def _sqrtdenest_rec(expr):
"""Helper that denests the square root of three or more surds.
It returns the denested expression; if it cannot be denested it
throws SqrtdenestStopIteration
Algorithm: expr.base is in the extension Q_m = Q(sqrt(r_1),..,sqrt(r_k));
split expr.base = a + b*sqrt(r_k), where `a` and `b` are on
Q_(m-1) = Q(sqrt(r_1),..,sqrt(r_(k-1))); then a**2 - b**2*r_k is
on Q_(m-1); denest sqrt(a**2 - b**2*r_k) and so on.
See [1], section 6.
Examples
========
>>> from diofant import sqrt
>>> from diofant.simplify.sqrtdenest import _sqrtdenest_rec
>>> _sqrtdenest_rec(sqrt(-72*sqrt(2) + 158*sqrt(5) + 498))
-sqrt(10) + sqrt(2) + 9 + 9*sqrt(5)
>>> w=-6*sqrt(55)-6*sqrt(35)-2*sqrt(22)-2*sqrt(14)+2*sqrt(77)+6*sqrt(10)+65
>>> _sqrtdenest_rec(sqrt(w))
-sqrt(11) - sqrt(7) + sqrt(2) + 3*sqrt(5)
"""
from diofant.simplify.radsimp import radsimp, rad_rationalize, split_surds
if not expr.is_Pow:
return sqrtdenest(expr)
if expr.base < 0:
return sqrt(-1) * _sqrtdenest_rec(sqrt(-expr.base))
g, a, b = split_surds(expr.base)
a = a * sqrt(g)
if a < b:
a, b = b, a
c2 = _mexpand(a**2 - b**2)
if len(c2.args) > 2:
g, a1, b1 = split_surds(c2)
a1 = a1 * sqrt(g)
if a1 < b1:
a1, b1 = b1, a1
c2_1 = _mexpand(a1**2 - b1**2)
c_1 = _sqrtdenest_rec(sqrt(c2_1))
d_1 = _sqrtdenest_rec(sqrt(a1 + c_1))
num, den = rad_rationalize(b1, d_1)
c = _mexpand(d_1 / sqrt(2) + num / (den * sqrt(2)))
else:
c = _sqrtdenest1(sqrt(c2))
if sqrt_depth(c) > 1:
raise SqrtdenestStopIteration
ac = a + c
if len(ac.args) >= len(expr.args):
if count_ops(ac) >= count_ops(expr.base):
raise SqrtdenestStopIteration
d = sqrtdenest(sqrt(ac))
if sqrt_depth(d) > 1:
raise SqrtdenestStopIteration
num, den = rad_rationalize(b, d)
r = d / sqrt(2) + num / (den * sqrt(2))
r = radsimp(r)
return _mexpand(r)
def _sqrt_symbolic_denest(a, b, r):
"""Given an expression, sqrt(a + b*sqrt(b)), return the denested
expression or None.
Algorithm:
If r = ra + rb*sqrt(rr), try replacing sqrt(rr) in ``a`` with
(y**2 - ra)/rb, and if the result is a quadratic, ca*y**2 + cb*y + cc, and
(cb + b)**2 - 4*ca*cc is 0, then sqrt(a + b*sqrt(r)) can be rewritten as
sqrt(ca*(sqrt(r) + (cb + b)/(2*ca))**2).
Examples
========
>>> from diofant.simplify.sqrtdenest import _sqrt_symbolic_denest, sqrtdenest
>>> from diofant import sqrt, Symbol
>>> from diofant.abc import x
>>> a, b, r = 16 - 2*sqrt(29), 2, -10*sqrt(29) + 55
>>> _sqrt_symbolic_denest(a, b, r)
sqrt(-2*sqrt(29) + 11) + sqrt(5)
If the expression is numeric, it will be simplified:
>>> w = sqrt(sqrt(sqrt(3) + 1) + 1) + 1 + sqrt(2)
>>> sqrtdenest(sqrt((w**2).expand()))
1 + sqrt(2) + sqrt(1 + sqrt(1 + sqrt(3)))
Otherwise, it will only be simplified if assumptions allow:
>>> w = w.subs(sqrt(3), sqrt(x + 3))
>>> sqrtdenest(sqrt((w**2).expand()))
sqrt((sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2))**2)
Notice that the argument of the sqrt is a square. If x is made positive
then the sqrt of the square is resolved:
>>> _.subs(x, Symbol('x', positive=True))
sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2)
"""
a, b, r = map(sympify, (a, b, r))
rval = _sqrt_match(r)
if not rval:
return
ra, rb, rr = rval
if rb:
y = Dummy('y', positive=True)
try:
newa = Poly(a.subs(sqrt(rr), (y**2 - ra) / rb), y)
except PolynomialError:
return
if newa.degree() == 2:
ca, cb, cc = newa.all_coeffs()
cb += b
if _mexpand(cb**2 - 4 * ca * cc).equals(0):
z = sqrt(ca * (sqrt(r) + cb / (2 * ca))**2)
if z.is_number:
z = _mexpand(Mul._from_args(z.as_content_primitive()))
return z
def _sqrtdenest1(expr, denester=True):
"""Return denested expr after denesting with simpler methods or, that
failing, using the denester."""
