Understanding N and Precision

• To: mathgroup at smc.vnet.net
• Subject: [mg71147] Understanding N and Precision
• From: Alain Cochard <alain at geophysik.uni-muenchen.de>
• Date: Thu, 9 Nov 2006 03:38:25 -0500 (EST)

```Hi.  I would like to understand the following behavior:

Mathematica 5.2 for Linux
-- Motif graphics initialized --

In[1]:= MatrixForm[{{exact1=Cos[3Pi/2+Pi/7], Precision[exact1]}, \
{exact2=Cos[40139127975 Pi/14], Precision[exact2]}}]

Out[1]//MatrixForm=     23 Pi
Cos[-----]
14               Infinity

40139127975 Pi
Cos[--------------]
14          Infinity

just checking:

In[2]:= FullSimplify[exact1-exact2]

Out[2]= 0

In[3]:= MatrixForm[{{float1=N[exact1], Precision[float1]},\
{float2=N[exact2], Precision[float2]}}]

Out[3]//MatrixForm= 0.433884           MachinePrecision

0.433883           MachinePrecision

So the N of supposedly(?) 2 identical numbers is different (although
the precision is indeed the same). That's what I would like to
understand most.

Also, if I specify the precision for N, it now gives the same.  E.g.:

In[4]:= MatrixForm[{{float1=N[exact1,3], Precision[float1]},\
{float2=N[exact2,3], Precision[float2]}}]

Out[4]//MatrixForm= 0.434   3.

0.434   3.

including the case where the precision asked is MachinePrecision:

In[5]:= MatrixForm[{{float1=N[exact1,\$MachinePrecision], Precision[float1]},\
{float2=N[exact2,\$MachinePrecision], Precision[float2]}}]

Out[5]//MatrixForm= 0.4338837391175581   15.9546

0.4338837391175581   15.9546

Why in this case does it gives a precision of "15.9546" and not
"MachinePrecision", as above, especially since

In[6]:= N[\$MachinePrecision,Infinity]

Out[6]:= 15.9546

Isn't N[x] equivalent to N[x,\$MachinePrecision]?

Thanks in advance for any tip.

Alain

```

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