- throw at atan()'s poles.
[ginac.git] / ginac / inifcns_trans.cpp
1 /** @file inifcns_trans.cpp
2  *
3  *  Implementation of transcendental (and trigonometric and hyperbolic)
4  *  functions. */
5
6 /*
7  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
8  *
9  *  This program is free software; you can redistribute it and/or modify
10  *  it under the terms of the GNU General Public License as published by
11  *  the Free Software Foundation; either version 2 of the License, or
12  *  (at your option) any later version.
13  *
14  *  This program is distributed in the hope that it will be useful,
15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
17  *  GNU General Public License for more details.
18  *
19  *  You should have received a copy of the GNU General Public License
20  *  along with this program; if not, write to the Free Software
21  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
22  */
23
24 #include <vector>
25 #include <stdexcept>
26
27 #include "inifcns.h"
28 #include "ex.h"
29 #include "constant.h"
30 #include "numeric.h"
31 #include "power.h"
32 #include "relational.h"
33 #include "symbol.h"
34 #include "pseries.h"
35 #include "utils.h"
36
37 #ifndef NO_NAMESPACE_GINAC
38 namespace GiNaC {
39 #endif // ndef NO_NAMESPACE_GINAC
40
41 //////////
42 // exponential function
43 //////////
44
45 static ex exp_evalf(const ex & x)
46 {
47         BEGIN_TYPECHECK
48                 TYPECHECK(x,numeric)
49         END_TYPECHECK(exp(x))
50         
51         return exp(ex_to_numeric(x)); // -> numeric exp(numeric)
52 }
53
54 static ex exp_eval(const ex & x)
55 {
56         // exp(0) -> 1
57         if (x.is_zero()) {
58                 return _ex1();
59         }
60         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
61         ex TwoExOverPiI=(_ex2()*x)/(Pi*I);
62         if (TwoExOverPiI.info(info_flags::integer)) {
63                 numeric z=mod(ex_to_numeric(TwoExOverPiI),_num4());
64                 if (z.is_equal(_num0()))
65                         return _ex1();
66                 if (z.is_equal(_num1()))
67                         return ex(I);
68                 if (z.is_equal(_num2()))
69                         return _ex_1();
70                 if (z.is_equal(_num3()))
71                         return ex(-I);
72         }
73         // exp(log(x)) -> x
74         if (is_ex_the_function(x, log))
75                 return x.op(0);
76         
77         // exp(float)
78         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
79                 return exp_evalf(x);
80         
81         return exp(x).hold();
82 }
83
84 static ex exp_deriv(const ex & x, unsigned deriv_param)
85 {
86         GINAC_ASSERT(deriv_param==0);
87
88         // d/dx exp(x) -> exp(x)
89         return exp(x);
90 }
91
92 REGISTER_FUNCTION(exp, eval_func(exp_eval).
93                        evalf_func(exp_evalf).
94                        derivative_func(exp_deriv));
95
96 //////////
97 // natural logarithm
98 //////////
99
100 static ex log_evalf(const ex & x)
101 {
102         BEGIN_TYPECHECK
103                 TYPECHECK(x,numeric)
104         END_TYPECHECK(log(x))
105         
106         return log(ex_to_numeric(x)); // -> numeric log(numeric)
107 }
108
109 static ex log_eval(const ex & x)
110 {
111         if (x.info(info_flags::numeric)) {
112                 if (x.is_equal(_ex0()))  // log(0) -> infinity
113                         throw(pole_error("log_eval(): log(0)",0));
114                 if (x.info(info_flags::real) && x.info(info_flags::negative))
115                         return (log(-x)+I*Pi);
116                 if (x.is_equal(_ex1()))  // log(1) -> 0
117                         return _ex0();
118                 if (x.is_equal(I))       // log(I) -> Pi*I/2
119                         return (Pi*I*_num1_2());
120                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
121                         return (Pi*I*_num_1_2());
122                 // log(float)
123                 if (!x.info(info_flags::crational))
124                         return log_evalf(x);
125         }
126         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
127         if (is_ex_the_function(x, exp)) {
128                 ex t = x.op(0);
129                 if (t.info(info_flags::numeric)) {
130                         numeric nt = ex_to_numeric(t);
131                         if (nt.is_real())
132                                 return t;
133                 }
134         }
135         
136         return log(x).hold();
137 }
138
139 static ex log_deriv(const ex & x, unsigned deriv_param)
140 {
141         GINAC_ASSERT(deriv_param==0);
142         
143         // d/dx log(x) -> 1/x
144         return power(x, _ex_1());
145 }
146
147 static ex log_series(const ex &arg,
148                      const relational &rel,
149                      int order,
150                      unsigned options)
151 {
