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Spatial Distances in NumKong

NumKong implements spatial distance functions for dense vectors: squared Euclidean distance, Euclidean distance, and angular (cosine) distance. These metrics are commonly used in nearest-neighbor search, clustering, and dimensionality reduction, and are implemented for every numeric type supported by the library.

Squared Euclidean distance measures the sum of squared element-wise differences:

$$ \text{sqeuclidean}(a, b) = \sum_{i=0}^{n-1} (a_i - b_i)^2 $$

Euclidean distance is the square root of the squared Euclidean distance:

$$ \text{euclidean}(a, b) = \sqrt{\sum_{i=0}^{n-1} (a_i - b_i)^2} $$

Angular distance (cosine distance) measures the angle between two vectors:

$$ \text{angular}(a, b) = 1 - \frac{\sum_{i=0}^{n-1} a_i \cdot b_i}{\sqrt{\sum_{i=0}^{n-1} a_i^2} \cdot \sqrt{\sum_{i=0}^{n-1} b_i^2}} $$

Reformulating as Python pseudocode:

import numpy as np

def sqeuclidean(a: np.ndarray, b: np.ndarray) -> float:
    return np.sum((a - b) ** 2)

def euclidean(a: np.ndarray, b: np.ndarray) -> float:
    return np.sqrt(np.sum((a - b) ** 2))

def angular(a: np.ndarray, b: np.ndarray) -> float:
    ab = np.dot(a, b)
    a2 = np.dot(a, a)
    b2 = np.dot(b, b)
    if a2 == 0 and b2 == 0: return 0
    if ab == 0: return 1
    return 1 - ab / (np.sqrt(a2) * np.sqrt(b2))

Input & Output Types

Input Type Output Type Description
f64 f64 64-bit IEEE 754 double precision
f32 f32 32-bit IEEE 754 single precision
f16 f32 16-bit IEEE 754 half precision, widened output
bf16 f32 16-bit brain float, widened output
e5m2 f32 8-bit Float8: 5 exponent, 2 mantissa bits
e4m3 f32 8-bit Float8: 4 exponent, 3 mantissa bits
e3m2 f32 8-bit MX format: 3 exponent, 2 mantissa bits
e2m3 f32 8-bit MX format: 2 exponent, 3 mantissa bits
i8 f32 8-bit signed integers
u8 f32 8-bit unsigned integers
i4 f32 4-bit signed integers, packed nibble pairs
u4 f32 4-bit unsigned integers, packed nibble pairs

Optimizations

Three-Accumulator Angular Pattern

nk_angular_f32_haswell, nk_angular_f32_skylake, nk_angular_f32_neon compute cosine distance as $1 - ab / (\sqrt{a^2} \cdot \sqrt{b^2})$, requiring three concurrent dot products in a single pass: $\sum a_i b_i$, $\sum a_i^2$, and $\sum b_i^2$. All spatial angular kernels interleave these three FMA streams so that each vector element is loaded once and immediately contributes to all three accumulators. This triples register pressure compared to a plain dot product — on Haswell with 16 YMM registers, three independent 4-register accumulator chains leave only 4 registers for temporaries. The single-pass design is essential because reading two vectors of length $n$ once costs $2n$ cache line fetches, while a three-pass approach would cost $6n$.

Reciprocal Square Root with Newton-Raphson Refinement

nk_angular_f32_haswell, nk_angular_f64_haswell, nk_angular_f32_neon, nk_angular_f64_neon compute the final normalization via in-hardware reciprocal square root estimates refined by Newton-Raphson iteration. The iteration formula is $x_{n+1} = x_n \cdot (3 - d \cdot x_n^2) / 2$, where $d$ is the value whose reciprocal square root is needed. NEON vrsqrte + vrsqrts performs one refinement step, reaching roughly 22 bits of precision. Haswell VRSQRT14 provides $2^{-14}$ relative error and one Newton-Raphson step doubles the precision to approximately 28 bits. Skylake VRSQRT28 achieves $2^{-28}$ accuracy directly, eliminating the need for a refinement step entirely. This reciprocal square root is needed for both euclidean distance ($\sqrt{d}$ via $d \cdot \text{rsqrt}(d)$) and angular distance ($1/\sqrt{a^2} \cdot 1/\sqrt{b^2}$).

Absolute Differences for Integer Types

nk_sqeuclidean_i8_haswell, nk_sqeuclidean_u8_haswell, nk_sqeuclidean_i8_icelake, nk_sqeuclidean_u8_icelake compute squared Euclidean distance by first obtaining element-wise absolute differences, then squaring and accumulating. For signed i8, XOR with 0x80 converts the range from [-128, 127] to unsigned [0, 255], then saturating subtract in both directions followed by OR gives $|a - b|$:

bias_a = _mm256_xor_si256(a, 0x80)
bias_b = _mm256_xor_si256(b, 0x80)
abs_diff = _mm256_or_si256(_mm256_subs_epu8(bias_a, bias_b), _mm256_subs_epu8(bias_b, bias_a))

For unsigned u8, the same saturating subtract trick works without the XOR bias. The absolute differences are then zero-extended via VPUNPCKLBW/VPUNPCKHBW (1 cycle, cheaper than VPMOVZXBW) and squared+accumulated via VPMADDWD, which computes $d_i^2 + d_{i+1}^2$ in one instruction.

Masked Neumaier Compensation on Skylake

nk_sqeuclidean_f64_skylake uses VGETEXP-based Neumaier TwoSum inside AVX-512 masked loops. The mask register tracks which lanes are active, handling tail elements when the vector length is not a multiple of the SIMD width. The compensation term accumulates the low-order rounding errors from each addition, and because the mask propagates through both the main sum and the compensation update, even the final partial iteration maintains full Neumaier accuracy. This avoids the need for a separate scalar tail loop that would otherwise lose the compensated error tracking.

