# Matrix Multiplication Revisited

In a recent post, I took a look at matrix multiplication in pure Java, to see if it can go faster than reported in SIMD Intrinsics on Managed Language Runtimes. I found faster implementations than the paper’s benchmarks implied was possible. Nevertheless, I found that there were some limitations in Hotspot’s autovectoriser that I didn’t expect to see, even in JDK10. Were these limitations somehow fundamental, or can other compilers do better with essentially the same input?

I took a look at the code generated by GCC’s autovectoriser to see what’s possible in C/C++ without resorting to complicated intrinsics. For a bit of fun, I went over to the dark side to squeeze out some a lot of extra performance, which gave inspiration to a simple vectorised Java implementation which can maintain intensity as matrix size increases.

### Background

The paper reports a 5x improvement in matrix multiplication throughput as a result of using LMS generated intrinsics. Using GCC as LMS’s backend, I easily reproduced very good throughput, but I found two Java implementations better than the paper’s baseline. The best performing Java implementation proposed in the paper was `blocked`. This post is not about the LMS benchmarks, but this code is this post’s inspiration.

``````
public void blocked(float[] a, float[] b, float[] c, int n) {
int BLOCK_SIZE = 8;
// GOOD: attempts to do as much work in submatrices
// GOOD: tries to avoid bringing data through cache multiple times
for (int kk = 0; kk < n; kk += BLOCK_SIZE) {
for (int jj = 0; jj < n; jj += BLOCK_SIZE) {
for (int i = 0; i < n; i++) {
for (int j = jj; j < jj + BLOCK_SIZE; ++j) {
// BAD: manual unrolling, bypasses optimisations
float sum = c[i * n + j];
for (int k = kk; k < kk + BLOCK_SIZE; ++k) {
// BAD: horizontal sums are inefficient
sum += a[i * n + k] * b[k * n + j];
}
c[i * n + j] = sum;
}
}
}
}
}
``````

I proposed the following implementation for improved cache efficiency and expected it to vectorise automatically.

``````public void fast(float[] a, float[] b, float[] c, int n) {
// GOOD: 2x faster than "blocked" - why?
int in = 0;
for (int i = 0; i < n; ++i) {
int kn = 0;
for (int k = 0; k < n; ++k) {
float aik = a[in + k];
// MIXED: passes over c[in:in+n] multiple times per k-value, "free" if n is small
// MIXED: reloads b[kn:kn+n] repeatedly for each i, bad if n is large, "free" if n is small
// BAD: doesn't vectorise but should
for (int j = 0; j < n; ++j) {
c[in + j] += aik * b[kn + j]; // sequential writes and reads, cache and vectoriser friendly
}
kn += n;
}
in += n;
}
}
``````

My code actually doesn’t vectorise, even in JDK10, which really surprised me because the inner loop vectorises if the offsets are always zero. In any case, there is a simple hack involving the use of buffers, which unfortunately thrashes the cache, but narrows the field significantly.

``````
public void fastBuffered(float[] a, float[] b, float[] c, int n) {
float[] bBuffer = new float[n];
float[] cBuffer = new float[n];
int in = 0;
for (int i = 0; i < n; ++i) {
int kn = 0;
for (int k = 0; k < n; ++k) {
float aik = a[in + k];
System.arraycopy(b, kn, bBuffer, 0, n);
saxpy(n, aik, bBuffer, cBuffer);
kn += n;
}
System.arraycopy(cBuffer, 0, c, in, n);
Arrays.fill(cBuffer, 0f);
in += n;
}
}
``````

I left the problem looking like this, with the “JDKX vectorised” lines using the algorithm above with a buffer hack:

### GCC Autovectorisation

The Java code is very easy to translate into C/C++. Before looking at performance I want to get an idea of what GCC’s autovectoriser does. I want to see the code generated at GCC optimisation level 3, with unrolled loops, FMA, and AVX2, which can be seen as follows:

`g++ -mavx2 -mfma -march=native -funroll-loops -O3 -S mmul.cpp`

The generated assembly code can be seen in full context here. Let’s look at the `mmul_saxpy` routine first:

``````static void mmul_saxpy(const int n, const float* left, const float* right, float* result) {
int in = 0;
for (int i = 0; i < n; ++i) {
int kn = 0;
for (int k = 0; k < n; ++k) {
float aik = left[in + k];
for (int j = 0; j < n; ++j) {
result[in + j] += aik * right[kn + j];
}
kn += n;
}
in += n;
}
}
``````

This routine uses SIMD instructions, which means in principle any other compiler could do this too. The inner loop has been unrolled, but this is only by virtue of the `-funroll-loops` flag. C2 does this sort of thing as standard, but only for hot loops. In general you might not want to unroll loops because of the impact on code size, and it’s great that a JIT compiler can decide only to do this when it’s profitable.

``````.L9:
vmovups  (%rdx,%rax), %ymm4
vmovaps  %ymm4, (%r11,%rax)
vmovups  32(%rdx,%rax), %ymm5
vmovaps  %ymm5, 32(%r11,%rax)
vmovups  64(%rdx,%rax), %ymm1
vmovaps  %ymm1, 64(%r11,%rax)
vmovups  96(%rdx,%rax), %ymm2
vmovaps  %ymm2, 96(%r11,%rax)
vmovups  128(%rdx,%rax), %ymm4
vmovaps  %ymm4, 128(%r11,%rax)
vmovups  160(%rdx,%rax), %ymm5
vmovaps  %ymm5, 160(%r11,%rax)
vmovups  192(%rdx,%rax), %ymm1
vmovaps  %ymm1, 192(%r11,%rax)
vmovups  224(%rdx,%rax), %ymm2
vmovaps  %ymm2, 224(%r11,%rax)
cmpl  %r10d, 24(%rsp)
ja  .L9
``````

