Skip to content

Commit c1d1e0e

Browse files
hulxvhulxv
authored
[libc][math][c23] Implement C23 math function atanpif16 (#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 #132212
1 parent 51163c5 commit c1d1e0e

File tree

15 files changed

+354
-1
lines changed

15 files changed

+354
-1
lines changed

‎libc/config/linux/x86_64/entrypoints.txt

Lines changed: 1 addition & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -707,6 +707,7 @@ if(LIBC_TYPES_HAS_FLOAT16)
707707
libc.src.math.asinpif16
708708
libc.src.math.atanf16
709709
libc.src.math.atanhf16
710+
libc.src.math.atanpif16
710711
libc.src.math.canonicalizef16
711712
libc.src.math.ceilf16
712713
libc.src.math.copysignf16

‎libc/docs/headers/math/index.rst

Lines changed: 1 addition & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -278,7 +278,7 @@ Higher Math Functions
278278
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
279279
| atanh | |check| | | | |check| | | 7.12.5.3 | F.10.2.3 |
280280
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
281-
| atanpi | | | | | | 7.12.4.10 | F.10.1.10 |
281+
| atanpi | | | | |check| | | 7.12.4.10 | F.10.1.10 |
282282
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
283283
| cbrt | |check| | |check| | | | | 7.12.7.1 | F.10.4.1 |
284284
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+

‎libc/include/math.yaml

Lines changed: 7 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -141,6 +141,13 @@ functions:
141141
arguments:
142142
- type: float
143143
- name: atanhf16
144+
standatds:
145+
- stdc
146+
return_type: _Float16
147+
arguments:
148+
- type: _Float16
149+
guard: LIBC_TYPES_HAS_FLOAT16
150+
- name: atanpif16
144151
standards:
145152
- stdc
146153
return_type: _Float16

‎libc/src/math/CMakeLists.txt

Lines changed: 2 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -73,6 +73,8 @@ add_math_entrypoint_object(atanh)
7373
add_math_entrypoint_object(atanhf)
7474
add_math_entrypoint_object(atanhf16)
7575

76+
add_math_entrypoint_object(atanpif16)
77+
7678
add_math_entrypoint_object(canonicalize)
7779
add_math_entrypoint_object(canonicalizef)
7880
add_math_entrypoint_object(canonicalizel)

‎libc/src/math/atanpif16.h

Lines changed: 21 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,21 @@
1+
//===-- Implementation header for atanpif16 ---------------------*- C++ -*-===//
2+
//
3+
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4+
// See https://llvm.org/LICENSE.txt for license information.
5+
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception.
6+
//
7+
//===----------------------------------------------------------------------===//
8+
9+
#ifndef LLVM_LIBC_SRC_MATH_ATANPIF16_H
10+
#define LLVM_LIBC_SRC_MATH_ATANPIF16_H
11+
12+
#include "src/__support/macros/config.h"
13+
#include "src/__support/macros/properties/types.h"
14+
15+
namespace LIBC_NAMESPACE_DECL {
16+
17+
float16 atanpif16(float16 x);
18+
19+
} // namespace LIBC_NAMESPACE_DECL
20+
21+
#endif // LLVM_LIBC_SRC_MATH_ASINF16_H

‎libc/src/math/generic/CMakeLists.txt

Lines changed: 19 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -4445,6 +4445,25 @@ add_entrypoint_object(
44454445
libc.src.__support.math.atanhf16
44464446
)
44474447

4448+
add_entrypoint_object(
4449+
atanpif16
4450+
SRCS
4451+
atanpif16.cpp
4452+
HDRS
4453+
../atanpif16.h
4454+
DEPENDS
4455+
libc.hdr.errno_macros
4456+
libc.hdr.fenv_macros
4457+
libc.src.__support.FPUtil.cast
4458+
libc.src.__support.FPUtil.fenv_impl
4459+
libc.src.__support.FPUtil.fp_bits
4460+
libc.src.__support.FPUtil.multiply_add
4461+
libc.src.__support.FPUtil.polyeval
4462+
libc.src.__support.FPUtil.sqrt
4463+
libc.src.__support.macros.optimization
4464+
libc.src.__support.macros.properties.types
4465+
)
4466+
44484467
add_entrypoint_object(
44494468
asinf
44504469
SRCS

