`root(Number, unsigned)` never returns for some inputs (Version: 3.4.0, develop 1d7669f)

[Platonenkov]

Issue Description

The Newton–Raphson loop in root(Number f, unsigned d) (src/libxrpl/basics/Number.cpp#L1593-L1602) stops only when the new value equals the previous one or the one before it:

} while (r != rm1 && r != rm2);

For d >= 3, rounding can make the iterates cycle through three values. Neither condition then holds, and the loop runs forever. power(Number const&, unsigned, unsigned) calls root and hangs on the same inputs.

This happens under every mantissa scale and every rounding mode. Under RoundingMode::ToNearest it affects about 0.45% of random inputs with d from 2 to 6; under the other modes, about 0.04–0.1%.

No root2 hang was found in 3.2 million random inputs across all scales and rounding modes. Only tests call root(f, d) and power(f, n, d); ledger code uses root2 and power(f, n).

Steps to Reproduce

Add to src/tests/libxrpl/basics/Number.cpp:

TEST(NumberTest, root_cycle)
{
    {
        NumberMantissaScaleGuard const sg(MantissaRange::MantissaScale::Large330);
        (void)root(Number{298}, 4);        // never returns
        (void)power(Number{298}, 1, 4);    // never returns
    }
    {
        NumberMantissaScaleGuard const sg(MantissaRange::MantissaScale::Small);
        (void)root(Number{147}, 3);        // never returns
    }
}

More inputs with the default rounding mode:

• LargeLegacy, Large320 and Large330: root(Number{298}, 4), root(Number{380}, 4) and root(Number{438}, 4).
• Small: root(Number{147}, 3), root(Number{573}, 4) and root(Number{612}, 4).

Expected Result

root returns a value (about 4.154837722659222 for root(298, 4)) or throws.

Actual Result

The call does not return. These are the iterates of r for root(Number{298}, 4) under Large330, where f is scaled to 0.0298:

 1  0.422698722585301542
 2  0.415666404778574261
 3  0.4154838925963071552
 4  0.4154837722659744575
 5  0.415483772265922183
 6  0.4154837722659221832
 7  0.4154837722659221835
 8  0.415483772265922183
 9  0.4154837722659221832
10  0.4154837722659221835

The iterates under Small for root(Number{147}, 3), where f is scaled to 0.147:

 4  0.5277632087904077
 5  0.5277632087904073
 6  0.527763208790408
 7  0.5277632087904077
 8  0.5277632087904073
 9  0.527763208790408

Environment

• Number.cpp and Number.h at develop 1d7669f528ce83113c1ef2d1c8ea4675ce19157d. The loop is identical in 3.4.0.
• Ubuntu 24.04.4 LTS, g++ 13.3.0, -std=c++23 -O2.

Supporting Files

None.