Remove 'level' argument of evalf().
[ginac.git] / ginac / power.cpp
1 /** @file power.cpp
2  *
3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2016 Johannes Gutenberg University Mainz, Germany
7  *
8  *  This program is free software; you can redistribute it and/or modify
9  *  it under the terms of the GNU General Public License as published by
10  *  the Free Software Foundation; either version 2 of the License, or
11  *  (at your option) any later version.
12  *
13  *  This program is distributed in the hope that it will be useful,
14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
16  *  GNU General Public License for more details.
17  *
18  *  You should have received a copy of the GNU General Public License
19  *  along with this program; if not, write to the Free Software
20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
21  */
22
23 #include "power.h"
24 #include "expairseq.h"
25 #include "add.h"
26 #include "mul.h"
27 #include "ncmul.h"
28 #include "numeric.h"
29 #include "constant.h"
30 #include "operators.h"
31 #include "inifcns.h" // for log() in power::derivative()
32 #include "matrix.h"
33 #include "indexed.h"
34 #include "symbol.h"
35 #include "lst.h"
36 #include "archive.h"
37 #include "utils.h"
38 #include "relational.h"
39 #include "compiler.h"
40
41 #include <iostream>
42 #include <limits>
43 #include <stdexcept>
44 #include <vector>
45 #include <algorithm>
46
47 namespace GiNaC {
48
49 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
50   print_func<print_dflt>(&power::do_print_dflt).
51   print_func<print_latex>(&power::do_print_latex).
52   print_func<print_csrc>(&power::do_print_csrc).
53   print_func<print_python>(&power::do_print_python).
54   print_func<print_python_repr>(&power::do_print_python_repr).
55   print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
56
57 //////////
58 // default constructor
59 //////////
60
61 power::power() { }
62
63 //////////
64 // other constructors
65 //////////
66
67 // all inlined
68
69 //////////
70 // archiving
71 //////////
72
73 void power::read_archive(const archive_node &n, lst &sym_lst)
74 {
75         inherited::read_archive(n, sym_lst);
76         n.find_ex("basis", basis, sym_lst);
77         n.find_ex("exponent", exponent, sym_lst);
78 }
79
80 void power::archive(archive_node &n) const
81 {
82         inherited::archive(n);
83         n.add_ex("basis", basis);
84         n.add_ex("exponent", exponent);
85 }
86
87 //////////
88 // functions overriding virtual functions from base classes
89 //////////
90
91 // public
92
93 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
94 {
95         // Ordinary output of powers using '^' or '**'
96         if (precedence() <= level)
97                 c.s << openbrace << '(';
98         basis.print(c, precedence());
99         c.s << powersymbol;
100         c.s << openbrace;
101         exponent.print(c, precedence());
102         c.s << closebrace;
103         if (precedence() <= level)
104                 c.s << ')' << closebrace;
105 }
106
107 void power::do_print_dflt(const print_dflt & c, unsigned level) const
108 {
109         if (exponent.is_equal(_ex1_2)) {
110
111                 // Square roots are printed in a special way
112                 c.s << "sqrt(";
113                 basis.print(c);
114                 c.s << ')';
115
116         } else
117                 print_power(c, "^", "", "", level);
118 }
119
120 void power::do_print_latex(const print_latex & c, unsigned level) const
121 {
122         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
123
124                 // Powers with negative numeric exponents are printed as fractions
125                 c.s << "\\frac{1}{";
126                 power(basis, -exponent).eval().print(c);
127                 c.s << '}';
128
129         } else if (exponent.is_equal(_ex1_2)) {
130
131                 // Square roots are printed in a special way
132                 c.s << "\\sqrt{";
133                 basis.print(c);
134                 c.s << '}';
135
136         } else
137                 print_power(c, "^", "{", "}", level);
138 }
139
140 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
141 {
142         // Optimal output of integer powers of symbols to aid compiler CSE.
143         // C.f. ISO/IEC 14882:2011, section 1.9 [intro execution], paragraph 15
144         // to learn why such a parenthesation is really necessary.
