[Rafael-SOWNet]

Part of #2195, item 3: the binomial coefficient, the falling and the rising factorial and the double factorial compile, in FastExpression and in the LINQ compiler.

Each was refused by both. They compile now in double precision, in Numerics/CombinatorialFunctions, on the interpreter's terms:

• A whole order is a product. For a negative order it is the reciprocal of one, with NaN where a factor of it vanishes.
• Any other order goes through the gamma function, with NaN at a pole of the numerator and 0 at one of the denominator, as Evaluate a binomial with a whole upper argument and a fractional lower one #2211 and A binomial at a pole of its denominator is zero #2212 made evaluation do.
• The double factorial has no value off the whole numbers, nor at a negative even one.

FastExpression has an instruction for each. The LINQ compiler has a Complex and a double definition in the generated MathAllMethods, widened to double first as the special functions are.

	2.6.0	now
binomial(x, 2), Compile("x").Call(5)	UncompilableNodeException	10
binomial(x, 1/2), Compile<double, double> at 3	UncompilableNodeException	2.037183271576257

fallingfactorial, risingfactorial and n!! are new since 2.6.0, and compile from the start.

Tests: CombinatorialCompilationTest compiles each of the three two-argument functions three ways: FastExpression, LINQ over Complex, and LINQ over double for real values. Each is compared with evaluation at every pair of 14 arguments: whole ones from -3 to 5, halves, 9/4, 1 + i and i/2. A pair evaluation leaves unevaluated is counted rather than claimed, and at least three quarters of the pairs are compared. n!! is compared the same way. UserInvalidExceptions pinned binomial(x, 2) as having no compiled form, and pins polygamma(1, x) instead. Full suite 17933/0 after that change, interactive trio, AOT smoke 68/0, all frameworks build.

Still open on #2195: gamma(a, z), polygamma and the newer special functions, the integer functions, and item 4.

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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[codecov]

Codecov Report

❌ Patch coverage is 92.66055% with 8 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.60%. Comparing base (e59e609) to head (117dfc3).

Files with missing lines	Patch %	Lines
...gouriMath/Functions/Compilation/IntoFE/Compiler.cs	78.94%	4 Missing ⚠️
...ces/AngouriMath/Numerics/CombinatorialFunctions.cs	94.73%	1 Missing and 2 partials ⚠️
...h/Functions/Compilation/IntoLinq/MathAllMethods.cs	93.75%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2213      +/-   ##
==========================================
+ Coverage   86.57%   86.60%   +0.02%     
==========================================
  Files         302      303       +1     
  Lines       67059    67168     +109     
  Branches    19472    19495      +23     
==========================================
+ Hits        58055    58168     +113     
+ Misses       4899     4895       -4     
  Partials     4105     4105              

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.15% <92.66%> (+0.02%)	⬆️
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
...ath/Functions/Compilation/IntoFE/FastExpression.cs	96.17% <100.00%> (+0.20%)	⬆️
...ctions/Compilation/IntoLinq/CompilationProtocol.cs	90.72% <100.00%> (+0.24%)	⬆️
...h/Functions/Compilation/IntoLinq/MathAllMethods.cs	70.11% <93.75%> (+1.13%)	⬆️
...ces/AngouriMath/Numerics/CombinatorialFunctions.cs	94.73% <94.73%> (ø)
...gouriMath/Functions/Compilation/IntoFE/Compiler.cs	86.53% <78.94%> (-0.50%)	⬇️

... and 9 files with indirect coverage changes

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