BUG: infer_date fails to infer semi-month frequencies

[rodrigolece]

Pandas version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

>>> import pandas as pd
>>> idx = pd.date_range("2017", freq="SMS", periods=4)
>>> idx
DatetimeIndex(['2017-01-01', '2017-01-15', '2017-02-01', '2017-02-15'],
              dtype='datetime64[ns]', freq='SMS-15')
>>> pd.infer_freq(idx)
# returns None
>>> idx = pd.date_range("2017", freq="SM", periods=4)
>>> pd.infer_freq(idx)
# returns None

Issue Description

The infer_freq method fails to infer the a Semi-Month frequency (both regular and start)

Expected Behavior

>>> idx = pd.date_range("2017", freq="SMS", periods=4)
>>> pd.infer_freq(idx)
'SMS'

Installed Versions

NSTALLED VERSIONS

commit : bb1f651
python : 3.9.7.final.0
python-bits : 64
OS : Linux
OS-release : 5.4.0-96-generic
Version : #109-Ubuntu SMP Wed Jan 12 16:49:16 UTC 2022
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : None
LANG : en_GB.UTF8
LOCALE : en_GB.UTF-8

pandas : 1.4.0
numpy : 1.20.1
pytz : 2021.3
dateutil : 2.8.2
pip : 21.2.4
setuptools : 58.0.4
Cython : None
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : 3.0.2
IPython : 7.22.0
pandas_datareader: None
bs4 : None
bottleneck : 1.3.2
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : 3.3.4
numba : None
numexpr : 2.8.1
odfpy : None
openpyxl : None
pandas_gbq : None
pyarrow : None
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : 1.6.2
sqlalchemy : None
tables : None
tabulate : 0.8.9
xarray : None
xlrd : None
xlwt : None
zstandard : None

[jbrockmendel]

pls give the issue an informative name

[rodrigolece]

Really sorry, I was more concerned about filling all the fields, copying the reproducible code etc and I forgot!