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[libc][math] Refactor atan2f implementation to header-only in src/__support/math folder. (#150993)

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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
2025-08-01 16:06:09 +03:00
committed by GitHub
parent 6533ad04ed
commit 9af2c21be4
10 changed files with 430 additions and 348 deletions
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1
libc/shared/math.h
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@@ -24,6 +24,7 @@
#include "math/asinhf16.h"
#include "math/atan.h"
#include "math/atan2.h"
#include "math/atan2f.h"
#include "math/atanf.h"
#include "math/atanf16.h"
#include "math/erff.h"

23
libc/shared/math/atan2f.h Normal file
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@@ -0,0 +1,23 @@
//===-- Shared atan2f 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_ATAN2F_H
#define LLVM_LIBC_SHARED_MATH_ATAN2F_H
#include "shared/libc_common.h"
#include "src/__support/math/atan2f.h"
namespace LIBC_NAMESPACE_DECL {
namespace shared {
using math::atan2f;
} // namespace shared
} // namespace LIBC_NAMESPACE_DECL
#endif // LLVM_LIBC_SHARED_MATH_ATAN2F_H

17
libc/src/__support/math/CMakeLists.txt
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@@ -213,6 +213,23 @@ add_header_library(
libc.src.__support.macros.optimization
)
add_header_library(
atan2f
HDRS
atan2f_float.h
atan2f.h
DEPENDS
.inv_trigf_utils
libc.src.__support.FPUtil.double_double
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.macros.config
libc.src.__support.macros.optimization
)
add_header_library(
atanf
HDRS