152         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
153         ex arg_pt;
154         bool must_expand_arg = false;
155         // maybe substitution of rel into arg fails because of a pole
156         try {
157                 arg_pt = arg.subs(rel);
158         } catch (pole_error) {
159                 must_expand_arg = true;
160         }
161         // or we are at the branch cut anyways
162         if (arg_pt.is_zero())
163                 must_expand_arg = true;
164         
165         if (must_expand_arg) {
166                 // method:
167                 // This is the branch point: Series expand the argument first, then
168                 // trivially factorize it to isolate that part which has constant
169                 // leading coefficient in this fashion:
170                 //   x^n + Order(x^(n+m))  ->  x^n * (1 + Order(x^m)).
171                 // Return a plain n*log(x) for the x^n part and series expand the
172                 // other part.  Add them together and reexpand again in order to have
173                 // one unnested pseries object.  All this also works for negative n.
174                 const pseries argser = ex_to_pseries(arg.series(rel, order, options));
175                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
176                 const ex point = rel.rhs();
177                 const int n = argser.ldegree(*s);
178                 epvector seq;
179                 seq.push_back(expair(n*log(*s-point), _ex0()));
180                 if (!argser.is_terminating() || argser.nops()!=1) {
181                         // in this case n more terms are needed
182                         ex newarg = ex_to_pseries(arg.series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
183                         return pseries(rel, seq).add_series(ex_to_pseries(log(newarg).series(rel, order, options)));
184                 } else  // it was a monomial
185                         return pseries(rel, seq);
186         }
187         if (!(options & series_options::suppress_branchcut) &&
188                  arg_pt.info(info_flags::negative)) {
189                 // method:
190                 // This is the branch cut: assemble the primitive series manually and
191                 // then add the corresponding complex step function.
192                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
193                 const ex point = rel.rhs();
194                 const symbol foo;
195                 ex replarg = series(log(arg), *s==foo, order, false).subs(foo==point);
196                 epvector seq;
197                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0()));
198                 seq.push_back(expair(Order(_ex1()), order));
199                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
200         }
201         throw do_taylor();  // caught by function::series()
202 }
203
204 REGISTER_FUNCTION(log, eval_func(log_eval).
205                        evalf_func(log_evalf).
206                        derivative_func(log_deriv).
207                        series_func(log_series));
208
209 //////////
210 // sine (trigonometric function)
211 //////////
212
213 static ex sin_evalf(const ex & x)
214 {
215         BEGIN_TYPECHECK
216            TYPECHECK(x,numeric)
217         END_TYPECHECK(sin(x))
218         
219         return sin(ex_to_numeric(x)); // -> numeric sin(numeric)
220 }
221
222 static ex sin_eval(const ex & x)
223 {
224         // sin(n/d*Pi) -> { all known non-nested radicals }
225         ex SixtyExOverPi = _ex60()*x/Pi;
226         ex sign = _ex1();
227         if (SixtyExOverPi.info(info_flags::integer)) {
228                 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
229                 if (z>=_num60()) {
230                         // wrap to interval [0, Pi)
231                         z -= _num60();
232                         sign = _ex_1();
233                 }
234                 if (z>_num30()) {
235                         // wrap to interval [0, Pi/2)
236                         z = _num60()-z;
237                 }
238                 if (z.is_equal(_num0()))  // sin(0)       -> 0
239                         return _ex0();
240                 if (z.is_equal(_num5()))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
241                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
242                 if (z.is_equal(_num6()))  // sin(Pi/10)   -> sqrt(5)/4-1/4
243                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
244                 if (z.is_equal(_num10())) // sin(Pi/6)    -> 1/2
245                         return sign*_ex1_2();
246                 if (z.is_equal(_num15())) // sin(Pi/4)    -> sqrt(2)/2
247                         return sign*_ex1_2()*power(_ex2(),_ex1_2());
248                 if (z.is_equal(_num18())) // sin(3/10*Pi) -> sqrt(5)/4+1/4
249                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
250                 if (z.is_equal(_num20())) // sin(Pi/3)    -> sqrt(3)/2
251                         return sign*_ex1_2()*power(_ex3(),_ex1_2());
252                 if (z.is_equal(_num25())) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
253                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
254                 if (z.is_equal(_num30())) // sin(Pi/2)    -> 1
255                         return sign*_ex1();
256         }
257         
258         if (is_ex_exactly_of_type(x, function)) {
259                 ex t = x.op(0);
260                 // sin(asin(x)) -> x
261                 if (is_ex_the_function(x, asin))
262                         return t;
263                 // sin(acos(x)) -> sqrt(1-x^2)
264                 if (is_ex_the_function(x, acos))
265                         return power(_ex1()-power(t,_ex2()),_ex1_2());
266                 // sin(atan(x)) -> x*(1+x^2)^(-1/2)
267                 if (is_ex_the_function(x, atan))
268                         return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
269         }
270         
271         // sin(float) -> float
272         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
273                 return sin_evalf(x);
274         
275         return sin(x).hold();
276 }
277
278 static ex sin_deriv(const ex & x, unsigned deriv_param)
279 {
280         GINAC_ASSERT(deriv_param==0);
281         
282         // d/dx sin(x) -> cos(x)
283         return cos(x);
284 }
285
286 REGISTER_FUNCTION(sin, eval_func(sin_eval).
287                        evalf_func(sin_evalf).
288                        derivative_func(sin_deriv));
289
290 //////////
291 // cosine (trigonometric function)
292 //////////
293
294 static ex cos_evalf(const ex & x)
295 {
296         BEGIN_TYPECHECK
297                 TYPECHECK(x,numeric)
298         END_TYPECHECK(cos(x))
299         
300         return cos(ex_to_numeric(x)); // -> numeric cos(numeric)
301 }
302
303 static ex cos_eval(const ex & x)
304 {
305         // cos(n/d*Pi) -> { all known non-nested radicals }
306         ex SixtyExOverPi = _ex60()*x/Pi;
307         ex sign = _ex1();
308         if (SixtyExOverPi.info(info_flags::integer)) {
309                 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
310                 if (z>=_num60()) {
311                         // wrap to interval [0, Pi)
312                         z = _num120()-z;
313                 }
314                 if (z>=_num30()) {
315                         // wrap to interval [0, Pi/2)
316                         z = _num60()-z;
317                         sign = _ex_1();
318                 }
319                 if (z.is_equal(_num0()))  // cos(0)       -> 1
320                         return sign*_ex1();
321                 if (z.is_equal(_num5()))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
322                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
323                 if (z.is_equal(_num10())) // cos(Pi/6)    -> sqrt(3)/2
324                         return sign*_ex1_2()*power(_ex3(),_ex1_2());
325                 if (z.is_equal(_num12())) // cos(Pi/5)    -> sqrt(5)/4+1/4
326                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
327                 if (z.is_equal(_num15())) // cos(Pi/4)    -> sqrt(2)/2
328                         return sign*_ex1_2()*power(_ex2(),_ex1_2());
329                 if (z.is_equal(_num20())) // cos(Pi/3)    -> 1/2
330                         return sign*_ex1_2();
331                 if (z.is_equal(_num24())) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
332                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
333                 if (z.is_equal(_num25())) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
334                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
335                 if (z.is_equal(_num30())) // cos(Pi/2)    -> 0
336                         return sign*_ex0();
337         }
338         
339         if (is_ex_exactly_of_type(x, function)) {
340                 ex t = x.op(0);
341                 // cos(acos(x)) -> x
342                 if (is_ex_the_function(x, acos))
343                         return t;
344                 // cos(asin(x)) -> (1-x^2)^(1/2)
345                 if (is_ex_the_function(x, asin))
346                         return power(_ex1()-power(t,_ex2()),_ex1_2());
347                 // cos(atan(x)) -> (1+x^2)^(-1/2)
348                 if (is_ex_the_function(x, atan))
349                         return power(_ex1()+power(t,_ex2()),_ex_1_2());
350         }
351         
352         // cos(float) -> float
353         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
354                 return cos_evalf(x);
355         
356         return cos(x).hold();
357 }
358
359 static ex cos_deriv(const ex & x, unsigned deriv_param)
360 {
361         GINAC_ASSERT(deriv_param==0);
362
363         // d/dx cos(x) -> -sin(x)
364         return _ex_1()*sin(x);
365 }
366
367 REGISTER_FUNCTION(cos, eval_func(cos_eval).
368                        evalf_func(cos_evalf).
369                        derivative_func(cos_deriv));
370
371 //////////
372 // tangent (trigonometric function)
373 //////////
374
375 static ex tan_evalf(const ex & x)
376 {
377         BEGIN_TYPECHECK
378            TYPECHECK(x,numeric)
379         END_TYPECHECK(tan(x)) // -> numeric tan(numeric)
380         
381         return tan(ex_to_numeric(x));
382 }
383
384 static ex tan_eval(const ex & x)
385 {
386         // tan(n/d*Pi) -> { all known non-nested radicals }
387         ex SixtyExOverPi = _ex60()*x/Pi;
388         ex sign = _ex1();
389         if (SixtyExOverPi.info(info_flags::integer)) {
390                 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num60());
391                 if (z>=_num60()) {
392                         // wrap to interval [0, Pi)
393                         z -= _num60();
394                 }
395                 if (z>=_num30()) {
396                         // wrap to interval [0, Pi/2)
397                         z = _num60()-z;
398                         sign = _ex_1();
399                 }
400                 if (z.is_equal(_num0()))  // tan(0)       -> 0
401                         return _ex0();
402                 if (z.is_equal(_num5()))  // tan(Pi/12)   -> 2-sqrt(3)
403                         return sign*(_ex2()-power(_ex3(),_ex1_2()));
404                 if (z.is_equal(_num10())) // tan(Pi/6)    -> sqrt(3)/3
405                         return sign*_ex1_3()*power(_ex3(),_ex1_2());
406                 if (z.is_equal(_num15())) // tan(Pi/4)    -> 1
407                         return sign*_ex1();
408                 if (z.is_equal(_num20())) // tan(Pi/3)    -> sqrt(3)
409                         return sign*power(_ex3(),_ex1_2());
410                 if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
411                         return sign*(power(_ex3(),_ex1_2())+_ex2());
412                 if (z.is_equal(_num30())) // tan(Pi/2)    -> infinity
413                         throw (pole_error("tan_eval(): simple pole",1));
414         }
415         
416         if (is_ex_exactly_of_type(x, function)) {
417                 ex t = x.op(0);
418                 // tan(atan(x)) -> x
419                 if (is_ex_the_function(x, atan))
420                         return t;
421                 // tan(asin(x)) -> x*(1+x^2)^(-1/2)
422                 if (is_ex_the_function(x, asin))
423                         return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
424                 // tan(acos(x)) -> (1-x^2)^(1/2)/x
425                 if (is_ex_the_function(x, acos))
426                         return power(t,_ex_1())*power(_ex1()-power(t,_ex2()),_ex1_2());
427         }
428         
429         // tan(float) -> float
430         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
431                 return tan_evalf(x);
432         }
433         
434         return tan(x).hold();
435 }
436
437 static ex tan_deriv(const ex & x, unsigned deriv_param)
438 {
439         GINAC_ASSERT(deriv_param==0);
440         
441         // d/dx tan(x) -> 1+tan(x)^2;
442         return (_ex1()+power(tan(x),_ex2()));
443 }
444
445 static ex tan_series(const ex &x,
446                      const relational &rel,
447                      int order,
448                      unsigned options)
449 {
450         // method:
451         // Taylor series where there is no pole falls back to tan_deriv.
452         // On a pole simply expand sin(x)/cos(x).
453         const ex x_pt = x.subs(rel);
454         if (!(2*x_pt/Pi).info(info_flags::odd))
455                 throw do_taylor();  // caught by function::series()
456         // if we got here we have to care for a simple pole
457         return (sin(x)/cos(x)).series(rel, order+2);
458 }
459
460 REGISTER_FUNCTION(tan, eval_func(tan_eval).
461                        evalf_func(tan_evalf).
462                        derivative_func(tan_deriv).
463                        series_func(tan_series));
464
465 //////////
466 // inverse sine (arc sine)
467 //////////
468
469 static ex asin_evalf(const ex & x)
470 {
471         BEGIN_TYPECHECK
472            TYPECHECK(x,numeric)
473         END_TYPECHECK(asin(x))
474         
475         return asin(ex_to_numeric(x)); // -> numeric asin(numeric)
476 }
477
478 static ex asin_eval(const ex & x)
479 {
480         if (x.info(info_flags::numeric)) {
481                 // asin(0) -> 0
482                 if (x.is_zero())
483                         return x;
484                 // asin(1/2) -> Pi/6
485                 if (x.is_equal(_ex1_2()))
486                         return numeric(1,6)*Pi;
487                 // asin(1) -> Pi/2
488                 if (x.is_equal(_ex1()))
489                         return _num1_2()*Pi;
490                 // asin(-1/2) -> -Pi/6
491                 if (x.is_equal(_ex_1_2()))
492                         return numeric(-1,6)*Pi;
493                 // asin(-1) -> -Pi/2
494                 if (x.is_equal(_ex_1()))
495                         return _num_1_2()*Pi;
496                 // asin(float) -> float
497                 if (!x.info(info_flags::crational))
498                         return asin_evalf(x);
499         }
500         
501         return asin(x).hold();
502 }
503
504 static ex asin_deriv(const ex & x, unsigned deriv_param)
505 {
506         GINAC_ASSERT(deriv_param==0);
507         
508         // d/dx asin(x) -> 1/sqrt(1-x^2)
509         return power(1-power(x,_ex2()),_ex_1_2());
510 }
511
512 REGISTER_FUNCTION(asin, eval_func(asin_eval).
513                         evalf_func(asin_evalf).
514                         derivative_func(asin_deriv));
515
516 //////////
517 // inverse cosine (arc cosine)
518 //////////
519
520 static ex acos_evalf(const ex & x)
521 {
522         BEGIN_TYPECHECK
523            TYPECHECK(x,numeric)
524         END_TYPECHECK(acos(x))
525         
526         return acos(ex_to_numeric(x)); // -> numeric acos(numeric)
527 }
528
529 static ex acos_eval(const ex & x)
530 {
531         if (x.info(info_flags::numeric)) {
532                 // acos(1) -> 0
533                 if (x.is_equal(_ex1()))
534                         return _ex0();
535                 // acos(1/2) -> Pi/3
536                 if (x.is_equal(_ex1_2()))
537                         return _ex1_3()*Pi;
538                 // acos(0) -> Pi/2
539                 if (x.is_zero())
540                         return _ex1_2()*Pi;
541                 // acos(-1/2) -> 2/3*Pi
542                 if (x.is_equal(_ex_1_2()))
543                         return numeric(2,3)*Pi;
544                 // acos(-1) -> Pi
545                 if (x.is_equal(_ex_1()))
546                         return Pi;
547                 // acos(float) -> float
548                 if (!x.info(info_flags::crational))
549                         return acos_evalf(x);
550         }
551         
552         return acos(x).hold();
553 }
554
555 static ex acos_deriv(const ex & x, unsigned deriv_param)
556 {
557         GINAC_ASSERT(deriv_param==0);
558         
559         // d/dx acos(x) -> -1/sqrt(1-x^2)
560         return _ex_1()*power(1-power(x,_ex2()),_ex_1_2());
561 }
562
563 REGISTER_FUNCTION(acos, eval_func(acos_eval).
564                         evalf_func(acos_evalf).