The `mmul_blocked` routine is compiled to quite convoluted assembly. It has a huge problem with the expression `right[k * n + j]`, which requires a gather and is almost guaranteed to create 8 cache misses per block for large matrices. Moreover, this inefficiency gets much worse with problem size.

``````static void mmul_blocked(const int n, const float* left, const float* right, float* result) {
int BLOCK_SIZE = 8;
for (int kk = 0; kk < n; kk += BLOCK_SIZE) {
for (int jj = 0; jj < n; jj += BLOCK_SIZE) {
for (int i = 0; i < n; i++) {
for (int j = jj; j < jj + BLOCK_SIZE; ++j) {
float sum = result[i * n + j];
for (int k = kk; k < kk + BLOCK_SIZE; ++k) {
sum += left[i * n + k] * right[k * n + j]; // second read here requires a gather
}
result[i * n + j] = sum;
}
}
}
}
}
``````

This compiles to assembly with the unrolled vectorised loop below:

``````.L114:
cmpq  %r10, %r9
setbe  %cl
cmpq  56(%rsp), %r8
setnb  %dl
orl  %ecx, %edx
cmpq  %r14, %r9
setbe  %cl
cmpq  64(%rsp), %r8
setnb  %r15b
orl  %ecx, %r15d
andl  %edx, %r15d
cmpq  %r11, %r9
setbe  %cl
cmpq  48(%rsp), %r8
setnb  %dl
orl  %ecx, %edx
andl  %r15d, %edx
cmpq  %rbx, %r9
setbe  %cl
cmpq  40(%rsp), %r8
setnb  %r15b
orl  %ecx, %r15d
andl  %edx, %r15d
cmpq  %rsi, %r9
setbe  %cl
cmpq  32(%rsp), %r8
setnb  %dl
orl  %ecx, %edx
andl  %r15d, %edx
cmpq  %rdi, %r9
setbe  %cl
cmpq  24(%rsp), %r8
setnb  %r15b
orl  %ecx, %r15d
andl  %edx, %r15d
cmpq  %rbp, %r9
setbe  %cl
cmpq  16(%rsp), %r8
setnb  %dl
orl  %ecx, %edx
andl  %r15d, %edx
cmpq  %r12, %r9
setbe  %cl
cmpq  8(%rsp), %r8
setnb  %r15b
orl  %r15d, %ecx
testb  %cl, %dl
je  .L111
leaq  32(%rax), %rdx
cmpq  %rdx, %r8
setnb  %cl
cmpq  %rax, %r9
setbe  %r15b
orb  %r15b, %cl
je  .L111
vmovups  (%r8), %ymm2
vmovups  %ymm0, (%r8)
``````

### Benchmarks

I implemented a suite of benchmarks to compare the implementations. You can run them, but since they measure throughput and intensity averaged over hundreds of iterations per matrix size, the full run will take several hours.

`g++ -mavx2 -mfma -march=native -funroll-loops -O3 mmul.cpp -o mmul.exe && ./mmul.exe > results.csv`

The `saxpy` routine wins, with `blocked` fading fast after a middling start.

name size throughput (ops/s) flops/cycle
blocked 64 22770.2 4.5916
saxpy 64 25638.4 5.16997
blocked 128 2736.9 4.41515
saxpy 128 4108.52 6.62783
blocked 192 788.132 4.29101
saxpy 192 1262.45 6.87346
blocked 256 291.728 3.76492
saxpy 256 521.515 6.73044
blocked 320 147.979 3.72997
saxpy 320 244.528 6.16362
blocked 384 76.986 3.35322
saxpy 384 150.441 6.55264
blocked 448 50.4686 3.4907
saxpy 448 95.0752 6.57594
blocked 512 30.0085 3.09821
saxpy 512 65.1842 6.72991
blocked 576 22.8301 3.35608
saxpy 576 44.871 6.59614
blocked 640 15.5007 3.12571
saxpy 640 32.3709 6.52757
blocked 704 12.2478 3.28726
saxpy 704 25.3047 6.79166
blocked 768 8.69277 3.02899
saxpy 768 19.8011 6.8997
blocked 832 7.29356 3.23122
saxpy 832 15.3437 6.7976
blocked 896 4.95207 2.74011
saxpy 896 11.9611 6.61836
blocked 960 3.4467 2.34571
saxpy 960 9.25535 6.29888
blocked 1024 2.02289 1.67082
saxpy 1024 6.87039 5.67463

With GCC autovectorisation, `saxpy` performs well, maintaining intensity as size increases, albeit well below the theoretical capacity. It would be nice if similar code could be JIT compiled in Java.

