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 exponent.info(info_flags::nonnegint) &&
 233                                basis.info(inf);
 234                 case info_flags::rational_function:
 235                         return exponent.info(info_flags::integer) &&
 236                                basis.info(inf);
 237                 case info_flags::algebraic:
 238                         return !exponent.info(info_flags::integer) ||
 239                                basis.info(inf);
 240                 case info_flags::expanded:
 241                         return (flags & status_flags::expanded);
 242                 case info_flags::positive:
 243                         return basis.info(info_flags::positive) && exponent.info(info_flags::real);
 244                 case info_flags::nonnegative:
 245                         return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
 246                                (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even));
 247                 case info_flags::has_indices: {
 248                         if (flags & status_flags::has_indices)
 249                                 return true;
 250                         else if (flags & status_flags::has_no_indices)
 251                                 return false;
 252                         else if (basis.info(info_flags::has_indices)) {
 253                                 setflag(status_flags::has_indices);
 254                                 clearflag(status_flags::has_no_indices);
 255                                 return true;
 256                         } else {
 257                                 clearflag(status_flags::has_indices);
 258                                 setflag(status_flags::has_no_indices);
 259                                 return false;
 260                         }
 261                 }
 262         }
 263         return inherited::info(inf);
 264 }
 265
 266 size_t power::nops() const
 267 {
 268         return 2;
 269 }
 270
 271 ex power::op(size_t i) const
 272 {
 273         GINAC_ASSERT(i<2);
 274
 275         return i==0 ? basis : exponent;
 276 }
 277
 278 ex power::map(map_function & f) const
 279 {
 280         const ex &mapped_basis = f(basis);
 281         const ex &mapped_exponent = f(exponent);
 282
 283         if (!are_ex_trivially_equal(basis, mapped_basis)
 284          || !are_ex_trivially_equal(exponent, mapped_exponent))
 285                 return dynallocate<power>(mapped_basis, mapped_exponent);
 286         else
 287                 return *this;
 288 }
 289
 290 bool power::is_polynomial(const ex & var) const
 291 {
 292         if (basis.is_polynomial(var)) {
 293                 if (basis.has(var))
 294                         // basis is non-constant polynomial in var
 295                         return exponent.info(info_flags::nonnegint);
 296                 else
 297                         // basis is constant in var
 298                         return !exponent.has(var);
 299         }
 300         // basis is a non-polynomial function of var
 301         return false;
 302 }
 303
 304 int power::degree(const ex & s) const
 305 {
 306         if (is_equal(ex_to<basic>(s)))
 307                 return 1;
 308         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 309                 if (basis.is_equal(s))
 310                         return ex_to<numeric>(exponent).to_int();
 311                 else
 312                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 313         } else if (basis.has(s))
 314                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 315         else
 316                 return 0;
 317 }
 318
 319 int power::ldegree(const ex & s) const
 320 {
 321         if (is_equal(ex_to<basic>(s)))
 322                 return 1;
 323         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 324                 if (basis.is_equal(s))
 325                         return ex_to<numeric>(exponent).to_int();
 326                 else
 327                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 328         } else if (basis.has(s))
 329                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 330         else
 331                 return 0;
 332 }
 333
 334 ex power::coeff(const ex & s, int n) const
 335 {
 336         if (is_equal(ex_to<basic>(s)))
 337                 return n==1 ? _ex1 : _ex0;
 338         else if (!basis.is_equal(s)) {
 339                 // basis not equal to s
 340                 if (n == 0)
 341                         return *this;
 342                 else
 343                         return _ex0;
 344         } else {
 345                 // basis equal to s
 346                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 347                         // integer exponent
 348                         int int_exp = ex_to<numeric>(exponent).to_int();
 349                         if (n == int_exp)
 350                                 return _ex1;
 351                         else
 352                                 return _ex0;
 353                 } else {
 354                         // non-integer exponents are treated as zero
 355                         if (n == 0)
 356                                 return *this;
 357                         else
 358                                 return _ex0;
 359                 }
 360         }
 361 }
 362
 363 /** Perform automatic term rewriting rules in this class.  In the following
 364  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 365  *  stand for such expressions that contain a plain number.
 366  *  - ^(x,0) -> 1  (also handles ^(0,0))
 367  *  - ^(x,1) -> x
 368  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 369  *  - ^(1,x) -> 1
 370  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 371  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 372  *  - ^(^(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!)
