[Rafael-SOWNet]

Over the complex numbers, the default codomain, an equation with x inside abs, sgn, conj, Re, Im or arg and outside it as well is now left unsolved. This is option 1 from #2139, and |x - 2| = |x - 3| was already left unsolved the same way (#1573).

Solve("x")	master and 2.6.0	now
x^2/abs(1-x^2) = x	{ 0, 0.618…, 1.618…, 1.618… + 1.7E-9i, 1.618… + 9.9E-8i }	{ x : x ^ 2 / abs(1 - x ^ 2) = x }
abs(x^2 - 4) = 3x	{ 1, 4, 4.0000000000000062… + 1.6E-7i }	{ x : abs(x ^ 2 - 4) = 3 * x }
x * abs(x) = 4	{ 2 }	{ x : x * abs(x) = 4 }

• Why the search can't answer these. Newton's method assumes a holomorphic function, and over the complex numbers |u| = ±u holds only where u is real, which a root needn't make it. So neither the search nor a split by sign can say it found every root, and the search's near-copies of a root came back as roots.
• The cost. The search's right answers go with the wrong ones, x * abs(x) = 4 among them.
• Over the reals each is still solved exactly, by cases, since Over the reals, a modulus of a polynomial is split at the polynomial's real roots #2178.
• Four of the six new cases fail on master. The suite, the trio, F#, AOT and the eight harnesses pass.

Fixes #2139.

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 86.77%. Comparing base (5d973db) to head (8ac2297).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2307      +/-   ##
==========================================
- Coverage   86.77%   86.77%   -0.01%     
==========================================
  Files         311      311              
  Lines       68025    68030       +5     
  Branches    19758    19762       +4     
==========================================
+ Hits        59028    59032       +4     
- Misses       4875     4876       +1     
  Partials     4122     4122              

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.32% <100.00%> (-0.01%)	⬇️
fsharp	32.97% <ø> (ø)
interactive	40.72% <ø> (ø)

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Files with missing lines	Coverage Δ
...Solvers/EquationSolver/AnalyticalEquationSolver.cs	96.29% <100.00%> (+0.05%)	⬆️
...Continuous/Solvers/EquationSolver/ModulusSolver.cs	74.68% <100.00%> (+0.99%)	⬆️

... and 2 files with indirect coverage changes

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[Happypig375]

Merely wrapping the source equation inside a conditional set isn't solving though.

[Rafael-SOWNet]

Agreed. #2309 solves them. With x = a + b i and each |u| a real t ≥ 0, the equation is a polynomial system in a, b and the ts. The Gröbner solver solves it exactly, and its real solutions are the roots. x^2/|1 - x^2| = x is { 0, (1 + sqrt(5))/2, (sqrt(5) - 1)/2 }, x |x| = 4 and x + sgn(x) = 3 are { 2 } again, and x^2 + 1 = |x| is ±i (sqrt(5) - 1)/2, which Newton didn't find.

Still left as a condition: infinitely many solutions (|x| = Re x is a ray), roots with no exact form (x^5 = |x| + 1), and arg.