BUG: 0/frame numeric ops buggy (GH9144)
closes #9144
closes #8445 

---

Here's the results from testing the vbenches related to DataFrames (I also added 6 vbenches).

```
Invoked with :
--ncalls: 10
--repeats: 10


-------------------------------------------------------------------------------
Test name                                    | head[ms] | base[ms] |  ratio   |
-------------------------------------------------------------------------------
frame_float_div_by_zero                      |   1.5656 |  23.4006 |   0.0669 |
frame_float_floor_by_zero                    |   3.1077 |  24.2786 |   0.1280 |
groupby_frame_nth_none                       |   1.4918 |   2.1483 |   0.6944 |
groupby_frame_nth_any                        |   4.0633 |   5.4162 |   0.7502 |
dataframe_resample_max_string                |   1.3970 |   1.5892 |   0.8791 |
stat_ops_frame_mean_int_axis_1               |   3.4506 |   3.9006 |   0.8846 |
stat_ops_frame_mean_float_axis_1             |   3.6501 |   4.1200 |   0.8859 |
dataframe_resample_max_numpy                 |   1.4134 |   1.5861 |   0.8911 |
frame_reindex_upcast                         |   5.6926 |   6.3748 |   0.8930 |
stat_ops_frame_sum_int_axis_1                |   3.2424 |   3.6136 |   0.8973 |
dataframe_resample_min_numpy                 |   1.4320 |   1.5937 |   0.8985 |
stat_ops_frame_mean_float_axis_0             |   3.5722 |   3.9361 |   0.9075 |
frame_ctor_dtindex_Nanox2                    |   0.8544 |   0.9404 |   0.9086 |
frame_dropna_axis1_any                       |  17.6609 |  19.4360 |   0.9087 |
frame_ctor_dtindex_Hourx1                    |   0.8695 |   0.9554 |   0.9101 |
frame_ctor_dtindex_Secondx1                  |   0.8586 |   0.9384 |   0.9150 |
frame_ctor_dtindex_Microx2                   |   0.8564 |   0.9341 |   0.9168 |
dataframe_resample_mean_string               |   1.8672 |   2.0302 |   0.9197 |
frame_ctor_dtindex_Nanox1                    |   0.8526 |   0.9242 |   0.9225 |
stat_ops_level_frame_sum_multiple            |   4.7174 |   5.0987 |   0.9252 |
frame_ctor_dtindex_Hourx2                    |   0.8645 |   0.9342 |   0.9254 |
frame_xs_row                                 |   0.0242 |   0.0261 |   0.9269 |
frame_ctor_dtindex_BMonthBeginx1             |   1.0908 |   1.1755 |   0.9280 |
stat_ops_frame_sum_float_axis_0              |   3.6003 |   3.8713 |   0.9300 |
eval_frame_mult_python_one_thread            |  12.8115 |  13.7715 |   0.9303 |
stat_ops_frame_sum_int_axis_0                |   3.2616 |   3.4839 |   0.9362 |
eval_frame_mult_python                       |  12.8203 |  13.6336 |   0.9403 |
dataframe_resample_min_string                |   1.4186 |   1.5066 |   0.9416 |
frame_shift_axis0                            |   7.3947 |   7.8502 |   0.9420 |
stat_ops_frame_sum_float_axis_1              |   3.5332 |   3.7481 |   0.9427 |
frame_ctor_dtindex_BYearEndx2                |   1.0797 |   1.1452 |   0.9428 |
frame_fillna_many_columns_pad                |   4.3826 |   4.6457 |   0.9434 |
frame_ctor_dtindex_Secondx2                  |   0.8635 |   0.9153 |   0.9434 |
frame_get_dtype_counts                       |   0.0601 |   0.0635 |   0.9467 |
frame_dropna_axis1_all                       |  34.1749 |  36.0852 |   0.9471 |
frame_float_equal                            |   2.1541 |   2.2727 |   0.9478 |
dataframe_resample_mean_numpy                |   1.8673 |   1.9664 |   0.9496 |
append_frame_single_mixed                    |   1.2900 |   1.3544 |   0.9525 |
eval_frame_add_python                        |  13.0182 |  13.6672 |   0.9525 |
eval_frame_and_python_one_thread             |  23.5453 |  24.5942 |   0.9574 |
frame_drop_dup_inplace                       |   1.8290 |   1.9081 |   0.9585 |
groupby_frame_cython_many_columns            |   2.4763 |   2.5806 |   0.9596 |
frame_get_numeric_data                       |   0.0697 |   0.0725 |   0.9612 |
eval_frame_and_python                        |  23.6922 |  24.6444 |   0.9614 |
frame_mask_bools                             |   8.5507 |   8.8782 |   0.9631 |
frame_shift_axis_1                           |  11.5263 |  11.9162 |   0.9673 |
eval_frame_chained_cmp_python                |  68.9274 |  71.2359 |   0.9676 |
frame_multi_and_st                           |  20.8511 |  21.5107 |   0.9693 |
eval_frame_add_python_one_thread             |  11.9996 |  12.3749 |   0.9697 |
frame_mask_floats                            |   5.7624 |   5.9394 |   0.9702 |
frame_multi_and_no_ne                        |  21.2904 |  21.9065 |   0.9719 |
frame_apply_axis_1                           |  45.9807 |  47.2289 |   0.9736 |
join_dataframe_integer_2key                  |   3.5381 |   3.6330 |   0.9739 |
frame_ctor_nested_dict_int64                 |  48.7418 |  50.0032 |   0.9748 |
eval_frame_chained_cmp_python_one_thread     |  67.2400 |  68.9405 |   0.9753 |
frame_to_csv_mixed                           | 404.9059 | 414.6069 |   0.9766 |
groupby_frame_singlekey_integer              |   1.4465 |   1.4805 |   0.9770 |
frame_dropna_axis1_all_mixed_dtypes          | 137.6919 | 140.7975 |   0.9779 |
frame_insert_500_columns_end                 |  65.0207 |  66.4810 |   0.9780 |
frame_drop_dup_na_inplace                    |   1.6683 |   1.7045 |   0.9788 |
frame_iteritems                              |  17.1694 |  17.5347 |   0.9792 |
stat_ops_level_frame_sum                     |   2.1278 |   2.1730 |   0.9792 |
reindex_frame_level_align                    |   0.6034 |   0.6153 |   0.9807 |
frame_count_level_axis0_mixed_dtypes_multi   |  73.6246 |  75.0608 |   0.9809 |
dataframe_reindex                            |   0.2569 |   0.2615 |   0.9824 |
frame_reindex_axis1                          |  44.7410 |  45.5110 |   0.9831 |
groupby_frame_median                         |   5.1782 |   5.2658 |   0.9834 |
frame_to_csv_date_formatting                 |   6.7534 |   6.8611 |   0.9843 |
frame_fancy_lookup_all                       |  11.3160 |  11.4933 |   0.9846 |
frame_ctor_dtindex_DateOffsetx2              |   0.7447 |   0.7562 |   0.9848 |
join_dataframe_integer_key                   |   1.2000 |   1.2166 |   0.9863 |
frame_reindex_columns                        |   0.2247 |   0.2276 |   0.9872 |
frame_from_records_generator                 |  41.6522 |  42.1751 |   0.9876 |
frame_ctor_dtindex_QuarterBeginx2            |   0.9039 |   0.9145 |   0.9884 |
frame_sort_index_by_columns                  |  24.9708 |  25.2553 |   0.9887 |
join_dataframe_index_single_key_bigger       |   8.9800 |   9.0787 |   0.9891 |
frame_apply_ref_by_name                      |   8.7795 |   8.8757 |   0.9892 |
