TL:DR: look for dependency chains, especially loop-carried ones. For a long-running loop, see which latency, front-end throughput, or back-end port contention/throughput is the worst bottleneck. That's how many cycles your loop probably takes per iteration, on average, if there are no cache misses or branch mispredicts.
Related: How many CPU cycles are needed for each assembly instruction? is a good introduction to throughput vs. latency on a per-instruction basis, and how what that means for sequences of multiple instructions.
This is called static (performance) analysis. Wikipedia says (https://en.wikipedia.org/wiki/List_of_performance_analysis_tools) that AMD's AMD CodeXL has a "static kernel analyzer" (i.e. for computational kernels, aka loops). I've never tried it.
Intel also has a free tool for analyzing how loops will go through the pipeline in Sandybridge-family CPUs: What is IACA and how do I use it?
IACA is not bad, but has bugs (e.g. wrong data for shld on Sandybridge, and last I checked, it doesn't know that Haswell/Skylake can keep indexed addressing modes micro-fused for some instructions. But maybe that will change now that Intel's added details on that to their optimization manual.) IACA is also unhelpful for counting front-end uops to see how close to a bottleneck you are (it likes to only give you unfused-domain uop counts).
Static analysis is often pretty good, but definitely check by profiling with performance counters. See Can x86's MOV really be "free"? Why can't I reproduce this at all? for an example of profiling a simple loop to investigate a microarchitectural feature.
Essential reading:
Agner Fog's microarch guide (chapter 2: Out of order exec) explains some of the basics of dependency chains and out-of-order execution. His "Optimizing Assembly" guide has more good introductory and advanced performance stuff.
The later chapters of his microarch guide cover the details of the pipelines in CPUs like Nehalem, Sandybridge, Haswell, K8/K10, Bulldozer, and Ryzen. (And Atom / Silvermont / Jaguar).
Agner Fog's instruction tables (spreadsheet or PDF) are also normally the best source for instruction latency / throughput / execution-port breakdowns.
David Kanter's microarch analysis docs are very good, with diagrams. e.g. https://www.realworldtech.com/sandy-bridge/, https://www.realworldtech.com/haswell-cpu/, and https://www.realworldtech.com/bulldozer/.
See also other performance links in the x86 tag wiki.
I also took a stab at explaining how a CPU core finds and exploits instruction-level parallelism in this answer, but I think you've already grasped those basics as far as it's relevant for tuning software. I did mention how SMT (Hyperthreading) works as a way to expose more ILP to a single CPU core, though.
In Intel terminology:
"issue" means to send a uop into the out-of-order part of the core; along with register-renaming, this is the last step in the front-end. The issue/rename stage is often the narrowest point in the pipeline, e.g. 4-wide on Intel since Core2. (With later uarches like Haswell and especially Skylake often actually coming very close to that in some real code, thanks to SKL's improved decoders and uop-cache bandwidth, as well as back-end and cache bandwidth improvements.) This is fused-domain uops: micro-fusion lets you send 2 uops through the front-end and only take up one ROB entry. (I was able to construct a loop on Skylake that sustains 7 unfused-domain uops per clock). See also http://blog.stuffedcow.net/2013/05/measuring-rob-capacity/ re: out-of-order window size.
"dispatch" means the scheduler sends a uop to an execution port. This happens as soon as all the inputs are ready, and the relevant execution port is available. How are x86 uops scheduled, exactly?. Scheduling happens in the "unfused" domain; micro-fused uops are tracked separately in the OoO scheduler (aka Reservation Station, RS).
A lot of other computer-architecture literature uses these terms in the opposite sense, but this is the terminology you will find in Intel's optimization manual, and the names of hardware performance counters like uops_issued.any or uops_dispatched_port.port_5.
exactly how long arbitrary arithmetical x86-64 assembly code will take
It depends on the surrounding code as well, because of OoO exec
Your final subps result doesn't have to be ready before the CPU starts running later instructions. Latency only matters for later instructions that need that value as an input, not for integer looping and whatnot.
Sometimes throughput is what matters, and out-of-order exec can hide the latency of multiple independent short dependency chains. (e.g. if you're doing the same thing to every element of a big array of multiple vectors, multiple cross products can be in flight at once.) You'll end up with multiple iterations in flight at once, even though in program order you finish all of one iteration before doing any of the next. (Software pipelining can help for high-latency loop bodies if OoO exec has a hard time doing all the reordering in HW.)
There are three major dimensions to analyze for a short block
You can approximately characterize a short block of non-branching code in terms of these three factors. Usually only one of them is the bottleneck for a given use-case. Often you're looking at a block that you will use as part of a loop, not as the whole loop body, but OoO exec normally works well enough that you can just add up these numbers for a couple different blocks, if they're not so long that OoO window size prevents finding all the ILP.
- latency from each input to the output(s). Look at which instructions are on the dependency chain from each input to each output. e.g. one choice might need one input to be ready sooner.
