Add up two arctangents, and name the angles arctan knows

[Rafael-SOWNet]

"arctan(1/2) + arctan(1/3)".Simplify()   // master: arctan(1/2) + arctan(1/3)
"arctan(1)".Simplify()                   // master: arctan(1)

It is pi/4, and neither half of that was reachable.

The addition formula

The tangent one read backwards: arctan(a) + arctan(b) = arctan((a + b)/(1 − ab)).

It holds as written only while ab < 1. Past that the sum leaves the range arctan answers in and the identity is off by a whole pi — arctan(2) + arctan(3) is 3pi/4, not arctan(−1). Both arguments are required to be numbers, which makes ab < 1 a question with an answer; the pairs past the boundary are left alone rather than answered wrongly.

arctan(1/2) + arctan(1/3)  →  pi/4
arctan(1/5) + arctan(1/8)  →  arctan(1/3)
arctan(2)   + arctan(3)    →  unchanged

Why arctan(1) is not a simplification candidate

The two rational values are handled where arctan is simplified, beside the values at plus and minus infinity that were already there — not in the pattern pool.

A candidate offered to the simplifier has to win on node count, and pi / 4 does not beat arctan(1). So a sum that came all the way to arctan(1) stopped one step short of the answer. The two irrational ones, arctan(sqrt(3)) and arctan(1/sqrt(3)), do win their contest and stay with the other trigonometric patterns.

Verification

Checked over 289 pairs either side of the boundary, negatives included: whatever a sum of two arctangents simplifies to is the same number it was. That check is in the tests.

17 tests, 8 of which fail without the change. 4026 unit tests and 127 F# tests on both target frameworks, none failing. 117-problem self-verifying corpus at 102/117 — simp:inverse-trig goes from 50% to 100% — with nothing wrong, in error or timing out. 1320 property checks over 151 expressions, none failing.

[codecov-commenter]

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Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 80.47%. Comparing base (90c00a8) to head (0738f2a).
⚠️ Report is 58 commits behind head on master.
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Additional details and impacted files

@@            Coverage Diff             @@
##           master     #692      +/-   ##
==========================================
- Coverage   80.99%   80.47%   -0.53%     
==========================================
  Files         155      156       +1     
  Lines       13687    12938     -749     
  Branches     1957     2123     +166     
==========================================
- Hits        11086    10412     -674     
+ Misses       1990     1919      -71     
+ Partials      611      607       -4     

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