stat_ops_frame_mean_int_axis_0               |   3.2407 |   3.2742 |   0.9898 |
indexing_dataframe_boolean                   |  82.9152 |  83.7444 |   0.9901 |
frame_add                                    |   3.7720 |   3.8059 |   0.9911 |
frame_ctor_dtindex_BYearBeginx2              |   1.0778 |   1.0870 |   0.9915 |
frame_drop_duplicates_na                     |  14.2417 |  14.3591 |   0.9918 |
frame_ctor_dtindex_CBMonthBeginx1            |   2.2563 |   2.2738 |   0.9923 |
frame_ctor_dtindex_Weekx2                    |   0.7799 |   0.7859 |   0.9924 |
join_dataframe_index_single_key_small        |   8.2365 |   8.2989 |   0.9925 |
frame_dropna_axis1_any_mixed_dtypes          | 124.3344 | 125.2386 |   0.9928 |
frame_ctor_list_of_dict                      |  42.8818 |  43.1860 |   0.9930 |
frame_ctor_dtindex_YearEndx2                 |   0.8680 |   0.8737 |   0.9935 |
append_frame_single_homogenous               |   0.8964 |   0.9022 |   0.9936 |
groupby_frame_apply                          |  22.1934 |  22.3310 |   0.9938 |
frame_mult_no_ne                             |   3.8051 |   3.8274 |   0.9942 |
frame_nonunique_equal                        |   7.3191 |   7.3515 |   0.9956 |
frame_ctor_dtindex_BMonthBeginx2             |   1.0844 |   1.0890 |   0.9958 |
frame_apply_lambda_mean                      |   3.8462 |   3.8609 |   0.9962 |
frame_ctor_nested_dict                       |  46.5123 |  46.6852 |   0.9963 |
frame_count_level_axis1_mixed_dtypes_multi   |  61.2177 |  61.4444 |   0.9963 |
frame_ctor_dtindex_BMonthEndx2               |   0.9260 |   0.9288 |   0.9970 |
frame_html_repr_trunc_mi                     |  22.1488 |  22.2066 |   0.9974 |
frame_multi_and                              |  21.2545 |  21.3085 |   0.9975 |
frame_add_st                                 |   3.7661 |   3.7741 |   0.9979 |
frame_ctor_dtindex_BYearBeginx1              |   1.0904 |   1.0926 |   0.9980 |
join_dataframe_index_multi                   |  13.3128 |  13.3311 |   0.9986 |
frame_getitem_single_column                  |  12.7089 |  12.7029 |   1.0005 |
frame_object_equal                           |   7.3267 |   7.3224 |   1.0006 |
frame_from_records_generator_nrows           |   0.5950 |   0.5940 |   1.0016 |
frame_to_string_floats                       |  15.5732 |  15.5458 |   1.0018 |
frame_add_no_ne                              |   3.8073 |   3.7999 |   1.0019 |
frame_ctor_dtindex_CBMonthBeginx2            |   1.9428 |   1.9389 |   1.0020 |
frame_ctor_dtindex_BusinessDayx2             |   0.8365 |   0.8348 |   1.0020 |
frame_ctor_dtindex_BDayx2                    |   0.8400 |   0.8381 |   1.0023 |
frame_float_mod                              |   2.4775 |   2.4717 |   1.0024 |
frame_reindex_axis0                          |  42.1150 |  42.0129 |   1.0024 |
frame_drop_duplicates                        |  13.3927 |  13.3401 |   1.0039 |
sparse_frame_constructor                     |   3.7919 |   3.7743 |   1.0047 |
frame_iloc_big                               |   0.1020 |   0.1015 |   1.0049 |
frame_interpolate_some_good_infer            |   1.8705 |   1.8610 |   1.0051 |
indexing_dataframe_boolean_st                |  85.3268 |  84.8837 |   1.0052 |
frame_dropna_axis0_any                       |  18.0998 |  18.0034 |   1.0054 |
frame_count_level_axis1_multi                |  57.8269 |  57.5086 |   1.0055 |
indexing_dataframe_boolean_rows              |   0.2319 |   0.2306 |   1.0056 |
indexing_dataframe_boolean_rows_object       |   0.3911 |   0.3880 |   1.0080 |
frame_dropna_axis0_any_mixed_dtypes          | 125.2823 | 124.2290 |   1.0085 |
frame_repr_wide                              |   8.4213 |   8.3428 |   1.0094 |
frame_apply_pass_thru                        |   2.7872 |   2.7612 |   1.0094 |
frame_dtypes                                 |   0.0732 |   0.0725 |   1.0099 |
frame_to_html_mixed                          | 120.9245 | 119.7213 |   1.0101 |
frame_ctor_dtindex_DateOffsetx1              |   0.7530 |   0.7452 |   1.0105 |
frame_ctor_dtindex_Dayx1                     |   0.8844 |   0.8746 |   1.0112 |
groupby_frame_apply_overhead                 |   5.2761 |   5.2173 |   1.0113 |
frame_ctor_dtindex_Millix1                   |   0.9011 |   0.8910 |   1.0113 |
frame_count_level_axis0_multi                |  43.2266 |  42.7371 |   1.0115 |
reindex_frame_level_reindex                  |   0.6016 |   0.5946 |   1.0117 |
frame_ctor_dtindex_BQuarterBeginx1           |   1.1092 |   1.0958 |   1.0122 |
frame_reindex_both_axes                      |  13.9118 |  13.7180 |   1.0141 |
join_dataframe_index_single_key_bigger_sort  |  11.0337 |  10.8757 |   1.0145 |
frame_ctor_dtindex_Weekx1                    |   0.7509 |   0.7395 |   1.0154 |
frame_ctor_dtindex_BMonthEndx1               |   0.9711 |   0.9547 |   1.0172 |
indexing_dataframe_boolean_no_ne             |  87.0329 |  85.4171 |   1.0189 |
frame_fancy_lookup                           |   2.0553 |   2.0135 |   1.0208 |
frame_mult_st                                |   3.8758 |   3.7959 |   1.0210 |
frame_repr_tall                              |  12.1078 |  11.8041 |   1.0257 |
frame_insert_100_columns_begin               |  24.3260 |  23.6877 |   1.0269 |
frame_ctor_dtindex_QuarterEndx2              |   1.0204 |   0.9927 |   1.0279 |
frame_iteritems_cached                       |   0.3542 |   0.3440 |   1.0297 |
frame_ctor_dtindex_Easterx2                  |   0.9328 |   0.9050 |   1.0307 |
frame_interpolate                            |  64.3447 |  62.4263 |   1.0307 |
frame_html_repr_trunc_si                     |  17.4936 |  16.9715 |   1.0308 |
frame_mult                                   |   3.9272 |   3.8050 |   1.0321 |
frame_dropna_axis0_all_mixed_dtypes          | 142.3560 | 137.7486 |   1.0334 |
frame_from_series                            |   0.0670 |   0.0648 |   1.0338 |
frame_apply_np_mean                          |   4.2445 |   4.0938 |   1.0368 |
frame_interpolate_some_good                  |   1.0546 |   1.0159 |   1.0381 |
frame_ctor_dtindex_MonthBeginx1              |   0.9428 |   0.9080 |   1.0383 |
frame_ctor_dtindex_Minutex1                  |   0.8931 |   0.8599 |   1.0386 |
frame_constructor_ndarray                    |   0.0554 |   0.0532 |   1.0408 |
frame_ctor_dtindex_BQuarterEndx2             |   1.0568 |   1.0139 |   1.0423 |
frame_ctor_dtindex_QuarterEndx1              |   1.0460 |   1.0035 |   1.0423 |
frame_getitem_single_column2                 |  12.9310 |  12.3968 |   1.0431 |
frame_ctor_dtindex_MonthBeginx2              |   0.9435 |   0.9042 |   1.0435 |
