ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2015 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(int level) const
 566 {
 567         ex ebasis;
 568         ex eexponent;
 569
 570         if (level==1) {
 571                 ebasis = basis;
 572                 eexponent = exponent;
 573         } else if (level == -max_recursion_level) {
 574                 throw(std::runtime_error("max recursion level reached"));
 575         } else {
 576                 ebasis = basis.evalf(level-1);
 577                 if (!is_exactly_a<numeric>(exponent))
 578                         eexponent = exponent.evalf(level-1);
 579                 else
 580                         eexponent = exponent;
 581         }
 582
 583         return dynallocate<power>(ebasis, eexponent);
 584 }
 585
 586 ex power::evalm() const
 587 {
 588         const ex ebasis = basis.evalm();
 589         const ex eexponent = exponent.evalm();
 590         if (is_a<matrix>(ebasis)) {
 591                 if (is_exactly_a<numeric>(eexponent)) {
 592                         return dynallocate<matrix>(ex_to<matrix>(ebasis).pow(eexponent));
 593                 }
 594         }
 595         return dynallocate<power>(ebasis, eexponent);
 596 }
 597
 598 bool power::has(const ex & other, unsigned options) const
 599 {
 600         if (!(options & has_options::algebraic))
 601                 return basic::has(other, options);
 602         if (!is_a<power>(other))
 603                 return basic::has(other, options);
 604         if (!exponent.info(info_flags::integer) ||
 605             !other.op(1).info(info_flags::integer))
 606                 return basic::has(other, options);
 607         if (exponent.info(info_flags::posint) &&
 608             other.op(1).info(info_flags::posint) &&
 609             ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
 610             basis.match(other.op(0)))
 611                 return true;
 612         if (exponent.info(info_flags::negint) &&
 613             other.op(1).info(info_flags::negint) &&
 614             ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
 615             basis.match(other.op(0)))
 616                 return true;
 617         return basic::has(other, options);
 618 }
 619
 620 // from mul.cpp
 621 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 622
 623 ex power::subs(const exmap & m, unsigned options) const
 624 {
 625         const ex &subsed_basis = basis.subs(m, options);
 626         const ex &subsed_exponent = exponent.subs(m, options);
 627
 628         if (!are_ex_trivially_equal(basis, subsed_basis)
 629          || !are_ex_trivially_equal(exponent, subsed_exponent))
 630                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 631
 632         if (!(options & subs_options::algebraic))
 633                 return subs_one_level(m, options);
 634
 635         for (auto & it : m) {
 636                 int nummatches = std::numeric_limits<int>::max();
 637                 exmap repls;
 638                 if (tryfactsubs(*this, it.first, nummatches, repls)) {
 639                         ex anum = it.second.subs(repls, subs_options::no_pattern);
 640                         ex aden = it.first.subs(repls, subs_options::no_pattern);
 641                         ex result = (*this) * pow(anum/aden, nummatches);
 642                         return (ex_to<basic>(result)).subs_one_level(m, options);
 643                 }
 644         }
 645
 646         return subs_one_level(m, options);
 647 }
 648
 649 ex power::eval_ncmul(const exvector & v) const
 650 {
 651         return inherited::eval_ncmul(v);
 652 }
 653
 654 ex power::conjugate() const
 655 {
 656         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
 657         // branch cut which runs along the negative real axis.
 658         if (basis.info(info_flags::positive)) {
 659                 ex newexponent = exponent.conjugate();
 660                 if (are_ex_trivially_equal(exponent, newexponent)) {
 661                         return *this;
 662                 }
 663                 return dynallocate<power>(basis, newexponent);
 664         }
 665         if (exponent.info(info_flags::integer)) {
 666                 ex newbasis = basis.conjugate();
 667                 if (are_ex_trivially_equal(basis, newbasis)) {
 668                         return *this;
 669                 }
 670                 return dynallocate<power>(newbasis, exponent);
 671         }
 672         return conjugate_function(*this).hold();
 673 }
 674
 675 ex power::real_part() const
 676 {
 677         // basis == a+I*b, exponent == c+I*d
 678         const ex a = basis.real_part();
 679         const ex c = exponent.real_part();
 680         if (basis.is_equal(a) && exponent.is_equal(c)) {
 681                 // Re(a^c)
 682                 return *this;
 683         }
 684
 685         const ex b = basis.imag_part();
 686         if (exponent.info(info_flags::integer)) {
 687                 // Re((a+I*b)^c)  w/  c ∈ ℤ
 688                 long N = ex_to<numeric>(c).to_long();
 689                 // Use real terms in Binomial expansion to construct
 690                 // Re(expand(pow(a+I*b, N))).
