Using rustymath from Rust

Add rustymath to your Cargo.toml:

[dependencies]
rustymath = "0.1"

On x86-64, build for a recent CPU, with -C target-cpu=x86-64-v3 or -C target-cpu=native. You can set it once in .cargo/config.toml:

[target.'cfg(target_arch = "x86_64")']
rustflags = ["-C", "target-cpu=x86-64-v3"]

Rust’s default x86-64 target has no fused multiply-add and no vector instruction for floor or trunc. Without the setting, rustymath multiplies and adds separately. Some of its loops also run one item at a time.

Each function comes in two forms. The free function, such as exp, is generic over f32, f64 and half::f16. The compiler inlines it into the loop that calls it and vectorizes that loop:

use half::f16;
use rustymath::exp;

let e64 = exp(1.0f64); // 2.718281828459045
let e32 = exp(1.0f32); // 2.718282
let e16 = exp(f16::from_f32(1.0)); // 2.719

The function object, such as Exp or Pow, has a scalar form one and a slice form many. Both run code that rustymath compiles itself. Your build settings don’t change their speed. Each argument of many is one item, which every result pairs with, or a slice with one item for each result. many appends the results to a vector and returns whether every item succeeded. A NaN result from arguments that aren’t NaN counts as a failure. An object implements Unary, Binary or Ternary at each item type it takes: f16, f32 and f64, complex numbers where the function has complex values, and integers for some, such as Gcd at i64. Code that is generic over the float type states the bound it needs, such as where Exp: Unary<F, Out = F>. The free functions need no bound.

use rustymath::{Arg, Binary, Pow};

let mut squares = Vec::new();
Pow.many(Arg::Many(&[1.5, 2.0, 3.0]), Arg::One(2.0), 3, &mut squares); // squares: [2.25, 4.0, 9.0]

kernels! gives an object’s kernels as function pointers, one at each dtype where the object has a form, for code that chooses a dtype at run time. Its documentation gives the forms it takes. Kernels::computes gives the dtype that an argument computes in. Kernels::unary, binary, ternary, run and product cast an argument there and run the kernel. run executes the accumulation selected by the Fold, Scan, RightFold or LeftScan probe:

use rustymath::buffer::{DType, FloatWidth, Width};
use rustymath::tolerant::{Equal, Tolerance};
use rustymath::{kernels, reduce::Sum, Exp};

let exp = kernels!(Unary Exp);
assert_eq!(exp.computes(DType::Integer(Width::I32)), Some(DType::Float(FloatWidth::F64)));
let equal = kernels!(Binary of Equal(Tolerance::new(1e-14).unwrap()));
assert!(equal.at(DType::Float(FloatWidth::F64)).is_some());
let sum = kernels!(Fold Sum);
assert_eq!(sum.computes(DType::Boolean), Some(DType::Integer(Width::I64)));

Use the slice forms of Sin, Cos and Tan for arrays. They reduce each item by multiples of π with a few multiply-adds, then give the rare items beyond 2^23 at f64, or 2^20 at f32, an exact reduction in integer arithmetic. The free sin takes a branch for those items. That branch stops a loop that calls it from vectorizing.

atan2(y, x) and its object Atan2 give the angle of the point (x, y), from -π to π. y comes first, as in f64::atan2.

The complex slice forms of Exp, Ln, Pow, Sin, Cos, Sinh and Cosh vectorize too. Each one splits a chunk of complex items into their real and imaginary parts and computes the results from the real slice forms. On macOS, the sines and cosines they need come from vForce, which computes both in one call. It then redoes the rare items with the scalar complex function: items with a tiny or non-finite part, and Pow’s special cases. Abs, Angle and Sqrt keep their loop over items, which runs faster than a split for their short formulas. Other complex slice forms, such as Lgamma’s, compute one item at a time.

divide(x, y) divides complex numbers, inline in the caller’s loop, as the free functions do. Two real arguments divide as reals, so a real zero divisor gives ±∞ or NaN with an imaginary part of 0. Its formula’s |y|² overflows beyond about 1e154 at f64 and 1e19 at f32.

