abs() is slow for complex, add abs2()

See the following code (run on Ubuntu 64bits):

In [1]: import numpy as np

In [2]: b = np.random.rand(500, 500) + 1j * np.random.rand(500, 500)

In [3]: %timeit np.sqrt(b.real**2 + b.imag**2)
100 loops, best of 3: 4.15 ms per loop

In [4]: %timeit np.abs(b)
100 loops, best of 3: 6.06 ms per loop

abs() is slow compared to the manual formula. Is there any drawback in using np.sqrt(b.real**2 + b.imag**2) (like possible overflow ?)

Also, it could be useful to add an abs2() function which would return the value of abs()**2 but should be really faster for complex:

In [5]: %timeit b.real**2 + b.imag**2
1000 loops, best of 3: 1.4 ms per loop

It is commonly used to compute the energy of a signal or to sort complex numbers by norm value, for example.

[kgryte]

A proposal has been made for adding abs2 to the array API specification. Were this proposal to be accepted, this is likely to affect the outcome of this issue and https://github.com/scipy/scipy/pull/15390.

[dlindelof]

I’ve implemented a first version of abs2() in https://github.com/scipy/scipy/pull/15390.