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The derivative of an answer at a large hyperbolic argument evaluates to NaN or loses its imaginary part #1944
Description
Found while checking Rubi 6.1.5, row 94 (1/sqrt(a + i a sinh(c + d x))) for 2.6.0. The answer is right. The library's own numerical evaluation of its derivative is not, at large arguments.
With a, c, d = 21/10, 19/10, 17/10 and x = 227283/1250 (so t = c + d x ≈ 311), the true value of the derivative is r(1 − i) with r ≈ 2.0185e-68. That is what mpmath gives at dps 400, and what the library gives when it evaluates the integrand itself.
The derivative of the answer, Differentiate(x) then substituted with exact rationals and evaluated, gives:
| digits | value |
|---|---|
| 50 | NaN |
| 300 | 1.5329…, which is wrong in magnitude too |
| 2000 | 2.0185…e-68: the real part is right and the imaginary part −2.0185…e-68 i is missing |
The relative difference at 2000 digits is exactly −1/2 + i/2. At x = 2 all three precisions agree with the integrand to the last digit.
The derivative holds e^(±t/2) and e^(±t) beside a + i a sinh(t), so large cancellations are expected. Losing a whole component at 2000 digits is not explained by that, though, and NaN at 50 digits should have been "could not evaluate". I have not narrowed it to a single subexpression.
Because of this, the Rubi harness reports this row as wrong, and anything that checks an answer by evaluating would do the same.
🤖 Generated with Claude Code
Activity
Make the rational tolerance only past the cheap check
Read a computed value as a rational only within half the working digi…
Rafael-SOWNet commentedon Oct 22, 2026
There were two causes, each fixed in its own PR.
1e-16, whatever the precision. At 2000 digits,ln(e^a + sqrt(2) e^(a/2) + 1)was exactlya, where it isa + 1.2e-67, and the derivative lost its imaginary part that way. The tolerance is now half the working digits, as it already was for a nearby integer.1e-136was0, a quotient by it was NaN (the NaN at 50 to 200 digits), and the derivative was1.53at 300 digits. Now a value is zero only as the residual of a difference, measured against its terms.The derivative at the issue's point, with both PRs:
2.0185e-68, with no imaginary part2.0185e-68 - 2.0185e-68 i1.53292.0185e-68 - 2.0185e-68 i2.0185e-68, with no imaginary part2.0185e-68 - 2.0185e-68 iWhat is left. The derivative's terms cancel in their 68th digit. At 100 digits that difference is below half of them, so it reads as exact cancellation. No absolute or relative floor can tell it apart from
sin(pi)'s residual. That needs an error bound carried through the arithmetic, which is the v3 number design in #1376, not another floor. With #2072 in, I'd close this issue and record the rest there.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Round a computed value onto zero only where its terms cancelled (#2072)