 373  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 374  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 375  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 376  */
 377 ex power::eval() const
 378 {
 379         if (flags & status_flags::evaluated)
 380                 return *this;
 381
 382         const numeric *num_basis = nullptr;
 383         const numeric *num_exponent = nullptr;
 384
 385         if (is_exactly_a<numeric>(basis)) {
 386                 num_basis = &ex_to<numeric>(basis);
 387         }
 388         if (is_exactly_a<numeric>(exponent)) {
 389                 num_exponent = &ex_to<numeric>(exponent);
 390         }
 391
 392         // ^(x,0) -> 1  (0^0 also handled here)
 393         if (exponent.is_zero()) {
 394                 if (basis.is_zero())
 395                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 396                 else
 397                         return _ex1;
 398         }
 399
 400         // ^(x,1) -> x
 401         if (exponent.is_equal(_ex1))
 402                 return basis;
 403
 404         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 405         if (basis.is_zero() && num_exponent) {
 406                 if ((num_exponent->real()).is_zero())
 407                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 408                 else if ((num_exponent->real()).is_negative())
 409                         throw (pole_error("power::eval(): division by zero",1));
 410                 else
 411                         return _ex0;
 412         }
 413
 414         // ^(1,x) -> 1
 415         if (basis.is_equal(_ex1))
 416                 return _ex1;
 417
 418         // power of a function calculated by separate rules defined for this function
 419         if (is_exactly_a<function>(basis))
 420                 return ex_to<function>(basis).power(exponent);
 421
 422         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 423         if (is_exactly_a<power>(basis) && basis.op(0).info(info_flags::positive) && basis.op(1).info(info_flags::real))
 424                 return dynallocate<power>(basis.op(0), basis.op(1) * exponent);
 425
 426         if ( num_exponent ) {
 427
 428                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 429                 // except if c1,c2 are rational, but c1^c2 is not)
 430                 if ( num_basis ) {
 431                         const bool basis_is_crational = num_basis->is_crational();
 432                         const bool exponent_is_crational = num_exponent->is_crational();
 433                         if (!basis_is_crational || !exponent_is_crational) {
 434                                 // return a plain float
 435                                 return dynallocate<numeric>(num_basis->power(*num_exponent));
 436                         }
 437
 438                         const numeric res = num_basis->power(*num_exponent);
 439                         if (res.is_crational()) {
 440                                 return res;
 441                         }
 442                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 443
 444                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 445                         if (basis_is_crational && exponent_is_crational
 446                             && num_exponent->is_real()
 447                             && !num_exponent->is_integer()) {
 448                                 const numeric n = num_exponent->numer();
 449                                 const numeric m = num_exponent->denom();
 450                                 numeric r;
 451                                 numeric q = iquo(n, m, r);
 452                                 if (r.is_negative()) {
 453                                         r += m;
 454                                         --q;
 455                                 }
 456                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 457                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 458                                                 // try it for numerator and denominator separately, in order to
 459                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 460                                                 const numeric bnum = num_basis->numer();
 461                                                 const numeric bden = num_basis->denom();
 462                                                 const numeric res_bnum = bnum.power(*num_exponent);
 463                                                 const numeric res_bden = bden.power(*num_exponent);
 464                                                 if (res_bnum.is_integer())
 465                                                         return dynallocate<mul>(dynallocate<power>(bden,-*num_exponent),res_bnum).setflag(status_flags::evaluated);
 466                                                 if (res_bden.is_integer())
 467                                                         return dynallocate<mul>(dynallocate<power>(bnum,*num_exponent),res_bden.inverse()).setflag(status_flags::evaluated);
 468                                         }
 469                                         return this->hold();
 470                                 } else {
 471                                         // assemble resulting product, but allowing for a re-evaluation,
 472                                         // because otherwise we'll end up with something like
 473                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 474                                         // instead of 7/16*7^(1/3).
 475                                         return pow(basis, r.div(m)) * pow(basis, q);
 476                                 }
 477                         }
 478                 }
 479
 480                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 481                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
 482                 // case c1==1 should not happen, see below!)
 483                 if (is_exactly_a<power>(basis)) {
 484                         const power & sub_power = ex_to<power>(basis);
 485                         const ex & sub_basis = sub_power.basis;
 486                         const ex & sub_exponent = sub_power.exponent;
 487                         if (is_exactly_a<numeric>(sub_exponent)) {
 488                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 489                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 490                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() ||
 491                                     (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
 492                                         return dynallocate<power>(sub_basis, num_sub_exponent.mul(*num_exponent));
 493                                 }
 494                         }
 495                 }
 496
 497                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 498                 if (num_exponent->is_integer() && is_exactly_a<mul>(basis)) {
 499                         return expand_mul(ex_to<mul>(basis), *num_exponent, false);
 500                 }
 501
 502                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 503                 if (num_exponent->is_integer() && is_exactly_a<add>(basis)) {
 504                         numeric icont = basis.integer_content();
 505                         const numeric lead_coeff =
 506                                 ex_to<numeric>(ex_to<add>(basis).seq.begin()->coeff).div(icont);
 507