145         if (exp == 1) {
146                 x.print(c);
147         } else if (exp == 2) {
148                 x.print(c);
149                 c.s << "*";
150                 x.print(c);
151         } else if (exp & 1) {
152                 x.print(c);
153                 c.s << "*";
154                 print_sym_pow(c, x, exp-1);
155         } else {
156                 c.s << "(";
157                 print_sym_pow(c, x, exp >> 1);
158                 c.s << ")*(";
159                 print_sym_pow(c, x, exp >> 1);
160                 c.s << ")";
161         }
162 }
163
164 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
165 {
166         if (exponent.is_equal(_ex_1)) {
167                 c.s << "recip(";
168                 basis.print(c);
169                 c.s << ')';
170                 return;
171         }
172         c.s << "expt(";
173         basis.print(c);
174         c.s << ", ";
175         exponent.print(c);
176         c.s << ')';
177 }
178
179 void power::do_print_csrc(const print_csrc & c, unsigned level) const
180 {
181         // Integer powers of symbols are printed in a special, optimized way
182         if (exponent.info(info_flags::integer) &&
183             (is_a<symbol>(basis) || is_a<constant>(basis))) {
184                 int exp = ex_to<numeric>(exponent).to_int();
185                 if (exp > 0)
186                         c.s << '(';
187                 else {
188                         exp = -exp;
189                         c.s << "1.0/(";
190                 }
191                 print_sym_pow(c, ex_to<symbol>(basis), exp);
192                 c.s << ')';
193
194         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
195         } else if (exponent.is_equal(_ex_1)) {
196                 c.s << "1.0/(";
197                 basis.print(c);
198                 c.s << ')';
199
200         // Otherwise, use the pow() function
201         } else {
202                 c.s << "pow(";
203                 basis.print(c);
204                 c.s << ',';
205                 exponent.print(c);
206                 c.s << ')';
207         }
208 }
209
210 void power::do_print_python(const print_python & c, unsigned level) const
211 {
212         print_power(c, "**", "", "", level);
213 }
214
215 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
216 {
217         c.s << class_name() << '(';
218         basis.print(c);
219         c.s << ',';
220         exponent.print(c);
221         c.s << ')';
222 }
223
224 bool power::info(unsigned inf) const
225 {
226         switch (inf) {
227                 case info_flags::polynomial:
228                 case info_flags::integer_polynomial:
229                 case info_flags::cinteger_polynomial:
230                 case info_flags::rational_polynomial:
231                 case info_flags::crational_polynomial:
232                         return basis.info(inf) && exponent.info(info_flags::nonnegint);
233                 case info_flags::rational_function:
234                         return basis.info(inf) && exponent.info(info_flags::integer);
235                 case info_flags::real:
236                         return basis.info(inf) && exponent.info(info_flags::integer);
237                 case info_flags::expanded:
238                         return (flags & status_flags::expanded);
239                 case info_flags::positive:
240                         return basis.info(info_flags::positive) && exponent.info(info_flags::real);
241                 case info_flags::nonnegative:
242                         return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
243                                (basis.info(info_flags::real) && exponent.info(info_flags::even));
244                 case info_flags::has_indices: {
245                         if (flags & status_flags::has_indices)
246                                 return true;
247                         else if (flags & status_flags::has_no_indices)
248                                 return false;
249                         else if (basis.info(info_flags::has_indices)) {
250                                 setflag(status_flags::has_indices);
251                                 clearflag(status_flags::has_no_indices);
252                                 return true;
253                         } else {
254                                 clearflag(status_flags::has_indices);
255                                 setflag(status_flags::has_no_indices);
256                                 return false;
257                         }
258                 }
259         }
260         return inherited::info(inf);
261 }
262
263 size_t power::nops() const
264 {
265         return 2;
266 }
267
268 ex power::op(size_t i) const
269 {
270         GINAC_ASSERT(i<2);
271
272         return i==0 ? basis : exponent;
273 }
274
275 ex power::map(map_function & f) const
276 {
277         const ex &mapped_basis = f(basis);
278         const ex &mapped_exponent = f(exponent);
279
280         if (!are_ex_trivially_equal(basis, mapped_basis)
281          || !are_ex_trivially_equal(exponent, mapped_exponent))
282                 return dynallocate<power>(mapped_basis, mapped_exponent);
283         else
284                 return *this;
285 }
286
287 bool power::is_polynomial(const ex & var) const
288 {
289         if (basis.is_polynomial(var)) {
290                 if (basis.has(var))
291                         // basis is non-constant polynomial in var
292                         return exponent.info(info_flags::nonnegint);
293                 else
294                         // basis is constant in var
295                         return !exponent.has(var);
296         }
297         // basis is a non-polynomial function of var
298         return false;
299 }
300
301 int power::degree(const ex & s) const
302 {
303         if (is_equal(ex_to<basic>(s)))
304                 return 1;
305         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
306                 if (basis.is_equal(s))
307                         return ex_to<numeric>(exponent).to_int();
308                 else
309                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
310         } else if (basis.has(s))
311                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
312         else
313                 return 0;
314 }
315
316 int power::ldegree(const ex & s) const 
317 {
318         if (is_equal(ex_to<basic>(s)))
319                 return 1;
320         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
321                 if (basis.is_equal(s))
322                         return ex_to<numeric>(exponent).to_int();
323                 else
324                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
325         } else if (basis.has(s))
326                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
327         else
328                 return 0;
329 }
330
331 ex power::coeff(const ex & s, int n) const
332 {
333         if (is_equal(ex_to<basic>(s)))
334                 return n==1 ? _ex1 : _ex0;
335         else if (!basis.is_equal(s)) {
336                 // basis not equal to s
337                 if (n == 0)
338                         return *this;
339                 else
340                         return _ex0;
341         } else {
342                 // basis equal to s
343                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
344                         // integer exponent
345                         int int_exp = ex_to<numeric>(exponent).to_int();
346                         if (n == int_exp)
347                                 return _ex1;
348                         else
349                                 return _ex0;
350                 } else {
351                         // non-integer exponents are treated as zero
352                         if (n == 0)
353                                 return *this;
354                         else
355                                 return _ex0;
356                 }
357         }
358 }
359
360 /** Perform automatic term rewriting rules in this class.  In the following
361  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
362  *  stand for such expressions that contain a plain number.
363  *  - ^(x,0) -> 1  (also handles ^(0,0))
364  *  - ^(x,1) -> x
365  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
366  *  - ^(1,x) -> 1
367  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
368  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
369  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0), case c1=1 should not happen, see below!)
