def __new__(cls, p, q=None): if q is None: if isinstance(p, Rational): return p if isinstance(p, basestring): try: # we might have a Real neg_pow, digits, expt = decimal.Decimal(p).as_tuple() p = [1, -1][neg_pow] * int("".join(str(x) for x in digits)) if expt > 0: rv = Rational(p*Pow(10, expt), 1) return Rational(p, Pow(10, -expt)) except decimal.InvalidOperation: import re f = re.match(' *([-+]? *[0-9]+)( */ *)([0-9]+)', p) if f: p, _, q = f.groups() return Rational(int(p), int(q)) else: raise ValueError('invalid literal: %s' % p) elif not isinstance(p, Basic): return Rational(S(p)) q = S.One if isinstance(q, Rational): p *= q.q q = q.p if isinstance(p, Rational): q *= p.q p = p.p p = int(p) q = int(q) if q == 0: if p == 0: if _errdict["divide"]: raise ValueError("Indeterminate 0/0") else: return S.NaN if p < 0: return S.NegativeInfinity return S.Infinity if q < 0: q = -q p = -p n = igcd(abs(p), q) if n > 1: p //= n q //= n if q == 1: return Integer(p) if p == 1 and q == 2: return S.Half obj = Expr.__new__(cls) obj.p = p obj.q = q #obj._args = (p, q) return obj
def _eval_power(b, e): # don't break up NC terms: (A*B)**3 != A**3*B**3, it is A*B*A*B*A*B cargs, nc = b.args_cnc(split_1=False) if e.is_Integer: return Mul(*[Pow(b, e, evaluate=False) for b in cargs]) * Pow(Mul._from_args(nc), e, evaluate=False) p = Pow(b, e, evaluate=False) if e.is_Rational or e.is_Float: return p._eval_expand_power_base() return p
def _eval_power(b, e): # don't break up NC terms: (A*B)**3 != A**3*B**3, it is A*B*A*B*A*B cargs, nc = b.args_cnc(split_1=False) if e.is_Integer: return Mul(*[Pow(b, e, evaluate=False) for b in cargs]) * \ Pow(Mul._from_args(nc), e, evaluate=False) p = Pow(b, e, evaluate=False) if e.is_Rational or e.is_Float: return p._eval_expand_power_base() return p
def breakup(eq): """break up powers assuming (not checking) that eq is a Mul: b**(Rational*e) -> b**e, Rational commutatives come back as a dictionary {b**e: Rational} noncommutatives come back as a list [(b**e, Rational)] """ (c, nc) = (dict(), list()) for (i, a) in enumerate( Mul.make_args(eq) or [eq]): # remove or [eq] after 2114 accepted a = powdenest(a) (b, e) = a.as_base_exp() if not e is S.One: (co, _) = e.as_coeff_mul() b = Pow(b, e / co) e = co if a.is_commutative: if b in c: # handle I and -1 like things where b, e for I is -1, 1/2 c[b] += e else: c[b] = e else: nc.append([b, e]) return (c, nc)
def rejoin(b, co): """ Put rational back with exponent; in general this is not ok, but since we took it from the exponent for analysis, it's ok to put it back. """ (b, e) = b.as_base_exp() return Pow(b, e * co)
def breakup(eq): """break up powers of eq when treated as a Mul: b**(Rational*e) -> b**e, Rational commutatives come back as a dictionary {b**e: Rational} noncommutatives come back as a list [(b**e, Rational)] """ (c, nc) = (defaultdict(int), list()) for a in Mul.make_args(eq): a = powdenest(a) (b, e) = base_exp(a) if e is not S.One: (co, _) = e.as_coeff_mul() b = Pow(b, e/co) e = co if a.is_commutative: c[b] += e else: nc.append([b, e]) return (c, nc)
def _eval_subs(self, old, new): from sympy import sign from sympy.simplify.simplify import powdenest if self == old: return new def fallback(): """Return this value when partial subs has failed.""" return self.__class__(*[s._eval_subs(old, new) for s in self.args]) def breakup(eq): """break up powers assuming (not checking) that eq is a Mul: b**(Rational*e) -> b**e, Rational commutatives come back as a dictionary {b**e: Rational} noncommutatives come back as a list [(b**e, Rational)] """ (c, nc) = (dict(), list()) for (i, a) in enumerate( Mul.make_args(eq) or [eq]): # remove or [eq] after 2114 accepted a = powdenest(a) (b, e) = a.as_base_exp() if not e is S.One: (co, _) = e.as_coeff_mul() b = Pow(b, e / co) e = co if a.is_commutative: if b in c: # handle I and -1 like things where b, e for I is -1, 1/2 c[b] += e else: c[b] = e else: nc.append([b, e]) return (c, nc) def rejoin(b, co): """ Put rational back with exponent; in general this is not ok, but since we took it from the exponent for analysis, it's ok to put it back. """ (b, e) = b.as_base_exp() return Pow(b, e * co) def ndiv(a, b): """if b divides a in an extractive way (like 1/4 divides 1/2 but not vice versa, and 2/5 does not divide 1/3) then return the integer number of times it divides, else return 0. """ if not b.q % a.q or not a.q % b.q: return int(a / b) return 0 if not old.is_Mul: return fallback() # handle the leading coefficient and use it to decide if anything # should even be started; we always know where to find the Rational # so it's a quick test coeff = S.One co_self = self.args[0] co_old = old.args[0] if co_old.is_Rational and co_self.is_Rational: co_xmul = co_self.extract_multiplicatively(co_old) elif co_old.is_Rational: co_xmul = None else: co_xmul = True if not co_xmul: return fallback() (c, nc) = breakup(self) (old_c, old_nc) = breakup(old) # update the coefficients if we had an extraction if getattr(co_xmul, 'is_Rational', False): c.pop(co_self) c[co_xmul] = S.One old_c.pop(co_old) # do quick tests to see if we can't succeed ok = True if ( # more non-commutative terms len(old_nc) > len(nc)): ok = False elif ( # more commutative terms len(old_c) > len(c)): ok = False elif ( # unmatched non-commutative bases set(_[0] for _ in old_nc).difference(set(_[0] for _ in nc))): ok = False elif ( # unmatched commutative terms set(old_c).difference(set(c))): ok = False elif ( # differences in sign any(sign(c[b]) != sign(old_c[b]) for b in old_c)): ok = False if not ok: return fallback() if not old_c: cdid = None else: rat = [] for (b, old_e) in old_c.items(): c_e = c[b] rat.append(ndiv(c_e, old_e)) if not rat[-1]: return fallback() cdid = min(rat) if not old_nc: ncdid = None for i in range(len(nc)): nc[i] = rejoin(*nc[i]) else: ncdid = 0 # number of nc replacements we did take = len(old_nc) # how much to look at each time limit = cdid or S.Infinity # max number that we can take failed = [] # failed terms will need subs if other terms pass i = 0 while limit and i + take <= len(nc): hit = False # the bases must be equivalent in succession, and # the powers must be extractively compatible on the # first and last factor but equal inbetween. rat = [] for j in range(take): if nc[i + j][0] != old_nc[j][0]: break elif j == 0: rat.append(ndiv(nc[i + j][1], old_nc[j][1])) elif j == take - 1: rat.append(ndiv(nc[i + j][1], old_nc[j][1])) elif nc[i + j][1] != old_nc[j][1]: break else: rat.append(1) j += 1 else: ndo = min(rat) if ndo: if take == 1: if cdid: ndo = min(cdid, ndo) nc[i] = Pow(new, ndo) * rejoin( nc[i][0], nc[i][1] - ndo * old_nc[0][1]) else: ndo = 1 # the left residual l = rejoin(nc[i][0], nc[i][1] - ndo * old_nc[0][1]) # eliminate all middle terms mid = new # the right residual (which may be the same as the middle if take == 2) ir = i + take - 1 r = (nc[ir][0], nc[ir][1] - ndo * old_nc[-1][1]) if r[1]: if i + take < len(nc): nc[i:i + take] = [l * mid, r] else: r = rejoin(*r) nc[i:i + take] = [l * mid * r] else: # there was nothing left on the right nc[i:i + take] = [l * mid] limit -= ndo ncdid += ndo hit = True if not hit: # do the subs on this failing factor failed.append(i) i += 1 else: if not ncdid: return fallback() # although we didn't fail, certain nc terms may have # failed so we rebuild them after attempting a partial # subs on them failed.extend(range(i, len(nc))) for i in failed: nc[i] = rejoin(*nc[i]).subs(old, new) # rebuild the expression if cdid is None: do = ncdid elif ncdid is None: do = cdid else: do = min(ncdid, cdid) margs = [] for b in c: if b in old_c: # calculate the new exponent e = c[b] - old_c[b] * do margs.append(rejoin(b, e)) else: margs.append(rejoin(b.subs(old, new), c[b])) if cdid and not ncdid: # in case we are replacing commutative with non-commutative, # we want the new term to come at the front just like the # rest of this routine margs = [Pow(new, cdid)] + margs return Mul(*margs) * Mul(*nc)
def _eval_power(b, e): if e.is_Number: if b.is_commutative: if e.is_Integer: # (a*b)**2 -> a**2 * b**2 return Mul(*[s**e for s in b.args]) if e.is_rational: coeff, rest = b.as_coeff_mul() unk = [] nonneg = [] neg = [] for bi in rest: if not bi.is_negative is None: #then we know the sign if bi.is_negative: neg.append(bi) else: nonneg.append(bi) else: unk.append(bi) if len(unk) == len(rest) or len(neg) == len(rest) == 1: # if all terms were unknown there is nothing to pull # out except maybe the coeff OR if there was only a # single negative term then it shouldn't be pulled out # either. if coeff < 0: coeff *= -1 if coeff is S.One: return None return Mul(Pow(coeff, e), Pow(b / coeff, e)) # otherwise return the new expression expanding out the # known terms; those that are not known can be expanded # out with separate() but this will introduce a lot of # "garbage" that is needed to keep one on the same branch # as the unexpanded expression. The negatives are brought # out with a negative sign added and a negative left behind # in the unexpanded terms. if neg: neg = [-w for w in neg] if len(neg) % 2 and not coeff.is_negative: unk.append(S.NegativeOne) if coeff.is_negative: coeff = -coeff unk.append(S.NegativeOne) return Mul(*[Pow(s, e) for s in nonneg + neg + [coeff]])* \ Pow(Mul(*unk), e) coeff, rest = b.as_coeff_mul() if coeff is not S.One: # (2*a)**3 -> 2**3 * a**3 return Mul(Pow(coeff, e), Mul(*[s**e for s in rest])) elif e.is_Integer: coeff, rest = b.as_coeff_mul() if coeff == S.One: return # the test below for even exponent needs coeff != 1 else: return Mul(Pow(coeff, e), Pow(Mul(*rest), e)) c, t = b.as_coeff_mul() if e.is_even and c.is_Number and c < 0: return Pow((Mul(-c, Mul(*t))), e)
def flatten(cls, seq): # apply associativity, separate commutative part of seq c_part = [] # out: commutative factors nc_part = [] # out: non-commutative factors nc_seq = [] coeff = S.One # standalone term # e.g. 3 * ... c_powers = [] # (base,exp) n # e.g. (x,n) for x num_exp = [] # (num-base, exp) y # e.g. (3, y) for ... * 3 * ... order_symbols = None # --- PART 1 --- # # "collect powers and coeff": # # o coeff # o c_powers # o num_exp # # NOTE: this is optimized for all-objects-are-commutative case for o in seq: # O(x) if o.is_Order: o, order_symbols = o.as_expr_variables(order_symbols) # Mul([...]) if o.is_Mul: if o.is_commutative: seq.extend(o.args) # XXX zerocopy? else: # NCMul can have commutative parts as well for q in o.args: if q.is_commutative: seq.append(q) else: nc_seq.append(q) # append non-commutative marker, so we don't forget to # process scheduled non-commutative objects seq.append(NC_Marker) continue # 3 elif o.is_Number: if o is S.NaN or coeff is S.ComplexInfinity and o is S.Zero: # we know for sure the result will be nan return [S.NaN], [], None elif coeff.is_Number: # it could be zoo coeff *= o if coeff is S.NaN: # we know for sure the result will be nan return [S.NaN], [], None continue elif o is S.ComplexInfinity: if not coeff or coeff is S.ComplexInfinity: # we know for sure the result will be nan return [S.NaN], [], None coeff = S.ComplexInfinity continue elif o.is_commutative: # e # o = b b, e = o.as_base_exp() # y # 3 if o.is_Pow and b.is_Number: # get all the factors with numeric base so they can be # combined below, but don't combine negatives unless # the exponent is an integer if b.is_positive or e.is_integer: num_exp.append((b, e)) continue # n n n # (-3 + y) -> (-1) * (3 - y) # # Give powers a chance to become a Mul if that's the # behavior obtained from Add._eval_power() if not Basic.keep_sign and b.is_Add and e.is_Number: cb = b._eval_power(e, terms=True) if cb: c, b = cb coeff *= c c_powers.append((b, e)) # NON-COMMUTATIVE # TODO: Make non-commutative exponents not combine automatically else: if o is not NC_Marker: nc_seq.append(o) # process nc_seq (if any) while nc_seq: o = nc_seq.pop(0) if not nc_part: nc_part.append(o) continue # b c b+c # try to combine last terms: a * a -> a o1 = nc_part.pop() b1, e1 = o1.as_base_exp() b2, e2 = o.as_base_exp() new_exp = e1 + e2 # Only allow powers to combine if the new exponent is # not an Add. This allow things like a**2*b**3 == a**5 # if a.is_commutative == False, but prohibits # a**x*a**y and x**a*x**b from combining (x,y commute). if b1 == b2 and (not new_exp.is_Add): o12 = b1**new_exp # now o12 could be a commutative object if o12.is_commutative: seq.append(o12) continue else: nc_seq.insert(0, o12) else: nc_part.append(o1) nc_part.append(o) # We do want a combined exponent if it would not be an Add, such as # y 2y 3y # x * x -> x # We determine this if two exponents have the same term in as_coeff_mul # # Unfortunately, this isn't smart enough to consider combining into # exponents that might already be adds, so things like: # z - y y # x * x will be left alone. This is because checking every possible # combination can slow things down. # gather exponents of common bases... # in c_powers new_c_powers = [] common_b = {} # b:e for b, e in c_powers: co = e.as_coeff_mul() common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0]) for b, d in common_b.items(): for di, li in d.items(): d[di] = Add(*li) for b, e in common_b.items(): for t, c in e.items(): new_c_powers.append((b, c * Mul(*t))) c_powers = new_c_powers # and in num_exp new_num_exp = [] common_b = {} # b:e for b, e in num_exp: co = e.as_coeff_mul() common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0]) for b, d in common_b.items(): for di, li in d.items(): d[di] = Add(*li) for b, e in common_b.items(): for t, c in e.items(): new_num_exp.append((b, c * Mul(*t))) num_exp = new_num_exp # --- PART 2 --- # # o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow) # o combine collected powers (2**x * 3**x -> 6**x) # with numeric base # ................................ # now we have: # - coeff: # - c_powers: (b, e) # - num_exp: (2, e) # 0 1 # x -> 1 x -> x for b, e in c_powers: if e is S.Zero: continue if e is S.One: if b.is_Number: coeff *= b else: c_part.append(b) elif e.is_Integer and b.is_Number: coeff *= Pow(b, e) else: c_part.append(Pow(b, e)) # x x x # 2 * 3 -> 6 inv_exp_dict = {} # exp:Mul(num-bases) x x # e.g. x:6 for ... * 2 * 3 * ... for b, e in num_exp: inv_exp_dict.setdefault(e, []).append(b) for e, b in inv_exp_dict.items(): inv_exp_dict[e] = Mul(*b) reeval = False for e, b in inv_exp_dict.items(): if e is S.Zero: continue if e is S.One: if b.is_Number: coeff *= b else: c_part.append(b) elif e.is_Integer and b.is_Number: coeff *= Pow(b, e) else: obj = b**e if obj.is_Mul: # We may have split out a number that needs to go in coeff # e.g., sqrt(6)*sqrt(2) == 2*sqrt(3). See issue 415. reeval = True if obj.is_Number: coeff *= obj else: c_part.append(obj) # oo, -oo if (coeff is S.Infinity) or (coeff is S.NegativeInfinity): new_c_part = [] coeff_sign = 1 for t in c_part: if t.is_positive: continue if t.is_negative: coeff_sign *= -1 continue new_c_part.append(t) c_part = new_c_part new_nc_part = [] for t in nc_part: if t.is_positive: continue if t.is_negative: coeff_sign *= -1 continue new_nc_part.append(t) nc_part = new_nc_part coeff *= coeff_sign # zoo if coeff is S.ComplexInfinity: # zoo might be # unbounded_real + bounded_im # bounded_real + unbounded_im # unbounded_real + unbounded_im # and non-zero real or imaginary will not change that status. c_part = [ c for c in c_part if not (c.is_nonzero and c.is_real is not None) ] nc_part = [ c for c in nc_part if not (c.is_nonzero and c.is_real is not None) ] # 0 elif coeff is S.Zero: # we know for sure the result will be 0 return [coeff], [], order_symbols elif coeff.is_Real: if coeff == Real(0): c_part, nc_part = [coeff], [] elif coeff == Real(1): # change it to One, so it doesn't get inserted to slot0 coeff = S.One # order commutative part canonically c_part.sort(Basic.compare) # current code expects coeff to be always in slot-0 if coeff is not S.One: c_part.insert(0, coeff) # we are done if len(c_part) == 2 and c_part[0].is_Number and c_part[1].is_Add: # 2*(1+a) -> 2 + 2 * a coeff = c_part[0] c_part = [Add(*[coeff * f for f in c_part[1].args])] if reeval: c_part, _, _ = Mul.flatten(c_part) return c_part, nc_part, order_symbols