 508                         const bool canonicalizable = lead_coeff.is_integer();
 509                         const bool unit_normal = lead_coeff.is_pos_integer();
 510                         if (canonicalizable && (! unit_normal))
 511                                 icont = icont.mul(*_num_1_p);
 512
 513                         if (canonicalizable && (icont != *_num1_p)) {
 514                                 const add& addref = ex_to<add>(basis);
 515                                 add & addp = dynallocate<add>(addref);
 516                                 addp.clearflag(status_flags::hash_calculated);
 517                                 addp.overall_coeff = ex_to<numeric>(addp.overall_coeff).div_dyn(icont);
 518                                 for (auto & i : addp.seq)
 519                                         i.coeff = ex_to<numeric>(i.coeff).div_dyn(icont);
 520
 521                                 const numeric c = icont.power(*num_exponent);
 522                                 if (likely(c != *_num1_p))
 523                                         return dynallocate<mul>(dynallocate<power>(addp, *num_exponent), c);
 524                                 else
 525                                         return dynallocate<power>(addp, *num_exponent);
 526                         }
 527                 }
 528
 529                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 530                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 531                 if (is_exactly_a<mul>(basis)) {
 532                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 533                         const mul & mulref = ex_to<mul>(basis);
 534                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 535                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 536                                 if (num_coeff.is_real()) {
 537                                         if (num_coeff.is_positive()) {
 538                                                 mul & mulp = dynallocate<mul>(mulref);
 539                                                 mulp.overall_coeff = _ex1;
 540                                                 mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
 541                                                 return dynallocate<mul>(dynallocate<power>(mulp, exponent),
 542                                                                         dynallocate<power>(num_coeff, *num_exponent));
 543                                         } else {
 544                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 545                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 546                                                         mul & mulp = dynallocate<mul>(mulref);
 547                                                         mulp.overall_coeff = _ex_1;
 548                                                         mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
 549                                                         return dynallocate<mul>(dynallocate<power>(mulp, exponent),
 550                                                                                 dynallocate<power>(abs(num_coeff), *num_exponent));
 551                                                 }
 552                                         }
 553                                 }
 554                         }
 555                 }
 556
 557                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 558                 if (num_exponent->is_pos_integer() &&
 559                     basis.return_type() != return_types::commutative &&
 560                     !is_a<matrix>(basis)) {
 561                         return ncmul(exvector(num_exponent->to_int(), basis));
 562                 }
 563         }
 564
 565         return this->hold();
 566 }
 567
 568 ex power::evalf(int level) const
 569 {
 570         ex ebasis;
 571         ex eexponent;
 572
 573         if (level==1) {
 574                 ebasis = basis;
 575                 eexponent = exponent;
 576         } else if (level == -max_recursion_level) {
 577                 throw(std::runtime_error("max recursion level reached"));
 578         } else {
 579                 ebasis = basis.evalf(level-1);
 580                 if (!is_exactly_a<numeric>(exponent))
 581                         eexponent = exponent.evalf(level-1);
 582                 else
 583                         eexponent = exponent;
 584         }
 585
 586         return dynallocate<power>(ebasis, eexponent);
 587 }
 588
 589 ex power::evalm() const
 590 {
 591         const ex ebasis = basis.evalm();
 592         const ex eexponent = exponent.evalm();
 593         if (is_a<matrix>(ebasis)) {
 594                 if (is_exactly_a<numeric>(eexponent)) {
 595                         return dynallocate<matrix>(ex_to<matrix>(ebasis).pow(eexponent));
 596                 }
 597         }
 598         return dynallocate<power>(ebasis, eexponent);
 599 }
 600
 601 bool power::has(const ex & other, unsigned options) const
 602 {
 603         if (!(options & has_options::algebraic))
 604                 return basic::has(other, options);
 605         if (!is_a<power>(other))
 606                 return basic::has(other, options);
 607         if (!exponent.info(info_flags::integer) ||
 608             !other.op(1).info(info_flags::integer))
 609                 return basic::has(other, options);
 610         if (exponent.info(info_flags::posint) &&
 611             other.op(1).info(info_flags::posint) &&
 612             ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
 613             basis.match(other.op(0)))
 614                 return true;
 615         if (exponent.info(info_flags::negint) &&
 616             other.op(1).info(info_flags::negint) &&
 617             ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
 618             basis.match(other.op(0)))
 619                 return true;
 620         return basic::has(other, options);
 621 }
 622
 623 // from mul.cpp
 624 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 625
 626 ex power::subs(const exmap & m, unsigned options) const
 627 {
 628         const ex &subsed_basis = basis.subs(m, options);
 629         const ex &subsed_exponent = exponent.subs(m, options);
 630
 631         if (!are_ex_trivially_equal(basis, subsed_basis)
 632          || !are_ex_trivially_equal(exponent, subsed_exponent))
 633                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 634
 635         if (!(options & subs_options::algebraic))
 636                 return subs_one_level(m, options);
 637
 638         for (auto & it : m) {
 639                 int nummatches = std::numeric_limits<int>::max();
 640                 exmap repls;
 641                 if (tryfactsubs(*this, it.first, nummatches, repls)) {
 642                         ex anum = it.second.subs(repls, subs_options::no_pattern);
 643                         ex aden = it.first.subs(repls, subs_options::no_pattern);
 644                         ex result = (*this) * pow(anum/aden, nummatches);
 645                         return (ex_to<basic>(result)).subs_one_level(m, options);
 646                 }
 647         }
 648
 649         return subs_one_level(m, options);
 650 }
 651
 652 ex power::eval_ncmul(const exvector & v) const
 653 {
 654         return inherited::eval_ncmul(v);
 655 }
 656
 657 ex power::conjugate() const
 658 {
 659         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
 660         // branch cut which runs along the negative real axis.
