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[libc][math][c23] Implement C23 math function atanpif16 (llvm#150400)
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 llvm#132212
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‎libc/config/linux/x86_64/entrypoints.txt‎

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@@ -707,6 +707,7 @@ if(LIBC_TYPES_HAS_FLOAT16)
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libc.src.math.asinpif16
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libc.src.math.atanf16
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libc.src.math.atanhf16
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libc.src.math.atanpif16
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libc.src.math.canonicalizef16
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libc.src.math.ceilf16
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libc.src.math.copysignf16

‎libc/docs/headers/math/index.rst‎

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@@ -278,7 +278,7 @@ Higher Math Functions
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+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
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| atanh | |check| | | | |check| | | 7.12.5.3 | F.10.2.3 |
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+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
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| atanpi | | | | | | 7.12.4.10 | F.10.1.10 |
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| atanpi | | | | |check| | | 7.12.4.10 | F.10.1.10 |
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+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
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| cbrt | |check| | |check| | | | | 7.12.7.1 | F.10.4.1 |
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+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+

‎libc/include/math.yaml‎

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@@ -141,6 +141,13 @@ functions:
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arguments:
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- type: float
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- name: atanhf16
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standatds:
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- stdc
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return_type: _Float16
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arguments:
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- type: _Float16
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guard: LIBC_TYPES_HAS_FLOAT16
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- name: atanpif16
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standards:
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- stdc
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return_type: _Float16

‎libc/src/math/CMakeLists.txt‎

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@@ -73,6 +73,8 @@ add_math_entrypoint_object(atanh)
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add_math_entrypoint_object(atanhf)
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add_math_entrypoint_object(atanhf16)
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add_math_entrypoint_object(atanpif16)
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add_math_entrypoint_object(canonicalize)
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add_math_entrypoint_object(canonicalizef)
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add_math_entrypoint_object(canonicalizel)

‎libc/src/math/atanpif16.h‎

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//===-- Implementation header for atanpif16 ---------------------*- C++ -*-===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception.
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//
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//===----------------------------------------------------------------------===//
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#ifndef LLVM_LIBC_SRC_MATH_ATANPIF16_H
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#define LLVM_LIBC_SRC_MATH_ATANPIF16_H
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#include "src/__support/macros/config.h"
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#include "src/__support/macros/properties/types.h"
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namespace LIBC_NAMESPACE_DECL {
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float16 atanpif16(float16 x);
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} // namespace LIBC_NAMESPACE_DECL
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#endif // LLVM_LIBC_SRC_MATH_ASINF16_H

‎libc/src/math/generic/CMakeLists.txt‎

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@@ -4445,6 +4445,25 @@ add_entrypoint_object(
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libc.src.__support.math.atanhf16
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)
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add_entrypoint_object(
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atanpif16
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SRCS
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atanpif16.cpp
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HDRS
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../atanpif16.h
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DEPENDS
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libc.hdr.errno_macros
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libc.hdr.fenv_macros
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libc.src.__support.FPUtil.cast
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libc.src.__support.FPUtil.fenv_impl
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libc.src.__support.FPUtil.fp_bits
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libc.src.__support.FPUtil.multiply_add
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libc.src.__support.FPUtil.polyeval
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libc.src.__support.FPUtil.sqrt
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libc.src.__support.macros.optimization
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libc.src.__support.macros.properties.types
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)
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add_entrypoint_object(
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asinf
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SRCS

