Fast cross product is not that fast

Relevant code: https://github.com/nipy/PySurfer/blob/a5a019ec5c6d25cdef92aca84a5e1ce1f9ba6fef/surfer/utils.py#L179-L213

Hi,

I was browsing through the code and I accidentally stumbled upon the function _fast_cross_3d referenced in the URL above. I don't think that this implementation is faster than numpy's implementation of the cross product - for example, please take a look at the following code:

import numpy as np

x = np.random.rand(1_000_000, 3)
y = np.random.rand(1, 3)

def cross_3d_pysurfer(x, y):
    return np.c_[x[:, 1] * y[:, 2] - x[:, 2] * y[:, 1],
                 x[:, 2] * y[:, 0] - x[:, 0] * y[:, 2],
                 x[:, 0] * y[:, 1] - x[:, 1] * y[:, 0]]

def cross_3d_numpy(x, y):
    return np.cross(x, y)

When I time it, I get the following results:

In [1]: %timeit cross_3d_pysurfer(x, y)
Out[1]: 34.7 ms ± 3.9 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

In [2]: %timeit cross_3d_numpy(x, y)
Out[2]: 29.4 ms ± 4.63 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

A potential, but very small speed up could be achieved by using np.stack instead of np.c_ because it has some cool compiled numpy's stuff which np.c_ lacks. For example:

def cross_3d_pysurfer_modified(x, y):
    return np.stack((x[:, 1] * y[:, 2] - x[:, 2] * y[:, 1],
                     x[:, 2] * y[:, 0] - x[:, 0] * y[:, 2],
                     x[:, 0] * y[:, 1] - x[:, 1] * y[:, 0])).T

In [3]: %timeit cross_3d_pysurfer_modified(x, y)
Out[3]: 28.4 ms ± 3.95 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

I didn't want to open a PR because I really don't know if this is even relevant anymore or if the difference in a few ms is that important, but still wanted to let you know :)

nipy / PySurfer

Fast cross product is not that fast #315