 691                 long NN = N > 0 ? N : -N;
 692                 ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
 693                 ex result = 0;
 694                 for (long n = 0; n <= NN; n += 2) {
 695                         ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
 696                         if (n % 4 == 0) {
 697                                 result += term;  // sign: I^n w/ n == 4*m
 698                         } else {
 699                                 result -= term;  // sign: I^n w/ n == 4*m+2
 700                         }
 701                 }
 702                 return result;
 703         }
 704
 705         // Re((a+I*b)^(c+I*d))
 706         const ex d = exponent.imag_part();
 707         return pow(abs(basis),c) * exp(-d*atan2(b,a)) * cos(c*atan2(b,a)+d*log(abs(basis)));
 708 }
 709
 710 ex power::imag_part() const
 711 {
 712         // basis == a+I*b, exponent == c+I*d
 713         const ex a = basis.real_part();
 714         const ex c = exponent.real_part();
 715         if (basis.is_equal(a) && exponent.is_equal(c)) {
 716                 // Im(a^c)
 717                 return 0;
 718         }
 719
 720         const ex b = basis.imag_part();
 721         if (exponent.info(info_flags::integer)) {
 722                 // Im((a+I*b)^c)  w/  c ∈ ℤ
 723                 long N = ex_to<numeric>(c).to_long();
 724                 // Use imaginary terms in Binomial expansion to construct
 725                 // Im(expand(pow(a+I*b, N))).
 726                 long p = N > 0 ? 1 : 3;  // modulus for positive sign
 727                 long NN = N > 0 ? N : -N;
 728                 ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
 729                 ex result = 0;
 730                 for (long n = 1; n <= NN; n += 2) {
 731                         ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
 732                         if (n % 4 == p) {
 733                                 result += term;  // sign: I^n w/ n == 4*m+p
 734                         } else {
 735                                 result -= term;  // sign: I^n w/ n == 4*m+2+p
 736                         }
 737                 }
 738                 return result;
 739         }
 740
 741         // Im((a+I*b)^(c+I*d))
 742         const ex d = exponent.imag_part();
 743         return pow(abs(basis),c) * exp(-d*atan2(b,a)) * sin(c*atan2(b,a)+d*log(abs(basis)));
 744 }
 745
 746 // protected
 747
 748 /** Implementation of ex::diff() for a power.
 749  *  @see ex::diff */
 750 ex power::derivative(const symbol & s) const
 751 {
 752         if (is_a<numeric>(exponent)) {
 753                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 754                 const epvector newseq = {expair(basis, exponent - _ex1), expair(basis.diff(s), _ex1)};
 755                 return dynallocate<mul>(std::move(newseq), exponent);
 756         } else {
 757                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 758                 return *this * (exponent.diff(s)*log(basis) + exponent*basis.diff(s)*pow(basis, _ex_1));
 759         }
 760 }
 761
 762 int power::compare_same_type(const basic & other) const
 763 {
 764         GINAC_ASSERT(is_exactly_a<power>(other));
 765         const power &o = static_cast<const power &>(other);
 766
 767         int cmpval = basis.compare(o.basis);
 768         if (cmpval)
 769                 return cmpval;
 770         else
 771                 return exponent.compare(o.exponent);
 772 }
 773
 774 unsigned power::return_type() const
 775 {
 776         return basis.return_type();
 777 }
 778
 779 return_type_t power::return_type_tinfo() const
 780 {
 781         return basis.return_type_tinfo();
 782 }
 783
 784 ex power::expand(unsigned options) const
 785 {
 786         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 787                 // A special case worth optimizing.