370  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
371  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
372  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
373  */
374 ex power::eval() const
375 {
376         if (flags & status_flags::evaluated)
377                 return *this;
378
379         const numeric *num_basis = nullptr;
380         const numeric *num_exponent = nullptr;
381
382         if (is_exactly_a<numeric>(basis)) {
383                 num_basis = &ex_to<numeric>(basis);
384         }
385         if (is_exactly_a<numeric>(exponent)) {
386                 num_exponent = &ex_to<numeric>(exponent);
387         }
388         
389         // ^(x,0) -> 1  (0^0 also handled here)
390         if (exponent.is_zero()) {
391                 if (basis.is_zero())
392                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
393                 else
394                         return _ex1;
395         }
396         
397         // ^(x,1) -> x
398         if (exponent.is_equal(_ex1))
399                 return basis;
400
401         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
402         if (basis.is_zero() && num_exponent) {
403                 if ((num_exponent->real()).is_zero())
404                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
405                 else if ((num_exponent->real()).is_negative())
406                         throw (pole_error("power::eval(): division by zero",1));
407                 else
408                         return _ex0;
409         }
410
411         // ^(1,x) -> 1
412         if (basis.is_equal(_ex1))
413                 return _ex1;
414
415         // power of a function calculated by separate rules defined for this function
416         if (is_exactly_a<function>(basis))
417                 return ex_to<function>(basis).power(exponent);
418
419         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
420         if (is_exactly_a<power>(basis) && basis.op(0).info(info_flags::positive) && basis.op(1).info(info_flags::real))
421                 return dynallocate<power>(basis.op(0), basis.op(1) * exponent);
422
423         if ( num_exponent ) {
424
425                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
426                 // except if c1,c2 are rational, but c1^c2 is not)
427                 if ( num_basis ) {
428                         const bool basis_is_crational = num_basis->is_crational();
429                         const bool exponent_is_crational = num_exponent->is_crational();
430                         if (!basis_is_crational || !exponent_is_crational) {
431                                 // return a plain float
432                                 return dynallocate<numeric>(num_basis->power(*num_exponent));
433                         }
434
435                         const numeric res = num_basis->power(*num_exponent);
436                         if (res.is_crational()) {
437                                 return res;
438                         }
439                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
440
441                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
442                         if (basis_is_crational && exponent_is_crational
443                             && num_exponent->is_real()
444                             && !num_exponent->is_integer()) {
445                                 const numeric n = num_exponent->numer();
446                                 const numeric m = num_exponent->denom();
447                                 numeric r;
448                                 numeric q = iquo(n, m, r);
449                                 if (r.is_negative()) {
450                                         r += m;
451                                         --q;
452                                 }
453                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
454                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
455                                                 // try it for numerator and denominator separately, in order to
456                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
457                                                 const numeric bnum = num_basis->numer();
458                                                 const numeric bden = num_basis->denom();
459                                                 const numeric res_bnum = bnum.power(*num_exponent);
460                                                 const numeric res_bden = bden.power(*num_exponent);
461                                                 if (res_bnum.is_integer())
462                                                         return dynallocate<mul>(dynallocate<power>(bden,-*num_exponent),res_bnum).setflag(status_flags::evaluated);
463                                                 if (res_bden.is_integer())
464                                                         return dynallocate<mul>(dynallocate<power>(bnum,*num_exponent),res_bden.inverse()).setflag(status_flags::evaluated);
465                                         }
466                                         return this->hold();
467                                 } else {
468                                         // assemble resulting product, but allowing for a re-evaluation,
469                                         // because otherwise we'll end up with something like
470                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
471                                         // instead of 7/16*7^(1/3).
472                                         return pow(basis, r.div(m)) * pow(basis, q);
473                                 }
474                         }
475                 }
476         
477                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
478                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
479                 // case c1==1 should not happen, see below!)
480                 if (is_exactly_a<power>(basis)) {
481                         const power & sub_power = ex_to<power>(basis);
482                         const ex & sub_basis = sub_power.basis;
483                         const ex & sub_exponent = sub_power.exponent;
484                         if (is_exactly_a<numeric>(sub_exponent)) {
485                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
486                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
487                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() ||
488                                     (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
489                                         return dynallocate<power>(sub_basis, num_sub_exponent.mul(*num_exponent));
490                                 }
491                         }
492                 }
493         
494                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
495                 if (num_exponent->is_integer() && is_exactly_a<mul>(basis)) {
496                         return expand_mul(ex_to<mul>(basis), *num_exponent, false);
497                 }
498
499                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
500                 if (num_exponent->is_integer() && is_exactly_a<add>(basis)) {
501                         numeric icont = basis.integer_content();
502                         const numeric lead_coeff = 
503                                 ex_to<numeric>(ex_to<add>(basis).seq.begin()->coeff).div(icont);
504
505                         const bool canonicalizable = lead_coeff.is_integer();
506                         const bool unit_normal = lead_coeff.is_pos_integer();
507                         if (canonicalizable && (! unit_normal))
508                                 icont = icont.mul(*_num_1_p);
509                         
510                         if (canonicalizable && (icont != *_num1_p)) {
511                                 const add& addref = ex_to<add>(basis);