351
libc/src/__support/math/atan2f.h Normal file
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@@ -0,0 +1,351 @@
//===-- Implementation header for atan2f ------------------------*- 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_ATAN2F_H
#define LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2F_H
#include "inv_trigf_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/multiply_add.h"
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \
defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT)
// We use float-float implementation to reduce size.
#include "atan2f_float.h"
#else
namespace LIBC_NAMESPACE_DECL {
namespace math {
namespace atan2f_internal {
#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
// Look up tables for accurate pass:
// atan(i/16) with i = 0..16, generated by Sollya with:
// > for i from 0 to 16 do {
// a = round(atan(i/16), D, RN);
// b = round(atan(i/16) - a, D, RN);
// print("{", b, ",", a, "},");
// };
static constexpr fputil::DoubleDouble ATAN_I[17] = {
{0.0, 0.0},
{-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
{-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
{0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
{0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
{-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
{-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
{-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
{0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
{-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
{-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
{0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
{0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
{0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
{-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
{-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
{0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
};
// Taylor polynomial, generated by Sollya with:
// > for i from 0 to 8 do {
// j = (-1)^(i + 1)/(2*i + 1);
// a = round(j, D, RN);
// b = round(j - a, D, RN);
// print("{", b, ",", a, "},");
// };
static constexpr fputil::DoubleDouble COEFFS[9] = {
{0.0, 1.0}, // 1
{-0x1.5555555555555p-56, -0x1.5555555555555p-2}, // -1/3
{-0x1.999999999999ap-57, 0x1.999999999999ap-3}, // 1/5
{-0x1.2492492492492p-57, -0x1.2492492492492p-3}, // -1/7
{0x1.c71c71c71c71cp-58, 0x1.c71c71c71c71cp-4}, // 1/9
{0x1.745d1745d1746p-59, -0x1.745d1745d1746p-4}, // -1/11
{-0x1.3b13b13b13b14p-58, 0x1.3b13b13b13b14p-4}, // 1/13
{-0x1.1111111111111p-60, -0x1.1111111111111p-4}, // -1/15
{0x1.e1e1e1e1e1e1ep-61, 0x1.e1e1e1e1e1e1ep-5}, // 1/17
};
// Veltkamp's splitting of a double precision into hi + lo, where the hi part is
// slightly smaller than an even split, so that the product of
// hi * (s1 * k + s2) is exact,
// where:
// s1, s2 are single precsion,
// 1/16 <= s1/s2 <= 1
// 1/16 <= k <= 1 is an integer.
// So the maximal precision of (s1 * k + s2) is:
// prec(s1 * k + s2) = 2 + log2(msb(s2)) - log2(lsb(k_d * s1))
// = 2 + log2(msb(s1)) + 4 - log2(lsb(k_d)) - log2(lsb(s1))
// = 2 + log2(lsb(s1)) + 23 + 4 - (-4) - log2(lsb(s1))
// = 33.
// Thus, the Veltkamp splitting constant is C = 2^33 + 1.
// This is used when FMA instruction is not available.
[[maybe_unused]] LIBC_INLINE static constexpr fputil::DoubleDouble
split_d(double a) {
fputil::DoubleDouble r{0.0, 0.0};
constexpr double C = 0x1.0p33 + 1.0;
double t1 = C * a;
double t2 = a - t1;
r.hi = t1 + t2;
r.lo = a - r.hi;
return r;
}
// Compute atan( num_d / den_d ) in double-double precision.
// num_d = min(|x|, |y|)
// den_d = max(|x|, |y|)
// q_d = num_d / den_d
// idx, k_d = round( 2^4 * num_d / den_d )
// final_sign = sign of the final result
// const_term = the constant term in the final expression.
LIBC_INLINE static float
atan2f_double_double(double num_d, double den_d, double q_d, int idx,
double k_d, double final_sign,
const fputil::DoubleDouble &const_term) {
fputil::DoubleDouble q;
double num_r = 0, den_r = 0;
if (idx != 0) {
// The following range reduction is accurate even without fma for
// 1/16 <= n/d <= 1.
// atan(n/d) - atan(idx/16) = atan((n/d - idx/16) / (1 + (n/d) * (idx/16)))
// = atan((n - d*(idx/16)) / (d + n*idx/16))
k_d *= 0x1.0p-4;
num_r = fputil::multiply_add(k_d, -den_d, num_d); // Exact
den_r = fputil::multiply_add(k_d, num_d, den_d); // Exact
q.hi = num_r / den_r;
} else {
// For 0 < n/d < 1/16, we just need to calculate the lower part of their
// quotient.
q.hi = q_d;
num_r = num_d;
den_r = den_d;
}
#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
q.lo = fputil::multiply_add(q.hi, -den_r, num_r) / den_r;
#else
// Compute `(num_r - q.hi * den_r) / den_r` accurately without FMA
// instructions.
fputil::DoubleDouble q_hi_dd = split_d(q.hi);
double t1 = fputil::multiply_add(q_hi_dd.hi, -den_r, num_r); // Exact
double t2 = fputil::multiply_add(q_hi_dd.lo, -den_r, t1);
q.lo = t2 / den_r;
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
// Taylor polynomial, evaluating using Horner's scheme:
// P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
// + x^17/17
// = x*(1 + x^2*(-1/3 + x^2*(1/5 + x^2*(-1/7 + x^2*(1/9 + x^2*
// *(-1/11 + x^2*(1/13 + x^2*(-1/15 + x^2 * 1/17))))))))
fputil::DoubleDouble q2 = fputil::quick_mult(q, q);
fputil::DoubleDouble p_dd =
fputil::polyeval(q2, COEFFS[0], COEFFS[1], COEFFS[2], COEFFS[3],
COEFFS[4], COEFFS[5], COEFFS[6], COEFFS[7], COEFFS[8]);
fputil::DoubleDouble r_dd =
fputil::add(const_term, fputil::multiply_add(q, p_dd, ATAN_I[idx]));
r_dd.hi *= final_sign;
r_dd.lo *= final_sign;
// Make sure the sum is normalized:
fputil::DoubleDouble rr = fputil::exact_add(r_dd.hi, r_dd.lo);
// Round to odd.
uint64_t rr_bits = cpp::bit_cast<uint64_t>(rr.hi);
if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) {
Sign hi_sign = fputil::FPBits<double>(rr.hi).sign();
Sign lo_sign = fputil::FPBits<double>(rr.lo).sign();
if (hi_sign == lo_sign) {
++rr_bits;
} else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) {
--rr_bits;
}
}