### Intel Intrinsics

To understand the problem space a bit better, I find out how fast matrix multiplication can get without domain expertise by handcrafting an algorithm with intrinsics. My laptop’s Skylake chip (turbo boost and hyperthreading disabled) is capable of 32 SP flops per cycle per core – Java and the LMS implementation previously fell a long way short of that. It was difficult getting beyond 4f/c with Java, and LMS peaked at almost 6f/c before quickly tailing off. GCC autovectorisation achieved and maintained 7f/c.

To start, I’ll take full advantage of the facility to align the matrices on 64 byte intervals, since I have 64B cache lines, though this might just be voodoo. I take the `saxpy` routine and replace its kernel with intrinsics. Because of the `-funroll-loops` option, this will get unrolled without effort.

``````static void mmul_saxpy_avx(const int n, const float* left, const float* right, float* result) {
int in = 0;
for (int i = 0; i < n; ++i) {
int kn = 0;
for (int k = 0; k < n; ++k) {
__m256 aik = _mm256_set1_ps(left[in + k]);
int j = 0;
for (; j < n; j += 8) {
}
for (; j < n; ++j) {
result[in + j] += left[in + k] * right[kn + j];
}
kn += n;
}
in += n;
}
}
``````

This code is actually not a lot faster, if at all, than the basic `saxpy` above: a lot of aggressive optimisations have already been applied.

### Combining Blocked and SAXPY

What makes `blocked` so poor is the gather and the cache miss, not the concept of blocking itself. A limiting factor for `saxpy` performance is that the ratio of loads to floating point operations is too high. With this in mind, I tried combining the blocking idea with `saxpy`, by implementing `saxpy` multiplications for smaller sub-matrices. This results in a different algorithm with fewer loads per floating point operation, and the inner two loops are swapped. It avoids the gather and the cache miss in `blocked`. Because the matrices are in row major format, I make the width of the blocks much larger than the height. Also, different heights and widths make sense depending on the size of the matrix, so I choose them dynamically. The design constraints are to avoid gathers and horizontal reduction.

``````static void mmul_tiled_avx(const int n, const float *left, const float *right, float *result) {
const int block_width = n >= 256 ? 512 : 256;
const int block_height = n >= 512 ? 8 : n >= 256 ? 16 : 32;
for (int row_offset = 0; row_offset < n; row_offset += block_height) {
for (int column_offset = 0; column_offset < n; column_offset += block_width) {
for (int i = 0; i < n; ++i) {
for (int j = column_offset; j < column_offset + block_width && j < n; j += 8) {
__m256 sum = _mm256_load_ps(result + i * n + j);
for (int k = row_offset; k < row_offset + block_height && k < n; ++k) {
sum = _mm256_fmadd_ps(_mm256_set1_ps(left[i * n + k]), _mm256_load_ps(right + k * n + j), sum);
}
_mm256_store_ps(result + i * n + j, sum);
}
}
}
}
}
``````

You will see in the benchmark results that this routine really doesn’t do very well compared to `saxpy`. Finally, I unroll it, which is profitable despite setting `-funroll-loops` because there is slightly more to this than an unroll. This is a sequence of vertical reductions which have no data dependencies.

``````static void mmul_tiled_avx_unrolled(const int n, const float *left, const float *right, float *result) {
const int block_width = n >= 256 ? 512 : 256;
const int block_height = n >= 512 ? 8 : n >= 256 ? 16 : 32;
for (int column_offset = 0; column_offset < n; column_offset += block_width) {
for (int row_offset = 0; row_offset < n; row_offset += block_height) {
for (int i = 0; i < n; ++i) {
for (int j = column_offset; j < column_offset + block_width && j < n; j += 64) {
__m256 sum1 = _mm256_load_ps(result + i * n + j);
__m256 sum2 = _mm256_load_ps(result + i * n + j + 8);
__m256 sum3 = _mm256_load_ps(result + i * n + j + 16);
__m256 sum4 = _mm256_load_ps(result + i * n + j + 24);
__m256 sum5 = _mm256_load_ps(result + i * n + j + 32);
__m256 sum6 = _mm256_load_ps(result + i * n + j + 40);
__m256 sum7 = _mm256_load_ps(result + i * n + j + 48);
__m256 sum8 = _mm256_load_ps(result + i * n + j + 56);
for (int k = row_offset; k < row_offset + block_height && k < n; ++k) {
__m256 multiplier = _mm256_set1_ps(left[i * n + k]);
sum2 = _mm256_fmadd_ps(multiplier, _mm256_load_ps(right + k * n + j + 8), sum2);
sum3 = _mm256_fmadd_ps(multiplier, _mm256_load_ps(right + k * n + j + 16), sum3);
sum4 = _mm256_fmadd_ps(multiplier, _mm256_load_ps(right + k * n + j + 24), sum4);
sum5 = _mm256_fmadd_ps(multiplier, _mm256_load_ps(right + k * n + j + 32), sum5);
sum6 = _mm256_fmadd_ps(multiplier, _mm256_load_ps(right + k * n + j + 40), sum6);
sum7 = _mm256_fmadd_ps(multiplier, _mm256_load_ps(right + k * n + j + 48), sum7);
sum8 = _mm256_fmadd_ps(multiplier, _mm256_load_ps(right + k * n + j + 56), sum8);
}
_mm256_store_ps(result + i * n + j, sum1);
_mm256_store_ps(result + i * n + j + 8, sum2);
_mm256_store_ps(result + i * n + j + 16, sum3);
_mm256_store_ps(result + i * n + j + 24, sum4);
_mm256_store_ps(result + i * n + j + 32, sum5);
_mm256_store_ps(result + i * n + j + 40, sum6);
_mm256_store_ps(result + i * n + j + 48, sum7);
_mm256_store_ps(result + i * n + j + 56, sum8);
}
}
}
}
}
``````

This final implementation is fast, and is probably as good as I am going to manage, without reading papers. This should be a CPU bound problem because the algorithm is O(n^3) whereas the problem size is O(n^2). But the flops/cycle decreases with problem size in all of these implementations. It’s possible that this could be amelioarated by a better dynamic tiling policy. I’m unlikely to be able to fix that.

