_sin_im), MPC_INEX (inex_cos_re, 0));

}

/* Fix an inexact overflow, when x is +Inf or -Inf:

   When rnd is towards zero, change x into the largest (in absolute value)

   floating-point number.

   Return the inexact flag. */

int

mpc_fix_inf (mpfr_t x, mpfr_rnd_t rnd)

{

  MPC_ASSERT (mpfr_inf_p (x));

  if (!MPC_IS_LIKE_RNDZ(rnd, MPFR_SIGNBIT(x)))

    return mpfr_sgn (x);

  else

    {

      if (mpfr_sgn (x) < 0)

        mpfr_nextabove (x);

      else

        mpfr_nextbelow (x);

      return -mpfr_sgn (x);

    }

}

/* Fix an inexact underflow, when x is +0 or -0:

   When rnd is away from zero, change x into the closest floating-point number.

   Return the inexact flag. */

int

mpc_fix_zero (mpfr_t x, mpfr_rnd_t rnd)

{

  if (!MPC_IS_LIKE_RNDA(rnd, MPFR_SIGNBIT(x)))

    return mpfr_signbit (x) == 0 ? -1 : 1;

  else

    {

      if (mpfr_signbit (x) == 0)

        {

          mpfr_nextabove (x);

          return 1;

        }

      else

        {

          mpfr_nextbelow (x);

          return -1;

        }

    }

}

int

mpc_sin_cos (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op,

   mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos)

   /* Feature not documented in the texinfo file: One of rop_sin or

      rop_cos may be NULL, in which case it is not computed, and the

      corresponding ternary inexact value is set to 0 (exact).       */

{

   if (!mpc_fin_p (op))

      return mpc_sin_cos_nonfinite (rop_sin, rop_cos, op, rnd_sin, rnd_cos);

   else if (mpfr_zero_p (mpc_imagref (op)))

      return mpc_sin_cos_real (rop_sin, rop_cos, op, rnd_sin, rnd_cos);

   else if (mpfr_zero_p (mpc_realref (op)))

      return mpc_sin_cos_imag (rop_sin, rop_cos, op, rnd_sin, rnd_cos);

   else {

      /* let op = a + i*b, then sin(op) = sin(a)*cosh(b) + i*cos(a)*sinh(b)

                           and  cos(op) = cos(a)*cosh(b) - i*sin(a)*sinh(b).

         For Re(sin(op)) (and analogously, the other parts), we use the

         following algorithm, with rounding to nearest for all operations

         and working precision w:

         (1) x = o(sin(a))

         (2) y = o(cosh(b))

         (3) r = o(x*y)

         then the error on r is at most 4 ulps, since we can write

         r = sin(a)*cosh(b)*(1+t)^3 with |t| <= 2^(-w),

         thus for w >= 2, r = sin(a)*cosh(b)*(1+4*t) with |t| <= 2^(-w),

         thus the relative error is bounded by 4*2^(-w) <= 4*ulp(r).

      */

      mpfr_t s, c, sh, ch, sch, csh;

      mpfr_prec_t prec;

      int ok;

      int inex_re, inex_im, inex_sin, inex_cos, loop = 0;

      prec = 2;

      if (rop_sin != NULL)

        {

          mp_prec_t er, ei;

          prec = MPC_MAX (prec, MPC_MAX_PREC (rop_sin));

          /* since the Taylor expansion of sin(x) at x=0 is x - x^3/6 + O(x^5),

             if x <= 2^(-p), then the second term/x is about 2^(-2p)/6, thus we

             need at least 2p+3 bits of precision. This is true only when x is

             exactly representable in the target precision. */

          if (MPC_MAX_PREC (op) <= prec)

            {

              er = mpfr_get_exp (mpc_realref (op));

              ei = mpfr_get_exp (mpc_imagref (op));

              /* consider the maximal exponent only */

              er = (er < ei) ? ei : er;

              if (er < 0)

                if (prec < 2 * (mp_prec_t) (-er) + 3)

                  prec = 2 * (mp_prec_t) (-er) + 3;

