[libc][math] Refactor acos implementation to header-only in src/__support/math folder. #148409
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…__support/math folder.
…_support/math folder.
…upport/math folder
…pport/math folder.
…upport/math folder.
…_support/math folder.
…port/math folder.
bassiounix
requested review from
rupprecht,
keith and
aaronmondal
as code owners
|
@llvm/pr-subscribers-libc Author: Muhammad Bassiouni (bassiounix) ChangesPart of #147386 in preparation for: https://discourse.llvm.org/t/rfc-make-clang-builtin-math-functions-constexpr-with-llvm-libc-to-support-c-23-constexpr-math-functions/86450 Please merge #148408 first @lntue Patch is 199.40 KiB, truncated to 20.00 KiB below, full version: https://github.com/llvm/llvm-project/pull/148409.diff 45 Files Affected:
diff --git a/libc/shared/math.h b/libc/shared/math.h
index e2950e075a81d..8dcfaf0352339 100644
--- a/libc/shared/math.h
+++ b/libc/shared/math.h
@@ -11,10 +11,18 @@
#include "libc_common.h"
+#include "math/acos.h"
+#include "math/exp.h"
+#include "math/exp10.h"
+#include "math/exp10f.h"
+#include "math/exp10f16.h"
#include "math/expf.h"
#include "math/expf16.h"
#include "math/frexpf.h"
#include "math/frexpf128.h"
#include "math/frexpf16.h"
+#include "math/ldexpf.h"
+#include "math/ldexpf128.h"
+#include "math/ldexpf16.h"
#endif // LLVM_LIBC_SHARED_MATH_H
diff --git a/libc/shared/math/acos.h b/libc/shared/math/acos.h
new file mode 100644
index 0000000000000..73c6b512e16f4
--- /dev/null
+++ b/libc/shared/math/acos.h
@@ -0,0 +1,23 @@
+//===-- Shared acos function ------------------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_ACOS_H
+#define LLVM_LIBC_SHARED_MATH_ACOS_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/acos.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::acos;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_ACOS_H
diff --git a/libc/shared/math/exp.h b/libc/shared/math/exp.h
new file mode 100644
index 0000000000000..7cdd6331e613a
--- /dev/null
+++ b/libc/shared/math/exp.h
@@ -0,0 +1,23 @@
+//===-- Shared exp function -------------------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_EXP_H
+#define LLVM_LIBC_SHARED_MATH_EXP_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/exp.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::exp;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_EXP_H
diff --git a/libc/shared/math/exp10.h b/libc/shared/math/exp10.h
new file mode 100644
index 0000000000000..3d36d9103705f
--- /dev/null
+++ b/libc/shared/math/exp10.h
@@ -0,0 +1,23 @@
+//===-- Shared exp10 function -----------------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_EXP10_H
+#define LLVM_LIBC_SHARED_MATH_EXP10_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/exp10.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::exp10;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_EXP10_H
diff --git a/libc/shared/math/exp10f.h b/libc/shared/math/exp10f.h
new file mode 100644
index 0000000000000..556e78ab3b7a7
--- /dev/null
+++ b/libc/shared/math/exp10f.h
@@ -0,0 +1,24 @@
+//===-- Shared exp10f function -----------------------------------*- C++
+//-*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_EXP10F_H
+#define LLVM_LIBC_SHARED_MATH_EXP10F_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/exp10f.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::exp10f;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_EXP10F_H
diff --git a/libc/shared/math/exp10f16.h b/libc/shared/math/exp10f16.h
new file mode 100644