It does make a huge difference being able to go very low level – handwritten intrinsics with GCC unlock awesome throughput – but it’s quite hard to actually get to the point where you can beat a good optimising compiler. Mind you, there are harder problems to solve this, and you may well be a domain expert.

The benchmark results summarise this best:

name size throughput (ops/s) flops/cycle
saxpy_avx 64 49225.7 9.92632
tiled_avx 64 33680.5 6.79165
tiled_avx_unrolled 64 127936 25.7981
saxpy_avx 128 5871.02 9.47109
tiled_avx 128 4210.07 6.79166
tiled_avx_unrolled 128 15997.6 25.8072
saxpy_avx 192 1603.84 8.73214
tiled_avx 192 1203.33 6.55159
tiled_avx_unrolled 192 4383.09 23.8638
saxpy_avx 256 633.595 8.17689
tiled_avx 256 626.157 8.0809
tiled_avx_unrolled 256 1792.52 23.1335
saxpy_avx 320 284.161 7.1626
tiled_avx 320 323.197 8.14656
tiled_avx_unrolled 320 935.571 23.5822
saxpy_avx 384 161.517 7.03508
tiled_avx 384 188.215 8.19794
tiled_avx_unrolled 384 543.235 23.6613
saxpy_avx 448 99.1987 6.86115
tiled_avx 448 118.588 8.2022
tiled_avx_unrolled 448 314 21.718
saxpy_avx 512 70.0296 7.23017
tiled_avx 512 73.2019 7.55769
tiled_avx_unrolled 512 197.815 20.4233
saxpy_avx 576 46.1944 6.79068
tiled_avx 576 50.6315 7.44294
tiled_avx_unrolled 576 126.045 18.5289
saxpy_avx 640 33.8209 6.81996
tiled_avx 640 37.0288 7.46682
tiled_avx_unrolled 640 92.784 18.7098
saxpy_avx 704 24.9096 6.68561
tiled_avx 704 27.7543 7.44912
tiled_avx_unrolled 704 69.0399 18.53
saxpy_avx 768 19.5158 6.80027
tiled_avx 768 21.532 7.50282
tiled_avx_unrolled 768 54.1763 18.8777
saxpy_avx 832 12.8635 5.69882
tiled_avx 832 14.6666 6.49766
tiled_avx_unrolled 832 37.9592 16.8168
saxpy_avx 896 12.0526 6.66899
tiled_avx 896 13.3799 7.40346
tiled_avx_unrolled 896 34.0838 18.8595
saxpy_avx 960 8.97193 6.10599
tiled_avx 960 10.1052 6.87725
tiled_avx_unrolled 960 21.0263 14.3098
saxpy_avx 1024 6.73081 5.55935
tiled_avx 1024 7.21214 5.9569
tiled_avx_unrolled 1024 12.7768 10.5531

### Can we do better in Java?

Writing genuinely fast code gives an indication of how little of the processor Java actually utilises, but is it possible to bring this knowledge over to Java? The `saxpy` based implementations in my previous post performed well for small to medium sized matrices. Once the matrices grow, however, they become too big to be allowed to pass through cache multiple times: we need hot, small cached data to be replenished from the larger matrix. Ideally we wouldn’t need to make any copies, but it seems that the autovectoriser doesn’t like offsets: `System.arraycopy` is a reasonably fast compromise. The basic sequential read pattern is validated: even native code requiring a gather does not perform well for this problem. The best effort C++ code translates almost verbatim into this Java code, which is quite fast for large matrices.

``````
public void tiled(float[] a, float[] b, float[] c, int n) {
final int bufferSize = 512;
final int width = Math.min(n, bufferSize);
final int height = Math.min(n, n >= 512 ? 8 : n >= 256 ? 16 : 32);
float[] sum = new float[bufferSize];
float[] vector = new float[bufferSize];
for (int rowOffset = 0; rowOffset < n; rowOffset += height) {
for (int columnOffset = 0; columnOffset < n; columnOffset += width) {
for (int i = 0; i < n; ++i) {
for (int j = columnOffset; j < columnOffset + width && j < n; j += width) {
int stride = Math.min(n - columnOffset, bufferSize);
// copy to give autovectorisation a hint
System.arraycopy(c, i * n + j, sum, 0, stride);
for (int k = rowOffset; k < rowOffset + height && k < n; ++k) {
float multiplier = a[i * n + k];
System.arraycopy(b, k * n  + j, vector, 0, stride);
for (int l = 0; l < stride; ++l) {
sum[l] = Math.fma(multiplier, vector[l], sum[l]);
}
}
System.arraycopy(sum, 0, c, i * n + j, stride);
}
}
}
}
}
``````

Benchmarking it using the same harness used in the previous post, the performance is ~10% higher for large arrays than my previous best effort. Still, the reality is that this is too slow to be useful. If you need to do linear algebra, use C/C++ for the time being!

