1/* mpc_exp -- exponential of a complex number.
2
3Copyright (C) 2002, 2009, 2010, 2011, 2012, 2020 INRIA
4
5This file is part of GNU MPC.
6
7GNU MPC is free software; you can redistribute it and/or modify it under
8the terms of the GNU Lesser General Public License as published by the
9Free Software Foundation; either version 3 of the License, or (at your
10option) any later version.
11
12GNU MPC is distributed in the hope that it will be useful, but WITHOUT ANY
13WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
14FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for
15more details.
16
17You should have received a copy of the GNU Lesser General Public License
18along with this program. If not, see http://www.gnu.org/licenses/ .
19*/
20
21#include "mpc-impl.h"
22
23int
24mpc_exp (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
25{
26  mpfr_t x, y, z;
27  mpfr_prec_t prec;
28  int ok = 0;
29  int inex_re, inex_im;
30  int saved_underflow, saved_overflow;
31  mpfr_exp_t saved_emin, saved_emax;
32
33  /* special values */
34  if (mpfr_nan_p (mpc_realref (op)) || mpfr_nan_p (mpc_imagref (op)))
35    /* NaNs
36       exp(nan +i*y) = nan -i*0   if y = -0,
37                       nan +i*0   if y = +0,
38                       nan +i*nan otherwise
39       exp(x+i*nan) =   +/-0 +/-i*0 if x=-inf,
40                      +/-inf +i*nan if x=+inf,
41                         nan +i*nan otherwise */
42    {
43      if (mpfr_zero_p (mpc_imagref (op)))
44        return mpc_set (rop, op, MPC_RNDNN);
45
46      if (mpfr_inf_p (mpc_realref (op)))
47        {
48          if (mpfr_signbit (mpc_realref (op)))
49            return mpc_set_ui_ui (rop, 0, 0, MPC_RNDNN);
50          else
51            {
52              mpfr_set_inf (mpc_realref (rop), +1);
53              mpfr_set_nan (mpc_imagref (rop));
54              return MPC_INEX(0, 0); /* Inf/NaN are exact */
55            }
56        }
57      mpfr_set_nan (mpc_realref (rop));
58      mpfr_set_nan (mpc_imagref (rop));
59      return MPC_INEX(0, 0); /* NaN is exact */
60    }
61
62  if (mpfr_zero_p (mpc_imagref(op)))
63    /* special case when the input is real
64       exp(x-i*0) = exp(x) -i*0, even if x is NaN
65       exp(x+i*0) = exp(x) +i*0, even if x is NaN */
66    {
67      inex_re = mpfr_exp (mpc_realref(rop), mpc_realref(op), MPC_RND_RE(rnd));
68      inex_im = mpfr_set (mpc_imagref(rop), mpc_imagref(op), MPC_RND_IM(rnd));
69      return MPC_INEX(inex_re, inex_im);
70    }
71
72  if (mpfr_zero_p (mpc_realref (op)))
73    /* special case when the input is imaginary  */
74    {
75      inex_re = mpfr_cos (mpc_realref (rop), mpc_imagref (op), MPC_RND_RE(rnd));
76      inex_im = mpfr_sin (mpc_imagref (rop), mpc_imagref (op), MPC_RND_IM(rnd));
77      return MPC_INEX(inex_re, inex_im);
78    }
79
80  if (mpfr_inf_p (mpc_realref (op)))
81    /* real part is an infinity,
82       exp(-inf +i*y) = 0*(cos y +i*sin y)
83       exp(+inf +i*y) = +/-inf +i*nan         if y = +/-inf
84                        +inf*(cos y +i*sin y) if 0 < |y| < inf */
85    {
86      mpfr_t n;
87
88      mpfr_init2 (n, 2);
89      if (mpfr_signbit (mpc_realref (op)))
90        mpfr_set_ui (n, 0, MPFR_RNDN);
91      else
92        mpfr_set_inf (n, +1);
93
94      if (mpfr_inf_p (mpc_imagref (op)))
95        {
96          int real_sign = mpfr_signbit (mpc_realref (op));
97          inex_re = mpfr_set (mpc_realref (rop), n, MPFR_RNDN);
98          if (real_sign)
99            inex_im = mpfr_set (mpc_imagref (rop), n, MPFR_RNDN);
100          else
101            {
102              mpfr_set_nan (mpc_imagref (rop));
103              inex_im = 0; /* NaN is exact */
104            }
105        }
106      else
107        {
108          mpfr_t c, s;
109          mpfr_init2 (c, 2);
