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    lavu/x86: add FFT assembly · 119a3f7e
    Lynne authored
    This commit adds a pure x86 assembly SIMD version of the FFT in libavutil/tx.
    The design of this pure assembly FFT is pretty unconventional.
    
    On the lowest level, instead of splitting the complex numbers into
    real and imaginary parts, we keep complex numbers together but split
    them in terms of parity. This saves a number of shuffles in each transform,
    but more importantly, it splits each transform into two independent
    paths, which we process using separate registers in parallel.
    This allows us to keep all units saturated and lets us use all available
    registers to avoid dependencies.
    Moreover, it allows us to double the granularity of our per-load permutation,
    skipping many expensive lookups and allowing us to use just 4 loads per register,
    rather than 8, or in case FMA3 (and by extension, AVX2), use the vgatherdpd
    instruction, which is at least as fast as 4 separate loads on old hardware,
    and quite a bit faster on modern CPUs).
    
    Higher up, we go for a bottom-up construction of large transforms, foregoing
    the traditional per-transform call-return recursion chains. Instead, we always
    start at the bottom-most basis transform (in this case, a 32-point transform),
    and continue constructing larger and larger transforms until we return to the
    top-most transform.
    This way, we only touch the stack 3 times per a complete target transform:
    once for the 1/2 length transform and two times for the 1/4 length transform.
    
    The combination algorithm we use is a standard Split-Radix algorithm,
    as used in our C code. Although a version with less operations exists
    (Steven G. Johnson and Matteo Frigo's "A modified split-radix FFT with fewer
    arithmetic operations", IEEE Trans. Signal Process. 55 (1), 111–119 (2007),
    which is the one FFTW uses), it only has 2% less operations and requires at least 4x
    the binary code (due to it needing 4 different paths to do a single transform).
    That version also has other issues which prevent it from being implemented
    with SIMD code as efficiently, which makes it lose the marginal gains it offered,
    and cannot be performed bottom-up, requiring many recursive call-return chains,
    whose overhead adds up.
    
    We go through a lot of effort to minimize load/stores by keeping as much in
    registers in between construcring transforms. This saves us around 32 cycles,
    on paper, but in reality a lot more due to load/store aliasing (a load from a
    memory location cannot be issued while there's a store pending, and there are
    only so many (2 for Zen 3) load/store units in a CPU).
    Also, we interleave coefficients during the last stage to save on a store+load
    per register.
    
    Each of the smallest, basis transforms (4, 8 and 16-point in our case)
    has been extremely optimized. Our 8-point transform is barely 20 instructions
    in total, beating our old implementation 8-point transform by 1 instruction.
    Our 2x8-point transform is 23 instructions, beating our old implementation by
    6 instruction and needing 50% less cycles. Our 16-point transform's combination
    code takes slightly more instructions than our old implementation, but makes up
    for it by requiring a lot less arithmetic operations.
    
    Overall, the transform was optimized for the timings of Zen 3, which at the
    time of writing has the most IPC from all documented CPUs. Shuffles were
    preferred over arithmetic operations due to their 1/0.5 latency/throughput.
    
    On average, this code is 30% faster than our old libavcodec implementation.
    It's able to trade blows with the previously-untouchable FFTW on small transforms,
    and due to its tiny size and better prediction, outdoes FFTW on larger transforms
    by 11% on the largest currently supported size.
    119a3f7e
tx.c 7.56 KB