def integer_nthroot(y, n):
"""
Return a tuple containing x = floor(y**(1/n))
and a boolean indicating whether the result is exact (that is,
whether x**n == y).
>>> from sympy import integer_nthroot
>>> integer_nthroot(16,2)
(4, True)
>>> integer_nthroot(26,2)
(5, False)
"""
y, n = int(y), int(n)
if y < 0:
raise ValueError("y must be nonnegative")
if n < 1:
raise ValueError("n must be positive")
if y in (0, 1):
return y, True
if n == 1:
return y, True
if n == 2:
x, rem = mpmath_sqrtrem(y)
return int(x), not rem
if n > y:
return 1, False
# Get initial estimate for Newton's method. Care must be taken to
# avoid overflow
try:
guess = int(y**(1./n) + 0.5)
except OverflowError:
exp = _log(y, 2)/n
if exp > 53:
shift = int(exp - 53)
guess = int(2.0**(exp - shift) + 1) << shift
else:
guess = int(2.0**exp)
#print n
if guess > 2**50:
# Newton iteration
xprev, x = -1, guess
while 1:
t = x**(n - 1)
#xprev, x = x, x - (t*x-y)//(n*t)
xprev, x = x, ((n - 1)*x + y//t)//n
#print n, x-xprev, abs(x-xprev) < 2
if abs(x - xprev) < 2:
break
else:
x = guess
# Compensate
t = x**n
while t < y:
x += 1
t = x**n
while t > y:
x -= 1
t = x**n
return x, t == y
def integer_nthroot(y, n):
"""
Return a tuple containing x = floor(y**(1/n))
and a boolean indicating whether the result is exact (that is,
whether x**n == y).
>>> from sympy import integer_nthroot
>>> integer_nthroot(16,2)
(4, True)
>>> integer_nthroot(26,2)
(5, False)
"""
y, n = int(y), int(n)
if y < 0:
raise ValueError("y must be nonnegative")
if n < 1:
raise ValueError("n must be positive")
if y in (0, 1):
return y, True
if n == 1:
return y, True
if n == 2:
x, rem = mpmath_sqrtrem(y)
return int(x), not rem
if n > y:
return 1, False
# Get initial estimate for Newton's method. Care must be taken to
# avoid overflow
try:
guess = int(y**(1./n) + 0.5)
except OverflowError:
exp = _log(y, 2)/n
if exp > 53:
shift = int(exp - 53)
guess = int(2.0**(exp - shift) + 1) << shift
else:
guess = int(2.0**exp)
#print n
if guess > 2**50:
# Newton iteration
xprev, x = -1, guess
while 1:
t = x**(n - 1)
#xprev, x = x, x - (t*x-y)//(n*t)
xprev, x = x, ((n - 1)*x + y//t)//n
#print n, x-xprev, abs(x-xprev) < 2
if abs(x - xprev) < 2:
break
else:
x = guess
# Compensate
t = x**n
while t < y:
x += 1
t = x**n
while t > y:
x -= 1
t = x**n
return x, t == y