from diofant.simplify.simplify import radsimp
if not is_sqrt(expr):
return expr
a = expr.base
if a.is_Atom:
return expr
val = _sqrt_match(a)
if not val:
return expr
a, b, r = val
# try a quick numeric denesting
d2 = _mexpand(a**2 - b**2 * r)
if d2.is_Rational:
if d2.is_positive:
z = _sqrt_numeric_denest(a, b, r, d2)
if z is not None:
return z
else:
# fourth root case
# sqrtdenest(sqrt(3 + 2*sqrt(3))) =
# sqrt(2)*3**(1/4)/2 + sqrt(2)*3**(3/4)/2
dr2 = _mexpand(-d2 * r)
dr = sqrt(dr2)
if dr.is_Rational:
z = _sqrt_numeric_denest(_mexpand(b * r), a, r, dr2)
if z is not None:
return z / root(r, 4)
else:
z = _sqrt_symbolic_denest(a, b, r)
if z is not None:
return z
if not denester or not is_algebraic(expr):
return expr
res = sqrt_biquadratic_denest(expr, a, b, r, d2)
if res:
return res
# now call to the denester
av0 = [a, b, r, d2]
z = _denester([radsimp(expr**2)], av0, 0, sqrt_depth(expr))[0]
if av0[1] is None:
return expr
if z is not None:
if sqrt_depth(z) == sqrt_depth(
expr) and count_ops(z) > count_ops(expr):
return expr
return z
return expr
def _minpoly_cos(ex, x):
"""
Returns the minimal polynomial of ``cos(ex)``
see http://mathworld.wolfram.com/TrigonometryAngles.html
"""
from diofant import sqrt
c, a = ex.args[0].as_coeff_Mul()
if a is pi:
if c.is_rational:
if c.p == 1:
if c.q == 7:
return 8 * x**3 - 4 * x**2 - 4 * x + 1
if c.q == 9:
return 8 * x**3 - 6 * x + 1
elif c.p == 2:
q = sympify(c.q)
if q.is_prime:
s = _minpoly_sin(ex, x)
return _mexpand(s.subs({x: sqrt((1 - x) / 2)}))
# for a = pi*p/q, cos(q*a) =T_q(cos(a)) = (-1)**p
n = int(c.q)
a = dup_chebyshevt(n, ZZ)
a = [x**(n - i) * a[i] for i in range(n + 1)]
r = Add(*a) - (-1)**c.p
_, factors = factor_list(r)
res = _choose_factor(factors, x, ex)
return res
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
def is_pell_transformation_ok(eq):
"""
Test whether X*Y, X, or Y terms are present in the equation
after transforming the equation using the transformation returned
by transformation_to_pell(). If they are not present we are good.
Moreover, coefficient of X**2 should be a divisor of coefficient of
Y**2 and the constant term.
"""
A, B = transformation_to_DN(eq)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
simplified = _mexpand(Subs(eq, (x, y), (u, v)).doit())
coeff = {val: key for key, val in (t.as_independent(X, Y)
for t in simplified.args)}
for term in [X*Y, X, Y]:
if term in coeff.keys():
return False
for term in [X**2, Y**2, Integer(1)]:
if term not in coeff.keys():
coeff[term] = Integer(0)
if coeff[X**2] != 0:
return isinstance(coeff[Y**2]/coeff[X**2], Integer) and isinstance(coeff[Integer(1)]/coeff[X**2], Integer)
return True
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
terms = factor_list(eq)[1]
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
okay = True
while len(s) and okay:
solution = s.pop()
okay = False
for term in terms:
subeq = term[0]
if simplify(_mexpand(Subs(subeq, var, solution).doit())) == 0:
okay = True
break
return okay
def test_factor_nc():
x, y = symbols('x,y')
k = symbols('k', integer=True)
n, m, o = symbols('n,m,o', commutative=False)
# mul and multinomial expansion is needed
e = x * (1 + y)**2
assert _mexpand(e) == x + x * 2 * y + x * y**2
def factor_nc_test(e):
ex = _mexpand(e)
assert ex.is_Add
f = factor_nc(ex)
assert not f.is_Add and _mexpand(f) == ex
factor_nc_test(x * (1 + y))
factor_nc_test(n * (x + 1))
factor_nc_test(n * (x + m))
factor_nc_test((x + m) * n)
factor_nc_test(n * m * (x * o + n * o * m) * n)
s = Sum(x, (x, 1, 2))
factor_nc_test(x * (1 + s))
factor_nc_test(x * (1 + s) * s)
factor_nc_test(x * (1 + sin(s)))
factor_nc_test((1 + n)**2)
factor_nc_test((x + n) * (x + m) * (x + y))
factor_nc_test(x * (n * m + 1))
factor_nc_test(x * (n * m + x))
factor_nc_test(x * (x * n * m + 1))
factor_nc_test(x * n * (x * m + 1))
factor_nc_test(x * (m * n + x * n * m))
factor_nc_test(n * (1 - m) * n**2)
factor_nc_test((n + m)**2)
factor_nc_test((n - m) * (n + m)**2)
factor_nc_test((n + m)**2 * (n - m))
factor_nc_test((m - n) * (n + m)**2 * (n - m))
assert factor_nc(n * (n + n * m)) == n**2 * (1 + m)
assert factor_nc(m * (m * n + n * m * n**2)) == m * (m + n * m * n) * n
eq = m * sin(n) - sin(n) * m
