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[libc][math] Refactor acos implementation to header-only in src/__support/math folder. (#148409)

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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
This commit is contained in:
Muhammad Bassiouni 2025-07-19 05:21:34 +03:00 committed by GitHub
parent b3c9ed151f
commit cfddb401db
No known key found for this signature in database
GPG Key ID: B5690EEEBB952194
9 changed files with 403 additions and 308 deletions
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1
libc/shared/math.h
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@ -11,6 +11,7 @@
#include "libc_common.h"
#include "math/acos.h"
#include "math/exp.h"
#include "math/exp10.h"
#include "math/exp10f.h"

23
libc/shared/math/acos.h Normal file
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@ -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

33
libc/src/__support/math/CMakeLists.txt
View File

@ -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

285
libc/src/__support/math/acos.h Normal file
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@ -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_DOUBLE
double h = fputil::multiply_add(v_hi, -v_hi, u);
#else
DoubleDouble v_hi_sq = fputil::exact_mult(v_hi, v_hi);
double h = (u - v_hi_sq.hi) - v_hi_sq.lo;
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
// Scale v_lo and v_hi by 2 from the formula:
// vh = v_hi * 2
// vl = 2*v_lo = h / v_hi.
double vh = v_hi * 2.0;
double vl = h / v_hi;
// Polynomial approximation:
// p ~ asin(sqrt(u))/sqrt(u)
unsigned idx = 0;
double err = vh * 0x1.0p-51;
DoubleDouble p = asin_eval(DoubleDouble{0.0, u}, idx, err);
// Perform computations in double-double arithmetic:
// asin(x) = pi/2 - (v_hi + v_lo) * (ASIN_COEFFS[idx][0] + p)
DoubleDouble r0 = fputil::quick_mult(DoubleDouble{vl, vh}, p);
double r_hi = 0, r_lo = 0;
if (xbits.is_pos()) {
r_hi = r0.hi;
r_lo = r0.lo;
} else {
DoubleDouble r = fputil::exact_add(PI.hi, -r0.hi);
r_hi = r.hi;
r_lo = (PI.lo - r0.lo) + r.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, we redo the computations in Float128.
// Recalculate mod 1/64.
idx = static_cast<unsigned>(fputil::nearest_integer(u * 0x1.0p6));
// After the first step of Newton-Raphson approximating v = sqrt(u), we have
// that:
// sqrt(u) = v_hi + h / (sqrt(u) + v_hi)
// v_lo = h / (2 * v_hi)
// With error:
// sqrt(u) - (v_hi + v_lo) = h * ( 1/(sqrt(u) + v_hi) - 1/(2*v_hi) )
// = -h^2 / (2*v * (sqrt(u) + v)^2).
// Since:
// (sqrt(u) + v_hi)^2 ~ (2sqrt(u))^2 = 4u,
// we can add another correction term to (v_hi + v_lo) that is:
// v_ll = -h^2 / (2*v_hi * 4u)
// = -v_lo * (h / 4u)
// = -vl * (h / 8u),
// making the errors:
// sqrt(u) - (v_hi + v_lo + v_ll) = O(h^3)
// well beyond 128-bit precision needed.
// Get the rounding error of vl = 2 * v_lo ~ h / vh
// Get full product of vh * vl
#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
double vl_lo = fputil::multiply_add(-v_hi, vl, h) / v_hi;
#else
DoubleDouble vh_vl = fputil::exact_mult(v_hi, vl);
double vl_lo = ((h - vh_vl.hi) - vh_vl.lo) / v_hi;
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
// vll = 2*v_ll = -vl * (h / (4u)).
double t = h * (-0.25) / u;
double vll = fputil::multiply_add(vl, t, vl_lo);
// m_v = -(v_hi + v_lo + v_ll).
Float128 m_v = fputil::quick_add(
Float128(vh), fputil::quick_add(Float128(vl), Float128(vll)));
m_v.sign = xbits.sign();
// Perform computations in Float128:
// acos(x) = (v_hi + v_lo + vll) * P(u) , when 0.5 <= x < 1,
// = pi - (v_hi + v_lo + vll) * P(u) , when -1 < x <= -0.5.
Float128 y_f128(fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, u));
Float128 p_f128 = asin_eval(y_f128, idx);
Float128 r_f128 = fputil::quick_mul(m_v, p_f128);
if (xbits.is_neg())
r_f128 = fputil::quick_add(PI_F128, r_f128);
return static_cast<double>(r_f128);
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
}
} // namespace math
} // namespace LIBC_NAMESPACE_DECL
#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ACOS_H

