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time.localtime(float("NaN")) raises a ValueError on x86_64 using the few compilers I have tested it with. (this makes sense)
>>> time.localtime(float("NaN"))
Traceback (most recent call last):
 File "<stdin>", line 1, in <module>
ValueError: (75, 'Value too large for defined data type')
On an arm and arm64 system, it does not and treats NaN as 0. (nonsense!)
>>> time.localtime(float("NaN"))
time.struct_time(tm_year=1970, tm_mon=1, tm_mday=1, tm_hour=0, tm_min=0, tm_sec=0, tm_wday=3, tm_yday=1, tm_isdst=0)
The root of this problem appears to be the (potentially questionable? I'll ask a C compiler person...) code in Python/pytime.c's (3.x) Modules/timemodule.c's (2.7) double to time_t conversion function.
I'm not sure what it does is supposed to be well defined behavior with NaN...
The easy fix is to add:
#include <math.h>
and
add || isnan(x) || isinf(x) to the check that raises a ValueError in
https://hg.python.org/cpython/file/4c903ceeb4d1/Python/pytime.c#l149 (3.x)
https://hg.python.org/cpython/file/2.7/Modules/timemodule.c#l102 (2.7)
Along with a relevant assertRaises(ValueError) unittest for NaN, inf and -inf in test_time.py.

> potentially questionable? I'll ask a C compiler person...
Questionable indeed. Attempting to cast a NaN to an integer type results in undefined behaviour. Unfortunately, so does attempting to cast any double value that's outside the range represented by the `time_t` type, so the questionability extends beyond just the NaN handling.

I like PR 3085 to raise explicitly a ValueError with an helpful error message.

New changeset 829dacce4fca60fc3c3367980e75e21dfcdbe6be by Victor Stinner (Han Lee) in branch 'master':
bpo-26669: Fix nan arg value error in pytime.c (#3085)
https://github.com/python/cpython/commit/829dacce4fca60fc3c3367980e75e21dfcdbe6be

For Python 2.7, you have at least to fix these 2 functions: parse_time_double_args(), _PyTime_DoubleToTimet().

New changeset a4baf1c543bca261c27e98ba296e42665f3cb872 by Mariatta (Miss Islington (bot)) in branch '3.6':
[3.6] bpo-26669: Fix nan arg value error in pytime.c (GH-3085) (GH-3467) 
https://github.com/python/cpython/commit/a4baf1c543bca261c27e98ba296e42665f3cb872

This has been backported to 3.6.
Is backport to 2.7 needed? If not we'll close the issue.

I wrote PR 11507 to raise ValueError rather than OverflowError for -inf and +inf.

Victor's PR 11507 is closed, what actions are going to be taken next? Close the issue as Mariatta said?

Python 2 users have this bug since 2010, I think that it's fine to not fix it. Python 2 support ends at the end of the year. I close the issue.
The issue has been fixed in Python 3.6 and newer.