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This PR implements `atanpif16(x)` which computes
$\frac{\arctan(x)}{\pi}$ for half-precision floating-point numbers using
polynomial approximation with domain reduction.
## Mathematical Implementation
The implementation uses a 15th-degree Taylor polynomial expansion of
$\frac{\arctan(x)}{\pi}$ that's computed using
[`python-sympy`](https://www.sympy.org/en/index.html) and it's accurate
in $|x| \in [0, 0.5)$:
$$
g(x) = \frac{\arctan(x)}{\pi} \approx
\begin{aligned}[t]
& 0.318309886183791x \\
& - 0.106103295394597x^3 \\
& + 0.0636619772367581x^5 \\
& - 0.0454728408833987x^7 \\
& + 0.0353677651315323x^9 \\
& - 0.0289372623803446x^{11} \\
& + 0.0244853758602916x^{13} \\
& - 0.0212206590789194x^{15} + O(x^{17})
\end{aligned}
$$
---
To ensure accuracy across all real inputs, the domain is divided into
three cases with appropriate transformations:
**Case 1: $|x| \leq 0.5$**
Direct polynomial evaluation:
$$\text{atanpi}(x) = \text{sign}(x) \cdot g(|x|)$$
**Case 2: $0.5 < |x| \leq 1$**
Double-angle reduction using:
$$\arctan(x) = 2\arctan\left(\frac{x}{1 + \sqrt{1 + x^2}}\right)$$
$$\text{atanpi}(x) = \text{sign}(x) \cdot 2g\left(\frac{|x|}{1 + \sqrt{1
+ x^2}}\right)$$
**Case 3: $|x| > 1$**
Reciprocal transformation using
$$\arctan(x) = \frac{\pi}{2} - \arctan\left(\frac{1}{x}\right) \
\text{for} \ x \gt 0$$
$$\text{atanpi}(x) = \text{sign}(x) \cdot \left(\frac{1}{2} -
g\left(\frac{1}{|x|}\right)\right)$$
Closes #132212