34
libc/src/math/generic/asin_utils.h → libc/src/__support/math/asin_utils.h
View File

@ -6,8 +6,8 @@
//
//===----------------------------------------------------------------------===//
#ifndef LLVM_LIBC_SRC_MATH_GENERIC_ASIN_UTILS_H
#define LLVM_LIBC_SRC_MATH_GENERIC_ASIN_UTILS_H
#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ASIN_UTILS_H
#define LLVM_LIBC_SRC___SUPPORT_MATH_ASIN_UTILS_H
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/double_double.h"
@ -16,7 +16,6 @@
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/integer_literals.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h"
namespace LIBC_NAMESPACE_DECL {
@ -25,10 +24,10 @@ namespace {
using DoubleDouble = fputil::DoubleDouble;
using Float128 = fputil::DyadicFloat<128>;
constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p1};
static constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p1};
constexpr DoubleDouble PI_OVER_TWO = {0x1.1a62633145c07p-54,
0x1.921fb54442d18p0};
static constexpr DoubleDouble PI_OVER_TWO = {0x1.1a62633145c07p-54,
0x1.921fb54442d18p0};
#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
@ -39,14 +38,14 @@ constexpr DoubleDouble PI_OVER_TWO = {0x1.1a62633145c07p-54,
// > dirtyinfnorm(asin(x)/x - P, [0, 0.5]);
// 0x1.1a71ef0a0f26a9fb7ed7e41dee788b13d1770db3dp-52
constexpr double ASIN_COEFFS[12] = {
static constexpr double ASIN_COEFFS[12] = {
0x1.0000000000000p0, 0x1.5555555556dcfp-3, 0x1.3333333082e11p-4,
0x1.6db6dd14099edp-5, 0x1.f1c69b35bf81fp-6, 0x1.6e97194225a67p-6,
0x1.1babddb82ce12p-6, 0x1.d55bd078600d6p-7, 0x1.33328959e63d6p-7,
0x1.2b5993bda1d9bp-6, -0x1.806aff270bf25p-7, 0x1.02614e5ed3936p-5,
};
LIBC_INLINE double asin_eval(double u) {
LIBC_INLINE static constexpr double asin_eval(double u) {
double u2 = u * u;
double c0 = fputil::multiply_add(u, ASIN_COEFFS[1], ASIN_COEFFS[0]);
double c1 = fputil::multiply_add(u, ASIN_COEFFS[3], ASIN_COEFFS[2]);
@ -124,7 +123,7 @@ LIBC_INLINE double asin_eval(double u) {
// > dirtyinfnorm(asin(x)/x - P, [-1/64, 1/64]);
// 0x1.999075402cafp-83
constexpr double ASIN_COEFFS[9][12] = {
static constexpr double ASIN_COEFFS[9][12] = {
{1.0, 0.0, 0x1.5555555555555p-3, 0x1.5555555555555p-57,
0x1.3333333333333p-4, 0x1.6db6db6db6db7p-5, 0x1.f1c71c71c71c7p-6,
0x1.6e8ba2e8ba2e9p-6, 0x1.1c4ec4ec4ec4fp-6, 0x1.c99999999999ap-7,
@ -164,8 +163,8 @@ constexpr double ASIN_COEFFS[9][12] = {
};
// We calculate the lower part of the approximation P(u).
LIBC_INLINE DoubleDouble asin_eval(const DoubleDouble &u, unsigned &idx,
double &err) {
LIBC_INLINE static DoubleDouble asin_eval(const DoubleDouble &u, unsigned &idx,
double &err) {
using fputil::multiply_add;
// k = round(u * 32).
double k = fputil::nearest_integer(u.hi * 0x1.0p5);
@ -239,7 +238,7 @@ LIBC_INLINE DoubleDouble asin_eval(const DoubleDouble &u, unsigned &idx,
// + (676039 x^24)/104857600 + (1300075 x^26)/226492416 +
// + (5014575 x^28)/973078528 + (9694845 x^30)/2080374784.
constexpr Float128 ASIN_COEFFS_F128[17][16] = {
static constexpr Float128 ASIN_COEFFS_F128[17][16] = {
{
{Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128},
{Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128},
@ -548,13 +547,14 @@ constexpr Float128 ASIN_COEFFS_F128[17][16] = {
},
};
constexpr Float128 PI_OVER_TWO_F128 = {
static constexpr Float128 PI_OVER_TWO_F128 = {
Sign::POS, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
constexpr Float128 PI_F128 = {Sign::POS, -126,
0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
static constexpr Float128 PI_F128 = {
Sign::POS, -126, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
LIBC_INLINE Float128 asin_eval(const Float128 &u, unsigned idx) {
LIBC_INLINE static constexpr Float128 asin_eval(const Float128 &u,
unsigned idx) {
return fputil::polyeval(u, ASIN_COEFFS_F128[idx][0], ASIN_COEFFS_F128[idx][1],
ASIN_COEFFS_F128[idx][2], ASIN_COEFFS_F128[idx][3],
ASIN_COEFFS_F128[idx][4], ASIN_COEFFS_F128[idx][5],
@ -571,4 +571,4 @@ LIBC_INLINE Float128 asin_eval(const Float128 &u, unsigned idx) {
} // namespace LIBC_NAMESPACE_DECL
#endif // LLVM_LIBC_SRC_MATH_GENERIC_ASIN_UTILS_H
#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ASIN_UTILS_H

28
libc/src/math/generic/CMakeLists.txt
View File

@ -4016,20 +4016,6 @@ add_entrypoint_object(
libc.src.__support.macros.properties.types
)
add_header_library(
asin_utils
HDRS
atan_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_entrypoint_object(
asin
SRCS
@ -4037,7 +4023,7 @@ add_entrypoint_object(
HDRS
../asin.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
@ -4092,17 +4078,7 @@ add_entrypoint_object(
HDRS
../acos.h
DEPENDS
.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
libc.src.__support.math.acos
)
add_entrypoint_object(

266
libc/src/math/generic/acos.cpp
View File

@ -7,272 +7,10 @@
//===----------------------------------------------------------------------===//
#include "src/math/acos.h"
#include "asin_utils.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.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
#include "src/__support/math/acos.h"
namespace LIBC_NAMESPACE_DECL {
using DoubleDouble = fputil::DoubleDouble;
using Float128 = fputil::DyadicFloat<128>;
LLVM_LIBC_FUNCTION(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;
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_DOUBLE
double h = fputil::multiply_add(v_hi, -v_hi, u);
#else
DoubleDouble v_hi_sq = fputil::exact_mult(v_hi, v_hi);
double h = (u - v_hi_sq.hi) - v_hi_sq.lo;
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
// Scale v_lo and v_hi by 2 from the formula:
// vh = v_hi * 2
// vl = 2*v_lo = h / v_hi.
double vh = v_hi * 2.0;
double vl = h / v_hi;
// Polynomial approximation:
// p ~ asin(sqrt(u))/sqrt(u)
unsigned idx;
double err = vh * 0x1.0p-51;
DoubleDouble p = asin_eval(DoubleDouble{0.0, u}, idx, err);
// Perform computations in double-double arithmetic:
// asin(x) = pi/2 - (v_hi + v_lo) * (ASIN_COEFFS[idx][0] + p)
DoubleDouble r0 = fputil::quick_mult(DoubleDouble{vl, vh}, p);
double r_hi, r_lo;
if (xbits.is_pos()) {
r_hi = r0.hi;
r_lo = r0.lo;
} else {
DoubleDouble r = fputil::exact_add(PI.hi, -r0.hi);
r_hi = r.hi;
r_lo = (PI.lo - r0.lo) + r.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, we redo the computations in Float128.
// Recalculate mod 1/64.
idx = static_cast<unsigned>(fputil::nearest_integer(u * 0x1.0p6));
// After the first step of Newton-Raphson approximating v = sqrt(u), we have
// that:
// sqrt(u) = v_hi + h / (sqrt(u) + v_hi)
// v_lo = h / (2 * v_hi)
// With error:
// sqrt(u) - (v_hi + v_lo) = h * ( 1/(sqrt(u) + v_hi) - 1/(2*v_hi) )
// = -h^2 / (2*v * (sqrt(u) + v)^2).
// Since:
// (sqrt(u) + v_hi)^2 ~ (2sqrt(u))^2 = 4u,
// we can add another correction term to (v_hi + v_lo) that is:
// v_ll = -h^2 / (2*v_hi * 4u)
// = -v_lo * (h / 4u)
// = -vl * (h / 8u),
// making the errors:
// sqrt(u) - (v_hi + v_lo + v_ll) = O(h^3)
// well beyond 128-bit precision needed.
// Get the rounding error of vl = 2 * v_lo ~ h / vh
// Get full product of vh * vl
#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
double vl_lo = fputil::multiply_add(-v_hi, vl, h) / v_hi;
#else
DoubleDouble vh_vl = fputil::exact_mult(v_hi, vl);
double vl_lo = ((h - vh_vl.hi) - vh_vl.lo) / v_hi;
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
// vll = 2*v_ll = -vl * (h / (4u)).
double t = h * (-0.25) / u;
double vll = fputil::multiply_add(vl, t, vl_lo);
// m_v = -(v_hi + v_lo + v_ll).
Float128 m_v = fputil::quick_add(
Float128(vh), fputil::quick_add(Float128(vl), Float128(vll)));
m_v.sign = xbits.sign();
// Perform computations in Float128:
// acos(x) = (v_hi + v_lo + vll) * P(u) , when 0.5 <= x < 1,
// = pi - (v_hi + v_lo + vll) * P(u) , when -1 < x <= -0.5.
Float128 y_f128(fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, u));
Float128 p_f128 = asin_eval(y_f128, idx);
Float128 r_f128 = fputil::quick_mul(m_v, p_f128);
if (xbits.is_neg())
r_f128 = fputil::quick_add(PI_F128, r_f128);
return static_cast<double>(r_f128);
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
}
LLVM_LIBC_FUNCTION(double, acos, (double x)) { return math::acos(x); }
} // namespace LIBC_NAMESPACE_DECL

2
libc/src/math/generic/asin.cpp
View File

@ -7,7 +7,6 @@
//===----------------------------------------------------------------------===//
#include "src/math/asin.h"
#include "asin_utils.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
@ -18,6 +17,7 @@
#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
#include "src/__support/math/asin_utils.h"
namespace LIBC_NAMESPACE_DECL {

39
utils/bazel/llvm-project-overlay/libc/BUILD.bazel
View File

@ -2077,6 +2077,38 @@ libc_support_library(
],
)
libc_support_library(
name = "__support_math_acos",
hdrs = ["src/__support/math/acos.h"],
deps = [
":__support_math_asin_utils",
":__support_fputil_double_double",
":__support_fputil_dyadic_float",
":__support_fputil_fenv_impl",
":__support_fputil_fp_bits",
":__support_fputil_multiply_add",
":__support_fputil_polyeval",
":__support_fputil_sqrt",
":__support_macros_optimization",
":__support_macros_properties_types",
":__support_macros_properties_cpu_features",
],
)
libc_support_library(
name = "__support_math_asin_utils",
hdrs = ["src/__support/math/asin_utils.h"],
deps = [
":__support_integer_literals",
":__support_fputil_double_double",
":__support_fputil_dyadic_float",
":__support_fputil_multiply_add",
":__support_fputil_nearest_integer",
":__support_fputil_polyeval",
":__support_macros_optimization",
],
)
libc_support_library(
name = "__support_math_exp_float_constants",
hdrs = ["src/__support/math/exp_float_constants.h"],
@ -2554,6 +2586,13 @@ libc_function(
################################ math targets ##################################
libc_math_function(
name = "acos",
additional_deps = [
":__support_math_acos",
],
)
libc_math_function(
name = "acosf",
additional_deps = [