def test_trig():
arg = FiniteSet(0, *(pi / x for x in (12, 6, 4, 3, 2, 1)))
arg += FiniteSet(*sum(((pi - x, pi + x, 2 * pi - x) for x in arg), ()))
arg += FiniteSet(*(-x for x in arg))
def emit(name, func):
for x in sorted(arg):
if func(x) != sp.zoo:
test(name, x, sp.simplify(sp.sqrtdenest(func(x))))
test(name, S.NaN, S.NaN)
def aemit(aname, func, domain):
for x, y in sorted(
(sp.simplify(sp.sqrtdenest(func(x))), x) for x in arg & domain):
test(aname, x, y)
test(aname, S.NaN, S.NaN)
emit("sin", sp.sin)
emit("cos", sp.cos)
emit("tan", sp.tan)
aemit("asin", sp.sin, Interval(-pi / 2, pi / 2))
aemit("acos", sp.cos, Interval(0, pi))
aemit("atan", sp.tan, Interval(-pi / 2, pi / 2, True, True))
for x in sorted(arg & Interval(-pi, pi, True, True)):
a = sp.simplify(sp.sqrtdenest(sp.sin(x)))
b = sp.simplify(sp.sqrtdenest(sp.cos(x)))
f = sp.sqrt(3) if \
sp.sqrt(3) == abs(sp.simplify(a/b).as_numer_denom()[0]) else 1
if b:
test("atan2", sp.simplify(a / abs(b)) * f, sp.sign(b) * f, x)
if a and a != b:
test("atan2", sp.sign(a) * f, sp.simplify(b / abs(a)) * f, x)
test("atan2", S.NaN, 1, S.NaN)
test("atan2", 1, S.NaN, S.NaN)
def test_trig():
arg = FiniteSet(0, *(pi/x for x in (12, 6, 4, 3, 2, 1)))
arg += FiniteSet(*sum(((pi-x, pi+x, 2*pi-x) for x in arg), ()))
arg += FiniteSet(*(-x for x in arg))
def emit(name, func):
for x in sorted(arg):
if func(x) != sp.zoo:
test(name, x, sp.simplify(sp.sqrtdenest(func(x))))
test(name, S.NaN, S.NaN)
def aemit(aname, func, domain):
for x, y in sorted((sp.simplify(sp.sqrtdenest(func(x))), x)
for x in arg & domain):
test(aname, x, y)
test(aname, S.NaN, S.NaN)
emit("sin", sp.sin)
emit("cos", sp.cos)
emit("tan", sp.tan)
aemit("asin", sp.sin, Interval(-pi/2, pi/2))
aemit("acos", sp.cos, Interval(0, pi))
aemit("atan", sp.tan, Interval(-pi/2, pi/2, True, True))
for x in sorted(arg & Interval(-pi, pi, True, True)):
a = sp.simplify(sp.sqrtdenest(sp.sin(x)))
b = sp.simplify(sp.sqrtdenest(sp.cos(x)))
f = sp.sqrt(3) if \
sp.sqrt(3) == abs(sp.simplify(a/b).as_numer_denom()[0]) else 1
if b:
test("atan2", sp.simplify(a/abs(b))*f, sp.sign(b)*f, x)
if a and a != b:
test("atan2", sp.sign(a)*f, sp.simplify(b/abs(a))*f, x)
test("atan2", S.NaN, 1, S.NaN)
test("atan2", 1, S.NaN, S.NaN)
def test_sqrtdenest():
d = {sqrt(5 + 2 * sqrt(6)): sqrt(2) + sqrt(3),
sqrt(sqrt(2)): sqrt(sqrt(2)),
sqrt(5+sqrt(7)): sqrt(5+sqrt(7)),
sqrt(3+sqrt(5+2*sqrt(7))):
sqrt(6+3*sqrt(7))/(sqrt(2)*(5+2*sqrt(7))**Rational(1,4)) +
3*(5+2*sqrt(7))**Rational(1,4)/(sqrt(2)*sqrt(6+3*sqrt(7))),
sqrt(3+2*sqrt(3)): 3**Rational(1,4)/sqrt(2)+3/(sqrt(2)*3**Rational(1,4))}
for i in d:
assert sqrtdenest(i) == d[i] or denester([i])[0] == d[i]
# this test caused a pattern recognition failure in sqrtdenest
# nest = sqrt(2) + sqrt(5) - sqrt(7)
nest = symbols('nest')
x0, x1, x2, x3, x4, x5, x6 = symbols('x:7')
l = sqrt(2) + sqrt(5)
r = sqrt(7) + nest
s = (l**2 - r**2).expand() + nest**2 # == nest**2
ok = solve(nest**4 - s**2, nest)[1] # this will change if results order changes
assert abs((l - r).subs(nest, ok).n()) < 1e-12
x0 = sqrt(3)
x2 = root(45*I*x0 - 28, 3)
x3 = 19/x2
x4 = x2 + x3
x5 = -x4 - 14
x6 = sqrt(-x5)
ans = -x0*x6/3 + x0*sqrt(-x4 + 28 - 6*sqrt(210)*x6/x5)/3
assert expand_mul(radsimp(ok) - ans) == 0
# issue 2554
eq = sqrt(sqrt(sqrt(2) + 2) + 2)
assert sqrtdenest(eq) == eq
def test_issue_2758():
from sympy.abc import x, y
z = sqrt(1/(4*sqrt(3) + 7) + 1)
ans = (sqrt(2) + sqrt(6))/(sqrt(3) + 2)
assert sqrtdenest(z) == ans
assert sqrtdenest(1 + z) == 1 + ans
assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == Integral(1 + ans, (x, 1, 2))
assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)
def test_issue_2758():
from sympy.abc import x, y
z = sqrt(1 / (4 * sqrt(3) + 7) + 1)
ans = (sqrt(2) + sqrt(6)) / (sqrt(3) + 2)
assert sqrtdenest(z) == ans
assert sqrtdenest(1 + z) == 1 + ans
assert sqrtdenest(Integral(z + 1,
(x, 1, 2))) == Integral(1 + ans, (x, 1, 2))
assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)
def test_sqrtdenest4():
# see Denest_en.pdf in https://github.com/sympy/sympy/issues/3192
z = sqrt(8 - r2*sqrt(5 - r5) - sqrt(3)*(1 + r5))
z1 = sqrtdenest(z)
c = sqrt(-r5 + 5)
z1 = ((-r15*c - r3*c + c + r5*c - r6 - r2 + r10 + sqrt(30))/4).expand()
assert sqrtdenest(z) == z1
z = sqrt(2*r2*sqrt(r2 + 2) + 5*r2 + 4*sqrt(r2 + 2) + 8)
assert sqrtdenest(z) == r2 + sqrt(r2 + 2) + 2
w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3)
z = sqrt((w**2).expand())
assert sqrtdenest(z) == w.expand()
def test_sqrtdenest4():
# see Denest_en.pdf in https://github.com/sympy/sympy/issues/3192
z = sqrt(8 - r2*sqrt(5 - r5) - sqrt(3)*(1 + r5))
z1 = sqrtdenest(z)
c = sqrt(-r5 + 5)
z1 = ((-r15*c - r3*c + c + r5*c - r6 - r2 + r10 + sqrt(30))/4).expand()
assert sqrtdenest(z) == z1
z = sqrt(2*r2*sqrt(r2 + 2) + 5*r2 + 4*sqrt(r2 + 2) + 8)
assert sqrtdenest(z) == r2 + sqrt(r2 + 2) + 2
w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3)
z = sqrt((w**2).expand())
assert sqrtdenest(z) == w.expand()
def test_sqrtdenest4():
# see Denest_en.pdf in http://code.google.com/p/sympy/issues/detail?id=93
z = sqrt(8 - r2*sqrt(5 - r5) - sqrt(3)*(1 + r5))
z1 = sqrtdenest(z)
c = sqrt(-r5 + 5)
z1 = ((-r15*c - r3*c + c + r5*c - r6 - r2 + r10 + sqrt(30))/4).expand()
assert sqrtdenest(z) == z1
z= sqrt(2*r2*sqrt(r2 + 2) + 5*r2 + 4*sqrt(r2 + 2) + 8)
assert sqrtdenest(z) == r2 + sqrt(r2 + 2) + 2
w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3)
z = sqrt((w**2).expand())
assert sqrtdenest(z) == w.expand()
def test_sqrtdenest4():
# see Denest_en.pdf in http://code.google.com/p/sympy/issues/detail?id=93
z = sqrt(8 - r2*sqrt(5 - r5) - sqrt(3)*(1 + r5))
z1 = sqrtdenest(z)
c = sqrt(-r5 + 5)
z1 = ((-r15*c - r3*c + c + r5*c - r6 - r2 + r10 + sqrt(30))/4).expand()
assert sqrtdenest(z) == z1
z= sqrt(2*r2*sqrt(r2 + 2) + 5*r2 + 4*sqrt(r2 + 2) + 8)
assert sqrtdenest(z) == r2 + sqrt(r2 + 2) + 2
w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3)
z = sqrt((w**2).expand())
assert sqrtdenest(z) == w.expand()
def test_sqrtdenest3_slow():
# Slow because of the equals, not the sqrtdenest
# Using == does not work as 7*(sqrt(-2*r29 + 11) + r5) is expanded
# automatically
n = sqrt(2 * r6 / 7 + 2 * r7 / 7 + 2 * sqrt(42) / 7 + 2)
d = sqrt(16 - 2 * r29 + 2 * sqrt(55 - 10 * r29))
assert sqrtdenest(n / d).equals(r7 * (1 + r6 + r7) /
(7 * (sqrt(-2 * r29 + 11) + r5)))
def test_sqrtdenest_fail():
# see Denest_en.pdf in http://code.google.com/p/sympy/issues/detail?id=93
# z = 4*sin(pi/60)
z = sqrt(8 - r2 * sqrt(5 - r5) - sqrt(3) * (1 + r5))
# z can be denested to an expression with depth 2; here are two forms
# (-sqrt(6) - r2 + r10 + sqrt(30))/4 - sqrt(5+r5 -5*r3/2 - sqrt(15)/2)
# r2/4*(-r3 - 1 + r5 + sqrt(15)) - (r3-1)*sqrt(5+r5)/2
z1 = sqrtdenest(z)
assert sqrt_depth(z1) == 2
def test_sqrtdenest_fail():
# see Denest_en.pdf in http://code.google.com/p/sympy/issues/detail?id=93
# z = 4*sin(pi/60)
z = sqrt(8 - r2*sqrt(5 - r5) - sqrt(3)*(1 + r5))
# z can be denested to an expression with depth 2; here are two forms
# (-sqrt(6) - r2 + r10 + sqrt(30))/4 - sqrt(5+r5 -5*r3/2 - sqrt(15)/2)
# r2/4*(-r3 - 1 + r5 + sqrt(15)) - (r3-1)*sqrt(5+r5)/2
z1 = sqrtdenest(z)
