WEBVTT

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A smooth polynomial should have finite derivatives.

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In Apple’s MLX, the second derivative can be NaN.

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Sum the cumulative products of three inputs.

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The result is a + ab + abc.

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At [0, 2, 3], the first gradient is correct.

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Differentiate again: one Hessian column becomes NaN.

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The backward graph still divides by the input.

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A division-free recurrence avoids that problem in these tests.

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The prototype passes 57 scenarios and 311 checks.

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44 scenarios failed before.

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This prototype costs O(N log N) work. Large-array performance is untested.

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The prior first-gradient fix is credited. Code and sources are linked.