3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
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.
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.
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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
29 #include "expairseq.h"
35 #include "operators.h"
36 #include "inifcns.h" // for log() in power::derivative()
46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
47 print_func<print_dflt>(&power::do_print_dflt).
48 print_func<print_latex>(&power::do_print_latex).
49 print_func<print_csrc>(&power::do_print_csrc).
50 print_func<print_python>(&power::do_print_python).
51 print_func<print_python_repr>(&power::do_print_python_repr))
53 typedef std::vector<int> intvector;
56 // default constructor
59 power::power() : inherited(TINFO_power) { }
71 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
73 n.find_ex("basis", basis, sym_lst);
74 n.find_ex("exponent", exponent, sym_lst);
77 void power::archive(archive_node &n) const
79 inherited::archive(n);
80 n.add_ex("basis", basis);
81 n.add_ex("exponent", exponent);
84 DEFAULT_UNARCHIVE(power)
87 // functions overriding virtual functions from base classes
92 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
94 // Ordinary output of powers using '^' or '**'
95 if (precedence() <= level)
96 c.s << openbrace << '(';
97 basis.print(c, precedence());
100 exponent.print(c, precedence());
102 if (precedence() <= level)
103 c.s << ')' << closebrace;
106 void power::do_print_dflt(const print_dflt & c, unsigned level) const
108 if (exponent.is_equal(_ex1_2)) {
110 // Square roots are printed in a special way
116 print_power(c, "^", "", "", level);
119 void power::do_print_latex(const print_latex & c, unsigned level) const
121 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
123 // Powers with negative numeric exponents are printed as fractions
125 power(basis, -exponent).eval().print(c);
128 } else if (exponent.is_equal(_ex1_2)) {
130 // Square roots are printed in a special way
136 print_power(c, "^", "{", "}", level);
139 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
141 // Optimal output of integer powers of symbols to aid compiler CSE.
142 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
143 // to learn why such a parenthesation is really necessary.
146 } else if (exp == 2) {
150 } else if (exp & 1) {
153 print_sym_pow(c, x, exp-1);
156 print_sym_pow(c, x, exp >> 1);
158 print_sym_pow(c, x, exp >> 1);
163 void power::do_print_csrc(const print_csrc & c, unsigned level) const
165 // Integer powers of symbols are printed in a special, optimized way
166 if (exponent.info(info_flags::integer)
167 && (is_a<symbol>(basis) || is_a<constant>(basis))) {
168 int exp = ex_to<numeric>(exponent).to_int();
173 if (is_a<print_csrc_cl_N>(c))
178 print_sym_pow(c, ex_to<symbol>(basis), exp);
181 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
182 } else if (exponent.is_equal(_ex_1)) {
183 if (is_a<print_csrc_cl_N>(c))
190 // Otherwise, use the pow() or expt() (CLN) functions
192 if (is_a<print_csrc_cl_N>(c))
203 void power::do_print_python(const print_python & c, unsigned level) const
205 print_power(c, "**", "", "", level);
208 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
210 c.s << class_name() << '(';
217 bool power::info(unsigned inf) const
220 case info_flags::polynomial:
221 case info_flags::integer_polynomial:
222 case info_flags::cinteger_polynomial:
223 case info_flags::rational_polynomial:
224 case info_flags::crational_polynomial:
225 return exponent.info(info_flags::nonnegint);
226 case info_flags::rational_function:
227 return exponent.info(info_flags::integer);
228 case info_flags::algebraic:
229 return (!exponent.info(info_flags::integer) ||
232 return inherited::info(inf);
235 size_t power::nops() const
240 ex power::op(size_t i) const
244 return i==0 ? basis : exponent;
247 ex power::map(map_function & f) const
249 const ex &mapped_basis = f(basis);
250 const ex &mapped_exponent = f(exponent);
252 if (!are_ex_trivially_equal(basis, mapped_basis)
253 || !are_ex_trivially_equal(exponent, mapped_exponent))
254 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
259 int power::degree(const ex & s) const
261 if (is_equal(ex_to<basic>(s)))
263 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
264 if (basis.is_equal(s))
265 return ex_to<numeric>(exponent).to_int();
267 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
268 } else if (basis.has(s))
269 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
274 int power::ldegree(const ex & s) const
276 if (is_equal(ex_to<basic>(s)))
278 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
279 if (basis.is_equal(s))
280 return ex_to<numeric>(exponent).to_int();
282 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
283 } else if (basis.has(s))
284 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
289 ex power::coeff(const ex & s, int n) const
291 if (is_equal(ex_to<basic>(s)))
292 return n==1 ? _ex1 : _ex0;
293 else if (!basis.is_equal(s)) {
294 // basis not equal to s
301 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
303 int int_exp = ex_to<numeric>(exponent).to_int();
309 // non-integer exponents are treated as zero
318 /** Perform automatic term rewriting rules in this class. In the following
319 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
320 * stand for such expressions that contain a plain number.
