def _sqrtdenest34(expr):
"""Helper that denests the square root of three or four surds.
Examples
========
>>> from sympy import sqrt
>>> from sympy.simplify.sqrtdenest import _sqrtdenest34
>>> _sqrtdenest34(sqrt(-72*sqrt(2) + 158*sqrt(5) + 498))
-sqrt(10) + sqrt(2) + 9 + 9*sqrt(5)
>>> _sqrtdenest34(sqrt(12 + 2*sqrt(6) + 2*sqrt(14) + 2*sqrt(21)))
sqrt(2) + sqrt(3) + sqrt(7)
References
==========
- D. J. Jeffrey and A. D. Rich, 'Symplifying Square Roots of Square Roots
by Denesting'
"""
from sympy.simplify.simplify import radsimp
if expr.base < 0:
return sqrt(-1) * _sqrtdenest34(sqrt(-expr.base))
# a should be > b so we sort; this also makes the process canonical
args = sorted(expr.base.args, reverse=True)
a = Add._from_args(args[:2])
b = Add._from_args(args[2:])
c = _sqrtdenest1(sqrt(_mexpand(a**2 - b**2)))
if sqrt_depth(c) > 1:
return expr
d = _sqrtdenest1(sqrt(a + c))
if sqrt_depth(d) > 1:
return expr
return _mexpand(radsimp((d / sqrt(2) + b / (sqrt(2) * d))))
def _sqrtdenest34(expr):
"""Helper that denests the square root of three or four surds.
Examples
========
>>> from sympy import sqrt
>>> from sympy.simplify.sqrtdenest import _sqrtdenest34
>>> _sqrtdenest34(sqrt(-72*sqrt(2) + 158*sqrt(5) + 498))
-sqrt(10) + sqrt(2) + 9 + 9*sqrt(5)
>>> _sqrtdenest34(sqrt(12 + 2*sqrt(6) + 2*sqrt(14) + 2*sqrt(21)))
sqrt(2) + sqrt(3) + sqrt(7)
References
==========
- D. J. Jeffrey and A. D. Rich, 'Symplifying Square Roots of Square Roots
by Denesting'
"""
from sympy.simplify.simplify import radsimp
if expr.base < 0:
return sqrt(-1)*_sqrtdenest34(sqrt(-expr.base))
# a should be > b so we sort; this also makes the process canonical
args = sorted(expr.base.args, reverse=True)
a = Add._from_args(args[:2])
b = Add._from_args(args[2:])
c = _sqrtdenest1(sqrt(_mexpand(a**2 - b**2)))
if sqrt_depth(c) > 1:
return expr
d = _sqrtdenest1(sqrt(a + c))
if sqrt_depth(d) > 1:
return expr
return _mexpand(radsimp((d/sqrt(2) + b/(sqrt(2)*d))))
def _convert_to_Add(eq, evaluate=True):
# Mimic Equality._eval_rewrite_as_Add.
args = eq.lhs, eq.rhs
L, R = args
if evaluate:
# allow cancellation of args
return L - R
args = Add.make_args(L) + Add.make_args(-R)
if evaluate is None:
# no cancellation, but canonical
return _unevaluated_Add(*args)
# no cancellation, not canonical
return Add._from_args(args)
def _sqrtdenest34(expr):
"""Helper that denests the square root of three or four surds.
Examples
========
>>> from sympy import sqrt
>>> from sympy.simplify.sqrtdenest import _sqrtdenest34
>>> _sqrtdenest34(sqrt(-72*sqrt(2) + 158*sqrt(5) + 498))
-sqrt(10) + sqrt(2) + 9 + 9*sqrt(5)
>>> _sqrtdenest34(sqrt(12 + 2*sqrt(6) + 2*sqrt(14) + 2*sqrt(21)))
sqrt(2) + sqrt(3) + sqrt(7)
References
==========
- D. J. Jeffrey and A. D. Rich, 'Symplifying Square Roots of Square Roots
by Denesting'
"""
from sympy.simplify.simplify import radsimp
if not is_sqrt(expr):
return expr
a_plus_b = expr.base
if not (a_plus_b.is_Add and len(a_plus_b.args) > 2):
return expr
if a_plus_b < 0:
return sqrt(-1)*_sqrtdenest34(sqrt(-a_plus_b))
args = a_plus_b.args
a = Add._from_args(args[:2])
b = Add._from_args(args[2:])
if a < b: # we want a > b; since a + b > 0 then a**2 > b**2, too
a, b = b, a
c = _sqrtdenest1(sqrt(_mexpand(a**2 - b**2)))
if sqrt_depth(c) > 1:
return expr
d = _sqrtdenest1(sqrt(a + c))
if sqrt_depth(d) > 1:
return expr
return _mexpand(radsimp((d/sqrt(2) + b/(sqrt(2)*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 sympy.functions.elementary.miscellaneous import sqrt
>>> from sympy.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)]
"""
p = _mexpand(p)
if p.is_Number:
res = (p, S.Zero, S.Zero)
elif p.is_Add:
pargs = list(p.args)
# 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)
if nmax[0] == 0:
res = []
else:
depth, _, i = nmax
r = pargs.pop(i)
a = Add._from_args(pargs)
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)
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 _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 sympy.functions.elementary.miscellaneous import sqrt
>>> from sympy.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)]
"""
p = _mexpand(p)
if p.is_Number:
res = (p, S.Zero, S.Zero)
elif p.is_Add:
pargs = list(p.args)
# 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)
if nmax[0] == 0:
res = []
else:
depth, _, i = nmax
r = pargs.pop(i)
a = Add._from_args(pargs)
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)
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 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 sympy.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))**numer(d.exp)
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 sympy.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 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 sympy.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))**numer(d.exp)
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 sympy.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)