       Is Put - Get cycle in Mathematica always deterministic?

• To: mathgroup at smc.vnet.net
• Subject: [mg119977] Is Put - Get cycle in Mathematica always deterministic?
• From: Alexey Popkov <lehin.p at gmail.com>
• Date: Sun, 3 Jul 2011 04:10:40 -0400 (EDT)

```In Mathematica as in other systems of computer math the numbers are
internally stored in binary form. However when exporting them with
such functions as Put and PutAppend they are converted into approximate
decimals.  When we import them back with such functions as Get they are restored
backward from this approximate decimal representation to binary form.  The
question is: whether the recovered number is always identical to the
original binary number and, if not always, in which cases it is not and how
great can be the difference? I am particularly interested in the Put - Get
cycle (on the same computer system).

I am aware of DumpSave which is guaranteed to restore original definitions
exactly but I am interested in exporting expressions in human-readable
format. I am quite satisfied with the default ASCII-based representation
created by Put and PutAppend.

The following two simple experiments show that restored numbers are
identical to the exported numbers in all binary RealDigits but their
Precisions may differ even in Equal sense.

test := (Put[list, "test.txt"];
list2 = Get["test.txt"];
{Order[list, list2] === 0,
Order[Total@Abs[list - list2], 0.] === 0,
Total[Order @@@ RealDigits[Transpose[{list, list2}], 2]],
Total[Order @@@ Map[Precision, Transpose[{list, list2}], {-1}]],
Total[1 - Boole[Equal @@@ Map[Precision, Transpose[{list, list2}],
{-1}]]]})

In:= list=RandomReal[NormalDistribution[],10000]^1001;
test
Out= {False,True,0,1,3}

In:=
list=RandomReal[NormalDistribution[],10000,WorkingPrecision->50]^1001;
test
Out= {False,False,0,-2174,1}

It seems the the Put - Get cycle always restores the original numbers
exactly and only Precision may be little different. But may be I am
missing something?

```

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