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: size flops/cycle
fastBuffered thrpt 1 10 53.331195 0.270526 ops/s 448 3.688688696
fastBuffered thrpt 1 10 34.365765 0.16641 ops/s 512 3.548072999
fastBuffered thrpt 1 10 26.128264 0.239719 ops/s 576 3.840914622
fastBuffered thrpt 1 10 19.044509 0.139197 ops/s 640 3.84031059
fastBuffered thrpt 1 10 14.312154 1.045093 ops/s 704 3.841312378
fastBuffered thrpt 1 10 7.772745 0.074598 ops/s 768 2.708411991
fastBuffered thrpt 1 10 6.276182 0.067338 ops/s 832 2.780495238
fastBuffered thrpt 1 10 4.8784 0.067368 ops/s 896 2.699343067
fastBuffered thrpt 1 10 4.068907 0.038677 ops/s 960 2.769160387
fastBuffered thrpt 1 10 2.568101 0.108612 ops/s 1024 2.121136502
tiled thrpt 1 10 56.495366 0.584872 ops/s 448 3.907540754
tiled thrpt 1 10 30.884954 3.221017 ops/s 512 3.188698735
tiled thrpt 1 10 15.580581 0.412654 ops/s 576 2.290381075
tiled thrpt 1 10 9.178969 0.841178 ops/s 640 1.850932038
tiled thrpt 1 10 12.229763 0.350233 ops/s 704 3.282408783
tiled thrpt 1 10 9.371032 0.330742 ops/s 768 3.265334889
tiled thrpt 1 10 7.727068 0.277969 ops/s 832 3.423271628
tiled thrpt 1 10 6.076451 0.30305 ops/s 896 3.362255222
tiled thrpt 1 10 4.916811 0.2823 ops/s 960 3.346215151
tiled thrpt 1 10 3.722623 0.26486 ops/s 1024 3.074720008

# Multiplying Matrices, Fast and Slow

I recently read a very interesting blog post about exposing Intel SIMD intrinsics via a fork of the Scala compiler (scala-virtualized), which reports multiplicative improvements in throughput over HotSpot JIT compiled code. The academic paper (SIMD Intrinsics on Managed Language Runtimes), which has been accepted at CGO 2018, proposes a powerful alternative to the traditional JVM approach of pairing dumb programmers with a (hopefully) smart JIT compiler. Lightweight Modular Staging (LMS) allows the generation of an executable binary from a high level representation: handcrafted representations of vectorised algorithms, written in a dialect of Scala, can be compiled natively and later invoked with a single JNI call. This approach bypasses C2 without incurring excessive JNI costs. The freely available benchmarks can be easily run to reproduce the results in the paper, which is an achievement in itself, but some of the Java implementations used as baselines look less efficient than they could be. This post is about improving the efficiency of the Java matrix multiplication the LMS generated code is benchmarked against. Despite finding edge cases where autovectorisation fails, I find it is possible to get performance comparable to LMS with plain Java (and a JDK upgrade).

Two implementations of Java matrix multiplication are provided in the NGen benchmarks: `JMMM.baseline` – a naive but cache unfriendly matrix multiplication – and `JMMM.blocked` which is supplied as an improvement. `JMMM.blocked` is something of a local maximum because it does manual loop unrolling: this actually removes the trigger for autovectorisation analysis. I provide a simple and cache-efficient Java implementation (with the same asymptotic complexity, the improvement is just technical) and benchmark these implementations using JDK8 and the soon to be released JDK10 separately.

``````public void fast(float[] a, float[] b, float[] c, int n) {
int in = 0;
for (int i = 0; i < n; ++i) {
int kn = 0;
for (int k = 0; k < n; ++k) {
float aik = a[in + k];
for (int j = 0; j < n; ++j) {
c[in + j] += aik * b[kn + j];
}
kn += n;
}
in += n;
}
}
``````