 661         if (basis.info(info_flags::positive)) {
 662                 ex newexponent = exponent.conjugate();
 663                 if (are_ex_trivially_equal(exponent, newexponent)) {
 664                         return *this;
 665                 }
 666                 return dynallocate<power>(basis, newexponent);
 667         }
 668         if (exponent.info(info_flags::integer)) {
 669                 ex newbasis = basis.conjugate();
 670                 if (are_ex_trivially_equal(basis, newbasis)) {
 671                         return *this;
 672                 }
 673                 return dynallocate<power>(newbasis, exponent);
 674         }
 675         return conjugate_function(*this).hold();
 676 }
 677
 678 ex power::real_part() const
 679 {
 680         // basis == a+I*b, exponent == c+I*d
 681         const ex a = basis.real_part();
 682         const ex c = exponent.real_part();
 683         if (basis.is_equal(a) && exponent.is_equal(c)) {
 684                 // Re(a^c)
 685                 return *this;
 686         }
 687
 688         const ex b = basis.imag_part();
 689         if (exponent.info(info_flags::integer)) {
 690                 // Re((a+I*b)^c)  w/  c ∈ ℤ
 691                 long N = ex_to<numeric>(c).to_long();
 692                 // Use real terms in Binomial expansion to construct
 693                 // Re(expand(pow(a+I*b, N))).
 694                 long NN = N > 0 ? N : -N;
 695                 ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
 696                 ex result = 0;
 697                 for (long n = 0; n <= NN; n += 2) {
 698                         ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
 699                         if (n % 4 == 0) {
 700                                 result += term;  // sign: I^n w/ n == 4*m
 701                         } else {
 702                                 result -= term;  // sign: I^n w/ n == 4*m+2
 703                         }
 704                 }
 705                 return result;
 706         }
 707
 708         // Re((a+I*b)^(c+I*d))
 709         const ex d = exponent.imag_part();
 710         return pow(abs(basis),c) * exp(-d*atan2(b,a)) * cos(c*atan2(b,a)+d*log(abs(basis)));
 711 }
 712
 713 ex power::imag_part() const
 714 {
 715         // basis == a+I*b, exponent == c+I*d
 716         const ex a = basis.real_part();
 717         const ex c = exponent.real_part();
 718         if (basis.is_equal(a) && exponent.is_equal(c)) {
 719                 // Im(a^c)
 720                 return 0;
 721         }
 722
 723         const ex b = basis.imag_part();
 724         if (exponent.info(info_flags::integer)) {
 725                 // Im((a+I*b)^c)  w/  c ∈ ℤ
 726                 long N = ex_to<numeric>(c).to_long();
 727                 // Use imaginary terms in Binomial expansion to construct
 728                 // Im(expand(pow(a+I*b, N))).
 729                 long p = N > 0 ? 1 : 3;  // modulus for positive sign
 730                 long NN = N > 0 ? N : -N;
 731                 ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
 732                 ex result = 0;
 733                 for (long n = 1; n <= NN; n += 2) {
 734                         ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
 735                         if (n % 4 == p) {
 736                                 result += term;  // sign: I^n w/ n == 4*m+p
 737                         } else {
 738                                 result -= term;  // sign: I^n w/ n == 4*m+2+p
 739                         }
 740                 }
 741                 return result;
 742         }
 743
 744         // Im((a+I*b)^(c+I*d))
 745         const ex d = exponent.imag_part();
 746         return pow(abs(basis),c) * exp(-d*atan2(b,a)) * sin(c*atan2(b,a)+d*log(abs(basis)));
 747 }
 748
 749 // protected
 750
 751 /** Implementation of ex::diff() for a power.
 752  *  @see ex::diff */
 753 ex power::derivative(const symbol & s) const
 754 {
 755         if (is_a<numeric>(exponent)) {
 756                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 757                 const epvector newseq = {expair(basis, exponent - _ex1), expair(basis.diff(s), _ex1)};
 758                 return dynallocate<mul>(std::move(newseq), exponent);
 759         } else {
 760                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 761                 return *this * (exponent.diff(s)*log(basis) + exponent*basis.diff(s)*pow(basis, _ex_1));
 762         }
 763 }
 764
 765 int power::compare_same_type(const basic & other) const
 766 {
 767         GINAC_ASSERT(is_exactly_a<power>(other));
 768         const power &o = static_cast<const power &>(other);
 769
 770         int cmpval = basis.compare(o.basis);
 771         if (cmpval)
 772                 return cmpval;
 773         else
 774                 return exponent.compare(o.exponent);
 775 }
 776
 777 unsigned power::return_type() const
 778 {
 779         return basis.return_type();
 780 }
 781
 782 return_type_t power::return_type_tinfo() const
 783 {
 784         return basis.return_type_tinfo();
 785 }
 786
 787 ex power::expand(unsigned options) const
 788 {
 789         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 790                 // A special case worth optimizing.
