# Why the nonsense number appears?

Discussion in 'Python' started by Johnny Lee, Oct 31, 2005.

1. ### Johnny LeeGuest

Hi,
Pls take a look at this code:

----------
>>> t1 = "1130748744"
>>> t2 = "461"
>>> t3 = "1130748744"
>>> t4 = "500"
>>> time1 = t1+"."+t2
>>> time2 = t3+"."+t4
>>> print time1, time2

1130748744.461 1130748744.500
>>> float(time2) - float(time1)

0.039000034332275391
>>>

Why are there so many nonsense tails? thanks for your help.

Regards,
Johnny

Johnny Lee, Oct 31, 2005

2. ### Daniel DittmarGuest

Johnny Lee wrote:
>>>>print time1, time2

>
> 1130748744.461 1130748744.500
>
>>>>float(time2) - float(time1)

>
> 0.039000034332275391
>
>
> Why are there so many nonsense tails? thanks for your help.

http://en.wikipedia.org/wiki/Floating_point#Problems_with_floating-point,
especially 'Rounding'. Or google for "gloating point precision" if you
need more details.

Daniel

Daniel Dittmar, Oct 31, 2005

3. ### Yu-Xi LimGuest

Johnny Lee wrote:
<snip>
> Why are there so many nonsense tails? thanks for your help.

I guess you were expecting 0.039? You first need to understand floating
point numbers:

http://docs.python.org/tut/node16.html

What you see are the effects of representation errors.

The solution is presented here:
http://www.python.org/peps/pep-0327.html

from decimal import *
t1 = "1130748744"
t2 = "461"
t3 = "1130748744"
t4 = "500"
time1 = t1+"."+t2
time2 = t3+"."+t4
print time1, time2
Decimal(time2) - Decimal(time1)

Yu-Xi Lim, Oct 31, 2005
4. ### Sybren StuvelGuest

Johnny Lee enlightened us with:
> Why are there so many nonsense tails? thanks for your help.

Because if the same reason you can't write 1/3 in decimal:

http://docs.python.org/tut/node16.html

Sybren
--
The problem with the world is stupidity. Not saying there should be a
capital punishment for stupidity, but why don't we just take the
safety labels off of everything and let the problem solve itself?
Frank Zappa

Sybren Stuvel, Oct 31, 2005
5. ### Ben O'SteenGuest

On Mon, October 31, 2005 9:39, Sybren Stuvel said:
> Johnny Lee enlightened us with:
>> Why are there so many nonsense tails? thanks for your help.

>
> Because if the same reason you can't write 1/3 in decimal:
>
> http://docs.python.org/tut/node16.html
>
> Sybren
> --
> The problem with the world is stupidity. Not saying there should be a
> capital punishment for stupidity, but why don't we just take the
> safety labels off of everything and let the problem solve itself?
> Frank Zappa
> --
> http://mail.python.org/mailman/listinfo/python-list
>

I think that the previous poster was asking something different. I think
he was asking something like this:

If

>>> t1 = 0.500
>>> t2 = 0.461
>>> print t1-t2

0.039

Then why:

>>> t1 += 12345678910
>>> t2 += 12345678910
>>> # Note, both t1 and t2 have been incremented by the same amount.
>>> print t1-t2

0.0389995574951

It appears Yu-Xi Lim beat me to the punch. Using decimal as opposed to
float sorts out this error as floats are not built to handle the size of
number used here.

Ben

Ben O'Steen, Oct 31, 2005
6. ### Sybren StuvelGuest

Ben O'Steen enlightened us with:
> I think that the previous poster was asking something different.

It all boils down to floating point inprecision.

> If
>
>>>> t1 = 0.500
>>>> t2 = 0.461
>>>> print t1-t2

> 0.039
>
> Then why:
>
>>>> t1 += 12345678910
>>>> t2 += 12345678910
>>>> # Note, both t1 and t2 have been incremented by the same amount.
>>>> print t1-t2

> 0.0389995574951

It's easier to explain in decimals. Just assume you only have memory
to keep three decimals. 12345678910.500 is internally stored as
something like 1.23456789105e10. Strip that to three decimals, and you
have 1.234e10. In that case, t1 - t2 = 1.234e10 - 1.234e10 = 0.

> Using decimal as opposed to float sorts out this error as floats are
> not built to handle the size of number used here.

They can handle the size just fine. What they can't handle is 1/1000th
precision when using numbers in the order of 1e10.

