Make x^y with Dual arguments with zero partials consistent with Real arguments

[devmotion]

Mathematically, a dual number with zero partials is just a regular number: Dual(v, 0) and v represent the same quantity. Hence derivative rules should return the same result regardless of whether an unperturbed argument is a Real or a Dual with zero partials.

Currently, this is not the case for ^ and NaNMath.pow if both arguments are Duals with the same tag:

julia> using ForwardDiff: Dual

julia> Dual(0.0, 0.0)^Dual(0.5, 1.0)  # same as `0.0^Dual(0.5, 1.0) == Dual(0.0, 0.0)`
Dual{Nothing}(0.0,NaN)

julia> Dual(0.0, 1.0)^Dual(0.0, 0.0)  # same as `Dual(0.0, 1.0)^0.0 == Dual(1.0, 0.0)`
Dual{Nothing}(1.0,NaN)

The second example returns NaN also in NaN-safe mode: there the NaN is not caused by a zero partial but by powval = 0 * 0^(-1).

This PR fixes both cases: if one of the arguments has zero partials, the Dual^Dual branch now defers to the Dual^Real or Real^Dual implementation (moved to the helper functions _pow_dual_base and _pow_dual_exponent, which are called directly: calling ^ again with value(y) would mix up the nesting levels if value(y) is itself a Dual with the same tag). Hence a Dual with zero partials gives exactly the same result as a Real, including the sign of zero partials (_mul_partials turned -0.0 partials into 0.0 with NaN-safe mode disabled). For consistency, the check in the Dual^Real branch is now also written as iszero(y) || isconstant(x) instead of y == zero(y) || iszero(partials(x)) (which is equivalent).

Additionally, the Dual^Real branch now computes the derivative before scaling the partials, i.e., (y * x^(y - 1)) * partials(x) instead of partials(x) * y * x^(y - 1). This rounds the same way as the Dual^Dual branch (the old version differed in the last bit for, e.g., Dual(1.3, 0.7)^2.7), and it scales the partials once instead of twice. As a consequence, results of x^y with a Real exponent may change in the last bit.

The zero partials in the Dual^Real branch now have the same type as the non-zero partials. Otherwise deferring to it would make, e.g., Dual{T,Float64}^Dual{T,BigInt} type-unstable (Float64 vs BigFloat partials), which it is not on master. As a side effect, Dual{T,Float64}^BigInt is now type-stable as well.

The new tests check with isequal that replacing an argument with a zero-partials Dual does not change the result.

With this PR, the derivatives at the special points agree with the Real case, SymPy, and WolframAlpha (default / NaN-safe mode where they differ):

	SymPy	WolframAlpha	master, Real	master, zero-partials Dual	this PR, zero-partials Dual
∂x^y/∂x at (0, 0)	0	0	0	NaN / NaN	0
∂x^y/∂x at (0, 0.5)	∞	∞	Inf	Inf	Inf
∂x^y/∂x at (0, -0.5)	-∞	-∞	-Inf	-Inf	-Inf
∂x^y/∂x at (2, 0)	0	0	0	0	0
∂x^y/∂y at (0, 0)	-∞	-∞	-Inf	NaN / -Inf	-Inf
∂x^y/∂y at (0, 0.5)	0	0	0	NaN / 0	0
∂x^y/∂y at (0, 1.5)	0	0	0	0	0
∂x^y/∂y at (0, -0.5)	-∞	-∞	-Inf	NaN / -Inf	-Inf

For x = 0, the SymPy and WolframAlpha values are the limits x -> 0+ of y * x^(y - 1) and x^y * log(x) (at (0, 0), ∂x^y/∂x is d/dx x^0 = 0). At (0, 0), ∂x^y/∂y = -∞ is also the limit of the one-sided difference quotients of y -> 0^y.

Note that this PR only makes the rules consistent for Duals whose partials are all zero. A Dual with only some zero partials still propagates NaNs in these directions in the default mode, so results depend on the chunk size, also with the existing check in the Dual^Real branch:

julia> f(v) = v[1]^0.5 + v[2];

julia> ForwardDiff.gradient(f, [0.0, 1.0], ForwardDiff.GradientConfig(f, [0.0, 1.0], ForwardDiff.Chunk{1}()))
2-element Vector{Float64}:
 Inf
   1.0

julia> ForwardDiff.gradient(f, [0.0, 1.0], ForwardDiff.GradientConfig(f, [0.0, 1.0], ForwardDiff.Chunk{2}()))
2-element Vector{Float64}:
 Inf
 NaN

That's not specific to ^ (e.g., sqrt(v[1]) + v[2] returns [Inf, NaN] for any chunk size), and only NaN-safe mode handles it consistently. See #857.

🤖 Generated with Claude Code

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Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 91.23%. Comparing base (a3c0f4f) to head (571ca6a).

Additional details and impacted files

@@           Coverage Diff           @@
##           master     #856   +/-   ##
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  Coverage   91.23%   91.23%           
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