elementary design

This page explains the elementary facade: which analytic traits it exposes, how their derivative rules are justified, and where they stop.

Design goal

Make analytic code written against the arithmetic traits differentiable without change, by exposing exactly the traits that Dual[T] implements with a correct derivative rule.

Mathematical background

Every rule has the form $f(a + b\varepsilon) = f(a) + f'(a)\,b\,\varepsilon$ (the dual design derives the chain rule from it). The derivatives used are the classical ones; for the inverse-function and base-change cases they follow from the chain rule:

$$\begin{aligned} \frac{d}{da}\sqrt a &= \frac{1}{2\sqrt a} &&\text{from } (\sqrt a)^2 = a, \\ \frac{d}{da}\,2^a &= \frac{d}{da}\,e^{a\ln 2} = 2^a \ln 2, \\ \frac{d}{da}\log_2 a &= \frac{d}{da}\frac{\ln a}{\ln 2} = \frac{1}{a\ln 2}, &\frac{d}{da}\log_{10} a &= \frac{1}{a \ln 10}, \\ \frac{d}{da}\tan a &= \frac{d}{da}\frac{\sin a}{\cos a} = \frac{\cos^2 a + \sin^2 a}{\cos^2 a} = \frac{1}{\cos^2 a} . \end{aligned}$$

Each formula holds on the open domain where $f$ is differentiable: $a > 0$ for the square root and the logarithms, $\cos a \ne 0$ for the tangent, and everywhere for $\exp$, $\sin$ and $\cos$.

Design decisions

Expose what Dual[T] implements

The facade re-exports Sqrt, SqrtChecked, Exponential, Logarithmic, Trigonometric and Constants, the analytic traits with an instance on Dual[T]. arithmetic also defines Hyperbolic, InverseTrigonometric, InverseHyperbolic, Cbrt and Power; Dual[T] has no instance for them yet, so they are not re-exported.

Instances need the whole trait

A trait instance must provide all its methods. Exponential contains exp2, whose rule needs $\ln 2$, so the instance needs Logarithmic and IntegralHomomorphism on T even if a caller only uses exp. The inherent method Dual::exp has the smaller bound Exponential + Mul for that case.

Constants have tangent zero

$\pi$, $\tau$ and $e$ do not depend on the differentiation variable, so the Constants instance embeds them with Dual::constant. This is the structure-preserving choice: the map $T \to T[\varepsilon]$, $c \mapsto c + 0\varepsilon$, is a ring homomorphism.

Correctness and invariants

• Each rule matches the derivative table above and is tested on Double (sin, exp and others in src/tests/dual_test.mbt).
• Outside the open domain, the result follows T: for Double, ln(0) is $-\infty$ with tangent $b/0$, and sqrt at $0$ has an infinite or NaN tangent.
• The tangent error is the error of T’s implementation of $f'(a)$ plus at most two roundings; see the dual design.

Alternatives rejected

• Re-exporting every analytic trait. Bounds such as T : Hyperbolic would type-check for T but fail to instantiate at Dual[T], which is confusing.
• Checked logarithms in this repository. arithmetic has no checked logarithm trait to implement; adding one locally would fork the error model.

Boundaries

• No hyperbolic, inverse trigonometric, inverse hyperbolic, cube-root or general power rules.
• No checked forms other than SqrtChecked.
• No branch-cut or complex-domain semantics; see luna-complex for complex functions.