565                         derivative_func(acos_deriv));
566
567 //////////
568 // inverse tangent (arc tangent)
569 //////////
570
571 static ex atan_evalf(const ex & x)
572 {
573         BEGIN_TYPECHECK
574                 TYPECHECK(x,numeric)
575         END_TYPECHECK(atan(x))
576         
577         return atan(ex_to_numeric(x)); // -> numeric atan(numeric)
578 }
579
580 static ex atan_eval(const ex & x)
581 {
582         if (x.info(info_flags::numeric)) {
583                 // atan(0) -> 0
584                 if (x.is_equal(_ex0()))
585                         return _ex0();
586                 // atan(1) -> Pi/4
587                 if (x.is_equal(_ex1()))
588                         return _ex1_4()*Pi;
589                 // atan(-1) -> -Pi/4
590                 if (x.is_equal(_ex_1()))
591                         return _ex_1_4()*Pi;
592                 if (x.is_equal(I) || x.is_equal(-I))
593                         throw (pole_error("atan_eval(): logarithmic pole",0));
594                 // atan(float) -> float
595                 if (!x.info(info_flags::crational))
596                         return atan_evalf(x);
597         }
598         
599         return atan(x).hold();
600 }
601
602 static ex atan_deriv(const ex & x, unsigned deriv_param)
603 {
604         GINAC_ASSERT(deriv_param==0);
605
606         // d/dx atan(x) -> 1/(1+x^2)
607         return power(_ex1()+power(x,_ex2()), _ex_1());
608 }
609
610 REGISTER_FUNCTION(atan, eval_func(atan_eval).
611                         evalf_func(atan_evalf).
612                         derivative_func(atan_deriv));
613
614 //////////
615 // inverse tangent (atan2(y,x))
616 //////////
617
618 static ex atan2_evalf(const ex & y, const ex & x)
619 {
620         BEGIN_TYPECHECK
621                 TYPECHECK(y,numeric)
622                 TYPECHECK(x,numeric)
623         END_TYPECHECK(atan2(y,x))
624         
625         return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric)
626 }
627
628 static ex atan2_eval(const ex & y, const ex & x)
629 {
630         if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
631                 x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
632                 return atan2_evalf(y,x);
633         }
634         
635         return atan2(y,x).hold();
636 }    
637
638 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
639 {
640         GINAC_ASSERT(deriv_param<2);
641         
642         if (deriv_param==0) {
643                 // d/dy atan(y,x)
644                 return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
645         }
646         // d/dx atan(y,x)
647         return -y*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
648 }
649
650 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
651                          evalf_func(atan2_evalf).
652                          derivative_func(atan2_deriv));
653
654 //////////
655 // hyperbolic sine (trigonometric function)
656 //////////
657
658 static ex sinh_evalf(const ex & x)
659 {
660         BEGIN_TYPECHECK
661            TYPECHECK(x,numeric)
662         END_TYPECHECK(sinh(x))
663         
664         return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric)
665 }
666
667 static ex sinh_eval(const ex & x)
668 {
669         if (x.info(info_flags::numeric)) {
670                 if (x.is_zero())  // sinh(0) -> 0
671                         return _ex0();        
672                 if (!x.info(info_flags::crational))  // sinh(float) -> float
673                         return sinh_evalf(x);
674         }
675         
676         if ((x/Pi).info(info_flags::numeric) &&
677                 ex_to_numeric(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
678                 return I*sin(x/I);
679         
680         if (is_ex_exactly_of_type(x, function)) {
681                 ex t = x.op(0);
682                 // sinh(asinh(x)) -> x
683                 if (is_ex_the_function(x, asinh))
684                         return t;
685                 // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2)
686                 if (is_ex_the_function(x, acosh))
687                         return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2());
688                 // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
689                 if (is_ex_the_function(x, atanh))
690                         return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
691         }
692         
693         return sinh(x).hold();
694 }
695
696 static ex sinh_deriv(const ex & x, unsigned deriv_param)
697 {
698         GINAC_ASSERT(deriv_param==0);
699         
700         // d/dx sinh(x) -> cosh(x)
701         return cosh(x);
702 }
703
704 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
705                         evalf_func(sinh_evalf).
