usearch

![SimSIMD banner](https://github.com/ashvardanian/ashvardanian/blob/master/repositories/SimSIMD.jpg?raw=true) Computing dot-products, similarity measures, and distances between low- and high-dimensional vectors is ubiquitous in Machine Learning, Scientific Computing, Geo-Spatial Analysis, and Information Retrieval. These algorithms generally have linear complexity in time, constant or linear complexity in space, and are data-parallel. In other words, it is easily parallelizable and vectorizable and often available in packages like BLAS (level 1) and LAPACK, as well as higher-level `numpy` and `scipy` Python libraries. Ironically, even with decades of evolution in compilers and numerical computing, [most libraries can be 3-200x slower than hardware potential][benchmarks] even on the most popular hardware, like 64-bit x86 and Arm CPUs. Moreover, most lack mixed-precision support, which is crucial for modern AI! The rare few that support minimal mixed precision, run only on one platform, and are vendor-locked, by companies like Intel and Nvidia. SimSIMD provides an alternative. 1️⃣ SimSIMD functions are practically as fast as `memcpy`. 2️⃣ Unlike BLAS, most kernels are designed for mixed-precision and bit-level operations. 3️⃣ SimSIMD often [ships more binaries than NumPy][compatibility] and has more backends than most BLAS implementations, and more high-level interfaces than most libraries. [benchmarks]: https://ashvardanian.com/posts/simsimd-faster-scipy [compatibility]: https://pypi.org/project/simsimd/#files <div> <a href="https://pepy.tech/project/simsimd"> <img alt="PyPI" src="https://static.pepy.tech/personalized-badge/simsimd?period=total&units=abbreviation&left_color=black&right_color=blue&left_text=SimSIMD%20Python%20installs" /> </a> <a href="https://www.npmjs.com/package/simsimd"> <img alt="npm" src="https://img.shields.io/npm/dy/simsimd?label=JavaScript%20NPM%20installs" /> </a> <a href="https://crates.io/crates/simsimd"> <img alt="rust" src="https://img.shields.io/crates/d/simsimd?label=Rust%20Crate%20installs" /> </a> <img alt="GitHub code size in bytes" src="https://img.shields.io/github/languages/code-size/ashvardanian/simsimd"> <a href="https://github.com/ashvardanian/SimSIMD/actions/workflows/release.yml"> <img alt="GitHub Actions Ubuntu" src="https://img.shields.io/github/actions/workflow/status/ashvardanian/SimSIMD/release.yml?branch=main&label=Ubuntu&logo=github&color=blue"> </a> <a href="https://github.com/ashvardanian/SimSIMD/actions/workflows/release.yml"> <img alt="GitHub Actions Windows" src="https://img.shields.io/github/actions/workflow/status/ashvardanian/SimSIMD/release.yml?branch=main&label=Windows&logo=windows&color=blue"> </a> <a href="https://github.com/ashvardanian/SimSIMD/actions/workflows/release.yml"> <img alt="GitHub Actions MacOS" src="https://img.shields.io/github/actions/workflow/status/ashvardanian/SimSIMD/release.yml?branch=main&label=MacOS&logo=apple&color=blue"> </a> <a href="https://github.com/ashvardanian/SimSIMD/actions/workflows/release.yml"> <img alt="GitHub Actions CentOS Linux" src="https://img.shields.io/github/actions/workflow/status/ashvardanian/SimSIMD/release.yml?branch=main&label=CentOS&logo=centos&color=blue"> </a> </div> ## Features __SimSIMD__ (Arabic: "سيمسيم دي") is a mixed-precision math library of __over 350 SIMD-optimized kernels__ extensively used in AI, Search, and DBMS workloads. Named after the iconic ["Open Sesame"](https://en.wikipedia.org/wiki/Open_sesame) command that opened doors to treasure in _Ali Baba and the Forty Thieves_, SimSimd can help you 10x the cost-efficiency of your computational pipelines. Implemented distance functions include: - Euclidean (L2) and Cosine (Angular) spatial distances for Vector Search. _[docs][docs-spatial]_ - Dot-Products for real & complex vectors for DSP & Quantum computing. _[docs][docs-dot]_ - Hamming (~ Manhattan) and Jaccard (~ Tanimoto) bit-level distances. _[docs][docs-binary]_ - Set Intersections for Sparse Vectors and Text Analysis. _[docs][docs-sparse]_ - Mahalanobis distance and Quadratic forms for Scientific Computing. _[docs][docs-curved]_ - Kullback-Leibler and Jensen–Shannon divergences for probability distributions. _[docs][docs-probability]_ - Fused-Multiply-Add (FMA) and Weighted Sums to replace BLAS level 1 functions. _[docs][docs-fma]_ - For Levenshtein, Needleman–Wunsch, and Smith-Waterman, check [StringZilla][stringzilla]. - 🔜 Haversine and Vincenty's formulae for Geospatial Analysis. [docs-spatial]: #cosine-similarity-reciprocal-square-root-and-newton-raphson-iteration [docs-curved]: #curved-spaces-mahalanobis-distance-and-bilinear-quadratic-forms [docs-sparse]: #set-intersection-galloping-and-binary-search [docs-binary]: https://github.com/ashvardanian/SimSIMD/pull/138 [docs-dot]: #complex-dot-products-conjugate-dot-products-and-complex-numbers [docs-probability]: #logarithms-in-kullback-leibler--jensenshannon-divergences [docs-fma]: #mixed-precision-in-fused-multiply-add-and-weighted-sums [scipy]: https://docs.scipy.org/doc/scipy/reference/spatial.distance.html#module-scipy.spatial.distance [numpy]: https://numpy.org/doc/stable/reference/generated/numpy.inner.html [stringzilla]: https://github.com/ashvardanian/stringzilla Moreover, SimSIMD... - handles `float64`, `float32`, `float16`, and `bfloat16` real & complex vectors. - handles `int8` integral, `int4` sub-byte, and `b8` binary vectors. - handles sparse `uint32` and `uint16` sets, and weighted sparse vectors. - is a zero-dependency [header-only C 99](#using-simsimd-in-c) library. - has [Python](#using-simsimd-in-python), [Rust](#using-simsimd-in-rust), [JS](#using-simsimd-in-javascript), and [Swift](#using-simsimd-in-swift) bindings. - has Arm backends for NEON, Scalable Vector Extensions (SVE), and SVE2. - has x86 backends for Haswell, Skylake, Ice Lake, Genoa, and Sapphire Rapids. - with both compile-time and runtime CPU feature detection easily integrates anywhere! Due to the high-level of fragmentation of SIMD support in different x86 CPUs, SimSIMD generally uses the names of select Intel CPU generations for its backends. They, however, also work on AMD CPUs. Intel Haswell is compatible with AMD Zen 1/2/3, while AMD Genoa Zen 4 covers AVX-512 instructions added to Intel Skylake and Ice Lake. You can learn more about the technical implementation details in the following blog-posts: - [Uses Horner's method for polynomial approximations, beating GCC 12 by 119x](https://ashvardanian.com/posts/gcc-12-vs-avx512fp16/). - [Uses Arm SVE and x86 AVX-512's masked loads to eliminate tail `for`-loops](https://ashvardanian.com/posts/simsimd-faster-scipy/#tails-of-the-past-the-significance-of-masked-loads). - [Substitutes LibC's `sqrt` with Newton Raphson iterations](https://github.com/ashvardanian/SimSIMD/releases/tag/v5.4.0). - [Uses Galloping and SVE2 histograms to intersect sparse vectors](https://ashvardanian.com/posts/simd-set-intersections-sve2-avx512/). - For Python: [avoids slow PyBind11, SWIG, & `PyArg_ParseTuple`](https://ashvardanian.com/posts/pybind11-cpython-tutorial/) [using faster calling convention](https://ashvardanian.com/posts/discount-on-keyword-arguments-in-python/). - For JavaScript: [uses typed arrays and NAPI for zero-copy calls](https://ashvardanian.com/posts/javascript-ai-vector-search/). ## Benchmarks <table style="width: 100%; text-align: center; table-layout: fixed;"> <colgroup> <col style="width: 33%;"> <col style="width: 33%;"> <col style="width: 33%;"> </colgroup> <tr> <th align="center">NumPy</th> <th align="center">C 99</th> <th align="center">SimSIMD</th> </tr>  <tr> <td colspan="4" align="center">cosine distances between 1536d vectors in <code>int8</code></td> </tr> <tr> <td align="center">  🚧 overflows </td> <td align="center">  x86: 10,548,600 ops/s arm: 11,379,300 ops/s </td> <td align="center">  x86: 16,151,800 ops/s arm: 13,524,000 ops/s </td> </tr> <tr> <td colspan="4" align="center">cosine distances between 1536d vectors in <code>bfloat16</code></td> </tr> <tr> <td align="center">  🚧 not supported </td> <td align="center">  x86: 119,835 ops/s arm: 403,909 ops/s </td> <td align="center">  x86: 9,738,540 ops/s arm: 4,881,900 ops/s </td> </tr> <tr> <td colspan="4" align="center">cosine distances between 1536d vectors in <code>float16</code></td> </tr> <tr> <td align="center">  x86: 40,481 ops/s arm: 21,451 ops/s </td> <td align="center">  x86: 501,310 ops/s arm: 871,963 ops/s </td> <td align="center">  x86: 7,627,600 ops/s arm: 3,316,810 ops/s </td> </tr> <tr> <td colspan="4" align="center">cosine distances between 1536d vectors in <code>float32</code></td> </tr> <tr> <td align="center">  x86: 253,902 ops/s arm: 46,394 ops/s </td> <td align="center">  x86: 882,484 ops/s arm: 399,661 ops/s </td> <td align="center">  x86: 8,202,910 ops/s arm: 3,400,620 ops/s </td> </tr> <tr> <td colspan="4" align="center">cosine distances between 1536d vectors in <code>float64</code></td> </tr> <tr> <td align="center">  x86: 212,421 ops/s arm: 52,904 ops/s </td> <td align="center">  x86: 839,301 ops/s arm: 837,126 ops/s </td> <td align="center">  x86: 1,538,530 ops/s arm: 1,678,920 ops/s </td> </tr>  <tr> <td colspan="4" align="center">eculidean distance between 1536d vectors in <code>int8</code></td> </tr> <tr> <td align="center">  x86: 252,113 ops/s arm: 177,443 ops/s </td> <td align="center">  x86: 6,690,110 ops/s arm: 4,114,160 ops/s </td> <td align="center">  x86: 18,989,000 ops/s arm: 18,878,200 ops/s </td> </tr> <tr> <td colspan="4" align="center">eculidean distance between 1536d vectors in <code>bfloat16</code></td> </tr> <tr> <td align="center">  🚧 not supported </td> <td align="center">  x86: 119,842 ops/s arm: 1,049,230 ops/s </td> <td align="center">  x86: 9,727,210 ops/s arm: 4,233,420 ops/s </td> </tr> <tr> <td colspan="4" align="center">eculidean distance between 1536d vectors in <code>float16</code></td> </tr> <tr> <td align="center">  x86: 54,621 ops/s arm: 71,793 ops/s </td> <td align="center">  x86: 196,413 ops/s arm: 911,370 ops/s </td> <td align="center">  x86: 19,466,800 ops/s arm: 3,522,760 ops/s </td> </tr> <tr> <td colspan="4" align="center">eculidean distance between 1536d vectors in <code>float32</code></td> </tr> <tr> <td align="center">  x86: 424,944 ops/s arm: 292,629 ops/s </td> <td align="center">  x86: 1,295,210 ops/s arm: 1,055,940 ops/s </td> <td align="center">  x86: 8,924,100 ops/s arm: 3,602,650 ops/s </td> </tr> <tr> <td colspan="4" align="center">eculidean distance between 1536d vectors in <code>float64</code></td> </tr> <tr> <td align="center">  x86: 334,929 ops/s arm: 237,505 ops/s </td> <td align="center">  x86: 1,215,190 ops/s arm: 905,782 ops/s </td> <td align="center">  x86: 1,701,740 ops/s arm: 1,735,840 ops/s </td> </tr>   </table> > For benchmarks we mostly use 1536-dimensional vectors, like the embeddings produced by the OpenAI Ada API. > The code was compiled with GCC 12, using glibc v2.35. > The benchmarks performed on Arm-based Graviton3 AWS `c7g` instances and `r7iz` Intel Sapphire Rapids. > Most modern Arm-based 64-bit CPUs will have similar relative speedups. > Variance withing x86 CPUs will be larger. Similar speedups are