[Rafael-SOWNet]

Part of #2195, item 3.

mobius, divisorsigma, primenu, primeomega, valuation, prime and primepi compile in FastExpression and in the LINQ compiler, over Complex and over double. prime and valuation were refused with UncompilableNodeException. The other five are new since 2.6.0.

Each uses the routine the interpreter uses:

function	routine
mobius, primenu, primeomega, divisorsigma	Primes.Factorization
valuation (to check that the base is prime)	IsPrime
prime	GetPrime
primepi	CountUpTo

So a compiled value is exact at a whole argument and NaN at any other, as evaluation gives. Two inputs the interpreter leaves as written are NaN when compiled: a number too large to factorise, and a prime past the millionth.

	2.6.0	now
"prime(x)".ToEntity().Compile("x").Call(10)	UncompilableNodeException	(29, 0)
"valuation(x, 2)".ToEntity().Compile("x").Call(12)	UncompilableNodeException	(2, 0)

NumberTheoreticCompilationTest compares both compilers with evaluation:

• each function of a factorisation at 68 arguments, among them 2^40, 600851475143, 5/2 and 1 + i;
• the divisor sigma at six orders, among them -1, 1/2 and i;
• the valuation at seven numbers and six bases, two of the bases not prime;
• the n-th prime up to the 10000th;
• the prime count up to 100000.

gcd and lcm are not in this PR. Evaluation takes them over the rationals, and every finite double is a rational, so a compiled gcd at 0.1 would be exact for the double nearest 0.1 and meaningless for 0.1. Whether they should compile only at whole numbers, or stay refused, is a separate decision.

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[codecov]

Codecov Report

❌ Patch coverage is 93.02326% with 9 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.64%. Comparing base (8f410b8) to head (c32ee42).

Files with missing lines	Patch %	Lines
...gouriMath/Functions/Compilation/IntoFE/Compiler.cs	76.66%	7 Missing ⚠️
...s/AngouriMath/Numerics/NumberTheoreticFunctions.cs	95.65%	0 Missing and 2 partials ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2217      +/-   ##
==========================================
+ Coverage   86.62%   86.64%   +0.01%     
==========================================
  Files         305      306       +1     
  Lines       67365    67494     +129     
  Branches    19550    19566      +16     
==========================================
+ Hits        58357    58480     +123     
- Misses       4897     4901       +4     
- Partials     4111     4113       +2     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.19% <93.02%> (+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
...ath/Functions/Compilation/IntoFE/FastExpression.cs	96.64% <100.00%> (+0.28%)	⬆️
...ctions/Compilation/IntoLinq/CompilationProtocol.cs	90.08% <100.00%> (-0.83%)	⬇️
...h/Functions/Compilation/IntoLinq/MathAllMethods.cs	73.46% <100.00%> (+2.04%)	⬆️
...s/AngouriMath/Numerics/NumberTheoreticFunctions.cs	95.65% <95.65%> (ø)
...gouriMath/Functions/Compilation/IntoFE/Compiler.cs	85.31% <76.66%> (-1.09%)	⬇️

... and 9 files with indirect coverage changes

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