Why does mulss take only 3 cycles on Haswell, different from Agner's instruction tables? (Unrolling FP loops with multiple accumulators)

Related:

• AVX2: Computing dot product of 512 float arrays has a good manually-vectorized dot-product loop using multiple accumulators with FMA intrinsics. The rest of the answer explains why that's a good thing, with cpu-architecture / asm details.
• Dot Product of Vectors with SIMD shows that with the right compiler options, some compilers will auto-vectorize that way.
• Loop unrolling to achieve maximum throughput with Ivy Bridge and Haswell another version of this Q&A with more focus on unrolling to hide latency (and bottleneck on throughput), less background on what that even means. And with examples using C intrinsics.

Look at your loop again: movss xmm1, src has no dependency on the old value of xmm1, because its destination is write-only. Each iteration's mulss is independent. Out-of-order execution can and does exploit that instruction-level parallelism, so you definitely don't bottleneck on mulss latency.

Optional reading: In computer architecture terms: register renaming avoids the WAR anti-dependency data hazard of reusing the same architectural register. (Some pipelining + dependency-tracking schemes before register renaming didn't solve all the problems, so the field of computer architecture makes a big deal out of different kinds of data hazards.

Register renaming with Tomasulo's algorithm makes everything go away except the actual true dependencies (read after write), so any instruction where the destination is not also a source register has no interaction with the dependency chain involving the old value of that register. (Except for false dependencies, like popcnt on Intel CPUs, and writing only part of a register without clearing the rest (like mov al, 5 or sqrtss xmm2, xmm1). Related: Why do x86-64 instructions on 32-bit registers zero the upper part of the full 64-bit register?).

Back to your code:

.L13:
    movss   xmm1, DWORD PTR [rdi+rax*4]  
    mulss   xmm1, DWORD PTR [rsi+rax*4]   
    add     rax, 1                       
    cmp     cx, ax
    addss   xmm0, xmm1
    jg      .L13

The loop-carried dependencies (from one iteration to the next) are each:

• xmm0, read and written by addss xmm0, xmm1, which has 3 cycle latency on Haswell.
• rax, read and written by add rax, 1. 1c latency, so it's not the critical-path.

It looks like you measured the execution time / cycle-count correctly, because the loop bottlenecks on the 3c addss latency.

This is expected: the serial dependency in a dot product is the addition into a single sum (aka the reduction), not the multiplies between vector elements. (Unrolling with multiple sum accumulator variables / registers can hide that latency.)

That is by far the dominant bottleneck for this loop, despite various minor inefficiencies:

short i produced the silly cmp cx, ax, which takes an extra operand-size prefix. Luckily, gcc managed to avoid actually doing add ax, 1, because signed-overflow is Undefined Behaviour in C. So the optimizer can assume it doesn't happen. (update: integer promotion rules make it different for short, so UB doesn't come into it, but gcc can still legally optimize. Pretty wacky stuff.)

If you'd compiled with -mtune=intel, or better, -march=haswell, gcc would have put the cmp and jg next to each other where they could macro-fuse.

I'm not sure why you have a * in your table on the cmp and add instructions. (update: I was purely guessing that you were using a notation like IACA does, but apparently you weren't). Neither of them fuse. The only fusion happening is micro-fusion of mulss xmm1, [rsi+rax*4].

And since it's a 2-operand ALU instruction with a read-modify-write destination register, it stays macro-fused even in the ROB on Haswell. (Sandybridge would un-laminate it at issue time.) Note that vmulss xmm1, xmm1, [rsi+rax*4] would un-laminate on Haswell, too.

None of this really matters, since you just totally bottleneck on FP-add latency, much slower than any uop-throughput limits. Without -ffast-math, there's nothing compilers can do. With -ffast-math, clang will usually unroll with multiple accumulators, and it will auto-vectorize so they will be vector accumulators. So you can probably saturate Haswell's throughput limit of 1 vector or scalar FP add per clock, if you hit in L1D cache.

