[Rafael-SOWNet]

Part of #2195, item 3: lambertw compiles. The PR also fixes a defect in #2202's evaluation that compiling it brought to light.

Compiling. Numerics/LambertWFunction is a double-precision kernel on the interpreter's branches, Corless et al.'s, as #2202 has them:

• a start, then Halley's iteration;
• the imaginary part held at 0 where the branch is real;
• -1 within rounding of the branch point, and NaN on a branch that is not whole.

FastExpression has an instruction for it, and LINQ has Complex and double definitions; over double a complex value is NaN. Evaluation now takes its double-precision start from the same kernel, which replaces InDoublePrecision.

The defect. Comparing the kernel with mpmath found evaluation wrong as well, because the two shared one start:

• W_0 on the real axis from -1 to -2/e was real. The start there was ln(1 + z), which is real where W_0 is not, so the iteration never left the axis: W_0(-0.975) was -6.61, where mpmath gives -0.3358 + 1.3221i.
• About -1, the same start is near ln 0, and the iteration reached another branch: W_0(-1.02) and W_0(-1.349 - 0.358i) came back as conjugates.

#2202's 700-point check against mpmath missed it, because random complex points almost never lie on the axis.

The starts now follow SciPy's lambertw regions on the principal branch:

• the branch-point series within 0.3 of -1/e;
• ln(1 + z) in a box right of -1 about the axis;
• the asymptotic start elsewhere.

The other branches are unchanged. lambertw is new since 2.6.0, so there is no 2.6.0 row; its entry says it compiles.

Checked against mpmath (dps 30) at 11,501 points, evaluated and compiled alike: the real axis from -5 to 0 in steps of 0.01 on branches -1, 0 and 1, and 10,000 random points on branches -3 to 3, a third of them near -1 and a tenth far out. All agree to 1e-9.

Tests:

• LambertWCompilationTest compares both compilers with evaluation on branches -2, -1, 0, 1, 3 and 1/2, at 19 points: the branch point, either side of it, the negative axis, far out, and off the axis.
• LambertWTest gains five mpmath rows at -0.8, -0.975, -1, -1.02 and -1.349 - 0.358i.
• Full suite 17940/0, interactive trio, AOT smoke 69/0, all frameworks build.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[codecov]

Codecov Report

❌ Patch coverage is 94.54545% with 3 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.58%. Comparing base (9164e15) to head (d580573).

Files with missing lines	Patch %	Lines
Sources/AngouriMath/Numerics/LambertWFunction.cs	95.12%	1 Missing and 1 partial ⚠️
...gouriMath/Functions/Compilation/IntoFE/Compiler.cs	80.00%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2214      +/-   ##
==========================================
- Coverage   86.59%   86.58%   -0.01%     
==========================================
  Files         303      304       +1     
  Lines       67168    67207      +39     
  Branches    19495    19509      +14     
==========================================
+ Hits        58162    58194      +32     
- Misses       4903     4907       +4     
- Partials     4103     4106       +3     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.14% <94.54%> (-0.01%)	⬇️
fsharp	32.97% <ø> (ø)
interactive	40.72% <ø> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

Files with missing lines	Coverage Δ
...ouriMath/Core/Entity/Continuous/Number/LambertW.cs	83.33% <100.00%> (-3.63%)	⬇️
...ath/Functions/Compilation/IntoFE/FastExpression.cs	96.22% <100.00%> (+0.04%)	⬆️
...ctions/Compilation/IntoLinq/CompilationProtocol.cs	90.77% <100.00%> (+0.05%)	⬆️
...h/Functions/Compilation/IntoLinq/MathAllMethods.cs	70.45% <100.00%> (+0.33%)	⬆️
...gouriMath/Functions/Compilation/IntoFE/Compiler.cs	86.75% <80.00%> (+0.21%)	⬆️
Sources/AngouriMath/Numerics/LambertWFunction.cs	95.12% <95.12%> (ø)

... and 8 files with indirect coverage changes

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