Tolerant comparison

The tolerant module has APL’s tolerant comparison: x equals y when |x - y| is less than the tolerance times the larger of |x| and |y|. Its objects, such as Equal, Less, Floor, Residue and Gcd, each hold a Tolerance. With Tolerance::EXACT, a slice form runs a loop of plain IEEE comparisons, chosen once per call.

use rustymath::{tolerant::{Equal, Tolerance}, Arg, Binary};

let mut same = Vec::new();
Equal(Tolerance::DEFAULT).many(Arg::Many(&[0.3, 0.1 + 0.2, 0.31]), Arg::One(0.3), 3, &mut same); // same: [true, true, false]

Other modules

  • number_theory: primality, factorisation, next and previous primes, Euler’s totient, and the enumeration and counting of primes, on a u64, a u128 or a BigUint. Each computes in the narrowest of those that holds the value. A function that can run for a long time also has a _with version, whose extra argument is a check that can stop it. factorial gives n! as a BigUint.
  • optimize: minimize finds the minimum of a function of an f32 or f64 slice, by Brent’s method for one variable, Nelder–Mead, Powell’s conjugate directions, L-BFGS with a gradient, or differential evolution within bounds. solve finds a zero of a function that gives a slice, or its least-squares solution, by Brent’s method for one equation in one variable and Levenberg–Marquardt otherwise. solve_linear solves a linear system, or its least squares, through the matrix’s products with vectors, by conjugate gradients, MINRES or restarted GMRES, with an optional preconditioner. solve_affine solves f(x) = 0 for an affine function the same way, through its values alone. Method::choose picks a method where the caller doesn’t. Functions are Objectives and Mappings: closures that return a Result, whose error stops the search, or types that also give a gradient, vector–Jacobian products or batches of points.
  • polynomial: evaluation over a slice by Horner’s rule, roots from the eigenvalues of the companion matrix, sorted by real part, then imaginary part, building from roots, derivatives and least-squares fitting. value and factored evaluate at one point. They, from_roots and derivative take any type with the field operations, such as exact rationals. So does decode, which gives the values of digits in a mixed radix.
  • reduce: sums, products, maximums and minimums over a slice, and their running versions. Reduction gives each one’s unit, its step for one more item, its fold of a slice, its merge of the folds of two neighbouring runs, and its scan, through Sum, Product, Max and Min. Its checked is the step, and whether the result fits. Float sums and products vectorize by combining the items in any order. An integer sum or product, and each running one, wraps where it doesn’t fit, and reports whether it fits. extremes gives the least and the greatest item in one pass. Alternating gives the right fold by - through Sum, and by ÷ through Product: the sum of the pairs’ differences, or the product of their ratios. A quotient of floats reports that it doesn’t fit where a running product leaves the normal range. Booleans fold by Max and Min as or and and, by Xor and Xnor as not-equal and equal, and by SumAs as a count of trues. Logical scans Booleans by any function of two Booleans, given as its truth table. SumAs<C> also sums integers, wrapping in the type C, which the caller picks to hold every sum: a byte, i16, i32 or i64, through the traits Term and Tally. along and scan_along fold and scan along the middle axis of an array of shape [outer, len, inner] held row by row: down the columns of a matrix, for example, with [1, rows, columns]. fold_into folds the same array one block at a time, for a caller that computes the array in blocks. A row of fewer than 16 items, such as an image’s 3 channels, has a loop of its own for each length. A running maximum or minimum of integers repeats its value over each block of 256 items that leaves it unchanged. along_mapped folds an array that a mapping lays over storage, without building it. sliding folds each window of neighbouring rows, and view_sliding_picks gives the row of each window’s first maximum or minimum.
  • linalg: inner products of row-major matrices through the Inner trait: the matrix product PlusTimes, which also takes complex matrices, the max-plus and min-plus products MaxPlus and MinPlus, the counts of equal and unequal items PlusEqual and PlusNotEqual, and the Boolean products OrAnd and AndOr. Each numeric pair also takes i64 matrices, whose arithmetic wraps. Inner::checked gives None where an integer result could overflow. They block for the caches, as GotoBLAS and BLIS do. Their innermost loops run on explicit vectors on AArch64, in wasm and on x86-64 with AVX2. Its decompositions of real and complex matrices, at f32, f64 and Complex with parts of either type, are lu, cholesky, qr, qr_complete and svd, with Lu::solve, Lu::inverse, Lu::lower, Lu::upper, Lu::determinant, Lu::log_determinant, Svd::solve, Svd::pseudoinverse, singular_values and the least-squares solver lstsq. solve and pinv give the least-norm solution and the pseudoinverse of any matrix, singular or not. rank counts the singular values above lstsq’s cutoff. eigh and hermitian_eigenvalues take Hermitian matrices, and eig and eigenvalues any square matrix. transpose_into transposes a matrix through 8×8 tiles, applying a conversion to each item. Scalar is the sealed trait of the four item types. linalg::exact works in any field, such as exact rationals: echelon, solve, pinv, determinant and Ryser’s permanent, each with a _with version.
  • exact: residue, gcd, lcm and binom of exact rationals, at any integer type, such as BigRational. linalg::exact solves matrices of them.
  • distribution: sixteen distributions. The objects Pdf, Cdf and Quantile hold a distribution and evaluate it over a slice. Each distribution also samples from a seeded generator.
  • array: Array, a Layout paired with the Buffer that holds one item for each of its positions, in row-major order.
  • create: filled arrays and integer or floating ranges, evenly spaced values and logarithmically spaced values.
  • stats: means, two-pass variance and standard deviation, weighted averages, elementwise p-norms, covariance, correlation and histograms. Axes prepares reductions over selected axes. Complex variance and norms return real values.
  • search: exact search and classification, stable counting/radix sorting, partitioning and quantiles. select places requested ranks without ordering the intervening runs. View-based entry points share the supported dtypes with BasedPL and Python. Ordering compares floats exactly, equates signed zeros and puts NaNs last ascending.
  • buffer: compact buffers. Buffer holds items of one element type, and View borrows them. DType names the type: Booleans, integers of four widths (Width), floats of three (FloatWidth), complex numbers of two (ComplexWidth), or characters. DType::common gives the dtype that NumPy promotes two dtypes to, and DType::inexact the dtype that a function of floats computes in. IntBuf, FloatBuf and ComplexBuf hold one family’s widths, with the views Ints, Floats and Complexes. The macros with_ints!, with_floats! and with_complexes! run code on a view’s slice, and with_width!, with_float_width! and with_complex_width! on a width’s element type. with_dtype! runs code on a DType’s element type. Element gives each element type’s fill, and of, view and buffer move between its slices and a View or a Buffer. Source reads an item of one type as another.
  • error: Error, which the functions on arrays return, with its Kind (Length, Rank, Index, Limit, Domain or Overflow) and a message.
  • layout: array shapes, axis keys and names, and their limits. Shape holds an array’s axis lengths. Keys names the positions on one axis. Layout pairs a shape with keys and a name for each axis. Layout::axis_named finds an axis by its name. position gives the position among an axis’s length that an index gives, counting from the end when it’s negative. Lanes describes an array’s items as lanes along one axis, with the offset of each item and the dims that rustymath’s axis functions take. element_count and generated_len count an array’s items, within MAX_RANK axes and MAX_GENERATED_ELEMENTS items. strides gives the distance between neighbouring items along each axis.
  • agreement: broadcasting by BPL’s rule. Agreement::new pairs the axes of two layouts: axes with the same name first, then the rest from the front. An axis of length 1 repeats along its partner, and an axis that both layouts key takes the union of their keys. The result is the paired layout and a Mapping for each argument, which gives the argument’s offset for each result position. blocks runs a kernel over two slices through an agreement, passing contiguous runs whole, and expanded reads one argument’s items for a range of result positions. Agreement is generic over its layout type, so a caller can carry its own data on layouts through map_layout. Frames pairs the frames of any number of arguments to a function of their trailing axes, such as the matrices of a stack, and gives each argument’s cell for each cell of the result.
  • random: rolls and deals that fill a slice from one generator.
  • gather: copying a slice’s items into a buffer by position: cycle, repeat, compress by a mask, take at indices, windows, walk along a pattern of Steps with fills, and scatter at offsets, each applying the caller’s conversion to each item. walk also takes Same, which copies whole slices of one type. Steps::reach gives the source offset of one item of a walk, and Steps::run the offset of a run of items that lie in order in the source. Steps::through composes two walks into one. Steps::per_axis gives a walk one entry for each axis of its result. Steps::walked also gives the source axis that each entry walks. Steps::windows_read gives what entries walking windows read along one source axis: where the windows start, and each entry’s length and move. Steps::windows_range gives the positions read where every entry moves by 1. Steps::windows_of recognises a copy of windows laid out as structure::Windows::plan lays them out without padding, and gives each axis’s frames, window size and move. Steps::spreads and Steps::rereads find walks that read an item more than once or read a fill.
  • structure: plans for structural operations. A Plan holds the result’s Layout and the Steps that walk reads it by. select applies independent axis selectors for positions, ranges, index arrays and inserted axes. Index arrays pair, all of them by broadcasting as .[] reads them, or those that share an axis name as ⌷ reads them. addressed gives the axis that each selector selects along. take takes or drops items, rotate rotates or reverses an axis, with one count or one for each lane, transpose sends axes to new positions, and catenate gives the layout of two arrays joined along an axis. Windows plans windows over leading axes, with fill around padded axes, and each window of a stencil. Each plan checks its arguments. Keys and names follow their positions.
  • signal: correlations and convolutions, which slide a kernel along an array’s leading axes, and overlap-add. correlation gives the plans that a correlation reads: the array with the fills its Mode adds, its windows, and the kernel, reversed for a convolution. correlate runs them through an Inner pair’s inner_mapped. Float and complex PlusTimes read the windows a block at a time. overlap_add adds each frame back where its window was read, the transpose of taking windows.
  • convert: int converts a whole float to an integer type. It gives None outside the type’s range, and for i64 and u32 beyond a magnitude of 2^51 - 1. float converts an integer to the nearest float. Loops that call either vectorize on every target. AVX2 and wasm lack a vector instruction between f64 and 64-bit integers. On those targets a loop that converts with as runs one item at a time.
  • timing: the machine’s timing lock, which keeps timings apart from builds and test runs. exclusive takes it for a timing and runs the calling thread’s calls on that thread alone while it’s held. shared takes it for a build or a test run. The browser build doesn’t have it.