512                                 add & addp = dynallocate<add>(addref);
513                                 addp.clearflag(status_flags::hash_calculated);
514                                 addp.overall_coeff = ex_to<numeric>(addp.overall_coeff).div_dyn(icont);
515                                 for (auto & i : addp.seq)
516                                         i.coeff = ex_to<numeric>(i.coeff).div_dyn(icont);
517
518                                 const numeric c = icont.power(*num_exponent);
519                                 if (likely(c != *_num1_p))
520                                         return dynallocate<mul>(dynallocate<power>(addp, *num_exponent), c);
521                                 else
522                                         return dynallocate<power>(addp, *num_exponent);
523                         }
524                 }
525
526                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
527                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
528                 if (is_exactly_a<mul>(basis)) {
529                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
530                         const mul & mulref = ex_to<mul>(basis);
531                         if (!mulref.overall_coeff.is_equal(_ex1)) {
532                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
533                                 if (num_coeff.is_real()) {
534                                         if (num_coeff.is_positive()) {
535                                                 mul & mulp = dynallocate<mul>(mulref);
536                                                 mulp.overall_coeff = _ex1;
537                                                 mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
538                                                 return dynallocate<mul>(dynallocate<power>(mulp, exponent),
539                                                                         dynallocate<power>(num_coeff, *num_exponent));
540                                         } else {
541                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
542                                                 if (!num_coeff.is_equal(*_num_1_p)) {
543                                                         mul & mulp = dynallocate<mul>(mulref);
544                                                         mulp.overall_coeff = _ex_1;
545                                                         mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
546                                                         return dynallocate<mul>(dynallocate<power>(mulp, exponent),
547                                                                                 dynallocate<power>(abs(num_coeff), *num_exponent));
548                                                 }
549                                         }
550                                 }
551                         }
552                 }
553
554                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
555                 if (num_exponent->is_pos_integer() &&
556                     basis.return_type() != return_types::commutative &&
557                     !is_a<matrix>(basis)) {
558                         return ncmul(exvector(num_exponent->to_int(), basis));
559                 }
560         }
561
562         return this->hold();
563 }
564
565 ex power::evalf() const
566 {
567         ex ebasis = basis.evalf();
568         ex eexponent;
569         
570         if (!is_exactly_a<numeric>(exponent))
571                 eexponent = exponent.evalf();
572         else
573                 eexponent = exponent;
574
575         return dynallocate<power>(ebasis, eexponent);
576 }
577
578 ex power::evalm() const
579 {
580         const ex ebasis = basis.evalm();
581         const ex eexponent = exponent.evalm();
582         if (is_a<matrix>(ebasis)) {
583                 if (is_exactly_a<numeric>(eexponent)) {
584                         return dynallocate<matrix>(ex_to<matrix>(ebasis).pow(eexponent));
585                 }
586         }
587         return dynallocate<power>(ebasis, eexponent);
588 }
589
590 bool power::has(const ex & other, unsigned options) const
591 {
592         if (!(options & has_options::algebraic))
593                 return basic::has(other, options);
594         if (!is_a<power>(other))
595                 return basic::has(other, options);
596         if (!exponent.info(info_flags::integer) ||
597             !other.op(1).info(info_flags::integer))
598                 return basic::has(other, options);
599         if (exponent.info(info_flags::posint) &&
600             other.op(1).info(info_flags::posint) &&
601             ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
602             basis.match(other.op(0)))
603                 return true;
604         if (exponent.info(info_flags::negint) &&
605             other.op(1).info(info_flags::negint) &&
606             ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
607             basis.match(other.op(0)))
608                 return true;
609         return basic::has(other, options);
610 }
611
612 // from mul.cpp
613 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
614
615 ex power::subs(const exmap & m, unsigned options) const
616 {       
617         const ex &subsed_basis = basis.subs(m, options);
618         const ex &subsed_exponent = exponent.subs(m, options);
619
620         if (!are_ex_trivially_equal(basis, subsed_basis)
621          || !are_ex_trivially_equal(exponent, subsed_exponent)) 
622                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
623
624         if (!(options & subs_options::algebraic))
625                 return subs_one_level(m, options);
626
627         for (auto & it : m) {
628                 int nummatches = std::numeric_limits<int>::max();
629                 exmap repls;
630                 if (tryfactsubs(*this, it.first, nummatches, repls)) {
631                         ex anum = it.second.subs(repls, subs_options::no_pattern);
632                         ex aden = it.first.subs(repls, subs_options::no_pattern);
633                         ex result = (*this) * pow(anum/aden, nummatches);
634                         return (ex_to<basic>(result)).subs_one_level(m, options);
635                 }
636         }
637
638         return subs_one_level(m, options);
639 }
640
641 ex power::eval_ncmul(const exvector & v) const
642 {
643         return inherited::eval_ncmul(v);
644 }
645
646 ex power::conjugate() const
647 {
648         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
649         // branch cut which runs along the negative real axis.
650         if (basis.info(info_flags::positive)) {
651                 ex newexponent = exponent.conjugate();
652                 if (are_ex_trivially_equal(exponent, newexponent)) {
653                         return *this;
654                 }
655                 return dynallocate<power>(basis, newexponent);
656         }
657         if (exponent.info(info_flags::integer)) {
658                 ex newbasis = basis.conjugate();
659                 if (are_ex_trivially_equal(basis, newbasis)) {
660                         return *this;
661                 }
662                 return dynallocate<power>(newbasis, exponent);
663         }
664         return conjugate_function(*this).hold();
665 }
666
667 ex power::real_part() const
668 {
669         // basis == a+I*b, exponent == c+I*d
670         const ex a = basis.real_part();
671         const ex c = exponent.real_part();
672         if (basis.is_equal(a) && exponent.is_equal(c)) {
673                 // Re(a^c)
674                 return *this;
675         }
676
677         const ex b = basis.imag_part();
678         if (exponent.info(info_flags::integer)) {
679                 // Re((a+I*b)^c)  w/  c ∈ ℤ
680                 long N = ex_to<numeric>(c).to_long();
681                 // Use real terms in Binomial expansion to construct
682                 // Re(expand(pow(a+I*b, N))).