‎libc/src/math/generic/atanpif16.cpp

Lines changed: 157 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,157 @@
1+
//===-- Half-precision atanpi function ------------------------------------===//
2+
//
3+
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4+
// See https://llvm.org/LICENSE.txt for license information.
5+
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6+
//
7+
//===----------------------------------------------------------------------===//
8+
9+
#include "src/math/atanpif16.h"
10+
#include "hdr/errno_macros.h"
11+
#include "hdr/fenv_macros.h"
12+
#include "src/__support/FPUtil/FEnvImpl.h"
13+
#include "src/__support/FPUtil/FPBits.h"
14+
#include "src/__support/FPUtil/PolyEval.h"
15+
#include "src/__support/FPUtil/cast.h"
16+
#include "src/__support/FPUtil/multiply_add.h"
17+
#include "src/__support/FPUtil/sqrt.h"
18+
#include "src/__support/macros/optimization.h"
19+
20+
namespace LIBC_NAMESPACE_DECL {
21+
22+
// Using Python's SymPy library, we can obtain the polynomial approximation of
23+
// arctan(x)/pi. The steps are as follows:
24+
// >>> from sympy import *
25+
// >>> import math
26+
// >>> x = symbols('x')
27+
// >>> print(series(atan(x)/math.pi, x, 0, 17))
28+
//
29+
// Output:
30+
// 0.318309886183791*x - 0.106103295394597*x**3 + 0.0636619772367581*x**5 -
31+
// 0.0454728408833987*x**7 + 0.0353677651315323*x**9 - 0.0289372623803446*x**11
32+
// + 0.0244853758602916*x**13 - 0.0212206590789194*x**15 + O(x**17)
33+
//
34+
// We will assign this degree-15 Taylor polynomial as g(x). This polynomial
35+
// approximation is accurate for arctan(x)/pi when |x| is in the range [0, 0.5].
36+
//
37+
//
38+
// To compute arctan(x) for all real x, we divide the domain into the following
39+
// cases:
40+
//
41+
// * Case 1: |x| <= 0.5
42+
// In this range, the direct polynomial approximation is used:
43+
// arctan(x)/pi = sign(x) * g(|x|)
44+
// or equivalently, arctan(x) = sign(x) * pi * g(|x|).
45+
//
46+
// * Case 2: 0.5 < |x| <= 1
47+
// We use the double-angle identity for the tangent function, specifically:
48+
// arctan(x) = 2 * arctan(x / (1 + sqrt(1 + x^2))).
49+
// Applying this, we have:
50+
// arctan(x)/pi = sign(x) * 2 * arctan(x')/pi,
51+
// where x' = |x| / (1 + sqrt(1 + x^2)).
52+
// Thus, arctan(x)/pi = sign(x) * 2 * g(x')
53+
//
54+
// When |x| is in (0.5, 1], the value of x' will always fall within the
55+
// interval [0.207, 0.414], which is within the accurate range of g(x).
56+
//
57+
// * Case 3: |x| > 1
58+
// For values of |x| greater than 1, we use the reciprocal transformation
59+
// identity:
60+
// arctan(x) = pi/2 - arctan(1/x) for x > 0.
61+
// For any x (real number), this generalizes to:
62+
// arctan(x)/pi = sign(x) * (1/2 - arctan(1/|x|)/pi).
63+
// Then, using g(x) for arctan(1/|x|)/pi:
64+
// arctan(x)/pi = sign(x) * (1/2 - g(1/|x|)).
65+
//
66+
// Note that if 1/|x| still falls outside the
67+
// g(x)'s primary range of accuracy (i.e., if 0.5 < 1/|x| <= 1), the rule
68+
// from Case 2 must be applied recursively to 1/|x|.
