[Rafael-SOWNet]

Part of #2195, item 3: a follow-up to #2213 and #2216.

At whole arguments, the compiled binomial, falling and rising factorials, double factorial and Catalan numbers are now evaluation's exact value, rounded once to a double.

The binomial took a whole lower argument as a product of ratios in doubles, (n - i)/(i + 1), and each ratio was rounded. So binomial(54, 45) came out 5317936259.999995 where evaluation gives 5317936260. Of the 2485 binomials with 0 <= k <= n < 70, 1173 were not the integer. Now it uses binomial(m, i + 1) = binomial(m, i) (m - i)/(i + 1), whose division leaves no remainder, over the shorter of k and m - k. A negative n goes through (-1)^k binomial(k - n - 1, k).

The factorial powers and the double factorial are products of whole factors, and were rounded at each step once a partial product passed 2^53. Each is now computed in doubles while every partial product is below 2^53, where every step is exact, and in EInteger past it. The result is rounded by EFloat, or by ERational for a negative order's reciprocal. A factor of 0 is checked for first, so a partial product past the range of the doubles stays past it, and the loop stops there.

The Catalan numbers (#2216) were computed exactly but rounded by (double)BigInteger, which truncates. 246 of the first 1000 came out a unit low in the last place: C_32 = 55534064877048200 compiled to 55534064877048190. They are rounded by EFloat now. Nothing else in the library rounds a BigInteger to a double for a result.

	master	now
"binomial(x, 45)".ToEntity().Compile<double, double>("x")(54)	5317936259.999995	5317936260
"binomial(x, 30)".ToEntity().Compile<double, double>("x")(60)	1.1826458156486146E+17	1.1826458156486142E+17
"catalannumber(x)".ToEntity().Compile("x").Call(32)	(55534064877048190, 0)	(55534064877048200, 0)
"x!!".ToEntity().Compile("x").Call(-301)	(2.6643404405685313E-307, 0)	(2.6643404405685293E-307, 0)

Each test compares FastExpression and the LINQ compiler with evaluation, as doubles and exactly:

• every two-argument function on a grid of 90 whole first arguments, up to 10^12, and 58 whole second arguments from -12 to 45;
• the binomial past 2^53 and past the doubles, among others at (60, 30), (1100, 550), (3000, 1500) and (-1, 4001);
• the double factorial from -301 to 320;
• the Catalan numbers from 0 to 1000.

The expected value is the exact rational, rounded once by ERational. ToNumerics goes through a decimal whose exponent has a floor, and reads 1/299!! as 0. On master's kernel, 9 of these tests fail.

The combinatorial entry in BREAKING-CHANGES.md, unreleased, now says so and gains the binomial(54, 45) row.

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 96.22642% with 2 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.65%. Comparing base (b21f383) to head (552d23a).

Files with missing lines	Patch %	Lines
...ces/AngouriMath/Numerics/CombinatorialFunctions.cs	96.22%	0 Missing and 2 partials ⚠️

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #2218   +/-   ##
=======================================
  Coverage   86.64%   86.65%           
=======================================
  Files         306      306           
  Lines       67494    67536   +42     
  Branches    19566    19587   +21     
=======================================
+ Hits        58482    58522   +40     
+ Misses       4898     4897    -1     
- Partials     4114     4117    +3     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.20% <96.22%> (+<0.01%)	⬆️
fsharp	32.97% <ø> (ø)
interactive	40.72% <ø> (ø)

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Files with missing lines	Coverage Δ
...ces/AngouriMath/Numerics/CombinatorialFunctions.cs	94.78% <96.22%> (-1.11%)	⬇️

... and 6 files with indirect coverage changes

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