            }

        }

      if (rop_cos != NULL)

         prec = MPC_MAX (prec, MPC_MAX_PREC (rop_cos));

      mpfr_init2 (s, 2);

      mpfr_init2 (c, 2);

      mpfr_init2 (sh, 2);

      mpfr_init2 (ch, 2);

      mpfr_init2 (sch, 2);

      mpfr_init2 (csh, 2);

      do {

         loop ++;

         ok = 1;

         prec += (loop <= 2) ? mpc_ceil_log2 (prec) + 5 : prec / 2;

         mpfr_set_prec (s, prec);

         mpfr_set_prec (c, prec);

         mpfr_set_prec (sh, prec);

         mpfr_set_prec (ch, prec);

         mpfr_set_prec (sch, prec);

         mpfr_set_prec (csh, prec);

         mpfr_sin_cos (s, c, mpc_realref(op), MPFR_RNDN);

         mpfr_sinh_cosh (sh, ch, mpc_imagref(op), MPFR_RNDN);

         if (rop_sin != NULL) {

            /* real part of sine */

            mpfr_mul (sch, s, ch, MPFR_RNDN);

            ok = (!mpfr_number_p (sch))

                  || mpfr_can_round (sch, prec - 2, MPFR_RNDN, MPFR_RNDZ,

                        MPC_PREC_RE (rop_sin)

                        + (MPC_RND_RE (rnd_sin) == MPFR_RNDN));

            if (ok) {

               /* imaginary part of sine */

               mpfr_mul (csh, c, sh, MPFR_RNDN);

               ok = (!mpfr_number_p (csh))

                     || mpfr_can_round (csh, prec - 2, MPFR_RNDN, MPFR_RNDZ,

                           MPC_PREC_IM (rop_sin)

                           + (MPC_RND_IM (rnd_sin) == MPFR_RNDN));

            }

         }

         if (rop_cos != NULL && ok) {

            /* real part of cosine */

            mpfr_mul (c, c, ch, MPFR_RNDN);

            ok = (!mpfr_number_p (c))

                  || mpfr_can_round (c, prec - 2, MPFR_RNDN, MPFR_RNDZ,

                        MPC_PREC_RE (rop_cos)

                        + (MPC_RND_RE (rnd_cos) == MPFR_RNDN));

            if (ok) {

               /* imaginary part of cosine */

               mpfr_mul (s, s, sh, MPFR_RNDN);

               mpfr_neg (s, s, MPFR_RNDN);

               ok = (!mpfr_number_p (s))

                     || mpfr_can_round (s, prec - 2, MPFR_RNDN, MPFR_RNDZ,

                           MPC_PREC_IM (rop_cos)

                           + (MPC_RND_IM (rnd_cos) == MPFR_RNDN));

            }

         }

      } while (ok == 0);

      if (rop_sin != NULL) {

         inex_re = mpfr_set (mpc_realref (rop_sin), sch, MPC_RND_RE (rnd_sin));

         if (mpfr_inf_p (sch))

           inex_re = mpc_fix_inf (mpc_realref (rop_sin), MPC_RND_RE (rnd_sin));

         inex_im = mpfr_set (mpc_imagref (rop_sin), csh, MPC_RND_IM (rnd_sin));

         if (mpfr_inf_p (csh))

           inex_im = mpc_fix_inf (mpc_imagref (rop_sin), MPC_RND_IM (rnd_sin));

         inex_sin = MPC_INEX (inex_re, inex_im);

      }

      else

         inex_sin = MPC_INEX (0,0); /* return exact if not computed */

      if (rop_cos != NULL) {

         inex_re = mpfr_set (mpc_realref (rop_cos), c, MPC_RND_RE (rnd_cos));

         if (mpfr_inf_p (c))

           inex_re = mpc_fix_inf (mpc_realref (rop_cos), MPC_RND_RE (rnd_cos));

         inex_im = mpfr_set (mpc_imagref (rop_cos), s, MPC_RND_IM (rnd_cos));

         if (mpfr_inf_p (s))

           inex_im = mpc_fix_inf (mpc_imagref (rop_cos), MPC_RND_IM (rnd_cos));

         inex_cos = MPC_INEX (inex_re, inex_im);

      }

      else

         inex_cos = MPC_INEX (0,0); /* return exact if not computed */

      mpfr_clear (s);

      mpfr_clear (c);

      mpfr_clear (sh);

      mpfr_clear (ch);

      mpfr_clear (sch);

      mpfr_clear (csh);

      return (MPC_INEX12 (inex_sin, inex_cos));

   }

}