often observed even when compared to BLAS and LAPACK libraries underlying most numerical computing libraries, including NumPy and SciPy in Python. Broader benchmarking results: - [Apple M2 Pro](https://ashvardanian.com/posts/simsimd-faster-scipy/#appendix-1-performance-on-apple-m2-pro). - [Intel Sapphire Rapids](https://ashvardanian.com/posts/simsimd-faster-scipy/#appendix-2-performance-on-4th-gen-intel-xeon-platinum-8480). - [AWS Graviton 3](https://ashvardanian.com/posts/simsimd-faster-scipy/#appendix-3-performance-on-aws-graviton-3). ## Using SimSIMD in Python The package is intended to replace the usage of `numpy.inner`, `numpy.dot`, and `scipy.spatial.distance`. Aside from drastic performance improvements, SimSIMD significantly improves accuracy in mixed precision setups. NumPy and SciPy, processing `int8`, `uint8` or `float16` vectors, will use the same types for accumulators, while SimSIMD can combine `int8` enumeration, `int16` multiplication, and `int32` accumulation to avoid overflows entirely. The same applies to processing `float16` and `bfloat16` values with `float32` precision. ### Installation Use the following snippet to install SimSIMD and list available hardware acceleration options available on your machine: ```sh pip install simsimd python -c "import simsimd; print(simsimd.get_capabilities())" # for hardware introspection python -c "import simsimd; help(simsimd)" # for documentation ``` With precompiled binaries, SimSIMD ships `.pyi` interface files for type hinting and static analysis. You can check all the available functions in [`python/annotations/__init__.pyi`](https://github.com/ashvardanian/SimSIMD/blob/main/python/annotations/__init__.pyi). ### One-to-One Distance ```py import simsimd import numpy as np vec1 = np.random.randn(1536).astype(np.float32) vec2 = np.random.randn(1536).astype(np.float32) dist = simsimd.cosine(vec1, vec2) ``` Supported functions include `cosine`, `inner`, `sqeuclidean`, `hamming`, `jaccard`, `kulbackleibler`, `jensenshannon`, and `intersect`. Dot products are supported for both real and complex numbers: ```py vec1 = np.random.randn(768).astype(np.float64) + 1j * np.random.randn(768).astype(np.float64) vec2 = np.random.randn(768).astype(np.float64) + 1j * np.random.randn(768).astype(np.float64) dist = simsimd.dot(vec1.astype(np.complex128), vec2.astype(np.complex128)) dist = simsimd.dot(vec1.astype(np.complex64), vec2.astype(np.complex64)) dist = simsimd.vdot(vec1.astype(np.complex64), vec2.astype(np.complex64)) # conjugate, same as `np.vdot` ``` Unlike SciPy, SimSIMD allows explicitly stating the precision of the input vectors, which is especially useful for mixed-precision setups. The `dtype` argument can be passed both by name and as a positional argument: ```py dist = simsimd.cosine(vec1, vec2, "int8") dist = simsimd.cosine(vec1, vec2, "float16") dist = simsimd.cosine(vec1, vec2, "float32") dist = simsimd.cosine(vec1, vec2, "float64") dist = simsimd.hamming(vec1, vec2, "bin8") ``` Binary distance functions are computed at a bit-level. Meaning a vector of 10x 8-bit integers will be treated as a sequence of 80 individual bits or dimensions. This differs from NumPy, that can't handle smaller-than-byte types, but you can still avoid the `bin8` argument by reinterpreting the vector as booleans: ```py vec1 = np.random.randint(2, size=80).astype(np.uint8).packbits().view(np.bool_) vec2 = np.random.randint(2, size=80).astype(np.uint8).packbits().view(np.bool_) hamming_distance = simsimd.hamming(vec1, vec2) jaccard_distance = simsimd.jaccard(vec1, vec2) ``` With other frameworks, like PyTorch, one can get a richer type-system than NumPy, but the lack of good CPython interoperability makes it hard to pass data without copies. Here is an example of using SimSIMD with PyTorch to compute the cosine similarity between two `bfloat16` vectors: ```py import numpy as np buf1 = np.empty(8, dtype=np.uint16) buf2 = np.empty(8, dtype=np.uint16) # View the same memory region with PyTorch and randomize it import torch vec1 = torch.asarray(memoryview(buf1), copy=False).view(torch.bfloat16) vec2 = torch.asarray(memoryview(buf2), copy=False).view(torch.bfloat16) torch.randn(8, out=vec1) torch.randn(8, out=vec2) # Both libs will look into the same memory buffers and report the same results dist_slow = 1 - torch.nn.functional.cosine_similarity(vec1, vec2, dim=0) dist_fast = simsimd.cosine(buf1, buf2, "bfloat16") ``` It also allows using SimSIMD for half-precision complex numbers, which NumPy does not support. For that, view data as continuous even-length `np.float16` vectors and override type-resolution with `complex32` string. ```py vec1 = np.random.randn(1536).astype(np.float16) vec2 = np.random.randn(1536).astype(np.float16) simd.dot(vec1, vec2, "complex32") simd.vdot(vec1, vec2, "complex32") ``` When dealing with sparse representations and integer sets, you can apply the `intersect` function to two 1-dimensional arrays of `uint16` or `uint32` integers: ```py from random import randint import numpy as np import simsimd as simd length1, length2 = randint(1, 100), randint(1, 100) vec1 = np.sort(np.random.randint(0, 1000, length1).astype(np.uint16)) vec2 = np.sort(np.random.randint(0, 1000, length2).astype(np.uint16)) slow_result = len(np.intersect1d(vec1, vec2)) fast_result = simd.intersect(vec1, vec2) assert slow_result == fast_result ``` ### One-to-Many Distances Every distance function can be used not only for one-to-one but also one-to-many and many-to-many distance calculations. For one-to-many: ```py vec1 = np.random.randn(1536).astype(np.float32) # rank 1 tensor