Slow inspect.getmodule

Bug report

inspect.getmodule is slow. It becomes slower as len(sys.modules) grows. It is slower with modules that don't resolve to valid file paths. This causes inspect.stack and others to be slow as well.

def run_test(module):
    # cols have stack depths
    # rows have len(sys.modules)

    def print_line(h, *vals):
        if not h:
            print("%5s" % h, *["%6i" % v for v in vals])
        else:
            print("%5s" % h, *["%6.1f" % v for v in vals])

    def add_modules(n):
        import random
        import sys
        for _ in range(n):
            sys.modules[f"foo_{random.randint(0,2**64)}"] = module

    def measure(depth):
        import inspect
        import timeit

        def nest(level):
            if level > 0:
                return nest(level-1)
            else:
                dur = timeit.timeit(lambda: inspect.stack(), number=1)
                return dur * 1000

        return nest(depth)

    import sys
    print_line("", *[2**e for e in range(7)])
    for i in range(1, 5):
        add_modules(n=10**i)
        times = [measure(depth=2**e) for e in range(7)]
        print_line(len(sys.modules), *times)

python 3.11.0a7+ optimized build ====

# sys has no __file__ and doesn't enter getmodule cache
>>> import sys
>>> run_test(sys)
           1      2      4      8     16     32     64
   87    2.1    0.8    1.0    1.3    2.0    3.3    6.1
  187    2.0    2.5    2.7    3.6    5.5    9.5   16.8
 1187   13.6   15.7   19.1   26.3   39.6   67.4  123.3
11187  134.1  150.3  183.4  252.0  383.5  750.3 1189.8
>>> exit()

>>> import random
>>> run_test(random)
           1      2      4      8     16     32     64
   87    2.1    0.7    0.9    1.2    1.7    2.9    5.6
  187    2.3    1.1    1.5    1.8    2.5    4.3    7.8
 1187   17.0    4.3    5.6    7.1   11.1   18.6   33.9
11187  164.6   37.4   45.7   61.6   94.7  161.0  297.7
>>> exit()

Tables show time in milliseconds for a single inspect.stack() call.
Rows show len(sys.modules)
Columns indicate stack depth

Your environment

• CPython versions tested on: 3.9, 3.11.0a7+
• Operating system and architecture: MacOS, Linux, x64

• PR: gh-92042

• PR: gh-154403

• PR: gh-156685

Start by reading the implementations of inspect.getmodule and inspect.stack, then reproduce the supplied benchmark with increasing sys.modules sizes and stack depths. Review the referenced PRs gh-92042, gh-154403, and gh-156685 before making changes; done means the reported scaling problem is addressed and the benchmark or relevant tests show the improvement.