Python epsilon is not the smallest number

Question

What does sys.float_info.epsilon return?

On my system I get:

>>> sys.float_info.epsilon
2.220446049250313e-16
>>> sys.float_info.epsilon / 2
1.1102230246251565e-16
>>> 0 < sys.float_info.epsilon / 2 < sys.float_info.epsilon
True

How is this possible?

EDIT:

You are all right, I thought epsilon does what min does. So I actually meant sys.float_info.min .

EDIT2

Everybody and especially John Kugelman, thanks for your answers!

Some playing around I did to clarify things to myself:

>>> float.hex(sys.float_info.epsilon)
'0x1.0000000000000p-52'
>>> float.hex(sys.float_info.min)
'0x1.0000000000000p-1022'
>>> float.hex(1 + a)
'0x1.0000000000001p+0'
>>> float.fromhex('0x0.0000000000001p+0') == sys.float_info.epsilon
True
>>> float.hex(sys.float_info.epsilon * sys.float_info.min)
'0x0.0000000000001p-1022'

So epsilon * min gives the number with the smallest positive significand (or mantissa) and the smallest exponent.

Answer 1

epsilon is the difference between 1 and the next representable float. That's not the same as the smallest float, which would be the closest number to 0 , not 1 .

There are two smallest floats, depending on your criteria. min is the smallest normalized float. The smallest subnormal float is min * epsilon .

>>> sys.float_info.min
2.2250738585072014e-308
>>> sys.float_info.min * sys.float_info.epsilon
5e-324

Note the distinction between normalized and subnormal floats: min is not actually the smallest float, it's just the smallest one with full precision. Subnormal numbers cover the range between 0 and min , but they lose a lot of precision. Notice that 5e-324 has only one significant digit. Subnormals are also much slower to work with, up to 100x slower than normalized floats.

>>> (sys.float_info.min * sys.float_info.epsilon) / 2
0.0
>>> 4e-324
5e-324
>>> 5e-325
0.0

These tests confirm that 5e-324 truly is the smallest float. Dividing by two underflows to 0.

See also: What is the range of values a float can have in Python?

Answer 2

You actually want sys.float_info.min ("minimum positive normalized float"), which on machine gives me .2250738585072014e-308 .

epsilon is:

difference between 1 and the least value greater than 1 that is representable as a float

See the docs for more info on the fields of sys.float_info .

Answer 3

Your last expression is possible, because for any real, positive number, 0 < num/2 < num .

From the docs :

difference between 1 and the least value greater than 1 that is representable as a float

Answer 4

sys.float_info is defined as

difference between 1 and the least value greater than 1 that is representable as a float

on this page .

Answer 5

The documentation defines sys.float_info.epsilon as the

difference between 1 and the least value greater than 1 that is representable as a float

However, the gap between successive floats is bigger for bigger floats, so the gap between epsilon and the next smaller float is a lot smaller than epsilon . In particular, the next smaller float is not 0.

Answer 6

Like every answer says, it's the difference between 1 and the next greatest value that can be represented, if you tried to add half of it to 1, you'll get 1 back

>>> (1 + (sys.float_info.epsilon/2)) == 1
True

Additionally if you try to add two thirds of it to 1 , you'll get the same value:

>>> (1 + sys.float_info.epsilon) == (1 + (sys.float_info.epsilon * (2./3)))
True