assert factor_nc(eq) == eq
eq = (sin(n) + x) * (cos(n) + x)
assert factor_nc(eq.expand()) == eq
# issue sympy/sympy#6534
assert (2 * n + 2 * m).factor() == 2 * (n + m)
# issue sympy/sympy#6701
assert factor_nc(n**k + n**(k + 1)) == n**k * (1 + n)
assert factor_nc((m * n)**k + (m * n)**(k + 1)) == (1 + m * n) * (m * n)**k
# issue sympy/sympy#6918
assert factor_nc(-n * (2 * x**2 + 2 * x)) == -2 * n * x * (x + 1)
assert factor_nc(1 + Mul(Expr(), Expr(), evaluate=False)) == 1 + Expr()**2
def test_factor_nc():
x, y = symbols('x,y')
k = symbols('k', integer=True)
n, m, o = symbols('n,m,o', commutative=False)
# mul and multinomial expansion is needed
e = x*(1 + y)**2
assert _mexpand(e) == x + x*2*y + x*y**2
def factor_nc_test(e):
ex = _mexpand(e)
assert ex.is_Add
f = factor_nc(ex)
assert not f.is_Add and _mexpand(f) == ex
factor_nc_test(x*(1 + y))
factor_nc_test(n*(x + 1))
factor_nc_test(n*(x + m))
factor_nc_test((x + m)*n)
factor_nc_test(n*m*(x*o + n*o*m)*n)
s = Sum(x, (x, 1, 2))
factor_nc_test(x*(1 + s))
factor_nc_test(x*(1 + s)*s)
factor_nc_test(x*(1 + sin(s)))
factor_nc_test((1 + n)**2)
factor_nc_test((x + n)*(x + m)*(x + y))
factor_nc_test(x*(n*m + 1))
factor_nc_test(x*(n*m + x))
factor_nc_test(x*(x*n*m + 1))
factor_nc_test(x*n*(x*m + 1))
factor_nc_test(x*(m*n + x*n*m))
factor_nc_test(n*(1 - m)*n**2)
factor_nc_test((n + m)**2)
factor_nc_test((n - m)*(n + m)**2)
factor_nc_test((n + m)**2*(n - m))
factor_nc_test((m - n)*(n + m)**2*(n - m))
assert factor_nc(n*(n + n*m)) == n**2*(1 + m)
assert factor_nc(m*(m*n + n*m*n**2)) == m*(m + n*m*n)*n
eq = m*sin(n) - sin(n)*m
assert factor_nc(eq) == eq
eq = (sin(n) + x)*(cos(n) + x)
assert factor_nc(eq.expand()) == eq
# issue sympy/sympy#6534
assert (2*n + 2*m).factor() == 2*(n + m)
# issue sympy/sympy#6701
assert factor_nc(n**k + n**(k + 1)) == n**k*(1 + n)
assert factor_nc((m*n)**k + (m*n)**(k + 1)) == (1 + m*n)*(m*n)**k
# issue sympy/sympy#6918
assert factor_nc(-n*(2*x**2 + 2*x)) == -2*n*x*(x + 1)
assert factor_nc(1 + Mul(Expr(), Expr(), evaluate=False)) == 1 + Expr()**2
def nthroot(expr, n, max_len=4, prec=15):
"""
compute a real nth-root of a sum of surds
Parameters
==========
expr : sum of surds
n : integer
max_len : maximum number of surds passed as constants to ``nsimplify``
Notes
=====
First ``nsimplify`` is used to get a candidate root; if it is not a
root the minimal polynomial is computed; the answer is one of its
roots.
Examples
========
>>> from diofant.simplify.simplify import nthroot
>>> from diofant import sqrt
>>> nthroot(90 + 34*sqrt(7), 3)
sqrt(7) + 3
"""
expr = sympify(expr)
n = sympify(n)
p = expr**Rational(1, n)
if not n.is_integer:
return p
if not _is_sum_surds(expr):
return p
surds = []
coeff_muls = [x.as_coeff_Mul() for x in expr.args]
for x, y in coeff_muls:
if not x.is_rational:
return p
if y is S.One:
continue
if not (y.is_Pow and y.exp == S.Half and y.base.is_integer):
return p
surds.append(y)
surds.sort()
surds = surds[:max_len]
if expr < 0 and n % 2 == 1:
p = (-expr)**Rational(1, n)
a = nsimplify(p, constants=surds)
res = a if _mexpand(a**n) == _mexpand(-expr) else p
return -res
a = nsimplify(p, constants=surds)
if _mexpand(a) is not _mexpand(p) and _mexpand(a**n) == _mexpand(expr):
return _mexpand(a)
expr = _nthroot_solve(expr, n, prec)
if expr is None:
return p
return expr
def rad_rationalize(num, den):
"""
Rationalize num/den by removing square roots in the denominator;
num and den are sum of terms whose squares are rationals
Examples
========
>>> from diofant import sqrt
>>> from diofant.simplify.radsimp import rad_rationalize
>>> rad_rationalize(sqrt(3), 1 + sqrt(2)/3)
(-sqrt(3) + sqrt(6)/3, -7/9)
"""
if not den.is_Add:
return num, den
g, a, b = split_surds(den)
a = a * sqrt(g)
num = _mexpand((a - b) * num)
den = _mexpand(a**2 - b**2)
return rad_rationalize(num, den)
def is_normal_transformation_ok(eq):
A = transformation_to_normal(eq)
X, Y, Z = A*Matrix([x, y, z])
simplified = _mexpand(Subs(eq, (x, y, z), (X, Y, Z)).doit())
coeff = dict(reversed(t.as_independent(*[X, Y, Z])) for t in simplified.args)
for term in [X*Y, Y*Z, X*Z]:
if term in coeff.keys():
return False
return True
def _sqrtdenest0(expr):
"""Returns expr after denesting its arguments."""