frame_ctor_dtindex_Microx1                   |   0.9086 |   0.8704 |   1.0439 |
frame_ctor_dtindex_CustomBusinessDayx2       |   0.8865 |   0.8490 |   1.0441 |
frame_ctor_dtindex_CustomBusinessDayx1       |   0.8898 |   0.8517 |   1.0447 |
frame_to_csv2                                |  82.5006 |  78.9100 |   1.0455 |
frame_ctor_dtindex_BQuarterBeginx2           |   1.1443 |   1.0945 |   1.0455 |
frame_ctor_dtindex_BQuarterEndx1             |   1.0723 |   1.0249 |   1.0462 |
frame_ctor_dtindex_Dayx2                     |   0.9089 |   0.8669 |   1.0485 |
frame_ctor_dtindex_CDayx1                    |   0.8952 |   0.8536 |   1.0487 |
frame_ctor_dtindex_Easterx1                  |   0.9384 |   0.8919 |   1.0521 |
frame_ctor_dtindex_YearBeginx1               |   0.8962 |   0.8514 |   1.0526 |
frame_float_div                              |   4.8441 |   4.6001 |   1.0530 |
frame_reindex_both_axes_ix                   |  14.6171 |  13.8707 |   1.0538 |
frame_ctor_dtindex_YearEndx1                 |   0.9240 |   0.8754 |   1.0555 |
frame_ctor_dtindex_QuarterBeginx1            |   0.9843 |   0.9323 |   1.0558 |
frame_ctor_dtindex_BDayx1                    |   0.8610 |   0.8153 |   1.0561 |
frame_ctor_dtindex_MonthEndx2                |   0.9667 |   0.9151 |   1.0564 |
frame_ctor_dtindex_Millix2                   |   0.9122 |   0.8625 |   1.0576 |
frame_to_csv                                 |  96.1645 |  90.8887 |   1.0580 |
frame_ctor_dtindex_CDayx2                    |   0.9058 |   0.8557 |   1.0585 |
frame_boolean_row_select                     |   0.1848 |   0.1741 |   1.0615 |
frame_loc_dups                               |   0.7166 |   0.6750 |   1.0616 |
frame_assign_timeseries_index                |   0.5794 |   0.5451 |   1.0629 |
frame_ctor_dtindex_Minutex2                  |   0.9103 |   0.8556 |   1.0639 |
frame_ctor_dtindex_CBMonthEndx1              |   3.0158 |   2.8304 |   1.0655 |
frame_apply_user_func                        |  57.2865 |  53.6327 |   1.0681 |
frame_ctor_dtindex_CBMonthEndx2              |   3.0658 |   2.8691 |   1.0686 |
frame_ctor_dtindex_BusinessDayx1             |   0.8704 |   0.8101 |   1.0744 |
frame_ctor_dtindex_YearBeginx2               |   0.9055 |   0.8417 |   1.0758 |
frame_ctor_dtindex_MonthEndx1                |   0.9801 |   0.9096 |   1.0775 |
frame_iloc_dups                              |   0.1877 |   0.1736 |   1.0813 |
frame_dropna_axis0_all                       |  31.8504 |  29.3697 |   1.0845 |
frame_fillna_inplace                         |   8.8447 |   8.0231 |   1.1024 |
frame_ctor_dtindex_BYearEndx1                |   1.2137 |   1.0994 |   1.1040 |
frame_isnull                                 |   0.6357 |   0.5477 |   1.1606 |
frame_xs_mi_ix                               |   2.3058 |   1.9489 |   1.1831 |
-------------------------------------------------------------------------------
Test name                                    | head[ms] | base[ms] |  ratio   |
-------------------------------------------------------------------------------

Ratio < 1.0 means the target commit is faster then the baseline.
Seed used: 1234

Target [e33f3bc] : BUG: Fix #9144 #8445  Fix how core.common._fill_zeros handles div and mod by zero
Base   [76195fb] : Merge pull request #9498 from jreback/consist
```

