When our Python code is too slow, like most others we switch to C and often get 100x speed boosts, just like when we replaced SciPy distance computations with SimSIMD. But imagine going 100x faster than C code!

It sounds crazy, especially for number-crunching tasks that are “data-parallel” and easy for compilers to optimize. In such spots the compiler will typically “unroll” the loop, vectorize the code, and use SIMD instructions to process multiple data elements in parallel. But I’ve found a simple case, where the GNU C Compiler (GCC 12) failed to do so, and lost to hand-written SIMD code by a factor of 119. Enough to make the friendly bull compiler cry. For the curious, the code is already part of my SimSIMD library on GitHub, and to be fair, today, its unimaginable that a compiler can properly optimize high-level code choosing from thousands of complex Assembly instructions, same way as a human can. So let’s see how we’ve got to a two-orders-of-magnitude speedup.

## Background#

Continuing with my work on USearch and the core SimSIMD library, I’ve turned my attention to speeding up the Jensen Shannon divergence, a symmetric variant of the Kullback-Leibler divergence, popular in stats, bio-informatics, and chem-informatics.

$$JS(P, Q) = \frac{1}{2} D(P || M) + \frac{1}{2} D(Q || M)$$

$$M = \frac{1}{2}(P + Q), D(P || Q) = \sum P(i) \cdot \log \left( \frac{P(i)}{Q(i)} \right)$$

It’s tougher to compute than the usual Cosine Similarity, mainly because it needs to crunch logarithms, which are pretty slow with LibC’s logf and most other math libraries. So, I reworked the logarithm part and the rest of the code using AVX-512 and AVX-512FP16 SIMD goodies for Intel’s Sapphire Rapids 2022 CPUs.

## Optimizations#

• Logarithm Computation. Instead of using multiple bitwise operations, I now use _mm512_getexp_ph and _mm512_getmant_ph to pull out the exponent and the mantissa of the floating-point number, which makes things simpler. I’ve also brought in Horner’s method for the polynomial approximation.
• Division Avoidance. To skip over the slow division operations, I’ve used reciprocal approximations - _mm512_rcp_ph for half-precision and _mm512_rcp14_ps for single-precision. Found out _mm512_rcp28_ps wasn’t needed for this work.
• Handling Zeros. I’m using _mm512_cmp_ph_mask to create a mask for near-zero values, which avoids having to add a small “epsilon” to every component, making it cleaner and more accurate.
• Parallel Accumulation. I now accumulate $KL(P||Q)$ and $KL(Q||P)$ in different registers, and use _mm512_maskz_fmadd_ph to replace separate addition and multiplication operations, which makes the calculation more efficient.

## Implementation#

The basic Jensen Shannon divergence implementation in C 99 appears as follows:

  1 2 3 4 5 6 7 8 9 10 11 12  inline static simsimd_f32_t simsimd_serial_f32_js(simsimd_f32_t const* a, simsimd_f32_t const* b, simsimd_size_t n) { simsimd_f32_t sum_a = 0; simsimd_f32_t sum_b = 0; for (simsimd_size_t i = 0; i < n; i++) { simsimd_f32_t m = (a[i] + b[i]) * 0.5f; if (m > 1e-6f) { sum_a += a[i] * logf(a[i] / m); sum_b += b[i] * logf(b[i] / m); } } return 0.5f * (sum_a + sum_b); } 

This is a straightforward implementation utilizing the log2f function from LibC. To enhance its speed, one approach is to eliminate the if (m > 1e-6f) branch which is there to prevent division by zero. Instead, we could add a small epsilon to every component:

 1 2 3  simsimd_f32_t m = (a[i] + b[i]) * 0.5f; sum_a += a[i] * logf((a[i] + 1e-6f) / (m + 1e-6f)); sum_b += b[i] * logf((b[i] + 1e-6f) / (m + 1e-6f)); 

However, the AVX-512FP16 implementation that I ended up with is quite different:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33  __attribute__((target("avx512f,avx512vl,avx512fp16"))) inline static simsimd_f32_t simsimd_avx512_f16_js(simsimd_f16_t const* a, simsimd_f16_t const* b, simsimd_size_t n) { __m512h sum_a_vec = _mm512_set1_ph((_Float16)0); __m512h sum_b_vec = _mm512_set1_ph((_Float16)0); __m512h epsilon_vec = _mm512_set1_ph((_Float16)1e-6f); for (simsimd_size_t i = 0; i < n; i += 32) { // Masked loads prevent reading beyond array limits __mmask32 mask = n - i >= 32 ? 0xFFFFFFFF : ((1u << (n - i)) - 1u); __m512h a_vec = _mm512_castsi512_ph(_mm512_maskz_loadu_epi16(mask, a + i)); __m512h b_vec = _mm512_castsi512_ph(_mm512_maskz_loadu_epi16(mask, b + i)); __m512h m_vec = _mm512_mul_ph(_mm512_add_ph(a_vec, b_vec), _mm512_set1_ph((_Float16)0.5f)); // Masking helps sidestep division by zero issues later __mmask32 nonzero_mask_a = _mm512_cmp_ph_mask(a_vec, epsilon_vec, _CMP_GE_OQ); __mmask32 nonzero_mask_b = _mm512_cmp_ph_mask(b_vec, epsilon_vec, _CMP_GE_OQ); __mmask32 nonzero_mask = nonzero_mask_a & nonzero_mask_b & mask; // Approximate the reciprocal of m to avoid two division operations __m512h m_recip_approx = _mm512_rcp_ph(m_vec); __m512h ratio_a_vec = _mm512_mul_ph(a_vec, m_recip_approx); __m512h ratio_b_vec = _mm512_mul_ph(b_vec, m_recip_approx); // Compute log2, adjusting with loge(2) to get the natural logarithm __m512h log_ratio_a_vec = simsimd_avx512_f16_log2(ratio_a_vec); __m512h log_ratio_b_vec = simsimd_avx512_f16_log2(ratio_b_vec); // FMA boosts efficiency by replacing separate mul and add operations sum_a_vec = _mm512_maskz_fmadd_ph(nonzero_mask, a_vec, log_ratio_a_vec, sum_a_vec); sum_b_vec = _mm512_maskz_fmadd_ph(nonzero_mask, b_vec, log_ratio_b_vec, sum_b_vec); } simsimd_f32_t log2_normalizer = 0.693147181f; return _mm512_reduce_add_ph(_mm512_add_ph(sum_a_vec, sum_b_vec)) * 0.5f * log2_normalizer; } 

This version leverages GCC attributes to ensure the compiler processes the AVX512-FP16 instructions, even on older CPUs. It calls the simsimd_avx512_f16_log2 function, inlined below:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17  __attribute__((target("avx512f,avx512vl,avx512fp16"))) inline __m512h simsimd_avx512_f16_log2(__m512h x) { // Extract the exponent and mantissa __m512h one = _mm512_set1_ph((_Float16)1); __m512h e = _mm512_getexp_ph(x); __m512h m = _mm512_getmant_ph(x, _MM_MANT_NORM_1_2, _MM_MANT_SIGN_src); // Compute the polynomial using Horner's method __m512h p = _mm512_set1_ph((_Float16)-3.4436006e-2f); p = _mm512_fmadd_ph(m, p, _mm512_set1_ph((_Float16)3.1821337e-1f)); p = _mm512_fmadd_ph(m, p, _mm512_set1_ph((_Float16)-1.2315303f)); p = _mm512_fmadd_ph(m, p, _mm512_set1_ph((_Float16)2.5988452f)); p = _mm512_fmadd_ph(m, p, _mm512_set1_ph((_Float16)-3.3241990f)); p = _mm512_fmadd_ph(m, p, _mm512_set1_ph((_Float16)3.1157899f)); return _mm512_add_ph(_mm512_mul_ph(p, _mm512_sub_ph(m, one)), e); } 

It computes the polynomial approximation of the logarithm, using Horner’s method, accumulating five components, and then adds the exponent back to the result.

## Benchmarks#

I carried out benchmarks at both Python and C++ layers, comparing auto-vectorization on GCC against our new implementation on an Intel Sapphire Rapids CPU on AWS. The newest LTS Ubuntu version is 22.04, which comes with GCC 11 and doesn’t support AVX512-FP16 intrinsics. So I’ve further upgraded the compiler on the machine to GCC 12 and compiled it with -O3, -march=native, and -ffast-math flags. Looking at the logs from the Google Benchmark suite, we see that the benchmark was executed on all cores of the 4-core instance, which might have tilted the scales in favor of the old non-vectorized solution, but the gap is still outrageous.

Single-Precision Math:

ImplementationPairs/sGigabytes/sSpeedupAbsolute ErrorRelative Error
serial_f32_js_1536d0.243 M/s2.98 G/s00
avx512_f32_js_1536d1.127 M/s13.84 G/s4.64x0.001345u

Half-Precision Math:

ImplementationPairs/sGigabytes/sSpeedupAbsolute ErrorRelative Error
serial_f16_js_1536d0.018 M/s0.11 G/s0.1230.035
avx2_f16_js_1536d0.547 M/s3.36 G/s30.39x0.0110.003
avx512_f16_js_1536d2.139 M/s13.14 G/s118.84x0.0700.020

Undoubtedly, the outcomes may fluctuate based on the vector size. Typically, I employ 1536 dimensions, which align with the size of OpenAI Ada embeddings, a standard in NLP tasks. But the most interesting applications of JS divergence would be far from NLP - closer to natural sciences.

The application I am most excited about - clustering of Billions of different protein sequences without any Multiple Sequence Alignment procedures. Sounds interesting? Check out the clustering capabilities of USearch 🤗