def flatten(cls, seq): """Return commutative, noncommutative and order arguments by combining related terms. ** Developer Notes ** * In an expression like ``a*b*c``, python process this through sympy as ``Mul(Mul(a, b), c)``. This can have undesirable consequences. - Sometimes terms are not combined as one would like: {c.f. http://code.google.com/p/sympy/issues/detail?id=1497} >>> from sympy import Mul, sqrt >>> from sympy.abc import x, y, z >>> 2*(x + 1) # this is the 2-arg Mul behavior 2*x + 2 >>> y*(x + 1)*2 2*y*(x + 1) >>> 2*(x + 1)*y # 2-arg result will be obtained first y*(2*x + 2) >>> Mul(2, x + 1, y) # all 3 args simultaneously processed 2*y*(x + 1) >>> 2*((x + 1)*y) # parentheses can control this behavior 2*y*(x + 1) Powers with compound bases may not find a single base to combine with unless all arguments are processed at once. Post-processing may be necessary in such cases. {c.f. http://code.google.com/p/sympy/issues/detail?id=2629} >>> a = sqrt(x*sqrt(y)) >>> a**3 (x*sqrt(y))**(3/2) >>> Mul(a,a,a) (x*sqrt(y))**(3/2) >>> a*a*a x*sqrt(y)*sqrt(x*sqrt(y)) >>> _.subs(a.base, z).subs(z, a.base) (x*sqrt(y))**(3/2) - If more than two terms are being multiplied then all the previous terms will be re-processed for each new argument. So if each of ``a``, ``b`` and ``c`` were :class:`Mul` expression, then ``a*b*c`` (or building up the product with ``*=``) will process all the arguments of ``a`` and ``b`` twice: once when ``a*b`` is computed and again when ``c`` is multiplied. Using ``Mul(a, b, c)`` will process all arguments once. * The results of Mul are cached according to arguments, so flatten will only be called once for ``Mul(a, b, c)``. If you can structure a calculation so the arguments are most likely to be repeats then this can save time in computing the answer. For example, say you had a Mul, M, that you wished to divide by ``d[i]`` and multiply by ``n[i]`` and you suspect there are many repeats in ``n``. It would be better to compute ``M*n[i]/d[i]`` rather than ``M/d[i]*n[i]`` since every time n[i] is a repeat, the product, ``M*n[i]`` will be returned without flattening -- the cached value will be returned. If you divide by the ``d[i]`` first (and those are more unique than the ``n[i]``) then that will create a new Mul, ``M/d[i]`` the args of which will be traversed again when it is multiplied by ``n[i]``. {c.f. http://code.google.com/p/sympy/issues/detail?id=2607} This consideration is moot if the cache is turned off. NB The validity of the above notes depends on the implementation details of Mul and flatten which may change at any time. Therefore, you should only consider them when your code is highly performance sensitive. """ # apply associativity, separate commutative part of seq c_part = [] # out: commutative factors nc_part = [] # out: non-commutative factors nc_seq = [] coeff = S.One # standalone term # e.g. 3 * ... c_powers = [] # (base,exp) n # e.g. (x,n) for x num_exp = [] # (num-base, exp) y # e.g. (3, y) for ... * 3 * ... neg1e = 0 # exponent on -1 extracted from Number-based Pow pnum_rat = {} # (num-base, Rat-exp) 1/2 # e.g. (3, 1/2) for ... * 3 * ... order_symbols = None # --- PART 1 --- # # "collect powers and coeff": # # o coeff # o c_powers # o num_exp # o neg1e # o pnum_rat # # NOTE: this is optimized for all-objects-are-commutative case for o in seq: # O(x) if o.is_Order: o, order_symbols = o.as_expr_variables(order_symbols) # Mul([...]) if o.is_Mul: if o.is_commutative: seq.extend(o.args) # XXX zerocopy? else: # NCMul can have commutative parts as well for q in o.args: if q.is_commutative: seq.append(q) else: nc_seq.append(q) # append non-commutative marker, so we don't forget to # process scheduled non-commutative objects seq.append(NC_Marker) continue # 3 elif o.is_Number: if o is S.NaN or coeff is S.ComplexInfinity and o is S.Zero: # we know for sure the result will be nan return [S.NaN], [], None elif coeff.is_Number: # it could be zoo coeff *= o if coeff is S.NaN: # we know for sure the result will be nan return [S.NaN], [], None continue elif o is S.ComplexInfinity: if not coeff or coeff is S.ComplexInfinity: # we know for sure the result will be nan return [S.NaN], [], None coeff = S.ComplexInfinity continue elif o.is_commutative: # e # o = b b, e = o.as_base_exp() # y # 3 if o.is_Pow and b.is_Number: # get all the factors with numeric base so they can be # combined below, but don't combine negatives unless # the exponent is an integer if e.is_Rational: if e.is_Integer: coeff *= Pow(b, e) # it is an unevaluated power continue elif e.is_negative: # also a sign of an unevaluated power seq.append(Pow(b, e)) continue elif b.is_negative: neg1e += e b = -b if b is not S.One: pnum_rat.setdefault(b, []).append(e) continue elif b.is_positive or e.is_integer: num_exp.append((b, e)) continue c_powers.append((b,e)) # NON-COMMUTATIVE # TODO: Make non-commutative exponents not combine automatically else: if o is not NC_Marker: nc_seq.append(o) # process nc_seq (if any) while nc_seq: o = nc_seq.pop(0) if not nc_part: nc_part.append(o) continue # b c b+c # try to combine last terms: a * a -> a o1 = nc_part.pop() b1,e1 = o1.as_base_exp() b2,e2 = o.as_base_exp() new_exp = e1 + e2 # Only allow powers to combine if the new exponent is # not an Add. This allow things like a**2*b**3 == a**5 # if a.is_commutative == False, but prohibits # a**x*a**y and x**a*x**b from combining (x,y commute). if b1==b2 and (not new_exp.is_Add): o12 = b1 ** new_exp # now o12 could be a commutative object if o12.is_commutative: seq.append(o12) continue else: nc_seq.insert(0, o12) else: nc_part.append(o1) nc_part.append(o) # We do want a combined exponent if it would not be an Add, such as # y 2y 3y # x * x -> x # We determine this if two exponents have the same term in as_coeff_mul # # Unfortunately, this isn't smart enough to consider combining into # exponents that might already be adds, so things like: # z - y y # x * x will be left alone. This is because checking every possible # combination can slow things down. # gather exponents of common bases... # in c_powers new_c_powers = [] common_b = {} # b:e for b, e in c_powers: co = e.as_coeff_mul() common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0]) for b, d in common_b.items(): for di, li in d.items(): d[di] = Add(*li) for b, e in common_b.items(): for t, c in e.items(): new_c_powers.append((b, c*Mul(*t))) c_powers = new_c_powers # and in num_exp new_num_exp = [] common_b = {} # b:e for b, e in num_exp: co = e.as_coeff_mul() common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0]) for b, d in common_b.items(): for di, li in d.items(): d[di] = Add(*li) for b, e in common_b.items(): for t, c in e.items(): new_num_exp.append((b,c*Mul(*t))) num_exp = new_num_exp # --- PART 2 --- # # o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow) # o combine collected powers (2**x * 3**x -> 6**x) # with numeric base # ................................ # now we have: # - coeff: # - c_powers: (b, e) # - num_exp: (2, e) # - pnum_rat: {(1/3, [1/3, 2/3, 1/4])} # 0 1 # x -> 1 x -> x for b, e in c_powers: if e is S.One: if b.is_Number: coeff *= b else: c_part.append(b) elif not e is S.Zero: c_part.append(Pow(b, e)) # x x x # 2 * 3 -> 6 inv_exp_dict = {} # exp:Mul(num-bases) x x # e.g. x:6 for ... * 2 * 3 * ... for b, e in num_exp: inv_exp_dict.setdefault(e, []).append(b) for e, b in inv_exp_dict.items(): inv_exp_dict[e] = Mul(*b) c_part.extend([Pow(b, e) for e, b in inv_exp_dict.iteritems() if e]) # b, e -> e, b # {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])} comb_e = {} for b, e in pnum_rat.iteritems(): comb_e.setdefault(Add(*e), []).append(b) del pnum_rat # process them, reducing exponents to values less than 1 # and updating coeff if necessary else adding them to # num_rat for further processing num_rat = [] for e, b in comb_e.iteritems(): b = Mul(*b) if e.q == 1: coeff *= Pow(b, e) continue if e.p > e.q: e_i, ep = divmod(e.p, e.q) coeff *= Pow(b, e_i) e = Rational(ep, e.q) num_rat.append((b, e)) del comb_e # extract gcd of bases in num_rat # 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4) pnew = {} i = 0 # steps through num_rat which may grow while i < len(num_rat): bi, ei = num_rat[i] grow = [] for j in range(i + 1, len(num_rat)): bj, ej = num_rat[j] g = igcd(bi, bj) if g != 1: # 4**r1*6**r2 -> 2**(r1+r2) * 2**r1 * 3**r2 # this might have a gcd with something else e = ei + ej if e.q == 1: coeff *= Pow(g, e) else: if e.p > e.q: e_i, ep = divmod(e.p, e.q) # change e in place coeff *= Pow(g, e_i) e = Rational(ep, e.q) grow.append((g, e)) # update the jth item num_rat[j] = (bj//g, ej) # update bi that we are checking with bi = bi//g if bi is S.One: break if bi is not S.One: obj = Pow(bi, ei) if obj.is_Number: coeff *= obj else: if obj.is_Mul: # sqrt(12) -> 2*sqrt(3) c, obj = obj.args # expecting only 2 args coeff *= c assert obj.is_Pow bi, ei = obj.args pnew.setdefault(ei, []).append(bi) num_rat.extend(grow) i += 1 # combine bases of the new powers for e, b in pnew.iteritems(): pnew[e] = Mul(*b) # see if there is a base with matching coefficient # that the -1 can be joined with if neg1e: p = Pow(S.NegativeOne, neg1e) if p.is_Number: coeff *= p else: if p.is_Mul: c, p = p.args coeff *= c assert p.is_Pow and p.base is S.NegativeOne neg1e = p.args[1] for e, b in pnew.iteritems(): if e == neg1e and b.is_positive: pnew[e] = -b break else: c_part.append(p) # add all the pnew powers c_part.extend([Pow(b, e) for e, b in pnew.iteritems()]) # oo, -oo if (coeff is S.Infinity) or (coeff is S.NegativeInfinity): new_c_part = [] coeff_sign = 1 for t in c_part: if t.is_positive: continue if t.is_negative: coeff_sign *= -1 continue new_c_part.append(t) c_part = new_c_part new_nc_part = [] for t in nc_part: if t.is_positive: continue if t.is_negative: coeff_sign *= -1 continue new_nc_part.append(t) nc_part = new_nc_part coeff *= coeff_sign # zoo if coeff is S.ComplexInfinity: # zoo might be # unbounded_real + bounded_im # bounded_real + unbounded_im # unbounded_real + unbounded_im # and non-zero real or imaginary will not change that status. c_part = [c for c in c_part if not (c.is_nonzero and c.is_real is not None)] nc_part = [c for c in nc_part if not (c.is_nonzero and c.is_real is not None)] # 0 elif coeff is S.Zero: # we know for sure the result will be 0 return [coeff], [], order_symbols # order commutative part canonically c_part.sort(key=cmp_to_key(Basic.compare)) # current code expects coeff to be always in slot-0 if coeff is not S.One: c_part.insert(0, coeff) # we are done if len(c_part)==2 and c_part[0].is_Number and c_part[1].is_Add: # 2*(1+a) -> 2 + 2 * a coeff = c_part[0] c_part = [Add(*[coeff*f for f in c_part[1].args])] return c_part, nc_part, order_symbols