batch1 = np.random.randn(1, 1536).astype(np.float32) # rank 2 tensor batch2 = np.random.randn(100, 1536).astype(np.float32) dist_rank1 = simsimd.cosine(vec1, batch2) dist_rank2 = simsimd.cosine(batch1, batch2) ``` ### Many-to-Many Distances All distance functions in SimSIMD can be used to compute many-to-many distances. For two batches of 100 vectors to compute 100 distances, one would call it like this: ```py batch1 = np.random.randn(100, 1536).astype(np.float32) batch2 = np.random.randn(100, 1536).astype(np.float32) dist = simsimd.cosine(batch1, batch2) ``` Input matrices must have identical shapes. This functionality isn't natively present in NumPy or SciPy, and generally requires creating intermediate arrays, which is inefficient and memory-consuming. ### Many-to-Many All-Pairs Distances One can use SimSIMD to compute distances between all possible pairs of rows across two matrices (akin to [`scipy.spatial.distance.cdist`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.distance.cdist.html)). The resulting object will have a type `DistancesTensor`, zero-copy compatible with NumPy and other libraries. For two arrays of 10 and 1,000 entries, the resulting tensor will have 10,000 cells: ```py import numpy as np from simsimd import cdist, DistancesTensor matrix1 = np.random.randn(1000, 1536).astype(np.float32) matrix2 = np.random.randn(10, 1536).astype(np.float32) distances: DistancesTensor = simsimd.cdist(matrix1, matrix2, metric="cosine") # zero-copy, managed by SimSIMD distances_array: np.ndarray = np.array(distances, copy=True) # now managed by NumPy ``` ### Element-wise Kernels SimSIMD also provides mixed-precision element-wise kernels, where the input vectors and the output have the same numeric type, but the intermediate accumulators are of a higher precision. ```py import numpy as np from simsimd import fma, wsum # Let's take two FullHD video frames first_frame = np.random.randn(1920 * 1024).astype(np.uint8) second_frame = np.random.randn(1920 * 1024).astype(np.uint8) average_frame = np.empty_like(first_frame) wsum(first_frame, second_frame, alpha=0.5, beta=0.5, out=average_frame) # Slow analog with NumPy: slow_average_frame = (0.5 * first_frame + 0.5 * second_frame).astype(np.uint8) ``` Similarly, the `fma` takes three arguments and computes the fused multiply-add operation. In applications like Machine Learning you may also benefit from using the "brain-float" format not natively supported by NumPy. In 3D Graphics, for example, we can use FMA to compute the [Phong shading model](https://en.wikipedia.org/wiki/Phong_shading): ```py # Assume a FullHD frame with random values for simplicity light_intensity = np.random.rand(1920 * 1080).astype(np.float16) # Intensity of light on each pixel diffuse_component = np.random.rand(1920 * 1080).astype(np.float16) # Diffuse reflectance on the surface specular_component = np.random.rand(1920 * 1080).astype(np.float16) # Specular reflectance for highlights output_color = np.empty_like(light_intensity) # Array to store the resulting color intensity # Define the scaling factors for diffuse and specular contributions alpha = 0.7 # Weight for the diffuse component beta = 0.3 # Weight for the specular component # Formula: color = alpha * light_intensity * diffuse_component + beta * specular_component fma(light_intensity, diffuse_component, specular_component, dtype="float16", # Optional, unless it can't be inferred from the input alpha=alpha, beta=beta, out=output_color) # Slow analog with NumPy for comparison slow_output_color = (alpha * light_intensity * diffuse_component + beta * specular_component).astype(np.float16) ``` ### Multithreading and Memory Usage By default, computations use a single CPU core. To override this behavior, use the `threads` argument. Set it to `0` to use all available CPU cores. Here is an example of dealing with large sets of binary vectors: ```py ndim = 1536 # OpenAI Ada embeddings matrix1 = np.packbits(np.random.randint(2, size=(10_000, ndim)).astype(np.uint8)) matrix2 = np.packbits(np.random.randint(2, size=(1_000, ndim)).astype(np.uint8)) distances = simsimd.cdist(matrix1, matrix2, metric="hamming", # Unlike SciPy, SimSIMD doesn't divide by the number of dimensions out_dtype="uint8", # so we can use `uint8` instead of `float64` to save memory. threads=0, # Use all CPU cores with OpenMP. dtype="bin8", # Override input argument type to `bin8` eight-bit words. ) ``` By default, the output distances will be stored in double-precision `float64` floating-point numbers. That behavior may not be space-efficient, especially if you are computing the hamming distance between short binary vectors, that will generally fit into 8x smaller `uint8` or `uint16` types. To override this behavior, use the `dtype` argument. ### Helper Functions You can turn specific backends on or off depending on the exact environment. A common case may be avoiding AVX-512 on older AMD CPUs and [Intel Ice Lake](https://travisdowns.github.io/blog/2020/08/19/icl-avx512-freq.html) CPUs to ensure the CPU doesn't change the frequency license and throttle performance. ```py $ simsimd.get_capabilities() > {'serial': True, 'neon': False, 'sve': False, 'neon_f16': False, 'sve_f16': False, 'neon_bf16': False, 'sve_bf16': False, 'neon_i8': False, 'sve_i8': False, 'haswell': True, 'skylake': True, 'ice': True, 'genoa': True, 'sapphire': True, 'turin': True} $ simsimd.disable_capability("sapphire") $ simsimd.enable_capability("sapphire") ``` ### Using Python API with USearch Want to use it in Python with [USearch](https://github.com/unum-cloud/usearch)? You can wrap the raw C function pointers SimSIMD backends into a `CompiledMetric` and pass it to USearch, similar to how it handles Numba's JIT-compiled code. ```py from usearch.index import Index, CompiledMetric, MetricKind, MetricSignature from simsimd import pointer_to_sqeuclidean, pointer_to_cosine, pointer_to_inner metric = CompiledMetric( pointer=pointer_to_cosine("f16"), kind=MetricKind.Cos, signature=MetricSignature.ArrayArraySize, ) index = Index(256, metric=metric) ``` ## Using SimSIMD in Rust To install, add the following to your `Cargo.toml`: ```toml [dependencies] simsimd = "..." ``` Before using the SimSIMD library, ensure you have imported the necessary traits and types into your Rust source file. The library provides several traits for different distance/similarity kinds - `SpatialSimilarity`, `BinarySimilarity`, and `ProbabilitySimilarity`. ### Spatial Similarity: Cosine and Euclidean Distances ```rust use simsimd::SpatialSimilarity; fn main() { let vector_a: Vec<f32> = vec![1.0, 2.0, 3.0]; let vector_b: Vec<f32> = vec![4.0, 5.0, 6.0]; // Compute the cosine similarity between vector_a and vector_b let cosine_similarity = f32::cosine(&vector_a, &vector_b) .expect("Vectors must be of the same length"); println!("Cosine Similarity: {}", cosine_similarity); // Compute the squared Euclidean distance between vector_a and vector_b let sq_euclidean_distance = f32::sqeuclidean(&vector_a, &vector_b) .expect("Vectors must be of the same length"); println!("Squared Euclidean Distance: {}", sq_euclidean_distance); } ``` Spatial similarity functions are available for `f64`, `f32`, `f16`, and `i8` types. ### Dot-Products: Inner and Complex Inner Products ```rust use simsimd::SpatialSimilarity; use simsimd::ComplexProducts; fn main() { let vector_a: Vec<f32> = vec![1.0, 2.0, 3.0, 4.0]; let vector_b: Vec<f32> = vec![5.0, 6.0, 7.0, 8.0]; // Compute the inner product between vector_a and vector_b let inner_product = SpatialSimilarity::dot(&vector_a, &vector_b) .expect("Vectors must be of the same length"); println!("Inner Product: {}", inner_product); // Compute the complex inner product between complex_vector_a and complex_vector_b let complex_inner_product = ComplexProducts::dot(&vector_a, &vector_b) .expect("Vectors must be of the same length"); let complex_conjugate_inner_product = ComplexProducts::vdot(&vector_a, &vector_b) .expect("Vectors must be of the same length"); println!("Complex Inner Product: {:?}", complex_inner_product); // -18, 69 println!("Complex C. Inner Product: {:?}", complex_conjugate_inner_product); // 70, -8 } ``` Complex inner products are available for `f64`, `f32`, and `f16` types. ### Probability Distributions: Jensen-Shannon and Kullback-Leibler Divergences ```rust use simsimd::SpatialSimilarity; fn main() { let vector_a: Vec<f32> = vec![1.0, 2.0, 3.0]; let vector_b: Vec<f32> = vec![4.0, 5.0, 6.0]; let cosine_similarity = f32::jensenshannon(&vector_a, &vector_b) .expect("Vectors must be of the same length"); println!("Cosine Similarity: {}", cosine_similarity); let sq_euclidean_distance = f32::kullbackleibler(&vector_a, &vector_b) .expect("Vectors must be of the same length"); println!("Squared Euclidean Distance: {}", sq_euclidean_distance); } ``` Probability similarity functions are available for `f64`, `f32`, and `f16` types. ### Binary Similarity: Hamming and Jaccard Distances Similar to spatial distances, one can compute bit-level distance functions between slices of unsigned integers: ```rust use simsimd::BinarySimilarity; fn main() { let vector_a = &[0b11110000, 0b00001111, 0b10101010]; let vector_b = &[0b11110000, 0b00001111, 0b01010101]; // Compute the Hamming distance between vector_a and vector_b let hamming_distance = u8::hamming(&vector_a, &vector_b) .expect("Vectors must be of the same length"); println!("Hamming Distance: {}", hamming_distance); // Compute the Jaccard distance between vector_a and vector_b let jaccard_distance = u8::jaccard(&vector_a, &vector_b) .expect("Vectors must be of the same length"); println!("Jaccard Distance: {}", jaccard_distance); } ``` Binary similarity functions are available only for `u8` types. ### Half-Precision Floating-Point Numbers Rust has no native support for half-precision floating-point numbers, but SimSIMD provides a `f16` type. It has no functionality - it is a `transparent` wrapper around `u16` and can be used with `half` or any other half-precision library. ```rust use simsimd::SpatialSimilarity; use simsimd::f16 as SimF16; use half::f16 as HalfF16; fn main() { let vector_a: Vec<HalfF16> = ... let vector_b: Vec<HalfF16> = ... let buffer_a: &[SimF16] = unsafe { std::slice::from_raw_parts(a_half.as_ptr() as *const SimF16, a_half.len()) }; let buffer_b: &[SimF16] = unsafe { std::slice::from_raw_parts(b_half.as_ptr() as *const SimF16, b_half.len()) }; // Compute the cosine similarity between vector_a and vector_b let cosine_similarity = SimF16::cosine(&vector_a, &vector_b) .expect("Vectors must be of the same length"); println!("Cosine Similarity: {}", cosine_similarity); } ``` ### Half-Precision Brain-Float Numbers The "brain-float-16" is a popular machine learning format. It's broadly supported in hardware and is very machine-friendly, but software support is still lagging behind. [Unlike NumPy](https://github.com/numpy/numpy/issues/19808), you can already use `bf16` datatype in SimSIMD. Luckily, to downcast `f32` to `bf16` you only have to drop the last 16 bits: ```py import numpy as np import simsimd as simd a = np.random.randn(ndim).astype(np.float32) b = np.random.randn(ndim).astype(np.float32) # NumPy doesn't natively support brain-float, so we need a trick! # Luckily, it's very easy to reduce the representation accuracy # by simply masking the low 16-bits of our 32-bit single-precision # numbers. We can also add `0x8000` to round the numbers. a_f32rounded = ((a.view(np.uint32) + 0x8000) & 0xFFFF0000).view(np.float32) b_f32rounded = ((b.view(np.uint32) + 0x8000) & 0xFFFF0000).view(np.float32) # To represent them as brain-floats, we need to drop the second half a_bf16 = np.right_shift(a_f32rounded.view(np.uint32), 16).astype(np.uint16) b_bf16 = np.right_shift(b_f32rounded.view(np.uint32), 16).astype(np.uint16) # Now we can compare the results expected = np.inner(a_f32rounded, b_f32rounded) result = simd.inner(a_bf16, b_bf16, "bf16") ``` ### Dynamic Dispatch in Rust SimSIMD provides a [dynamic dispatch](#dynamic-dispatch) mechanism to select the most advanced micro-kernel for the current CPU. You can query supported backends and use the `SimSIMD::capabilities` function to select the best one. ```rust println!("uses neon: {}", capabilities::uses_neon()); println!("uses sve: {}", capabilities::uses_sve()); println!("uses haswell: {}", capabilities::uses_haswell()); println!("uses skylake: {}", capabilities::uses_skylake()); println!("uses ice: {}", capabilities::uses_ice()); println!("uses genoa: {}", capabilities::uses_genoa()); println!("uses sapphire: {}", capabilities::uses_sapphire()); println!("uses turin: {}", capabilities::uses_turin()); ``` ## Using SimSIMD in JavaScript To install, choose one of the following options depending on your environment: - `npm install --save simsimd` - `yarn add simsimd` - `pnpm add simsimd` - `bun install simsimd` The package is distributed with prebuilt binaries, but if your platform is not supported, you can build the package from the source via `npm run build`. This will automatically happen unless you install the package with the `--ignore-scripts` flag or use Bun. After you install it, you will be able to call the SimSIMD functions on various `TypedArray` variants: ```js const { sqeuclidean, cosine, inner, hamming, jaccard } = require('simsimd'); const vectorA = new Float32Array([1.0, 2.0, 3.0]); const vectorB = new Float32Array([4.0, 5.0, 6.0]); const distance = sqeuclidean(vectorA, vectorB); console.log('Squared Euclidean Distance:', distance); ``` Other numeric types and precision levels are supported as well. For double-precision floating-point numbers, use `Float64Array`: ```js const vectorA = new Float64Array([1.0, 2.0, 3.0]); const vectorB = new Float64Array([4.0, 5.0, 6.0]); const distance = cosine(vectorA, vectorB); ``` When doing machine learning and vector search with high-dimensional vectors you may want to quantize them to 8-bit integers. You may want to project values from the $[-1, 1]$ range to the $[-127, 127]$ range and then cast them to `Int8Array`: ```js const quantizedVectorA = new Int8Array(vectorA.map(v => (v * 127))); const quantizedVectorB = new Int8Array(vectorB.map(v => (v * 127))); const distance = cosine(quantizedVectorA, quantizedVectorB); ``` A more extreme quantization case would be to use binary vectors. You can map all positive values to `1` and all negative values and zero to `0`, packing eight values into a single byte. After that, Hamming and Jaccard distances can be computed. ```js const { toBinary, hamming } = require('simsimd'); const binaryVectorA = toBinary(vectorA); const binaryVectorB = toBinary(vectorB); const distance = hamming(binaryVectorA, binaryVectorB); ``` ## Using SimSIMD in Swift To install, simply add the following dependency to your `Package.swift`: ```swift dependencies: [ .package(url: "https://github.com/ashvardanian/simsimd") ] ``` The package provides the most common spatial metrics for `Int8`, `Float16`, `Float32`, and `Float64` vectors. ```swift import SimSIMD let vectorA: [Int8] = [1, 2, 3] let vectorB: [Int8] = [4, 5, 6] let cosineSimilarity = vectorA.cosine(vectorB) // Computes the cosine similarity let dotProduct = vectorA.dot(vectorB) // Computes the dot product let sqEuclidean = vectorA.sqeuclidean(vectorB) // Computes the squared Euclidean distance ``` ## Using SimSIMD in C For integration within a CMake-based project, add the following segment to your `CMakeLists.txt`: ```cmake FetchContent_Declare( simsimd GIT_REPOSITORY https://github.com/ashvardanian/simsimd.git GIT_SHALLOW TRUE ) FetchContent_MakeAvailable(simsimd) ``` After that, you can use the SimSIMD library in your C code in several ways. Simplest of all, you can include the headers, and the compiler will automatically select the most recent CPU extensions that SimSIMD will use. ```c #include <simsimd/simsimd.h> int main() { simsimd_f32_t vector_a[1536]; simsimd_f32_t vector_b[1536]; simsimd_kernel_punned_t distance_function = simsimd_metric_punned( simsimd_metric_cos_k, // Metric kind, like the angular cosine distance simsimd_datatype_f32_k, // Data type, like: f16, f32, f64, i8, b8, and complex variants simsimd_cap_any_k); // Which CPU capabilities are we allowed to use simsimd_distance_t distance; distance_function(vector_a, vector_b, 1536, &distance); return 0; } ``` ### Dynamic Dispatch in C To avoid hard-coding the backend, you can rely on `c/lib.c` to prepackage all possible backends in one binary, and select the most recent CPU features at runtime. That feature of the C library is called [dynamic dispatch](#dynamic-dispatch) and is extensively used in the Python, JavaScript, and Rust bindings. To test which CPU features are available on the machine at runtime, use the following APIs: ```c int uses_dynamic_dispatch = simsimd_uses_dynamic_dispatch(); // Check if dynamic dispatch was enabled simsimd_capability_t capabilities = simsimd_capabilities(); // Returns a bitmask int uses_neon = simsimd_uses_neon(); int uses_sve = simsimd_uses_sve(); int uses_haswell = simsimd_uses_haswell(); int uses_skylake = simsimd_uses_skylake(); int uses_ice = simsimd_uses_ice(); int uses_genoa = simsimd_uses_genoa(); int uses_sapphire = simsimd_uses_sapphire(); ``` To override compilation settings and switch between runtime and compile-time dispatch, define the following macro: ```c #define SIMSIMD_DYNAMIC_DISPATCH 1 // or 0 ``` ### Spatial Distances: Cosine and Euclidean Distances ```c #include <simsimd/simsimd.h> int main() { simsimd_i8_t i8[1536]; simsimd_i8_t u8[1536]; simsimd_f64_t f64s[1536]; simsimd_f32_t f32s[1536]; simsimd_f16_t f16s[1536]; simsimd_bf16_t bf16s[1536]; simsimd_distance_t distance; // Cosine distance between two vectors simsimd_cos_i8(i8s, i8s, 1536, &distance); simsimd_cos_u8(u8s, u8s, 1536, &distance); simsimd_cos_f16(f16s, f16s, 1536, &distance); simsimd_cos_f32(f32s, f32s, 1536, &distance); simsimd_cos_f64(f64s, f64s, 1536, &distance); simsimd_cos_bf16(bf16s, bf16s, 1536, &distance); // Euclidean distance between two vectors simsimd_l2sq_i8(i8s, i8s, 1536, &distance); simsimd_l2sq_u8(u8s, u8s, 1536, &distance); simsimd_l2sq_f16(f16s, f16s, 1536, &distance); simsimd_l2sq_f32(f32s, f32s, 1536, &distance); simsimd_l2sq_f64(f64s, f64s, 1536, &distance); simsimd_l2sq_bf16(bf16s, bf16s, 1536, &distance); return 0; } ``` ### Dot-Products: Inner and Complex Inner Products ```c #include <simsimd/simsimd.h> int main() { // SimSIMD provides "sized" type-aliases without relying on `stdint.h` simsimd_i8_t i8[1536]; simsimd_i8_t u8[1536]; simsimd_f16_t f16s[1536]; simsimd_f32_t f32s[1536]; simsimd_f64_t f64s[1536]; simsimd_bf16_t bf16s[1536]; simsimd_distance_t product; // Inner product between two real vectors simsimd_dot_i8(i8s, i8s, 1536, &product); simsimd_dot_u8(u8s, u8s, 1536, &product); simsimd_dot_f16(f16s, f16s, 1536, &product); simsimd_dot_f32(f32s, f32s, 1536, &product); simsimd_dot_f64(f64s, f64s, 1536, &product); simsimd_dot_bf16(bf16s, bf16s, 1536, &product); // SimSIMD provides complex types with `real` and `imag` fields simsimd_f64c_t f64s[768]; simsimd_f32c_t f32s[768]; simsimd_f16c_t f16s[768]; simsimd_bf16c_t bf16s[768]; simsimd_distance_t products[2]; // real and imaginary parts // Complex inner product between two vectors simsimd_dot_f16c(f16cs, f16cs, 768, &products[0]); simsimd_dot_f32c(f32cs, f32cs, 768, &products[0]); simsimd_dot_f64c(f64cs, f64cs, 768, &products[0]); simsimd_dot_bf16c(bf16cs, bf16cs, 768, &products[0]); // Complex conjugate inner product between two vectors simsimd_vdot_f16c(f16cs, f16cs, 768, &products[0]); simsimd_vdot_f32c(f32cs, f32cs, 768, &products[0]); simsimd_vdot_f64c(f64cs, f64cs, 768, &products[0]); simsimd_vdot_bf16c(bf16cs, bf16cs, 768, &products[0]); return 0; } ``` ### Binary Distances: Hamming and Jaccard Distances ```c #include <simsimd/simsimd.h> int main() { simsimd_b8_t b8s[1536 / 8]; // 8 bits per word simsimd_distance_t distance; simsimd_hamming_b8(b8s, b8s, 1536 / 8, &distance); simsimd_jaccard_b8(b8s, b8s, 1536 / 8, &distance); return 0; } ``` ### Probability Distributions: Jensen-Shannon and Kullback-Leibler Divergences ```c #include <simsimd/simsimd.h> int main() { simsimd_f64_t f64s[1536]; simsimd_f32_t f32s[1536]; simsimd_f16_t f16s[1536]; simsimd_distance_t divergence; // Jensen-Shannon divergence between two vectors simsimd_js_f16(f16s, f16s, 1536, &divergence); simsimd_js_f32(f32s, f32s, 1536, &divergence); simsimd_js_f64(f64s, f64s, 1536, &divergence); // Kullback-Leibler divergence between two vectors simsimd_kl_f16(f16s, f16s, 1536, &divergence); simsimd_kl_f32(f32s, f32s, 1536, &divergence); simsimd_kl_f64(f64s, f64s, 1536, &divergence); return 0; } ``` ### Half-Precision Floating-Point Numbers If you aim to utilize the `_Float16` functionality with SimSIMD, ensure your development environment is compatible with C 11. For other SimSIMD functionalities, C 99 compatibility will suffice. To explicitly disable half-precision support, define the following macro before imports: ```c #define SIMSIMD_NATIVE_F16 0 // or 1 #define SIMSIMD_NATIVE_BF16 0 // or 1 #include <simsimd/simsimd.h> ``` ### Compilation Settings and Debugging `SIMSIMD_DYNAMIC_DISPATCH`: > By default, SimSIMD is a header-only library. > But if you are running on different generations of devices, it makes sense to pre-compile the library for all supported generations at once, and dispatch at runtime. > This flag does just that and is used to produce the `simsimd.so` shared library, as well as the Python and other bindings. For Arm: `SIMSIMD_TARGET_NEON`, `SIMSIMD_TARGET_SVE`, `SIMSIMD_TARGET_SVE2`, `SIMSIMD_TARGET_NEON_F16`, `SIMSIMD_TARGET_SVE_F16`, `SIMSIMD_TARGET_NEON_BF16`, `SIMSIMD_TARGET_SVE_BF16`. For x86: (`SIMSIMD_TARGET_HASWELL`, `SIMSIMD_TARGET_SKYLAKE`, `SIMSIMD_TARGET_ICE`, `SIMSIMD_TARGET_GENOA`, `SIMSIMD_TARGET_SAPPHIRE`, `SIMSIMD_TARGET_TURIN`, `SIMSIMD_TARGET_SIERRA`. > By default, SimSIMD automatically infers the target architecture and pre-compiles as many kernels as possible. > In some cases, you may want to explicitly disable some of the kernels. > Most often it's due to compiler support issues, like the lack of some recent intrinsics or low-precision numeric types. > In other cases, you may want to disable some kernels to speed up the compilation process and trim the binary size. `SIMSIMD_SQRT`, `SIMSIMD_RSQRT`, `SIMSIMD_LOG`: > By default, for __non__-SIMD backends, SimSIMD