`N[3^200] ` in formats as `PrecisionReal` instead of `MachinePrecision`

[rocky]

Description

Numeric conversion N is converting to a PrecisionReal.

How to Reproduce and Output Given

Here is what we give:

In[1]:= N[3^200]
Out[1]= 2.65613988875875×10^95

In[16]:= N[Sqrt[19]]
Out[16]= 4.3588989435406735522369

We should get:

In[1]:= N[3^200]
Out[1]= 2.65614×10^95

In[2]:= N[Sqrt[19]]
Out[2]= 4.3589

Priority

Probably an easy fix. Should do before release.

https://github.com/Mathics3/mathics-core/blob/master/mathics/format/form/outputform.py#L752

        digits = dps(n.get_precision()) if isinstance(n, PrecisionReal) else 6

seems to be the offending line. The precision of the number is not the same thing as the display precision, whose default is 6. This is implicitly described in https://reference.wolfram.com/language/howto/ChangeTheFormatOfNumbers.html and https://reference.wolfram.com/language/ref/DefaultPrintPrecision.html

We might give thought on how to handle the DefaultPrintPrecision option.

Additional context

Noticed in going over discrepancies in a video on Mathematica, comparing output with what we have.

[mmatera]

The difference is not in the use of DefaultPrintPrecision, but in how the numbers are evaluated.
For example, in WMA,

In[1]:= N[3^200]                                                                

                  95
Out[1]= 2.65614 10

In[2]:= N[3^200] //Precision                                                    

Out[2]= MachinePrecision

In[3]:= N[3^200] //Head                                                         

Out[3]= Real

meaning that N[3^200] is evaluated as a MachineReal number, while if we force to evaluate a PrecisionReal

In[4]:= N[3^200, 15]                                                            

                           95
Out[4]= 2.65613988875875 10

we get all the decimal figures.

In Mathics3,

In[1]:= N[3^200] //Precision
Out[1]= 15.

In[2]:= N[3^200] //Head
Out[2]= Real

On the other hand, if we work with a MachineReal number instead of an Integer input,

In[1]:= N[3.^200] 
Out[1]= 2.65614×10^95

which actually sounds like a more consistent behavior...

On the other hand, for N[Sqrt[19], 200] I get in WMA 12.2.0:

In[1]:=  N[Sqrt[19], 200]                                                       

Out[1]= 4.35889894354067355223698198385961565913700392523244493689034413815955\
 
>    7328203158085656159155851944526905658621298274213629583992783826117012156\
 
>    5608364174699009777529188794058900619967156631207402231024

with the 200 decimal figures, the same output that I get in Mathics3:

In[1]:= N[Sqrt[19], 200]
Out[1]= 4.3588989435406735522369819838596156591370039252324449368903441381595573282031580856561591558519445269056586212982742136295839927838261170121565608364174699009777529188794058900619967156631207402231024

[rocky]

The difference is not in the use of DefaultPrintPrecision, but in how the numbers are evaluated. For example, in WMA,

Ok. Thanks for the investigation and correction. I've updated the bug report.

[mmatera]

In any case, DefaultPrintPrecision is an option of NumberForm that we are not handling, and we could easily do, I guess.

[rocky]

In any case, DefaultPrintPrecision is an option of NumberForm that we are not handling, and we could easily do, I guess.

See also #1741. Don't you love it when trying to work out or understand one bug, you see two more?

[mmatera]

In any case, DefaultPrintPrecision is an option of NumberForm that we are not handling, and we could easily do, I guess.

See also #1741. Don't you love it when trying to work out or understand one bug, you see two more?

This is not a bug, but just a missing feature :).

[rocky]

Fixed via #1744. Thanks to @mmatera for his education and guidance on this.