如果参数大小大于 8192，为什么 numpy.sin 会返回不同的结果?

问题描述

我发现当参数大小为 <= 8192 和 > 8192 时，numpy.sin 的行为不同.不同之处在于性能和返回的值.有人能解释一下这种效果吗?

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?

例如，让我们计算 sin(pi/4):

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

超过 8192 个元素限制后，计算速度提高了 3 倍以上，并给出了不同的结果:最后一位数字变为 5 而不是 6.

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.

当我尝试以其他方式计算相同的值时:

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

C++ std::sin(Visual Studio 2017，Win32 平台)给出 0.7071067811865475；
C++ std::sin(Visual Studio 2017，x64 平台)给出 0.70710678118654756；
math.sin 给出 0.7071067811865476，这是合乎逻辑的，因为我使用的是 64 位 Python.

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.

我在 NumPy 文档及其代码中都找不到任何解释.

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

更新 #2:很难相信，但是将 sin 替换为 sqrt 给出了这样的结果:

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

更新:np.show_config() 输出:

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']

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