With FMA being 5c latency and 0.5c throughput on Haswell, you would need 10 accumulators to keep 10 FMAs in flight and max out FMA throughput by keeping p0/p1 saturated with FMAs. (Skylake reduced FMA latency to 4 cycles, and runs multiply, add, and FMA on the FMA units. So it actually has higher add latency than Haswell.)

(You're bottlenecked on loads, because you need two loads for every FMA. In other cases, you can actually gain add throughput by replacing some a vaddps instruction with an FMA with a multiplier of 1.0. This means more latency to hide, so it's best in a more complex algorithm where you have an add that's not on the critical path in the first place.)

Re: uops per port:

there are 1.19 uops per loop in the port 5, it is much more than expect 0.5, is it the matter about the uops dispatcher trying to make uops on every port same

Yes, something like that.

The uops are not assigned randomly, or somehow evenly distributed across every port they could run on. You assumed that the add and cmp uops would distribute evenly across p0156, but that's not the case.

The issue stage assigns uops to ports based on how many uops are already waiting for that port. Since addss can only run on p1 (and it's the loop bottleneck), there are usually a lot of p1 uops issued but not executed. So few other uops will ever be scheduled to port1. (This includes mulss: most of the mulss uops will end up scheduled to port 0.)

Taken-branches can only run on port 6. Port 5 doesn't have any uops in this loop that can only run there, so it ends up attracting a lot of the many-port uops.

The scheduler (which picks unfused-domain uops out of the Reservation Station) isn't smart enough to run critical-path-first, so this is assignment algorithm reduces resource-conflict latency (other uops stealing port1 on cycles when an addss could have run). It's also useful in cases where you bottleneck on the throughput of a given port.

Scheduling of already-assigned uops is normally oldest-ready first, as I understand it. This simple algorithm is hardly surprising, since it has to pick a uop with its inputs ready for each port from a 60-entry RS every clock cycle, without melting your CPU. The out-of-order machinery that finds and exploits the ILP is one of the significant power costs in a modern CPU, comparable to the execution units that do the actual work.

Related / more details: How are x86 uops scheduled, exactly?

More performance analysis stuff:

Other than cache misses / branch mispredicts, the three main possible bottlenecks for CPU-bound loops are:

• dependency chains (like in this case)
• front-end throughput (max of 4 fused-domain uops issued per clock on Haswell)
• execution port bottlenecks, like if lots of uops need p0/p1, or p2/p3, like in your unrolled loop. Count unfused-domain uops for specific ports. Generally you can assuming best-case distribution, with uops that can run on other ports not stealing the busy ports very often, but it does happen some.

A loop body or short block of code can be approximately characterized by 3 things: fused-domain uop count, unfused-domain count of which execution units it can run on, and total critical-path latency assuming best-case scheduling for its critical path. (Or latencies from each of input A/B/C to the output...)

For example of doing all three to compare a few short sequences, see my answer on What is the efficient way to count set bits at a position or lower?

For short loops, modern CPUs have enough out-of-order execution resources (physical register file size so renaming doesn't run out of registers, ROB size) to have enough iterations of a loop in-flight to find all the parallelism. But as dependency chains within loops get longer, eventually they run out. See Measuring Reorder Buffer Capacity for some details on what happens when a CPU runs out of registers to rename onto.

See also lots of performance and reference links in the x86 tag wiki.

Tuning your FMA loop:

Yes, dot-product on Haswell will bottleneck on L1D throughput at only half the throughput of the FMA units, since it takes two loads per multiply+add.

If you were doing B[i] = x * A[i] + y; or sum(A[i]^2), you could saturate FMA throughput.

It looks like you're still trying to avoid register reuse even in write-only cases like the destination of a vmovaps load, so you ran out of registers after unrolling by 8. That's fine, but could matter for other cases.