 791                 setflag(status_flags::expanded);
 792                 return *this;
 793         }
 794
 795         // (x*p)^c -> x^c * p^c, if p>0
 796         // makes sense before expanding the basis
 797         if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
 798                 const mul &m = ex_to<mul>(basis);
 799                 exvector prodseq;
 800                 epvector powseq;
 801                 prodseq.reserve(m.seq.size() + 1);
 802                 powseq.reserve(m.seq.size() + 1);
 803                 bool possign = true;
 804
 805                 // search for positive/negative factors
 806                 for (auto & cit : m.seq) {
 807                         ex e=m.recombine_pair_to_ex(cit);
 808                         if (e.info(info_flags::positive))
 809                                 prodseq.push_back(pow(e, exponent).expand(options));
 810                         else if (e.info(info_flags::negative)) {
 811                                 prodseq.push_back(pow(-e, exponent).expand(options));
 812                                 possign = !possign;
 813                         } else
 814                                 powseq.push_back(cit);
 815                 }
 816
 817                 // take care on the numeric coefficient
 818                 ex coeff=(possign? _ex1 : _ex_1);
 819                 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
 820                         prodseq.push_back(pow(m.overall_coeff, exponent));
 821                 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
 822                         prodseq.push_back(pow(-m.overall_coeff, exponent));
 823                 else
 824                         coeff *= m.overall_coeff;
 825
 826                 // If positive/negative factors are found, then extract them.
 827                 // In either case we set a flag to avoid the second run on a part
 828                 // which does not have positive/negative terms.
 829                 if (prodseq.size() > 0) {
 830                         ex newbasis = dynallocate<mul>(std::move(powseq), coeff);
 831                         ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
 832                         return dynallocate<mul>(std::move(prodseq)) * pow(newbasis, exponent);
 833                 } else
 834                         ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
 835         }
 836
 837         const ex expanded_basis = basis.expand(options);
 838         const ex expanded_exponent = exponent.expand(options);
 839
 840         // x^(a+b) -> x^a * x^b
 841         if (is_exactly_a<add>(expanded_exponent)) {
 842                 const add &a = ex_to<add>(expanded_exponent);
 843                 exvector distrseq;
 844                 distrseq.reserve(a.seq.size() + 1);
 845                 for (auto & cit : a.seq) {
 846                         distrseq.push_back(pow(expanded_basis, a.recombine_pair_to_ex(cit)));
 847                 }
 848
 849                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 850                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 851                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 852                         long int_exponent = num_exponent.to_int();
 853                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 854                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 855                         else
 856                                 distrseq.push_back(pow(expanded_basis, a.overall_coeff));
 857                 } else
 858                         distrseq.push_back(pow(expanded_basis, a.overall_coeff));
 859
 860                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 861                 ex r = dynallocate<mul>(distrseq);
 862                 return r.expand(options);
 863         }
 864
 865         if (!is_exactly_a<numeric>(expanded_exponent) ||
 866                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 867                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 868                         return this->hold();
 869                 } else {
 870                         return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
 871                 }
 872         }
 873
 874         // integer numeric exponent
 875         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 876         long int_exponent = num_exponent.to_long();
 877
 878         // (x+y)^n, n>0
 879         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 880                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 881
 882         // (x*y)^n -> x^n * y^n
 883         if (is_exactly_a<mul>(expanded_basis))
 884                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 885
 886         // cannot expand further
 887         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 888                 return this->hold();
 889         else
 890                 return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
 891 }
 892
 893 //////////
 894 // new virtual functions which can be overridden by derived classes
 895 //////////
 896
 897 // none
 898
 899 //////////
 900 // non-virtual functions in this class
 901 //////////
 902
 903 namespace {  // anonymous namespace for power::expand_add() helpers
 904
 905 /** Helper class to generate all bounded combinatorial partitions of an integer
 906  *  n with exactly m parts (including zero parts) in non-decreasing order.
 907  */
 908 class partition_generator {
 909 private:
 910         // Partitions n into m parts, not including zero parts.