def as_content_primitive(self, radical=False): """Return the tuple (R, self/R) where R is the positive Rational extracted from self. If radical is True (default is False) then common radicals will be removed and included as a factor of the primitive expression. **Examples** >>> from sympy import sqrt >>> (3 + 3*sqrt(2)).as_content_primitive() (3, 1 + sqrt(2)) Radical content can also be factored out of the primitive: >>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2)*(1 + 2*sqrt(5))) See docstring of Expr.as_content_primitive for more examples. """ con, prim = Add(*[_keep_coeff(*a.as_content_primitive(radical=radical)) for a in self.args]).primitive() if radical and prim.is_Add: # look for common radicals that can be removed args = prim.args rads = [] common_q = None for m in args: term_rads = defaultdict(list) for ai in Mul.make_args(m): if ai.is_Pow: b, e = ai.as_base_exp() if e.is_Rational and b.is_Integer and b > 0: term_rads[e.q].append(int(b)**e.p) if not term_rads: break if common_q is None: common_q = set(term_rads.keys()) else: common_q = common_q & set(term_rads.keys()) if not common_q: break rads.append(term_rads) else: # process rads # keep only those in common_q for r in rads: for q in r.keys(): if q not in common_q: r.pop(q) for q in r: r[q] = prod(r[q]) # find the gcd of bases for each q G = [] for q in common_q: g = reduce(igcd, [r[q] for r in rads], 0) if g != 1: G.append(Pow(g, Rational(1, q))) if G: G = Mul(*G) args = [ai/G for ai in args] prim = G*Add(*args) return con, prim
def flatten(cls, seq): """ Takes the sequence "seq" of nested Adds and returns a flatten list. Returns: (commutative_part, noncommutative_part, order_symbols) Applies associativity, all terms are commutable with respect to addition. ** Developer Notes ** See Mul.flatten """ rv = None if len(seq) == 2: a, b = seq if b.is_Rational: a, b = b, a assert a if a.is_Rational: if b.is_Mul: # if it's an unevaluated 2-arg, expand it c, t = b.as_coeff_Mul() if t.is_Add: h, t = t.as_coeff_Add() bargs = [c*ti for ti in Add.make_args(t)] bargs.sort(key=hash) ch = c*h if ch: bargs.insert(0, ch) b = Add._from_args(bargs) if b.is_Add: bargs = list(b.args) if bargs[0].is_Number: bargs[0] += a if not bargs[0]: bargs.pop(0) else: bargs.insert(0, a) rv = bargs, [], None elif b.is_Mul: rv = [a, b], [], None if rv: if all(s.is_commutative for s in rv[0]): return rv return [], rv[0], None terms = {} # term -> coeff # e.g. x**2 -> 5 for ... + 5*x**2 + ... coeff = S.Zero # standalone term (Number or zoo will always be in slot 0) # e.g. 3 + ... order_factors = [] for o in seq: # O(x) if o.is_Order: for o1 in order_factors: if o1.contains(o): o = None break if o is None: continue order_factors = [o]+[o1 for o1 in order_factors if not o.contains(o1)] continue # 3 or NaN elif o.is_Number: if o is S.NaN or coeff is S.ComplexInfinity and o.is_bounded is False: # we know for sure the result will be nan return [S.NaN], [], None if coeff.is_Number: coeff += o if coeff is S.NaN: # we know for sure the result will be nan return [S.NaN], [], None continue elif o is S.ComplexInfinity: if coeff.is_bounded is False: # we know for sure the result will be nan return [S.NaN], [], None coeff = S.ComplexInfinity continue # Add([...]) elif o.is_Add: # NB: here we assume Add is always commutative seq.extend(o.args) # TODO zerocopy? continue # Mul([...]) elif o.is_Mul: c, s = o.as_coeff_Mul() # 3*... # unevaluated 2-arg Mul, but we always unfold it so # it can combine with other terms (just like is done # with the Pow below) if c.is_Number and s.is_Add: seq.extend([c*a for a in s.args]) continue # check for unevaluated Pow, e.g. 2**3 or 2**(-1/2) elif o.is_Pow: b, e = o.as_base_exp() if b.is_Number and (e.is_Integer or (e.is_Rational and e.is_negative)): seq.append(Pow(b, e)) continue c, s = S.One, o else: # everything else c = S.One s = o # now we have: # o = c*s, where # # c is a Number # s is an expression with number factor extracted # let's collect terms with the same s, so e.g. # 2*x**2 + 3*x**2 -> 5*x**2 if s in terms: terms[s] += c else: terms[s] = c # now let's construct new args: # [2*x**2, x**3, 7*x**4, pi, ...] newseq = [] noncommutative = False for s,c in terms.items(): # 0*s if c is S.Zero: continue # 1*s elif c is S.One: newseq.append(s) # c*s else: if s.is_Mul: # Mul, already keeps its arguments in perfect order. # so we can simply put c in slot0 and go the fast way. cs = s._new_rawargs(*((c,) + s.args)) newseq.append(cs) else: # alternatively we have to call all Mul's machinery (slow) newseq.append(Mul(c,s)) noncommutative = noncommutative or not s.is_commutative # oo, -oo if coeff is S.Infinity: newseq = [f for f in newseq if not (f.is_nonnegative or f.is_real and (f.is_bounded or f.is_finite or f.is_infinitesimal))] elif coeff is S.NegativeInfinity: newseq = [f for f in newseq if not (f.is_nonpositive or f.is_real and (f.is_bounded or f.is_finite or f.is_infinitesimal))] if coeff is S.ComplexInfinity: # zoo might be # unbounded_real + bounded_im # bounded_real + unbounded_im # unbounded_real + unbounded_im # addition of a bounded real or imaginary number won't be able to # change the zoo nature; if unbounded a NaN condition could result if # the unbounded symbol had sign opposite of the unbounded portion of zoo, # e.g. unbounded_real - unbounded_real newseq = [c for c in newseq if not (c.is_bounded and c.is_real is not None)] # process O(x) if order_factors: newseq2 = [] for t in newseq: for o in order_factors: # x + O(x) -> O(x) if o.contains(t): t = None break # x + O(x**2) -> x + O(x**2) if t is not None: newseq2.append(t) newseq = newseq2 + order_factors # 1 + O(1) -> O(1) for o in order_factors: if o.contains(coeff): coeff = S.Zero break # order args canonically # Currently we sort things using hashes, as it is quite fast. A better # solution is not to sort things at all - but this needs some more # fixing. NOTE: this is used in primitive, too, so if it changes # here it should be changed there. newseq.sort(key=hash) # current code expects coeff to be always in slot-0 if coeff is not S.Zero: newseq.insert(0, coeff) # we are done if noncommutative: return [], newseq, None else: return newseq, [], None
def __rpow__(self, other): return Pow(other, self)