321 * - ^(x,0) -> 1 (also handles ^(0,0))
323 * - ^(0,c) -> 0 or exception (depending on the real part of c)
325 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
326 * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
327 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
328 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
329 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
331 * @param level cut-off in recursive evaluation */
332 ex power::eval(int level) const
334 if ((level==1) && (flags & status_flags::evaluated))
336 else if (level == -max_recursion_level)
337 throw(std::runtime_error("max recursion level reached"));
339 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
340 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
342 bool basis_is_numerical = false;
343 bool exponent_is_numerical = false;
344 const numeric *num_basis;
345 const numeric *num_exponent;
347 if (is_exactly_a<numeric>(ebasis)) {
348 basis_is_numerical = true;
349 num_basis = &ex_to<numeric>(ebasis);
351 if (is_exactly_a<numeric>(eexponent)) {
352 exponent_is_numerical = true;
353 num_exponent = &ex_to<numeric>(eexponent);
356 // ^(x,0) -> 1 (0^0 also handled here)
357 if (eexponent.is_zero()) {
358 if (ebasis.is_zero())
359 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
365 if (eexponent.is_equal(_ex1))
368 // ^(0,c1) -> 0 or exception (depending on real value of c1)
369 if (ebasis.is_zero() && exponent_is_numerical) {
370 if ((num_exponent->real()).is_zero())
371 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
372 else if ((num_exponent->real()).is_negative())
373 throw (pole_error("power::eval(): division by zero",1));
379 if (ebasis.is_equal(_ex1))
382 if (exponent_is_numerical) {
384 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
385 // except if c1,c2 are rational, but c1^c2 is not)
386 if (basis_is_numerical) {
387 const bool basis_is_crational = num_basis->is_crational();
388 const bool exponent_is_crational = num_exponent->is_crational();
389 if (!basis_is_crational || !exponent_is_crational) {
390 // return a plain float
391 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
392 status_flags::evaluated |
393 status_flags::expanded);
396 const numeric res = num_basis->power(*num_exponent);
397 if (res.is_crational()) {
400 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
402 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
403 if (basis_is_crational && exponent_is_crational
404 && num_exponent->is_real()
405 && !num_exponent->is_integer()) {
406 const numeric n = num_exponent->numer();
407 const numeric m = num_exponent->denom();
409 numeric q = iquo(n, m, r);
410 if (r.is_negative()) {
414 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
415 if (num_basis->is_rational() && !num_basis->is_integer()) {
416 // try it for numerator and denominator separately, in order to
417 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
418 const numeric bnum = num_basis->numer();
419 const numeric bden = num_basis->denom();
420 const numeric res_bnum = bnum.power(*num_exponent);
421 const numeric res_bden = bden.power(*num_exponent);
422 if (res_bnum.is_integer())
423 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
424 if (res_bden.is_integer())
425 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
429 // assemble resulting product, but allowing for a re-evaluation,
430 // because otherwise we'll end up with something like
431 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
432 // instead of 7/16*7^(1/3).
433 ex prod = power(*num_basis,r.div(m));
434 return prod*power(*num_basis,q);
439 // ^(^(x,c1),c2) -> ^(x,c1*c2)
440 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
441 // case c1==1 should not happen, see below!)