@jreback, my code is failing because I had assumed that we want 0/0 to be NaN, so implemented that as part of the bugfix, but there's [a test](https://github.com/pydata/pandas/blob/master/pandas/tests/test_frame.py#L5147) that thinks 0/0 should be inf, which seems mistaken.

Should I go ahead and change that test? 
 see here: http://pandas.pydata.org/pandas-docs/stable/whatsnew.html#id31

also IIRC there was some discussion w.r.t. to 0/0 (but might be in another issue). pls see if you can locate it. 
 Hmmm... I couldn't find any 0/0 discussion (there was a discussion in #3590 about mod 0 though), but in the release notes you linked to for v.0.12.0, it said "Fix modulo and integer division on Series,DataFrames to act similary to float dtypes to return np.nan or np.inf as appropriate" and it gave 4 examples.  I think the fix (PR #3600) worked except for the third example as demonstrated here:

```
>>> p = pd.DataFrame({ 'first' : [4,5,8], 'second' : [0,0,3] })

>>> q = p.astype('float')

>>> print(p%0,'\n', q%0)
   first  second
0    NaN     NaN
1    NaN     NaN
2    NaN     NaN 
    first  second
0    NaN     NaN
1    NaN     NaN
2    NaN     NaN

>>> print(p%p,'\n', q%q)
   first  second
0      0     NaN
1      0     NaN
2      0       0 
    first  second
0      0     NaN
1      0     NaN
2      0       0

>>> print(p/p,'\n', q/q)    # Inconsistent!
   first    second
0      1       inf
1      1       inf
2      1  1.000000 
    first  second
0      1     NaN
1      1     NaN
2      1       1

>>> print(p/0,'\n', q/0)
   first  second
0    inf     inf
1    inf     inf
2    inf     inf 
    first  second
0    inf     inf
1    inf     inf
2    inf     inf
```

The results are the same for Python 2.

This PR would correct this.  
(_Edit: Actually, my PR fails to correct the inconsistency that's also present with `print(p//p,'\n', q//q)` and `print(p//0,'\n', q//0)`, but I'll wait to hear what you think I should do before looking into how to change that._)

I see NumPy has some strange things going on (although it sounds like they're discussing fixing this in numpy/numpy#899), for example, 

```
>>> x = np.int_(0)
>>> x / x
nan
>>> x // x
0
```

but I haven't ever seen 0/0 being infinite.  I think we should change that... 

BTW, I'll also write a release note once we decide how to handle this.
 cc @seth-p

iIIRC didn't we have s discussions about this?

what are your thoughts here?
 I agree that `0/0`, `0//0`, and `0.0/0.0`, should be `NaN` and not `inf`.

Is the issue with `0//0` that `NaN` is actually a float, and so an integer series or DataFrame column cannot contain `NaN`? So in cases where the result must be an integer (are there such cases?), I'm not sure what we want `0//0` to be -- `0` seems about as good as anything, I suppose. If there are no such cases, then never mind...

[A completely separate issue is that `NaN` is used both for (a) results that are not numbers (e.g. `0.0/0.0`), and (b) missing values (e.g. `None`). I could see a reasonable argument for representing those differently.]
 missing and undefined values are both represented by nan

I know others have different ways of doing this - bottoms line is pandss chose a long time ago to only have a single nan represent both - mainly this a perf issue (to have multiple missing value types)
 @Garrett-R go ahead snd make the change to make 0/0 be nan (both floats and ints)
eill be an api change so needs a small example / mention in the API page
 @seth-p, right, integer types don't support `NaN`, so you have to do one of the following for `0//0`:
- choose an arbitrary value (NumPy's current behavior)
- throw an exception (ex: C++ and Python)
- cast to a float (this is actually what Pandas already does and what they're discussing doing in numpy/numpy#899)
  
  `assert  (Series([0]) // Series([0])).dtype == np.float_`
  `assert  (Series([0]) // Series([1])).dtype == np.int_`

Given that we're already casting to float, and assuming no one's planning on changing that behavior, we might as well make it `NaN`.  @jreback, cool, I'll get on it!
 @Garrett-R the int/float issue is already solved. Pandas handles this correctly. That said I am thinking about this again. 