Performance

The following performance tables are produced by manually re-running nk_test and nk_bench included internal tools to measure both accuracy and throughput at different input shapes. The input size is controlled by the NK_DENSE_DIMENSIONS environment variable and set to 256, 1024, and 4096 elements. The throughput is measured in GB/s as the number of input bytes per second. Accuracy is reported as mean ULP (units in last place) unless noted otherwise — the average number of representable floating-point values between the result and the exact answer. Each kernel runs for at least 20 seconds per configuration. Benchmark threads are pinned to specific cores; on machines with heterogeneous core types (e.g., Apple P/E cores), only the fastest cores are used. Workloads that significantly degrade CPU frequencies (Intel AMX, Apple SME) run in separate passes to avoid affecting throughput measurements of other kernels.

Intel Sapphire Rapids

Native

Kernel 256 1024 4096
f64 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f64_serial 8.00 gb/s, 0.1 ulp 8.32 gb/s, 0 ulp 8.13 gb/s, 0 ulp
nk_euclidean_f64_serial 7.81 gb/s, 0.6 ulp 7.95 gb/s, 0.5 ulp 8.34 gb/s, 0.5 ulp
nk_angular_f64_serial 2.80 gb/s, 0 ulp 3.03 gb/s, 0 ulp 3.18 gb/s, 0 ulp
nk_sqeuclidean_f64_skylake 32.4 gb/s, 0.4 ulp 30.6 gb/s, 0.7 ulp 22.2 gb/s, 1.3 ulp
nk_euclidean_f64_skylake 31.7 gb/s, 0.3 ulp 29.4 gb/s, 0.4 ulp 22.9 gb/s, 0.7 ulp
nk_angular_f64_skylake 26.5 gb/s, 0 ulp 26.8 gb/s, 0 ulp 17.8 gb/s, 0 ulp
f32 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f32_serial 4.01 gb/s, 0 ulp 4.06 gb/s, 0 ulp 4.19 gb/s, 0 ulp
nk_euclidean_f32_serial 3.99 gb/s, 0.1 ulp 4.07 gb/s, 0.1 ulp 4.11 gb/s, 0.1 ulp
nk_angular_f32_serial 1.29 gb/s, 0 ulp 1.41 gb/s, 0 ulp 1.53 gb/s, 0 ulp
nk_sqeuclidean_f32_skylake 36.5 gb/s, 0 ulp 27.0 gb/s, 0 ulp 23.2 gb/s, 0 ulp
nk_euclidean_f32_skylake 36.4 gb/s, 0.1 ulp 28.1 gb/s, 0.1 ulp 26.7 gb/s, 0.1 ulp
nk_angular_f32_skylake 24.3 gb/s, 0 ulp 23.2 gb/s, 0 ulp 22.5 gb/s, 0 ulp
bf16 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_bf16_serial 0.582 gb/s, 0 ulp 0.358 gb/s, 0 ulp 0.390 gb/s, 0 ulp
nk_euclidean_bf16_serial 0.569 gb/s, 0.5 ulp 0.373 gb/s, 0.5 ulp 0.372 gb/s, 0.4 ulp
nk_angular_bf16_serial 0.455 gb/s, 0 ulp 0.241 gb/s, 0 ulp 0.259 gb/s, 0 ulp
nk_sqeuclidean_bf16_haswell 27.7 gb/s, 0.5 ulp 14.0 gb/s, 7.5 ulp 11.8 gb/s, 27 ulp
nk_euclidean_bf16_haswell 23.3 gb/s, 0.3 ulp 13.4 gb/s, 4.1 ulp 12.0 gb/s, 15 ulp
nk_angular_bf16_haswell 20.1 gb/s, 0 ulp 13.4 gb/s, 0 ulp 10.6 gb/s, 0.2 ulp
nk_sqeuclidean_bf16_genoa 50.1 gb/s, 0.3 ulp 21.0 gb/s, 0.5 ulp 20.5 gb/s, 10 ulp
nk_euclidean_bf16_genoa 48.3 gb/s, 0.2 ulp 23.1 gb/s, 0.3 ulp 20.4 gb/s, 5.8 ulp
nk_angular_bf16_genoa 36.4 gb/s, 0 ulp 22.4 gb/s, 0 ulp 21.0 gb/s, 0.1 ulp
f16 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f16_serial 0.950 gb/s, 0.1 ulp 0.872 gb/s, 0.1 ulp 0.864 gb/s, 0.1 ulp
nk_euclidean_f16_serial 0.934 gb/s, 0.5 ulp 0.913 gb/s, 0.5 ulp 0.906 gb/s, 0.5 ulp
nk_angular_f16_serial 0.881 gb/s, 0 ulp 0.531 gb/s, 0 ulp 0.543 gb/s, 0 ulp
nk_sqeuclidean_f16_haswell 29.8 gb/s, 0.4 ulp 14.8 gb/s, 1.4 ulp 11.8 gb/s, 5.2 ulp