706                         derivative_func(sinh_deriv));
707
708 //////////
709 // hyperbolic cosine (trigonometric function)
710 //////////
711
712 static ex cosh_evalf(const ex & x)
713 {
714         BEGIN_TYPECHECK
715            TYPECHECK(x,numeric)
716         END_TYPECHECK(cosh(x))
717         
718         return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric)
719 }
720
721 static ex cosh_eval(const ex & x)
722 {
723         if (x.info(info_flags::numeric)) {
724                 if (x.is_zero())  // cosh(0) -> 1
725                         return _ex1();
726                 if (!x.info(info_flags::crational))  // cosh(float) -> float
727                         return cosh_evalf(x);
728         }
729         
730         if ((x/Pi).info(info_flags::numeric) &&
731                 ex_to_numeric(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
732                 return cos(x/I);
733         
734         if (is_ex_exactly_of_type(x, function)) {
735                 ex t = x.op(0);
736                 // cosh(acosh(x)) -> x
737                 if (is_ex_the_function(x, acosh))
738                         return t;
739                 // cosh(asinh(x)) -> (1+x^2)^(1/2)
740                 if (is_ex_the_function(x, asinh))
741                         return power(_ex1()+power(t,_ex2()),_ex1_2());
742                 // cosh(atanh(x)) -> (1-x^2)^(-1/2)
743                 if (is_ex_the_function(x, atanh))
744                         return power(_ex1()-power(t,_ex2()),_ex_1_2());
745         }
746         
747         return cosh(x).hold();
748 }
749
750 static ex cosh_deriv(const ex & x, unsigned deriv_param)
751 {
752         GINAC_ASSERT(deriv_param==0);
753         
754         // d/dx cosh(x) -> sinh(x)
755         return sinh(x);
756 }
757
758 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
759                         evalf_func(cosh_evalf).
760                         derivative_func(cosh_deriv));
761
762
763 //////////
764 // hyperbolic tangent (trigonometric function)
765 //////////
766
767 static ex tanh_evalf(const ex & x)
768 {
769         BEGIN_TYPECHECK
770            TYPECHECK(x,numeric)
771         END_TYPECHECK(tanh(x))
772         
773         return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric)
774 }
775
776 static ex tanh_eval(const ex & x)
777 {
778         if (x.info(info_flags::numeric)) {
779                 if (x.is_zero())  // tanh(0) -> 0
780                         return _ex0();
781                 if (!x.info(info_flags::crational))  // tanh(float) -> float
782                         return tanh_evalf(x);
783         }
784         
785         if ((x/Pi).info(info_flags::numeric) &&
786                 ex_to_numeric(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
787                 return I*tan(x/I);
788         
789         if (is_ex_exactly_of_type(x, function)) {
790                 ex t = x.op(0);
791                 // tanh(atanh(x)) -> x
792                 if (is_ex_the_function(x, atanh))
793                         return t;
794                 // tanh(asinh(x)) -> x*(1+x^2)^(-1/2)
795                 if (is_ex_the_function(x, asinh))
796                         return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
797                 // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
798                 if (is_ex_the_function(x, acosh))
799                         return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2())*power(t,_ex_1());
800         }
801         
802         return tanh(x).hold();
803 }
804
805 static ex tanh_deriv(const ex & x, unsigned deriv_param)
806 {
807         GINAC_ASSERT(deriv_param==0);
808         
809         // d/dx tanh(x) -> 1-tanh(x)^2
810         return _ex1()-power(tanh(x),_ex2());
811 }
812
813 static ex tanh_series(const ex &x,
814                       const relational &rel,
815                       int order,
816                       unsigned options)
817 {
818         // method:
819         // Taylor series where there is no pole falls back to tanh_deriv.
820         // On a pole simply expand sinh(x)/cosh(x).
821         const ex x_pt = x.subs(rel);
822         if (!(2*I*x_pt/Pi).info(info_flags::odd))
823                 throw do_taylor();  // caught by function::series()
824         // if we got here we have to care for a simple pole
825         return (sinh(x)/cosh(x)).series(rel, order+2);
826 }
827
828 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
829                         evalf_func(tanh_evalf).