def flatten(cls, seq): """Return commutative, noncommutative and order arguments by combining related terms. Notes ===== * In an expression like ``a*b*c``, python process this through sympy as ``Mul(Mul(a, b), c)``. This can have undesirable consequences. - Sometimes terms are not combined as one would like: {c.f. http://code.google.com/p/sympy/issues/detail?id=1497} >>> from sympy import Mul, sqrt >>> from sympy.abc import x, y, z >>> 2*(x + 1) # this is the 2-arg Mul behavior 2*x + 2 >>> y*(x + 1)*2 2*y*(x + 1) >>> 2*(x + 1)*y # 2-arg result will be obtained first y*(2*x + 2) >>> Mul(2, x + 1, y) # all 3 args simultaneously processed 2*y*(x + 1) >>> 2*((x + 1)*y) # parentheses can control this behavior 2*y*(x + 1) Powers with compound bases may not find a single base to combine with unless all arguments are processed at once. Post-processing may be necessary in such cases. {c.f. http://code.google.com/p/sympy/issues/detail?id=2629} >>> a = sqrt(x*sqrt(y)) >>> a**3 (x*sqrt(y))**(3/2) >>> Mul(a,a,a) (x*sqrt(y))**(3/2) >>> a*a*a x*sqrt(y)*sqrt(x*sqrt(y)) >>> _.subs(a.base, z).subs(z, a.base) (x*sqrt(y))**(3/2) - If more than two terms are being multiplied then all the previous terms will be re-processed for each new argument. So if each of ``a``, ``b`` and ``c`` were :class:`Mul` expression, then ``a*b*c`` (or building up the product with ``*=``) will process all the arguments of ``a`` and ``b`` twice: once when ``a*b`` is computed and again when ``c`` is multiplied. Using ``Mul(a, b, c)`` will process all arguments once. * The results of Mul are cached according to arguments, so flatten will only be called once for ``Mul(a, b, c)``. If you can structure a calculation so the arguments are most likely to be repeats then this can save time in computing the answer. For example, say you had a Mul, M, that you wished to divide by ``d[i]`` and multiply by ``n[i]`` and you suspect there are many repeats in ``n``. It would be better to compute ``M*n[i]/d[i]`` rather than ``M/d[i]*n[i]`` since every time n[i] is a repeat, the product, ``M*n[i]`` will be returned without flattening -- the cached value will be returned. If you divide by the ``d[i]`` first (and those are more unique than the ``n[i]``) then that will create a new Mul, ``M/d[i]`` the args of which will be traversed again when it is multiplied by ``n[i]``. {c.f. http://code.google.com/p/sympy/issues/detail?id=2607} This consideration is moot if the cache is turned off. NB -- The validity of the above notes depends on the implementation details of Mul and flatten which may change at any time. Therefore, you should only consider them when your code is highly performance sensitive. Removal of 1 from the sequence is already handled by AssocOp.__new__. """ rv = None if len(seq) == 2: a, b = seq if b.is_Rational: a, b = b, a assert not a is S.One if a and a.is_Rational: r, b = b.as_coeff_Mul() a *= r if b.is_Mul: bargs, nc = b.args_cnc() rv = bargs, nc, None if a is not S.One: bargs.insert(0, a) elif b.is_Add and b.is_commutative: if a is S.One: rv = [b], [], None else: r, b = b.as_coeff_Add() bargs = [_keep_coeff(a, bi) for bi in Add.make_args(b)] bargs.sort(key=hash) ar = a*r if ar: bargs.insert(0, ar) bargs = [Add._from_args(bargs)] rv = bargs, [], None if rv: return rv # apply associativity, separate commutative part of seq c_part = [] # out: commutative factors nc_part = [] # out: non-commutative factors nc_seq = [] coeff = S.One # standalone term # e.g. 3 * ... iu = [] # ImaginaryUnits, I c_powers = [] # (base,exp) n # e.g. (x,n) for x num_exp = [] # (num-base, exp) y # e.g. (3, y) for ... * 3 * ... neg1e = 0 # exponent on -1 extracted from Number-based Pow pnum_rat = {} # (num-base, Rat-exp) 1/2 # e.g. (3, 1/2) for ... * 3 * ... order_symbols = None # --- PART 1 --- # # "collect powers and coeff": # # o coeff # o c_powers # o num_exp # o neg1e # o pnum_rat # # NOTE: this is optimized for all-objects-are-commutative case for o in seq: # O(x) if o.is_Order: o, order_symbols = o.as_expr_variables(order_symbols) # Mul([...]) if o.is_Mul: if o.is_commutative: seq.extend(o.args) # XXX zerocopy? else: # NCMul can have commutative parts as well for q in o.args: if q.is_commutative: seq.append(q) else: nc_seq.append(q) # append non-commutative marker, so we don't forget to # process scheduled non-commutative objects seq.append(NC_Marker) continue # 3 elif o.is_Number: if o is S.NaN or coeff is S.ComplexInfinity and o is S.Zero: # we know for sure the result will be nan return [S.NaN], [], None elif coeff.is_Number: # it could be zoo coeff *= o if coeff is S.NaN: # we know for sure the result will be nan return [S.NaN], [], None continue elif o is S.ComplexInfinity: if not coeff: # 0 * zoo = NaN return [S.NaN], [], None if coeff is S.ComplexInfinity: # zoo * zoo = zoo return [S.ComplexInfinity], [], None coeff = S.ComplexInfinity continue elif o is S.ImaginaryUnit: iu.append(o) continue elif o.is_commutative: # e # o = b b, e = o.as_base_exp() # y # 3 if o.is_Pow and b.is_Number: # get all the factors with numeric base so they can be # combined below, but don't combine negatives unless # the exponent is an integer if e.is_Rational: if e.is_Integer: coeff *= Pow(b, e) # it is an unevaluated power continue elif e.is_negative: # also a sign of an unevaluated power seq.append(Pow(b, e)) continue elif b.is_negative: neg1e += e b = -b if b is not S.One: pnum_rat.setdefault(b, []).append(e) continue elif b.is_positive or e.is_integer: num_exp.append((b, e)) continue c_powers.append((b, e)) # NON-COMMUTATIVE # TODO: Make non-commutative exponents not combine automatically else: if o is not NC_Marker: nc_seq.append(o) # process nc_seq (if any) while nc_seq: o = nc_seq.pop(0) if not nc_part: nc_part.append(o) continue # b c b+c # try to combine last terms: a * a -> a o1 = nc_part.pop() b1, e1 = o1.as_base_exp() b2, e2 = o.as_base_exp() new_exp = e1 + e2 # Only allow powers to combine if the new exponent is # not an Add. This allow things like a**2*b**3 == a**5 # if a.is_commutative == False, but prohibits # a**x*a**y and x**a*x**b from combining (x,y commute). if b1 == b2 and (not new_exp.is_Add): o12 = b1 ** new_exp # now o12 could be a commutative object if o12.is_commutative: seq.append(o12) continue else: nc_seq.insert(0, o12) else: nc_part.append(o1) nc_part.append(o) # handle the ImaginaryUnits if iu: if len(iu) == 1: c_powers.append((iu[0], S.One)) else: # a product of I's has one of 4 values; select that value # based on the length of iu: # len(iu) % 4 of (0, 1, 2, 3) has a corresponding value of # (1, I,-1,-I) niu = len(iu) % 4 if niu % 2: c_powers.append((S.ImaginaryUnit, S.One)) if niu in (2, 3): coeff = -coeff # We do want a combined exponent if it would not be an Add, such as # y 2y 3y # x * x -> x # We determine if two exponents have the same term by using # as_coeff_Mul. # # Unfortunately, this isn't smart enough to consider combining into # exponents that might already be adds, so things like: # z - y y # x * x will be left alone. This is because checking every possible # combination can slow things down. # gather exponents of common bases... def _gather(c_powers): new_c_powers = [] common_b = {} # b:e for b, e in c_powers: co = e.as_coeff_Mul() common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0]) for b, d in common_b.items(): for di, li in d.items(): d[di] = Add(*li) for b, e in common_b.items(): for t, c in e.items(): new_c_powers.append((b, c*t)) return new_c_powers # in c_powers c_powers = _gather(c_powers) # and in num_exp num_exp = _gather(num_exp) # --- PART 2 --- # # o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow) # o combine collected powers (2**x * 3**x -> 6**x) # with numeric base # ................................ # now we have: # - coeff: # - c_powers: (b, e) # - num_exp: (2, e) # - pnum_rat: {(1/3, [1/3, 2/3, 1/4])} # 0 1 # x -> 1 x -> x for b, e in c_powers: if e is S.One: if b.is_Number: coeff *= b else: c_part.append(b) elif e is not S.Zero: c_part.append(Pow(b, e)) # x x x # 2 * 3 -> 6 inv_exp_dict = {} # exp:Mul(num-bases) x x # e.g. x:6 for ... * 2 * 3 * ... for b, e in num_exp: inv_exp_dict.setdefault(e, []).append(b) for e, b in inv_exp_dict.items(): inv_exp_dict[e] = Mul(*b) c_part.extend([Pow(b, e) for e, b in inv_exp_dict.iteritems() if e]) # b, e -> e' = sum(e), b # {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])} comb_e = {} for b, e in pnum_rat.iteritems(): comb_e.setdefault(Add(*e), []).append(b) del pnum_rat # process them, reducing exponents to values less than 1 # and updating coeff if necessary else adding them to # num_rat for further processing num_rat = [] for e, b in comb_e.iteritems(): b = Mul(*b) if e.q == 1: coeff *= Pow(b, e) continue if e.p > e.q: e_i, ep = divmod(e.p, e.q) coeff *= Pow(b, e_i) e = Rational(ep, e.q) num_rat.append((b, e)) del comb_e # extract gcd of bases in num_rat # 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4) pnew = defaultdict(list) i = 0 # steps through num_rat which may grow while i < len(num_rat): bi, ei = num_rat[i] grow = [] for j in range(i + 1, len(num_rat)): bj, ej = num_rat[j] g = _rgcd(bi, bj) if g is not S.One: # 4**r1*6**r2 -> 2**(r1+r2) * 2**r1 * 3**r2 # this might have a gcd with something else e = ei + ej if e.q == 1: coeff *= Pow(g, e) else: if e.p > e.q: e_i, ep = divmod(e.p, e.q) # change e in place coeff *= Pow(g, e_i) e = Rational(ep, e.q) grow.append((g, e)) # update the jth item num_rat[j] = (bj/g, ej) # update bi that we are checking with bi = bi/g if bi is S.One: break if bi is not S.One: obj = Pow(bi, ei) if obj.is_Number: coeff *= obj else: # changes like sqrt(12) -> 2*sqrt(3) for obj in Mul.make_args(obj): if obj.is_Number: coeff *= obj else: assert obj.is_Pow bi, ei = obj.args pnew[ei].append(bi) num_rat.extend(grow) i += 1 # combine bases of the new powers for e, b in pnew.iteritems(): pnew[e] = Mul(*b) # see