From: Fabian Seoane <fab...@fseoane.net>
implemented factoring on negative roots, now (-8)**Rational(1,3) returns
2*(-1)**Rational(1,3). This is important, for example, in solve, so that
it doesen't return uneecesarry long roots
removed unnecesary nesting, renamed variables b -> base, e -> exp
---
sympy/core/numbers.py | 189 ++++++++++++++++++++------------------
sympy/core/power.py | 7 +-
sympy/core/tests/test_eval.py | 6 +
sympy/core/tests/test_numbers.py | 4 -
4 files changed, 109 insertions(+), 97 deletions(-)
diff --git a/sympy/core/numbers.py b/sympy/core/numbers.py
index 865bce9..73daa51 100644
--- a/sympy/core/numbers.py
+++ b/sympy/core/numbers.py
@@ -894,100 +894,107 @@ class Integer(Rational):
def _eval_is_odd(self):
return bool(self.p % 2)
- def _eval_power(b, e):
- if isinstance(e, Number):
- if e is S.NaN: return S.NaN
- if isinstance(e, Real):
- return b._eval_evalf(e._prec) ** e
- if e.is_negative:
- # (3/4)**-2 -> (4/3)**2
- ne = -e
- if ne is S.One:
- return Rational(1, b.p)
- return Rational(1, b.p) ** ne
- if e is S.Infinity:
- if b.p > 1:
- # (3)**oo -> oo
- return S.Infinity
- if b.p < -1:
- # (-3)**oo -> oo + I*oo
- return S.Infinity + S.Infinity * S.ImaginaryUnit
- return S.Zero
- if isinstance(e, Integer):
- # (4/3)**2 -> 4**2 / 3**2
- return Integer(b.p ** e.p)
- if isinstance(e, Rational):
- if b == -1: # any one has tested this ???
- # calculate the roots of -1
- if e.q.is_odd:
- return -1
- r = cos(pi/e.q) + S.ImaginaryUnit*sin(pi/e.q)
- return r**e.p
- if b >= 0:
- x, xexact = integer_nthroot(b.p, e.q)
- if xexact:
- res = Integer(x ** abs(e.p))
- if e >= 0:
- return res
- else:
- return 1/res
- else:
- if b > 2**32: #Prevent from factorizing too big
integers:
- for i in xrange(2, e.q//2 + 1): #OLD CODE
- if e.q % i == 0:
- x, xexact = integer_nthroot(b.p, i)
- if xexact:
- return Integer(x)**(e * i)
- # Try to get some part of the base out, if
exponent > 1
- if e.p > e.q:
- i = e.p // e.q
- r = e.p % e.q
- return b**i * b**Rational(r, e.q)
- return
-
-
- dict = b.factors()
-
- out_int = 1
- sqr_int = 1
- sqr_gcd = 0
-
- sqr_dict = {}
-
- for prime,exponent in dict.iteritems():
- exponent *= e.p
- div_e = exponent // e.q
- div_m = 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.p and out_int == 1:
- return None
-
- return out_int * Pow(sqr_int , Rational(sqr_gcd, e.q))
+ def _eval_power(base, exp):
+ """
+ Tries to do some simplifications on base ** exp, where base is
+ an instance of Rational
+
+ 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
+ """
+
+ MAX_INT_FACTOR = 4294967296 # Prevent from factorizing too big integers
+ if not isinstance(exp, Number):
+ c,t = base.as_coeff_terms()
+ if exp.is_even and isinstance(c, Number) and c < 0:
+ return (-c * Mul(*t)) ** exp
+ if exp is S.NaN: return S.NaN
+ if isinstance(exp, Real):
+ return base._eval_evalf(exp._prec) ** exp
+ if exp.is_negative:
+ # (3/4)**-2 -> (4/3)**2
+ ne = -exp
+ if ne is S.One:
+ return Rational(1, base.p)
+ return Rational(1, base.p) ** ne
+ if exp is S.Infinity:
+ if base.p > 1:
+ # (3)**oo -> oo
+ return S.Infinity
+ if base.p < -1:
+ # (-3)**oo -> oo + I*oo
+ return S.Infinity + S.Infinity * S.ImaginaryUnit
+ return S.Zero
+ if isinstance(exp, Integer):