 911         // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
 912         // FXT library)
 913         struct mpartition2
 914         {
 915                 // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
 916                 std::vector<int> x;
 917                 int n;   // n>0
 918                 int m;   // 0<m<=n
 919                 mpartition2(unsigned n_, unsigned m_)
 920                   : x(m_+1), n(n_), m(m_)
 921                 {
 922                         for (int k=1; k<m; ++k)
 923                                 x[k] = 1;
 924                         x[m] = n - m + 1;
 925                 }
 926                 bool next_partition()
 927                 {
 928                         int u = x[m];  // last element
 929                         int k = m;
 930                         int s = u;
 931                         while (--k) {
 932                                 s += x[k];
 933                                 if (x[k] + 2 <= u)
 934                                         break;
 935                         }
 936                         if (k==0)
 937                                 return false;  // current is last
 938                         int f = x[k] + 1;
 939                         while (k < m) {
 940                                 x[k] = f;
 941                                 s -= f;
 942                                 ++k;
 943                         }
 944                         x[m] = s;
 945                         return true;
 946                 }
 947         } mpgen;
 948         int m;  // number of parts 0<m<=n
 949         mutable std::vector<int> partition;  // current partition
 950 public:
 951         partition_generator(unsigned n_, unsigned m_)
 952           : mpgen(n_, 1), m(m_), partition(m_)
 953         { }
 954         // returns current partition in non-decreasing order, padded with zeros
 955         const std::vector<int>& current() const
 956         {
 957                 for (int i = 0; i < m - mpgen.m; ++i)
 958                         partition[i] = 0;  // pad with zeros
 959
 960                 for (int i = m - mpgen.m; i < m; ++i)
 961                         partition[i] = mpgen.x[i - m + mpgen.m + 1];
 962
 963                 return partition;
 964         }
 965         bool next()
 966         {
 967                 if (!mpgen.next_partition()) {
 968                         if (mpgen.m == m || mpgen.m == mpgen.n)
 969                                 return false;  // current is last
 970                         // increment number of parts
 971                         mpgen = mpartition2(mpgen.n, mpgen.m + 1);
 972                 }
 973                 return true;
 974         }
 975 };
 976
 977 /** Helper class to generate all compositions of a partition of an integer n,
 978  *  starting with the compositions which has non-decreasing order.
 979  */
 980 class composition_generator {
 981 private:
 982         // Generates all distinct permutations of a multiset.
 983         // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
 984         // Multiset Permutations using a Constant Number of Variables by Prefix
 985         // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
 986         struct coolmulti {
 987                 // element of singly linked list
 988                 struct element {
 989                         int value;
 990                         element* next;
 991                         element(int val, element* n)
 992                           : value(val), next(n) {}
 993                         ~element()
 994                         {   // recurses down to the end of the singly linked list
 995                                 delete next;
 996                         }
 997                 };
 998                 element *head, *i, *after_i;
 999                 // NB: Partition must be sorted in non-decreasing order.
1000                 explicit coolmulti(const std::vector<int>& partition)
1001                   : head(nullptr), i(nullptr), after_i(nullptr)
1002                 {
1003                         for (unsigned n = 0; n < partition.size(); ++n) {
1004                                 head = new element(partition[n], head);
1005                                 if (n <= 1)
1006                                         i = head;
1007                         }
1008                         after_i = i->next;
1009                 }
1010                 ~coolmulti()
1011                 {   // deletes singly linked list
1012                         delete head;
1013                 }
1014                 void next_permutation()
1015                 {
1016                         element *before_k;
1017                         if (after_i->next != nullptr && i->value >= after_i->next->value)
1018                                 before_k = after_i;
1019                         else
1020                                 before_k = i;
1021                         element *k = before_k->next;
1022                         before_k->next = k->next;
1023                         k->next = head;
1024                         if (k->value < head->value)
1025                                 i = k;
1026                         after_i = i->next;
1027                         head = k;
1028                 }
1029                 bool finished() const
1030                 {
1031                         return after_i->next == nullptr && after_i->value >= head->value;
1032                 }
1033         } cmgen;
1034         bool atend;  // needed for simplifying iteration over permutations
1035         bool trivial;  // likewise, true if all elements are equal
1036         mutable std::vector<int> composition;  // current compositions
1037 public:
1038         explicit composition_generator(const std::vector<int>& partition)
1039           : cmgen(partition), atend(false), trivial(true), composition(partition.size())
1040         {
1041                 for (unsigned i=1; i<partition.size(); ++i)
1042                         trivial = trivial && (partition[0] == partition[i]);
1043         }
1044         const std::vector<int>& current() const
1045         {
1046                 coolmulti::element* it = cmgen.head;
1047                 size_t i = 0;
1048                 while (it != nullptr) {
1049                         composition[i] = it->value;
1050                         it = it->next;
1051                         ++i;
1052                 }
1053                 return composition;
1054         }
1055         bool next()
1056         {
1057                 // This ugly contortion is needed because the original coolmulti
1058                 // algorithm requires code duplication of the payload procedure,
1059                 // one before the loop and one inside it.
1060                 if (trivial || atend)
1061                         return false;
1062                 cmgen.next_permutation();
1063                 atend = cmgen.finished();
1064                 return true;
1065         }
1066 };
1067
1068 /** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
1069  *  where n = p1+p2+...+pk, i.e. p is a partition of n.
1070  */
1071 const numeric
1072 multinomial_coefficient(const std::vector<int> & p)
1073 {
1074         numeric n = 0, d = 1;
1075         for (auto & it : p) {
1076                 n += numeric(it);
1077                 d *= factorial(numeric(it));
1078         }
1079         return factorial(numeric(n)) / d;
1080 }
1081
1082 }  // anonymous namespace
1083
1084
1085 /** expand a^n where a is an add and n is a positive integer.
1086  *  @see power::expand */
1087 ex power::expand_add(const add & a, long n, unsigned options)
1088 {
1089         // The special case power(+(x,...y;x),2) can be optimized better.