if there is a base with matching coefficient # that the -1 can be joined with if neg1e: p = Pow(S.NegativeOne, neg1e) if p.is_Number: coeff *= p else: c, p = p.as_coeff_Mul() coeff *= c if p.is_Pow and p.base is S.NegativeOne: neg1e = p.exp for e, b in pnew.iteritems(): if e == neg1e and b.is_positive: pnew[e] = -b break else: c_part.append(p) # add all the pnew powers c_part.extend([Pow(b, e) for e, b in pnew.iteritems()]) # oo, -oo if (coeff is S.Infinity) or (coeff is S.NegativeInfinity): def _handle_for_oo(c_part, coeff_sign): new_c_part = [] for t in c_part: if t.is_positive: continue if t.is_negative: coeff_sign *= -1 continue new_c_part.append(t) return new_c_part, coeff_sign c_part, coeff_sign = _handle_for_oo(c_part, 1) nc_part, coeff_sign = _handle_for_oo(nc_part, coeff_sign) coeff *= coeff_sign # zoo if coeff is S.ComplexInfinity: # zoo might be # unbounded_real + bounded_im # bounded_real + unbounded_im # unbounded_real + unbounded_im # and non-zero real or imaginary will not change that status. c_part = [c for c in c_part if not (c.is_nonzero and c.is_real is not None)] nc_part = [c for c in nc_part if not (c.is_nonzero and c.is_real is not None)] # 0 elif coeff is S.Zero: # we know for sure the result will be 0 return [coeff], [], order_symbols # order commutative part canonically c_part.sort(key=cls._args_sortkey) # current code expects coeff to be always in slot-0 if coeff is not S.One: c_part.insert(0, coeff) # we are done if len(c_part) == 2 and c_part[0].is_Number and c_part[1].is_Add: # 2*(1+a) -> 2 + 2 * a coeff = c_part[0] c_part = [Add(*[coeff*f for f in c_part[1].args])] return c_part, nc_part, order_symbols
def _eval_power(base, exp): """ Tries to do some simplifications on base ** exp, where base is an instance of Integer Returns None if no further simplifications can be done When exponent is a fraction (so we have for example a square root), we try to find the simplest possible representation, so that - 4**Rational(1,2) becomes 2 - (-4)**Rational(1,2) becomes 2*I We will """ if exp is S.NaN: return S.NaN if base is S.One: return S.One if base is S.NegativeOne: return if exp is S.Infinity: if base.p > S.One: return S.Infinity if base.p == -1: return S.NaN # cases 0, 1 are done in their respective classes return S.Infinity + S.ImaginaryUnit * S.Infinity if not isinstance(exp, Number): # simplify when exp is even # (-2) ** k --> 2 ** k c,t = base.as_coeff_terms() if exp.is_even and isinstance(c, Number) and c < 0: return (-c * Mul(*t)) ** exp if not isinstance(exp, Rational): return if exp is S.Half and base < 0: # we extract I for this special case since everyone is doing so return S.ImaginaryUnit * Pow(-base, exp) if exp < 0: # invert base and change sign on exponent if base < 0: return -(S.NegativeOne) ** ((exp.p % exp.q) / S(exp.q)) * Rational(1, -base) ** (-exp) else: return Rational(1, base.p) ** (-exp) # see if base is a perfect root, sqrt(4) --> 2 x, xexact = integer_nthroot(abs(base.p), exp.q) if xexact: # if it's a perfect root we've finished result = Integer(x ** abs(exp.p)) if exp < 0: result = 1/result if base < 0: result *= (-1)**exp return result # The following is an algorithm where we collect perfect roots # from the factors of base if base > 4294967296: # Prevent from factorizing too big integers return None dict = base.factors() out_int = 1 sqr_int = 1 sqr_gcd = 0 sqr_dict = {} for prime,exponent in dict.iteritems(): exponent *= exp.p div_e = exponent // exp.q div_m = exponent % exp.q if div_e > 0: out_int *= prime**div_e if div_m > 0: sqr_dict[prime] = div_m for p,ex in sqr_dict.iteritems(): if sqr_gcd == 0: sqr_gcd = ex else: sqr_gcd = igcd(sqr_gcd, ex) for k,v in sqr_dict.iteritems(): sqr_int *= k**(v // sqr_gcd) if sqr_int == base.p and out_int == 1: result = None else: result = out_int * Pow(sqr_int , Rational(sqr_gcd, exp.q)) return result
def _eval_subs(self, old, new): from sympy import sign, multiplicity from sympy.simplify.simplify import powdenest, fraction if not old.is_Mul: return None if old.args[0] == -1: return self._subs(-old, -new) def base_exp(a): # if I and -1 are in a Mul, they get both end up with # a -1 base (see issue 3322); all we want here are the # true Pow or exp separated into base and exponent if a.is_Pow or a.func is C.exp: return a.as_base_exp() return a, S.One def breakup(eq): """break up powers of eq when treated as a Mul: b**(Rational*e) -> b**e, Rational commutatives come back as a dictionary {b**e: Rational} noncommutatives come back as a list [(b**e, Rational)] """ (c, nc) = (defaultdict(int), list()) for a in Mul.make_args(eq): a = powdenest(a) (b, e) = base_exp(a) if e is not S.One: (co, _) = e.as_coeff_mul() b = Pow(b, e/co) e = co if a.is_commutative: c[b] += e else: nc.append([b, e]) return (c, nc) def rejoin(b, co): """ Put rational back with exponent; in general this is not ok, but since we took it from the exponent for analysis, it's ok to put it back. """ (b, e) = base_exp(b) return Pow(b, e*co) def ndiv(a, b): """if b divides a in an extractive way (like 1/4 divides 1/2 but not vice versa, and 2/5 does not divide 1/3) then return the integer number of times it divides, else return 0. """ if not b.q % a.q or not a.q % b.q: return int(a/b) return 0 # give Muls in the denominator a chance to be changed (see issue 2552) # rv will be the default return value rv = None n, d = fraction(self) if d is not S.One: self2 = n._subs(old, new)/d._subs(old, new) if not self2.is_Mul: return self2._subs(old, new) if self2 != self: self = rv = self2 # Now continue with regular substitution. # handle the leading coefficient and use it to decide if anything # should even be started; we always know where to find the Rational # so it's a quick test co_self = self.args[0] co_old = old.args[0] co_xmul = None if co_old.is_Rational and co_self.is_Rational: # if coeffs are the same there will be no updating to do # below after breakup() step; so skip (and keep co_xmul=None) if co_old != co_self: co_xmul = co_self.extract_multiplicatively(co_old) elif co_old.is_Rational: return rv # break self and old into factors (c, nc) = breakup(self) (old_c, old_nc) = breakup(old) # update the coefficients if we had an extraction # e.g. if co_self were 2*(3/35*x)**2 and co_old = 3/5 # then co_self in c is replaced by (3/5)**2 and co_residual # is 2*(1/7)**2 if co_xmul and co_xmul.is_Rational: n_old, d_old = co_old.as_numer_denom() n_self, d_self = co_self.as_numer_denom() def _multiplicity(p, n): p = abs(p) if p is S.One: return S.Infinity return multiplicity(p, abs(n)) mult = S(min(_multiplicity(n_old, n_self), _multiplicity(d_old, d_self))) c.pop(co_self) c[co_old] = mult co_residual = co_self/co_old**mult else: co_residual = 1 # do quick tests to see if we can't succeed ok = True if len(old_nc) > len(nc): # more non-commutative terms ok = False elif len(old_c) > len(c): # more commutative terms ok = False elif set(i[0] for i in old_nc).difference(set(i[0] for i in nc)): # unmatched non-commutative bases ok = False elif set(old_c).difference(set(c)): # unmatched commutative terms ok = False elif any(sign(c[b]) != sign(old_c[b]) for b in old_c): # differences in sign ok = False if not ok: return rv if not old_c: cdid = None else: rat = [] for (b, old_e) in old_c.items(): c_e = c[b] rat.append(ndiv(c_e, old_e)) if not rat[-1]: return rv cdid = min(rat) if not old_nc: ncdid = None for i in range(len(nc)): nc[i] = rejoin(*nc[i]) else: ncdid = 0 # number of nc replacements we did take = len(old_nc) # how much to look at each time limit = cdid or S.Infinity # max number that we can take failed = [] # failed terms will need subs if other terms pass i = 0 while limit and i + take <= len(nc): hit = False # the bases must be equivalent in succession, and # the powers must be extractively compatible on the # first and last factor but equal inbetween. rat = [] for j in range(take): if nc[i + j][0] != old_nc[j][0]: break elif j == 0: rat.append(ndiv(nc[i + j][1], old_nc[j][1])) elif j == take - 1: rat.append(ndiv(nc[i + j][1], old_nc[j][1])) elif nc[i + j][1] != old_nc[j][1]: break else: rat.append(1) j += 1 else: ndo = min(rat) if ndo: if take == 1: if cdid: ndo = min(cdid, ndo) nc[i] = Pow(new, ndo)*rejoin(nc[i][0], nc[i][1] - ndo*old_nc[0][1]) else: ndo = 1 # the left residual l = rejoin(nc[i][0], nc[i][1] - ndo* old_nc[0][1]) # eliminate all middle terms mid = new # the right residual (which may be the same as the middle if take == 2) ir = i + take - 1 r = (nc[ir][0], nc[ir][1] - ndo* old_nc[-1][1]) if r[1]: if i + take < len(nc): nc[i:i + take] = [l*mid, r] else: r = rejoin(*r) nc[i:i + take] = [l*mid*r] else: # there was nothing left on the right nc[i:i + take] = [l*mid] limit -= ndo ncdid += ndo hit = True if not hit: # do the subs on this failing factor failed.append(i) i += 1 else: if not ncdid: return rv # although we didn't fail, certain nc terms may have # failed so we rebuild them after attempting a partial # subs on them failed.extend(range(i, len(nc))) for i in failed: nc[i] = rejoin(*nc[i]).subs(old, new) # rebuild the expression if cdid is None: do = ncdid elif ncdid is None: do = cdid else: do = min(ncdid, cdid) margs = [] for b in c: if b in old_c: # calculate the new exponent e = c[b] - old_c[b]*do margs.append(rejoin(b, e)) else: margs.append(rejoin(b.subs(old, new), c[b])) if cdid and not ncdid: # in case we are replacing commutative with non-commutative, # we want the new term to come at the front just like the # rest of this routine margs = [Pow(new, cdid)] + margs return co_residual*Mul(*margs)*Mul(*nc)
