Why is numpy’s einsum faster than numpy’s built in functions?

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First off, there’s been a lot of past discussion about this on the numpy list. For example, see:
http://numpy-discussion.10968.n7.nabble.com/poor-performance-of-sum-with-sub-machine-word-integer-types-td41.html
http://numpy-discussion.10968.n7.nabble.com/odd-performance-of-sum-td3332.html

Some of boils down to the fact that einsum is new, and is presumably trying to be better about cache alignment and other memory access issues, while many of the older numpy functions focus on a easily portable implementation over a heavily optimized one. I’m just speculating, there, though.

However, some of what you’re doing isn’t quite an “apples-to-apples” comparison.

In addition to what @Jamie already said, sum uses a more appropriate accumulator for arrays

For example, sum is more careful about checking the type of the input and using an appropriate accumulator. For example, consider the following:

In [1]: x = 255 * np.ones(100, dtype=np.uint8)

In [2]: x
Out[2]:
array([255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255,
       255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255,
       255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255,
       255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255,
       255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255,
       255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255,
       255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255,
       255, 255, 255, 255, 255, 255, 255, 255, 255], dtype=uint8)

Note that the sum is correct:

In [3]: x.sum()
Out[3]: 25500

While einsum will give the wrong result:

In [4]: np.einsum('i->', x)
Out[4]: 156

But if we use a less limited dtype, we’ll still get the result you’d expect:

In [5]: y = 255 * np.ones(100)

In [6]: np.einsum('i->', y)
Out[6]: 25500.0

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