return static_cast<float>(cpp::bit_cast<double>(rr_bits));
}
#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
} // namespace atan2f_internal
// There are several range reduction steps we can take for atan2(y, x) as
// follow:
// * Range reduction 1: signness
// atan2(y, x) will return a number between -PI and PI representing the angle
// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
// In particular, we have that:
// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
// Since atan function is odd, we can use the formula:
// atan(-u) = -atan(u)
// to adjust the above conditions a bit further:
// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
// Which can be simplified to:
// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
// = sign(y) * (pi - atan( |y|/|x| )) if x < 0
// * Range reduction 2: reciprocal
// Now that the argument inside atan is positive, we can use the formula:
// atan(1/x) = pi/2 - atan(x)
// to make the argument inside atan <= 1 as follow:
// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
// * Range reduction 3: look up table.
// After the previous two range reduction steps, we reduce the problem to
// compute atan(u) with 0 <= u <= 1, or to be precise:
// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
// An accurate polynomial approximation for the whole [0, 1] input range will
// require a very large degree. To make it more efficient, we reduce the input
// range further by finding an integer idx such that:
// | n/d - idx/16 | <= 1/32.
// In particular,
// idx := 2^-4 * round(2^4 * n/d)
// Then for the fast pass, we find a polynomial approximation for:
// atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16)
// For the accurate pass, we use the addition formula:
// atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (n*idx)/(16*d)) )
// = atan( (n - d * idx/16)/(d + n * idx/16) )
// And finally we use Taylor polynomial to compute the RHS in the accurate pass:
// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 -
// - u^15/15 + u^17/17
// It's error in double-double precision is estimated in Sollya to be:
// > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
// + x^17/17;
// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
// 0x1.aec6f...p-100
// which is about rounding errors of double-double (2^-104).
LIBC_INLINE static constexpr float atan2f(float y, float x) {
using namespace atan2f_internal;
using namespace inv_trigf_utils_internal;
using FPBits = typename fputil::FPBits<float>;
constexpr double IS_NEG[2] = {1.0, -1.0};
constexpr double PI = 0x1.921fb54442d18p1;
constexpr double PI_LO = 0x1.1a62633145c07p-53;
constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1;
constexpr double PI_OVER_2 = 0x1.921fb54442d18p0;
constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1;
// Adjustment for constant term:
// CONST_ADJ[x_sign][y_sign][recip]
constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = {
{{{0.0, 0.0}, {-PI_LO / 2, -PI_OVER_2}},
{{-0.0, -0.0}, {-PI_LO / 2, -PI_OVER_2}}},
{{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}},
{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}}}};
FPBits x_bits(x), y_bits(y);
bool x_sign = x_bits.sign().is_neg();
bool y_sign = y_bits.sign().is_neg();
x_bits.set_sign(Sign::POS);
y_bits.set_sign(Sign::POS);
uint32_t x_abs = x_bits.uintval();
uint32_t y_abs = y_bits.uintval();
uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs;
uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs;
float num_f = FPBits(min_abs).get_val();
float den_f = FPBits(max_abs).get_val();
double num_d = static_cast<double>(num_f);
double den_d = static_cast<double>(den_f);
if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || num_d == 0.0)) {
if (x_bits.is_nan() || y_bits.is_nan()) {
if (x_bits.is_signaling_nan() || y_bits.is_signaling_nan())
fputil::raise_except_if_required(FE_INVALID);
return FPBits::quiet_nan().get_val();
}
double x_d = static_cast<double>(x);
double y_d = static_cast<double>(y);
size_t x_except = (x_d == 0.0) ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1);
size_t y_except = (y_d == 0.0) ? 0 : (y_abs == 0x7f80'0000 ? 2 : 1);
// Exceptional cases:
// EXCEPT[y_except][x_except][x_is_neg]
// with x_except & y_except:
// 0: zero
// 1: finite, non-zero
// 2: infinity
constexpr double EXCEPTS[3][3][2] = {
{{0.0, PI}, {0.0, PI}, {0.0, PI}},
{{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}},
{{PI_OVER_2, PI_OVER_2},
{PI_OVER_2, PI_OVER_2},
{PI_OVER_4, THREE_PI_OVER_4}},
};
double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign];
return static_cast<float>(r);
}
bool recip = x_abs < y_abs;
double final_sign = IS_NEG[(x_sign != y_sign) != recip];
fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
double q_d = num_d / den_d;
double k_d = fputil::nearest_integer(q_d * 0x1.0p4);
int idx = static_cast<int>(k_d);
double r = 0.0;
#ifdef LIBC_MATH_HAS_SMALL_TABLES
double p = atan_eval_no_table(num_d, den_d, k_d * 0x1.0p-4);
r = final_sign * (p + (const_term.hi + ATAN_K_OVER_16[idx]));
#else
q_d = fputil::multiply_add(k_d, -0x1.0p-4, q_d);
double p = atan_eval(q_d, idx);
r = final_sign *
fputil::multiply_add(q_d, p, const_term.hi + ATAN_COEFFS[idx][0]);
#endif // LIBC_MATH_HAS_SMALL_TABLES
#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
return static_cast<float>(r);
#else
constexpr uint32_t LOWER_ERR = 4;
// Mask sticky bits in double precision before rounding to single precision.
constexpr uint32_t MASK =
mask_trailing_ones<uint32_t, fputil::FPBits<double>::SIG_LEN -
FPBits::SIG_LEN - 1>();
constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR;
uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(r)) & MASK;
// Ziv's rounding test.
if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR))
return static_cast<float>(r);
return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign,
const_term);
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
}
} // namespace math
} // namespace LIBC_NAMESPACE_DECL
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2F_H