assert sqrt_depth(z1) == 2
def test_issue_19914():
a = Integer(-8)
b = Integer(-1)
r = Integer(63)
d2 = a * a - b * b * r
assert _sqrt_numeric_denest(a, b, r, d2) == \
sqrt(14)*I/2 + 3*sqrt(2)*I/2
assert sqrtdenest(sqrt(-8 -
sqrt(63))) == sqrt(14) * I / 2 + 3 * sqrt(2) * I / 2
def test_sqrtdenest():
d = {sqrt(5 + 2 * sqrt(6)): sqrt(2) + sqrt(3),
sqrt(sqrt(2)): sqrt(sqrt(2)),
sqrt(5+sqrt(7)): sqrt(5+sqrt(7)),
sqrt(3+sqrt(5+2*sqrt(7))):
sqrt(6+3*sqrt(7))/(sqrt(2)*(5+2*sqrt(7))**Rational(1,4)) +
3*(5+2*sqrt(7))**Rational(1,4)/(sqrt(2)*sqrt(6+3*sqrt(7))),
sqrt(3+2*sqrt(3)): 3**Rational(1,4)/sqrt(2)+3/(sqrt(2)*3**Rational(1,4))}
for i in d:
assert sqrtdenest(i) == d[i] or denester([i])[0] == d[i]
def test_sqrtdenest():
d = {sqrt(5 + 2 * sqrt(6)): sqrt(2) + sqrt(3),
sqrt(sqrt(2)): sqrt(sqrt(2)),
sqrt(5+sqrt(7)): sqrt(5+sqrt(7)),
sqrt(3+sqrt(5+2*sqrt(7))):
sqrt(6+3*sqrt(7))/(sqrt(2)*(5+2*sqrt(7))**Rational(1,4)) +
3*(5+2*sqrt(7))**Rational(1,4)/(sqrt(2)*sqrt(6+3*sqrt(7))),
sqrt(3+2*sqrt(3)): 3**Rational(1,4)/sqrt(2)+3/(sqrt(2)*3**Rational(1,4))}
for i in d:
assert sqrtdenest(i) == d[i]
def test_sqrtdenest():
d = {sqrt(5 + 2 * r6): r2 + r3,
sqrt(5. + 2 * r6): sqrt(5. + 2 * r6),
sqrt(5. + 4*sqrt(5 + 2 * r6)): sqrt(5.0 + 4*r2 + 4*r3),
sqrt(r2): sqrt(r2),
sqrt(5 + r7): sqrt(5 + r7),
sqrt(3 + sqrt(5 + 2*r7)):
3*r2*(5 + 2*r7)**(S(1)/4)/(2*sqrt(6 + 3*r7)) +
r2*sqrt(6 + 3*r7)/(2*(5 + 2*r7)**(S(1)/4)),
sqrt(3 + 2*r3): 3**(S(3)/4)*(r6/2 + 3*r2/2)/3}
for i in d:
assert sqrtdenest(i) == d[i]
def test_sqrtdenest():
d = {sqrt(5 + 2 * r6): r2 + r3,
sqrt(5. + 2 * r6): sqrt(5. + 2 * r6),
sqrt(5. + 4*sqrt(5 + 2 * r6)): sqrt(5.0 + 4*r2 + 4*r3),
sqrt(r2): sqrt(r2),
sqrt(5 + r7): sqrt(5 + r7),
sqrt(3 + sqrt(5 + 2*r7)):
3*r2*(5 + 2*r7)**Rational(1, 4)/(2*sqrt(6 + 3*r7)) +
r2*sqrt(6 + 3*r7)/(2*(5 + 2*r7)**Rational(1, 4)),
sqrt(3 + 2*r3): 3**Rational(3, 4)*(r6/2 + 3*r2/2)/3}
for i in d:
assert sqrtdenest(i) == d[i]
def test_issue_5857():
from sympy.abc import x, y
z = sqrt(1 / (4 * r3 + 7) + 1)
ans = (r2 + r6) / (r3 + 2)
assert sqrtdenest(z) == ans
assert sqrtdenest(1 + z) == 1 + ans
assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == Integral(1 + ans, (x, 1, 2))
assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)
ans = (r2 + r6) / (r3 + 2)
assert sqrtdenest(z) == ans
assert sqrtdenest(1 + z) == 1 + ans
assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == Integral(1 + ans, (x, 1, 2))
assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)
def simplifier_racines(expression):
try:
if getattr(expression, "is_Pow", False):
return sqrtdenest(expression)
elif getattr(expression, "is_Mul", False):
return reduce(lambda x,y:x*y, [simplifier_racines(fact) for fact in expression.args], 1)
elif getattr(expression, "is_Add", False):
return reduce(lambda x,y:x+y, [simplifier_racines(term) for term in expression.args], 0)
return expression
except TypeError:
if param.debug:
print "Warning: error occured during expression denesting:", expression
return expression
def test_sqrtdenest_four_terms():
assert sqrtdenest(sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 33)) == \
-r2 + r3 + 2*r7
assert sqrtdenest(sqrt(468*r3 + 3024*r2 + 2912*r6 + 19735)) == \
9*r3 + 26 + 56*r6
assert sqrtdenest(sqrt(-28*r7 - 14*r5 + 4*sqrt(35) + 82)) == \
-7 + r5 + 2*r7
assert sqrtdenest(sqrt(6*r2/11 + 2*sqrt(22)/11 + 6*sqrt(11)/11 + 2)) == \
sqrt(11)*(r2 + 3 + sqrt(11))/11
z = sqrt(-490 * r3 - 98 * sqrt(115) - 98 * sqrt(345) - 2107)
assert sqrtdenest(z) == sqrt(-1) * (7 * r5 + 7 * sqrt(15) + 7 * sqrt(23))