683                 long NN = N > 0 ? N : -N;
684                 ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
685                 ex result = 0;
686                 for (long n = 0; n <= NN; n += 2) {
687                         ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
688                         if (n % 4 == 0) {
689                                 result += term;  // sign: I^n w/ n == 4*m
690                         } else {
691                                 result -= term;  // sign: I^n w/ n == 4*m+2
692                         }
693                 }
694                 return result;
695         }
696
697         // Re((a+I*b)^(c+I*d))
698         const ex d = exponent.imag_part();
699         return pow(abs(basis),c) * exp(-d*atan2(b,a)) * cos(c*atan2(b,a)+d*log(abs(basis)));
700 }
701
702 ex power::imag_part() const
703 {
704         // basis == a+I*b, exponent == c+I*d
705         const ex a = basis.real_part();
706         const ex c = exponent.real_part();
707         if (basis.is_equal(a) && exponent.is_equal(c)) {
708                 // Im(a^c)
709                 return 0;
710         }
711
712         const ex b = basis.imag_part();
713         if (exponent.info(info_flags::integer)) {
714                 // Im((a+I*b)^c)  w/  c ∈ ℤ
715                 long N = ex_to<numeric>(c).to_long();
716                 // Use imaginary terms in Binomial expansion to construct
717                 // Im(expand(pow(a+I*b, N))).
718                 long p = N > 0 ? 1 : 3;  // modulus for positive sign
719                 long NN = N > 0 ? N : -N;
720                 ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
721                 ex result = 0;
722                 for (long n = 1; n <= NN; n += 2) {
723                         ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
724                         if (n % 4 == p) {
725                                 result += term;  // sign: I^n w/ n == 4*m+p
726                         } else {
727                                 result -= term;  // sign: I^n w/ n == 4*m+2+p
728                         }
729                 }
730                 return result;
731         }
732
733         // Im((a+I*b)^(c+I*d))
734         const ex d = exponent.imag_part();
735         return pow(abs(basis),c) * exp(-d*atan2(b,a)) * sin(c*atan2(b,a)+d*log(abs(basis)));
736 }
737
738 // protected
739
740 /** Implementation of ex::diff() for a power.
741  *  @see ex::diff */
742 ex power::derivative(const symbol & s) const
743 {
744         if (is_a<numeric>(exponent)) {
745                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
746                 const epvector newseq = {expair(basis, exponent - _ex1), expair(basis.diff(s), _ex1)};
747                 return dynallocate<mul>(std::move(newseq), exponent);
748         } else {
749                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
750                 return *this * (exponent.diff(s)*log(basis) + exponent*basis.diff(s)*pow(basis, _ex_1));
751         }
752 }
753
754 int power::compare_same_type(const basic & other) const
755 {
756         GINAC_ASSERT(is_exactly_a<power>(other));
757         const power &o = static_cast<const power &>(other);
758
759         int cmpval = basis.compare(o.basis);
760         if (cmpval)
761                 return cmpval;
762         else
763                 return exponent.compare(o.exponent);
764 }
765
766 unsigned power::return_type() const
767 {
768         return basis.return_type();
769 }
770
771 return_type_t power::return_type_tinfo() const
772 {
773         return basis.return_type_tinfo();
774 }
775
776 ex power::expand(unsigned options) const
777 {
778         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
779                 // A special case worth optimizing.
780                 setflag(status_flags::expanded);
781                 return *this;
782         }
783
784         // (x*p)^c -> x^c * p^c, if p>0
785         // makes sense before expanding the basis
786         if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
787                 const mul &m = ex_to<mul>(basis);
788                 exvector prodseq;
789                 epvector powseq;
790                 prodseq.reserve(m.seq.size() + 1);
791                 powseq.reserve(m.seq.size() + 1);
792                 bool possign = true;
793
794                 // search for positive/negative factors
795                 for (auto & cit : m.seq) {
796                         ex e=m.recombine_pair_to_ex(cit);
797                         if (e.info(info_flags::positive))
798                                 prodseq.push_back(pow(e, exponent).expand(options));
799                         else if (e.info(info_flags::negative)) {
800                                 prodseq.push_back(pow(-e, exponent).expand(options));
801                                 possign = !possign;
802                         } else
803                                 powseq.push_back(cit);
804                 }
805
806                 // take care on the numeric coefficient
807                 ex coeff=(possign? _ex1 : _ex_1);
808                 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
809                         prodseq.push_back(pow(m.overall_coeff, exponent));
810                 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
811                         prodseq.push_back(pow(-m.overall_coeff, exponent));
812                 else
813                         coeff *= m.overall_coeff;
814
815                 // If positive/negative factors are found, then extract them.
816                 // In either case we set a flag to avoid the second run on a part
817                 // which does not have positive/negative terms.
818                 if (prodseq.size() > 0) {
819                         ex newbasis = dynallocate<mul>(std::move(powseq), coeff);
820                         ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
821                         return dynallocate<mul>(std::move(prodseq)) * pow(newbasis, exponent);
822                 } else
823                         ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
824         }
825
826         const ex expanded_basis = basis.expand(options);
827         const ex expanded_exponent = exponent.expand(options);
828         
829         // x^(a+b) -> x^a * x^b
830         if (is_exactly_a<add>(expanded_exponent)) {
831                 const add &a = ex_to<add>(expanded_exponent);
832                 exvector distrseq;
833                 distrseq.reserve(a.seq.size() + 1);
834                 for (auto & cit : a.seq) {
835                         distrseq.push_back(pow(expanded_basis, a.recombine_pair_to_ex(cit)));
836                 }
837                 
838                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
839                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
840                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
841                         long int_exponent = num_exponent.to_int();
842                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
843                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
844                         else
845                                 distrseq.push_back(pow(expanded_basis, a.overall_coeff));
846                 } else
847                         distrseq.push_back(pow(expanded_basis, a.overall_coeff));
848                 
849                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
850                 ex r = dynallocate<mul>(distrseq);
851                 return r.expand(options);
852         }
853         
854         if (!is_exactly_a<numeric>(expanded_exponent) ||
855                 !ex_to<numeric>(expanded_exponent).is_integer()) {
856                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
857                         return this->hold();
858                 } else {
859                         return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
860                 }
861         }
862         
863         // integer numeric exponent
864         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
865         long int_exponent = num_exponent.to_long();
866         
867         // (x+y)^n, n>0
868         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
869                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
870         
871         // (x*y)^n -> x^n * y^n
872         if (is_exactly_a<mul>(expanded_basis))
873                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
874         
875         // cannot expand further
876         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
877                 return this->hold();
878         else
879                 return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
880 }
881
882 //////////
883 // new virtual functions which can be overridden by derived classes
884 //////////
885
886 // none
887
888 //////////
889 // non-virtual functions in this class
890 //////////
891
892 namespace {  // anonymous namespace for power::expand_add() helpers
893
894 /** Helper class to generate all bounded combinatorial partitions of an integer
895  *  n with exactly m parts (including zero parts) in non-decreasing order.