442 if (is_exactly_a<power>(ebasis)) {
443 const power & sub_power = ex_to<power>(ebasis);
444 const ex & sub_basis = sub_power.basis;
445 const ex & sub_exponent = sub_power.exponent;
446 if (is_exactly_a<numeric>(sub_exponent)) {
447 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
448 GINAC_ASSERT(num_sub_exponent!=numeric(1));
449 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
450 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
454 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
455 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
456 return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
459 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
460 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
461 if (is_exactly_a<mul>(ebasis)) {
462 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
463 const mul & mulref = ex_to<mul>(ebasis);
464 if (!mulref.overall_coeff.is_equal(_ex1)) {
465 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
466 if (num_coeff.is_real()) {
467 if (num_coeff.is_positive()) {
468 mul *mulp = new mul(mulref);
469 mulp->overall_coeff = _ex1;
470 mulp->clearflag(status_flags::evaluated);
471 mulp->clearflag(status_flags::hash_calculated);
472 return (new mul(power(*mulp,exponent),
473 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
475 GINAC_ASSERT(num_coeff.compare(_num0)<0);
476 if (!num_coeff.is_equal(_num_1)) {
477 mul *mulp = new mul(mulref);
478 mulp->overall_coeff = _ex_1;
479 mulp->clearflag(status_flags::evaluated);
480 mulp->clearflag(status_flags::hash_calculated);
481 return (new mul(power(*mulp,exponent),
482 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
489 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
490 if (num_exponent->is_pos_integer() &&
491 ebasis.return_type() != return_types::commutative &&
492 !is_a<matrix>(ebasis)) {
493 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
497 if (are_ex_trivially_equal(ebasis,basis) &&
498 are_ex_trivially_equal(eexponent,exponent)) {
501 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
502 status_flags::evaluated);
505 ex power::evalf(int level) const
512 eexponent = exponent;
513 } else if (level == -max_recursion_level) {
514 throw(std::runtime_error("max recursion level reached"));
516 ebasis = basis.evalf(level-1);
517 if (!is_exactly_a<numeric>(exponent))
518 eexponent = exponent.evalf(level-1);
520 eexponent = exponent;
523 return power(ebasis,eexponent);
526 ex power::evalm() const
528 const ex ebasis = basis.evalm();
529 const ex eexponent = exponent.evalm();
530 if (is_a<matrix>(ebasis)) {
531 if (is_exactly_a<numeric>(eexponent)) {
532 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
535 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
539 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
541 ex power::subs(const exmap & m, unsigned options) const
543 const ex &subsed_basis = basis.subs(m, options);
544 const ex &subsed_exponent = exponent.subs(m, options);
546 if (!are_ex_trivially_equal(basis, subsed_basis)
547 || !are_ex_trivially_equal(exponent, subsed_exponent))
548 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
550 if (!(options & subs_options::algebraic))
551 return subs_one_level(m, options);
553 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
554 int nummatches = std::numeric_limits<int>::max();
556 if (tryfactsubs(*this, it->first, nummatches, repls))
557 return (ex_to<basic>((*this) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches))).subs_one_level(m, options);
560 return subs_one_level(m, options);
563 ex power::eval_ncmul(const exvector & v) const
565 return inherited::eval_ncmul(v);
568 ex power::conjugate() const
570 ex newbasis = basis.conjugate();
571 ex newexponent = exponent.conjugate();
572 if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
575 return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
580 /** Implementation of ex::diff() for a power.
582 ex power::derivative(const symbol & s) const
584 if (exponent.info(info_flags::real)) {
585 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
588 newseq.push_back(expair(basis, exponent - _ex1));
589 newseq.push_back(expair(basis.diff(s), _ex1));
590 return mul(newseq, exponent);
592 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
594 add(mul(exponent.diff(s), log(basis)),
595 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
599 int power::compare_same_type(const basic & other) const
601 GINAC_ASSERT(is_exactly_a<power>(other));
602 const power &o = static_cast<const power &>(other);
604 int cmpval = basis.compare(o.basis);
608 return exponent.compare(o.exponent);
611 unsigned power::return_type() const
613 return basis.return_type();
616 unsigned power::return_type_tinfo() const
618 return basis.return_type_tinfo();
621 ex power::expand(unsigned options) const
623 if (options == 0 && (flags & status_flags::expanded))
626 const ex expanded_basis = basis.expand(options);
627 const ex expanded_exponent = exponent.expand(options);
629 // x^(a+b) -> x^a * x^b
630 if (is_exactly_a<add>(expanded_exponent)) {
631 const add &a = ex_to<add>(expanded_exponent);
633 distrseq.reserve(a.seq.size() + 1);
634 epvector::const_iterator last = a.seq.end();
635 epvector::const_iterator cit = a.seq.begin();
637 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
641 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
642 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
643 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
644 int int_exponent = num_exponent.to_int();
645 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
646 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
648 distrseq.push_back(power(expanded_basis, a.overall_coeff));
650 distrseq.push_back(power(expanded_basis, a.overall_coeff));
652 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
653 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
654 return r.expand(options);
657 if (!is_exactly_a<numeric>(expanded_exponent) ||
658 !ex_to<numeric>(expanded_exponent).is_integer()) {
659 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
662 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
666 // integer numeric exponent
667 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
668 int int_exponent = num_exponent.to_int();
671 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
672 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
674 // (x*y)^n -> x^n * y^n
675 if (is_exactly_a<mul>(expanded_basis))
676 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
678 // cannot expand further
679 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
682 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
686 // new virtual functions which can be overridden by derived classes
692 // non-virtual functions in this class
695 /** expand a^n where a is an add and n is a positive integer.
696 * @see power::expand */
697 ex power::expand_add(const add & a, int n, unsigned options) const
700 return expand_add_2(a, options);
702 const size_t m = a.nops();
704 // The number of terms will be the number of combinatorial compositions,
705 // i.e. the number of unordered arrangement of m nonnegative integers
706 // which sum up to n. It is frequently written as C_n(m) and directly
707 // related with binomial coefficients:
708 result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
710 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
711 intvector upper_limit(m-1);
714 for (size_t l=0; l<m-1; ++l) {
723 for (l=0; l<m-1; ++l) {
724 const ex & b = a.op(l);
725 GINAC_ASSERT(!is_exactly_a<add>(b));
726 GINAC_ASSERT(!is_exactly_a<power>(b) ||
727 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
728 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
729 !is_exactly_a<add>(ex_to<power>(b).basis) ||
730 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
731 !is_exactly_a<power>(ex_to<power>(b).basis));
732 if (is_exactly_a<mul>(b))
733 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
735 term.push_back(power(b,k[l]));
738 const ex & b = a.op(l);
739 GINAC_ASSERT(!is_exactly_a<add>(b));
740 GINAC_ASSERT(!is_exactly_a<power>(b) ||
741 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
742 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
743 !is_exactly_a<add>(ex_to<power>(b).basis) ||
744 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
745 !is_exactly_a<power>(ex_to<power>(b).basis));
746 if (is_exactly_a<mul>(b))
747 term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
749 term.push_back(power(b,n-k_cum[m-2]));
751 numeric f = binomial(numeric(n),numeric(k[0]));
752 for (l=1; l<m-1; ++l)
753 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
757 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
761 while ((l>=0) && ((++k[l])>upper_limit[l])) {
767 // recalc k_cum[] and upper_limit[]
768 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
770 for (size_t i=l+1; i<m-1; ++i)
771 k_cum[i] = k_cum[i-1]+k[i];
773 for (size_t i=l+1; i<m-1; ++i)
774 upper_limit[i] = n-k_cum[i-1];
777 return (new add(result))->setflag(status_flags::dynallocated |
778 status_flags::expanded);
782 /** Special case of power::expand_add. Expands a^2 where a is an add.
783 * @see power::expand_add */
784 ex power::expand_add_2(const add & a, unsigned options) const
787 size_t a_nops = a.nops();
788 sum.reserve((a_nops*(a_nops+1))/2);
789 epvector::const_iterator last = a.seq.end();
791 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
792 // first part: ignore overall_coeff and expand other terms
793 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
794 const ex & r = cit0->rest;
795 const ex & c = cit0->coeff;
797 GINAC_ASSERT(!is_exactly_a<add>(r));
798 GINAC_ASSERT(!is_exactly_a<power>(r) ||
799 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
800 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
801 !is_exactly_a<add>(ex_to<power>(r).basis) ||
802 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
803 !is_exactly_a<power>(ex_to<power>(r).basis));
805 if (c.is_equal(_ex1)) {
806 if (is_exactly_a<mul>(r)) {
807 sum.push_back(expair(expand_mul(ex_to<mul>(r), _num2, options, true),
810 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
814 if (is_exactly_a<mul>(r)) {
815 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), _num2, options, true),
816 ex_to<numeric>(c).power_dyn(_num2)));
818 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
819 ex_to<numeric>(c).power_dyn(_num2)));
823 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
824 const ex & r1 = cit1->rest;
825 const ex & c1 = cit1->coeff;
826 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
827 _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
831 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
833 // second part: add terms coming from overall_factor (if != 0)
834 if (!a.overall_coeff.is_zero()) {
835 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
837 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
840 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
843 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
845 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
848 /** Expand factors of m in m^n where m is a mul and n is and integer.
849 * @see power::expand */
850 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
852 GINAC_ASSERT(n.is_integer());
858 distrseq.reserve(m.seq.size());
859 bool need_reexpand = false;
861 epvector::const_iterator last = m.seq.end();
862 epvector::const_iterator cit = m.seq.begin();
864 if (is_exactly_a<numeric>(cit->rest)) {
865 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
867 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
868 // since n is an integer
869 numeric new_coeff = ex_to<numeric>(cit->coeff).mul(n);
870 if (from_expand && is_exactly_a<add>(cit->rest) && new_coeff.is_pos_integer()) {
871 // this happens when e.g. (a+b)^(1/2) gets squared and
872 // the resulting product needs to be reexpanded
873 need_reexpand = true;
875 distrseq.push_back(expair(cit->rest, new_coeff));
880 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
882 return ex(result).expand(options);
884 return result.setflag(status_flags::expanded);