# Floating point multiplication in python

Christophe Chong cdchong at stanford.edu
Thu Sep 8 02:37:51 CEST 2011

```And then we learned in class what happens when you're calculating "0.1" with
different precision in the industry.

http://www.ima.umn.edu/~arnold/disasters/patriot.html

Beware.

On Tue, Sep 6, 2011 at 3:14 AM, Thomas Rachel <
nutznetz-0c1b6768-bfa9-48d5-a470-7603bd3aa915 at spamschutz.glglgl.de> wrote:

> Am 06.09.2011 07:57 schrieb xyz:
>
>> hi all:
>>
>> As we know ,  1.1 * 1.1 is 1.21 .
>> But in python ,I got following :
>>
>>  1.1 * 1.1
>>>>>
>>>> 1.2100000000000002
>>
>> why python get wrong result? Who can tell me  where's the
>> 0.0000000000000002 from?
>>
>
> 1.1 does not fit in a binary floating point number. It is approximated - in
> binary! - as 1.000110011001100110011 ... (periodically).
>
> Note that, while in the decimal system we normally use, only numbers which
> have other components in the denominator than 2 or 5 are periodically, in
> the binary systems only components with 2 are allowed in order not to be
> periodically.
>
> Example: 3.453 is not periodically, because it is 3453/100 and 100 has only
> the factors 2 and 5, each twice.
>
> 1/3 = .3333333... is periodically, because it has the factor 3. The same
> applies to 1/6, which has 2 and 3 as factors. The latter destroys the
> non-periodical behaviour.
>
> As said, in the dual system, only the 2 is allowed.
>
> .5 (10) = 2** -1 = .1 (2).
> .25 (10) = 2 ** -2 = .01 (2).
> .75 (10) = their sum = .11 (2).
>
> But .1 (1/10) is more complicated, -2 would be as well.
>
> As the IEEE floating point representation is limited, there is a slight
> error value which makes the stored value differ from the intended one.
>
> Look here:
>
>
>
>  x=(1,0,0,0,1,1,0,0,1,1,0,0,1,**1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,**
>>>> 1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,**0,1,1,0,0,1,1,0,0,1,1,0,0,1,1)
>>>> a=0
>>>> for n,i in enumerate(x): a += i*2**-n; print a, a-1.1, i*2**-n, a-olda
>>>>
>>> ...
> 1 -0.1 1 1
> 1.0 -0.1 0.0 0.0
> 1.0 -0.1 0.0 0.0
> 1.0 -0.1 0.0 0.0
> 1.0625 -0.0375 0.0625 0.0625
> 1.09375 -0.00625 0.03125 0.03125
> 1.09375 -0.00625 0.0 0.0
> 1.09375 -0.00625 0.0 0.0
> 1.09765625 -0.00234375 0.00390625 0.00390625
> 1.099609375 -0.000390625 0.001953125 0.001953125
> 1.099609375 -0.000390625 0.0 0.0
> 1.099609375 -0.000390625 0.0 0.0
> 1.09985351562 -0.000146484375 0.000244140625 0.000244140625
> 1.09997558594 -2.44140625001e-05 0.0001220703125 0.0001220703125
> 1.09997558594 -2.44140625001e-05 0.0 0.0
> 1.09997558594 -2.44140625001e-05 0.0 0.0
> 1.09999084473 -9.15527343759e-06 1.52587890625e-05 1.52587890625e-05
> 1.09999847412 -1.52587890634e-06 7.62939453125e-06 7.62939453125e-06
> 1.09999847412 -1.52587890634e-06 0.0 0.0
> 1.09999847412 -1.52587890634e-06 0.0 0.0
> 1.0999994278 -5.72204589933e-07 9.53674316406e-07 9.53674316406e-07
> 1.09999990463 -9.53674317294e-08 4.76837158203e-07 4.76837158203e-07
> 1.09999990463 -9.53674317294e-08 0.0 0.0
> 1.09999990463 -9.53674317294e-08 0.0 0.0
> 1.09999996424 -3.57627869541e-08 5.96046447754e-08 5.96046447754e-08
> 1.09999999404 -5.96046456636e-09 2.98023223877e-08 2.98023223877e-08
> 1.09999999404 -5.96046456636e-09 0.0 0.0
> 1.09999999404 -5.96046456636e-09 0.0 0.0
> 1.09999999776 -2.23517426789e-09 3.72529029846e-09 3.72529029846e-09
> 1.09999999963 -3.72529118664e-10 1.86264514923e-09 1.86264514923e-09
> 1.09999999963 -3.72529118664e-10 0.0 0.0
> 1.09999999963 -3.72529118664e-10 0.0 0.0
> 1.09999999986 -1.3969847501e-10 2.32830643654e-10 2.32830643654e-10
> 1.09999999998 -2.32831531832e-11 1.16415321827e-10 1.16415321827e-10
> 1.09999999998 -2.32831531832e-11 0.0 0.0
> 1.09999999998 -2.32831531832e-11 0.0 0.0
> 1.09999999999 -8.73123795486e-12 1.45519152284e-11 1.45519152284e-11
> 1.1 -1.45528034068e-12 7.27595761418e-12 7.27595761418e-12
> 1.1 -1.45528034068e-12 0.0 0.0
> 1.1 -1.45528034068e-12 0.0 0.0
> 1.1 -5.45785638906e-13 9.09494701773e-13 9.09494701773e-13
> 1.1 -9.10382880193e-14 4.54747350886e-13 4.54747350886e-13
> 1.1 -9.10382880193e-14 0.0 0.0
> 1.1 -9.10382880193e-14 0.0 0.0
> 1.1 -3.41948691585e-14 5.68434188608e-14 5.68434188608e-14
> 1.1 -5.77315972805e-15 2.84217094304e-14 2.84217094304e-14
> 1.1 -5.77315972805e-15 0.0 0.0
> 1.1 -5.77315972805e-15 0.0 0.0
> 1.1 -2.22044604925e-15 3.5527136788e-15 3.5527136788e-15
> 1.1 -4.4408920985e-16 1.7763568394e-15 1.7763568394e-15
> 1.1 -4.4408920985e-16 0.0 0.0
> 1.1 -4.4408920985e-16 0.0 0.0
> 1.1 -2.22044604925e-16 2.22044604925e-16 2.22044604925e-16
> 1.1 0.0 1.11022302463e-16 2.22044604925e-16
> 1.1 0.0 0.0 0.0
> 1.1 0.0 0.0 0.0
> 1.1 0.0 1.38777878078e-17 0.0
> 1.1 0.0 6.93889390391e-18 0.0
>
> So here we have reached the point where the maximum precision is reached -
> a doesn't change anymore, although it should. The error is about 1E-16.
>
> Now if you multiply two values with an error, the error also propagates
> into the result - PLUs the result can have its own error source - in the
> same order of magnitude.
>
> (a+e) * (a+e) = a*a + 2*a*e + e*e. So your new error term is 2*a*e + e*e or
> (2*a + e) * e.
>
> --
> http://mail.python.org/**mailman/listinfo/python-list<http://mail.python.org/mailman/listinfo/python-list>
>

--
Christophe Chong
Department of Economics
Stanford University
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