+ # (4/3)**2 -> 4**2 / 3**2
+ return Integer(base.p ** exp.p)
+ if exp == S.Half and base.is_negative:
+ # sqrt(-2) -> I*sqrt(2)
+ return S.ImaginaryUnit * ((-base)**exp)
+ if isinstance(exp, Rational):
+ # 4**Rational(1,2) -> 2
+ result = None
+ x, xexact = integer_nthroot(abs(base.p), exp.q)
+ if xexact:
+ res = Integer(x ** abs(exp.p))
+ if exp >= 0:
+ result = res
else:
- if e.q == 2:
- return S.ImaginaryUnit ** e.p * (-b)**e
+ result = 1/res
+ else:
+ if base > MAX_INT_FACTOR:
+ for i in xrange(2, exp.q//2 + 1): #OLD CODE
+ if exp.q % i == 0:
+ x, xexact = integer_nthroot(abs(base.p), i)
+ if xexact:
+ result = Integer(x)**(e * i)
+ break
+ # Try to get some part of the base out, if exponent > 1
+ if exp.p > exp.q:
+ i = exp.p // exp.q
+ r = exp.p % exp.q
+ result = base**i * b**Rational(r, exp.q)
+ return
+
+ 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:
- return None
-
- c,t = b.as_coeff_terms()
- if e.is_even and isinstance(c, Number) and c < 0:
- return (-c * Mul(*t)) ** e
+ 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
+ if result is not None:
+ if base.is_positive:
+ return result
+ elif exp.q == 2:
+ return result * S.ImaginaryUnit
+ else:
+ return result * ((-1)**exp)
+ else:
+ if exp == S.Half and base.is_negative:
+ return S.ImaginaryUnit * Pow(-base, exp)
def _eval_is_prime(self):
if self.p < 0:
diff --git a/sympy/core/power.py b/sympy/core/power.py
index 893f614..aefdd4c 100644
--- a/sympy/core/power.py
+++ b/sympy/core/power.py
@@ -141,9 +141,12 @@ class Pow(Basic):
def _eval_is_integer(self):
c1 = self.base.is_integer
- if c1 is None: return
c2 = self.exp.is_integer
- if c2 is None: return
+ if c1 is None or c2 is None:
+ return None
+ if not c1:
+ if self.exp.is_nonnegative:
+ return False
if c1 and c2:
if self.exp.is_nonnegative or self.exp.is_positive:
return True
diff --git a/sympy/core/tests/test_eval.py b/sympy/core/tests/test_eval.py
index 15e5fc4..1e9eb67 100644
--- a/sympy/core/tests/test_eval.py
+++ b/sympy/core/tests/test_eval.py
@@ -34,12 +34,18 @@ def test_pow_eval():
assert sqrt(-4) == 2*I
assert sqrt( 4) == 2
assert (8)**Rational(1,3) == 2
+ assert (-8)**Rational(1,3) == 2*((-1)**Rational(1,3))
assert sqrt(-2) == I*sqrt(2)
assert (-1)**Rational(1,3) != I
assert (-10)**Rational(1,3) != I*((10)**Rational(1,3))
assert (-2)**Rational(1,4) != (2)**Rational(1,4)
+ assert 64**(Rational(1)/3) == 4
+ assert 64**(Rational(2)/3) == 16
+ assert 24*64**(-Rational(1)/2) == 3
+ assert (-27)**Rational(1,3) == 3*(-1)**Rational(1,3)
+
assert (cos(2) / tan(2))**2 == (cos(2) / tan(2))**2
@XFAIL
diff --git a/sympy/core/tests/test_numbers.py b/sympy/core/tests/test_numbers.py
index 44f4bf5..894101c 100644
--- a/sympy/core/tests/test_numbers.py
+++ b/sympy/core/tests/test_numbers.py
@@ -172,10 +172,6 @@ def test_powers():
assert integer_nthroot(c2+1, 2) == (c, False)
assert integer_nthroot(c2-1, 2) == (c-1, False)
- assert 64**(Rational(1)/3)==4
- assert 64**(Rational(2)/3)==16
- assert 24*64**(-Rational(1)/2)==3
-
assert Rational(5**3, 8**3)**Rational(4,3) == Rational(5**4, 8**4)
assert Rational(-4,7)**Rational(1,2) == I*Rational(4,7)**Rational(1,2)
--
1.6.0.2
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