# Maths to the rescue

I had a bit of a think about the expression I am evaluating and I noticed I could use logarithms to perform the calculation instead of multiplies. And actually, as I don't use the value itself other than as a threshold, the summation result can be used directly anyway - it just wont give as sharp peaks.

First, changed the inner loop to perform a summation of two possible values, in 16 bits. This removed the need for 32-bit masks, and the floating point multiplies (not that they were holding anything up). This took 2 classifiers down from 26 instructions to 18, and a test-case from 100 to 70ms. It also freed 3 quad registers but I haven't yet investigated whether this will let me do more (actually it will let me take the loop down to 16 instructions as I have 2 immediate loads in there).

However, this morning I had a closer look at the summation I now have.

r = sum m { testbit ? p : n } (1)

The bit testing code can't be further improved, but perhaps the select and sum can?

First, subtract n from the expression and move it outside of the select, and then separate the summation.

r = sum m { (testbit ? (p - n) : 0) - n } = sum m { testbit ? (p - n) : 0 } - sum m { n } = sum m { testbit ? (p - n) : 0 } - mn (2)

Which leaves with a slightly simpler internal summation and a constant. This expression can be implemented without a select instruction but since we have select in NEON it doesn't gain any instructions on it's own.

But this is where a neat trick of binary arithmetic comes in to play. Unlike C logic which uses 0 and 1 for false and true, the result of a bit test in SIMD code (and opencl c on vectors) is 0 for false, and ~0 for true. But in two's compliment signed binary, ~0 is also equal to -1.

So ... if the expression can be converted into a sum of -1's, then all one needs to do is directly add up the result of the bit tests, and avoid the select altogether. It is rather trivial to convert (2) into this form (if I have my summation identities on target anyway).

r = sum m { (n-p) (testbit ? -1 : 0) } - mn = (n-p) sum m { testbit ? -1 : 0 } - mn = A sum m { testbit ? -1 : 0 } + B (3)

Where A and B are constants.

Actually there is another benefit here, as the summand is either 0 or -1 the range of the working sum is far more limited. Because of the size of the template I couldn't get away with 8 bit arithmetic without the danger of overflow - if I was using normal 8 bit two's complement arithmetic. But NEON to the rescue again, it has saturating instructions which will not overflow. It may still lead to a slightly incorrect answer - but i'm not interested in the absolute result but the relative one, and i think it should suffice.

I have yet to code this up, but if it works this will get the two-classifier case down to only 12 instructions - for 8 pixels. Less than one instruction per classifier per template, or about one when including the data loading as well.

And as this all now runs in double registers instead of quad registers, and I don't need n or p in the inner loop, it freeds up a whole swag of registers. These can be used to unroll more of the loop or share more of the data, for example by calculating more than 8 results per pass, more than one classifier at once, and so on. This might squeeze out a tad more juice out of it, but i'm not expecting much more here.

**Update:** So I coded it, it works. I'm calculating 32 locations at once, and there's still a good few registers free. Test case down from 70ms to 50ms.

Even better, I noticed the 'limited dual issue' capability of the NEON processor could be exploited in a couple of cases. Interleave the vtbl with some shifts: down to 43ms. Interleave the vqadd's with some vext's: down to under 41ms. Noice.

For comparison, an integer version in C (gcc, -O2) takes about 700ms.

Time for beer.