24
libc/src/math/generic/atan2f_float.h → libc/src/__support/math/atan2f_float.h
View File

@@ -1,4 +1,4 @@
//===-- Single-precision atan2f function ----------------------------------===//
//===-- Single-precision atan2f float function ----------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
@@ -6,18 +6,21 @@
//
//===----------------------------------------------------------------------===//
#ifndef LIBC_SRC___SUPPORT_MATH_ATAN2F_FLOAT_H
#define LIBC_SRC___SUPPORT_MATH_ATAN2F_FLOAT_H
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/FPUtil/rounding_mode.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "src/math/atan2f.h"
namespace LIBC_NAMESPACE_DECL {
namespace {
namespace math {
namespace atan2f_internal {
using FloatFloat = fputil::FloatFloat;
@@ -27,7 +30,7 @@ using FloatFloat = fputil::FloatFloat;
// b = round(atan(i/16) - a, SG, RN);
// print("{", b, ",", a, "},");
// };
constexpr FloatFloat ATAN_I[17] = {
static constexpr FloatFloat ATAN_I[17] = {
{0.0f, 0.0f},
{-0x1.1a6042p-30f, 0x1.ff55bcp-5f},
{-0x1.54f424p-30f, 0x1.fd5baap-4f},
@@ -57,7 +60,7 @@ constexpr FloatFloat ATAN_I[17] = {
// For x = x_hi + x_lo, fully expand the polynomial and drop any terms less than
// ulp(x_hi^3 / 3) gives us:
// P(x) ~ x_hi - x_hi^3/3 + x_lo * (1 - x_hi^2)
FloatFloat atan_eval(const FloatFloat &x) {
LIBC_INLINE static constexpr FloatFloat atan_eval(const FloatFloat &x) {
FloatFloat p;
p.hi = x.hi;
float x_hi_sq = x.hi * x.hi;
@@ -70,7 +73,7 @@ FloatFloat atan_eval(const FloatFloat &x) {
return p;
}
} // anonymous namespace
} // namespace atan2f_internal
// There are several range reduction steps we can take for atan2(y, x) as
// follow:
@@ -121,7 +124,8 @@ FloatFloat atan_eval(const FloatFloat &x) {
// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
// 0x1.995...p-28.
LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
LIBC_INLINE static constexpr float atan2f(float y, float x) {
using namespace atan2f_internal;
using FPBits = typename fputil::FPBits<float>;
constexpr float IS_NEG[2] = {1.0f, -1.0f};
constexpr FloatFloat ZERO = {0.0f, 0.0f};
@@ -234,4 +238,8 @@ LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
return final_sign * r.hi;
}
} // namespace math
} // namespace LIBC_NAMESPACE_DECL
#endif // LIBC_SRC___SUPPORT_MATH_ATAN2F_FLOAT_H