Sybren
--
The problem with the world is stupidity. Not saying there should be a
capital punishment for stupidity, but why don't we just take the
safety labels off of everything and let the problem solve itself?
Frank Zappa

Sybren Stuvel, Oct 31, 2005
7. ### Ben O'SteenGuest

On Mon, October 31, 2005 10:23, Sybren Stuvel said:
> Ben O'Steen enlightened us with:
>> Using decimal as opposed to float sorts out this error as floats are
>> not built to handle the size of number used here.

>
> They can handle the size just fine. What they can't handle is 1/1000th
> precision when using numbers in the order of 1e10.
>

I used the word 'size' here incorrectly, I intended to mean 'length'
rather than numerical value. Sorry for the confusion

> Sybren
> --
> The problem with the world is stupidity. Not saying there should be a
> capital punishment for stupidity, but why don't we just take the
> safety labels off of everything and let the problem solve itself?
> Frank Zappa
> --
> http://mail.python.org/mailman/listinfo/python-list
>

Ben O'Steen, Oct 31, 2005
8. ### Steve HorsleyGuest

Ben O'Steen wrote:
> On Mon, October 31, 2005 10:23, Sybren Stuvel said:
>> Ben O'Steen enlightened us with:
>>> Using decimal as opposed to float sorts out this error as floats are
>>> not built to handle the size of number used here.

>> They can handle the size just fine. What they can't handle is 1/1000th
>> precision when using numbers in the order of 1e10.
>>

>
> I used the word 'size' here incorrectly, I intended to mean 'length'
> rather than numerical value. Sorry for the confusion
>

Sybren is right. The problem is not the length or the size, it's
the fact that 0.039 cannot be represented exactly in binary, in
just the same way that 1/3 cannot be represented exactly in
decimal. They both give recurring numbers. If you truncate those
recurring numbers to a finite number of digits, you lose
precision. And this shows up when you convert the inaccurate
number from binary into decimal representation where an exact
representation IS possible.

Steve

Steve Horsley, Oct 31, 2005
9. ### Dan BishopGuest

Steve Horsley wrote:
> Ben O'Steen wrote:
> > On Mon, October 31, 2005 10:23, Sybren Stuvel said:
> >> Ben O'Steen enlightened us with:
> >>> Using decimal as opposed to float sorts out this error as floats are
> >>> not built to handle the size of number used here.
> >> They can handle the size just fine. What they can't handle is 1/1000th
> >> precision when using numbers in the order of 1e10.

> >
> > I used the word 'size' here incorrectly, I intended to mean 'length'
> > rather than numerical value. Sorry for the confusion

>
> Sybren is right. The problem is not the length or the size, it's
> the fact that 0.039 cannot be represented exactly in binary, in
> just the same way that 1/3 cannot be represented exactly in
> decimal. They both give recurring numbers. If you truncate those
> recurring numbers to a finite number of digits, you lose
> precision. And this shows up when you convert the inaccurate
> number from binary into decimal representation where an exact
> representation IS possible.

That's A source of error, but it's only part of the story. The
double-precision binary representation of 0.039 is 5620492334958379 *
2**(-57), which is in error by 1/18014398509481984000. By contrast,
Johnny Lee's answer is in error by 9/262144000, which is more than 618
billion times the error of simply representing 0.039 in floating point
-- a loss of 39 bits.

The problem here is catastrophic cancellation.

1130748744.500 ~= 4742703982051328 * 2**(-22)
1130748744.461 ~= 4742703981887750 * 2**(-22)

Subtracting gives 163578 * 2**(-22), which has only 18 significant bits.

Dan Bishop, Nov 1, 2005
10. ### Steve HorsleyGuest

Dan Bishop wrote:

> That's A source of error, but it's only part of the story. The
> double-precision binary representation of 0.039 is 5620492334958379 *
> 2**(-57), which is in error by 1/18014398509481984000. By contrast,
> Johnny Lee's answer is in error by 9/262144000, which is more than 618
> billion times the error of simply representing 0.039 in floating point
> -- a loss of 39 bits.
>
> The problem here is catastrophic cancellation.
>
> 1130748744.500 ~= 4742703982051328 * 2**(-22)
> 1130748744.461 ~= 4742703981887750 * 2**(-22)
>
> Subtracting gives 163578 * 2**(-22), which has only 18 significant bits.
>

Hmm. Good point.

Steve Horsley, Nov 1, 2005