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Re: N[] MachinePrecision vs. Arbitrary Precision

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These are the same:

N[Sin[Exp[100]]]
Sin[Exp[100.]]
RealDigits /@ {%, %%}

-0.999105

-0.999105

{{{9, 9, 9, 1, 0, 5, 0, 1, 3, 0, 7, 7, 4, 3, 9, 3},
   0}, {{9, 9, 9, 1, 0, 5, 0, 1, 3, 0, 7, 7, 4, 3, 9, 3}, 0}}

But they're not precise. How could they be?

D[Sin[Exp[x]], x]
% /. x -> 100 // N

E^x Cos[E^x]

-1.13704*10^42

You need 43 digits of accuracy in the argument (100.) to get ONE digit  
accuracy in Sin[Exp[100.]], and machine precision doesn't give that much.

$MachinePrecision

15.9546

Using precision arithmetic gives better results (for this problem):

N[Sin[Exp[100]], 1]
N[Sin[Exp[100]], 11]
N[Sin[Exp[100]], 50]

0.1

0.14219812366

0.14219812365823863777245030647868525633473760956388

And that's because it's completely different arithmetic. (We mortals are  
not privy to the details, which may be just as well.)

Bobby

On Wed, 27 Apr 2011 04:39:25 -0500, Rafael Dunn  
<worthless.trash.junk@gmail.com> wrote:

> Mathematica 8.0.1.0, Mac OSX x86
>
> In:=
> N[Sin[Exp[100]]]
> N[Sin[Exp[100]], 1]
> N[Sin[Exp[100]], 11]
>
> Out:=
> -0.999105
> 0.1
> 0.14219812366
>
> I realize that N[exp] = N[exp, MachinePrecision].  This indicates when
> N[] is asked to calculate to MachinePrecision, it produces an incorrect
> result in this example.  However, when I use N[exp,n] where n <
> MachinePrecision, I get a correct output.  Is this because N[exp,n] uses
> ExtraPrecision, while N[exp] does not?  If I set MaxExtraPrecision =
> 0, then N[exp,n] produces a result of 0 with the note that there are no
> significant digits to display until I have made n high enough for
> Mathematica to be able to internally compute it.
>


-- 
DrMajorBob@yahoo.com

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Re: N[] MachinePrecision vs. Arbitrary Precision DrMajorBob <btreat1@austin.rr.com> - 2011-04-28 10:33 +0000

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