‎libc/src/math/generic/atanpif16.cpp‎

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//===-- Half-precision atanpi function ------------------------------------===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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#include "src/math/atanpif16.h"
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#include "hdr/errno_macros.h"
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#include "hdr/fenv_macros.h"
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#include "src/__support/FPUtil/FEnvImpl.h"
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#include "src/__support/FPUtil/FPBits.h"
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#include "src/__support/FPUtil/PolyEval.h"
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#include "src/__support/FPUtil/cast.h"
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#include "src/__support/FPUtil/multiply_add.h"
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#include "src/__support/FPUtil/sqrt.h"
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#include "src/__support/macros/optimization.h"
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namespace LIBC_NAMESPACE_DECL {
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// Using Python's SymPy library, we can obtain the polynomial approximation of
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// arctan(x)/pi. The steps are as follows:
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// >>> from sympy import *
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// >>> import math
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// >>> x = symbols('x')
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// >>> print(series(atan(x)/math.pi, x, 0, 17))
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//
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// Output:
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// 0.318309886183791*x - 0.106103295394597*x**3 + 0.0636619772367581*x**5 -
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// 0.0454728408833987*x**7 + 0.0353677651315323*x**9 - 0.0289372623803446*x**11
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// + 0.0244853758602916*x**13 - 0.0212206590789194*x**15 + O(x**17)
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//
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// We will assign this degree-15 Taylor polynomial as g(x). This polynomial
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// approximation is accurate for arctan(x)/pi when |x| is in the range [0, 0.5].
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//
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//
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// To compute arctan(x) for all real x, we divide the domain into the following
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// cases:
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//
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// * Case 1: |x| <= 0.5
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// In this range, the direct polynomial approximation is used:
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// arctan(x)/pi = sign(x) * g(|x|)
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// or equivalently, arctan(x) = sign(x) * pi * g(|x|).
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//
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// * Case 2: 0.5 < |x| <= 1
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// We use the double-angle identity for the tangent function, specifically:
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// arctan(x) = 2 * arctan(x / (1 + sqrt(1 + x^2))).
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// Applying this, we have:
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// arctan(x)/pi = sign(x) * 2 * arctan(x')/pi,
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// where x' = |x| / (1 + sqrt(1 + x^2)).
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// Thus, arctan(x)/pi = sign(x) * 2 * g(x')
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//
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// When |x| is in (0.5, 1], the value of x' will always fall within the
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// interval [0.207, 0.414], which is within the accurate range of g(x).
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//
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// * Case 3: |x| > 1
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// For values of |x| greater than 1, we use the reciprocal transformation
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// identity:
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// arctan(x) = pi/2 - arctan(1/x) for x > 0.
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// For any x (real number), this generalizes to:
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// arctan(x)/pi = sign(x) * (1/2 - arctan(1/|x|)/pi).
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// Then, using g(x) for arctan(1/|x|)/pi:
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// arctan(x)/pi = sign(x) * (1/2 - g(1/|x|)).
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//
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// Note that if 1/|x| still falls outside the
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// g(x)'s primary range of accuracy (i.e., if 0.5 < 1/|x| <= 1), the rule
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// from Case 2 must be applied recursively to 1/|x|.
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LLVM_LIBC_FUNCTION(float16, atanpif16, (float16 x)) {
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using FPBits = fputil::FPBits<float16>;
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FPBits xbits(x);
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bool is_neg = xbits.is_neg();
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auto signed_result = [is_neg](double r) -> float16 {
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return fputil::cast<float16>(is_neg ? -r : r);
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};
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if (LIBC_UNLIKELY(xbits.is_inf_or_nan())) {
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if (xbits.is_nan()) {
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if (xbits.is_signaling_nan()) {
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fputil::raise_except_if_required(FE_INVALID);
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return FPBits::quiet_nan().get_val();
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}
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return x;
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}
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// atanpi(±∞) = ±0.5
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return signed_result(0.5);
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}
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if (LIBC_UNLIKELY(xbits.is_zero()))
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return x;
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double x_abs = fputil::cast<double>(xbits.abs().get_val());
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if (LIBC_UNLIKELY(x_abs == 1.0))
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return signed_result(0.25);
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// evaluate atan(x)/pi using polynomial approximation, valid for |x| <= 0.5
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constexpr auto atanpi_eval = [](double x) -> double {
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// polynomial coefficients for atan(x)/pi taylor series
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// generated using sympy: series(atan(x)/pi, x, 0, 17)
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constexpr static double POLY_COEFFS[] = {
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0x1.45f306dc9c889p-2, // x^1: 1/pi
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-0x1.b2995e7b7b60bp-4, // x^3: -1/(3*pi)
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0x1.04c26be3b06ccp-4, // x^5: 1/(5*pi)
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-0x1.7483758e69c08p-5, // x^7: -1/(7*pi)
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0x1.21bb945252403p-5, // x^9: 1/(9*pi)
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-0x1.da1bace3cc68ep-6, // x^11: -1/(11*pi)
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0x1.912b1c2336cf2p-6, // x^13: 1/(13*pi)
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-0x1.5bade52f95e7p-6, // x^15: -1/(15*pi)
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};
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double x_sq = x * x;
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return x * fputil::polyeval(x_sq, POLY_COEFFS[0], POLY_COEFFS[1],
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POLY_COEFFS[2], POLY_COEFFS[3], POLY_COEFFS[4],
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POLY_COEFFS[5], POLY_COEFFS[6], POLY_COEFFS[7]);
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};
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// Case 1: |x| <= 0.5 - Direct polynomial evaluation
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if (LIBC_LIKELY(x_abs <= 0.5)) {
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double result = atanpi_eval(x_abs);
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return signed_result(result);
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}
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// case 2: 0.5 < |x| <= 1 - use double-angle reduction
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// atan(x) = 2 * atan(x / (1 + sqrt(1 + x^2)))
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// so atanpi(x) = 2 * atanpi(x') where x' = x / (1 + sqrt(1 + x^2))
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if (x_abs <= 1.0) {
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double x_abs_sq = x_abs * x_abs;
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double sqrt_term = fputil::sqrt<double>(1.0 + x_abs_sq);
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double x_prime = x_abs / (1.0 + sqrt_term);
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double result = 2.0 * atanpi_eval(x_prime);
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return signed_result(result);
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}
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// case 3: |x| > 1 - use reciprocal transformation
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// atan(x) = pi/2 - atan(1/x) for x > 0
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// so atanpi(x) = 1/2 - atanpi(1/x)
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double x_recip = 1.0 / x_abs;
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double result;
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// if 1/|x| > 0.5, we need to apply Case 2 transformation to 1/|x|
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if (x_recip > 0.5) {
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double x_recip_sq = x_recip * x_recip;
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double sqrt_term = fputil::sqrt<double>(1.0 + x_recip_sq);
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double x_prime = x_recip / (1.0 + sqrt_term);
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result = fputil::multiply_add(-2.0, atanpi_eval(x_prime), 0.5);
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} else {
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// direct evaluation since 1/|x| <= 0.5
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result = 0.5 - atanpi_eval(x_recip);
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}
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return signed_result(result);
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}
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} // namespace LIBC_NAMESPACE_DECL

