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Re: Puzzled by IntegerPart

  • To: mathgroup at smc.vnet.net
  • Subject: [mg114641] Re: Puzzled by IntegerPart
  • From: Bill Rowe <readnews at sbcglobal.net>
  • Date: Sat, 11 Dec 2010 01:54:41 -0500 (EST)

On 12/10/10 at 2:29 AM, sebhofer at gmail.com (Sebastian) wrote:

>On Dec 9, 12:00 pm, Themis Matsoukas <tmatsou... at me.com> wrote:
>>IntegerPart[100*1.15]

>>114

>>???

>I was going to say: Mathematica seems to floor numeric quantities
>(which would kindof make sense to me), e.g. In[86]:=
>In[86]:= IntegerPart[100*1.13]
>IntegerPart[100*(1.13 + $MachineEpsilon)]

>Out[86]= 112

>Out[87]= 113

>In[88]:= Floor[100*1.13]
>Floor[100*(1.13 + $MachineEpsilon)]

>Out[88]= 112

>Out[89]= 113

>On the other hand...
>In[90]:= IntegerPart[100*1.12]
>IntegerPart[100*(1.12 + $MachineEpsilon)]
>
>Out[90]= 112

>Out[91]= 112

>In[92]:= Floor[100*1.12]
>Floor[100*(1.12 + $MachineEpsilon)]

>Out[92]= 112

>Out[93]= 112 ... so I'm also puzzled.

The puzzle is explained by looking at how these values are
represented and understanding what the functions act on.

That is

In[1]:= RealDigits[1.12] == RealDigits[1.12 + $MachineEpsilon]

Out[1]= True

So, any function working on the decimal digits clearly gets the
same result for both 1.12 and 1.12+$MachineEpsilon

But note,

In[2]:= RealDigits[1.12, 2] == RealDigits[1.12 +
$MachineEpsilon, 2]

Out[2]= False

That is adding $MachineEpsilon does change the binary
representation but does so at bits that not used in the decimal
representation for this particular value. However

In[3]:= RealDigits[1.13] == RealDigits[1.13 + $MachineEpsilon]

Out[3]= False

So, functions working on the decimal representation for 1.13
will yield a different result when $MachineEpsilon is added.

The behavior depends on the precise value of the nearest machine
number to whatever decimal input you make.

There are a several tutorials in the documentation that expound
on this further. If you want to research this further a starting
point would be:

tutorial/MachinePrecisionNumbers
ComputerArithmetic/tutorial/ComputerArithmetic



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