Which is more accurate, x**.5 or math.sqrt(x)?

Question

I recently discovered that x**.5 and math.sqrt(x) do not always produce the same result in Python:

Python 2.6.1 (r261:67517, Dec 4 2008, 16:51:00) [MSC v.1500 32 bit (Intel)] on win32 >>> 8885558**.5 - math.sqrt(8885558) -4.5474735088646412e-13 

Checking all integers below 10**7, the two methods produced different results for almost exactly 0.1% of the samples, with the size of the error increasing (slowly) for larger numbers.

So the question is, which method is more accurate?

Answer 1

Neither one is more accurate, they both diverge from the actual answer in equal parts:

>>> (8885558**0.5)**2 8885557.9999999981 >>> sqrt(8885558)**2 8885558.0000000019  >>> 2**1023.99999999999 1.7976931348498497e+308  >>> (sqrt(2**1023.99999999999))**2 1.7976931348498495e+308 >>> ((2**1023.99999999999)**0.5)**2 1.7976931348498499e+308  >>> ((2**1023.99999999999)**0.5)**2 - 2**1023.99999999999 1.9958403095347198e+292 >>> (sqrt(2**1023.99999999999))**2 - 2**1023.99999999999 -1.9958403095347198e+292 

http://mail.python.org/pipermail/python-list/2003-November/238546.html

The math module wraps the platform C library math functions of the same names; math.pow() is most useful if you need (or just want) high compatibility with C extensions calling C's pow().

__builtin__.pow() is the implementation of Python's infix ** operator, and deals with complex numbers, unbounded integer powers, and modular exponentiation too (the C pow() doesn't handle any of those).

** is more complete. math.sqrt is probably just the C implementation of sqrt which is probably related to pow.