z = sqrt(-4 * sqrt(14) - 2 * r6 + 4 * sqrt(21) + 34)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(-8 * r2 - 2 * r5 + 18)) == -r10 + 1 + r2 + r5
assert sqrtdenest(sqrt(8*r2 + 2*r5 - 18)) == \
sqrt(-1)*(-r10 + 1 + r2 + r5)
assert sqrtdenest(sqrt(8*r2/3 + 14*r5/3 + S(154)/9)) == \
-r10/3 + r2 + r5 + 3
assert sqrtdenest(sqrt(sqrt(2*r6 + 5) + sqrt(2*r7 + 8))) == \
sqrt(1 + r2 + r3 + r7)
assert _sqrtdenest34(sqrt(4*sqrt(15) + 8*r5 + 12*r3 + 24)) == \
1 + r3 + r5 + sqrt(15)
def test_sqrtdenest_four_terms():
assert sqrtdenest(sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 33)) == \
-r2 + r3 + 2*r7
assert sqrtdenest(sqrt(-28*r7 - 14*r5 + 4*sqrt(35) + 82)) == \
-7 + r5 + 2*r7
assert sqrtdenest(sqrt(6*r2/11 + 2*sqrt(22)/11 + 6*sqrt(11)/11 + 2)) == \
sqrt(11)*(r2 + 3 + sqrt(11))/11
assert sqrtdenest(sqrt(468*r3 + 3024*r2 + 2912*r6 + 19735)) == \
9*r3 + 26 + 56*r6
z = sqrt(-490*r3 - 98*sqrt(115) - 98*sqrt(345) - 2107)
assert sqrtdenest(z) == sqrt(-1)*(7*r5 + 7*sqrt(15) + 7*sqrt(23))
z = sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 34)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(-8*r2 - 2*r5 + 18)) == -r10 + 1 + r2 + r5
assert sqrtdenest(sqrt(8*r2 + 2*r5 - 18)) == \
sqrt(-1)*(-r10 + 1 + r2 + r5)
assert sqrtdenest(sqrt(8*r2/3 + 14*r5/3 + S(154)/9)) == \
-r10/3 + r2 + r5 + 3
assert sqrtdenest(sqrt(sqrt(2*r6 + 5) + sqrt(2*r7 + 8))) == \
sqrt(1 + r2 + r3 + r7)
assert _sqrtdenest34(sqrt(4*sqrt(15) + 8*r5 + 12*r3 + 24)) == \
1 + r3 + r5 + sqrt(15)
def test_sqrtdenest():
d = {
sqrt(5 + 2 * sqrt(6)):
sqrt(2) + sqrt(3),
sqrt(sqrt(2)):
sqrt(sqrt(2)),
sqrt(5 + sqrt(7)):
sqrt(5 + sqrt(7)),
sqrt(3 + sqrt(5 + 2 * sqrt(7))):
sqrt(6 + 3 * sqrt(7)) / (sqrt(2) *
(5 + 2 * sqrt(7))**Rational(1, 4)) + 3 *
(5 + 2 * sqrt(7))**Rational(1, 4) / (sqrt(2) * sqrt(6 + 3 * sqrt(7))),
sqrt(3 + 2 * sqrt(3)):
3**Rational(1, 4) / sqrt(2) + 3 / (sqrt(2) * 3**Rational(1, 4))
}
for i in d:
assert sqrtdenest(i) == d[i] or denester([i])[0] == d[i]
# this test caused a pattern recognition failure in sqrtdenest
# nest = sqrt(2) + sqrt(5) - sqrt(7)
nest = symbols('nest')
x0, x1, x2, x3, x4, x5, x6 = symbols('x:7')
l = sqrt(2) + sqrt(5)
r = sqrt(7) + nest
s = (l**2 - r**2).expand() + nest**2 # == nest**2
ok = solve(nest**4 - s**2,
nest)[1] # this will change if results order changes
assert abs((l - r).subs(nest, ok).n()) < 1e-12
x0 = sqrt(3)
x2 = root(45 * I * x0 - 28, 3)
x3 = 19 / x2
x4 = x2 + x3
x5 = -x4 - 14
x6 = sqrt(-x5)
ans = -x0 * x6 / 3 + x0 * sqrt(-x4 + 28 - 6 * sqrt(210) * x6 / x5) / 3
assert expand_mul(radsimp(ok) - ans) == 0
# issue 2554
eq = sqrt(sqrt(sqrt(2) + 2) + 2)
assert sqrtdenest(eq) == eq
def test_issue_2758():
from sympy.abc import x, y
z = sqrt(1/(4*r3 + 7) + 1)
ans = (r2 + r6)/(r3 + 2)
assert sqrtdenest(z) == ans
assert sqrtdenest(1 + z) == 1 + ans
assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == \
Integral(1 + ans, (x, 1, 2))
assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)
ans = (r2 + r6)/(r3 + 2)
assert sqrtdenest(z) == ans
assert sqrtdenest(1 + z) == 1 + ans
assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == \
Integral(1 + ans, (x, 1, 2))
assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)
def simplifier_racines(expression):
try:
if getattr(expression, "is_Pow", False):
return sqrtdenest(expression)
elif getattr(expression, "is_Mul", False):
return reduce(lambda x,y:x*y, [simplifier_racines(fact) for fact in expression.args], 1)
elif getattr(expression, "is_Add", False):
return reduce(lambda x,y:x+y, [simplifier_racines(term) for term in expression.args], 0)
return expression
except Exception:
# sqrtdenest() renvoie assez souvent des erreurs.
if param.debug:
print("Warning: error occured during expression denesting:", expression)
return expression
def simplifier_racines(expression):
try:
if getattr(expression, "is_Pow", False):
return sqrtdenest(expression)
elif getattr(expression, "is_Mul", False):
return reduce(
lambda x, y: x * y,
[simplifier_racines(fact) for fact in expression.args], 1)
elif getattr(expression, "is_Add", False):
return reduce(
lambda x, y: x + y,
[simplifier_racines(term) for term in expression.args], 0)
return expression
except TypeError:
if param.debug:
print "Warning: error occured during expression denesting:", expression
return expression
def test_sqrtdenest3():
z = sqrt(13 - 2*r10 + 2*r2*sqrt(-2*r10 + 11))
assert sqrtdenest(z) == -1 + r2 + r10
assert sqrtdenest(z, max_iter=1) == -1 + sqrt(2) + sqrt(10)
z = sqrt(sqrt(r2 + 2) + 2)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(-2*r10 + 4*r2*sqrt(-2*r10 + 11) + 20)) == \
sqrt(-2*r10 - 4*r2 + 8*r5 + 20)
assert sqrtdenest(sqrt((112 + 70*r2) + (46 + 34*r2)*r5)) == \
r10 + 5 + 4*r2 + 3*r5
z = sqrt(5 + sqrt(2*r6 + 5)*sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
r = sqrt(-2*r29 + 11)
assert sqrtdenest(z) == sqrt(r2*r + r3*r + r10 + r15 + 5)
def test_sqrtdenest3():
z = sqrt(13 - 2*r10 + 2*r2*sqrt(-2*r10 + 11))
assert sqrtdenest(z) == -1 + r2 + r10
assert sqrtdenest(z, max_iter=1) == -1 + sqrt(2) + sqrt(10)
n = sqrt(2*r6/7 + 2*r7/7 + 2*sqrt(42)/7 + 2)
d = sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))
assert sqrtdenest(n/d).equals(
r7*(1 + r6 + r7)/(7*(sqrt(-2*r29 + 11) + r5)))
z = sqrt(sqrt(r2 + 2) + 2)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(-2*r10 + 4*r2*sqrt(-2*r10 + 11) + 20)) == \
sqrt(-2*r10 - 4*r2 + 8*r5 + 20)
assert sqrtdenest(sqrt((112 + 70*r2) + (46 + 34*r2)*r5)) == \
r10 + 5 + 4*r2 + 3*r5
z = sqrt(5 + sqrt(2*r6 + 5)*sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
r = sqrt(-2*r29 + 11)
assert sqrtdenest(z) == sqrt(r2*r + r3*r + r10 + r15 + 5)
def test_sqrtdenest3():
z = sqrt(13 - 2*r10 + 2*r2*sqrt(-2*r10 + 11))
assert sqrtdenest(z) == -1 + r2 + r10
assert sqrtdenest(z, max_iter=1) == -1 + sqrt(2) + sqrt(10)
n = sqrt(2*r6/7 + 2*r7/7 + 2*sqrt(42)/7 + 2)
d = sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))
assert sqrtdenest(n/d).equals(
r7*(1 + r6 + r7)/(7*(sqrt(-2*r29 + 11) + r5)))
z = sqrt(sqrt(r2 + 2) + 2)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(-2*r10 + 4*r2*sqrt(-2*r10 + 11) + 20)) == \
sqrt(-2*r10 - 4*r2 + 8*r5 + 20)
assert sqrtdenest(sqrt((112 + 70*r2) + (46 + 34*r2)*r5)) == \
r10 + 5 + 4*r2 + 3*r5
z = sqrt(5 + sqrt(2*r6 + 5)*sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
r = sqrt(-2*r29 + 11)
assert sqrtdenest(z) == sqrt(r2*r + r3*r + r10 + r15 + 5)
def test_sqrt_symbolic_denest():
x = Symbol('x')
z = sqrt(((1 + sqrt(sqrt(2 + x) + 3))**2).expand())
assert sqrtdenest(z) == sqrt((1 + sqrt(sqrt(2 + x) + 3))**2)
z = sqrt(((1 + sqrt(sqrt(2 + cos(1)) + 3))**2).expand())
assert sqrtdenest(z) == 1 + sqrt(sqrt(2 + cos(1)) + 3)
z = ((1 + cos(2))**4 + 1).expand()
assert sqrtdenest(z) == z
z = sqrt(((1 + sqrt(sqrt(2 + cos(3*x)) + 3))**2 + 1).expand())
assert sqrtdenest(z) == z
c = cos(3)
c2 = c**2
assert sqrtdenest(sqrt(2*sqrt(1 + r3)*c + c2 + 1 + r3*c2)) == \
-1 - sqrt(1 + r3)*c
ra = sqrt(1 + r3)
z = sqrt(20*ra*sqrt(3 + 3*r3) + 12*r3*ra*sqrt(3 + 3*r3) + 64*r3 + 112)
assert sqrtdenest(z) == z
def test_sqrt_symbolic_denest():
x = Symbol('x')
z = sqrt(((1 + sqrt(sqrt(2 + x) + 3))**2).expand())
assert sqrtdenest(z) == sqrt((1 + sqrt(sqrt(2 + x) + 3))**2)