12
libc/src/math/generic/CMakeLists.txt
View File

@@ -4058,18 +4058,8 @@ add_entrypoint_object(
atan2f.cpp
HDRS
../atan2f.h
atan2f_float.h
DEPENDS
libc.hdr.fenv_macros
libc.src.__support.FPUtil.double_double
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.macros.optimization
libc.src.__support.math.inv_trigf_utils
libc.src.__support.math.atan2f
)
add_entrypoint_object(

328
libc/src/math/generic/atan2f.cpp
View File

@@ -7,336 +7,12 @@
//===----------------------------------------------------------------------===//
#include "src/math/atan2f.h"
#include "hdr/fenv_macros.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/multiply_add.h"
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/FPUtil/rounding_mode.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "src/__support/math/inv_trigf_utils.h"
#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \
defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT)
// We use float-float implementation to reduce size.
#include "src/math/generic/atan2f_float.h"
#else
#include "src/__support/math/atan2f.h"
namespace LIBC_NAMESPACE_DECL {
namespace {
#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
// Look up tables for accurate pass:
// atan(i/16) with i = 0..16, generated by Sollya with:
// > for i from 0 to 16 do {
// a = round(atan(i/16), D, RN);
// b = round(atan(i/16) - a, D, RN);
// print("{", b, ",", a, "},");
// };
constexpr fputil::DoubleDouble ATAN_I[17] = {
{0.0, 0.0},
{-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
{-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
{0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
{0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
{-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
{-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
{-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
{0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
{-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
{-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
{0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
{0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
{0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
{-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
{-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
{0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
};
// Taylor polynomial, generated by Sollya with:
// > for i from 0 to 8 do {
// j = (-1)^(i + 1)/(2*i + 1);
// a = round(j, D, RN);
// b = round(j - a, D, RN);
// print("{", b, ",", a, "},");
// };
constexpr fputil::DoubleDouble COEFFS[9] = {
{0.0, 1.0}, // 1
{-0x1.5555555555555p-56, -0x1.5555555555555p-2}, // -1/3
{-0x1.999999999999ap-57, 0x1.999999999999ap-3}, // 1/5
{-0x1.2492492492492p-57, -0x1.2492492492492p-3}, // -1/7
{0x1.c71c71c71c71cp-58, 0x1.c71c71c71c71cp-4}, // 1/9
{0x1.745d1745d1746p-59, -0x1.745d1745d1746p-4}, // -1/11
{-0x1.3b13b13b13b14p-58, 0x1.3b13b13b13b14p-4}, // 1/13
{-0x1.1111111111111p-60, -0x1.1111111111111p-4}, // -1/15
{0x1.e1e1e1e1e1e1ep-61, 0x1.e1e1e1e1e1e1ep-5}, // 1/17
};
// Veltkamp's splitting of a double precision into hi + lo, where the hi part is
// slightly smaller than an even split, so that the product of
// hi * (s1 * k + s2) is exact,
// where:
// s1, s2 are single precsion,
// 1/16 <= s1/s2 <= 1
// 1/16 <= k <= 1 is an integer.
// So the maximal precision of (s1 * k + s2) is:
// prec(s1 * k + s2) = 2 + log2(msb(s2)) - log2(lsb(k_d * s1))
// = 2 + log2(msb(s1)) + 4 - log2(lsb(k_d)) - log2(lsb(s1))
// = 2 + log2(lsb(s1)) + 23 + 4 - (-4) - log2(lsb(s1))
// = 33.
// Thus, the Veltkamp splitting constant is C = 2^33 + 1.
// This is used when FMA instruction is not available.
[[maybe_unused]] constexpr fputil::DoubleDouble split_d(double a) {
fputil::DoubleDouble r{0.0, 0.0};
constexpr double C = 0x1.0p33 + 1.0;
double t1 = C * a;
double t2 = a - t1;
r.hi = t1 + t2;
r.lo = a - r.hi;
return r;
}
// Compute atan( num_d / den_d ) in double-double precision.
// num_d = min(|x|, |y|)
// den_d = max(|x|, |y|)
// q_d = num_d / den_d
// idx, k_d = round( 2^4 * num_d / den_d )
// final_sign = sign of the final result
// const_term = the constant term in the final expression.
float atan2f_double_double(double num_d, double den_d, double q_d, int idx,
double k_d, double final_sign,
const fputil::DoubleDouble &const_term) {
fputil::DoubleDouble q;
double num_r, den_r;
if (idx != 0) {
// The following range reduction is accurate even without fma for
// 1/16 <= n/d <= 1.
// atan(n/d) - atan(idx/16) = atan((n/d - idx/16) / (1 + (n/d) * (idx/16)))
// = atan((n - d*(idx/16)) / (d + n*idx/16))
k_d *= 0x1.0p-4;
num_r = fputil::multiply_add(k_d, -den_d, num_d); // Exact
den_r = fputil::multiply_add(k_d, num_d, den_d); // Exact
q.hi = num_r / den_r;
} else {
// For 0 < n/d < 1/16, we just need to calculate the lower part of their
// quotient.
q.hi = q_d;
num_r = num_d;
den_r = den_d;
}
#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
q.lo = fputil::multiply_add(q.hi, -den_r, num_r) / den_r;
#else
// Compute `(num_r - q.hi * den_r) / den_r` accurately without FMA
// instructions.
fputil::DoubleDouble q_hi_dd = split_d(q.hi);
double t1 = fputil::multiply_add(q_hi_dd.hi, -den_r, num_r); // Exact
double t2 = fputil::multiply_add(q_hi_dd.lo, -den_r, t1);
q.lo = t2 / den_r;
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
// Taylor polynomial, evaluating using Horner's scheme:
// P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
// + x^17/17
// = x*(1 + x^2*(-1/3 + x^2*(1/5 + x^2*(-1/7 + x^2*(1/9 + x^2*
// *(-1/11 + x^2*(1/13 + x^2*(-1/15 + x^2 * 1/17))))))))
fputil::DoubleDouble q2 = fputil::quick_mult(q, q);
fputil::DoubleDouble p_dd =
fputil::polyeval(q2, COEFFS[0], COEFFS[1], COEFFS[2], COEFFS[3],
COEFFS[4], COEFFS[5], COEFFS[6], COEFFS[7], COEFFS[8]);
fputil::DoubleDouble r_dd =
fputil::add(const_term, fputil::multiply_add(q, p_dd, ATAN_I[idx]));
r_dd.hi *= final_sign;
r_dd.lo *= final_sign;
// Make sure the sum is normalized:
fputil::DoubleDouble rr = fputil::exact_add(r_dd.hi, r_dd.lo);
// Round to odd.
uint64_t rr_bits = cpp::bit_cast<uint64_t>(rr.hi);
if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) {
Sign hi_sign = fputil::FPBits<double>(rr.hi).sign();
Sign lo_sign = fputil::FPBits<double>(rr.lo).sign();
if (hi_sign == lo_sign) {
++rr_bits;
} else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) {
--rr_bits;
}
}
return static_cast<float>(cpp::bit_cast<double>(rr_bits));
}
#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
} // anonymous namespace
// There are several range reduction steps we can take for atan2(y, x) as
// follow:
// * Range reduction 1: signness
// atan2(y, x) will return a number between -PI and PI representing the angle
// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
// In particular, we have that:
// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
// Since atan function is odd, we can use the formula:
// atan(-u) = -atan(u)
// to adjust the above conditions a bit further:
// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
// Which can be simplified to:
// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
// = sign(y) * (pi - atan( |y|/|x| )) if x < 0
// * Range reduction 2: reciprocal
// Now that the argument inside atan is positive, we can use the formula:
// atan(1/x) = pi/2 - atan(x)
// to make the argument inside atan <= 1 as follow:
// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
// * Range reduction 3: look up table.
// After the previous two range reduction steps, we reduce the problem to
// compute atan(u) with 0 <= u <= 1, or to be precise:
// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
// An accurate polynomial approximation for the whole [0, 1] input range will
// require a very large degree. To make it more efficient, we reduce the input
// range further by finding an integer idx such that:
// | n/d - idx/16 | <= 1/32.
// In particular,
// idx := 2^-4 * round(2^4 * n/d)
// Then for the fast pass, we find a polynomial approximation for:
// atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16)
// For the accurate pass, we use the addition formula:
// atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (n*idx)/(16*d)) )
// = atan( (n - d * idx/16)/(d + n * idx/16) )
// And finally we use Taylor polynomial to compute the RHS in the accurate pass:
// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 -
// - u^15/15 + u^17/17
// It's error in double-double precision is estimated in Sollya to be:
// > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
// + x^17/17;
// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
// 0x1.aec6f...p-100
// which is about rounding errors of double-double (2^-104).
LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
using namespace inv_trigf_utils_internal;
using FPBits = typename fputil::FPBits<float>;
constexpr double IS_NEG[2] = {1.0, -1.0};
constexpr double PI = 0x1.921fb54442d18p1;
constexpr double PI_LO = 0x1.1a62633145c07p-53;
constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1;
constexpr double PI_OVER_2 = 0x1.921fb54442d18p0;
constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1;
// Adjustment for constant term:
// CONST_ADJ[x_sign][y_sign][recip]
constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = {
{{{0.0, 0.0}, {-PI_LO / 2, -PI_OVER_2}},
{{-0.0, -0.0}, {-PI_LO / 2, -PI_OVER_2}}},
{{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}},
{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}}}};
FPBits x_bits(x), y_bits(y);
bool x_sign = x_bits.sign().is_neg();
bool y_sign = y_bits.sign().is_neg();
x_bits.set_sign(Sign::POS);
y_bits.set_sign(Sign::POS);
uint32_t x_abs = x_bits.uintval();
uint32_t y_abs = y_bits.uintval();
uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs;
uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs;
float num_f = FPBits(min_abs).get_val();
float den_f = FPBits(max_abs).get_val();
double num_d = static_cast<double>(num_f);
double den_d = static_cast<double>(den_f);
if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || num_d == 0.0)) {
if (x_bits.is_nan() || y_bits.is_nan()) {
if (x_bits.is_signaling_nan() || y_bits.is_signaling_nan())
fputil::raise_except_if_required(FE_INVALID);
return FPBits::quiet_nan().get_val();
}
double x_d = static_cast<double>(x);
double y_d = static_cast<double>(y);
size_t x_except = (x_d == 0.0) ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1);
size_t y_except = (y_d == 0.0) ? 0 : (y_abs == 0x7f80'0000 ? 2 : 1);
// Exceptional cases:
// EXCEPT[y_except][x_except][x_is_neg]
// with x_except & y_except:
// 0: zero
// 1: finite, non-zero
// 2: infinity
constexpr double EXCEPTS[3][3][2] = {
{{0.0, PI}, {0.0, PI}, {0.0, PI}},
{{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}},
{{PI_OVER_2, PI_OVER_2},
{PI_OVER_2, PI_OVER_2},
{PI_OVER_4, THREE_PI_OVER_4}},
};
double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign];
return static_cast<float>(r);
}
bool recip = x_abs < y_abs;
double final_sign = IS_NEG[(x_sign != y_sign) != recip];
fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
double q_d = num_d / den_d;
double k_d = fputil::nearest_integer(q_d * 0x1.0p4);
int idx = static_cast<int>(k_d);
double r;
#ifdef LIBC_MATH_HAS_SMALL_TABLES
double p = atan_eval_no_table(num_d, den_d, k_d * 0x1.0p-4);
r = final_sign * (p + (const_term.hi + ATAN_K_OVER_16[idx]));
#else
q_d = fputil::multiply_add(k_d, -0x1.0p-4, q_d);
double p = atan_eval(q_d, idx);
r = final_sign *
fputil::multiply_add(q_d, p, const_term.hi + ATAN_COEFFS[idx][0]);
#endif // LIBC_MATH_HAS_SMALL_TABLES
#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
return static_cast<float>(r);
#else
constexpr uint32_t LOWER_ERR = 4;
// Mask sticky bits in double precision before rounding to single precision.
constexpr uint32_t MASK =
mask_trailing_ones<uint32_t, fputil::FPBits<double>::SIG_LEN -
FPBits::SIG_LEN - 1>();
constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR;
uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(r)) & MASK;
// Ziv's rounding test.
if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR))
return static_cast<float>(r);
return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign,
const_term);
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
return math::atan2f(y, x);
}
} // namespace LIBC_NAMESPACE_DECL
#endif

1
libc/test/shared/CMakeLists.txt
View File

@@ -20,6 +20,7 @@ add_fp_unittest(
libc.src.__support.math.asinhf16
libc.src.__support.math.atan
libc.src.__support.math.atan2
libc.src.__support.math.atan2f
libc.src.__support.math.atanf
libc.src.__support.math.atanf16
libc.src.__support.math.erff

1
libc/test/shared/shared_math_test.cpp
View File

@@ -45,6 +45,7 @@ TEST(LlvmLibcSharedMathTest, AllFloat) {
EXPECT_FP_EQ(0x0p+0f, LIBC_NAMESPACE::shared::acoshf(1.0f));
EXPECT_FP_EQ(0x0p+0f, LIBC_NAMESPACE::shared::asinf(0.0f));
EXPECT_FP_EQ(0x0p+0f, LIBC_NAMESPACE::shared::asinhf(0.0f));
EXPECT_FP_EQ(0x0p+0f, LIBC_NAMESPACE::shared::atan2f(0.0f, 0.0f));
EXPECT_FP_EQ(0x0p+0f, LIBC_NAMESPACE::shared::atanf(0.0f));
EXPECT_FP_EQ(0x0p+0f, LIBC_NAMESPACE::shared::erff(0.0f));
EXPECT_FP_EQ(0x1p+0f, LIBC_NAMESPACE::shared::exp10f(0.0f));

20
utils/bazel/llvm-project-overlay/libc/BUILD.bazel
View File

@@ -2284,6 +2284,22 @@ libc_support_library(
],
)
libc_support_library(
name = "__support_math_atan2f",
hdrs = ["src/__support/math/atan2f.h"],
deps = [
":__support_fputil_fenv_impl",
":__support_fputil_fp_bits",
":__support_fputil_polyeval",
":__support_fputil_double_double",
":__support_fputil_multiply_add",
":__support_fputil_nearest_integer",
":__support_macros_config",
":__support_macros_optimization",
":__support_math_inv_trigf_utils",
],
)
libc_support_library(
name = "__support_math_atanf",
hdrs = ["src/__support/math/atanf.h"],
@@ -2946,9 +2962,7 @@ libc_math_function(
libc_math_function(
name = "atan2f",
additional_deps = [
":__support_fputil_double_double",
":__support_fputil_nearest_integer",
":__support_math_inv_trigf_utils",
":__support_math_atan2f",
],
)