896  */
897 class partition_generator {
898 private:
899         // Partitions n into m parts, not including zero parts.
900         // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
901         // FXT library)
902         struct mpartition2
903         {
904                 // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
905                 std::vector<int> x;
906                 int n;   // n>0
907                 int m;   // 0<m<=n
908                 mpartition2(unsigned n_, unsigned m_)
909                   : x(m_+1), n(n_), m(m_)
910                 {
911                         for (int k=1; k<m; ++k)
912                                 x[k] = 1;
913                         x[m] = n - m + 1;
914                 }
915                 bool next_partition()
916                 {
917                         int u = x[m];  // last element
918                         int k = m;
919                         int s = u;
920                         while (--k) {
921                                 s += x[k];
922                                 if (x[k] + 2 <= u)
923                                         break;
924                         }
925                         if (k==0)
926                                 return false;  // current is last
927                         int f = x[k] + 1;
928                         while (k < m) {
929                                 x[k] = f;
930                                 s -= f;
931                                 ++k;
932                         }
933                         x[m] = s;
934                         return true;
935                 }
936         } mpgen;
937         int m;  // number of parts 0<m<=n
938         mutable std::vector<int> partition;  // current partition
939 public:
940         partition_generator(unsigned n_, unsigned m_)
941           : mpgen(n_, 1), m(m_), partition(m_)
942         { }
943         // returns current partition in non-decreasing order, padded with zeros
944         const std::vector<int>& current() const
945         {
946                 for (int i = 0; i < m - mpgen.m; ++i)
947                         partition[i] = 0;  // pad with zeros
948
949                 for (int i = m - mpgen.m; i < m; ++i)
950                         partition[i] = mpgen.x[i - m + mpgen.m + 1];
951
952                 return partition;
953         }
954         bool next()
955         {
956                 if (!mpgen.next_partition()) {
957                         if (mpgen.m == m || mpgen.m == mpgen.n)
958                                 return false;  // current is last
959                         // increment number of parts
960                         mpgen = mpartition2(mpgen.n, mpgen.m + 1);
961                 }
962                 return true;
963         }
964 };
965
966 /** Helper class to generate all compositions of a partition of an integer n,
967  *  starting with the compositions which has non-decreasing order.
968  */
969 class composition_generator {
970 private:
971         // Generates all distinct permutations of a multiset.
972         // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
973         // Multiset Permutations using a Constant Number of Variables by Prefix
974         // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
975         struct coolmulti {
976                 // element of singly linked list
977                 struct element {
978                         int value;
979                         element* next;
980                         element(int val, element* n)
981                           : value(val), next(n) {}
982                         ~element()
983                         {   // recurses down to the end of the singly linked list
984                                 delete next;
985                         }
986                 };
987                 element *head, *i, *after_i;
988                 // NB: Partition must be sorted in non-decreasing order.
989                 explicit coolmulti(const std::vector<int>& partition)
990                   : head(nullptr), i(nullptr), after_i(nullptr)
991                 {
992                         for (unsigned n = 0; n < partition.size(); ++n) {
993                                 head = new element(partition[n], head);
994                                 if (n <= 1)
995                                         i = head;
996                         }
997                         after_i = i->next;
998                 }
999                 ~coolmulti()
1000                 {   // deletes singly linked list
1001                         delete head;
1002                 }
1003                 void next_permutation()
1004                 {
1005                         element *before_k;
1006                         if (after_i->next != nullptr && i->value >= after_i->next->value)
1007                                 before_k = after_i;
1008                         else
1009                                 before_k = i;
1010                         element *k = before_k->next;
1011                         before_k->next = k->next;
1012                         k->next = head;
1013                         if (k->value < head->value)
1014                                 i = k;
1015                         after_i = i->next;
1016                         head = k;
1017                 }
1018                 bool finished() const
1019                 {
1020                         return after_i->next == nullptr && after_i->value >= head->value;
1021                 }
1022         } cmgen;
1023         bool atend;  // needed for simplifying iteration over permutations
1024         bool trivial;  // likewise, true if all elements are equal
1025         mutable std::vector<int> composition;  // current compositions
1026 public:
1027         explicit composition_generator(const std::vector<int>& partition)
1028           : cmgen(partition), atend(false), trivial(true), composition(partition.size())
1029         {
1030                 for (unsigned i=1; i<partition.size(); ++i)
1031                         trivial = trivial && (partition[0] == partition[i]);
1032         }
1033         const std::vector<int>& current() const
1034         {
1035                 coolmulti::element* it = cmgen.head;
1036                 size_t i = 0;
1037                 while (it != nullptr) {
1038                         composition[i] = it->value;
1039                         it = it->next;
1040                         ++i;
1041                 }
1042                 return composition;
1043         }
1044         bool next()
1045         {
1046                 // This ugly contortion is needed because the original coolmulti
1047                 // algorithm requires code duplication of the payload procedure,
1048                 // one before the loop and one inside it.