if is_sqrt(expr):
n, d = expr.as_numer_denom()
if d is S.One: # n is a square root
if n.base.is_Add:
args = sorted(n.base.args, key=default_sort_key)
if len(args) > 2 and all((x**2).is_Integer for x in args):
try:
return _sqrtdenest_rec(n)
except SqrtdenestStopIteration:
pass
expr = sqrt(_mexpand(Add(*[_sqrtdenest0(x) for x in args])))
return _sqrtdenest1(expr)
else:
n, d = [_sqrtdenest0(i) for i in (n, d)]
return n / d
if isinstance(expr, Expr):
args = expr.args
if args:
return expr.func(*[_sqrtdenest0(a) for a in args])
return expr
def _nthroot_solve(p, n, prec):
"""
helper function for ``nthroot``
It denests ``p**Rational(1, n)`` using its minimal polynomial
"""
from diofant.polys.numberfields import _minimal_polynomial_sq
from diofant.solvers import solve
while n % 2 == 0:
p = sqrtdenest(sqrt(p))
n = n // 2
if n == 1:
return p
pn = p**Rational(1, n)
x = Symbol('x')
f = _minimal_polynomial_sq(p, n, x)
if f is None:
return
sols = solve(f, x)
for sol in sols:
if abs(sol - pn).n() < 1. / 10**prec:
sol = sqrtdenest(sol)
if _mexpand(sol**n) == p:
return sol
def factor_nc_test(e):
ex = _mexpand(e)
assert ex.is_Add
f = factor_nc(ex)
assert not f.is_Add and _mexpand(f) == ex
def handle(expr):
# Handle first reduces to the case
# expr = 1/d, where d is an add, or d is base**p/2.
# We do this by recursively calling handle on each piece.
from diofant.simplify.simplify import nsimplify
n, d = fraction(expr)
if expr.is_Atom or (d.is_Atom and n.is_Atom):
return expr
elif not n.is_Atom:
n = n.func(*[handle(a) for a in n.args])
return _unevaluated_Mul(n, handle(1 / d))
elif n is not S.One:
return _unevaluated_Mul(n, handle(1 / d))
elif d.is_Mul:
return _unevaluated_Mul(*[handle(1 / d) for d in d.args])
# By this step, expr is 1/d, and d is not a mul.
if not symbolic and d.free_symbols:
return expr
if ispow2(d):
d2 = sqrtdenest(sqrt(d.base))**fraction(d.exp)[0]
if d2 != d:
return handle(1 / d2)
elif d.is_Pow and (d.exp.is_integer or d.base.is_positive):
# (1/d**i) = (1/d)**i
return handle(1 / d.base)**d.exp
if not (d.is_Add or ispow2(d)):
return 1 / d.func(*[handle(a) for a in d.args])
# handle 1/d treating d as an Add (though it may not be)
keep = True # keep changes that are made
# flatten it and collect radicals after checking for special
# conditions
d = _mexpand(d)
# did it change?
if d.is_Atom:
return 1 / d
# is it a number that might be handled easily?
if d.is_number:
_d = nsimplify(d)
if _d.is_Number and _d.equals(d):
return 1 / _d
while True:
# collect similar terms
collected = defaultdict(list)
for m in Add.make_args(d): # d might have become non-Add
p2 = []
other = []
for i in Mul.make_args(m):
if ispow2(i, log2=True):
p2.append(i.base if i.exp is S.Half else i.base**(
2 * i.exp))
elif i is S.ImaginaryUnit:
p2.append(S.NegativeOne)
else:
other.append(i)
collected[tuple(ordered(p2))].append(Mul(*other))
rterms = list(ordered(list(collected.items())))
rterms = [(Mul(*i), Add(*j)) for i, j in rterms]
nrad = len(rterms) - (1 if rterms[0][0] is S.One else 0)
if nrad < 1:
break
elif nrad > max_terms:
# there may have been invalid operations leading to this point
# so don't keep changes, e.g. this expression is troublesome
# in collecting terms so as not to raise the issue of 2834:
# r = sqrt(sqrt(5) + 5)
# eq = 1/(sqrt(5)*r + 2*sqrt(5)*sqrt(-sqrt(5) + 5) + 5*r)
keep = False
break
if len(rterms) > 4:
# in general, only 4 terms can be removed with repeated squaring
# but other considerations can guide selection of radical terms
# so that radicals are removed
if all(
[x.is_Integer and (y**2).is_Rational for x, y in rterms]):
nd, d = rad_rationalize(
S.One,
Add._from_args([sqrt(x) * y for x, y in rterms]))
n *= nd
else:
# is there anything else that might be attempted?
keep = False
break
from diofant.simplify.powsimp import powsimp, powdenest
num = powsimp(_num(rterms))
n *= num
d *= num
d = powdenest(_mexpand(d), force=symbolic)
if d.is_Atom:
break
if not keep:
return expr
return _unevaluated_Mul(n, 1 / d)
def _sqrt_match(p):
"""Return [a, b, r] for p.match(a + b*sqrt(r)) where, in addition to
matching, sqrt(r) also has then maximal sqrt_depth among addends of p.
Examples
========
>>> from diofant.functions.elementary.miscellaneous import sqrt
>>> from diofant.simplify.sqrtdenest import _sqrt_match
>>> _sqrt_match(1 + sqrt(2) + sqrt(2)*sqrt(3) + 2*sqrt(1+sqrt(5)))
[1 + sqrt(2) + sqrt(6), 2, 1 + sqrt(5)]
"""
from diofant.simplify.radsimp import split_surds
p = _mexpand(p)
if p.is_Number:
res = (p, S.Zero, S.Zero)
elif p.is_Add:
pargs = sorted(p.args, key=default_sort_key)
if all((x**2).is_Rational for x in pargs):
r, b, a = split_surds(p)
res = a, b, r
return list(res)
# to make the process canonical, the argument is included in the tuple
# so when the max is selected, it will be the largest arg having a
# given depth
v = [(sqrt_depth(x), x, i) for i, x in enumerate(pargs)]
nmax = max(v, key=default_sort_key)
if nmax[0] == 0:
res = []
else:
# select r
depth, _, i = nmax
r = pargs.pop(i)
v.pop(i)
b = S.One
if r.is_Mul:
bv = []
rv = []
for x in r.args:
if sqrt_depth(x) < depth:
bv.append(x)
else:
rv.append(x)
b = Mul._from_args(bv)
r = Mul._from_args(rv)
# collect terms comtaining r
a1 = []
b1 = [b]
for x in v:
if x[0] < depth:
a1.append(x[1])
else:
x1 = x[1]
if x1 == r:
b1.append(1)
else:
if x1.is_Mul:
x1args = list(x1.args)
if r in x1args:
x1args.remove(r)
b1.append(Mul(*x1args))
else:
a1.append(x[1])
else:
a1.append(x[1])
a = Add(*a1)
b = Add(*b1)
res = (a, b, r**2)
else:
b, r = p.as_coeff_Mul()
if is_sqrt(r):
res = (S.Zero, b, r**2)
else:
res = []
return list(res)
def simplify(expr, ratio=1.7, measure=count_ops, fu=False):
"""
Simplifies the given expression.
Simplification is not a well defined term and the exact strategies
this function tries can change in the future versions of Diofant. If
your algorithm relies on "simplification" (whatever it is), try to
determine what you need exactly - is it powsimp()?, radsimp()?,
together()?, logcombine()?, or something else? And use this particular
function directly, because those are well defined and thus your algorithm
will be robust.
Nonetheless, especially for interactive use, or when you don't know
anything about the structure of the expression, simplify() tries to apply
intelligent heuristics to make the input expression "simpler". For
example:
>>> from diofant import simplify, cos, sin
>>> from diofant.abc import x, y
>>> a = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)
>>> a
(x**2 + x)/(x*sin(y)**2 + x*cos(y)**2)
>>> simplify(a)
x + 1
Note that we could have obtained the same result by using specific
simplification functions:
>>> from diofant import trigsimp, cancel
>>> trigsimp(a)
(x**2 + x)/x
>>> cancel(_)
x + 1
In some cases, applying :func:`simplify` may actually result in some more
complicated expression. The default ``ratio=1.7`` prevents more extreme
cases: if (result length)/(input length) > ratio, then input is returned
unmodified. The ``measure`` parameter lets you specify the function used
to determine how complex an expression is. The function should take a
single argument as an expression and return a number such that if
expression ``a`` is more complex than expression ``b``, then
``measure(a) > measure(b)``. The default measure function is
:func:`~diofant.core.function.count_ops`, which returns the total number of operations in the
expression.
For example, if ``ratio=1``, ``simplify`` output can't be longer
than input.
::
>>> from diofant import sqrt, simplify, count_ops, oo
>>> root = 1/(sqrt(2)+3)
Since ``simplify(root)`` would result in a slightly longer expression,
root is returned unchanged instead::
>>> simplify(root, ratio=1) == root
True
If ``ratio=oo``, simplify will be applied anyway::
>>> count_ops(simplify(root, ratio=oo)) > count_ops(root)
True
Note that the shortest expression is not necessary the simplest, so
setting ``ratio`` to 1 may not be a good idea.
Heuristically, the default value ``ratio=1.7`` seems like a reasonable
choice.
You can easily define your own measure function based on what you feel
should represent the "size" or "complexity" of the input expression. Note
that some choices, such as ``lambda expr: len(str(expr))`` may appear to be
good metrics, but have other problems (in this case, the measure function
may slow down simplify too much for very large expressions). If you don't
know what a good metric would be, the default, ``count_ops``, is a good
one.
For example:
>>> from diofant import symbols, log
>>> a, b = symbols('a b', positive=True)
>>> g = log(a) + log(b) + log(a)*log(1/b)
>>> h = simplify(g)
>>> h
log(a*b**(-log(a) + 1))
>>> count_ops(g)
8
>>> count_ops(h)
5
So you can see that ``h`` is simpler than ``g`` using the count_ops metric.
