Root the reciprocal in integers for a negative-degree RootN [patch]

[matt-edmondson]

Fixes #142

Before: RootN(x, -n, d) was DivideApproximation(One, RootN(x, n, d + 2), d + 2, d). That rounds three times, and two guard digits can't absorb two of those roundings near a boundary. So RootN(3, -3, 10) returned 0.6933612743, where the correct answer is …744. The issue's sweep found 27 of 3,588 cases wrong, including some at 50 digits.

After: the negative path takes the root of the reciprocal directly in integers, as the issue's preferred fix describes:

• The new PositiveReciprocalRootN computes q = floor(10^k / s). k is chosen so that q has at least n·(d+2) digits and n divides the exponent -k - e.
• It then takes IntegerRootN(q, n). Since floor(floor(y)^(1/n)) = floor(y^(1/n)), the root is the true value truncated, and one ReduceSignificance(d) rounds it correctly.
• The result is treated as exact only when 10^k % s == 0 and root^n == q. That matches how the positive path treats exact roots.
• Negative x keeps the old behaviour: an even degree throws ArgumentOutOfRangeException, and an odd degree carries the sign. Zero still throws DivideByZeroException. A degree of -1 uses q directly, without a root.

Tests:

• TestNegativeDegreeRootRoundsOnceFromTheTrueValue pins the issue's table: RootN(3, -3, 10) == 0.6933612744, RootN(20, -2, 10) == 0.2236067977, plus (47, -2, 5), (41, -3, 5), (0.5, -3, 8), (1.5, -3, 6), and a negative odd case.
• TestNegativeDegreeRootsAgreeWithAWideReciprocalAcrossASweep covers the issue's differential sweep: k = 2..300, n ∈ {-2, -3, -5}, d ∈ {5, 10, 20, 50}. It compares each result with 1 / RootN(k, n, d+40) rounded to d. That reference goes through the positive path and Divide, so it shares no code with the path under test.
• On main, both tests fail (RootN(3, -3, 10) and RootN(47, -2, 5) are wrong). With this change both pass. Full suite: 442 passed, 0 failed.

This doesn't touch #127 (FractionalPow double rounding). That one goes through exp(y·ln x), so the integer-root approach doesn't carry over directly.

🤖 Generated with Claude Code

https://claude.ai/code/session_01VL7qukbkUpkVRArT45bBFb

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Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
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92.9% Coverage on New Code
0.0% Duplication on New Code

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