830                         derivative_func(tanh_deriv).
831                         series_func(tanh_series));
832
833 //////////
834 // inverse hyperbolic sine (trigonometric function)
835 //////////
836
837 static ex asinh_evalf(const ex & x)
838 {
839         BEGIN_TYPECHECK
840            TYPECHECK(x,numeric)
841         END_TYPECHECK(asinh(x))
842         
843         return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric)
844 }
845
846 static ex asinh_eval(const ex & x)
847 {
848         if (x.info(info_flags::numeric)) {
849                 // asinh(0) -> 0
850                 if (x.is_zero())
851                         return _ex0();
852                 // asinh(float) -> float
853                 if (!x.info(info_flags::crational))
854                         return asinh_evalf(x);
855         }
856         
857         return asinh(x).hold();
858 }
859
860 static ex asinh_deriv(const ex & x, unsigned deriv_param)
861 {
862         GINAC_ASSERT(deriv_param==0);
863         
864         // d/dx asinh(x) -> 1/sqrt(1+x^2)
865         return power(_ex1()+power(x,_ex2()),_ex_1_2());
866 }
867
868 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
869                          evalf_func(asinh_evalf).
870                          derivative_func(asinh_deriv));
871
872 //////////
873 // inverse hyperbolic cosine (trigonometric function)
874 //////////
875
876 static ex acosh_evalf(const ex & x)
877 {
878         BEGIN_TYPECHECK
879            TYPECHECK(x,numeric)
880         END_TYPECHECK(acosh(x))
881         
882         return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric)
883 }
884
885 static ex acosh_eval(const ex & x)
886 {
887         if (x.info(info_flags::numeric)) {
888                 // acosh(0) -> Pi*I/2
889                 if (x.is_zero())
890                         return Pi*I*numeric(1,2);
891                 // acosh(1) -> 0
892                 if (x.is_equal(_ex1()))
893                         return _ex0();
894                 // acosh(-1) -> Pi*I
895                 if (x.is_equal(_ex_1()))
896                         return Pi*I;
897                 // acosh(float) -> float
898                 if (!x.info(info_flags::crational))
899                         return acosh_evalf(x);
900         }
901         
902         return acosh(x).hold();
903 }
904
905 static ex acosh_deriv(const ex & x, unsigned deriv_param)
906 {
907         GINAC_ASSERT(deriv_param==0);
908         
909         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
910         return power(x+_ex_1(),_ex_1_2())*power(x+_ex1(),_ex_1_2());
911 }
912
913 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
914                          evalf_func(acosh_evalf).
915                          derivative_func(acosh_deriv));
916
917 //////////
918 // inverse hyperbolic tangent (trigonometric function)
919 //////////
920
921 static ex atanh_evalf(const ex & x)
922 {
923         BEGIN_TYPECHECK
924            TYPECHECK(x,numeric)
925         END_TYPECHECK(atanh(x))
926         
927         return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric)
928 }
929
930 static ex atanh_eval(const ex & x)
931 {
932         if (x.info(info_flags::numeric)) {
933                 // atanh(0) -> 0
934                 if (x.is_zero())
935                         return _ex0();
936                 // atanh({+|-}1) -> throw
937                 if (x.is_equal(_ex1()) || x.is_equal(_ex_1()))
938                         throw (pole_error("atanh_eval(): logarithmic pole",0));
939                 // atanh(float) -> float
940                 if (!x.info(info_flags::crational))
941                         return atanh_evalf(x);
942         }
943         
944         return atanh(x).hold();
945 }
946
947 static ex atanh_deriv(const ex & x, unsigned deriv_param)
948 {
949         GINAC_ASSERT(deriv_param==0);
950         
951         // d/dx atanh(x) -> 1/(1-x^2)
952         return power(_ex1()-power(x,_ex2()),_ex_1());
953 }
954
955 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
956                          evalf_func(atanh_evalf).
957                          derivative_func(atanh_deriv));
958
959 #ifndef NO_NAMESPACE_GINAC
960 } // namespace GiNaC
961 #endif // ndef NO_NAMESPACE_GINAC