- total uop count (for front-end throughput bottlenecks), fused-domain on Intel CPUs. e.g. Core2 and later can in theory issue/rename 4 fused-domain uops per clock into the out-of-order scheduler/ROB. Sandybridge-family can often achieve that in practice with the uop cache and loop buffer, especially Skylake with its improved decoders and uop-cache throughput.
- uop count for each back-end execution port (unfused domain). e.g. shuffle-heavy code will often bottleneck on port 5 on Intel CPUs. Intel usually only publishes throughput numbers, not port breakdowns, which is why you have to look at Agner Fog's tables (or IACA output) to do anything meaningful if you're not just repeating the same instruction a zillion times.
Generally you can assuming best-case scheduling/distribution, with uops that can run on other ports not stealing the busy ports very often, but it does happen some. (How are x86 uops scheduled, exactly?)
Looking at CPI is not sufficient; two CPI=1 instructions might or might not compete for the same execution port. If they don't, they can execute in parallel. e.g. Haswell can only run psadbw on port 0 (5c latency, 1c throughput, i.e. CPI=1) but it's a single uop so a mix of 1 psadbw + 3 add instructions could sustain 4 instructions per clock. There are vector ALUs on 3 different ports in Intel CPUs, with some operations replicated on all 3 (e.g. booleans) and some only on one port (e.g. shifts before Skylake).
Sometimes you can come up with a couple different strategies, one maybe lower latency but costing more uops. A classic example is multiplying by constants like imul eax, ecx, 10 (1 uop, 3c latency on Intel) vs. lea eax, [rcx + rcx*4] / add eax,eax (2 uops, 2c latency). Modern compilers tend to choose 2 LEA vs. 1 IMUL, although clang up to 3.7 favoured IMUL unless it could get the job done with only a single other instruction.
See What is the efficient way to count set bits at a position or lower? for an example of static analysis for a few different ways to implement a function.
See also Why does mulss take only 3 cycles on Haswell, different from Agner's instruction tables? (Unrolling FP loops with multiple accumulators) (which ended up being way more detailed than you'd guess from the question title) for another summary of static analysis, and some neat stuff about unrolling with multiple accumulators for a reduction.
Every (?) functional unit is pipelined
The divider is pipelined in recent CPUs, but not fully pipelined. (FP divide is single-uop, though, so if you do one divps mixed in with dozens of mulps / addps, it can have negligible throughput impact if latency doesn't matter: Floating point division vs floating point multiplication. rcpps + a Newton iteration is worse throughput and about the same latency.
Everything else is fully pipelined on mainstream Intel CPUs; multi-cycle (reciprocal) throughput for a single uop. (variable-count integer shifts like shl eax, cl have lower-than-expected throughput for their 3 uops, because they create a dependency through the flag-merging uops. But if you break that dependency through FLAGS with an add or something, you can get better throughput and latency.)
On AMD before Ryzen, the integer multiplier is also only partially pipelined. e.g. Bulldozer's imul ecx, edx is only 1 uop, but with 4c latency, 2c throughput.
Xeon Phi (KNL) also has some not-fully-pipelined shuffle instructions, but it tends to bottleneck on the front-end (instruction decode), not the back-end, and does have a small buffer + OoO exec capability to hide back-end bubbles.
If it is a floating-point instruction, every floating-point instruction before it has been issued (floating-point instructions have static instruction re-ordering)
No.
Maybe you read that for Silvermont, which doesn't do OoO exec for FP/SIMD, only integer (with a small ~20 uop window). Maybe some ARM chips are like that, too, with simpler schedulers for NEON? I don't know much about ARM uarch details.
The mainstream big-core microarchitectures like P6 / SnB-family, and all AMD OoO chips, do OoO exec for SIMD and FP instructions the same as for integer. AMD CPUs use a separate scheduler, but Intel uses a unified scheduler so its full size can be applied to finding ILP in integer or FP code, whichever is currently running.
Even the silvermont-based Knight's Landing (in Xeon Phi) does OoO exec for SIMD.
x86 is generally not very sensitive to instruction ordering, but uop scheduling doesn't do critical-path analysis. So it could sometimes help to put instructions on the critical path first, so they aren't stuck waiting with their inputs ready while other instructions run on that port, leading to a bigger stall later when we get to instructions that need the result of the critical path. (i.e. that's why it is the critical path.)
My attempt to predict the latency for Haswell looks something like this:
Yup, that looks right. shufps runs on port 5, addps runs on p1, mulps runs on p0 or p1. Skylake drops the dedicated FP-add unit and runs SIMD FP add/mul/FMA on the FMA units on p0/p1, all with 4c latency (up/down from 3/5/5 in Haswell, or 3/3/5 in Broadwell).
This is a good example of why keeping a whole XYZ direction vector in a SIMD vector usually sucks. Keeping an array of X, an array of Y, and an array of Z, would let you do 4 cross products in parallel without any shuffles.
The SSE tag wiki has a link to these slides: SIMD at Insomniac Games (GDC 2015) which covers that array-of-structs vs. struct-of-arrays issues for 3D vectors, and why it's often a mistake to always try to SIMD a single operation instead of using SIMD to do multiple operations in parallel.