With JDK 1.8.0_131, the “fast” implementation is only 2x faster than the blocked algorithm; this is nowhere near fast enough to match LMS. In fact, LMS does a lot better than 5x blocked (6x-8x) on my Skylake laptop at 2.6GHz, and performs between 2x and 4x better than the improved implementation. Flops / Cycle is calculated as `size ^ 3 * 2 / CPU frequency Hz`.

```====================================================
Benchmarking MMM.jMMM.fast (JVM implementation)
----------------------------------------------------
Size (N) | Flops / Cycle
----------------------------------------------------
8 | 0.4994459272
32 | 1.0666533335
64 | 0.9429120397
128 | 0.9692385519
192 | 0.9796619688
256 | 1.0141446247
320 | 0.9894415771
384 | 1.0046245750
448 | 1.0221353392
512 | 0.9943527764
576 | 0.9952093603
640 | 0.9854689714
704 | 0.9947153752
768 | 1.0197765248
832 | 1.0479691069
896 | 1.0060121097
960 | 0.9937347412
1024 | 0.9056494897
====================================================

====================================================
Benchmarking MMM.nMMM.blocked (LMS generated)
----------------------------------------------------
Size (N) | Flops / Cycle
----------------------------------------------------
8 | 0.2500390686
32 | 3.9999921875
64 | 4.1626523901
128 | 4.4618695374
192 | 3.9598982956
256 | 4.3737341517
320 | 4.2412225389
384 | 3.9640163416
448 | 4.0957167537
512 | 3.3801071278
576 | 4.1869326167
640 | 3.8225244883
704 | 3.8648224140
768 | 3.5240611589
832 | 3.7941562681
896 | 3.1735179981
960 | 2.5856903789
1024 | 1.7817152313
====================================================

====================================================
Benchmarking MMM.jMMM.blocked (JVM implementation)
----------------------------------------------------
Size (N) | Flops / Cycle
----------------------------------------------------
8 | 0.3333854248
32 | 0.6336670915
64 | 0.5733484649
128 | 0.5987433798
192 | 0.5819900921
256 | 0.5473562109
320 | 0.5623263520
384 | 0.5583823292
448 | 0.5657882256
512 | 0.5430879470
576 | 0.5269635678
640 | 0.5595204791
704 | 0.5297557807
768 | 0.5493631388
832 | 0.5471832673
896 | 0.4769554752
960 | 0.4985080443
1024 | 0.4014589400
====================================================
```

JDK10 is about to be released so it’s worth looking at the effect of recent improvements to C2, including better use of AVX2 and support for vectorised FMA. Since LMS depends on scala-virtualized, which currently only supports Scala 2.11, the LMS implementation cannot be run with a more recent JDK so its performance running in JDK10 could only be extrapolated. Since its raison d’ĂȘtre is to bypass C2, it could be reasonably assumed it is insulated from JVM performance improvements (or regressions). Measurements of floating point operations per cycle provide a sensible comparison, in any case.

Moving away from ScalaMeter, I created a JMH benchmark to see how matrix multiplication behaves in JDK10.

``````@OutputTimeUnit(TimeUnit.SECONDS)
@State(Scope.Benchmark)
public class MMM {

@Param({"8", "32", "64", "128", "192", "256", "320", "384", "448", "512" , "576", "640", "704", "768", "832", "896", "960", "1024"})
int size;

private float[] a;
private float[] b;
private float[] c;

@Setup(Level.Trial)
public void init() {
a = DataUtil.createFloatArray(size * size);
b = DataUtil.createFloatArray(size * size);
c = new float[size * size];
}

@Benchmark
public void fast(Blackhole bh) {
fast(a, b, c, size);
bh.consume(c);
}

@Benchmark
public void baseline(Blackhole bh) {
baseline(a, b, c, size);
bh.consume(c);
}

@Benchmark
public void blocked(Blackhole bh) {
blocked(a, b, c, size);
bh.consume(c);
}

//
// Baseline implementation of a Matrix-Matrix-Multiplication
//
public void baseline (float[] a, float[] b, float[] c, int n){
for (int i = 0; i < n; i += 1) {
for (int j = 0; j < n; j += 1) {
float sum = 0.0f;
for (int k = 0; k < n; k += 1) {
sum += a[i * n + k] * b[k * n + j];
}
c[i * n + j] = sum;
}
}
}

//
// Blocked version of MMM, reference implementation available at:
// http://csapp.cs.cmu.edu/2e/waside/waside-blocking.pdf
//
public void blocked(float[] a, float[] b, float[] c, int n) {
int BLOCK_SIZE = 8;
for (int kk = 0; kk < n; kk += BLOCK_SIZE) {
for (int jj = 0; jj < n; jj += BLOCK_SIZE) {
for (int i = 0; i < n; i++) {
for (int j = jj; j < jj + BLOCK_SIZE; ++j) {
float sum = c[i * n + j];
for (int k = kk; k < kk + BLOCK_SIZE; ++k) {
sum += a[i * n + k] * b[k * n + j];
}
c[i * n + j] = sum;
}
}
}
}
}

public void fast(float[] a, float[] b, float[] c, int n) {
int in = 0;
for (int i = 0; i < n; ++i) {
int kn = 0;
for (int k = 0; k < n; ++k) {
float aik = a[in + k];
for (int j = 0; j < n; ++j) {
c[in + j] = Math.fma(aik,  b[kn + j], c[in + j]);
}
kn += n;
}
in += n;
}
}
}
``````

