README.md

Announcements

New release (September 2026)

After months of experimenting, learning (finding about Levi Civita fields etc.), building and testing different implementations, here is the new release that contains the accumulated findings. This release contains results of trying out different approaches and results of numeruos experiments. The more experimental stuff still relies to external support, oracles etc. (as it should), the more tested features are tending to become more and more self reliant with additional iterations (eg, derivations and integrals.).

What it tries to achieve:

• thinning the reliance on external libraries, trying to express as much as we can through composite tooling
• performance enhancements
• isolation end elimination of trucation errors
• add more depth, reach and precision to the toolkit by adding the transseries support for initial experimentation (can of worms)

What it adds:

• refinements to zero handling edge cases
• adds float dimensions, so we can finally take a square roots on composites with exact precision and remain composite
• adds experimental support for vector dimensions which enables taking log of an composite number
• tons of edge case bugfixes (especially for integration)

Note of caution: this is still highly experimental and most likely (for sure) still contains some misconceptions and a lot of edge cases and other bugs. The purpose of the library is to showcase what is possible and to serve as a baseline for further exploration.

Library release (April 2026)

The first proper pypy library based on this experimental features has been released. A standalone tool to evaluate Python functions at points where they're undefined and get exact limit values if they exists.

• https://github.com/FWDhr/composite-resolve

Composite Machine

Automatic calculus via dimensional arithmetic... and a bit more.

A data structure that implements a number system, letting different parts of non-standard (and standard) math work together.

For example, it gives you derivatives, integrals, and limits as a side effect of normal computation. No symbolic engine, no computation graph, no tape. Tag a number, do your math, read the results off the dimensional coefficients.

1 − 1 ≠ 0

The residue is infinitesimal, structured, and it contains the derivative of every operation that produced it.

Alpha stage. Research code. The math works, performance doesn't (yet). AGPL-licensed. A PyTorch/CUDA backend is available under commercial license."

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What's this

Numbers are sparse dicts mapping dimensions to coefficients. A dimension is an integer, or a vector over an iterated-logarithm basis when log-scale terms are in play. Dimension 0 is the value. Negative dimensions store derivative info. Multiply dimensions - turns out that's the same thing as the product rule and chain rule, just expressed as data structure operations.

from composite.composite_lib import R, ZERO

x = R(3) + ZERO          # 3 + infinitesimal seed
result = x ** 4           # just compute normally

result.st()               # 81  - the value, f(3)
result.d(1)               # 108 - first derivative
result.d(2)               # 108 - second derivative
result.d(3)               # 72  - third derivative
result.d(4)               # 24  - fourth derivative

One evaluation. All derivatives fall out. No separate differentiation pass.

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Background

The derivative computation part builds on well-known work: Clifford's dual numbers (1873), Wengert's forward-mode AD (1964), Rall's Taylor arithmetic (1981), Griewank's framework (2000).

The number system has a separate and older lineage. A sparse map from exponents to coefficients, with non-integer exponents and finitely many terms below any given one, is the shape of the Levi-Civita field (Levi-Civita, 1892–1898) - the smallest non-Archimedean ordered field extension of the reals that is real-closed and Cauchy-complete. Letting an exponent be a vector ordered lexicographically instead of a single number gives Hahn series (Hahn, 1907), which is what the iterated-logarithm basis here amounts to: dimensions valued in an ordered group, compared componentwise. The scale those vectors index - x, log x, log log x, ranked by eventual dominance - is du Bois-Reymond's Infinitärcalcül as set out in Hardy's Orders of Infinity (1910), and the Hardy fields built on it. Expansions that mix powers, exponentials and iterated logs are transseries (Écalle, 1992; van der Hoeven, 2006). The infinitesimals themselves are made rigorous by Robinson's non-standard analysis (1966), and the surreals (Conway, 1976) contain the Levi-Civita field as a subfield.

Computing in such a field, rather than reasoning about it, also has prior art. Berz framed automatic differentiation as non-Archimedean analysis (1992), and Shamseddine and Berz developed numerical analysis directly on the Levi-Civita field, including derivatives of functions where classical AD breaks down. Sergeyev's grossone (2003 onward) is the closest in representation: a positional numeral system in powers of an infinite unit ①, with the infinitesimal ①⁻¹, used on an "Infinity Computer" for exact higher-order differentiation, ODE solvers and lexicographic optimization - the same records as the dimensions here, written in a different notation. Grossone keeps the ordinary zero (0·① = 0, ① − ① = 0); this library does not, and that is where the two part ways. The overlap is worth stating plainly: the algebra here is not new, and where this library's structures coincide with those, the credit is theirs.

What this library explores is a different algebraic context for that mechanism. Higher-order terms are preserved instead of truncated. Subtraction retains provenance instead of collapsing to zero. Multiplication by zero shifts structure instead of destroying it. The idea is that if you stop throwing away information at each step, calculus operations become extractable from the algebra.

Does this generalize to everything? Open question. The test suite covers a wide range of standard problems and the results match. Finding the boundaries is the point of this project.

For the theoretical framing, see the paper.

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How it compares

Breadth in one structure, at a cost that depends entirely on the shape of the problem. The numbers under Performance are measured, not asserted, and they do not all point the same way.

• vs PyTorch/JAX - They give first-order gradients, fast, and vectorised across a batch. This gives every order from one evaluation, plus limits and integration. Neither is a backend here, so no throughput ratio against them is quoted - the measurements below are against NumPy and SymPy, which are what this actually runs on.
• vs SymPy - SymPy is symbolic, this is numerical. On the cases measured this is the faster of the two: 3–70x on indeterminate limits (both exact) and ~6400x on a Taylor expansion to order 8, agreeing to 2.5e-15.
• vs mpmath - mpmath is arbitrary-precision and carries the special-function library this does not (gamma, zeta, Bessel). On derivatives the two agree exactly: the 4th derivative of x⁴eˣ at 1 matches to all 15 digits. The difference is method - mpmath samples and extrapolates, so a limit is only as good as the extrapolation converges. On six harder limits it returned 0.99962 for xˣ as x→0⁺ and −2.7e−8 for x²·ln x, where reading the standard part off the algebra gives both exactly.
• vs dual numbers - Classic dual numbers give you one derivative (epsilon squared is zero). Here epsilon squared is kept, so you get all orders.

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Examples

Derivatives

from composite.composite_lib import derivative, nth_derivative, all_derivatives, exp

derivative(lambda x: x ** 2, at=3)               # 6.0
nth_derivative(lambda x: x ** 5, n=3, at=2)      # 240.0
all_derivatives(lambda x: exp(x), at=0, up_to=5) # [1, 1, 1, 1, 1, 1]

Limits

No L'Hôpital. Plug in the infinitesimal, read the standard part.

from composite.composite_lib import limit, sin, R

limit(lambda x: sin(x) / x, as_x_to=0)                  # 1.0
limit(lambda x: (x**2 - R(4)) / (x - R(2)), as_x_to=2)  # 4.0

Integration

from composite.composite_lib import integrate, exp

integrate(lambda x: x ** 2, 0, 1)              # 0.333...
integrate(lambda x: exp(-x), 0, float('inf'))  # 1.0
integrate(lambda x, y: x * y, (0, 1), (0, 1))  # 0.25

Division by zero

ZERO isn't Python's 0 - it's a structural infinitesimal, coefficient 1 at dimension −1. Operations on it are well-defined and reversible:

from composite.composite_lib import ZERO, R

(ZERO / ZERO).st()                          # 1.0
(R(5) * ZERO / ZERO).st()                   # 5.0
(R(7) * ZERO * ZERO / ZERO / ZERO).st()     # 7.0

Multivariable

from composite.composite_multivar import gradient_at, laplacian_at

gradient_at(lambda x, y: x**2 + y**2, [3, 4])  # [6, 8]
laplacian_at(lambda x, y: x**2 + y**2, [3, 4]) # 4

Complex analysis

from composite.composite_extended import residue, convergence_radius

residue(lambda z: 1 / z, at=0)                  # 1.0
convergence_radius(lambda z: 1 / (1 - z), at=0) # 1.0

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Modules

• composite_lib.py - Core engine. Composite class, all arithmetic, transcendentals, derivatives, limits, integration.
• composite_multivar.py - Multivariable calculus. MC class, partial derivatives, gradient, Hessian, Jacobian, Laplacian, divergence, curl.
• composite_extended.py - Complex analysis. Complex composites, residues, poles, contour integrals, asymptotics, ODE solver.
• composite_vector.py - Vector calculus. Triple integrals, line integrals, surface integrals.
• backends/ - Interchangeable storage for the dimension map: dict, sparse-dense, vector-dimension, dense-series.