 788                 setflag(status_flags::expanded);
 789                 return *this;
 790         }
 791
 792         // (x*p)^c -> x^c * p^c, if p>0
 793         // makes sense before expanding the basis
 794         if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
 795                 const mul &m = ex_to<mul>(basis);
 796                 exvector prodseq;
 797                 epvector powseq;
 798                 prodseq.reserve(m.seq.size() + 1);
 799                 powseq.reserve(m.seq.size() + 1);
 800                 bool possign = true;
 801
 802                 // search for positive/negative factors
 803                 for (auto & cit : m.seq) {
 804                         ex e=m.recombine_pair_to_ex(cit);
 805                         if (e.info(info_flags::positive))
 806                                 prodseq.push_back(pow(e, exponent).expand(options));
 807                         else if (e.info(info_flags::negative)) {
 808                                 prodseq.push_back(pow(-e, exponent).expand(options));
 809                                 possign = !possign;
 810                         } else
 811                                 powseq.push_back(cit);
 812                 }
 813
 814                 // take care on the numeric coefficient
 815                 ex coeff=(possign? _ex1 : _ex_1);
 816                 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
 817                         prodseq.push_back(pow(m.overall_coeff, exponent));
 818                 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
 819                         prodseq.push_back(pow(-m.overall_coeff, exponent));
 820                 else
 821                         coeff *= m.overall_coeff;
 822
 823                 // If positive/negative factors are found, then extract them.
 824                 // In either case we set a flag to avoid the second run on a part
 825                 // which does not have positive/negative terms.
 826                 if (prodseq.size() > 0) {
 827                         ex newbasis = dynallocate<mul>(std::move(powseq), coeff);
 828                         ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
 829                         return dynallocate<mul>(std::move(prodseq)) * pow(newbasis, exponent);
 830                 } else
 831                         ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
 832         }
 833
 834         const ex expanded_basis = basis.expand(options);
 835         const ex expanded_exponent = exponent.expand(options);
 836
 837         // x^(a+b) -> x^a * x^b
 838         if (is_exactly_a<add>(expanded_exponent)) {
 839                 const add &a = ex_to<add>(expanded_exponent);
 840                 exvector distrseq;
 841                 distrseq.reserve(a.seq.size() + 1);
 842                 for (auto & cit : a.seq) {
 843                         distrseq.push_back(pow(expanded_basis, a.recombine_pair_to_ex(cit)));
 844                 }
 845
 846                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 847                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 848                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 849                         long int_exponent = num_exponent.to_int();
 850                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 851                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 852                         else
 853                                 distrseq.push_back(pow(expanded_basis, a.overall_coeff));
 854                 } else
 855                         distrseq.push_back(pow(expanded_basis, a.overall_coeff));
 856
 857                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 858                 ex r = dynallocate<mul>(distrseq);
 859                 return r.expand(options);
 860         }
 861
 862         if (!is_exactly_a<numeric>(expanded_exponent) ||
 863                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 864                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 865                         return this->hold();
 866                 } else {
 867                         return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
 868                 }
 869         }
 870
 871         // integer numeric exponent
 872         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 873         long int_exponent = num_exponent.to_long();
 874
 875         // (x+y)^n, n>0
 876         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 877                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 878
 879         // (x*y)^n -> x^n * y^n
 880         if (is_exactly_a<mul>(expanded_basis))
 881                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 882
 883         // cannot expand further
 884         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 885                 return this->hold();
 886         else
 887                 return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
 888 }
 889
 890 //////////
 891 // new virtual functions which can be overridden by derived classes
 892 //////////
 893
 894 // none
 895
 896 //////////
 897 // non-virtual functions in this class
 898 //////////
 899
 900 namespace {  // anonymous namespace for power::expand_add() helpers
 901
 902 /** Helper class to generate all bounded combinatorial partitions of an integer
 903  *  n with exactly m parts (including zero parts) in non-decreasing order.
 904  */
 905 class partition_generator {
 906 private:
 907         // Partitions n into m parts, not including zero parts.
 908         // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
 909         // FXT library)
 910         struct mpartition2
 911         {
 912                 // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
 913                 std::vector<int> x;
 914                 int n;   // n>0
 915                 int m;   // 0<m<=n
 916                 mpartition2(unsigned n_, unsigned m_)
 917                   : x(m_+1), n(n_), m(m_)
 918                 {
 919                         for (int k=1; k<m; ++k)
 920                                 x[k] = 1;
 921                         x[m] = n - m + 1;
 922                 }
 923                 bool next_partition()
 924                 {
 925                         int u = x[m];  // last element
 926                         int k = m;
 927                         int s = u;
 928                         while (--k) {
 929                                 s += x[k];
 930                                 if (x[k] + 2 <= u)
 931                                         break;
 932                         }
 933                         if (k==0)
 934                                 return false;  // current is last
 935                         int f = x[k] + 1;
 936                         while (k < m) {
 937                                 x[k] = f;
 938                                 s -= f;
 939                                 ++k;
 940                         }
 941                         x[m] = s;
 942                         return true;
 943                 }
 944         } mpgen;
 945         int m;  // number of parts 0<m<=n
 946         mutable std::vector<int> partition;  // current partition
 947 public:
 948         partition_generator(unsigned n_, unsigned m_)
 949           : mpgen(n_, 1), m(m_), partition(m_)
 950         { }
 951         // returns current partition in non-decreasing order, padded with zeros
 952         const std::vector<int>& current() const
 953         {
 954                 for (int i = 0; i < m - mpgen.m; ++i)
 955                         partition[i] = 0;  // pad with zeros
 956
 957                 for (int i = m - mpgen.m; i < m; ++i)
 958                         partition[i] = mpgen.x[i - m + mpgen.m + 1];
 959
 960                 return partition;
 961         }
 962         bool next()
 963         {
 964                 if (!mpgen.next_partition()) {
 965                         if (mpgen.m == m || mpgen.m == mpgen.n)
 966                                 return false;  // current is last
 967                         // increment number of parts
 968                         mpgen = mpartition2(mpgen.n, mpgen.m + 1);
 969                 }
 970                 return true;
 971         }
 972 };
 973
 974 /** Helper class to generate all compositions of a partition of an integer n,
 975  *  starting with the compositions which has non-decreasing order.
 976  */
 977 class composition_generator {
 978 private:
 979         // Generates all distinct permutations of a multiset.
 980         // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
 981         // Multiset Permutations using a Constant Number of Variables by Prefix
 982         // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
 983         struct coolmulti {
 984                 // element of singly linked list
 985                 struct element {
 986                         int value;
 987                         element* next;
 988                         element(int val, element* n)
 989                           : value(val), next(n) {}
 990                         ~element()
 991                         {   // recurses down to the end of the singly linked list
 992                                 delete next;
 993                         }
 994                 };
 995                 element *head, *i, *after_i;
 996                 // NB: Partition must be sorted in non-decreasing order.
 997                 explicit coolmulti(const std::vector<int>& partition)
 998                   : head(nullptr), i(nullptr), after_i(nullptr)
 999                 {
1000                         for (unsigned n = 0; n < partition.size(); ++n) {
1001                                 head = new element(partition[n], head);
1002                                 if (n <= 1)
1003                                         i = head;
1004                         }
1005                         after_i = i->next;
1006                 }
1007                 ~coolmulti()
1008                 {   // deletes singly linked list
1009                         delete head;
1010                 }
1011                 void next_permutation()
1012                 {
1013                         element *before_k;
1014                         if (after_i->next != nullptr && i->value >= after_i->next->value)
1015                                 before_k = after_i;
1016                         else
1017                                 before_k = i;
1018                         element *k = before_k->next;
1019                         before_k->next = k->next;
1020                         k->next = head;
1021                         if (k->value < head->value)
1022                                 i = k;
1023                         after_i = i->next;
1024                         head = k;
1025                 }
1026                 bool finished() const
1027                 {
1028                         return after_i->next == nullptr && after_i->value >= head->value;
1029                 }
1030         } cmgen;
1031         bool atend;  // needed for simplifying iteration over permutations
1032         bool trivial;  // likewise, true if all elements are equal
1033         mutable std::vector<int> composition;  // current compositions
1034 public:
1035         explicit composition_generator(const std::vector<int>& partition)
1036           : cmgen(partition), atend(false), trivial(true), composition(partition.size())
1037         {
1038                 for (unsigned i=1; i<partition.size(); ++i)
1039                         trivial = trivial && (partition[0] == partition[i]);
1040         }
1041         const std::vector<int>& current() const
1042         {
1043                 coolmulti::element* it = cmgen.head;
1044                 size_t i = 0;
1045                 while (it != nullptr) {
1046                         composition[i] = it->value;
1047                         it = it->next;
1048                         ++i;
1049                 }
1050                 return composition;
1051         }
1052         bool next()
1053         {
1054                 // This ugly contortion is needed because the original coolmulti
1055                 // algorithm requires code duplication of the payload procedure,
1056                 // one before the loop and one inside it.