z = sqrt(((1 + sqrt(sqrt(2 + cos(1)) + 3))**2).expand())
assert sqrtdenest(z) == 1 + sqrt(sqrt(2 + cos(1)) + 3)
z = ((1 + cos(2))**4 + 1).expand()
assert sqrtdenest(z) == z
z = sqrt(((1 + sqrt(sqrt(2 + cos(3*x)) + 3))**2 + 1).expand())
assert sqrtdenest(z) == z
c = cos(3)
c2 = c**2
assert sqrtdenest(sqrt(2*sqrt(1 + r3)*c + c2 + 1 + r3*c2)) == \
-1 - sqrt(1 + r3)*c
ra = sqrt(1 + r3)
z = sqrt(20*ra*sqrt(3 + 3*r3) + 12*r3*ra*sqrt(3 + 3*r3) + 64*r3 + 112)
assert sqrtdenest(z) == z
def test_issue_2554():
assert sqrtdenest(sqrt(2 + sqrt(2 + sqrt(2)))) == sqrt(2 +
sqrt(2 + sqrt(2)))
def emit(name, func):
for x in sorted(arg):
if func(x) != sp.zoo:
test(name, x, sp.simplify(sp.sqrtdenest(func(x))))
test(name, S.NaN, S.NaN)
def aemit(aname, func, domain):
for x, y in sorted((sp.simplify(sp.sqrtdenest(func(x))), x)
for x in arg & domain):
test(aname, x, y)
test(aname, S.NaN, S.NaN)
def test_issue3142():
z = sqrt(-320 + 32 * sqrt(5) + 64 * sqrt(15))
assert sqrtdenest(z) == z
def test_sqrtdenest_fail():
# when this doesn't fail, put the correct value at line 55
z = sqrt(8 - r2 * sqrt(5 - r5) - 3 * (1 + r5))
assert sqrtdenest(z) != z
def test_sqrtdenest2():
assert sqrtdenest(sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))) == \
r5 + sqrt(11 - 2*r29)
e = sqrt(-r5 + sqrt(-2 * r29 + 2 * sqrt(-10 * r29 + 55) + 16))
assert sqrtdenest(e) == root(-2 * r29 + 11, 4)
r = sqrt(1 + r7)
assert sqrtdenest(sqrt(1 + r)) == sqrt(1 + r)
e = sqrt(((1 + sqrt(1 + 2 * sqrt(3 + r2 + r5)))**2).expand())
assert sqrtdenest(e) == 1 + sqrt(1 + 2 * sqrt(r2 + r5 + 3))
assert sqrtdenest(sqrt(5*r3 + 6*r2)) == \
sqrt(2)*root(3, 4) + root(3, 4)**3
assert sqrtdenest(sqrt(((1 + r5 + sqrt(1 + r3))**2).expand())) == \
1 + r5 + sqrt(1 + r3)
assert sqrtdenest(sqrt(((1 + r5 + r7 + sqrt(1 + r3))**2).expand())) == \
1 + sqrt(1 + r3) + r5 + r7
e = sqrt(((1 + cos(2) + cos(3) + sqrt(1 + r3))**2).expand())
assert sqrtdenest(e) == cos(3) + cos(2) + 1 + sqrt(1 + r3)
e = sqrt(-2 * r10 + 2 * r2 * sqrt(-2 * r10 + 11) + 14)
assert sqrtdenest(e) == sqrt(-2 * r10 - 2 * r2 + 4 * r5 + 14)
# check that the result is not more complicated than the input
z = sqrt(-2 * r29 + cos(2) + 2 * sqrt(-10 * r29 + 55) + 16)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(r6 + sqrt(15))) == sqrt(r6 + sqrt(15))
z = sqrt(15 - 2 * sqrt(31) + 2 * sqrt(55 - 10 * r29))
assert sqrtdenest(z) == z
def test_sqrtdenest2():
assert sqrtdenest(sqrt(16-2*sqrt(29)+2*sqrt(55-10*sqrt(29)))) == \
sqrt(5) + sqrt(11-2*sqrt(29))
def test_sqrtdenest2():
assert sqrtdenest(sqrt(16-2*sqrt(29)+2*sqrt(55-10*sqrt(29)))) == \
sqrt(5) + sqrt(11-2*sqrt(29))
def test_issue_12420():
I = S.ImaginaryUnit
assert _sqrt_match(4 + I) == []
assert sqrtdenest((3 - sqrt(2)*sqrt(4 + 3*I) + 3*I)/2) == I
e = 3 - sqrt(2)*sqrt(4 + I) + 3*I
assert sqrtdenest(e) == e
def test_sqrtdenest2():
assert sqrtdenest(sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))) == \
r5 + sqrt(11 - 2*r29)
e = sqrt(-r5 + sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
assert sqrtdenest(e) == root(-2*r29 + 11, 4)
r = sqrt(1 + r7)
assert sqrtdenest(sqrt(1 + r)) == sqrt(1 + r)
e = sqrt(((1 + sqrt(1 + 2*sqrt(3 + r2 + r5)))**2).expand())
assert sqrtdenest(e) == 1 + sqrt(1 + 2*sqrt(r2 + r5 + 3))
assert sqrtdenest(sqrt(5*r3 + 6*r2)) == \
root(3, 4)**3*(r6 + 3)/3
assert sqrtdenest(sqrt(((1 + r5 + sqrt(1 + r3))**2).expand())) == \
1 + r5 + sqrt(1 + r3)
assert sqrtdenest(sqrt(((1 + r5 + r7 + sqrt(1 + r3))**2).expand())) == \
1 + sqrt(1 + r3) + r5 + r7
e = sqrt(((1 + cos(2) + cos(3) + sqrt(1 + r3))**2).expand())
assert sqrtdenest(e) == cos(3) + cos(2) + 1 + sqrt(1 + r3)
e = sqrt(-2*r10 + 2*r2*sqrt(-2*r10 + 11) + 14)