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What works

Stable:

• Full arithmetic with dimensional convolution and deconvolution
• Integer and real-exponent powers
• Transcendentals - sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, ln, sqrt, erf, erfc, normal_cdf
• All-order derivatives from a single evaluation
• Algebraic limits including indeterminate forms and limits at infinity
• Definite, improper, and adaptive integration with error estimates
• Vector dimensions over an iterated-logarithm basis - log, loglog and deeper scales as ordinary arithmetic, at any depth, with the transcendentals accepting log-axis arguments
• Completeness tracking - every value carries the highest order it is complete to, propagated through each operation
• TracedComposite for step-by-step operation logging

Experimental:

• Multivariable calculus (MC class, partial derivatives, differential operators)
• Vector calculus (line integrals, surface integrals, triple integrals)
• Complex analysis (residues, contour integrals, analytic continuation, convergence radius)
• ODE solver via RK4 with composite evaluation

Not yet implemented or highly experimental:

• Inverse hyperbolics (asinh, acosh, atanh)
• Fourier, Laplace, Z transforms
• Special functions (Bessel, gamma)
• Optimization routines

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Performance

No single ratio - it depends on what you ask for. Measured on CPython/macOS, float64, one thread, NumPy 1.26.4, SymPy 1.14.0.

Derivatives. Tag a number, evaluate 1/(1+x) once, read all ten derivatives off the result:

Faster and exact. One evaluation carries every order, while a stencil needs a fresh step size per order and has run out of digits by order 10.

A single first derivative goes the other way: NumPy does it in 0.2 µs against 35 µs here, roughly 140x. The crossover is around the third derivative.

Batches. NumPy vectorises and this does not - four orders of magnitude per point. Pythorch and CUDA backends not present here support batching.

Sparse grids. An explicit PDE whose active front stays at 121 cells: 24x faster than a dense NumPy grid at 200,000 cells, 260x at 2,000,000. Composite time is flat; the dense grid pays for the whole domain whether anything is happening in it or not.

vs SymPy. Indeterminate limits: 3–70x faster, both exact. A Taylor expansion to order 8: 0.6 ms against 3.7 s, about 6000x, agreeing to 15 digits.

Backend selection. use_dense_series() for calculus - anything with transcendentals builds a dense, contiguous series, and it is ~1.6x faster there than the alternatives. use_dict() for pure-arithmetic jets, where there is no series to lay out (~2x faster on 1/(1+x)). The default use_sparse_dense() is the grid backend: several times slower on either kind of jet, and the one to keep for sparse supports over a large domain.

A high-performance backend using PyTorch with CUDA and MPS acceleration is available under commercial license. Contact tmilovan@fwd.hr.

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Installation

git clone https://github.com/tmilovan/composite-machine.git
cd composite-machine
pip install -e .

Python 3.7+. NumPy is optional (used for FFT-accelerated multiplication).

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Testing

pip install -e .          # so `import composite` resolves to this checkout
python -m pytest tests/   # everything

Or run a single suite directly:

PYTHONPATH=. python tests/test_standalone.py               # core + paper theorems
PYTHONPATH=. python tests/test_limits.py                   # limits, all classes
PYTHONPATH=. python tests/test_integration_comprehensive.py # every integral form

PYTHONPATH=. matters: running a test as a bare script puts tests/ on the import path rather than the repo root, so composite resolves to whatever is installed instead of the working copy.

839 tests across thirteen suites, all passing.

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Paper

Milovan, T. (2026). Provenance-Preserving Arithmetic: A Unified Framework for Automatic Calculus. Zenodo.

https://doi.org/10.5281/zenodo.18528788

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Docs

• Tutorial - Get started quickly
• API Reference - Complete function docs
• Implementation Guide - How it works internally
• Examples - Code snippets for common tasks
• Roadmap (DRAFT) - What's next
• Zero Rules v2 - What a zero coefficient means, and how it behaves

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Contributing

Contributions welcome. Useful areas:

• Special functions - Bessel, gamma, etc.
• Bug reports and edge cases
• Docs and examples

Process: open an issue first, fork, add tests, PR.

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Citation

If you use this in research:

Milovan, T. (2026). Composite Machine: Automatic Calculus via Dimensional Arithmetic. https://github.com/tmilovan/composite-machine

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License

Code: AGPL-3.0. Free for open-source, research, and personal use. Commercial licensing available - contact tmilovan@fwd.hr.

Paper: CC BY 4.0.

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Toni Milovan · Pula, Croatia · tmilovan@fwd.hr