1057                 if (trivial || atend)
1058                         return false;
1059                 cmgen.next_permutation();
1060                 atend = cmgen.finished();
1061                 return true;
1062         }
1063 };
1064
1065 /** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
1066  *  where n = p1+p2+...+pk, i.e. p is a partition of n.
1067  */
1068 const numeric
1069 multinomial_coefficient(const std::vector<int> & p)
1070 {
1071         numeric n = 0, d = 1;
1072         for (auto & it : p) {
1073                 n += numeric(it);
1074                 d *= factorial(numeric(it));
1075         }
1076         return factorial(numeric(n)) / d;
1077 }
1078
1079 }  // anonymous namespace
1080
1081
1082 /** expand a^n where a is an add and n is a positive integer.
1083  *  @see power::expand */
1084 ex power::expand_add(const add & a, long n, unsigned options)
1085 {
1086         // The special case power(+(x,...y;x),2) can be optimized better.
1087         if (n==2)
1088                 return expand_add_2(a, options);
1089
1090         // method:
1091         //
1092         // Consider base as the sum of all symbolic terms and the overall numeric
1093         // coefficient and apply the binomial theorem:
1094         // S = power(+(x,...,z;c),n)
1095         //   = power(+(+(x,...,z;0);c),n)
1096         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1097         // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
1098         // The multinomial theorem is computed by an outer loop over all
1099         // partitions of the exponent and an inner loop over all compositions of
1100         // that partition. This method makes the expansion a combinatorial
1101         // problem and allows us to directly construct the expanded sum and also
1102         // to re-use the multinomial coefficients (since they depend only on the
1103         // partition, not on the composition).
1104         //
1105         // multinomial power(+(x,y,z;0),3) example:
1106         // partition : compositions                : multinomial coefficient
1107         // [0,0,3]   : [3,0,0],[0,3,0],[0,0,3]     : 3!/(3!*0!*0!) = 1
1108         // [0,1,2]   : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
1109         // [1,1,1]   : [1,1,1]                     : 3!/(1!*1!*1!) = 6
1110         //  =>  (x + y + z)^3 =
1111         //        x^3 + y^3 + z^3
1112         //      + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
1113         //      + 6*x*y*z
1114         //
1115         // multinomial power(+(x,y,z;0),4) example:
1116         // partition : compositions                : multinomial coefficient
1117         // [0,0,4]   : [4,0,0],[0,4,0],[0,0,4]     : 4!/(4!*0!*0!) = 1
1118         // [0,1,3]   : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
1119         // [0,2,2]   : [2,2,0],[2,0,2],[0,2,2]     : 4!/(2!*2!*0!) = 6
1120         // [1,1,2]   : [2,1,1],[1,2,1],[1,1,2]     : 4!/(2!*1!*1!) = 12
1121         // (no [1,1,1,1] partition since it has too many parts)
1122         //  =>  (x + y + z)^4 =
1123         //        x^4 + y^4 + z^4
1124         //      + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
1125         //      + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
1126         //      + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
1127         //
1128         // Summary:
1129         // r = 0
1130         // for k from 0 to n:
1131         //     f = c^(n-k)*binomial(n,k)
1132         //     for p in all partitions of n with m parts (including zero parts):
1133         //         h = f * multinomial coefficient of p
1134         //         for c in all compositions of p:
1135         //             t = 1
1136         //             for e in all elements of c:
1137         //                 t = t * a[e]^e
1138         //             r = r + h*t
1139         // return r
1140
1141         epvector result;
1142         // The number of terms will be the number of combinatorial compositions,
1143         // i.e. the number of unordered arrangements of m nonnegative integers
1144         // which sum up to n.  It is frequently written as C_n(m) and directly
1145         // related with binomial coefficients: binomial(n+m-1,m-1).