69+
70+
LLVM_LIBC_FUNCTION(float16, atanpif16, (float16 x)) {
71+
using FPBits = fputil::FPBits<float16>;
72+
73+
FPBits xbits(x);
74+
bool is_neg = xbits.is_neg();
75+
76+
auto signed_result = [is_neg](double r) -> float16 {
77+
return fputil::cast<float16>(is_neg ? -r : r);
78+
};
79+
80+
if (LIBC_UNLIKELY(xbits.is_inf_or_nan())) {
81+
if (xbits.is_nan()) {
82+
if (xbits.is_signaling_nan()) {
83+
fputil::raise_except_if_required(FE_INVALID);
84+
return FPBits::quiet_nan().get_val();
85+
}
86+
return x;
87+
}
88+
// atanpi(±∞) = ±0.5
89+
return signed_result(0.5);
90+
}
91+
92+
if (LIBC_UNLIKELY(xbits.is_zero()))
93+
return x;
94+
95+
double x_abs = fputil::cast<double>(xbits.abs().get_val());
96+
97+
if (LIBC_UNLIKELY(x_abs == 1.0))
98+
return signed_result(0.25);
99+
100+
// evaluate atan(x)/pi using polynomial approximation, valid for |x| <= 0.5
101+
constexpr auto atanpi_eval = [](double x) -> double {
102+
// polynomial coefficients for atan(x)/pi taylor series
103+
// generated using sympy: series(atan(x)/pi, x, 0, 17)
104+
constexpr static double POLY_COEFFS[] = {
105+
0x1.45f306dc9c889p-2, // x^1: 1/pi
106+
-0x1.b2995e7b7b60bp-4, // x^3: -1/(3*pi)
107+
0x1.04c26be3b06ccp-4, // x^5: 1/(5*pi)
108+
-0x1.7483758e69c08p-5, // x^7: -1/(7*pi)
109+
0x1.21bb945252403p-5, // x^9: 1/(9*pi)
110+
-0x1.da1bace3cc68ep-6, // x^11: -1/(11*pi)
111+
0x1.912b1c2336cf2p-6, // x^13: 1/(13*pi)
112+
-0x1.5bade52f95e7p-6, // x^15: -1/(15*pi)
113+
};
114+
double x_sq = x * x;
115+
return x * fputil::polyeval(x_sq, POLY_COEFFS[0], POLY_COEFFS[1],
116+
POLY_COEFFS[2], POLY_COEFFS[3], POLY_COEFFS[4],
117+
POLY_COEFFS[5], POLY_COEFFS[6], POLY_COEFFS[7]);
118+
};
119+
120+
// Case 1: |x| <= 0.5 - Direct polynomial evaluation
121+
if (LIBC_LIKELY(x_abs <= 0.5)) {
122+
double result = atanpi_eval(x_abs);
123+
return signed_result(result);
124+
}
125+
126+
// case 2: 0.5 < |x| <= 1 - use double-angle reduction
127+
// atan(x) = 2 * atan(x / (1 + sqrt(1 + x^2)))
128+
// so atanpi(x) = 2 * atanpi(x') where x' = x / (1 + sqrt(1 + x^2))
129+
if (x_abs <= 1.0) {
130+
double x_abs_sq = x_abs * x_abs;
131+
double sqrt_term = fputil::sqrt<double>(1.0 + x_abs_sq);
132+
double x_prime = x_abs / (1.0 + sqrt_term);
133+
double result = 2.0 * atanpi_eval(x_prime);
134+
return signed_result(result);
135+
}
136+
137+
// case 3: |x| > 1 - use reciprocal transformation
138+
// atan(x) = pi/2 - atan(1/x) for x > 0
139+
// so atanpi(x) = 1/2 - atanpi(1/x)
140+
double x_recip = 1.0 / x_abs;
141+
double result;
142+
143+
// if 1/|x| > 0.5, we need to apply Case 2 transformation to 1/|x|
144+
if (x_recip > 0.5) {
145+
double x_recip_sq = x_recip * x_recip;
146+
double sqrt_term = fputil::sqrt<double>(1.0 + x_recip_sq);
147+
double x_prime = x_recip / (1.0 + sqrt_term);
148+
result = fputil::multiply_add(-2.0, atanpi_eval(x_prime), 0.5);
149+
} else {
150+
// direct evaluation since 1/|x| <= 0.5
151+
result = 0.5 - atanpi_eval(x_recip);
152+
}
153+
154+
return signed_result(result);
155+
}
156+
157+
} // namespace LIBC_NAMESPACE_DECL

‎libc/test/src/math/CMakeLists.txt

Lines changed: 11 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -2275,6 +2275,17 @@ add_fp_unittest(
22752275
libc.src.math.atanhf16
22762276
)
22772277

2278+
add_fp_unittest(
2279+
atanpif16_test
2280+
NEED_MPFR
2281+
SUITE
2282+
libc-math-unittests
2283+
SRCS
2284+
atanpif16_test.cpp
2285+
DEPENDS
2286+
libc.src.math.atanpif16
2287+
)
2288+
22782289
add_fp_unittest(
22792290
fmul_test
22802291
NEED_MPFR

‎libc/test/src/math/atanpif16_test.cpp

Lines changed: 40 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,40 @@
1+
//===-- Exhaustive test for atanpif16 -------------------------------------===//
2+
//
3+
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4+
// See https://llvm.org/LICENSE.txt for license information.
5+
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6+
//
7+
//===----------------------------------------------------------------------===//
8+
9+
#include "src/math/atanpif16.h"
10+
#include "test/UnitTest/FPMatcher.h"
11+
#include "test/UnitTest/Test.h"
12+
#include "utils/MPFRWrapper/MPFRUtils.h"
13+
14+
using LlvmLibcAtanpif16Test = LIBC_NAMESPACE::testing::FPTest<float16>;
15+
16+
namespace mpfr = LIBC_NAMESPACE::testing::mpfr;
17+
18+
// Range: [0, Inf]
19+
static constexpr uint16_t POS_START = 0x0000U;
20+
static constexpr uint16_t POS_STOP = 0x7c00U;
21+
22+
// Range: [-Inf, 0]
23+
static constexpr uint16_t NEG_START = 0x8000U;
24+
static constexpr uint16_t NEG_STOP = 0xfc00U;
25+
26+
TEST_F(LlvmLibcAtanpif16Test, PositiveRange) {
27+
for (uint16_t v = POS_START; v <= POS_STOP; ++v) {
28+
float16 x = FPBits(v).get_val();
29+
EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Atanpi, x,
30+
LIBC_NAMESPACE::atanpif16(x), 0.5);
31+
}
32+
}
33+
34+
TEST_F(LlvmLibcAtanpif16Test, NegativeRange) {
35+
for (uint16_t v = NEG_START; v <= NEG_STOP; ++v) {
36+
float16 x = FPBits(v).get_val();
37+
EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Atanpi, x,
38+
LIBC_NAMESPACE::atanpif16(x), 0.5);
39+
}
40+
}

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

Lines changed: 12 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -4603,6 +4603,18 @@ add_fp_unittest(
46034603
libc.src.__support.FPUtil.cast
46044604
)
46054605

4606+
add_fp_unittest(
4607+
atanpif16_test
4608+
NEED_MPFR
4609+
SUITE
4610+
libc-math-unittests
4611+
SRCS
4612+
atanpif16_test.cpp
4613+
DEPENDS
4614+
libc.src.math.atanpif16
4615+
libc.src.errno.errno
4616+
)
4617+
46064618
add_fp_unittest(
46074619
asinhf_test
46084620
SUITE

0 commit comments

Comments
 (0)