index 0000000000000..8acdbdb7c70a1
--- /dev/null
+++ b/libc/shared/math/exp10f16.h
@@ -0,0 +1,29 @@
+//===-- Shared exp10f16 function --------------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_EXP10F_H
+#define LLVM_LIBC_SHARED_MATH_EXP10F_H
+
+#include "include/llvm-libc-macros/float16-macros.h"
+
+#ifdef LIBC_TYPES_HAS_FLOAT16
+
+#include "shared/libc_common.h"
+#include "src/__support/math/exp10f16.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::exp10f16;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LIBC_TYPES_HAS_FLOAT16
+
+#endif // LLVM_LIBC_SHARED_MATH_EXP10F_H
diff --git a/libc/shared/math/ldexpf.h b/libc/shared/math/ldexpf.h
new file mode 100644
index 0000000000000..497933c47321f
--- /dev/null
+++ b/libc/shared/math/ldexpf.h
@@ -0,0 +1,23 @@
+//===-- Shared ldexpf function ----------------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_LDEXPF_H
+#define LLVM_LIBC_SHARED_MATH_LDEXPF_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/ldexpf.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::ldexpf;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_LDEXPF_H
diff --git a/libc/shared/math/ldexpf128.h b/libc/shared/math/ldexpf128.h
new file mode 100644
index 0000000000000..d4066beb809c7
--- /dev/null
+++ b/libc/shared/math/ldexpf128.h
@@ -0,0 +1,29 @@
+//===-- Shared ldexpf128 function -------------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_LDEXPF128_H
+#define LLVM_LIBC_SHARED_MATH_LDEXPF128_H
+
+#include "include/llvm-libc-types/float128.h"
+
+#ifdef LIBC_TYPES_HAS_FLOAT128
+
+#include "shared/libc_common.h"
+#include "src/__support/math/ldexpf128.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::ldexpf128;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LIBC_TYPES_HAS_FLOAT128
+
+#endif // LLVM_LIBC_SHARED_MATH_LDEXPF128_H
diff --git a/libc/shared/math/ldexpf16.h b/libc/shared/math/ldexpf16.h
new file mode 100644
index 0000000000000..d4e39c449943e
--- /dev/null
+++ b/libc/shared/math/ldexpf16.h
@@ -0,0 +1,31 @@
+//===-- Shared ldexpf16 function --------------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_ldexpf16_H
+#define LLVM_LIBC_SHARED_MATH_ldexpf16_H
+
+#include "include/llvm-libc-macros/float16-macros.h"
+
+#ifdef LIBC_TYPES_HAS_FLOAT16
+
+#include "shared/libc_common.h"
+#include "src/__support/math/ldexpf16.h"
+
+namespace LIBC_NAMESPACE_DECL {
+
+namespace shared {
+
+using math::ldexpf16;
+
+} // namespace shared
+
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LIBC_TYPES_HAS_FLOAT16
+
+#endif // LLVM_LIBC_SHARED_MATH_ldexpf16_H
diff --git a/libc/src/__support/FPUtil/double_double.h b/libc/src/__support/FPUtil/double_double.h
index c27885aadc028..8e54e845de493 100644
--- a/libc/src/__support/FPUtil/double_double.h
+++ b/libc/src/__support/FPUtil/double_double.h
@@ -151,8 +151,8 @@ LIBC_INLINE DoubleDouble quick_mult(double a, const DoubleDouble &b) {
}
template <size_t SPLIT_B = 27>
-LIBC_INLINE DoubleDouble quick_mult(const DoubleDouble &a,
- const DoubleDouble &b) {
+LIBC_INLINE constexpr DoubleDouble quick_mult(const DoubleDouble &a,
+ const DoubleDouble &b) {
DoubleDouble r = exact_mult<double, SPLIT_B>(a.hi, b.hi);
double t1 = multiply_add(a.hi, b.lo, r.lo);
double t2 = multiply_add(a.lo, b.hi, t1);
diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt
index 0baf12a26a9d5..4a29c2975d523 100644
--- a/libc/src/__support/math/CMakeLists.txt
+++ b/libc/src/__support/math/CMakeLists.txt
@@ -1,3 +1,36 @@
+add_header_library(
+ acos
+ HDRS
+ acos.h
+ DEPENDS
+ .asin_utils
+ libc.src.__support.math.asin_utils