The special functions gamma, lgamma, erf, erfc, their inverses, the incomplete gamma and beta functions and Lambert W have free functions and objects at the crate root, as the elementary functions do.

Add, Subtract and Multiply are objects at the crate root, for every float, complex and integer type. Their integer forms, and Abs’s, fail where a result doesn’t fit its type. A caller that keeps integers exact, as basedpl does, takes another path for those results.

Accuracy and speed

The table gives nanoseconds per item for a loop over 65,536 inputs on an Apple M5 Max, in October 2026, beside the standard library’s version of each function. A row named after an object times its slice form. Pow.one and Root.one time the scalar forms. To print the table on your own machine, run RUSTYMATH_COSTS=1 cargo test --test speed.

Function f64 std f64 f32 std f32
Exp 0.84 3.92 0.33 0.91
Expm1 1.07 2.70 0.48 3.10
Ln 1.36 1.62 0.62 1.34
Log1p 1.80 2.31 0.85 1.39
Log2 1.36 1.62 0.61 1.27
Log10 1.38 1.74 0.62 1.27
Sin 0.87 4.95 0.36 4.68
Cos 0.91 5.12 0.40 5.01
Tan 1.14 2.58 0.45 4.68
Asin 0.94 4.55 0.34 3.83
Acos 1.03 6.03 0.37 5.15
Atan 0.89 6.04 0.34 2.16
Sinh 1.26 2.47 0.56 2.37
Cosh 0.95 2.27 0.38 2.30
Tanh 1.24 3.44 0.53 2.28
Asinh 1.50 2.56 0.69 1.87
Acosh 1.59 2.17 0.68 1.56
Atanh 1.40 3.78 0.62 1.62
Cbrt 1.11 1.39 0.89 1.85
Pow 2.92 7.28 1.61 2.06
Pow.one 4.11 7.18 2.27 1.97
Hypot 0.55 1.21 0.27 0.87
Root 3.38 4.24 1.76 1.96
Root.one 4.23 4.25 2.36 2.09
Maximum 0.19 0.19 0.10 0.09
reduce::sum 0.06 0.49 0.03 0.46
reduce::max 0.06 0.11 0.03 0.05
reduce::cumsum 0.15 1.94 0.15 2.61
reduce::cummax 0.15 2.53 0.07 2.54
polynomial::evaluate 0.51 2.60 0.22 2.60

An f16 computes in f32 and rounds once. Special values, such as infinities, NaN and signed zeros, give IEEE’s results. The tests compare each function with a correctly rounded reference at every f16 input, at samples of f32 and f64 inputs, and at published hard cases.

Where the target has a fused multiply-add instruction, as on AArch64 and on x86-64 with FMA, the functions use it. Elsewhere, as in wasm, they multiply and add separately, because a call to a software fma would stop the loop vectorizing. FUSED says which the target uses. The modules fused and unfused hold every function again, with the choice made. A loop compiled with FMA for itself, such as an AVX2 version chosen when your program runs, calls fused. The 4-ulp bound holds either way.

At f64, pow and root compute ln |x| and its product with y as double words, through Arm Optimized Routines’ tables. The result stays within 4 ulps even where y ln |x| nears the overflow threshold. A power or root whose exact result is a float, such as 15^4 or ∛27, comes out exactly. Their scalar forms take one branch for special values, as Arm’s scalar pow does. Their slice forms compute every item the fast way, then redo only the rare items with special values. At f32 and f16, pow and root compute in f64 and round once. There the free pow and root have no branch. A loop that calls them vectorizes.