python中epsilon什么意思_Python的epsilon值是否正确？ (Is Python's epsilon value correct?)

2014-04-21 02:41:18

0

According to Wikipedia:

Machine epsilon is defined as the smallest number that, when added to one, yields a result different from one

In Python, epsilon can be found using sys.float_info.epsilon and returns a value equivalent to 2^-52. However, I can add any number greater than 2^-53 to 1 and still get a result different to one.

But by the above definition of epsilon, adding any value less than epsilon to one should give one.

Does this mean that sys.float_info.epsilon is returning an incorrect value, or that Python is using some other definition of epsilon?

The following code illustrates this, with the floating point numbers printed out in hex format.

import sys

import numpy

print 'epsilon=%g' % sys.float_info.epsilon

# output: 2.22045e-16

epsilon = sys.float_info.epsilon

print 'epsilon(hex) = %s' % float.hex(epsilon)

# output: 0x1.0000000000000p-52

one = numpy.float64(1.0)

delta = float.fromhex('0x1.fffffffffffffp-53')

print 'delta = %s' % float.hex(delta)

print 'epsilon - delta = %s' % (float.hex(epsilon-delta))

#output: 0x1.0000000000000p-105

print 'n1.0 + epsilon = %s' % (float.hex(one+numpy.float64(epsilon)))

#output: 0x1.0000000000001p+0

print 'n1.0 + delta = %s' % (float.hex(one+numpy.float64(delta)))

#output: 0x1.0000000000001p+0

# since delta is smaller than epsilon, I expected 0x1.0000000000001p+0

delta1 = float.fromhex('0x1.0000000000001p-53')

print 'n1.0 + %s = %s' % (float.hex(delta1), float.hex(one+delta1))

#output: 0x1.0000000000001p+0

# since delta is smaller than epsilon, I expected 0x1.0000000000001p+0

delta2 = float.fromhex('0x1.0000000000000p-53')

# note: delta2 = epsilon / 2.0

print 'n1.0 + %s = %s' % (float.hex(delta2), float.hex(one+delta2))

# 0x1.0000000000000p+0

The resulting output is

epsilon=2.22045e-16

epsilon(hex) = 0x1.0000000000000p-52

delta = 0x1.fffffffffffffp-53

epsilon - delta = 0x1.0000000000000p-105

1.0 + epsilon = 0x1.0000000000001p+0

1.0 + delta = 0x1.0000000000001p+0

1.0 + 0x1.0000000000001p-53 = 0x1.0000000000001p+0

1.0 + 0x1.0000000000000p-53 = 0x1.0000000000000p+0