Consider the following piece of code, which generates some (potentially) huge, multi-dimensional array and performs numpy.tensordot with it (whether we multiply the same or two different arrays here, does not really matter).

import time
import numpy
L, N = 6, 4
shape = (2*L)*[N,]
A = numpy.arange(numpy.prod(shape)).reshape(shape)
A = A % 256 - 128 # [-127,+127]
axes=(range(1,2*L,2), range(0,2*L,2))
def run(dtype, repeat=1):
A_ = A.astype(dtype)
t = time.time()
for i in range(repeat):
numpy.tensordot(A_, A_, axes)
t = time.time() - t
print(dtype, ' \t%8.2f sec\t%8.2f MB' %(t, A_.nbytes/1e6))

Now we can compare the performance for different data types, e.g.:

run(numpy.float64)
run(numpy.int64)

Since the array only consists of small integer numbers, I would like to save some memory by using dtype=int8. However, this slows down the matrix multiplication A LOT.

## Here are some test cases

The first one, is the important one for my use case. The others are just for reference. Using Numpy 1.13.1 and Python 3.4.2

### Large array

L, N = 6, 4; A.size = 4**12 = 16777216
<class 'numpy.float64'> 59.58 sec 134.22 MB
<class 'numpy.float32'> 44.19 sec 67.11 MB
<class 'numpy.int16'> 711.16 sec 33.55 MB
<class 'numpy.int8'> 647.40 sec 16.78 MB

Same array with different data types. Memory decreases as expected. But why the large differences in the CPU time? If anything I would expect int to be faster than float.

### Large array with different shape

L, N = 1, 4**6; A.size = (4**6)**2 = 16777216
<class 'numpy.float64'> 57.95 sec 134.22 MB
<class 'numpy.float32'> 42.84 sec 67.11 MB

The shape doesn't seem to have a large effect.

### Not so large array

L, N = 5, 4
<class 'numpy.float128'> 10.91 sec 16.78 MB
<class 'numpy.float64'> 0.98 sec 8.39 MB
<class 'numpy.float32'> 0.90 sec 4.19 MB
<class 'numpy.float16'> 9.80 sec 2.10 MB
<class 'numpy.int64'> 8.84 sec 8.39 MB
<class 'numpy.int32'> 5.55 sec 4.19 MB
<class 'numpy.int16'> 2.23 sec 2.10 MB
<class 'numpy.int8'> 1.82 sec 1.05 MB

Smaller values, but same weird trend.

### small array, lots of repetitions

L, N = 2, 4; A.size = 4**4 = 256; repeat=1000000

<class 'numpy.float128'> 17.92 sec 4.10 KB
<class 'numpy.float64'> 14.20 sec 2.05 KB
<class 'numpy.float32'> 12.21 sec 1.02 KB
<class 'numpy.float16'> 41.72 sec 0.51 KB
<class 'numpy.int64'> 14.21 sec 2.05 KB
<class 'numpy.int32'> 14.26 sec 1.02 KB
<class 'numpy.int16'> 13.88 sec 0.51 KB
<class 'numpy.int8'> 13.03 sec 0.26 KB

Other than float16 being much slower, everything is fine here.

Why is int8 for very large arrays so much slower? Is there any way around this? Saving memory becomes increasingly important for larger arrays!