assert sqrtdenest(e) == sqrt(-2*r10 - 2*r2 + 4*r5 + 14)
# check that the result is not more complicated than the input
z= sqrt(-2*r29 + cos(2) + 2*sqrt(-10*r29 + 55) + 16)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(r6 + sqrt(15))) == sqrt(r6 + sqrt(15))
# no assertion error when the 'r's are not the same in _denester
z = sqrt(15 - 2*sqrt(31) + 2*sqrt(55 - 10*r29))
assert sqrtdenest(z) == z
# currently cannot denest this; check one does not get a wrong answer
z = sqrt(8 - r2*sqrt(5 - r5) - 3*(1 + r5))
assert (sqrtdenest(z) - z).evalf() < 1.0e-100
def test_issue_18041():
e = -sqrt(-2 + 2 * sqrt(3) * I)
assert sqrtdenest(e) == -1 - sqrt(3) * I
def test_issue_12420():
assert sqrtdenest((3 - sqrt(2) * sqrt(4 + 3 * I) + 3 * I) / 2) == I
e = 3 - sqrt(2) * sqrt(4 + I) + 3 * I
assert sqrtdenest(e) == e
def test_C16():
test = sqrtdenest(sqrt(10 + 2 * sqrt(6) + 2 * sqrt(10) + 2 * sqrt(15)))
good = sqrt(2) + sqrt(3) + sqrt(5)
assert test == good
def test_C14():
assert sqrtdenest(sqrt(2*sqrt(3) + 4)) == 1 + sqrt(3)
def test_sqrtdenest_fail():
# when this doesn't fail, put the correct value at line 55
z = sqrt(8 - r2*sqrt(5 - r5) - 3*(1 + r5))
assert sqrtdenest(z) != z
def test_issue3142():
z = sqrt( -320 + 32*sqrt(5) + 64*r15)
assert sqrtdenest(z) == z
def test_issue_5653():
assert sqrtdenest(
sqrt(2 + sqrt(2 + sqrt(2)))) == sqrt(2 + sqrt(2 + sqrt(2)))
def test_sqrtdenest2():
assert sqrtdenest(sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))) == \
r5 + sqrt(11 - 2*r29)
e = sqrt(-r5 + sqrt(-2 * r29 + 2 * sqrt(-10 * r29 + 55) + 16))
assert sqrtdenest(e) == root(-2 * r29 + 11, 4)
r = sqrt(1 + r7)
assert sqrtdenest(sqrt(1 + r)) == sqrt(1 + r)
e = sqrt(((1 + sqrt(1 + 2 * sqrt(3 + r2 + r5)))**2).expand())
assert sqrtdenest(e) == 1 + sqrt(1 + 2 * sqrt(r2 + r5 + 3))
assert sqrtdenest(sqrt(5*r3 + 6*r2)) == \
r2*root(3, 4) + root(3, 4)**3
assert sqrtdenest(sqrt(((1 + r5 + sqrt(1 + r3))**2).expand())) == \
1 + r5 + sqrt(1 + r3)
assert sqrtdenest(sqrt(((1 + r5 + r7 + sqrt(1 + r3))**2).expand())) == \
1 + sqrt(1 + r3) + r5 + r7
e = sqrt(((1 + cos(2) + cos(3) + sqrt(1 + r3))**2).expand())
assert sqrtdenest(e) == cos(3) + cos(2) + 1 + sqrt(1 + r3)
e = sqrt(-2 * r10 + 2 * r2 * sqrt(-2 * r10 + 11) + 14)
assert sqrtdenest(e) == sqrt(-2 * r10 - 2 * r2 + 4 * r5 + 14)
# check that the result is not more complicated than the input
z = sqrt(-2 * r29 + cos(2) + 2 * sqrt(-10 * r29 + 55) + 16)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(r6 + sqrt(15))) == sqrt(r6 + sqrt(15))
# no assertion error when the 'r's are not the same in _denester
z = sqrt(15 - 2 * sqrt(31) + 2 * sqrt(55 - 10 * r29))
assert sqrtdenest(z) == z
# currently cannot denest this; check one does not get a wrong answer
z = sqrt(8 - r2 * sqrt(5 - r5) - 3 * (1 + r5))
assert (sqrtdenest(z) - z).evalf() < 1.0e-100
def test_issue_6241():
z = sqrt(-320 + 32 * sqrt(5) + 64 * r15)
assert sqrtdenest(z) == z
def test_C15():
test = sqrtdenest(sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2))))))
good = sqrt(2) + 3
assert test == good
def test_issue_5653():
assert sqrtdenest(sqrt(2 + sqrt(2 + sqrt(2)))) == sqrt(2 +
sqrt(2 + sqrt(2)))
def test_issue_2554():
assert sqrtdenest(sqrt(2 + sqrt(2 + sqrt(2)))) == sqrt(2 + sqrt(2 + sqrt(2)))
def test_sqrt_ratcomb():
assert sqrtdenest(sqrt(1 + r3) + sqrt(3 + 3 * r3) - sqrt(10 + 6 * r3)) == 0
def test_sqrtdenest2():
assert sqrtdenest(sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))) == \
r5 + sqrt(11 - 2*r29)
e = sqrt(-r5 + sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
assert sqrtdenest(e) == root(-2*r29 + 11, 4)