+ libc.src.__support.FPUtil.double_double
+ libc.src.__support.FPUtil.dyadic_float
+ libc.src.__support.FPUtil.fenv_impl
+ libc.src.__support.FPUtil.fp_bits
+ libc.src.__support.FPUtil.multiply_add
+ libc.src.__support.FPUtil.polyeval
+ libc.src.__support.FPUtil.sqrt
+ libc.src.__support.macros.optimization
+ libc.src.__support.macros.properties.types
+ libc.src.__support.macros.properties.cpu_features
+)
+
+add_header_library(
+ asin_utils
+ HDRS
+ asin_utils.h
+ DEPENDS
+ libc.src.__support.integer_literals
+ libc.src.__support.FPUtil.double_double
+ libc.src.__support.FPUtil.dyadic_float
+ libc.src.__support.FPUtil.multiply_add
+ libc.src.__support.FPUtil.nearest_integer
+ libc.src.__support.FPUtil.polyeval
+ libc.src.__support.macros.optimization
+)
+
add_header_library(
exp_float_constants
HDRS
@@ -82,3 +115,153 @@ add_header_library(
DEPENDS
libc.src.__support.FPUtil.manipulation_functions
)
+
+add_header_library(
+ ldexpf128
+ HDRS
+ ldexpf128.h
+ DEPENDS
+ libc.src.__support.macros.properties.types
+ libc.src.__support.FPUtil.manipulation_functions
+ libc.include.llvm-libc-types.float128
+)
+
+add_header_library(
+ ldexpf16
+ HDRS
+ ldexpf16.h
+ DEPENDS
+ libc.src.__support.macros.properties.types
+ libc.src.__support.FPUtil.manipulation_functions
+ libc.include.llvm-libc-macros.float16_macros
+)
+
+add_header_library(
+ ldexpf
+ HDRS
+ ldexpf.h
+ DEPENDS
+ libc.src.__support.FPUtil.manipulation_functions
+)
+
+add_header_library(
+ exp_constants
+ HDRS
+ exp_constants.h
+ DEPENDS
+ libc.src.__support.FPUtil.triple_double
+)
+
+add_header_library(
+ exp_utils
+ HDRS
+ exp_utils.h
+ DEPENDS
+ libc.src.__support.CPP.optional
+ libc.src.__support.CPP.bit
+ libc.src.__support.FPUtil.fp_bits
+)
+
+add_header_library(
+ exp
+ HDRS
+ exp.h
+ DEPENDS
+ .exp_constants
+ .exp_utils
+ libc.src.__support.CPP.bit
+ libc.src.__support.CPP.optional
+ libc.src.__support.FPUtil.dyadic_float
+ libc.src.__support.FPUtil.fenv_impl
+ libc.src.__support.FPUtil.fp_bits
+ libc.src.__support.FPUtil.multiply_add
+ libc.src.__support.FPUtil.nearest_integer
+ libc.src.__support.FPUtil.polyeval
+ libc.src.__support.FPUtil.rounding_mode
+ libc.src.__support.FPUtil.triple_double
+ libc.src.__support.integer_literals
+ libc.src.__support.macros.optimization
+)
+
+add_header_library(
+ exp10
+ HDRS
+ exp10.h
+ DEPENDS
+ .exp_constants
+ .exp_utils
+ libc.src.__support.CPP.bit
+ libc.src.__support.CPP.optional
+ libc.src.__support.FPUtil.dyadic_float
+ libc.src.__support.FPUtil.fenv_impl
+ libc.src.__support.FPUtil.fp_bits
+ libc.src.__support.FPUtil.multiply_add
+ libc.src.__support.FPUtil.nearest_integer
+ libc.src.__support.FPUtil.polyeval
+ libc.src.__support.FPUtil.rounding_mode
+ libc.src.__support.FPUtil.triple_double
+ libc.src.__support.integer_literals
+ libc.src.__support.macros.optimization
+)
+
+add_header_library(
+ exp10f_utils
+ HDRS
+ exp10f_utils.h
+ DEPENDS
+ libc.src.__support.FPUtil.basic_operations
+ libc.src.__support.FPUtil.fenv_impl
+ libc.src.__support.FPUtil.multiply_add
+ libc.src.__support.FPUtil.nearest_integer
+ libc.src.__support.FPUtil.polyeval
+ libc.src.__support.common
+ libc.src.__support.math.exp_utils
+)
+
+add_header_library(
+ exp10f
+ HDRS
+ exp10f.h
+ DEPENDS
+ .exp10f_utils
+ libc.src.__support.macros.config
+ libc.src.__support.FPUtil.fenv_impl
+ libc.src.__support.FPUtil.fp_bits
+ libc.src.__support.FPUtil.multiply_add
+ libc.src.__support.FPUtil.rounding_mode
+ libc.src.__support.macros.optimization
+)
+
+add_header_library(
+ exp10_float16_constants
