Why does numpy.sin return a different result if the argument size is greater than 8192?

Question

I discovered that numpy.sin behaves differently when the argument size is <= 8192 and when it is > 8192. The difference is in both performance and values returned. Can someone explain this effect?

For example, let's calculate sin(pi/4):

x = np.pi*0.25
for n in range(8191, 8195):
    xx = np.repeat(x, n)
    %timeit np.sin(xx)
    print(n, np.sin(xx)[0])

64.7 µs ± 194 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
8191 0.7071067811865476
64.6 µs ± 166 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
8192 0.7071067811865476
20.1 µs ± 189 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
8193 0.7071067811865475
21.8 µs ± 13.4 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
8194 0.7071067811865475

After crossing the 8192 elements limit the calculations become more than 3 times faster and give a different result: the last digit becomes 5 instead of 6.

When I tried to calculate the same value in other ways I obtained:

• C++ std::sin (Visual Studio 2017, Win32 platform) gives 0.7071067811865475;
• C++ std::sin (Visual Studio 2017, x64 platform) gives 0.70710678118654756;
• math.sin gives 0.7071067811865476, which is logical because I used 64-bit Python.

I couldn't find any explanation in the NumPy documentation, nor in its code.

Update #2: It is hard to believe, but replacing sin by sqrt gives this:

44.2 µs ± 751 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
8191 0.8862269254527579
44.1 µs ± 543 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
8192 0.8862269254527579
10.3 µs ± 105 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
8193 0.886226925452758
10.4 µs ± 4.41 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
8194 0.886226925452758

Update: np.show_config() output:

mkl_info:
    libraries = ['mkl_rt']
    library_dirs = ['C:/GNU/Anaconda3\\Library\\lib']
    define_macros = [('SCIPY_MKL_H', None), ('HAVE_CBLAS', None)]
    include_dirs = ['C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\include', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\lib', 'C:/GNU/Anaconda3\\Library\\include']
blas_mkl_info:
    libraries = ['mkl_rt']
    library_dirs = ['C:/GNU/Anaconda3\\Library\\lib']
    define_macros = [('SCIPY_MKL_H', None), ('HAVE_CBLAS', None)]
    include_dirs = ['C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\include', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\lib', 'C:/GNU/Anaconda3\\Library\\include']
blas_opt_info:
    libraries = ['mkl_rt']
    library_dirs = ['C:/GNU/Anaconda3\\Library\\lib']
    define_macros = [('SCIPY_MKL_H', None), ('HAVE_CBLAS', None)]
    include_dirs = ['C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\include', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\lib', 'C:/GNU/Anaconda3\\Library\\include']
lapack_mkl_info:
    libraries = ['mkl_rt']
    library_dirs = ['C:/GNU/Anaconda3\\Library\\lib']
    define_macros = [('SCIPY_MKL_H', None), ('HAVE_CBLAS', None)]
    include_dirs = ['C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\include', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\lib', 'C:/GNU/Anaconda3\\Library\\include']
lapack_opt_info:
    libraries = ['mkl_rt']
    library_dirs = ['C:/GNU/Anaconda3\\Library\\lib']
    define_macros = [('SCIPY_MKL_H', None), ('HAVE_CBLAS', None)]
    include_dirs = ['C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\include', 'C:\\Program Files (x86)\\IntelSWTools\\compilers_and_libraries_2019.0.117\\windows\\mkl\\lib', 'C:/GNU/Anaconda3\\Library\\include']

Answer 1

As @WarrenWeckesser wrote, "it's almost certainly an Anaconda & Intel MKL issue; cf. https://github.com/numpy/numpy/issues/11448 and https://github.com/ContinuumIO/anaconda-issues/issues/9129".

And unfortunately, the only way to solve the issue under Windows is to uninstall Anaconda and use another distribution with MKL-free numpy. I used python-3.6.6-amd64 from https://www.python.org/ and installed everything else via pip, including numpy 1.14.5. I even managed to make Spyder work (had to downgrade PyQt5 to 5.11.3, it refused to launch on >= 5.12).

Now np.sin(xx) is consistently 0.7071067811865476 (67.1 µs at n = 8192) and np.sqrt(xx) 0.8862269254527579 (16.4 µs). A bit slower, but perfectly reproducible.