1049                 if (trivial || atend)
1050                         return false;
1051                 cmgen.next_permutation();
1052                 atend = cmgen.finished();
1053                 return true;
1054         }
1055 };
1056
1057 /** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
1058  *  where n = p1+p2+...+pk, i.e. p is a partition of n.
1059  */
1060 const numeric
1061 multinomial_coefficient(const std::vector<int> & p)
1062 {
1063         numeric n = 0, d = 1;
1064         for (auto & it : p) {
1065                 n += numeric(it);
1066                 d *= factorial(numeric(it));
1067         }
1068         return factorial(numeric(n)) / d;
1069 }
1070
1071 }  // anonymous namespace
1072
1073
1074 /** expand a^n where a is an add and n is a positive integer.
1075  *  @see power::expand */
1076 ex power::expand_add(const add & a, long n, unsigned options)
1077 {
1078         // The special case power(+(x,...y;x),2) can be optimized better.
1079         if (n==2)
1080                 return expand_add_2(a, options);
1081
1082         // method:
1083         //
1084         // Consider base as the sum of all symbolic terms and the overall numeric
1085         // coefficient and apply the binomial theorem:
1086         // S = power(+(x,...,z;c),n)
1087         //   = power(+(+(x,...,z;0);c),n)
1088         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1089         // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
1090         // The multinomial theorem is computed by an outer loop over all
1091         // partitions of the exponent and an inner loop over all compositions of
1092         // that partition. This method makes the expansion a combinatorial
1093         // problem and allows us to directly construct the expanded sum and also
1094         // to re-use the multinomial coefficients (since they depend only on the
1095         // partition, not on the composition).
1096         // 
1097         // multinomial power(+(x,y,z;0),3) example:
1098         // partition : compositions                : multinomial coefficient
1099         // [0,0,3]   : [3,0,0],[0,3,0],[0,0,3]     : 3!/(3!*0!*0!) = 1
1100         // [0,1,2]   : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
1101         // [1,1,1]   : [1,1,1]                     : 3!/(1!*1!*1!) = 6
1102         //  =>  (x + y + z)^3 =
1103         //        x^3 + y^3 + z^3
1104         //      + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
1105         //      + 6*x*y*z
1106         //
1107         // multinomial power(+(x,y,z;0),4) example:
1108         // partition : compositions                : multinomial coefficient
1109         // [0,0,4]   : [4,0,0],[0,4,0],[0,0,4]     : 4!/(4!*0!*0!) = 1
1110         // [0,1,3]   : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
1111         // [0,2,2]   : [2,2,0],[2,0,2],[0,2,2]     : 4!/(2!*2!*0!) = 6
1112         // [1,1,2]   : [2,1,1],[1,2,1],[1,1,2]     : 4!/(2!*1!*1!) = 12
1113         // (no [1,1,1,1] partition since it has too many parts)
1114         //  =>  (x + y + z)^4 =
1115         //        x^4 + y^4 + z^4
1116         //      + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
1117         //      + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
1118         //      + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
1119         //
1120         // Summary:
1121         // r = 0
1122         // for k from 0 to n:
1123         //     f = c^(n-k)*binomial(n,k)
1124         //     for p in all partitions of n with m parts (including zero parts):
1125         //         h = f * multinomial coefficient of p
1126         //         for c in all compositions of p:
1127         //             t = 1
1128         //             for e in all elements of c:
1129         //                 t = t * a[e]^e
1130         //             r = r + h*t
1131         // return r
1132
1133         epvector result;
1134         // The number of terms will be the number of combinatorial compositions,
1135         // i.e. the number of unordered arrangements of m nonnegative integers
1136         // which sum up to n.  It is frequently written as C_n(m) and directly
1137         // related with binomial coefficients: binomial(n+m-1,m-1).
1138         size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
1139         if (!a.overall_coeff.is_zero()) {
1140                 // the result's overall_coeff is one of the terms
1141                 --result_size;
1142         }
1143         result.reserve(result_size);
1144
1145         // Iterate over all terms in binomial expansion of
1146         // S = power(+(x,...,z;c),n)
1147         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1148         for (int k = 1; k <= n; ++k) {
1149                 numeric binomial_coefficient;  // binomial(n,k)*c^(n-k)
1150                 if (a.overall_coeff.is_zero()) {
1151                         // degenerate case with zero overall_coeff:
1152                         // apply multinomial theorem directly to power(+(x,...z;0),n)
1153                         binomial_coefficient = 1;
1154                         if (k < n) {
1155                                 continue;
1156                         }
1157                 } else {
1158                         binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
1159                 }
1160
1161                 // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
1162                 // Iterate over all partitions of k with exactly as many parts as
1163                 // there are symbolic terms in the basis (including zero parts).
1164                 partition_generator partitions(k, a.seq.size());
1165                 do {
1166                         const std::vector<int>& partition = partitions.current();
1167                         // All monomials of this partition have the same number of terms and the same coefficient.
1168                         const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; });
1169                         const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
1170
1171                         // Iterate over all compositions of the current partition.