However, we may not like how ``simplify`` (in this case, using
``logcombine``) has created the ``b**(log(1/a) + 1)`` term. A simple way
to reduce this would be to give more weight to powers as operations in
``count_ops``. We can do this by using the ``visual=True`` option:
>>> print(count_ops(g, visual=True))
2*ADD + DIV + 4*LOG + MUL
>>> print(count_ops(h, visual=True))
2*LOG + MUL + POW + SUB
>>> from diofant import Symbol, S
>>> def my_measure(expr):
... POW = Symbol('POW')
... # Discourage powers by giving POW a weight of 10
... count = count_ops(expr, visual=True).subs(POW, 10)
... # Every other operation gets a weight of 1 (the default)
... count = count.replace(Symbol, type(S.One))
... return count
>>> my_measure(g)
8
>>> my_measure(h)
14
>>> 15./8 > 1.7 # 1.7 is the default ratio
True
>>> simplify(g, measure=my_measure)
-log(a)*log(b) + log(a) + log(b)
Note that because ``simplify()`` internally tries many different
simplification strategies and then compares them using the measure
function, we get a completely different result that is still different
from the input expression by doing this.
"""
expr = sympify(expr)
try:
return expr._eval_simplify(ratio=ratio, measure=measure)
except AttributeError:
pass
original_expr = expr = signsimp(expr)
from diofant.simplify.hyperexpand import hyperexpand
from diofant.functions.special.bessel import BesselBase
from diofant import Sum, Product
if not isinstance(expr, Basic) or not expr.args: # XXX: temporary hack
return expr
if not isinstance(expr, (Add, Mul, Pow, exp_polar)):
return expr.func(*[
simplify(x, ratio=ratio, measure=measure, fu=fu) for x in expr.args
])
# TODO: Apply different strategies, considering expression pattern:
# is it a purely rational function? Is there any trigonometric function?...
# See also https://github.com/sympy/sympy/pull/185.
def shorter(*choices):
'''Return the choice that has the fewest ops. In case of a tie,
the expression listed first is selected.'''
if not has_variety(choices):
return choices[0]
return min(choices, key=measure)
expr = bottom_up(expr, lambda w: w.normal())
expr = Mul(*powsimp(expr).as_content_primitive())
_e = cancel(expr)
expr1 = shorter(_e, _mexpand(_e).cancel()) # issue 6829
expr2 = shorter(together(expr, deep=True), together(expr1, deep=True))
if ratio is S.Infinity:
expr = expr2
else:
expr = shorter(expr2, expr1, expr)
if not isinstance(expr, Basic): # XXX: temporary hack
return expr
expr = factor_terms(expr, sign=False)
# hyperexpand automatically only works on hypergeometric terms
expr = hyperexpand(expr)
expr = piecewise_fold(expr)
if expr.has(BesselBase):
expr = besselsimp(expr)
if expr.has(TrigonometricFunction) and not fu or expr.has(
HyperbolicFunction):
expr = trigsimp(expr, deep=True)
if expr.has(log):
expr = shorter(expand_log(expr, deep=True), logcombine(expr))
if expr.has(CombinatorialFunction, gamma):
expr = combsimp(expr)
if expr.has(Sum):
expr = sum_simplify(expr)
if expr.has(Product):
expr = product_simplify(expr)
short = shorter(powsimp(expr, combine='exp', deep=True), powsimp(expr),
expr)
short = shorter(short, factor_terms(short),
expand_power_exp(expand_mul(short)))
if (short.has(TrigonometricFunction, HyperbolicFunction, exp_polar)
or any(a.base is S.Exp1 for a in short.atoms(Pow))):
short = exptrigsimp(short, simplify=False)
# get rid of hollow 2-arg Mul factorization
hollow_mul = Transform(
lambda x: Mul(*x.args), lambda x: x.is_Mul and len(x.args) == 2 and x.
args[0].is_Number and x.args[1].is_Add and x.is_commutative)
expr = short.xreplace(hollow_mul)
numer, denom = expr.as_numer_denom()
if denom.is_Add:
n, d = fraction(radsimp(1 / denom, symbolic=False, max_terms=1))
if n is not S.One:
expr = (numer * n).expand() / d
if expr.could_extract_minus_sign():
n, d = fraction(expr)
if d != 0:
expr = signsimp(-n / (-d))
if measure(expr) > ratio * measure(original_expr):
expr = original_expr
return expr
def diop_simplify(eq):
return _mexpand(powsimp(_mexpand(eq)))
def test_mexpand():
assert _mexpand(None) is None
assert _mexpand(1) is Integer(1)
assert _mexpand(x*(x + 1)**2) == (x*(x + 1)**2).expand()
def _denester(nested, av0, h, max_depth_level):
"""Denests a list of expressions that contain nested square roots.
Algorithm based on <http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf>.
It is assumed that all of the elements of 'nested' share the same
bottom-level radicand. (This is stated in the paper, on page 177, in
the paragraph immediately preceding the algorithm.)
When evaluating all of the arguments in parallel, the bottom-level
radicand only needs to be denested once. This means that calling
_denester with x arguments results in a recursive invocation with x+1
arguments; hence _denester has polynomial complexity.
However, if the arguments were evaluated separately, each call would
result in two recursive invocations, and the algorithm would have
exponential complexity.
This is discussed in the paper in the middle paragraph of page 179.
"""
from diofant.simplify.simplify import radsimp
if h > max_depth_level:
return None, None
if av0[1] is None:
return None, None
if (av0[0] is None
and all(n.is_Number for n in nested)): # no arguments are nested
for f in _subsets(len(nested)): # test subset 'f' of nested
p = _mexpand(Mul(*[nested[i] for i in range(len(f)) if f[i]]))
if f.count(1) > 1 and f[-1]:
p = -p
sqp = sqrt(p)
if sqp.is_Rational:
return sqp, f # got a perfect square so return its square root.
# Otherwise, return the radicand from the previous invocation.
return sqrt(nested[-1]), [0] * len(nested)
else:
R = None
if av0[0] is not None:
values = [av0[:2]]
R = av0[2]
nested2 = [av0[3], R]
av0[0] = None
else:
values = list(filter(None, [_sqrt_match(expr) for expr in nested]))
for v in values:
if v[2]: # Since if b=0, r is not defined
if R is not None:
if R != v[2]:
av0[1] = None
return None, None
else:
R = v[2]
if R is None:
# return the radicand from the previous invocation
return sqrt(nested[-1]), [0] * len(nested)
nested2 = [
_mexpand(v[0]**2) - _mexpand(R * v[1]**2) for v in values
] + [R]
d, f = _denester(nested2, av0, h + 1, max_depth_level)
if not f:
return None, None
if not any(f[i] for i in range(len(nested))):
v = values[-1]
return sqrt(v[0] + _mexpand(v[1] * d)), f
else:
p = Mul(*[nested[i] for i in range(len(nested)) if f[i]])
v = _sqrt_match(p)
if 1 in f and f.index(1) < len(nested) - 1 and f[len(nested) - 1]:
v[0] = -v[0]
v[1] = -v[1]
if not f[len(nested)]: # Solution denests with square roots
vad = _mexpand(v[0] + d)
if vad <= 0:
# return the radicand from the previous invocation.
return sqrt(nested[-1]), [0] * len(nested)
if not (sqrt_depth(vad) <= sqrt_depth(R) + 1 or
(vad**2).is_Number):
av0[1] = None
return None, None
sqvad = _sqrtdenest1(sqrt(vad), denester=False)
if not (sqrt_depth(sqvad) <= sqrt_depth(R) + 1):
av0[1] = None
return None, None
sqvad1 = radsimp(1 / sqvad)
res = _mexpand(sqvad / sqrt(2) +
(v[1] * sqrt(R) * sqvad1 / sqrt(2)))
return res, f
else: # Solution requires a fourth root
s2 = _mexpand(v[1] * R) + d
if s2 <= 0:
return sqrt(nested[-1]), [0] * len(nested)
FR, s = root(_mexpand(R), 4), sqrt(s2)
return _mexpand(s / (sqrt(2) * FR) + v[0] * FR /
(sqrt(2) * s)), f
def minimal_polynomial(ex, x=None, **args):
"""
Computes the minimal polynomial of an algebraic element.
Parameters
==========
ex : algebraic element expression
x : independent variable of the minimal polynomial
compose : boolean, optional
If ``True`` (default), the minimal polynomial of the subexpressions
of ``ex`` are computed, then the arithmetic operations on them are
performed using the resultant and factorization. Else a bottom-up
algorithm is used with ``groebner``. The default algorithm
stalls less frequently.
polys : boolean, optional
if ``True`` returns a ``Poly`` object (the default is ``False``).
domain : Domain, optional
If no ground domain is given, it will be generated automatically
from the expression.
Examples
========
>>> from diofant import minimal_polynomial, sqrt, solve, QQ
>>> from diofant.abc import x, y
>>> minimal_polynomial(sqrt(2), x)
x**2 - 2
>>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
x - sqrt(2)
>>> minimal_polynomial(sqrt(2) + sqrt(3), x)
x**4 - 10*x**2 + 1
>>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
x**3 + x + 3
>>> minimal_polynomial(sqrt(y), x)
x**2 - y
"""
from diofant.polys.polytools import degree
from diofant.polys.domains import FractionField
from diofant.core.basic import preorder_traversal
compose = args.get('compose', True)
polys = args.get('polys', False)
dom = args.get('domain', None)
ex = sympify(ex)
if ex.is_number:
# not sure if it's always needed but try it for numbers (issue 8354)
ex = _mexpand(ex, recursive=True)
for expr in preorder_traversal(ex):
if expr.is_AlgebraicNumber:
compose = False
break
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not dom:
dom = FractionField(QQ, list(
ex.free_symbols)) if ex.free_symbols else QQ
if hasattr(dom, 'symbols') and x in dom.symbols:
raise GeneratorsError(
"the variable %s is an element of the ground domain %s" % (x, dom))
if compose:
result = _minpoly_compose(ex, x, dom)
result = result.primitive()[1]
c = result.coeff(x**degree(result, x))
if c.is_negative:
result = expand_mul(-result)
return cls(result, x, field=True) if polys else result.collect(x)
if not dom.is_QQ:
raise NotImplementedError("groebner method only works for QQ")
result = _minpoly_groebner(ex, x, cls)
return cls(result, x, field=True) if polys else result.collect(x)
def factor_nc_test(e):
ex = _mexpand(e)
assert ex.is_Add
f = factor_nc(ex)
assert not f.is_Add and _mexpand(f) == ex
def _separate_sq(p):
"""
helper function for ``_minimal_polynomial_sq``
It selects a rational ``g`` such that the polynomial ``p``
consists of a sum of terms whose surds squared have gcd equal to ``g``
and a sum of terms with surds squared prime with ``g``;
then it takes the field norm to eliminate ``sqrt(g)``
See simplify.simplify.split_surds and polytools.sqf_norm.