1090         if (n==2)
1091                 return expand_add_2(a, options);
1092
1093         // method:
1094         //
1095         // Consider base as the sum of all symbolic terms and the overall numeric
1096         // coefficient and apply the binomial theorem:
1097         // S = power(+(x,...,z;c),n)
1098         //   = power(+(+(x,...,z;0);c),n)
1099         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1100         // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
1101         // The multinomial theorem is computed by an outer loop over all
1102         // partitions of the exponent and an inner loop over all compositions of
1103         // that partition. This method makes the expansion a combinatorial
1104         // problem and allows us to directly construct the expanded sum and also
1105         // to re-use the multinomial coefficients (since they depend only on the
1106         // partition, not on the composition).
1107         //
1108         // multinomial power(+(x,y,z;0),3) example:
1109         // partition : compositions                : multinomial coefficient
1110         // [0,0,3]   : [3,0,0],[0,3,0],[0,0,3]     : 3!/(3!*0!*0!) = 1
1111         // [0,1,2]   : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
1112         // [1,1,1]   : [1,1,1]                     : 3!/(1!*1!*1!) = 6
1113         //  =>  (x + y + z)^3 =
1114         //        x^3 + y^3 + z^3
1115         //      + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
1116         //      + 6*x*y*z
1117         //
1118         // multinomial power(+(x,y,z;0),4) example:
1119         // partition : compositions                : multinomial coefficient
1120         // [0,0,4]   : [4,0,0],[0,4,0],[0,0,4]     : 4!/(4!*0!*0!) = 1
1121         // [0,1,3]   : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
1122         // [0,2,2]   : [2,2,0],[2,0,2],[0,2,2]     : 4!/(2!*2!*0!) = 6
1123         // [1,1,2]   : [2,1,1],[1,2,1],[1,1,2]     : 4!/(2!*1!*1!) = 12
1124         // (no [1,1,1,1] partition since it has too many parts)
1125         //  =>  (x + y + z)^4 =
1126         //        x^4 + y^4 + z^4
1127         //      + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
1128         //      + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
1129         //      + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
1130         //
1131         // Summary:
1132         // r = 0
1133         // for k from 0 to n:
1134         //     f = c^(n-k)*binomial(n,k)
1135         //     for p in all partitions of n with m parts (including zero parts):
1136         //         h = f * multinomial coefficient of p
1137         //         for c in all compositions of p:
1138         //             t = 1
1139         //             for e in all elements of c:
1140         //                 t = t * a[e]^e
1141         //             r = r + h*t
1142         // return r
1143
1144         epvector result;
1145         // The number of terms will be the number of combinatorial compositions,
1146         // i.e. the number of unordered arrangements of m nonnegative integers
1147         // which sum up to n.  It is frequently written as C_n(m) and directly
1148         // related with binomial coefficients: binomial(n+m-1,m-1).
1149         size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
1150         if (!a.overall_coeff.is_zero()) {
1151                 // the result's overall_coeff is one of the terms
1152                 --result_size;
1153         }
1154         result.reserve(result_size);
1155
1156         // Iterate over all terms in binomial expansion of
1157         // S = power(+(x,...,z;c),n)
1158         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1159         for (int k = 1; k <= n; ++k) {
1160                 numeric binomial_coefficient;  // binomial(n,k)*c^(n-k)
1161                 if (a.overall_coeff.is_zero()) {
1162                         // degenerate case with zero overall_coeff:
1163                         // apply multinomial theorem directly to power(+(x,...z;0),n)
1164                         binomial_coefficient = 1;
1165                         if (k < n) {
1166                                 continue;
1167                         }
1168                 } else {
1169                         binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
1170                 }
1171
1172                 // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
1173                 // Iterate over all partitions of k with exactly as many parts as
1174                 // there are symbolic terms in the basis (including zero parts).
1175                 partition_generator partitions(k, a.seq.size());
1176                 do {
1177                         const std::vector<int>& partition = partitions.current();
1178                         // All monomials of this partition have the same number of terms and the same coefficient.
1179                         const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; });
1180                         const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
1181
1182                         // Iterate over all compositions of the current partition.
1183                         composition_generator compositions(partition);
1184                         do {
1185                                 const std::vector<int>& exponent = compositions.current();
1186                                 epvector monomial;
1187                                 monomial.reserve(msize);
1188                                 numeric factor = coeff;
1189                                 for (unsigned i = 0; i < exponent.size(); ++i) {
1190                                         const ex & r = a.seq[i].rest;
1191                                         GINAC_ASSERT(!is_exactly_a<add>(r));
1192                                         GINAC_ASSERT(!is_exactly_a<power>(r) ||
1193                                                      !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1194                                                      !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1195                                                      !is_exactly_a<add>(ex_to<power>(r).basis) ||
1196                                                      !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1197                                                      !is_exactly_a<power>(ex_to<power>(r).basis));
1198                                         GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
1199                                         const numeric & c = ex_to<numeric>(a.seq[i].coeff);
1200                                         if (exponent[i] == 0) {
1201                                                 // optimize away
1202                                         } else if (exponent[i] == 1) {
1203                                                 // optimized
1204                                                 monomial.push_back(expair(r, _ex1));
1205                                                 if (c != *_num1_p)
1206                                                         factor = factor.mul(c);
1207                                         } else { // general case exponent[i] > 1
1208                                                 monomial.push_back(expair(r, exponent[i]));
1209                                                 if (c != *_num1_p)
1210                                                         factor = factor.mul(c.power(exponent[i]));
1211                                         }
1212                                 }
1213                                 result.push_back(expair(mul(monomial).expand(options), factor));
1214                         } while (compositions.next());
1215                 } while (partitions.next());
1216         }
1217
1218         GINAC_ASSERT(result.size() == result_size);
1219         if (a.overall_coeff.is_zero()) {
1220                 return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1221         } else {
1222                 return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)).setflag(status_flags::expanded);
1223         }
1224 }
1225
1226
1227 /** Special case of power::expand_add. Expands a^2 where a is an add.