110          mpfr_init2 (s, 2);
111
112          mpfr_sin_cos (s, c, mpc_imagref (op), MPFR_RNDN);
113          inex_re = mpfr_copysign (mpc_realref (rop), n, c, MPFR_RNDN);
114          inex_im = mpfr_copysign (mpc_imagref (rop), n, s, MPFR_RNDN);
115
116          mpfr_clear (s);
117          mpfr_clear (c);
118        }
119
120      mpfr_clear (n);
121      return MPC_INEX(inex_re, inex_im);
122    }
123
124  if (mpfr_inf_p (mpc_imagref (op)))
125    /* real part is finite non-zero number, imaginary part is an infinity */
126    {
127      mpfr_set_nan (mpc_realref (rop));
128      mpfr_set_nan (mpc_imagref (rop));
129      return MPC_INEX(0, 0); /* NaN is exact */
130    }
131
132  saved_emin = mpfr_get_emin ();
133  saved_emax = mpfr_get_emax ();
134  mpfr_set_emin (mpfr_get_emin_min ());
135  mpfr_set_emax (mpfr_get_emax_max ());
136
137  /* from now on, both parts of op are regular numbers */
138
139  prec = MPC_MAX_PREC(rop)
140         + MPC_MAX (MPC_MAX (-mpfr_get_exp (mpc_realref (op)), 0),
141                   -mpfr_get_exp (mpc_imagref (op)));
142    /* When op is close to 0, then exp is close to 1+Re(op), while
143       cos is close to 1-Im(op); to decide on the ternary value of exp*cos,
144       we need a high enough precision so that none of exp or cos is
145       computed as 1. */
146  mpfr_init2 (x, 2);
147  mpfr_init2 (y, 2);
148  mpfr_init2 (z, 2);
149
150  /* save the underflow or overflow flags from MPFR */
151  saved_underflow = mpfr_underflow_p ();
152  saved_overflow = mpfr_overflow_p ();
153
154  do
155    {
156      prec += prec / 2 + mpc_ceil_log2 (prec) + 5;
157
158      mpfr_set_prec (x, prec);
159      mpfr_set_prec (y, prec);
160      mpfr_set_prec (z, prec);
161
162      /* FIXME: x may overflow so x.y does overflow too, while Re(exp(op))
163         could be represented in the precision of rop. */
164      mpfr_clear_overflow ();
165      mpfr_clear_underflow ();
166      mpfr_exp (x, mpc_realref(op), MPFR_RNDN); /* error <= 0.5ulp */
167      mpfr_sin_cos (z, y, mpc_imagref(op), MPFR_RNDN); /* errors <= 0.5ulp */
168      mpfr_mul (y, y, x, MPFR_RNDN); /* error <= 2ulp */
169      ok = mpfr_overflow_p () || mpfr_zero_p (x)
170        || mpfr_can_round (y, prec - 2, MPFR_RNDN, MPFR_RNDZ,
171                       MPC_PREC_RE(rop) + (MPC_RND_RE(rnd) == MPFR_RNDN));
172      if (ok) /* compute imaginary part */
173        {
174          mpfr_mul (z, z, x, MPFR_RNDN);
175          ok = mpfr_overflow_p () || mpfr_zero_p (x)
176            || mpfr_can_round (z, prec - 2, MPFR_RNDN, MPFR_RNDZ,
177                       MPC_PREC_IM(rop) + (MPC_RND_IM(rnd) == MPFR_RNDN));
178        }
179    }
180  while (ok == 0);
181
182  inex_re = mpfr_set (mpc_realref(rop), y, MPC_RND_RE(rnd));
183  inex_im = mpfr_set (mpc_imagref(rop), z, MPC_RND_IM(rnd));
184  if (mpfr_overflow_p ())
185    {
186      inex_re = mpc_fix_inf (mpc_realref(rop), MPC_RND_RE(rnd));
187      inex_im = mpc_fix_inf (mpc_imagref(rop), MPC_RND_IM(rnd));
188    }
189  else if (mpfr_underflow_p ())
190    {
191      inex_re = mpc_fix_zero (mpc_realref(rop), MPC_RND_RE(rnd));
192      inex_im = mpc_fix_zero (mpc_imagref(rop), MPC_RND_IM(rnd));
193    }
194
195  mpfr_clear (x);
196  mpfr_clear (y);
197  mpfr_clear (z);
198
199  /* restore underflow and overflow flags from MPFR */
200  if (saved_underflow)
201    mpfr_set_underflow ();
202  if (saved_overflow)
203    mpfr_set_overflow ();
204
205  /* restore the exponent range, and check the range of results */
206  mpfr_set_emin (saved_emin);
207  mpfr_set_emax (saved_emax);
208  inex_re = mpfr_check_range (mpc_realref (rop), inex_re, MPC_RND_RE (rnd));
209  inex_im = mpfr_check_range (mpc_imagref (rop), inex_im, MPC_RND_IM (rnd));
210
211  return MPC_INEX(inex_re, inex_im);
212}
213