1146         size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
1147         if (!a.overall_coeff.is_zero()) {
1148                 // the result's overall_coeff is one of the terms
1149                 --result_size;
1150         }
1151         result.reserve(result_size);
1152
1153         // Iterate over all terms in binomial expansion of
1154         // S = power(+(x,...,z;c),n)
1155         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1156         for (int k = 1; k <= n; ++k) {
1157                 numeric binomial_coefficient;  // binomial(n,k)*c^(n-k)
1158                 if (a.overall_coeff.is_zero()) {
1159                         // degenerate case with zero overall_coeff:
1160                         // apply multinomial theorem directly to power(+(x,...z;0),n)
1161                         binomial_coefficient = 1;
1162                         if (k < n) {
1163                                 continue;
1164                         }
1165                 } else {
1166                         binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
1167                 }
1168
1169                 // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
1170                 // Iterate over all partitions of k with exactly as many parts as
1171                 // there are symbolic terms in the basis (including zero parts).
1172                 partition_generator partitions(k, a.seq.size());
1173                 do {
1174                         const std::vector<int>& partition = partitions.current();
1175                         // All monomials of this partition have the same number of terms and the same coefficient.
1176                         const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; });
1177                         const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
1178
1179                         // Iterate over all compositions of the current partition.
1180                         composition_generator compositions(partition);
1181                         do {
1182                                 const std::vector<int>& exponent = compositions.current();
1183                                 epvector monomial;
1184                                 monomial.reserve(msize);
1185                                 numeric factor = coeff;
1186                                 for (unsigned i = 0; i < exponent.size(); ++i) {
1187                                         const ex & r = a.seq[i].rest;
1188                                         GINAC_ASSERT(!is_exactly_a<add>(r));
1189                                         GINAC_ASSERT(!is_exactly_a<power>(r) ||
1190                                                      !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1191                                                      !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1192                                                      !is_exactly_a<add>(ex_to<power>(r).basis) ||
1193                                                      !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1194                                                      !is_exactly_a<power>(ex_to<power>(r).basis));
1195                                         GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
1196                                         const numeric & c = ex_to<numeric>(a.seq[i].coeff);
1197                                         if (exponent[i] == 0) {
1198                                                 // optimize away
1199                                         } else if (exponent[i] == 1) {
1200                                                 // optimized
1201                                                 monomial.push_back(expair(r, _ex1));
1202                                                 if (c != *_num1_p)
1203                                                         factor = factor.mul(c);
1204                                         } else { // general case exponent[i] > 1
1205                                                 monomial.push_back(expair(r, exponent[i]));
1206                                                 if (c != *_num1_p)
1207                                                         factor = factor.mul(c.power(exponent[i]));
1208                                         }
1209                                 }
1210                                 result.push_back(expair(mul(monomial).expand(options), factor));
1211                         } while (compositions.next());
1212                 } while (partitions.next());
1213         }
1214
1215         GINAC_ASSERT(result.size() == result_size);
1216         if (a.overall_coeff.is_zero()) {
1217                 return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1218         } else {
1219                 return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)).setflag(status_flags::expanded);
1220         }
1221 }
1222
1223
1224 /** Special case of power::expand_add. Expands a^2 where a is an add.
