Sigmoid Function

The problem, from first principles

A linear score can be any real number, but a binary prediction needs a value that can be read as confidence for one of two outcomes. Sigmoid converts scores to that probability-like scale while retaining their ordering. This implementation also has to survive scores too extreme for a direct exponential.

From theory to code

Implement sigmoid(z) elementwise. Theory gives the transform; the implementation first bounds the input so its exponential remains finite.

Constraints

• z may have any NumPy array shape; preserve that shape.
• Return a finite numeric array that is monotonic in z.
• Return 0.5 for zero.
• Clip z to [-500, 500] before exponentiating.
• Do not use Python loops or mutate the caller's array.

Hints

Open one at a time. Each gives away a little more than the last.