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: size Ratio to blocked Flops/Cycle
baseline thrpt 1 10 1228544.82 38793.17392 ops/s 8 1.061598336 0.483857652
baseline thrpt 1 10 22973.03402 1012.043446 ops/s 32 1.302266947 0.57906183
baseline thrpt 1 10 2943.088879 221.57475 ops/s 64 1.301414733 0.593471609
baseline thrpt 1 10 358.010135 9.342801 ops/s 128 1.292889618 0.577539747
baseline thrpt 1 10 105.758366 4.275503 ops/s 192 1.246415143 0.575804515
baseline thrpt 1 10 41.465557 1.112753 ops/s 256 1.430003946 0.535135851
baseline thrpt 1 10 20.479081 0.462547 ops/s 320 1.154267894 0.516198866
baseline thrpt 1 10 11.686685 0.263476 ops/s 384 1.186535349 0.509027985
baseline thrpt 1 10 7.344184 0.269656 ops/s 448 1.166421127 0.507965526
baseline thrpt 1 10 3.545153 0.108086 ops/s 512 0.81796657 0.366017216
baseline thrpt 1 10 3.789384 0.130934 ops/s 576 1.327168294 0.557048123
baseline thrpt 1 10 1.981957 0.040136 ops/s 640 1.020965271 0.399660104
baseline thrpt 1 10 1.76672 0.036386 ops/s 704 1.168272442 0.474179037
baseline thrpt 1 10 1.01026 0.049853 ops/s 768 0.845514112 0.352024966
baseline thrpt 1 10 1.115814 0.03803 ops/s 832 1.148752171 0.494331667
baseline thrpt 1 10 0.703561 0.110626 ops/s 896 0.938435436 0.389298235
baseline thrpt 1 10 0.629896 0.052448 ops/s 960 1.081741651 0.428685898
baseline thrpt 1 10 0.407772 0.019079 ops/s 1024 1.025356561 0.336801424
blocked thrpt 1 10 1157259.558 49097.48711 ops/s 8 1 0.455782226
blocked thrpt 1 10 17640.8025 1226.401298 ops/s 32 1 0.444656782
blocked thrpt 1 10 2261.453481 98.937035 ops/s 64 1 0.456020355
blocked thrpt 1 10 276.906961 22.851857 ops/s 128 1 0.446704605
blocked thrpt 1 10 84.850033 4.441454 ops/s 192 1 0.461968485
blocked thrpt 1 10 28.996813 7.585551 ops/s 256 1 0.374219842
blocked thrpt 1 10 17.742052 0.627629 ops/s 320 1 0.447208892
blocked thrpt 1 10 9.84942 0.367603 ops/s 384 1 0.429003641
blocked thrpt 1 10 6.29634 0.402846 ops/s 448 1 0.435490676
blocked thrpt 1 10 4.334105 0.384849 ops/s 512 1 0.447472097
blocked thrpt 1 10 2.85524 0.199102 ops/s 576 1 0.419726816
blocked thrpt 1 10 1.941258 0.10915 ops/s 640 1 0.391453182
blocked thrpt 1 10 1.51225 0.076621 ops/s 704 1 0.40588053
blocked thrpt 1 10 1.194847 0.063147 ops/s 768 1 0.416344283
blocked thrpt 1 10 0.971327 0.040421 ops/s 832 1 0.430320551
blocked thrpt 1 10 0.749717 0.042997 ops/s 896 1 0.414837526
blocked thrpt 1 10 0.582298 0.016725 ops/s 960 1 0.39629231
blocked thrpt 1 10 0.397688 0.043639 ops/s 1024 1 0.328472491
fast thrpt 1 10 1869676.345 76416.50848 ops/s 8 1.615606743 0.736364837
fast thrpt 1 10 48485.47216 1301.926828 ops/s 32 2.748484496 1.222132271
fast thrpt 1 10 6431.341657 153.905413 ops/s 64 2.843897392 1.296875098
fast thrpt 1 10 840.601821 45.998723 ops/s 128 3.035683242 1.356053685
fast thrpt 1 10 260.386996 13.022418 ops/s 192 3.068790745 1.417684611
fast thrpt 1 10 107.895708 6.584674 ops/s 256 3.720950575 1.392453537
fast thrpt 1 10 56.245336 2.729061 ops/s 320 3.170170846 1.417728592
fast thrpt 1 10 32.917996 2.196624 ops/s 384 3.342125323 1.433783932
fast thrpt 1 10 20.960189 2.077684 ops/s 448 3.328948087 1.449725854
fast thrpt 1 10 14.005186 0.7839 ops/s 512 3.231390564 1.445957112
fast thrpt 1 10 8.827584 0.883654 ops/s 576 3.091713481 1.297675056
fast thrpt 1 10 7.455607 0.442882 ops/s 640 3.840605937 1.503417416
fast thrpt 1 10 5.322894 0.464362 ops/s 704 3.519850554 1.428638807
fast thrpt 1 10 4.308522 0.153846 ops/s 768 3.605919419 1.501303934
fast thrpt 1 10 3.375274 0.106715 ops/s 832 3.474910097 1.495325228
fast thrpt 1 10 2.320152 0.367881 ops/s 896 3.094703735 1.28379924
fast thrpt 1 10 2.057478 0.150198 ops/s 960 3.533376381 1.400249889
fast thrpt 1 10 1.66255 0.181116 ops/s 1024 4.180538513 1.3731919

Interestingly, the blocked algorithm is now the worst native JVM implementation. The code generated by C2 got a lot faster, but peaks at 1.5 flops/cycle, which still doesn’t compete with LMS. Why? Taking a look at the assembly, it’s clear that the autovectoriser choked on the array offsets and produced scalar SSE2 code, just like the implementations in the paper. I wasn’t expecting this.

```vmovss  xmm5,dword ptr [rdi+rcx*4+10h]
vmovss  dword ptr [rdi+rcx*4+10h],xmm5
```

Is this the end of the story? No, with some hacks and the cost of array allocation and a copy or two, autovectorisation can be tricked into working again to generate faster code:

``````
public void fast(float[] a, float[] b, float[] c, int n) {
float[] bBuffer = new float[n];
float[] cBuffer = new float[n];
int in = 0;
for (int i = 0; i < n; ++i) {
int kn = 0;
for (int k = 0; k < n; ++k) {
float aik = a[in + k];
System.arraycopy(b, kn, bBuffer, 0, n);
saxpy(n, aik, bBuffer, cBuffer);
kn += n;
}
System.arraycopy(cBuffer, 0, c, in, n);
Arrays.fill(cBuffer, 0f);
in += n;
}
}

private void saxpy(int n, float aik, float[] b, float[] c) {
for (int i = 0; i < n; ++i) {
c[i] += aik * b[i];
}
}
``````

Adding this hack into the NGen benchmark (back in JDK 1.8.0_131) I get closer to the LMS generated code, and beat it beyond L3 cache residency (6MB). LMS is still faster when both matrices fit in L3 concurrently, but by percentage points rather than a multiple. The cost of the hacky array buffers gives the game up for small matrices.

```====================================================
Benchmarking MMM.jMMM.fast (JVM implementation)
----------------------------------------------------
Size (N) | Flops / Cycle
----------------------------------------------------
8 | 0.2500390686
32 | 0.7710872405
64 | 1.1302489072
128 | 2.5113453810
192 | 2.9525859816
256 | 3.1180920385
320 | 3.1081563593
384 | 3.1458423577
448 | 3.0493148252
512 | 3.0551158263
576 | 3.1430376938
640 | 3.2169923048
704 | 3.1026513283
768 | 2.4190053777
832 | 3.3358586705
896 | 3.0755689237
960 | 2.9996690697
1024 | 2.2935654309
====================================================

====================================================
Benchmarking MMM.nMMM.blocked (LMS generated)
----------------------------------------------------
Size (N) | Flops / Cycle
----------------------------------------------------
8 | 1.0001562744
32 | 5.3330416826
64 | 5.8180867784
128 | 5.1717318641
192 | 5.1639907462
256 | 4.3418618628
320 | 5.2536572701
384 | 4.0801359215
448 | 4.1337007093
512 | 3.2678160754
576 | 3.7973028890
640 | 3.3557513664
704 | 4.0103133240
768 | 3.4188362575
832 | 3.2189488327
896 | 3.2316685219
960 | 2.9985655539
1024 | 1.7750946796
====================================================
```

With the benchmark below I calculate flops/cycle with improved JDK10 autovectorisation.

``````
@Benchmark
public void fastBuffered(Blackhole bh) {
fastBuffered(a, b, c, size);
bh.consume(c);
}

public void fastBuffered(float[] a, float[] b, float[] c, int n) {
float[] bBuffer = new float[n];
float[] cBuffer = new float[n];
int in = 0;
for (int i = 0; i < n; ++i) {
int kn = 0;
for (int k = 0; k < n; ++k) {
float aik = a[in + k];
System.arraycopy(b, kn, bBuffer, 0, n);
saxpy(n, aik, bBuffer, cBuffer);
kn += n;
}
System.arraycopy(cBuffer, 0, c, in, n);
Arrays.fill(cBuffer, 0f);
in += n;
}
}

private void saxpy(int n, float aik, float[] b, float[] c) {
for (int i = 0; i < n; ++i) {
c[i] = Math.fma(aik, b[i], c[i]);
}
}
``````

Just as in the modified NGen benchmark, this starts paying off once the matrices have 64 rows and columns. Finally, and it took an upgrade and a hack, I breached 4 Flops per cycle:

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: size Flops / Cycle
fastBuffered thrpt 1 10 1047184.034 63532.95095 ops/s 8 0.412429404
fastBuffered thrpt 1 10 58373.56367 3239.615866 ops/s 32 1.471373026
fastBuffered thrpt 1 10 12099.41654 497.33988 ops/s 64 2.439838038
fastBuffered thrpt 1 10 2136.50264 105.038006 ops/s 128 3.446592911
fastBuffered thrpt 1 10 673.470622 102.577237 ops/s 192 3.666730488
fastBuffered thrpt 1 10 305.541519 25.959163 ops/s 256 3.943181586
fastBuffered thrpt 1 10 158.437372 6.708384 ops/s 320 3.993596774
fastBuffered thrpt 1 10 88.283718 7.58883 ops/s 384 3.845306266
fastBuffered thrpt 1 10 58.574507 4.248521 ops/s 448 4.051345968
fastBuffered thrpt 1 10 37.183635 4.360319 ops/s 512 3.839002314
fastBuffered thrpt 1 10 29.949884 0.63346 ops/s 576 4.40270151
fastBuffered thrpt 1 10 20.715833 4.175897 ops/s 640 4.177331789
fastBuffered thrpt 1 10 10.824837 0.902983 ops/s 704 2.905333492
fastBuffered thrpt 1 10 8.285254 1.438701 ops/s 768 2.886995686
fastBuffered thrpt 1 10 6.17029 0.746537 ops/s 832 2.733582608
fastBuffered thrpt 1 10 4.828872 1.316901 ops/s 896 2.671937962
fastBuffered thrpt 1 10 3.6343 1.293923 ops/s 960 2.473381573
fastBuffered thrpt 1 10 2.458296 0.171224 ops/s 1024 2.030442485

The code generated for the core of the loop looks better now:

```vmovdqu ymm1,ymmword ptr [r13+r11*4+10h]