+ HDRS
+ exp10_float16_constants.h
+ DEPENDS
+ libc.src.__support.CPP.array
+)
+
+add_header_library(
+ exp10f16_utils
+ HDRS
+ exp10f16_utils.h
+ DEPENDS
+ .expf16_utils
+ .exp10_float16_constants
+ libc.src.__support.FPUtil.fp_bits
+)
+
+add_header_library(
+ exp10f16
+ HDRS
+ exp10f16.h
+ DEPENDS
+ .exp10f16_utils
+ libc.src.__support.FPUtil.fp_bits
+ src.__support.FPUtil.FEnvImpl
+ src.__support.FPUtil.FPBits
+ src.__support.FPUtil.cast
+ src.__support.FPUtil.rounding_mode
+ src.__support.FPUtil.except_value_utils
+ src.__support.macros.optimization
+ src.__support.macros.properties.cpu_features
+)
diff --git a/libc/src/__support/math/acos.h b/libc/src/__support/math/acos.h
new file mode 100644
index 0000000000000..a7287f11aa302
--- /dev/null
+++ b/libc/src/__support/math/acos.h
@@ -0,0 +1,285 @@
+//===-- Implementation header for acos --------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ACOS_H
+#define LLVM_LIBC_SRC___SUPPORT_MATH_ACOS_H
+
+#include "asin_utils.h"
+#include "src/__support/FPUtil/FEnvImpl.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/double_double.h"
+#include "src/__support/FPUtil/dyadic_float.h"
+#include "src/__support/FPUtil/multiply_add.h"
+#include "src/__support/FPUtil/sqrt.h"
+#include "src/__support/macros/config.h"
+#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
+#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
+
+namespace LIBC_NAMESPACE_DECL {
+
+namespace math {
+
+using DoubleDouble = fputil::DoubleDouble;
+using Float128 = fputil::DyadicFloat<128>;
+
+static constexpr double acos(double x) {
+ using FPBits = fputil::FPBits<double>;
+
+ FPBits xbits(x);
+ int x_exp = xbits.get_biased_exponent();
+
+ // |x| < 0.5.
+ if (x_exp < FPBits::EXP_BIAS - 1) {
+ // |x| < 2^-55.
+ if (LIBC_UNLIKELY(x_exp < FPBits::EXP_BIAS - 55)) {
+ // When |x| < 2^-55, acos(x) = pi/2
+#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS)
+ return PI_OVER_TWO.hi;
+#else
+ // Force the evaluation and prevent constant propagation so that it
+ // is rounded correctly for FE_UPWARD rounding mode.
+ return (xbits.abs().get_val() + 0x1.0p-160) + PI_OVER_TWO.hi;
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+ }
+
+#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+ // acos(x) = pi/2 - asin(x)
+ // = pi/2 - x * P(x^2)
+ double p = asin_eval(x * x);
+ return PI_OVER_TWO.hi + fputil::multiply_add(-x, p, PI_OVER_TWO.lo);
+#else
+ unsigned idx = 0;
+ DoubleDouble x_sq = fputil::exact_mult(x, x);
+ double err = xbits.abs().get_val() * 0x1.0p-51;
+ // Polynomial approximation:
+ // p ~ asin(x)/x
+ DoubleDouble p = asin_eval(x_sq, idx, err);
+ // asin(x) ~ x * p
+ DoubleDouble r0 = fputil::exact_mult(x, p.hi);
+ // acos(x) = pi/2 - asin(x)
+ // ~ pi/2 - x * p
+ // = pi/2 - x * (p.hi + p.lo)
+ double r_hi = fputil::multiply_add(-x, p.hi, PI_OVER_TWO.hi);
+ // Use Dekker's 2SUM algorithm to compute the lower part.
+ double r_lo = ((PI_OVER_TWO.hi - r_hi) - r0.hi) - r0.lo;
+ r_lo = fputil::multiply_add(-x, p.lo, r_lo + PI_OVER_TWO.lo);
+
+ // Ziv's accuracy test.
+
+ double r_upper = r_hi + (r_lo + err);
+ double r_lower = r_hi + (r_lo - err);
+
+ if (LIBC_LIKELY(r_upper == r_lower))
+ return r_upper;
+
+ // Ziv's accuracy test failed, perform 128-bit calculation.
+
+ // Recalculate mod 1/64.
+ idx = static_cast<unsigned>(fputil::nearest_integer(x_sq.hi * 0x1.0p6));
+
+ // Get x^2 - idx/64 exactly. When FMA is available, double-double
+ // multiplication will be correct for all rounding modes. Otherwise we use
+ // Float128 directly.
+ Float128 x_f128(x);
+
+#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
+ // u = x^2 - idx/64
+ Float128 u_hi(
+ fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, x_sq.hi));
+ Float128 u = fputil::quick_add(u_hi, Float128(x_sq.lo));
+#else
+ Float128 x_sq_f128 = fputil::quick_mul(x_f128, x_f128);
+ Float128 u = fputil::quick_add(
+ x_sq_f128, Float128(static_cast<double>(idx) * (-0x1.0p-6)));
+#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
+
+ Float128 p_f128 = asin_eval(u, idx);
+ // Flip the sign of x_f128 to perform subtraction.
+ x_f128.sign = x_f128.sign.negate();
+ Float128 r =
+ fputil::quick_add(PI_OVER_TWO_F128, fputil::quick_mul(x_f128, p_f128));
+
+ return static_cast<double>(r);
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+ }
+ // |x| >= 0.5
+
+ double x_abs = xbits.abs().get_val();
+
+ // Maintaining the sign:
+ constexpr double SIGN[2] = {1.0, -1.0};
+ double x_sign = SIGN[xbits.is_neg()];
+ // |x| >= 1
+ if (LIBC_UNLIKELY(x_exp >= FPBits::EXP_BIAS)) {
+ // x = +-1, asin(x) = +- pi/2
+ if (x_abs == 1.0) {
+ // x = 1, acos(x) = 0,
+ // x = -1, acos(x) = pi
+ return x == 1.0 ? 0.0 : fputil::multiply_add(-x_sign, PI.hi, PI.lo);
+ }
+ // |x| > 1, return NaN.
+ if (xbits.is_quiet_nan())
+ return x;
+
+ // Set domain error for non-NaN input.
+ if (!xbits.is_nan())
+ fputil::set_errno_if_required(EDOM);
+
+ fputil::raise_except_if_required(FE_INVALID);
+ return FPBits::quiet_nan().get_val();
+ }
+
+ // When |x| >= 0.5, we perform range reduction as follow:
+ //
+ // When 0.5 <= x < 1, let:
+ // y = acos(x)
+ // We will use the double angle formula:
+ // cos(2y) = 1 - 2 sin^2(y)
+ // and the complement angle identity:
+ // x = cos(y) = 1 - 2 sin^2 (y/2)
+ // So:
+ // sin(y/2) = sqrt( (1 - x)/2 )
+ // And hence:
+ // y/2 = asin( sqrt( (1 - x)/2 ) )
+ // Equivalently:
+ // acos(x) = y = 2 * asin( sqrt( (1 - x)/2 ) )
+ // Let u = (1 - x)/2, then:
+ // acos(x) = 2 * asin( sqrt(u) )
+ // Moreover, since 0.5 <= x < 1:
+ // 0 < u <= 1/4, and 0 < sqrt(u) <= 0.5,
+ // And hence we can reuse the same polynomial approximation of asin(x) when
+ // |x| <= 0.5:
+ // acos(x) ~ 2 * sqrt(u) * P(u).
+ //
+ // When -1 < x <= -0.5, we reduce to the previous case using the formula:
+ // acos(x) = pi - acos(-x)
+ // = pi - 2 * asin ( sqrt( (1 + x)/2 ) )
+ // ~ pi - 2 * sqrt(u) * P(u),
+ // where u = (1 - |x|)/2.
+
+ // u = (1 - |x|)/2
+ double u = fputil::multiply_add(x_abs, -0.5, 0.5);
+ // v_hi + v_lo ~ sqrt(u).
+ // Let:
+ // h = u - v_hi^2 = (sqrt(u) - v_hi) * (sqrt(u) + v_hi)
+ // Then:
+ // sqrt(u) = v_hi + h / (sqrt(u) + v_hi)
+ // ~ v_hi + h / (2 * v_hi)
+ // So we can use:
+ // v_lo = h / (2 * v_hi).
+ double v_hi = fputil::sqrt<double>(u);
+
+#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+ constexpr DoubleDouble CONST_TERM[2] = {{0.0, 0.0}, PI};
+ DoubleDouble const_term = CONST_TERM[xbits.is_neg()];
+
+ double p = asin_eval(u);
+ double scale = x_sign * 2.0 * v_hi;
+ double r = const_term.hi + fputil::multiply_add(scale, p, const_term.lo);
+ return r;
+#else
+
+#ifdef LIBC_TARGET_CPU_HAS_FMA_...
[truncated]
|
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Part of #147386
in preparation for: https://discourse.llvm.org/t/rfc-make-clang-builtin-math-functions-constexpr-with-llvm-libc-to-support-c-23-constexpr-math-functions/86450
Please merge #148408 first