[Rafael-SOWNet]

Part of #1944. Follows #2066.

The defect. A number an evaluation produced was rounded onto zero below half the working digits outright, 1e-50 at the default hundred, whatever had produced it. On 2.6.0:

• e^(-155) was 0 at a hundred digits.
• The modulus of a complex number near 1e-136 was 0, and a quotient by that modulus was NaN.
• The derivative of a right antiderivative of 1/sqrt(a + i a sinh(c + d x)) at x = 227283/1250 was NaN at up to two hundred digits. At three hundred digits it was 1.5329…, where it is 2.0185e-68 (1 - i).

The change. A residual of cancellation is a difference below half the working digits of its terms. That is now the only value rounded onto zero by size.

• Sums. A sum of opposite signs is measured against one of its terms.
• Operations that cancel nothing. A product, a quotient, a power, a root, an exponential or a modulus is never rounded onto zero.
• Cosines and sines. Their residual is measured against the modulus it multiplies.
• Unchanged. sin(pi), e^(i pi) + 1 and sqrt(2) sqrt(3) - sqrt(6) are still 0. Incorrect work with large values in the matrix #561's matrix times its inverse is still the identity.
• Division. A quotient of decimals is divided to the working precision's places, as before, and divided again to its digits where that lost them. To places, 3.5e-68 kept thirty-three digits, and a quotient below 1e-100 was 0.

What it does not do. The derivative above is right from two hundred digits. Its terms cancel in their sixty-eighth digit, so below about twice that many digits the difference is under half of them and reads as a residual. At a hundred digits it comes out as the real part without the imaginary one, where it was NaN. An evaluation that cancels more digits than half the working precision cannot be told apart from exact cancellation without carrying an error bound through the arithmetic.

Allocation. It matches master's: 12,731 B/op against 12,743 on the kernel gate's polynomial at a hundred digits, and 12,077 against 12,077 at fifteen. The scale is either term, which costs nothing, where comparing the two rescaled a mantissa. A quotient is divided again only when it lost digits, where dividing every quotient to digits cost 680 bytes more each.

AResidualIsZeroOnlyWhereItCancelledTest, against mpmath, covers:

• small products, powers and exponentials at a hundred digits;
• a small quotient with its imaginary part;
• the root of a small negative;
• residuals that are still exact;
• the derivative at 200 and 300 digits;
• the integrand at 50 and 100 digits.

Results:

• Unit suite: passes (16,765).
• Corpus: the full Rubi corpus, run against the 2.6.0 run on the same harness (c614b2c8), is 48,305 of 48,305, with no row moved and no answer's allocation grown.
• Build: the library builds for netstandard2.0.
• BREAKING-CHANGES: the entry gives the values on 2.6.0 and now.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura