Python arithmetic with small numbers

Question

I am getting the following unexpected result when I do arithmetic with small numbers in Python:

>>> sys.float_info
sys.float_info(max=1.797693134862         


        

Answer 1

You have epsilon=2.220446049250313e-16 so it is normal that (1. - (1.e-17) ) = 1 because 1.e-17 < epsilon.

Answer 2

it should be able to handle "large" small numbers like 1e-17, shouldn't it?

Not necessarily (it depends on the numbers). A float cannot exactly represent either 1e-17 or 1-(1e-17). In the case of the latter, the nearest number that it can represent is 1.

I suggest you read What Every Computer Scientist Should Know About Floating-Point Arithmetic.

Answer 3

First, let's review what epsilon really is in the return value of sys.float_info.

Epsilon (or

Answer 4

If you need this level of precision, consider the Decimal module

>>> decimal.Decimal(1.0)-decimal.Decimal('1.0e-17')
Decimal('0.999999999999999990')
>>> decimal.Decimal(1.0)-decimal.Decimal('1.0e-17')<decimal.Decimal(1.0)
True

And:

>>> decimal.Decimal(1.0)-decimal.Decimal('1.0e-17')<1.0
True

Careful with the last one though because you can get conversion errors.

Others have suggested What Every Computer Scientist Should Know About Floating-Point Arithmetic. and I also recommend Don’t Store That in a Float

Answer 5

you can handle those. note that

>>> 1.e-17 == 0
False

and

>>> 1.e-17 + 1.e-18
1.1e-17

you simply cannot handle 1-1e-17, because the mantissa won't fit in the finite precision

Answer 6

>>> from decimal import Decimal
>>> Decimal('1')-Decimal(str(10**-17)) < Decimal('1')
True

Use the decimal module, for that kind of accuracy!