def flatten(cls, seq): # apply associativity, separate commutative part of seq c_part = [] # out: commutative factors nc_part = [] # out: non-commutative factors nc_seq = [] coeff = S.One # standalone term # e.g. 3 * ... c_powers = [] # (base,exp) n # e.g. (x,n) for x num_exp = [] # (num-base, exp) y # e.g. (3, y) for ... * 3 * ... neg1e = 0 # exponent on -1 extracted from Number-based Pow pnum_rat = {} # (num-base, Rat-exp) 1/2 # e.g. (3, 1/2) for ... * 3 * ... order_symbols = None # --- PART 1 --- # # "collect powers and coeff": # # o coeff # o c_powers # o num_exp # o neg1e # o pnum_rat # # NOTE: this is optimized for all-objects-are-commutative case for o in seq: # O(x) if o.is_Order: o, order_symbols = o.as_expr_variables(order_symbols) # Mul([...]) if o.is_Mul: if o.is_commutative: seq.extend(o.args) # XXX zerocopy? else: # NCMul can have commutative parts as well for q in o.args: if q.is_commutative: seq.append(q) else: nc_seq.append(q) # append non-commutative marker, so we don't forget to # process scheduled non-commutative objects seq.append(NC_Marker) continue # 3 elif o.is_Number: if o is S.NaN or coeff is S.ComplexInfinity and o is S.Zero: # we know for sure the result will be nan return [S.NaN], [], None elif coeff.is_Number: # it could be zoo coeff *= o if coeff is S.NaN: # we know for sure the result will be nan return [S.NaN], [], None continue elif o is S.ComplexInfinity: if not coeff or coeff is S.ComplexInfinity: # we know for sure the result will be nan return [S.NaN], [], None coeff = S.ComplexInfinity continue elif o.is_commutative: # e # o = b b, e = o.as_base_exp() # y # 3 if o.is_Pow and b.is_Number: # get all the factors with numeric base so they can be # combined below, but don't combine negatives unless # the exponent is an integer if e.is_Rational: if e.is_Integer: coeff *= Pow(b, e) # it is an unevaluated power continue elif e.is_negative: # also a sign of an unevaluated power seq.append(Pow(b, e)) continue elif b.is_negative: neg1e += e b = -b if b is not S.One: pnum_rat.setdefault(b, []).append(e) continue elif b.is_positive or e.is_integer: num_exp.append((b, e)) continue c_powers.append((b, e)) # NON-COMMUTATIVE # TODO: Make non-commutative exponents not combine automatically else: if o is not NC_Marker: nc_seq.append(o) # process nc_seq (if any) while nc_seq: o = nc_seq.pop(0) if not nc_part: nc_part.append(o) continue # b c b+c # try to combine last terms: a * a -> a o1 = nc_part.pop() b1, e1 = o1.as_base_exp() b2, e2 = o.as_base_exp() new_exp = e1 + e2 # Only allow powers to combine if the new exponent is # not an Add. This allow things like a**2*b**3 == a**5 # if a.is_commutative == False, but prohibits # a**x*a**y and x**a*x**b from combining (x,y commute). if b1 == b2 and (not new_exp.is_Add): o12 = b1**new_exp # now o12 could be a commutative object if o12.is_commutative: seq.append(o12) continue else: nc_seq.insert(0, o12) else: nc_part.append(o1) nc_part.append(o) # We do want a combined exponent if it would not be an Add, such as # y 2y 3y # x * x -> x # We determine this if two exponents have the same term in as_coeff_mul # # Unfortunately, this isn't smart enough to consider combining into # exponents that might already be adds, so things like: # z - y y # x * x will be left alone. This is because checking every possible # combination can slow things down. # gather exponents of common bases... # in c_powers new_c_powers = [] common_b = {} # b:e for b, e in c_powers: co = e.as_coeff_mul() common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0]) for b, d in common_b.items(): for di, li in d.items(): d[di] = Add(*li) for b, e in common_b.items(): for t, c in e.items(): new_c_powers.append((b, c * Mul(*t))) c_powers = new_c_powers # and in num_exp new_num_exp = [] common_b = {} # b:e for b, e in num_exp: co = e.as_coeff_mul() common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0]) for b, d in common_b.items(): for di, li in d.items(): d[di] = Add(*li) for b, e in common_b.items(): for t, c in e.items(): new_num_exp.append((b, c * Mul(*t))) num_exp = new_num_exp # --- PART 2 --- # # o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow) # o combine collected powers (2**x * 3**x -> 6**x) # with numeric base # ................................ # now we have: # - coeff: # - c_powers: (b, e) # - num_exp: (2, e) # - pnum_rat: {(1/3, [1/3, 2/3, 1/4])} # 0 1 # x -> 1 x -> x for b, e in c_powers: if e is S.One: if b.is_Number: coeff *= b else: c_part.append(b) elif not e is S.Zero: c_part.append(Pow(b, e)) # x x x # 2 * 3 -> 6 inv_exp_dict = {} # exp:Mul(num-bases) x x # e.g. x:6 for ... * 2 * 3 * ... for b, e in num_exp: inv_exp_dict.setdefault(e, []).append(b) for e, b in inv_exp_dict.items(): inv_exp_dict[e] = Mul(*b) c_part.extend([Pow(b, e) for e, b in inv_exp_dict.iteritems() if e]) # b, e -> e, b # {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])} comb_e = {} for b, e in pnum_rat.iteritems(): comb_e.setdefault(Add(*e), []).append(b) del pnum_rat # process them, reducing exponents to values less than 1 # and updating coeff if necessary else adding them to # num_rat for further processing num_rat = [] for e, b in comb_e.iteritems(): b = Mul(*b) if e.q == 1: coeff *= Pow(b, e) continue if e.p > e.q: e_i, ep = divmod(e.p, e.q) coeff *= Pow(b, e_i) e = Rational(ep, e.q) num_rat.append((b, e)) del comb_e # extract gcd of bases in num_rat # 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4) pnew = {} i = 0 # steps through num_rat which may grow while i < len(num_rat): bi, ei = num_rat[i] grow = [] for j in range(i + 1, len(num_rat)): bj, ej = num_rat[j] g = igcd(bi, bj) if g != 1: # 4**r1*6**r2 -> 2**(r1+r2) * 2**r1 * 3**r2 # this might have a gcd with something else e = ei + ej if e.q == 1: coeff *= Pow(g, e) else: if e.p > e.q: e_i, ep = divmod(e.p, e.q) # change e in place coeff *= Pow(g, e_i) e = Rational(ep, e.q) grow.append((g, e)) # update the jth item num_rat[j] = (bj // g, ej) # update bi that we are checking with bi = bi // g if bi is S.One: break if bi is not S.One: obj = Pow(bi, ei) if obj.is_Number: coeff *= obj else: if obj.is_Mul: # 12**(1/2) -> 2*sqrt(3) c, obj = obj.args # expecting only 2 args coeff *= c assert obj.is_Pow bi, ei = obj.args pnew.setdefault(ei, []).append(bi) num_rat.extend(grow) i += 1 # combine bases of the new powers for e, b in pnew.iteritems(): pnew[e] = Mul(*b) # see if there is a base with matching coefficient # that the -1 can be joined with if neg1e: p = Pow(S.NegativeOne, neg1e) if p.is_Number: coeff *= p else: if p.is_Mul: c, p = p.args coeff *= c assert p.is_Pow and p.base is S.NegativeOne neg1e = p.args[1] for e, b in pnew.iteritems(): if e == neg1e and b.is_positive: pnew[e] = -b break else: c_part.append(p) # add all the pnew powers c_part.extend([Pow(b, e) for e, b in pnew.iteritems()]) # oo, -oo if (coeff is S.Infinity) or (coeff is S.NegativeInfinity): new_c_part = [] coeff_sign = 1 for t in c_part: if t.is_positive: continue if t.is_negative: coeff_sign *= -1 continue new_c_part.append(t) c_part = new_c_part new_nc_part = [] for t in nc_part: if t.is_positive: continue if t.is_negative: coeff_sign *= -1 continue new_nc_part.append(t) nc_part = new_nc_part coeff *= coeff_sign # zoo if coeff is S.ComplexInfinity: # zoo might be # unbounded_real + bounded_im # bounded_real + unbounded_im # unbounded_real + unbounded_im # and non-zero real or imaginary will not change that status. c_part = [ c for c in c_part if not (c.is_nonzero and c.is_real is not None) ] nc_part = [ c for c in nc_part if not (c.is_nonzero and c.is_real is not None) ] # 0 elif coeff is S.Zero: # we know for sure the result will be 0 return [coeff], [], order_symbols # order commutative part canonically c_part.sort(key=cmp_to_key(Basic.compare)) # current code expects coeff to be always in slot-0 if coeff is not S.One: c_part.insert(0, coeff) # we are done if len(c_part) == 2 and c_part[0].is_Number and c_part[1].is_Add: # 2*(1+a) -> 2 + 2 * a coeff = c_part[0] c_part = [Add(*[coeff * f for f in c_part[1].args])] return c_part, nc_part, order_symbols