So now `1/0` -> `inf`, but `0/0` -> `NaN`. Isn't this inconsistent?
 @jreback, I believe `1/0` -> `inf` and `0/0` -> `NaN` is conventional (assuming we're casting to floats).  I wanted to try and find it in the IEEE floating point arithmetic document, but I think it's hidden behind an evil paywall.

Anyway, just check out the different languages:

---

**C++**

```
#include <iostream>
int main() {
    float x=0.0;
    std::cout << x/x << std::endl;
    std::cout << 1.0/x << std::endl;
}
```

outputs

```
-nan
inf
```

(the negative is irrelevant)

---

**Julia**

```
julia> 0/0
NaN

julia> 1/0
Inf
```

---

**R**

```
> 1/0
[1] NaN
> 0/0
[1] Inf
```

---

**Mathematica**

```
0/0
Indeterminate
1/0
ComplexInfinity
```
 pls add a release note (maybe short example as well), just to make it obvious
put in api change section in 0.16.0
 Sorry for the delay.  I've added a release note.

I also modified one more thing.  For modulus, there's an issue:

```
In [1]: s = pd.Series([0,1])

In [2]: s % 0
Out[2]: 
0   NaN
1   NaN
dtype: float64

In [3]: 0 % s        # Should give a NaN for 0%0
Out[3]: 
0    0
1    0
dtype: int64
```

The modified behavior is:

```
In [1]: s = pd.Series([0,1])

In [2]: s % 0
Out[2]: 
0   NaN
1   NaN
dtype: float64

In [3]: 0 % s
Out[3]: 
0   NaN
1     0
dtype: float64
```
 this should also fix #8445 yes? (if so, pls add that as a test as well), if not, see if its simple to extend to fix.
 Thanks for explaining the float issue.  I undid that change as suggested.

---

Yes, it fixes #8445.  I've added that to the commit message.

---

You're not gonna like this but I added more special casing.  It was to correct some new unexpected behavior I found:  `Series([-1,1]) / 0` gave signed infinities, while `Series([-1,1]) // 0` gave only positive infinities.  I believe there's no way to get around the special casing.  I've added the appropriate test.

---

As for the slow division, that was existent before this PR.  But good call on checking into that.  I found that the `_fill_zeros` methods is unnecessary if the result is already a float since then everything will already be good.  Therefore, this PR improves the truediv operation (or any Series/DataFrame operation resulting in a float) runtime by up to a factor of 10.

**old behavior**

In [1]: xx = pd.DataFrame(np.random.random((10000, 10000)))

In [2]: %timeit xx / 0
1 loops, best of 3: 1.59 s per loop

**new behavior**

In [1]: xx = pd.DataFrame(np.random.random((10000, 10000)))

In [2]: %timeit xx / 0
10 loops, best of 3: 165 ms per loop
 pls add a release note for #8445 (you can simply add it on where you have #9144)

ok on the special casing, sometimes it is unavoidable.

can you add a couple of vbenches (though use a smaller matrix, maybe 1000x1000), the proportion should still be the same; do for float/int results (for floor/div), e.g. add 4 or so (or more if you think are necessary, e.g. maybe for module too). name then consistently and add somewhere in the vbench suite.
help on vbenches here: https://github.com/pydata/pandas/wiki/Performance-Testing.

These are mainly to prevent performance regressions if/when things are changed in the future. Post the results in the top of the PR as well.

thxs
 @jreback, how's it looking now?

Also, I've edited my first comment in the PR to include the results of the vbenches (including the ones I added).  Is this what you meant by "the top of the PR"?
 @Garrett-R only show the relevant vbenches and compare vs current master

e.g. bring your master up to date, then
`.\test_perf.sh -b master -t HEAD -r frame_`

they top of the PR is correct (just replace it with the revised results).
 @Garrett-R minor doc changes, and revise the vbench, otherwise looks good to go.
 @jreback, I've made the suggested doc changes and also revised the vbenches.
 @Garrett-R thanks for this! nice work!