nk_euclidean_f16_haswell 22.9 gb/s, 0.3 ulp 12.9 gb/s, 0.8 ulp 10.6 gb/s, 2.8 ulp
nk_angular_f16_haswell 19.9 gb/s, 0.1 ulp 17.5 gb/s, 0.1 ulp 16.1 gb/s, 0.1 ulp
e5m2 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e5m2_serial 0.955 gb/s, 0 ulp 1.01 gb/s, 0 ulp 1.02 gb/s, 0 ulp
nk_euclidean_e5m2_serial 0.954 gb/s, 0.5 ulp 0.985 gb/s, 0.5 ulp 1.03 gb/s, 0.5 ulp
nk_angular_e5m2_serial 0.336 gb/s, 0 ulp 0.385 gb/s, 0 ulp 0.407 gb/s, 0 ulp
nk_sqeuclidean_e5m2_skylake 4.44 gb/s, 0 ulp 4.65 gb/s, 0 ulp 5.80 gb/s, 0 ulp
nk_euclidean_e5m2_skylake 4.34 gb/s, 0 ulp 4.65 gb/s, 0 ulp 5.88 gb/s, 0 ulp
nk_angular_e5m2_skylake 3.83 gb/s, 0 ulp 4.39 gb/s, 0 ulp 6.10 gb/s, 0 ulp
nk_sqeuclidean_e5m2_genoa 7.12 gb/s, 0 ulp 8.07 gb/s, 0 ulp 8.05 gb/s, 0 ulp
nk_euclidean_e5m2_genoa 7.01 gb/s, 0 ulp 6.97 gb/s, 0 ulp 8.16 gb/s, 0 ulp
nk_angular_e5m2_genoa 6.33 gb/s, 0 ulp 6.79 gb/s, 0 ulp 7.99 gb/s, 0 ulp
e4m3 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e4m3_serial 0.569 gb/s, 0 ulp 0.606 gb/s, 0 ulp 0.609 gb/s, 0 ulp
nk_euclidean_e4m3_serial 0.587 gb/s, 0.5 ulp 0.602 gb/s, 0.5 ulp 0.578 gb/s, 0.5 ulp
nk_angular_e4m3_serial 0.326 gb/s, 0 ulp 0.196 gb/s, 0 ulp 0.366 gb/s, 0 ulp
nk_sqeuclidean_e4m3_skylake 3.84 gb/s, 0 ulp 3.62 gb/s, 0 ulp 3.95 gb/s, 0.2 ulp
nk_euclidean_e4m3_skylake 3.48 gb/s, 0 ulp 3.69 gb/s, 0 ulp 3.33 gb/s, 0.2 ulp
nk_angular_e4m3_skylake 4.22 gb/s, 0 ulp 3.38 gb/s, 0 ulp 4.54 gb/s, 0 ulp
nk_sqeuclidean_e4m3_icelake 10.2 gb/s, 0 ulp 12.0 gb/s, 0 ulp 12.0 gb/s, 0.2 ulp
nk_euclidean_e4m3_icelake 10.3 gb/s, 0 ulp 11.8 gb/s, 0 ulp 11.9 gb/s, 0.2 ulp
nk_angular_e4m3_icelake 8.78 gb/s, 0 ulp 11.3 gb/s, 0 ulp 11.9 gb/s, 0 ulp
e3m2 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e3m2_serial 1.01 gb/s, 0 ulp 0.971 gb/s, 0 ulp 1.03 gb/s, 0 ulp
nk_euclidean_e3m2_serial 0.997 gb/s, 0.5 ulp 0.990 gb/s, 0.5 ulp 0.999 gb/s, 0.4 ulp
nk_angular_e3m2_serial 0.332 gb/s, 0 ulp 0.361 gb/s, 0 ulp 0.437 gb/s, 0 ulp
nk_sqeuclidean_e3m2_skylake 4.47 gb/s, 0 ulp 5.46 gb/s, 0 ulp 5.04 gb/s, 0 ulp
nk_euclidean_e3m2_skylake 4.34 gb/s, 0 ulp 6.20 gb/s, 0 ulp 5.10 gb/s, 0 ulp
nk_angular_e3m2_skylake 3.79 gb/s, 0 ulp 4.41 gb/s, 0 ulp 4.82 gb/s, 0 ulp
nk_sqeuclidean_e3m2_genoa 8.79 gb/s, 0 ulp 9.52 gb/s, 0 ulp 10.6 gb/s, 0 ulp
nk_euclidean_e3m2_genoa 8.68 gb/s, 0 ulp 9.01 gb/s, 0 ulp 12.8 gb/s, 0 ulp
nk_angular_e3m2_genoa 6.89 gb/s, 0 ulp 9.30 gb/s, 0 ulp 10.3 gb/s, 0 ulp
nk_sqeuclidean_e3m2_icelake 21.2 gb/s, 0 ulp 22.1 gb/s, 0 ulp 21.9 gb/s, 0 ulp
nk_euclidean_e3m2_icelake 21.2 gb/s, 0 ulp 22.9 gb/s, 0 ulp 21.2 gb/s, 0 ulp
nk_angular_e3m2_icelake 14.1 gb/s, 0 ulp 18.0 gb/s, 0 ulp 17.6 gb/s, 0 ulp
e2m3 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e2m3_serial 0.964 gb/s, 0 ulp 0.981 gb/s, 0 ulp 1.03 gb/s, 0 ulp
nk_euclidean_e2m3_serial 0.979 gb/s, 0.5 ulp 0.966 gb/s, 0.5 ulp 1.02 gb/s, 0.5 ulp
nk_angular_e2m3_serial 0.347 gb/s, 0 ulp 0.389 gb/s, 0 ulp 0.418 gb/s, 0 ulp
nk_sqeuclidean_e2m3_skylake 4.58 gb/s, 0 ulp 4.65 gb/s, 0 ulp 5.08 gb/s, 0 ulp
nk_euclidean_e2m3_skylake 4.48 gb/s, 0 ulp 4.39 gb/s, 0 ulp 4.96 gb/s, 0 ulp
nk_angular_e2m3_skylake 3.94 gb/s, 0 ulp 4.25 gb/s, 0 ulp 4.90 gb/s, 0 ulp
