SIMD Fundamentals. Part II: AoS, SoA, Gather/Scatter - Oh my!

Last time we looked at a very basic example of a data parallel problem: adding two arrays. Unfortunately, data parallelism is not always so easy to extract from existing code bases and often requires considerable effort. Today we will therefore address a slightly more complicated problem and move step by step from a pure scalar to a fully vectorized version.

The problem and its SISD solution

Imagine we wanted to compute the squared ℓ²-norm of a set of m 3-vectors u in R³, what is really just the dot product of each vector x with itself: ‖x‖² = x² + y² + z²

In good OO fashion, you'd probably first define a Vec3f data type, in its simplest form like so (F# syntax)

Our vector field u can then simply be represented by an array of these structs and the code that performs the computation of the dot products is

Load the three components, square them, add up the squares and store the result in dp:

 

For SSE2, that's a total of three MULSS and two ADDSS instructions, so the one dot product we get out is worth 5 FLOPs. Now we would of course like to make use of our shiny SIMD hardware to accelerate the computation. There are a few options of how we can approach this:

Pseudo-SIMD: Bolting SIMD on top of AoS

Our current data layout is an "array of structures" (AoS). As we shall see further below, AoS isn't a particular good choice for vectorization and SIMDifying such code won't yield peak performance. Yet it can be done and depending on the used SIMD instruction set it may even provide some speed-up.

Here's the idea: Instead of loading the three scalar components into separate scalar registers, we store all of them in a single SIMD register. In case of SSE2, n = 4 for single precision floats, so one of the elements in the register is a dummy value that we can ignore.  Now we can multiply the SIMD register with itself to get the squared components (plus the ignored fourth) one. Then we need to horizontally add up the three squares that we are interested in; in pseudo-F#:

Graphically:

Although we were able to replace three MULSS with a single MULPS and therefore express 5 FLOPs through only 3 arithmetic instructions, there are multiple issues with this idea: 

1. We need three scalar loads plus shuffling operations to place the three components in a single SIMD register.
2. We effectively only use 3 of the 4 vector lanes, as we operate on Vec3f, not Vec4f structs. This problem becomes even more severe with wider vector paths.
3. We use vector operations only for multiplication, but not for addition.
4. Horizontal operations like adding the components are comparatively slow, require shuffling the components around and extracting a single scalar (the result).
5. We still only compute one dot product per iteration.

Another example for this kind of pseudo-vectorization can be found in Microsoft's SIMD sample pack (ray tracer sample). Yet, for all the reasons mentioned above, you should avoid this approach whenever possible.

On-the-fly vectorization

If you recall our "Hello, world!" example from Part I, then you may also remember that vectorizing our loop meant that we computed n results per iteration instead of one. And that is what we have to achieve for our current dot-product example as well: We need to compute n dot products in each iteration. How can we accomplish this?

Imagine our data (array of Vec3f structs) to form a 3×m matrix (three rows, m columns). Each column represents a Vec3f, each row the x, y or z components of all Vec3fs. In the previous section, we tried to vectorize along the columns by parallelizing the computation of a single dot product—and largely failed.

The reason is that the data parallelism in this example can really be found along the rows of our imagined matrix, as each dot product can be computed independently and thus in parallel from each other. Furthermore m, the number of Vec3fs, is typically much larger then n and so we don't face problems in utilizing the full SIMD width. The wider the SIMD vector is, the more dot products we can compute per iteration.

As with our example from last time, the vectorized version of the algorithm is actually very similar to the plain scalar one. That's why vectorizing a data-parallel problem isn't really hard once you get the idea. The only difference is that we don't handle individual floats, but chunks of n floats:

1. Load n x values and square them
2. Load n y values and square them
3. Load n z values and square them
4. Add the n x, y and z values
5. Store the n resulting dot products

In pseudo-F# the algorithm is now

Graphically:

That good part about this version is that it uses vector arithmetic instructions throughout, performing n times the work of its scalar counterparts, performing a total of 20 FLOPs each iteration.

The bad part is the one labeled "scalar loads & shuffling" in the picture above: Before we can compute the n results, we have to gather n x, y and z values, but our data is still laid out in memory as an AoS, i.e. ordered [xyzxyzxyzxyzxyz...]. Loading logically successive [x0 x1 x2 x3 ...] values thus requires indexed/indirect load instructions (vector gather). SIMD instruction sets without gather/scatter support, like SSE2, have to load the data using n conventional scalar loads and appropriately place them in the vector registers, hurting performance considerably.

Approaching nirvana: Structure of arrays (SoA)

To avoid this drawback, maybe we should just store the data in a SIMD-friendly way, as a Structure of Arrays (SoA): Instead of modeling the vector field u as a number of Vec3f structs, we consider it to be an object of three float arrays:

This is indeed very similar to switching from a column-major to row-major matrix layout, because it changes our data layout from [xyzxyzxyz...]. to [xxxxx...yyyyy...zzzzz...].

Observe how this implementation differs from the previous one only in that we index the components instead of the vectors:

And yet it saves us a lot of loads and shuffling, resulting in pure, 100% vectorized SIMD code:

As this version really only replaces the scalar instructions with vector equivalents but executes 20 FLOPs in each iteration, we should now indeed get about 4x the performance of the scalar version and a much more favourable arithmetic-to-branch/load/store ratio.

In fact, once the hard part of the work is done (switching from AoS to an SoA data layout), many compilers can even automatically vectorize loops operating on that kind of data.

Conclusions

Vectorizing OO code needs some getting used to, the SoA view of the world may be a bit alien to non-APL programmers. If you can foresee the need for vectorizing your code at some point in the future, it may be wise to use an SoA layout from the get-go instead of having to rewrite half of your code later on. Experience certainly helps in identifying data-parallel problems; but as a rule of thumb good candidates for extracting data parallelism are those problems where thinking in terms of operations that apply to a whole array/field of items comes naturally.

In part III we are going to discuss concrete implementations of the above concepts using F#, RyuJIT and System.Numerics.Vectors and compare the performance of the different versions—once they sorted out this issue.

SIMD Fundamentals. Part I: From SISD to SIMD

For a long time, .NET developers didn't have to care about SIMD programming, as it was simply not available to them (except maybe for Mono's Mono.Simd). Only recently Microsoft introduced a new JIT compiler RyuJIT for .NET that, in conjunction with a special library System.Numerics.Vectors, offers access to the SIMD hardware of modern CPUs. The goal of this three-part series of articles is therefore to introduce the fundamental ideas of SIMD to .NET developers who want to make use of all the power today's CPUs provide.

Parallelism in modern CPUs

In recent years CPUs gained performance mainly by increasing parallelism on all levels of execution. The advent of x86-CPUs with multiple cores established true thread level parallelism in the PC world. Solving the "multi-core problem", the question of how to distribute workloads between different threads in order to use all that parallel hardware goodness—ideally (semi-)automatically, suddenly became and continues to be one of the predominant goals of current hard- and software related research.

Years before the whole multi-core issue started, CPUs already utilized another kind of parallelism to increase performance: Instruction level parallelism (ILP) was exploited via pipelining, super-scalar and out-of-order execution and other advanced techniques. The nice thing about this kind of parallelism is that your average Joe Developer doesn't has to think about it, it's all handled by smart compilers and smart processors.

But then in 1997 Intel introduced the P55C and with it a new 64-bit-wide SIMD instruction set called MMX ("multimedia extension"; after all, "multimedia" was "the cloud" of the 90s). MMX made available a level of parallelism new to most PC developers: data level parallelism (DLP). Contrary to ILP however, DLP requires specifically designed code to be of any good. This was true for MMX and it remains to be true for its successors like Intel's 256-bit-wide AVX2. Just as with multi threading, programmers need to understand how to use those capabilities properly in order to exploit the tremendous amounts of floating point horse power.

From SISD to SIMD

Whenever I learn new concepts, I like to first think of examples and generalize afterwards (inductive reasoning). In my experience that's true for most people, so let us therefore start with the "Hello, world!" of array programming: Suppose you have two floating point arrays, say xs and ys of length m and, for some reason, you want to add those arrays component-wise and store the result in an array zs. The conventional "scalar" or SISD (Single Instruction Single Data, see Flynn's taxonomy) way of doing this looks like this (C# syntax)

for (var i = 0; i < m; ++i) {
    zs[i] = xs[i] + ys[i];
}

Fetch two floats, add them up and store the result in zs[i]. Easy:

For this specific example, adding SIMD (Single Instruction Multiple Data) is almost trivial: For simplicity, let us further assume that m is evenly divisible by n, the vector lane width. Now instead of performing one addition per iteration, we add up xs and ys in chunks of n items, in pseudo-C# with an imagined array range expression syntax

for (var i = 0; i < m / n; i += n) {
    zs[i:(i+n-1)] = xs[i:(i+n-1)] + ys[i:(i+n-1)];
}

We fetch n items from xs and n items from ys using vector load instructions add them up using a single vector add and store the n results in zs. Compared to the scalar version, we need to decode fewer instructions and perform n times the work in each iteration. Think of each SIMD register as a window, n values wide into an arbitrarily long stream (array) of scalar values. Each SIMD instructions modifies n scalar values of that stream at once (that's where the speed-up comes from) and moves on to the next chunk of n values:

So SIMD really is just a form of array processing, where you think in terms of applying one operation (single instruction) to a lot of different elements (multiple data), loop-unrolling on steroids. And that's already really all you need to know about SIMD, basically.

Yet, this example is perhaps the most obvious data parallel case one could think of. It gets a whole lot more interesting once you add OOP and the data layout this paradigm propagates to the mix. We will look into the issue of vectorizing typical OOP code in the next part of this series.

NativeInterop 2.4.0

NativeInterop v2.4.0 is ready and awaiting your download from NuGet. This version brings a revised version of Buffer.Copy tuned for .NET 4.6/RyuJIT: By using a 16 byte block copy, the JITter can generate movq (x86) or movdqu (x64) instructions for further improved performance. Enjoy!

Now I just have to figure out, where the larger performance drop for large data sizes comes from compared to the other methods. memcpy somehow reaches approx. 20 GB/s beyond the L3 cache.

Btw., I have a further post on SIMD programming with .NET almost ready, but I can't publish it yet due to problems with the current version of System.Numerics.Vectors in combination with .NET 4.6/VS 2015 RC. Hope that'll get fixed soon!