New kernel for reciprocal

[Ka-zam]

New kernel that calculates the reciprocal:

volk_profile -R rec -i 10000

RUN_VOLK_TESTS: volk_32f_reciprocal_32f(131071,10000)
generic completed in 119.092 ms
a_avx2_fma completed in 94.7562 ms
a_avx completed in 97.1054 ms
a_avx_div completed in 118.053 ms
u_avx2_fma completed in 97.3939 ms
u_avx completed in 98.4622 ms
u_avx_div completed in 129.018 ms
Best aligned arch: a_avx2_fma
Best unaligned arch: u_avx2_fma

[jdemel]

This looks like an interesting kernel. I have a few comments.
My biggest worry would be to see this kernel fail randomly with some 1.0f / 1e-xx value.

[Ka-zam]

This looks like an interesting kernel. I have a few comments. My biggest worry would be to see this kernel fail randomly with some 1.0f / 1e-xx value.

I ran the code for some special values:

	x	x		generic		div		rcp		avx2_fma		avx
-0.000000e+00	 -0x0p+0	    -inf	    -inf	    -inf	    -nan	    -nan
+0.000000e+00	 +0x0p+0	    +inf	    +inf	    +inf	    -nan	    -nan
+3.673420e-40	+0x1p-131	    +inf	    +inf	    +inf	    -inf	    -inf
+5.877472e-39	+0x1p-127	+1.701412e+38	+1.701412e+38	    +inf	    -inf	    -inf
+1.175494e-38	+0x1.fffffp-127	+8.507063e+37	+8.507063e+37	    +inf	    -inf	    -inf
+1.175494e-38	+0x1p-126	+8.507059e+37	+8.507059e+37	+8.504982e+37	+8.507059e+37	+8.507059e+37
+4.253530e+37	+0x1p+125	+2.350989e-38	+2.350989e-38	+2.350415e-38	+2.350989e-38	+2.350989e-38
    +inf	    +inf	+0.000000e+00	+0.000000e+00	+0.000000e+00	    -nan	    -nan

div is the _mm256_div_ps instruction
rcp is the _mm256_rcp_ps instruction

So, the proposed implementation fails for special values more or less spectactularly. However, it's 20% faster for valid inputs and more accurate than _mm256_rcp_ps.

_mm256_rcp_ps is 10% faster still but not as accurate in the valid domain.

[Ka-zam]

I did some more experiments and tried to add checks for nan, 0.0 and inf. Turns out it's not worth it and then it's better to just use div_ps()

So, 30% faster but

• returns nan instead of inf for 0.0 input
• returns nan instead of 0.0 for inf input

[jdemel]

How do we prevent that this kernel is constantly causing random CI failures? Values close to 0.0f are likely in our tests. We'd need to make sure that we don't see flaky kernel issues raised.

Also, the inf vs nan issue needs to be discussed. If I'd use this kernel I'd expect that 1.0f / inf = 0.0f. This is not consistent across all implementations. This would probably cause issues in the future.
We had a similar issue in #622 and eventually had to remove some kernel implementations #658 . I'm worried this kernel will see the same fate. We should find a solution for this issue before we merge it.

[Ka-zam]

Values close to 0.f are no problem, it's only a very slight problem for subnormals where there's decreased accuracy.

inf vs nan is more of a problem. Perhaps that can be looked at but the problem is we can't do too much work before the div_ps( 1.f / x ) instruction is faster.

I will investigate.