Also, using ymm8-15 can slightly increase code-size if it means a 3-byte VEX prefix is needed instead of 2-byte. Fun fact: vpxor ymm7,ymm7,ymm8 needs a 3-byte VEX while vpxor ymm8,ymm8,ymm7 only needs a 2-byte VEX prefix. For commutative ops, sort source regs from high to low.

Our load bottleneck means the best-case FMA throughput is half the max, so we need at least 5 vector accumulators to hide their latency. 8 is good, so there's plenty of slack in the dependency chains to let them catch up after any delays from unexpected latency or competition for p0/p1. 7 or maybe even 6 would be fine, too: your unroll factor doesn't have to be a power of 2.

Unrolling by exactly 5 would mean that you're also right at the bottleneck for dependency chains. Any time an FMA doesn't run in the exact cycle its input is ready means a lost cycle in that dependency chain. This can happen if a load is slow (e.g. it misses in L1 cache and has to wait for L2), or if loads complete out of order and an FMA from another dependency chain steals the port this FMA was scheduled for. (Remember that scheduling happens at issue time, so the uops sitting in the scheduler are either port0 FMA or port1 FMA, not an FMA that can take whichever port is idle).

If you leave some slack in the dependency chains, out-of-order execution can "catch up" on the FMAs, because they won't be bottlenecked on throughput or latency, just waiting for load results. @Forward found (in an update to the question) that unrolling by 5 reduced performance from 93% of L1D throughput to 89.5% for this loop.

My guess is that unroll by 6 (one more than the minimum to hide the latency) would be ok here, and get about the same performance as unroll by 8. If we were closer to maxing out FMA throughput (rather than just bottlenecked on load throughput), one more than the minimum might not be enough.

update: @Forward's experimental test shows my guess was wrong. There isn't a big difference between unroll5 and unroll6. Also, unroll15 is twice as close as unroll8 to the theoretical max throughput of 2x 256b loads per clock. Measuring with just independent loads in the loop, or with independent loads and register-only FMA, would tell us how much of that is due to interaction with the FMA dependency chain. Even the best case won't get perfect 100% throughput, if only because of measurement errors and disruption due to timer interrupts. (Linux perf measures only user-space cycles unless you run it as root, but time still includes time spent in interrupt handlers. This is why your CPU frequency might be reported as 3.87GHz when run as non-root, but 3.900GHz when run as root and measuring cycles instead of cycles:u.)

We aren't bottlenecked on front-end throughput, but we can reduce the fused-domain uop count by avoiding indexed addressing modes for non-mov instructions. Fewer is better and makes this more hyperthreading-friendly when sharing a core with something other than this.

The simple way is just to do two pointer-increments inside the loop. The complicated way is a neat trick of indexing one array relative to the other:

;; input pointers for x[] and y[] in rdi and rsi
;; size_t n  in rdx

    ;;; zero ymm1..8, or load+vmulps into them

    add             rdx, rsi             ; end_y
    ; lea rdx, [rdx+rsi-252]  to break out of the unrolled loop before going off the end, with odd n

    sub             rdi, rsi             ; index x[] relative to y[], saving one pointer increment

.unroll8:
    vmovaps         ymm0, [rdi+rsi]            ; *px, actually py[xy_offset]
    vfmadd231ps     ymm1, ymm0, [rsi]          ; *py

    vmovaps         ymm0,       [rdi+rsi+32]   ; write-only reuse of ymm0
    vfmadd231ps     ymm2, ymm0, [rsi+32]

    vmovaps         ymm0,       [rdi+rsi+64]
    vfmadd231ps     ymm3, ymm0, [rsi+64]

    vmovaps         ymm0,       [rdi+rsi+96]
    vfmadd231ps     ymm4, ymm0, [rsi+96]

    add             rsi, 256       ; pointer-increment here
                                   ; so the following instructions can still use disp8 in their addressing modes: [-128 .. +127] instead of disp32
                                   ; smaller code-size helps in the big picture, but not for a micro-benchmark

