{
// Optimal output of integer powers of symbols to aid compiler CSE.
// C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
- // to learn why such a hack is really necessary.
+ // to learn why such a parenthisation is really necessary.
if (exp == 1) {
x.print(c);
} else if (exp == 2) {
int power::degree(const ex & s) const
{
- if (is_exactly_of_type(*exponent.bp,numeric)) {
- if (basis.is_equal(s)) {
- if (ex_to<numeric>(exponent).is_integer())
- return ex_to<numeric>(exponent).to_int();
- else
- return 0;
- } else
+ if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
+ if (basis.is_equal(s))
+ return ex_to<numeric>(exponent).to_int();
+ else
return basis.degree(s) * ex_to<numeric>(exponent).to_int();
}
return 0;
int power::ldegree(const ex & s) const
{
- if (is_exactly_of_type(*exponent.bp,numeric)) {
- if (basis.is_equal(s)) {
- if (ex_to<numeric>(exponent).is_integer())
- return ex_to<numeric>(exponent).to_int();
- else
- return 0;
- } else
+ if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
+ if (basis.is_equal(s))
+ return ex_to<numeric>(exponent).to_int();
+ else
return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
}
return 0;
return _ex0();
} else {
// basis equal to s
- if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
+ if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
// integer exponent
int int_exp = ex_to<numeric>(exponent).to_int();
if (n == int_exp)
}
}
+/** Perform automatic term rewriting rules in this class. In the following
+ * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
+ * stand for such expressions that contain a plain number.
+ * - ^(x,0) -> 1 (also handles ^(0,0))
+ * - ^(x,1) -> x
+ * - ^(0,c) -> 0 or exception (depending on the real part of c)
+ * - ^(1,x) -> 1
+ * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
+ * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
+ * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
+ * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
+ * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
+ *
+ * @param level cut-off in recursive evaluation */
ex power::eval(int level) const
{
- // simplifications: ^(x,0) -> 1 (0^0 handled here)
- // ^(x,1) -> x
- // ^(0,c1) -> 0 or exception (depending on real value of c1)
- // ^(1,x) -> 1
- // ^(c1,c2) -> *(c1^n,c1^(c2-n)) (c1, c2 numeric(), 0<(c2-n)<1 except if c1,c2 are rational, but c1^c2 is not)
- // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
- // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
- // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
- // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
-
debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
if ((level==1) && (flags & status_flags::evaluated))
bool basis_is_numerical = false;
bool exponent_is_numerical = false;
- numeric * num_basis;
- numeric * num_exponent;
+ const numeric *num_basis;
+ const numeric *num_exponent;
- if (is_exactly_of_type(*ebasis.bp,numeric)) {
+ if (is_ex_exactly_of_type(ebasis, numeric)) {
basis_is_numerical = true;
- num_basis = static_cast<numeric *>(ebasis.bp);
+ num_basis = &ex_to<numeric>(ebasis);
}
- if (is_exactly_of_type(*eexponent.bp,numeric)) {
+ if (is_ex_exactly_of_type(eexponent, numeric)) {
exponent_is_numerical = true;
- num_exponent = static_cast<numeric *>(eexponent.bp);
+ num_exponent = &ex_to<numeric>(eexponent);
}
- // ^(x,0) -> 1 (0^0 also handled here)
+ // ^(x,0) -> 1 (0^0 also handled here)
if (eexponent.is_zero()) {
if (ebasis.is_zero())
throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
// ^(x,1) -> x
if (eexponent.is_equal(_ex1()))
return ebasis;
-
- // ^(0,c1) -> 0 or exception (depending on real value of c1)
+
+ // ^(0,c1) -> 0 or exception (depending on real value of c1)
if (ebasis.is_zero() && exponent_is_numerical) {
if ((num_exponent->real()).is_zero())
throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
else
return _ex0();
}
-
+
// ^(1,x) -> 1
if (ebasis.is_equal(_ex1()))
return _ex1();
-
+
if (exponent_is_numerical) {
- // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
+ // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
// except if c1,c2 are rational, but c1^c2 is not)
if (basis_is_numerical) {
- bool basis_is_crational = num_basis->is_crational();
- bool exponent_is_crational = num_exponent->is_crational();
- numeric res = num_basis->power(*num_exponent);
-
- if ((!basis_is_crational || !exponent_is_crational)
- || res.is_crational()) {
+ const bool basis_is_crational = num_basis->is_crational();
+ const bool exponent_is_crational = num_exponent->is_crational();
+ if (!basis_is_crational || !exponent_is_crational) {
+ // return a plain float
+ return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
+ status_flags::evaluated |
+ status_flags::expanded);
+ }
+
+ const numeric res = num_basis->power(*num_exponent);
+ if (res.is_crational()) {
return res;
}
GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
- // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
+ // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
if (basis_is_crational && exponent_is_crational
- && num_exponent->is_real()
- && !num_exponent->is_integer()) {
- numeric n = num_exponent->numer();
- numeric m = num_exponent->denom();
+ && num_exponent->is_real()
+ && !num_exponent->is_integer()) {
+ const numeric n = num_exponent->numer();
+ const numeric m = num_exponent->denom();
numeric r;
numeric q = iquo(n, m, r);
if (r.is_negative()) {
- r = r.add(m);
- q = q.sub(_num1());
+ r += m;
+ --q;
}
- if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
+ if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
+ if (num_basis->is_rational() && !num_basis->is_integer()) {
+ // try it for numerator and denominator separately, in order to
+ // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
+ const numeric bnum = num_basis->numer();
+ const numeric bden = num_basis->denom();
+ const numeric res_bnum = bnum.power(*num_exponent);
+ const numeric res_bden = bden.power(*num_exponent);
+ if (res_bnum.is_integer())
+ return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
+ if (res_bden.is_integer())
+ return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
+ }
return this->hold();
- else {
- epvector res;
- res.push_back(expair(ebasis,r.div(m)));
- return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
+ } else {
+ // assemble resulting product, but allowing for a re-evaluation,
+ // because otherwise we'll end up with something like
+ // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
+ // instead of 7/16*7^(1/3).
+ ex prod = power(*num_basis,r.div(m));
+ return prod*power(*num_basis,q);
}
}
}
return expand_mul(ex_to<mul>(ebasis), *num_exponent);
}
- // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
- // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
+ // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
+ // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
if (is_ex_exactly_of_type(ebasis,mul)) {
GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
const mul & mulref = ex_to<mul>(ebasis);
const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
if (num_coeff.is_real()) {
if (num_coeff.is_positive()) {
- mul * mulp = new mul(mulref);
+ mul *mulp = new mul(mulref);
mulp->overall_coeff = _ex1();
mulp->clearflag(status_flags::evaluated);
mulp->clearflag(status_flags::hash_calculated);
} else {
GINAC_ASSERT(num_coeff.compare(_num0())<0);
if (num_coeff.compare(_num_1())!=0) {
- mul * mulp = new mul(mulref);
+ mul *mulp = new mul(mulref);
mulp->overall_coeff = _ex_1();
mulp->clearflag(status_flags::evaluated);
mulp->clearflag(status_flags::hash_calculated);
}
if (are_ex_trivially_equal(ebasis,basis) &&
- are_ex_trivially_equal(eexponent,exponent)) {
+ are_ex_trivially_equal(eexponent,exponent)) {
return this->hold();
}
return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
- status_flags::evaluated);
+ status_flags::evaluated);
}
ex power::evalf(int level) const
{
debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
-
+
ex ebasis;
ex eexponent;
throw(std::runtime_error("max recursion level reached"));
} else {
ebasis = basis.evalf(level-1);
- if (!is_ex_exactly_of_type(eexponent,numeric))
+ if (!is_ex_exactly_of_type(exponent,numeric))
eexponent = exponent.evalf(level-1);
else
eexponent = exponent;
&& are_ex_trivially_equal(exponent, subsed_exponent))
return basic::subs(ls, lr, no_pattern);
else
- return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
+ return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern);
}
ex power::simplify_ncmul(const exvector & v) const
int power::compare_same_type(const basic & other) const
{
- GINAC_ASSERT(is_exactly_of_type(other, power));
+ GINAC_ASSERT(is_exactly_a<power>(other));
const power &o = static_cast<const power &>(other);
int cmpval = basis.compare(o.basis);
if (options == 0 && (flags & status_flags::expanded))
return *this;
- ex expanded_basis = basis.expand(options);
- ex expanded_exponent = exponent.expand(options);
+ const ex expanded_basis = basis.expand(options);
+ const ex expanded_exponent = exponent.expand(options);
// x^(a+b) -> x^a * x^b
if (is_ex_exactly_of_type(expanded_exponent, add)) {
term.reserve(m+1);
for (l=0; l<m-1; l++) {
const ex & b = a.op(l);
- GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
- GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
- !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
+ GINAC_ASSERT(!is_exactly_a<add>(b));
+ GINAC_ASSERT(!is_exactly_a<power>(b) ||
+ !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
!ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
+ !is_exactly_a<add>(ex_to<power>(b).basis) ||
+ !is_exactly_a<mul>(ex_to<power>(b).basis) ||
+ !is_exactly_a<power>(ex_to<power>(b).basis));
if (is_ex_exactly_of_type(b,mul))
term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
else
}
const ex & b = a.op(l);
- GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
- GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
- !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
+ GINAC_ASSERT(!is_exactly_a<add>(b));
+ GINAC_ASSERT(!is_exactly_a<power>(b) ||
+ !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
!ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
+ !is_exactly_a<add>(ex_to<power>(b).basis) ||
+ !is_exactly_a<mul>(ex_to<power>(b).basis) ||
+ !is_exactly_a<power>(ex_to<power>(b).basis));
if (is_ex_exactly_of_type(b,mul))
term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
else
const ex & r = cit0->rest;
const ex & c = cit0->coeff;
- GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
- GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
- !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
+ GINAC_ASSERT(!is_exactly_a<add>(r));
+ GINAC_ASSERT(!is_exactly_a<power>(r) ||
+ !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
!ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
- !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
- !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
- !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
+ !is_exactly_a<add>(ex_to<power>(r).basis) ||
+ !is_exactly_a<mul>(ex_to<power>(r).basis) ||
+ !is_exactly_a<power>(ex_to<power>(r).basis));
if (are_ex_trivially_equal(c,_ex1())) {
if (is_ex_exactly_of_type(r,mul)) {
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