The branches a simplification went down and came back from (#273)

[Rafael-SOWNet]

Closes #273.

The issue asks for two things:

1. record steps while solving something — already here, as RewriteRecording and DerivationPath
2. reverse to the branch's root whenever we fail to solve something within a method — this

No new recording was needed

PathFrom searches the recorded edges for a chain from the input to the result and throws the rest away. The rest is exactly the branches, and each already carries the expression it left from — the branch's root, in the issue's words. DerivationPath.Abandoned hands them over beside the Steps it kept.

What it shows

Simplifying (x + 1)^2 keeps one step and abandons seven:

kept       CanonicalOrder     (x + 1) ^ 2        ->  (1 + x) ^ 2
abandoned  AsPolynomial       (x + 1) ^ 2        ->  x ^ 2 + 2 * x + 1
abandoned  Expand             (1 + x) ^ 2        ->  x ^ 2 + 2 * x + 1
abandoned  CanonicalOrder     x ^ 2 + 2 * x + 1  ->  1 + x ^ 2 + 2 * x
abandoned  Common             (1 + x) ^ 2        ->  (x + 1) ^ 2

The search expands the square and comes back, because the unexpanded form rates better. A Steps list on its own reads as though the library had walked straight to the answer.

Deduplicated, and that is not tidying

The simplifier runs the same passes over the same expressions at every level of its candidate search, so the raw edges are mostly one rewrite recorded over and over:

	raw	distinct
x^(-1)/(y/z)	425	13
(x + 1)^2	177	7
sin(x)^2 + cos(x)^2	2	0

That last row is the one that mattered. Both raw entries were duplicates of the kept steps, so the branch list read as though the search had rejected the very steps it went on to keep. A list of 425 is a record of how often the search was asked, not of where it went — so entries are deduplicated against each other and against the kept chain, and a test asserts both.

Not an error list

A step here is a road not taken, usually to a perfectly correct expression that simply rated worse than the one kept. The documentation says so rather than leaving a reader to infer that Abandoned means "went wrong".

Tests

Seven, and the interesting ones are the negative shapes: no kept step also appears as abandoned, no branch is reported twice, every branch leaves from an expression the search had actually reached, and expressions that were their own answer abandon nothing rather than "unknown".

One member added to the public surface, PublicApi.txt regenerated. Nothing removed:

+AngouriMath.Core.Transformations.DerivationPath.Abandoned { } : IReadOnlyList<DerivationStep>

Failed: 0, Passed: 9228 on net10.0.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura