# Confused by (apparent) inconsistent precision

$$e^{\pi \sqrt{163}} \approx 262537412640768743.99999999999925$$

E^(Pi Sqrt[163.0])
N[E^(Pi Sqrt[163.0]), 35]
NumberForm[E^(Pi Sqrt[163.]), 35]


returns

2.6253741*10^17
2.6253741*10^17
2.625374126407682*10^17


That's not the 35 digits I expected. OTOH,

N[Pi* E, 35]


returns 35 digits,

8.5397342226735670654635508695465745


but then

NumberForm[Pi*E*1., 35]


again doesn't:

8.53973422267357


So I'm confused. Why doesn't one N[] what the other one does? In the documentation:

NumberForm[ expr, n ]
prints with approximate real numbers in expr given to n-digit precision.

I read this three times, slowly, went for a cup of tea, and read it again. But 15 isn't 35, or is it?

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It is, for sufficiently large values of... Okay, the issue, as stated in the response below, is that you cannot get more correct digits after the fact. To obtain 35 digits, use N[] on an exact input, not one that already contains approximate numbers of lower precision. – Daniel Lichtblau Sep 14 '12 at 14:31
@Daniel - Thanks for the feedback. Are you referring to my 163.0 instead of 163? To be honest, I don't know why I wrote it that way :-/ – stevenvh Sep 14 '12 at 14:35
Yes, I meant 163.0 was, in effect, polluting the input with machine precision. – Daniel Lichtblau Sep 14 '12 at 14:36

N does not raise the precision of approximate numbers in its input

163.0 (or 163., or 163) is a machine precision number, and Mathematica will not fake a higher precision when a certain number of digits are requested with N.

See this answer and this tutorial for more.

These questions may also be of interest:

Converting to machine precision

Annoying display truncation of numerical results

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@stevenvh In keeping with Mr.Wizard's advice, you may wish to compare using 163` (without the decimal point) in your code. – David Carraher Sep 14 '12 at 13:58