nk_sqeuclidean_e2m3_genoa 9.62 gb/s, 0 ulp 10.9 gb/s, 0 ulp 10.8 gb/s, 0 ulp
nk_euclidean_e2m3_genoa 8.45 gb/s, 0 ulp 9.80 gb/s, 0 ulp 10.3 gb/s, 0 ulp
nk_angular_e2m3_genoa 7.21 gb/s, 0 ulp 10.1 gb/s, 0 ulp 10.4 gb/s, 0 ulp
nk_sqeuclidean_e2m3_icelake 50.7 gb/s, 0 ulp 42.6 gb/s, 0 ulp 31.0 gb/s, 0 ulp
nk_euclidean_e2m3_icelake 50.2 gb/s, 0 ulp 44.3 gb/s, 0 ulp 31.0 gb/s, 0 ulp
nk_angular_e2m3_icelake 27.2 gb/s, 0 ulp 34.9 gb/s, 0 ulp 30.5 gb/s, 0 ulp
i8 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_i8_serial 34.0 gb/s 18.4 gb/s 16.5 gb/s
nk_euclidean_i8_serial 29.0 gb/s, 0.4 ulp 18.0 gb/s, 0.4 ulp 15.6 gb/s, 0.4 ulp
nk_angular_i8_serial 7.88 gb/s, 0 ulp 6.31 gb/s, 0 ulp 6.12 gb/s, 0 ulp
nk_sqeuclidean_i8_haswell 38.4 gb/s 17.9 gb/s 18.4 gb/s
nk_euclidean_i8_haswell 35.6 gb/s, 0 ulp 17.0 gb/s, 0 ulp 15.5 gb/s, 0 ulp
nk_angular_i8_haswell 20.3 gb/s, 0.1 ulp 12.9 gb/s, 0 ulp 11.9 gb/s, 0 ulp
nk_sqeuclidean_i8_icelake 60.2 gb/s 24.5 gb/s 23.5 gb/s
nk_euclidean_i8_icelake 59.0 gb/s, 0 ulp 23.0 gb/s, 0 ulp 22.3 gb/s, 0 ulp
nk_angular_i8_icelake 25.2 gb/s, 0.1 ulp 18.4 gb/s, 0 ulp 20.5 gb/s, 0 ulp
nk_sqeuclidean_i8_alder 33.4 gb/s 17.4 gb/s 17.6 gb/s
nk_euclidean_i8_alder 31.9 gb/s, 0 ulp 19.1 gb/s, 0 ulp 17.8 gb/s, 0 ulp
nk_angular_i8_alder 26.2 gb/s, 0.1 ulp 17.1 gb/s, 0 ulp 17.8 gb/s, 0 ulp
u8 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_u8_serial 11.7 gb/s 8.77 gb/s 7.07 gb/s
nk_euclidean_u8_serial 11.6 gb/s, 0.5 ulp 8.31 gb/s, 0.5 ulp 8.36 gb/s, 0.6 ulp
nk_angular_u8_serial 7.95 gb/s, 0.4 ulp 6.68 gb/s, 0.4 ulp 5.88 gb/s, 0.4 ulp
nk_sqeuclidean_u8_haswell 45.4 gb/s 17.7 gb/s 18.5 gb/s
nk_euclidean_u8_haswell 38.9 gb/s, 0 ulp 18.8 gb/s, 0 ulp 19.3 gb/s, 0 ulp
nk_angular_u8_haswell 21.9 gb/s, 0.7 ulp 11.7 gb/s, 0.6 ulp 13.4 gb/s, 0.5 ulp
nk_sqeuclidean_u8_icelake 70.1 gb/s 28.8 gb/s 21.0 gb/s
nk_euclidean_u8_icelake 66.4 gb/s, 0 ulp 27.6 gb/s, 0 ulp 23.5 gb/s, 0 ulp
nk_angular_u8_icelake 28.9 gb/s, 0.7 ulp 21.2 gb/s, 0.6 ulp 21.5 gb/s, 0.5 ulp
nk_sqeuclidean_u8_alder 32.2 gb/s 17.5 gb/s 19.0 gb/s
nk_euclidean_u8_alder 31.3 gb/s, 0 ulp 17.0 gb/s, 0 ulp 19.6 gb/s, 0 ulp
nk_angular_u8_alder 26.5 gb/s, 0.7 ulp 17.1 gb/s, 0.6 ulp 17.5 gb/s, 0.5 ulp
i4 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_i4_serial 15.4 gb/s 16.5 gb/s 15.6 gb/s
nk_euclidean_i4_serial 12.2 gb/s, 0.5 ulp 15.6 gb/s, 0.5 ulp 15.2 gb/s, 0.6 ulp
nk_angular_i4_serial 5.60 gb/s, 0.4 ulp 6.42 gb/s, 0.4 ulp 6.69 gb/s, 0.4 ulp
nk_sqeuclidean_i4_icelake 23.6 gb/s 51.5 gb/s 29.3 gb/s
nk_euclidean_i4_icelake 20.6 gb/s, 0 ulp 45.2 gb/s, 0 ulp 28.9 gb/s, 0 ulp
nk_angular_i4_icelake 5.14 gb/s, 0.7 ulp 18.0 gb/s, 0.6 ulp 17.6 gb/s, 0.5 ulp
u4 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_u4_serial 15.6 gb/s 17.3 gb/s 15.8 gb/s
nk_euclidean_u4_serial 12.0 gb/s, 0.5 ulp 15.9 gb/s, 0.5 ulp 15.3 gb/s, 0.6 ulp
nk_angular_u4_serial 5.20 gb/s, 0.4 ulp 6.63 gb/s, 0.4 ulp 7.01 gb/s, 0.4 ulp
nk_sqeuclidean_u4_icelake 22.7 gb/s 23.7 gb/s 24.5 gb/s
nk_euclidean_u4_icelake 20.9 gb/s, 0 ulp 18.8 gb/s, 0 ulp 24.1 gb/s, 0 ulp
nk_angular_u4_icelake 9.32 gb/s, 0.7 ulp 27.4 gb/s, 0.6 ulp 24.2 gb/s, 0.5 ulp

WASM

Measured with Wasmtime v42 (Cranelift backend).

Kernel 256 1024 4096
f64 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f64_serial 2.97 gb/s, 0.1 ulp 3.16 gb/s, 0 ulp 0.02 gb/s, 0 ulp
nk_euclidean_f64_serial 0.104 gb/s, 0.6 ulp 1.06 gb/s, 0.6 ulp 0.33 gb/s, 0.5 ulp
nk_angular_f64_serial 1.91 gb/s, 0.1 ulp 1.93 gb/s, 0 ulp 0.18 gb/s, 0 ulp
nk_sqeuclidean_f64_v128relaxed 1.23 gb/s, 1.3 ulp 1.87 gb/s, 2.5 ulp 0.15 gb/s, 5.0 ulp
nk_euclidean_f64_v128relaxed 0.315 gb/s, 0.7 ulp 2.21 gb/s, 1.4 ulp 0.03 gb/s, 2.8 ulp
nk_angular_f64_v128relaxed 1.14 gb/s, 0.1 ulp 0.928 gb/s, 0.1 ulp 0.26 gb/s, 0.1 ulp
f32 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f32_serial 0.657 gb/s, 0 ulp 0.928 gb/s, 0 ulp 0.06 gb/s, 0 ulp
nk_euclidean_f32_serial 0.757 gb/s, 0.1 ulp 0.914 gb/s, 0.1 ulp 0.05 gb/s, 0.1 ulp
nk_angular_f32_serial 0.882 gb/s, 0 ulp 0.902 gb/s, 0 ulp 0.26 gb/s, 0 ulp
nk_sqeuclidean_f32_v128relaxed 2.87 gb/s, 0.7 ulp 3.03 gb/s, 1.3 ulp 1.77 gb/s, 2.6 ulp
nk_euclidean_f32_v128relaxed 1.83 gb/s, 0.4 ulp 3.00 gb/s, 0.7 ulp 0.22 gb/s, 1.4 ulp
nk_angular_f32_v128relaxed 3.37 gb/s, 0 ulp 0.991 gb/s, 0 ulp 0.19 gb/s, 0 ulp
bf16 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_bf16_serial 1.89 gb/s, 0 ulp 1.09 gb/s, 0 ulp 0.31 gb/s, 0 ulp
nk_euclidean_bf16_serial 2.02 gb/s, 0.6 ulp 2.13 gb/s, 0.5 ulp 0.29 gb/s, 0.5 ulp
nk_angular_bf16_serial 0.399 gb/s, 0 ulp 0.308 gb/s, 0 ulp 0.11 gb/s, 0 ulp
nk_sqeuclidean_bf16_v128relaxed 2.10 gb/s, 0.9 ulp 1.94 gb/s, 12.6 ulp 0.17 gb/s, 20.8 ulp
nk_euclidean_bf16_v128relaxed 2.08 gb/s, 0.5 ulp 2.22 gb/s, 7.0 ulp 0.13 gb/s, 11.4 ulp
nk_angular_bf16_v128relaxed 1.08 gb/s, 0 ulp 2.09 gb/s, 0.2 ulp 0.20 gb/s, 0.6 ulp
f16 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f16_serial 1.10 gb/s, 0.1 ulp 1.13 gb/s, 0.1 ulp 0.20 gb/s, 0.1 ulp
nk_euclidean_f16_serial 1.17 gb/s, 0.6 ulp 1.16 gb/s, 0.6 ulp 0.26 gb/s, 0.5 ulp
nk_angular_f16_serial 0.363 gb/s, 0 ulp 0.372 gb/s, 0 ulp 0.06 gb/s, 0 ulp
nk_sqeuclidean_f16_v128relaxed 1.12 gb/s, 0.9 ulp 0.633 gb/s, 3.6 ulp 0.03 gb/s, 9.7 ulp
nk_euclidean_f16_v128relaxed 0.806 gb/s, 0.5 ulp 0.991 gb/s, 2.0 ulp 0.09 gb/s, 5.4 ulp
nk_angular_f16_v128relaxed 1.79 gb/s, 0.1 ulp 0.976 gb/s, 0.1 ulp 0.00 gb/s, 0.1 ulp
e5m2 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e5m2_serial 0.713 gb/s, 0 ulp 0.689 gb/s, 0 ulp 0.16 gb/s, 0 ulp
nk_euclidean_e5m2_serial 0.637 gb/s, 0.5 ulp 0.736 gb/s, 0.5 ulp 0.12 gb/s, 0.5 ulp
nk_angular_e5m2_serial 0.169 gb/s, 0 ulp 0.162 gb/s, 0 ulp 0.17 gb/s, 0 ulp
e4m3 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e4m3_serial 0.374 gb/s, 0 ulp 0.383 gb/s, 0 ulp 0.09 gb/s, 0 ulp
nk_euclidean_e4m3_serial 0.374 gb/s, 0.5 ulp 0.360 gb/s, 0.5 ulp 0.09 gb/s, 0.5 ulp
nk_angular_e4m3_serial 0.162 gb/s, 0 ulp 0.166 gb/s, 0 ulp 0.17 gb/s, 0 ulp
e3m2 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e3m2_serial 0.712 gb/s, 0 ulp 0.744 gb/s, 0 ulp 0.17 gb/s, 0 ulp
nk_euclidean_e3m2_serial 0.709 gb/s, 0.5 ulp 0.759 gb/s, 0.5 ulp 0.17 gb/s, 0.5 ulp
nk_angular_e3m2_serial 0.152 gb/s, 0 ulp 0.165 gb/s, 0 ulp 0.17 gb/s, 0 ulp
e2m3 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e2m3_serial 0.702 gb/s, 0 ulp 0.760 gb/s, 0 ulp 0.13 gb/s, 0 ulp
nk_euclidean_e2m3_serial 0.650 gb/s, 0.5 ulp 0.753 gb/s, 0.5 ulp 0.15 gb/s, 0.5 ulp
nk_angular_e2m3_serial 0.158 gb/s, 0 ulp 0.168 gb/s, 0 ulp 0.17 gb/s, 0 ulp
i8 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_i8_serial 0.327 gb/s 0.328 gb/s 0.09 gb/s
nk_euclidean_i8_serial 2.93 gb/s, 0.5 ulp 0.174 gb/s, 0.4 ulp 0.14 gb/s, 0.4 ulp
nk_angular_i8_serial 1.23 gb/s, 0 ulp 0.946 gb/s, 0 ulp 0.10 gb/s, 0 ulp
nk_sqeuclidean_i8_v128relaxed 1.84 gb/s 0.736 gb/s 0.08 gb/s
nk_euclidean_i8_v128relaxed 1.36 gb/s, 0 ulp 0.805 gb/s, 0 ulp 0.21 gb/s, 0 ulp
nk_angular_i8_v128relaxed 1.80 gb/s, 0 ulp 2.79 gb/s, 0 ulp 0.14 gb/s, 0 ulp
u8 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_u8_serial 0.528 gb/s 0.496 gb/s 0.30 gb/s
nk_euclidean_u8_serial 0.00982 gb/s, 0.5 ulp 0.311 gb/s, 0.5 ulp 0.04 gb/s, 0.6 ulp
nk_angular_u8_serial 0.813 gb/s, 0.5 ulp 1.46 gb/s, 0.4 ulp 0.29 gb/s, 0.5 ulp
nk_sqeuclidean_u8_v128relaxed 3.05 gb/s 1.68 gb/s 0.28 gb/s
nk_euclidean_u8_v128relaxed 2.52 gb/s, 0 ulp 1.70 gb/s, 0 ulp 0.09 gb/s, 0 ulp
nk_angular_u8_v128relaxed 2.47 gb/s, 526M ulp 1.91 gb/s, 501M ulp 0.09 gb/s, 443M ulp
i4 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_i4_serial 1.91 gb/s 1.94 gb/s 0.30 gb/s
nk_euclidean_i4_serial 1.76 gb/s, 0.5 ulp 1.90 gb/s, 0.5 ulp 0.02 gb/s, 0.0 ulp
nk_angular_i4_serial 1.28 gb/s, 0.5 ulp 1.34 gb/s, 0.5 ulp 0.10 gb/s, 0.5 ulp
u4 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_u4_serial 2.91 gb/s 3.00 gb/s 0.09 gb/s
nk_euclidean_u4_serial 2.78 gb/s, 0.5 ulp 3.01 gb/s, 0.5 ulp 0.10 gb/s, 0.0 ulp
nk_angular_u4_serial 1.84 gb/s, 0.5 ulp 2.03 gb/s, 0.5 ulp 0.21 gb/s, 0.5 ulp

Apple M5

Native

Kernel 256 1024 4096
f64 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f64_serial 12.4 gb/s, 0.1 ulp 12.8 gb/s, 0 ulp 12.8 gb/s, 0 ulp
nk_euclidean_f64_serial 12.7 gb/s, 0.6 ulp 12.9 gb/s, 0.5 ulp 12.6 gb/s, 0.5 ulp
nk_angular_f64_serial 8.42 gb/s, 0 ulp 8.57 gb/s, 0 ulp 8.30 gb/s, 0 ulp
nk_sqeuclidean_f64_neon 50.6 gb/s, 1.3 ulp 40.0 gb/s, 2.6 ulp 36.1 gb/s, 5.1 ulp
nk_euclidean_f64_neon 48.4 gb/s, 0.7 ulp 38.7 gb/s, 1.4 ulp 35.1 gb/s, 2.8 ulp
nk_angular_f64_neon 33.3 gb/s, 0.1 ulp 33.4 gb/s, 0 ulp 32.4 gb/s, 0 ulp
f32 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f32_serial 6.32 gb/s, 0 ulp 6.25 gb/s, 0 ulp 6.30 gb/s, 0 ulp
nk_euclidean_f32_serial 6.31 gb/s, 0.1 ulp 6.37 gb/s, 0.1 ulp 6.41 gb/s, 0.1 ulp
nk_angular_f32_serial 4.03 gb/s, 0 ulp 4.06 gb/s, 0 ulp 4.07 gb/s, 0 ulp
nk_sqeuclidean_f32_neon 25.3 gb/s, 0.1 ulp 19.1 gb/s, 0 ulp 17.5 gb/s, 0 ulp
nk_euclidean_f32_neon 25.0 gb/s, 0.1 ulp 20.8 gb/s, 0.1 ulp 18.6 gb/s, 0.1 ulp
nk_angular_f32_neon 22.2 gb/s, 0 ulp 17.3 gb/s, 0 ulp 16.6 gb/s, 0 ulp
bf16 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_bf16_serial 3.19 gb/s, 0 ulp 3.16 gb/s, 0 ulp 3.14 gb/s, 0 ulp
nk_euclidean_bf16_serial 3.16 gb/s, 0.5 ulp 3.08 gb/s, 0.5 ulp 3.11 gb/s, 0.5 ulp
nk_angular_bf16_serial 1.88 gb/s, 0 ulp 1.91 gb/s, 0 ulp 1.93 gb/s, 0 ulp
nk_sqeuclidean_bf16_neonbfdot 35.0 gb/s, 0.9 ulp 22.7 gb/s, 13 ulp 18.8 gb/s, 21 ulp
nk_euclidean_bf16_neonbfdot 33.4 gb/s, 0.5 ulp 23.0 gb/s, 7.0 ulp 18.6 gb/s, 12 ulp
nk_angular_bf16_neonbfdot 23.8 gb/s, 0 ulp 32.7 gb/s, 0.1 ulp 35.9 gb/s, 0 ulp
f16 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f16_serial 3.09 gb/s, 0.1 ulp 3.16 gb/s, 0.1 ulp 3.10 gb/s, 0.1 ulp
nk_euclidean_f16_serial 3.13 gb/s, 0.6 ulp 3.14 gb/s, 0.5 ulp 3.11 gb/s, 0.5 ulp
nk_angular_f16_serial 1.84 gb/s, 0 ulp 1.92 gb/s, 0 ulp 1.88 gb/s, 0 ulp
nk_sqeuclidean_f16_neonhalf 34.7 gb/s, 0.9 ulp 21.5 gb/s, 3.6 ulp 18.3 gb/s, 9.7 ulp
nk_euclidean_f16_neonhalf 32.7 gb/s, 0.5 ulp 21.7 gb/s, 2.0 ulp 18.4 gb/s, 5.3 ulp
nk_angular_f16_neonhalf 25.2 gb/s, 0.1 ulp 19.6 gb/s, 0.1 ulp 17.3 gb/s, 0.1 ulp
e5m2 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e5m2_serial 2.09 gb/s, 0 ulp 2.08 gb/s, 0 ulp 2.10 gb/s, 0 ulp
nk_euclidean_e5m2_serial 2.06 gb/s, 0.5 ulp 2.10 gb/s, 0.5 ulp 2.05 gb/s, 0.5 ulp
nk_angular_e5m2_serial 0.921 gb/s, 0 ulp 0.956 gb/s, 0 ulp 0.938 gb/s, 0 ulp
nk_sqeuclidean_e5m2_neon 18.2 gb/s, 0 ulp 12.8 gb/s, 0 ulp 9.84 gb/s, 0 ulp
nk_euclidean_e5m2_neon 18.0 gb/s, 0.5 ulp 11.8 gb/s, 0.5 ulp 9.33 gb/s, 0.5 ulp
nk_angular_e5m2_neon 13.7 gb/s, 0 ulp 10.9 gb/s, 0 ulp 9.83 gb/s, 0 ulp