may use `libc` functions like `sqrt` and `log`. > Those are generally very accurate, but slow, and introduce a dependency on the C standard library. > To avoid that you can override those definitions with your custom implementations, like: `#define SIMSIMD_RSQRT(x) (1 / sqrt(x))`. ## Algorithms & Design Decisions 📚 In general there are a few principles that SimSIMD follows: - Avoid loop unrolling. - Never allocate memory. - Never throw exceptions or set `errno`. - Keep all function arguments the size of the pointer. - Avoid returning from public interfaces, use out-arguments instead. - Don't over-optimize for old CPUs and single- and double-precision floating-point numbers. - Prioritize mixed-precision and integer operations, and new ISA extensions. - Prefer saturated arithmetic and avoid overflows. Possibly, in the future: - Best effort computation silencing `NaN` components in low-precision inputs. - Detect overflows and report the distance with a "signaling" `NaN`. Last, but not the least - don't build unless there is a demand for it. So if you have a specific use-case, please open an issue or a pull request, and ideally, bring in more users with similar needs. ### Cosine Similarity, Reciprocal Square Root, and Newton-Raphson Iteration The cosine similarity is the most common and straightforward metric used in machine learning and information retrieval. Interestingly, there are multiple ways to shoot yourself in the foot when computing it. The cosine similarity is the inverse of the cosine distance, which is the cosine of the angle between two vectors. ```math \text{CosineSimilarity}(a, b) = \frac{a \cdot b}{\|a\| \cdot \|b\|} ``` ```math \text{CosineDistance}(a, b) = 1 - \frac{a \cdot b}{\|a\| \cdot \|b\|} ``` In NumPy terms, SimSIMD implementation is similar to: ```python import numpy as np def cos_numpy(a: np.ndarray, b: np.ndarray) -> float: ab, a2, b2 = np.dot(a, b), np.dot(a, a), np.dot(b, b) # Fused in SimSIMD if a2 == 0 and b2 == 0: result = 0 # Same in SciPy elif ab == 0: result = 1 # Division by zero error in SciPy else: result = 1 - ab / (sqrt(a2) * sqrt(b2)) # Bigger rounding error in SciPy return result ``` In SciPy, however, the cosine distance is computed as `1 - ab / np.sqrt(a2 * b2)`. It handles the edge case of a zero and non-zero argument pair differently, resulting in a division by zero error. It's not only less efficient, but also less accurate, given how the reciprocal square roots are computed. The C standard library provides the `sqrt` function, which is generally very accurate, but slow. The `rsqrt` in-hardware implementations are faster, but have different accuracy characteristics. - SSE `rsqrtps` and AVX `vrsqrtps`: $1.5 \times 2^{-12}$ maximal relative error. - AVX-512 `vrsqrt14pd` instruction: $2^{-14}$ maximal relative error. - NEON `frsqrte` instruction has no documented error bounds, but [can be][arm-rsqrt] $2^{-3}$. [arm-rsqrt]: https://gist.github.com/ashvardanian/5e5cf585d63f8ab6d240932313c75411 To overcome the limitations of the `rsqrt` instruction, SimSIMD uses the Newton-Raphson iteration to refine the initial estimate for high-precision floating-point numbers. It can be defined as: ```math x_{n+1} = x_n \cdot (3 - x_n \cdot x_n) / 2 ``` On 1536-dimensional inputs on Intel Sapphire Rapids CPU a single such iteration can result in a 2-3 orders of magnitude relative error reduction: | Datatype | NumPy Error | SimSIMD w/out Iteration | SimSIMD | | :--------- | ------------------: | ----------------------: | ------------------: | | `bfloat16` | 1.89e-08 ± 1.59e-08 | 3.07e-07 ± 3.09e-07 | 3.53e-09 ± 2.70e-09 | | `float16` | 1.67e-02 ± 1.44e-02 | 2.68e-05 ± 1.95e-05 | 2.02e-05 ± 1.39e-05 | | `float32` | 2.21e-08 ± 1.65e-08 | 3.47e-07 ± 3.49e-07 | 3.77e-09 ± 2.84e-09 | | `float64` | 0.00e+00 ± 0.00e+00 | 3.80e-07 ± 4.50e-07 | 1.35e-11 ± 1.85e-11 | ### Curved Spaces, Mahalanobis Distance, and Bilinear Quadratic Forms The Mahalanobis distance is a generalization of the Euclidean distance, which takes into account the covariance of the data. It's very similar in its form to the bilinear form, which is a generalization of the dot product. ```math \text{BilinearForm}(a, b, M) = a^T M b ``` ```math \text{Mahalanobis}(a, b, M) = \sqrt{(a - b)^T M^{-1} (a - b)} ``` Bilinear Forms can be seen as one of the most important linear algebraic operations, surprisingly missing in BLAS and LAPACK. They are versatile and appear in various domains: - In Quantum Mechanics, the expectation value of an observable $A$ in a state $\psi$ is given by $\langle \psi | A | \psi \rangle$, which is a bilinear form. - In Machine Learning, in Support Vector Machines (SVMs), bilinear forms define kernel functions that measure similarity between data points. - In Differential Geometry, the metric tensor, which defines distances and angles on a manifold, is a bilinear form on the tangent space. - In Economics, payoff functions in certain Game Theoretic problems can be modeled as bilinear forms of players' strategies. - In Physics, interactions between electric and magnetic fields can be expressed using bilinear forms. Broad applications aside, the lack of a specialized primitive for bilinear forms in BLAS and LAPACK means significant performance overhead. A $vector * matrix * vector$ product is a scalar, whereas its constituent parts ($vector * matrix$ and $matrix * vector$) are vectors: - They need memory to be stored in: $O(n)$ allocation. - The data will be written to memory and read back, wasting CPU cycles. SimSIMD doesn't produce intermediate vector results, like `a @ M @ b`, but com