    vmovaps         ymm0,       [rdi+rsi+128-256]  ; be pedantic in the source about compensating for the pointer-increment
    vfmadd231ps     ymm5, ymm0, [rsi+128-256]
    vmovaps         ymm0,       [rdi+rsi+160-256]
    vfmadd231ps     ymm6, ymm0, [rsi+160-256]
    vmovaps         ymm0,       [rdi+rsi-64]       ; or not
    vfmadd231ps     ymm7, ymm0, [rsi-64]
    vmovaps         ymm0,       [rdi+rsi-32]
    vfmadd231ps     ymm8, ymm0, [rsi-32]

    cmp             rsi, rdx
    jb              .unroll8                 ; } while(py < endy);

Using a non-indexed addressing mode as the memory operand for vfmaddps lets it stay micro-fused in the out-of-order core, instead of being un-laminated at issue. Micro fusion and addressing modes

So my loop is 18 fused-domain uops for 8 vectors. Yours takes 3 fused-domain uops for each vmovaps + vfmaddps pair, instead of 2, because of un-lamination of indexed addressing modes. Both of them still of course have 2 unfused-domain load uops (port2/3) per pair, so that's still the bottleneck.

Fewer fused-domain uops lets out-of-order execution see more iterations ahead, potentially helping it absorb cache misses better. It's a minor thing when we're bottlenecked on an execution unit (load uops in this case) even with no cache misses, though. But with hyperthreading, you only get every other cycle of front-end issue bandwidth unless the other thread is stalled. If it's not competing too much for load and p0/1, fewer fused-domain uops will let this loop run faster while sharing a core. (e.g. maybe the other hyper-thread is running a lot of port5 / port6 and store uops?)

Since un-lamination happens after the uop-cache, your version doesn't take extra space in the uop cache. A disp32 with each uop is ok, and doesn't take extra space. But bulkier code-size means the uop-cache is less likely to pack as efficiently, since you'll hit 32B boundaries before uop cache lines are full more often. (Actually, smaller code doesn't guarantee better either. Smaller instructions could lead to filling a uop cache line and needing one entry in another line before crossing a 32B boundary.) This small loop can run from the loopback buffer (LSD), so fortunately the uop-cache isn't a factor.

Then after the loop: Efficient cleanup is the hard part of efficient vectorization for small arrays that might not be a multiple of the unroll factor or especially the vector width

    ...
    jb

    ;; If `n` might not be a multiple of 4x 8 floats, put cleanup code here
    ;; to do the last few ymm or xmm vectors, then scalar or an unaligned last vector + mask.

    ; reduce down to a single vector, with a tree of dependencies
    vaddps          ymm1, ymm2, ymm1
    vaddps          ymm3, ymm4, ymm3
    vaddps          ymm5, ymm6, ymm5
    vaddps          ymm7, ymm8, ymm7

    vaddps          ymm0, ymm3, ymm1
    vaddps          ymm1, ymm7, ymm5

    vaddps          ymm0, ymm1, ymm0

    ; horizontal within that vector, low_half += high_half until we're down to 1
    vextractf128    xmm1, ymm0, 1
    vaddps          xmm0, xmm0, xmm1
    vmovhlps        xmm1, xmm0, xmm0        
    vaddps          xmm0, xmm0, xmm1
    vmovshdup       xmm1, xmm0
    vaddss          xmm0, xmm1
    ; this is faster than 2x vhaddps

    vzeroupper    ; important if returning to non-AVX-aware code after using ymm regs.
    ret           ; with the scalar result in xmm0

For more about the horizontal sum at the end, see Fastest way to do horizontal SSE vector sum (or other reduction). The two 128b shuffles I used don't even need an immediate control byte, so it saves 2 bytes of code size vs. the more obvious shufps. (And 4 bytes of code-size vs. vpermilps, because that opcode always needs a 3-byte VEX prefix as well as an immediate). AVX 3-operand stuff is very nice compared the SSE, especially when writing in C with intrinsics so you can't as easily pick a cold register to movhlps into.