e4m3 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e4m3_serial 1.07 gb/s, 0 ulp 1.12 gb/s, 0 ulp 1.11 gb/s, 0 ulp
nk_euclidean_e4m3_serial 1.01 gb/s, 0.5 ulp 1.12 gb/s, 0.5 ulp 1.09 gb/s, 0.5 ulp
nk_angular_e4m3_serial 0.711 gb/s, 0 ulp 0.732 gb/s, 0 ulp 0.729 gb/s, 0 ulp
nk_sqeuclidean_e4m3_neon 4.29 gb/s, 0.2 ulp 4.36 gb/s, 0.2 ulp 4.33 gb/s, 0.2 ulp
nk_euclidean_e4m3_neon 4.20 gb/s, 0.5 ulp 4.11 gb/s, 0.5 ulp 4.17 gb/s, 0.5 ulp
nk_angular_e4m3_neon 4.13 gb/s, 0 ulp 4.21 gb/s, 0 ulp 4.16 gb/s, 0 ulp
e3m2 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e3m2_serial 1.95 gb/s, 0 ulp 2.15 gb/s, 0 ulp 2.08 gb/s, 0 ulp
nk_euclidean_e3m2_serial 1.97 gb/s, 0.5 ulp 2.18 gb/s, 0.5 ulp 2.09 gb/s, 0.5 ulp
nk_angular_e3m2_serial 0.900 gb/s, 0 ulp 0.985 gb/s, 0 ulp 0.943 gb/s, 0 ulp
nk_sqeuclidean_e3m2_neon 4.73 gb/s, 0 ulp 5.19 gb/s, 0 ulp 5.03 gb/s, 0 ulp
nk_euclidean_e3m2_neon 4.78 gb/s, 0 ulp 5.23 gb/s, 0 ulp 5.05 gb/s, 0 ulp
nk_angular_e3m2_neon 4.24 gb/s, 0 ulp 4.85 gb/s, 0 ulp 4.73 gb/s, 0 ulp
e2m3 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_e2m3_serial 1.98 gb/s, 0 ulp 2.20 gb/s, 0 ulp 2.11 gb/s, 0 ulp
nk_euclidean_e2m3_serial 1.91 gb/s, 0.5 ulp 2.16 gb/s, 0.5 ulp 2.09 gb/s, 0.4 ulp
nk_angular_e2m3_serial 0.885 gb/s, 0 ulp 0.985 gb/s, 0 ulp 0.953 gb/s, 0 ulp
nk_sqeuclidean_e2m3_neon 4.67 gb/s, 0 ulp 5.06 gb/s, 0 ulp 5.07 gb/s, 0 ulp
nk_euclidean_e2m3_neon 4.84 gb/s, 0 ulp 5.17 gb/s, 0 ulp 4.98 gb/s, 0 ulp
nk_angular_e2m3_neon 4.45 gb/s, 0 ulp 4.88 gb/s, 0 ulp 4.73 gb/s, 0 ulp
i8 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_i8_serial 61.6 gb/s 75.2 gb/s 64.9 gb/s
nk_euclidean_i8_serial 43.5 gb/s 54.0 gb/s 64.3 gb/s
nk_angular_i8_serial 55.8 gb/s 63.7 gb/s 49.6 gb/s
nk_sqeuclidean_i8_neonsdot 89.1 gb/s 85.9 gb/s 58.8 gb/s
nk_euclidean_i8_neonsdot 86.9 gb/s 78.9 gb/s 57.4 gb/s
nk_angular_i8_neonsdot 66.5 gb/s 68.9 gb/s 50.9 gb/s
u8 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_u8_serial 62.9 gb/s 77.2 gb/s 66.5 gb/s
nk_euclidean_u8_serial 45.7 gb/s 52.3 gb/s 61.4 gb/s
nk_angular_u8_serial 17.8 gb/s 18.5 gb/s 16.0 gb/s
nk_sqeuclidean_u8_neonsdot 91.7 gb/s 83.1 gb/s 56.6 gb/s
nk_euclidean_u8_neonsdot 87.9 gb/s 79.3 gb/s 56.5 gb/s
nk_angular_u8_neonsdot 68.0 gb/s 64.8 gb/s 49.5 gb/s
i4 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_i4_serial 22.9 gb/s 25.2 gb/s 25.1 gb/s
nk_euclidean_i4_serial 20.1 gb/s 23.6 gb/s 24.2 gb/s
nk_angular_i4_serial 9.11 gb/s 10.4 gb/s 10.4 gb/s
u4 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_u4_serial 26.7 gb/s 26.8 gb/s 22.4 gb/s
nk_euclidean_u4_serial 20.9 gb/s 22.5 gb/s 21.2 gb/s
nk_angular_u4_serial 9.00 gb/s 9.57 gb/s 9.62 gb/s

WASM

Measured with Wasmtime v43 (Cranelift backend).

