Root finding

halleyCubic

b79313f5-2181-5013-bd21-e59a7222dce5

Teaching overview

In FRM-like/1, halleyCubic initializes z from pixel and replaces very small |z| with (0.00001,0). The loop builds f, fp, fpp, and denom; it keeps z when denom has squared size below 1e-10, otherwise computes numer and assigns z - numer/denom. The pinned julia profile uses 96 iterations.

Source walkthrough

frm-v1:c9f11ef69b73ea2c6489025f83e6b53cabebc1df1e399f282b8ce44e7da4b05f:init/0
Initialization: init/0 sets z from pixel.
frm-v1:c9f11ef69b73ea2c6489025f83e6b53cabebc1df1e399f282b8ce44e7da4b05f:loop/0
Iteration: loop/0 sets z2 = z*z; later loop nodes build f, fp, fpp, and denom.
frm-v1:c9f11ef69b73ea2c6489025f83e6b53cabebc1df1e399f282b8ce44e7da4b05f:loop/6
Branch: loop/6 keeps z when denom has squared size below 1e-10; otherwise its else path computes numer and assigns z - numer/denom.
frm-v1:c9f11ef69b73ea2c6489025f83e6b53cabebc1df1e399f282b8ce44e7da4b05f:bailout
Termination: bailout tests |z - zPrev| >= 0.000001.

Syntax features

sequence

The cited annotations connect z = pixel, the first z2 construction, and the change-based bailout.

branch

The cited branch keeps z for a squared denom size below 1e-10; otherwise it computes numer and assigns z - numer/denom.

Parameter experiment

Steps

Use the pinned julia profile with all controls fixed. Run 48 iterations and then 96 iterations, and record whether the denom guard or the z - numer/denom update changes the observed iteration counts.

What to observe

The larger cap may expose additional guarded or unguarded Halley updates; record whether the iteration counts differ.

Exercise

State the two outcomes of the Halley denom-size guard.

Completion check

Your response names all 4 annotations and gives both keeping z and z - numer/denom.

Sources and rights

This teaching unit is grounded in the pinned runtime source revision.

rightsStatus is project-owned; implementationBasis is project-owned; publicationDecision is publish; decisionReason is publish-project-owned-native-recipe. This does not make a broader claim about any third-party original.

Formula Record

halleyCubic

This canonical FractalPark implementation is published and runnable.

Canonical source

Language: frm-like/1
Standard library: 1
Profile mode: parameter-plane
Profile center: 0, 0
Profile zoom: 0.35
Profile iterations: 96
Profile evidence: family

Parameters

No declared parameters

Open in Explorer

Canonical source

This verified canonical revision is read-only. Remix creates a separate editable fork.

Loading canonical source…

Source and implementation

Formula ID: b79313f5-2181-5013-bd21-e59a7222dce5
Canonical name: halleyCubic
Original name: halleyCubic
Historical source: FractalPark
Original resource: public/formula-library/v1/runtime/published/definitions/c9f11ef69b73ea2c6489025f83e6b53cabebc1df1e399f282b8ce44e7da4b05f.frm Repository
Current implementation: Project-owned

This is a FractalPark project-owned implementation. The linked file is the pinned canonical Definition.

Rights or attribution concern?

Send the Formula ID and supporting evidence. The maintainer can hold or withdraw the implementation while the claim is reviewed. contact@fractalpark.com