1228  *  @see power::expand_add */
1229 ex power::expand_add_2(const add & a, unsigned options)
1230 {
1231         epvector result;
1232         size_t result_size = (a.nops() * (a.nops()+1)) / 2;
1233         if (!a.overall_coeff.is_zero()) {
1234                 // the result's overall_coeff is one of the terms
1235                 --result_size;
1236         }
1237         result.reserve(result_size);
1238
1239         epvector::const_iterator last = a.seq.end();
1240
1241         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1242         // first part: ignore overall_coeff and expand other terms
1243         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1244                 const ex & r = cit0->rest;
1245                 const ex & c = cit0->coeff;
1246
1247                 GINAC_ASSERT(!is_exactly_a<add>(r));
1248                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1249                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1250                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1251                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
1252                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1253                              !is_exactly_a<power>(ex_to<power>(r).basis));
1254
1255                 if (c.is_equal(_ex1)) {
1256                         if (is_exactly_a<mul>(r)) {
1257                                 result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1258                                                         _ex1));
1259                         } else {
1260                                 result.push_back(expair(dynallocate<power>(r, _ex2),
1261                                                         _ex1));
1262                         }
1263                 } else {
1264                         if (is_exactly_a<mul>(r)) {
1265                                 result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1266                                                         ex_to<numeric>(c).power_dyn(*_num2_p)));
1267                         } else {
1268                                 result.push_back(expair(dynallocate<power>(r, _ex2),
1269                                                         ex_to<numeric>(c).power_dyn(*_num2_p)));
1270                         }
1271                 }
1272
1273                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1274                         const ex & r1 = cit1->rest;
1275                         const ex & c1 = cit1->coeff;
1276                         result.push_back(expair(mul(r,r1).expand(options),
1277                                                 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1278                 }
1279         }
1280
1281         // second part: add terms coming from overall_coeff (if != 0)
1282         if (!a.overall_coeff.is_zero()) {
1283                 for (auto & i : a.seq)
1284                         result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1285         }
1286
1287         GINAC_ASSERT(result.size() == result_size);
1288
1289         if (a.overall_coeff.is_zero()) {
1290                 return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1291         } else {
1292                 return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)).setflag(status_flags::expanded);
1293         }
1294 }
1295
1296 /** Expand factors of m in m^n where m is a mul and n is an integer.
1297  *  @see power::expand */
1298 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand)
1299 {
1300         GINAC_ASSERT(n.is_integer());
1301
1302         if (n.is_zero()) {
1303                 return _ex1;
1304         }
1305
1306         // do not bother to rename indices if there are no any.
1307         if (!(options & expand_options::expand_rename_idx) &&
1308             m.info(info_flags::has_indices))
1309                 options |= expand_options::expand_rename_idx;
1310         // Leave it to multiplication since dummy indices have to be renamed
1311         if ((options & expand_options::expand_rename_idx) &&
1312             (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1313                 ex result = m;
1314                 exvector va = get_all_dummy_indices(m);
1315                 sort(va.begin(), va.end(), ex_is_less());
1316
1317                 for (int i=1; i < n.to_int(); i++)
1318                         result *= rename_dummy_indices_uniquely(va, m);
1319                 return result;
1320         }
1321
1322         epvector distrseq;
1323         distrseq.reserve(m.seq.size());
1324         bool need_reexpand = false;
1325
1326         for (auto & cit : m.seq) {
1327                 expair p = m.combine_pair_with_coeff_to_pair(cit, n);
1328                 if (from_expand && is_exactly_a<add>(cit.rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1329                         // this happens when e.g. (a+b)^(1/2) gets squared and
1330                         // the resulting product needs to be reexpanded
1331                         need_reexpand = true;
1332                 }
1333                 distrseq.push_back(p);
1334         }
1335
1336         const mul & result = dynallocate<mul>(std::move(distrseq), ex_to<numeric>(m.overall_coeff).power_dyn(n));
1337         if (need_reexpand)
1338                 return ex(result).expand(options);
1339         if (from_expand)
1340                 return result.setflag(status_flags::expanded);
1341         return result;
1342 }
1343
1344 GINAC_BIND_UNARCHIVER(power);
1345
1346 } // namespace GiNaC