1172                         composition_generator compositions(partition);
1173                         do {
1174                                 const std::vector<int>& exponent = compositions.current();
1175                                 epvector monomial;
1176                                 monomial.reserve(msize);
1177                                 numeric factor = coeff;
1178                                 for (unsigned i = 0; i < exponent.size(); ++i) {
1179                                         const ex & r = a.seq[i].rest;
1180                                         GINAC_ASSERT(!is_exactly_a<add>(r));
1181                                         GINAC_ASSERT(!is_exactly_a<power>(r) ||
1182                                                      !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1183                                                      !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1184                                                      !is_exactly_a<add>(ex_to<power>(r).basis) ||
1185                                                      !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1186                                                      !is_exactly_a<power>(ex_to<power>(r).basis));
1187                                         GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
1188                                         const numeric & c = ex_to<numeric>(a.seq[i].coeff);
1189                                         if (exponent[i] == 0) {
1190                                                 // optimize away
1191                                         } else if (exponent[i] == 1) {
1192                                                 // optimized
1193                                                 monomial.push_back(expair(r, _ex1));
1194                                                 if (c != *_num1_p)
1195                                                         factor = factor.mul(c);
1196                                         } else { // general case exponent[i] > 1
1197                                                 monomial.push_back(expair(r, exponent[i]));
1198                                                 if (c != *_num1_p)
1199                                                         factor = factor.mul(c.power(exponent[i]));
1200                                         }
1201                                 }
1202                                 result.push_back(expair(mul(monomial).expand(options), factor));
1203                         } while (compositions.next());
1204                 } while (partitions.next());
1205         }
1206
1207         GINAC_ASSERT(result.size() == result_size);
1208         if (a.overall_coeff.is_zero()) {
1209                 return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1210         } else {
1211                 return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)).setflag(status_flags::expanded);
1212         }
1213 }
1214
1215
1216 /** Special case of power::expand_add. Expands a^2 where a is an add.
1217  *  @see power::expand_add */
1218 ex power::expand_add_2(const add & a, unsigned options)
1219 {
1220         epvector result;
1221         size_t result_size = (a.nops() * (a.nops()+1)) / 2;
1222         if (!a.overall_coeff.is_zero()) {
1223                 // the result's overall_coeff is one of the terms
1224                 --result_size;
1225         }
1226         result.reserve(result_size);
1227
1228         epvector::const_iterator last = a.seq.end();
1229
1230         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1231         // first part: ignore overall_coeff and expand other terms
1232         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1233                 const ex & r = cit0->rest;
1234                 const ex & c = cit0->coeff;
1235                 
1236                 GINAC_ASSERT(!is_exactly_a<add>(r));
1237                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1238                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1239                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1240                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
1241                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1242                              !is_exactly_a<power>(ex_to<power>(r).basis));
1243                 
1244                 if (c.is_equal(_ex1)) {
1245                         if (is_exactly_a<mul>(r)) {
1246                                 result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1247                                                         _ex1));
1248                         } else {
1249                                 result.push_back(expair(dynallocate<power>(r, _ex2),
1250                                                         _ex1));
1251                         }
1252                 } else {
1253                         if (is_exactly_a<mul>(r)) {
1254                                 result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1255                                                         ex_to<numeric>(c).power_dyn(*_num2_p)));
1256                         } else {
1257                                 result.push_back(expair(dynallocate<power>(r, _ex2),
1258                                                         ex_to<numeric>(c).power_dyn(*_num2_p)));
1259                         }
1260                 }
1261
1262                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1263                         const ex & r1 = cit1->rest;
1264                         const ex & c1 = cit1->coeff;
1265                         result.push_back(expair(mul(r,r1).expand(options),
1266                                                 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1267                 }
1268         }
1269         
1270         // second part: add terms coming from overall_coeff (if != 0)
1271         if (!a.overall_coeff.is_zero()) {
1272                 for (auto & i : a.seq)
1273                         result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1274         }
1275
1276         GINAC_ASSERT(result.size() == result_size);
1277
1278         if (a.overall_coeff.is_zero()) {
1279                 return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1280         } else {
1281                 return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)).setflag(status_flags::expanded);
1282         }
1283 }
1284
1285 /** Expand factors of m in m^n where m is a mul and n is an integer.
1286  *  @see power::expand */
1287 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand)
1288 {
1289         GINAC_ASSERT(n.is_integer());
1290
1291         if (n.is_zero()) {
1292                 return _ex1;
1293         }
1294
1295         // do not bother to rename indices if there are no any.
1296         if (!(options & expand_options::expand_rename_idx) &&
1297             m.info(info_flags::has_indices))
1298                 options |= expand_options::expand_rename_idx;
1299         // Leave it to multiplication since dummy indices have to be renamed
1300         if ((options & expand_options::expand_rename_idx) &&
1301             (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1302                 ex result = m;
1303                 exvector va = get_all_dummy_indices(m);
1304                 sort(va.begin(), va.end(), ex_is_less());
1305
1306                 for (int i=1; i < n.to_int(); i++)
1307                         result *= rename_dummy_indices_uniquely(va, m);
1308                 return result;
1309         }
1310
1311         epvector distrseq;
1312         distrseq.reserve(m.seq.size());
1313         bool need_reexpand = false;
1314
1315         for (auto & cit : m.seq) {
1316                 expair p = m.combine_pair_with_coeff_to_pair(cit, n);
1317                 if (from_expand && is_exactly_a<add>(cit.rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1318                         // this happens when e.g. (a+b)^(1/2) gets squared and
1319                         // the resulting product needs to be reexpanded
1320                         need_reexpand = true;
1321                 }
1322                 distrseq.push_back(p);
1323         }
1324
1325         const mul & result = dynallocate<mul>(std::move(distrseq), ex_to<numeric>(m.overall_coeff).power_dyn(n));
1326         if (need_reexpand)
1327                 return ex(result).expand(options);
1328         if (from_expand)
1329                 return result.setflag(status_flags::expanded);
1330         return result;
1331 }
1332
1333 GINAC_BIND_UNARCHIVER(power);
1334
1335 } // namespace GiNaC