Examples
========
>>> from diofant import sqrt
>>> from diofant.abc import x
>>> p= -x + sqrt(2) + sqrt(3) + sqrt(7)
>>> p = _separate_sq(p); p
-x**2 + 2*sqrt(3)*x + 2*sqrt(7)*x - 2*sqrt(21) - 8
>>> p = _separate_sq(p); p
-x**4 + 4*sqrt(7)*x**3 - 32*x**2 + 8*sqrt(7)*x + 20
>>> p = _separate_sq(p); p
-x**8 + 48*x**6 - 536*x**4 + 1728*x**2 - 400
"""
from diofant.utilities.iterables import sift
def is_sqrt(expr):
return expr.is_Pow and expr.exp is S.Half
# p = c1*sqrt(q1) + ... + cn*sqrt(qn) -> a = [(c1, q1), .., (cn, qn)]
a = []
for y in p.args:
if not y.is_Mul:
if is_sqrt(y):
a.append((S.One, y**2))
elif y.is_Atom:
a.append((y, S.One))
elif y.is_Pow and y.exp.is_integer:
a.append((y, S.One))
else:
raise NotImplementedError
continue
sifted = sift(y.args, is_sqrt)
a.append((Mul(*sifted[False]), Mul(*sifted[True])**2))
a.sort(key=lambda z: z[1])
if a[-1][1] is S.One:
# there are no surds
return p
surds = [z for y, z in a]
for i in range(len(surds)):
if surds[i] != 1:
break
g, b1, b2 = _split_gcd(*surds[i:])
a1 = []
a2 = []
for y, z in a:
if z in b1:
a1.append(y * z**S.Half)
else:
a2.append(y * z**S.Half)
p1 = Add(*a1)
p2 = Add(*a2)
p = _mexpand(p1**2) - _mexpand(p2**2)
return p
def sqrt_biquadratic_denest(expr, a, b, r, d2):
"""denest expr = sqrt(a + b*sqrt(r))
where a, b, r are linear combinations of square roots of
positive rationals on the rationals (SQRR) and r > 0, b != 0,
d2 = a**2 - b**2*r > 0
If it cannot denest it returns None.
ALGORITHM
Search for a solution A of type SQRR of the biquadratic equation
4*A**4 - 4*a*A**2 + b**2*r = 0 (1)
sqd = sqrt(a**2 - b**2*r)
Choosing the sqrt to be positive, the possible solutions are
A = sqrt(a/2 +/- sqd/2)
Since a, b, r are SQRR, then a**2 - b**2*r is a SQRR,
so if sqd can be denested, it is done by
_sqrtdenest_rec, and the result is a SQRR.
Similarly for A.
Examples of solutions (in both cases a and sqd are positive):
Example of expr with solution sqrt(a/2 + sqd/2) but not
solution sqrt(a/2 - sqd/2):
expr = sqrt(-sqrt(15) - sqrt(2)*sqrt(-sqrt(5) + 5) - sqrt(3) + 8)
a = -sqrt(15) - sqrt(3) + 8; sqd = -2*sqrt(5) - 2 + 4*sqrt(3)
Example of expr with solution sqrt(a/2 - sqd/2) but not
solution sqrt(a/2 + sqd/2):
w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3)
expr = sqrt((w**2).expand())
a = 4*sqrt(6) + 8*sqrt(2) + 47 + 28*sqrt(3)
sqd = 29 + 20*sqrt(3)
Define B = b/2*A; eq.(1) implies a = A**2 + B**2*r; then
expr**2 = a + b*sqrt(r) = (A + B*sqrt(r))**2
Examples
========
>>> from diofant import sqrt
>>> from diofant.simplify.sqrtdenest import _sqrt_match, sqrt_biquadratic_denest
>>> z = sqrt((2*sqrt(2) + 4)*sqrt(2 + sqrt(2)) + 5*sqrt(2) + 8)
>>> a, b, r = _sqrt_match(z**2)
>>> d2 = a**2 - b**2*r
>>> sqrt_biquadratic_denest(z, a, b, r, d2)
sqrt(2) + sqrt(sqrt(2) + 2) + 2
"""
from diofant.simplify.radsimp import radsimp, rad_rationalize
if r <= 0 or d2 < 0 or not b or sqrt_depth(expr.base) < 2:
return
for x in (a, b, r):
for y in x.args:
y2 = y**2
if not y2.is_Integer or not y2.is_positive:
return
sqd = _mexpand(sqrtdenest(sqrt(radsimp(d2))))
if sqrt_depth(sqd) > 1:
return
x1, x2 = [a / 2 + sqd / 2, a / 2 - sqd / 2]
# look for a solution A with depth 1
for x in (x1, x2):
A = sqrtdenest(sqrt(x))
if sqrt_depth(A) > 1:
continue
Bn, Bd = rad_rationalize(b, _mexpand(2 * A))
B = Bn / Bd
z = A + B * sqrt(r)
if z < 0:
z = -z
return _mexpand(z)
return
def test_mexpand():
assert _mexpand(None) is None
assert _mexpand(1) is Integer(1)
assert _mexpand(x * (x + 1)**2) == (x * (x + 1)**2).expand()
def diop_simplify(eq):
return _mexpand(powsimp(_mexpand(eq)))
def test_mexpand():
from diofant.abc import x
assert _mexpand(None) is None
assert _mexpand(1) is S.One
assert _mexpand(x * (x + 1)**2) == (x * (x + 1)**2).expand()