r = sqrt(1 + r7)
assert sqrtdenest(sqrt(1 + r)) == sqrt(1 + r)
e = sqrt(((1 + sqrt(1 + 2*sqrt(3 + r2 + r5)))**2).expand())
assert sqrtdenest(e) == 1 + sqrt(1 + 2*sqrt(r2 + r5 + 3))
assert sqrtdenest(sqrt(5*r3 + 6*r2)) == \
sqrt(2)*root(3, 4) + root(3, 4)**3
assert sqrtdenest(sqrt(((1 + r5 + sqrt(1 + r3))**2).expand())) == \
1 + r5 + sqrt(1 + r3)
assert sqrtdenest(sqrt(((1 + r5 + r7 + sqrt(1 + r3))**2).expand())) == \
1 + sqrt(1 + r3) + r5 + r7
e = sqrt(((1 + cos(2) + cos(3) + sqrt(1 + r3))**2).expand())
assert sqrtdenest(e) == cos(3) + cos(2) + 1 + sqrt(1 + r3)
e = sqrt(-2*r10 + 2*r2*sqrt(-2*r10 + 11) + 14)
assert sqrtdenest(e) == sqrt(-2*r10 - 2*r2 + 4*r5 + 14)
# check that the result is not more complicated than the input
z= sqrt(-2*r29 + cos(2) + 2*sqrt(-10*r29 + 55) + 16)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(r6 + sqrt(15))) == sqrt(r6 + sqrt(15))
z = sqrt(15 - 2*sqrt(31) + 2*sqrt(55 - 10*r29))
assert sqrtdenest(z) == z
def test_C16():
test = sqrtdenest(sqrt(10 + 2*sqrt(6) + 2*sqrt(10) + 2*sqrt(15)))
good = sqrt(2) + sqrt(3) + sqrt(5)
assert test == good
def test_sqrtdenest_rec():
assert sqrtdenest(sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 33)) == \
-r2 + r3 + 2*r7
assert sqrtdenest(sqrt(-28*r7 - 14*r5 + 4*sqrt(35) + 82)) == \
-7 + r5 + 2*r7
assert sqrtdenest(sqrt(6*r2/11 + 2*sqrt(22)/11 + 6*sqrt(11)/11 + 2)) == \
sqrt(11)*(r2 + 3 + sqrt(11))/11
assert sqrtdenest(sqrt(468*r3 + 3024*r2 + 2912*r6 + 19735)) == \
9*r3 + 26 + 56*r6
z = sqrt(-490*r3 - 98*sqrt(115) - 98*sqrt(345) - 2107)
assert sqrtdenest(z) == sqrt(-1)*(7*r5 + 7*r15 + 7*sqrt(23))
z = sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 34)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(-8*r2 - 2*r5 + 18)) == -r10 + 1 + r2 + r5
assert sqrtdenest(sqrt(8*r2 + 2*r5 - 18)) == \
sqrt(-1)*(-r10 + 1 + r2 + r5)
assert sqrtdenest(sqrt(8*r2/3 + 14*r5/3 + S(154)/9)) == \
-r10/3 + r2 + r5 + 3
assert sqrtdenest(sqrt(sqrt(2*r6 + 5) + sqrt(2*r7 + 8))) == \
sqrt(1 + r2 + r3 + r7)
assert sqrtdenest(sqrt(4*r15 + 8*r5 + 12*r3 + 24)) == 1 + r3 + r5 + r15
w = 1 + r2 + r3 + r5 + r7
assert sqrtdenest(sqrt((w**2).expand())) == w
z = sqrt((w**2).expand() + 1)
assert sqrtdenest(z) == z
z = sqrt(2*r10 + 6*r2 + 4*r5 + 12 + 10*r15 + 30*r3)
assert sqrtdenest(z) == z
def test_sqrtdenest_rec():
assert sqrtdenest(sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 33)) == \
-r2 + r3 + 2*r7
assert sqrtdenest(sqrt(-28*r7 - 14*r5 + 4*sqrt(35) + 82)) == \
-7 + r5 + 2*r7
assert sqrtdenest(sqrt(6*r2/11 + 2*sqrt(22)/11 + 6*sqrt(11)/11 + 2)) == \
sqrt(11)*(r2 + 3 + sqrt(11))/11
assert sqrtdenest(sqrt(468*r3 + 3024*r2 + 2912*r6 + 19735)) == \
9*r3 + 26 + 56*r6
z = sqrt(-490 * r3 - 98 * sqrt(115) - 98 * sqrt(345) - 2107)
assert sqrtdenest(z) == sqrt(-1) * (7 * r5 + 7 * r15 + 7 * sqrt(23))
z = sqrt(-4 * sqrt(14) - 2 * r6 + 4 * sqrt(21) + 34)
assert sqrtdenest(z) == z
assert sqrtdenest(sqrt(-8 * r2 - 2 * r5 + 18)) == -r10 + 1 + r2 + r5
assert sqrtdenest(sqrt(8*r2 + 2*r5 - 18)) == \
sqrt(-1)*(-r10 + 1 + r2 + r5)
assert sqrtdenest(sqrt(8*r2/3 + 14*r5/3 + Rational(154, 9))) == \
-r10/3 + r2 + r5 + 3
assert sqrtdenest(sqrt(sqrt(2*r6 + 5) + sqrt(2*r7 + 8))) == \
sqrt(1 + r2 + r3 + r7)
assert sqrtdenest(sqrt(4 * r15 + 8 * r5 + 12 * r3 +
24)) == 1 + r3 + r5 + r15
w = 1 + r2 + r3 + r5 + r7
assert sqrtdenest(sqrt((w**2).expand())) == w
z = sqrt((w**2).expand() + 1)
assert sqrtdenest(z) == z
z = sqrt(2 * r10 + 6 * r2 + 4 * r5 + 12 + 10 * r15 + 30 * r3)
assert sqrtdenest(z) == z
def test_sqrt_ratcomb():
assert sqrtdenest(sqrt(1 + r3) + sqrt(3 + 3*r3) - sqrt(10 + 6*r3)) == 0