1225  *  @see power::expand_add */
1226 ex power::expand_add_2(const add & a, unsigned options)
1227 {
1228         epvector result;
1229         size_t result_size = (a.nops() * (a.nops()+1)) / 2;
1230         if (!a.overall_coeff.is_zero()) {
1231                 // the result's overall_coeff is one of the terms
1232                 --result_size;
1233         }
1234         result.reserve(result_size);
1235
1236         epvector::const_iterator last = a.seq.end();
1237
1238         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1239         // first part: ignore overall_coeff and expand other terms
1240         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1241                 const ex & r = cit0->rest;
1242                 const ex & c = cit0->coeff;
1243
1244                 GINAC_ASSERT(!is_exactly_a<add>(r));
1245                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1246                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1247                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1248                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
1249                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1250                              !is_exactly_a<power>(ex_to<power>(r).basis));
1251
1252                 if (c.is_equal(_ex1)) {
1253                         if (is_exactly_a<mul>(r)) {
1254                                 result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1255                                                         _ex1));
1256                         } else {
1257                                 result.push_back(expair(dynallocate<power>(r, _ex2),
1258                                                         _ex1));
1259                         }
1260                 } else {
1261                         if (is_exactly_a<mul>(r)) {
1262                                 result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1263                                                         ex_to<numeric>(c).power_dyn(*_num2_p)));
1264                         } else {
1265                                 result.push_back(expair(dynallocate<power>(r, _ex2),
1266                                                         ex_to<numeric>(c).power_dyn(*_num2_p)));
1267                         }
1268                 }
1269
1270                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1271                         const ex & r1 = cit1->rest;
1272                         const ex & c1 = cit1->coeff;
1273                         result.push_back(expair(mul(r,r1).expand(options),
1274                                                 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1275                 }
1276         }
1277
1278         // second part: add terms coming from overall_coeff (if != 0)
1279         if (!a.overall_coeff.is_zero()) {
1280                 for (auto & i : a.seq)
1281                         result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1282         }
1283
1284         GINAC_ASSERT(result.size() == result_size);
1285
1286         if (a.overall_coeff.is_zero()) {
1287                 return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1288         } else {
1289                 return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)).setflag(status_flags::expanded);
1290         }
1291 }
1292
1293 /** Expand factors of m in m^n where m is a mul and n is an integer.
1294  *  @see power::expand */
1295 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand)
1296 {
1297         GINAC_ASSERT(n.is_integer());
1298
1299         if (n.is_zero()) {
1300                 return _ex1;
1301         }
1302
1303         // do not bother to rename indices if there are no any.
1304         if (!(options & expand_options::expand_rename_idx) &&
1305             m.info(info_flags::has_indices))
1306                 options |= expand_options::expand_rename_idx;
1307         // Leave it to multiplication since dummy indices have to be renamed
1308         if ((options & expand_options::expand_rename_idx) &&
1309             (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1310                 ex result = m;
1311                 exvector va = get_all_dummy_indices(m);
1312                 sort(va.begin(), va.end(), ex_is_less());
1313
1314                 for (int i=1; i < n.to_int(); i++)
1315                         result *= rename_dummy_indices_uniquely(va, m);
1316                 return result;
1317         }
1318
1319         epvector distrseq;
1320         distrseq.reserve(m.seq.size());
1321         bool need_reexpand = false;
1322
1323         for (auto & cit : m.seq) {
1324                 expair p = m.combine_pair_with_coeff_to_pair(cit, n);
1325                 if (from_expand && is_exactly_a<add>(cit.rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1326                         // this happens when e.g. (a+b)^(1/2) gets squared and
1327                         // the resulting product needs to be reexpanded
1328                         need_reexpand = true;
1329                 }
1330                 distrseq.push_back(p);
1331         }
1332
1333         const mul & result = dynallocate<mul>(std::move(distrseq), ex_to<numeric>(m.overall_coeff).power_dyn(n));
1334         if (need_reexpand)
1335                 return ex(result).expand(options);
1336         if (from_expand)
1337                 return result.setflag(status_flags::expanded);
1338         return result;
1339 }
1340
1341 GINAC_BIND_UNARCHIVER(power);
1342
1343 } // namespace GiNaC