‎libc/test/src/math/CMakeLists.txt‎

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libc.src.math.atanhf16
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)
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add_fp_unittest(
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atanpif16_test
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NEED_MPFR
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SUITE
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libc-math-unittests
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SRCS
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atanpif16_test.cpp
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DEPENDS
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libc.src.math.atanpif16
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)
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add_fp_unittest(
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fmul_test
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NEED_MPFR

‎libc/test/src/math/atanpif16_test.cpp‎

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//===-- Exhaustive test for atanpif16 -------------------------------------===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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#include "src/math/atanpif16.h"
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#include "test/UnitTest/FPMatcher.h"
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#include "test/UnitTest/Test.h"
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#include "utils/MPFRWrapper/MPFRUtils.h"
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using LlvmLibcAtanpif16Test = LIBC_NAMESPACE::testing::FPTest<float16>;
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namespace mpfr = LIBC_NAMESPACE::testing::mpfr;
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// Range: [0, Inf]
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static constexpr uint16_t POS_START = 0x0000U;
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static constexpr uint16_t POS_STOP = 0x7c00U;
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// Range: [-Inf, 0]
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static constexpr uint16_t NEG_START = 0x8000U;
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static constexpr uint16_t NEG_STOP = 0xfc00U;
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TEST_F(LlvmLibcAtanpif16Test, PositiveRange) {
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for (uint16_t v = POS_START; v <= POS_STOP; ++v) {
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float16 x = FPBits(v).get_val();
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EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Atanpi, x,
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LIBC_NAMESPACE::atanpif16(x), 0.5);
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}
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}
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TEST_F(LlvmLibcAtanpif16Test, NegativeRange) {
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for (uint16_t v = NEG_START; v <= NEG_STOP; ++v) {
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float16 x = FPBits(v).get_val();
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EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Atanpi, x,
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LIBC_NAMESPACE::atanpif16(x), 0.5);
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}
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}

‎libc/test/src/math/smoke/CMakeLists.txt‎

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@@ -4603,6 +4603,18 @@ add_fp_unittest(
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libc.src.__support.FPUtil.cast
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)
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add_fp_unittest(
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atanpif16_test
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NEED_MPFR
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SUITE
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libc-math-unittests
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SRCS
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atanpif16_test.cpp
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DEPENDS
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libc.src.math.atanpif16
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libc.src.errno.errno
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)
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add_fp_unittest(
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asinhf_test
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SUITE

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