Kernel 256 1024 4096
f64 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f64_serial 20.3 gb/s, 0.1 ulp 19.3 gb/s, 0 ulp 20.3 gb/s, 0 ulp
nk_euclidean_f64_serial 20.0 gb/s, 0.6 ulp 19.4 gb/s, 0.6 ulp 20.3 gb/s, 0.5 ulp
nk_angular_f64_serial 9.28 gb/s, 0 ulp 8.83 gb/s, 0 ulp 9.29 gb/s, 0 ulp
nk_sqeuclidean_f64_v128relaxed 48.2 gb/s, 1.3 ulp 35.7 gb/s, 2.6 ulp 37.0 gb/s, 5.0 ulp
nk_euclidean_f64_v128relaxed 50.1 gb/s, 0.7 ulp 36.5 gb/s, 1.4 ulp 36.7 gb/s, 2.8 ulp
nk_angular_f64_v128relaxed 31.5 gb/s, 0.1 ulp 22.3 gb/s, 0.1 ulp 22.4 gb/s, 0.1 ulp
f32 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f32_serial 8.90 gb/s, 0 ulp 8.54 gb/s, 0 ulp 8.86 gb/s, 0 ulp
nk_euclidean_f32_serial 8.86 gb/s, 0.1 ulp 8.58 gb/s, 0.1 ulp 8.86 gb/s, 0.1 ulp
nk_angular_f32_serial 4.33 gb/s, 0 ulp 4.17 gb/s, 0 ulp 4.30 gb/s, 0 ulp
nk_sqeuclidean_f32_v128relaxed 20.4 gb/s, 0.7 ulp 17.9 gb/s, 1.3 ulp 18.3 gb/s, 2.6 ulp
nk_euclidean_f32_v128relaxed 20.3 gb/s, 0.4 ulp 18.0 gb/s, 0.7 ulp 18.3 gb/s, 1.4 ulp
nk_angular_f32_v128relaxed 19.7 gb/s, 0 ulp 17.8 gb/s, 0 ulp 18.5 gb/s, 0 ulp
bf16 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_bf16_serial 5.16 gb/s, 0 ulp 4.88 gb/s, 0 ulp 5.10 gb/s, 0 ulp
nk_euclidean_bf16_serial 5.12 gb/s, 0.6 ulp 4.89 gb/s, 0.5 ulp 5.10 gb/s, 0.5 ulp
nk_angular_bf16_serial 2.24 gb/s, 0 ulp 2.15 gb/s, 0 ulp 2.24 gb/s, 0 ulp
nk_sqeuclidean_bf16_v128relaxed 39.9 gb/s, 0.9 ulp 27.0 gb/s, 13 ulp 20.3 gb/s, 21 ulp
nk_euclidean_bf16_v128relaxed 38.6 gb/s, 0.5 ulp 27.1 gb/s, 7.0 ulp 21.2 gb/s, 12 ulp
nk_angular_bf16_v128relaxed 27.9 gb/s, 0 ulp 22.6 gb/s, 0.2 ulp 20.5 gb/s, 0.6 ulp
f16 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_f16_serial 3.22 gb/s, 0.1 ulp 3.06 gb/s, 0.1 ulp 3.09 gb/s, 0.1 ulp
nk_euclidean_f16_serial 3.19 gb/s, 0.6 ulp 2.92 gb/s, 0.5 ulp 3.26 gb/s, 0.5 ulp
nk_angular_f16_serial 2.33 gb/s, 0 ulp 2.21 gb/s, 0 ulp 2.32 gb/s, 0 ulp
nk_sqeuclidean_f16_v128relaxed 11.2 gb/s, 0.9 ulp 11.0 gb/s, 3.6 ulp 11.8 gb/s, 9.6 ulp
nk_euclidean_f16_v128relaxed 11.3 gb/s, 0.5 ulp 11.0 gb/s, 2.0 ulp 11.8 gb/s, 5.3 ulp
nk_angular_f16_v128relaxed 9.41 gb/s, 0.1 ulp 9.56 gb/s, 0.1 ulp 10.4 gb/s, 0.1 ulp
i8 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_i8_serial 14.9 gb/s 14.7 gb/s 16.5 gb/s
nk_euclidean_i8_serial 14.7 gb/s, 0.5 ulp 14.8 gb/s, 0.4 ulp 16.3 gb/s, 0.4 ulp
nk_angular_i8_serial 8.06 gb/s, 0 ulp 8.42 gb/s, 0 ulp 10.7 gb/s, 0 ulp
nk_sqeuclidean_i8_v128relaxed 30.7 gb/s 22.9 gb/s 18.0 gb/s
nk_euclidean_i8_v128relaxed 27.4 gb/s, 0 ulp 22.6 gb/s, 0 ulp 17.9 gb/s, 0 ulp
nk_angular_i8_v128relaxed 17.2 gb/s, 0 ulp 18.1 gb/s, 0 ulp 19.7 gb/s, 0 ulp
u8 ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░ ░░░░░░░░░░░░░░░░░░░░░░░░
nk_sqeuclidean_u8_serial 14.8 gb/s 14.5 gb/s 16.3 gb/s
nk_euclidean_u8_serial 14.5 gb/s, 0.5 ulp 14.5 gb/s, 0.5 ulp 16.0 gb/s, 0.6 ulp
nk_angular_u8_serial 7.86 gb/s, 0.5 ulp 8.25 gb/s, 0.5 ulp 10.7 gb/s, 0.4 ulp
nk_sqeuclidean_u8_v128relaxed 33.2 gb/s 24.6 gb/s 18.3 gb/s
nk_euclidean_u8_v128relaxed 28.6 gb/s, 0 ulp 23.7 gb/s, 0 ulp 18.2 gb/s, 0 ulp
nk_angular_u8_v128relaxed 14.1 gb/s, 0 ulp 15.0 gb/s, 0 ulp 16.1 gb/s, 0 ulp