[devmotion]

Fixes #866.

For dual numbers x and y, the partials of x / y are computed as (a - (aval / bval) * b) / bval instead of inv(bval) * a - (aval / (bval * bval)) * b. With nested duals, bval is a dual number, so dividing by bval * bval divides by a dual number again, and the n-th derivative divides by y^(2^n). This overflows or underflows long before the derivatives do. For example, the third derivative of x / (x^2 + x^3) is -Inf at x = 1e-30 instead of about -6e120, and the gradient of a Laplace approximation in #866 is -Inf instead of about -386.29. Dividing by bval twice avoids the square. The same issue affects first-order derivatives when y^2 over- or underflows: the derivative of 2x / x is -Inf at x = 1e-200 and 2e-200 at x = 1e200 (instead of 0).

The change is extracted from #858, where it is unrelated to the NaN-safe mode.

Tests:

• test/DerivativeTest.jl: first derivatives of 2x / x and zero(x) / x at 1e-200, 1e-310 and 1e200 (from Enable NaN-safe mode by default #858), and third derivatives of x / (x^2 + x^3) at 1e-20, 1e-30 and 1e-40 (from Third-order derivatives of x / y are Inf or NaN for small y, e.g. in gradients of Laplace approximations #866).
• test/PartialsTest.jl: the reference expression of _div_partials (from Enable NaN-safe mode by default #858).

I bumped the version to 1.4.7.

This pull request was written with the assistance of generative AI (Claude Code).

🤖 Generated with Claude Code

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 91.17%. Comparing base (37a61e1) to head (62a8985).

Additional details and impacted files

@@           Coverage Diff           @@
##           master     #867   +/-   ##
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  Coverage   91.17%   91.17%           
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  Files          11       11           
  Lines        1077     1077           
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  Hits          982      982           
  Misses         95       95           

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