def __rdiv__(self, other): return Mul(other, Pow(self, S.NegativeOne))
def __div__(self, other): return Mul(self, Pow(other, S.NegativeOne))
def flatten(cls, seq): # apply associativity, separate commutative part of seq c_part = [] # out: commutative factors nc_part = [] # out: non-commutative factors nc_seq = [] coeff = S.One # standalone term # e.g. 3 * ... c_powers = [] # (base,exp) n # e.g. (x,n) for x num_exp = [] # (num-base, exp) y # e.g. (3, y) for ... * 3 * ... order_symbols = None # --- PART 1 --- # # "collect powers and coeff": # # o coeff # o c_powers # o num_exp # # NOTE: this is optimized for all-objects-are-commutative case for o in seq: # O(x) if o.is_Order: o, order_symbols = o.as_expr_symbols(order_symbols) # Mul([...]) if o.is_Mul: if o.is_commutative: seq.extend(o.args) # XXX zerocopy? else: # NCMul can have commutative parts as well for q in o.args: if q.is_commutative: seq.append(q) else: nc_seq.append(q) # append non-commutative marker, so we don't forget to # process scheduled non-commutative objects seq.append(NC_Marker) continue # 3 elif o.is_Number: coeff *= o continue elif o.is_commutative: # e # o = b b, e = o.as_base_exp() # y # 3 if o.is_Pow and b.is_Number: # get all the factors with numeric base so they can be # combined below num_exp.append((b, e)) continue # n n n # (-3 + y) -> (-1) * (3 - y) if b.is_Add and e.is_Number: #found factor (x+y)**number; split off initial coefficient c, t = b.as_coeff_terms() #last time I checked, Add.as_coeff_terms returns One or NegativeOne #but this might change if c.is_negative and not e.is_integer: # extracting root from negative number: ignore sign if c is not S.NegativeOne: # make c positive (probably never occurs) coeff *= (-c)**e assert len(t) == 1, ` t ` b = -t[0] #else: ignoring sign from NegativeOne: nothing to do! elif c is not S.One: coeff *= c**e assert len(t) == 1, ` t ` b = t[0] #else: c is One, so pass c_powers.append((b, e)) # NON-COMMUTATIVE # TODO: Make non-commutative exponents not combine automatically else: if o is not NC_Marker: nc_seq.append(o) # process nc_seq (if any) while nc_seq: o = nc_seq.pop(0) if not nc_part: nc_part.append(o) continue # b c b+c # try to combine last terms: a * a -> a o1 = nc_part.pop() b1, e1 = o1.as_base_exp() b2, e2 = o.as_base_exp() if b1 == b2: o12 = b1**(e1 + e2) # now o12 could be a commutative object if o12.is_commutative: seq.append(o12) continue else: nc_seq.insert(0, o12) else: nc_part.append(o1) nc_part.append(o) # We do want a combined exponent if it would not be an Add, such as # y 2y 3y # x * x -> x # We determine this if two exponents have the same term in as_coeff_terms # # Unfortunately, this isn't smart enough to consider combining into # exponents that might already be adds, so thing like: # z - y y # x * x will be left alone. This is because checking every possible # combination can slow things down. new_c_powers = [] common_b = {} # b:e # First gather exponents of common bases for b, e in c_powers: co = e.as_coeff_terms() if b in common_b: if co[1] in common_b[b]: common_b[b][co[1]] += co[0] else: common_b[b][co[1]] = co[0] else: common_b[b] = {co[1]: co[0]} for b, e, in common_b.items(): for t, c in e.items(): new_c_powers.append((b, c * Mul(*t))) c_powers = new_c_powers # And the same for numeric bases new_num_exp = [] common_b = {} # b:e for b, e in num_exp: co = e.as_coeff_terms() if b in common_b: if co[1] in common_b[b]: common_b[b][co[1]] += co[0] else: common_b[b][co[1]] = co[0] else: common_b[b] = {co[1]: co[0]} for b, e, in common_b.items(): for t, c in e.items(): new_num_exp.append((b, c * Mul(*t))) num_exp = new_num_exp # --- PART 2 --- # # o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow) # o combine collected powers (2**x * 3**x -> 6**x) # with numeric base # ................................ # now we have: # - coeff: # - c_powers: (b, e) # - num_exp: (2, e) # 0 1 # x -> 1 x -> x for b, e in c_powers: if e is S.Zero: continue if e is S.One: if b.is_Number: coeff *= b else: c_part.append(b) elif e.is_Integer and b.is_Number: coeff *= b**e else: c_part.append(Pow(b, e)) # x x x # 2 * 3 -> 6 inv_exp_dict = {} # exp:Mul(num-bases) x x # e.g. x:6 for ... * 2 * 3 * ... for b, e in num_exp: if e in inv_exp_dict: inv_exp_dict[e] *= b else: inv_exp_dict[e] = b for e, b in inv_exp_dict.items(): if e is S.Zero: continue if e is S.One: if b.is_Number: coeff *= b else: c_part.append(b) elif e.is_Integer and b.is_Number: coeff *= b**e else: obj = b**e if obj.is_Number: coeff *= obj else: c_part.append(obj) # oo, -oo if (coeff is S.Infinity) or (coeff is S.NegativeInfinity): new_c_part = [] for t in c_part: if t.is_positive: continue if t.is_negative: coeff = -coeff continue new_c_part.append(t) c_part = new_c_part new_nc_part = [] for t in nc_part: if t.is_positive: continue if t.is_negative: coeff = -coeff continue new_nc_part.append(t) nc_part = new_nc_part # 0, nan elif (coeff is S.Zero) or (coeff is S.NaN): # we know for sure the result will be the same as coeff (0 or nan) return [coeff], [], order_symbols elif coeff.is_Real: if coeff == Real(0): c_part, nc_part = [coeff], [] elif coeff == Real(1): # change it to One, so it doesn't get inserted to slot0 coeff = S.One # order commutative part canonically c_part.sort(Basic.compare) # current code expects coeff to be always in slot-0 if coeff is not S.One: c_part.insert(0, coeff) # we are done if len(c_part) == 2 and c_part[0].is_Number and c_part[1].is_Add: # 2*(1+a) -> 2 + 2 * a coeff = c_part[0] c_part = [Add(*[coeff * f for f in c_part[1].args])] return c_part, nc_part, order_symbols
def _eval_power(b, e): """ Tries to do some simplifications on b ** e, where b is an instance of Integer Returns None if no further simplifications can be done When exponent is a fraction (so we have for example a square root), we try to find a simpler representation by factoring the argument up to factors of 2**15, e.g. - 4**Rational(1,2) becomes 2 - (-4)**Rational(1,2) becomes 2*I - (2**(3+7)*3**(6+7))**Rational(1,7) becomes 6*18**(3/7) Further simplification would require a special call to factorint on the argument which is not done here for sake of speed. """ from sympy import perfect_power if e is S.NaN: return S.NaN if b is S.One: return S.One if b is S.NegativeOne: return if e is S.Infinity: if b > S.One: return S.Infinity if b is S.NegativeOne: return S.NaN # cases for 0 and 1 are done in their respective classes return S.Infinity + S.ImaginaryUnit * S.Infinity if not isinstance(e, Number): # simplify when exp is even # (-2) ** k --> 2 ** k c, t = b.as_coeff_mul() if e.is_even and isinstance(c, Number) and c < 0: return (-c*Mul(*t))**e if not isinstance(e, Rational): return if e is S.Half and b < 0: # we extract I for this special case since everyone is doing so return S.ImaginaryUnit*Pow(-b, e) if e < 0: # invert base and change sign on exponent ne = -e if b < 0: if e.q != 1: return -(S.NegativeOne)**((e.p % e.q) / S(e.q)) * Rational(1, -b)**ne else: return (S.NegativeOne)**ne*Rational(1, -b)**ne else: return Rational(1, b)**ne # see if base is a perfect root, sqrt(4) --> 2 b_pos = int(abs(b)) x, xexact = integer_nthroot(b_pos, e.q) if xexact: # if it's a perfect root we've finished result = Integer(x ** abs(e.p)) if b < 0: result *= (-1)**e return result # The following is an algorithm where we collect perfect roots # from the factors of base. # if it's not an nth root, it still might be a perfect power p = perfect_power(b_pos) if p: dict = {p[0]: p[1]} else: dict = Integer(b_pos).factors(limit=2**15) # now process the dict of factors if b.is_negative: dict[-1] = 1 out_int = 1 sqr_int = 1 sqr_gcd = 0 sqr_dict = {} for prime, exponent in dict.iteritems(): exponent *= e.p div_e, div_m = divmod(exponent, e.q) if div_e > 0: out_int *= prime**div_e if div_m > 0: sqr_dict[prime] = div_m for p, ex in sqr_dict.iteritems(): if sqr_gcd == 0: sqr_gcd = ex else: sqr_gcd = igcd(sqr_gcd, ex) for k, v in sqr_dict.iteritems(): sqr_int *= k**(v//sqr_gcd) if sqr_int == b and out_int == 1: result = None else: result = out_int*Pow(sqr_int , Rational(sqr_gcd, e.q)) return result
def __pow__(self, other): return Pow(self, other)