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//===-- Quad-precision atan2 function -------------------------------------===//
//
// 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
//
//===----------------------------------------------------------------------===//
#include "src/math/atan2f128.h"
#include "atan_utils.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/dyadic_float.h"
#include "src/__support/FPUtil/multiply_add.h"
#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" // LIBC_UNLIKELY
#include "src/__support/macros/properties/types.h"
#include "src/__support/uint128.h"
namespace LIBC_NAMESPACE_DECL {
namespace {
using Float128 = fputil::DyadicFloat<128>;
static constexpr Float128 ZERO = {Sign::POS, 0, 0_u128};
static constexpr Float128 MZERO = {Sign::NEG, 0, 0_u128};
static constexpr Float128 PI = {Sign::POS, -126,
0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
static constexpr Float128 MPI = {Sign::NEG, -126,
0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
static constexpr Float128 PI_OVER_2 = {
Sign::POS, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
static constexpr Float128 MPI_OVER_2 = {
Sign::NEG, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
static constexpr Float128 PI_OVER_4 = {
Sign::POS, -128, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
static constexpr Float128 THREE_PI_OVER_4 = {
Sign::POS, -128, 0x96cbe3f9'990e91a7'9394c9e8'a0a5159d_u128};
// Adjustment for constant term:
// CONST_ADJ[x_sign][y_sign][recip]
static constexpr Float128 CONST_ADJ[2][2][2] = {
{{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}},
{{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}};
} // 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/64 | <= 1/128.
// In particular,
// idx := round(2^6 * n/d)
// Then for the fast pass, we find a polynomial approximation for:
// atan( n/d ) ~ atan( idx/64 ) + (n/d - idx/64) * Q(n/d - idx/64)
// For the accurate pass, we use the addition formula:
// atan( n/d ) - atan( idx/64 ) = atan( (n/d - idx/64)/(1 + (n*idx)/(64*d)) )
// = atan( (n - d*(idx/64))/(d + n*(idx/64)) )
// And for the fast pass, we use degree-13 minimax polynomial to compute the
// RHS:
// atan(u) ~ P(u) = u - c_3 * u^3 + c_5 * u^5 - c_7 * u^7 + c_9 *u^9 -
// - c_11 * u^11 + c_13 * u^13
// with absolute errors bounded by:
// |atan(u) - P(u)| < 2^-121
// and relative errors bounded by:
// |(atan(u) - P(u)) / P(u)| < 2^-114.
LLVM_LIBC_FUNCTION(float128, atan2f128, (float128 y, float128 x)) {
using FPBits = fputil::FPBits<float128>;
using Float128 = fputil::DyadicFloat<128>;
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 = x_bits.abs();
y_bits = y_bits.abs();
UInt128 x_abs = x_bits.uintval();
UInt128 y_abs = y_bits.uintval();
bool recip = x_abs < y_abs;
UInt128 min_abs = recip ? x_abs : y_abs;
UInt128 max_abs = !recip ? x_abs : y_abs;
unsigned min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
unsigned max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
Float128 num(FPBits(min_abs).get_val());
Float128 den(FPBits(max_abs).get_val());
// Check for exceptional cases, whether inputs are 0, inf, nan, or close to
// overflow, or close to underflow.
if (LIBC_UNLIKELY(max_exp >= 0x7fffU || min_exp == 0U)) {
if (x_bits.is_nan() || y_bits.is_nan())
return FPBits::quiet_nan().get_val();
unsigned x_except = x == 0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);
unsigned y_except = y == 0 ? 0 : (FPBits(y_abs).is_inf() ? 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 Float128 EXCEPTS[3][3][2] = {
{{ZERO, PI}, {ZERO, PI}, {ZERO, PI}},
{{PI_OVER_2, PI_OVER_2}, {ZERO, ZERO}, {ZERO, PI}},
{{PI_OVER_2, PI_OVER_2},
{PI_OVER_2, PI_OVER_2},
{PI_OVER_4, THREE_PI_OVER_4}},
};
if ((x_except != 1) || (y_except != 1)) {
Float128 r = EXCEPTS[y_except][x_except][x_sign];
if (y_sign)
r.sign = r.sign.negate();
return static_cast<float128>(r);
}
}
bool final_sign = ((x_sign != y_sign) != recip);
Float128 const_term = CONST_ADJ[x_sign][y_sign][recip];
int exp_diff = den.exponent - num.exponent;
// We have the following bound for normalized n and d:
// 2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1).
if (LIBC_UNLIKELY(exp_diff > FPBits::FRACTION_LEN + 2)) {
if (final_sign)
const_term.sign = const_term.sign.negate();
return static_cast<float128>(const_term);
}
// Take 24 leading bits of num and den to convert to float for fast division.
// We also multiply the numerator by 64 using integer addition directly to the
// exponent field.
float num_f =
cpp::bit_cast<float>(static_cast<uint32_t>(num.mantissa >> 104) +
(6U << fputil::FPBits<float>::FRACTION_LEN));
float den_f = cpp::bit_cast<float>(
static_cast<uint32_t>(den.mantissa >> 104) +
(static_cast<uint32_t>(exp_diff) << fputil::FPBits<float>::FRACTION_LEN));
float k = fputil::nearest_integer(num_f / den_f);
unsigned idx = static_cast<unsigned>(k);
// k_f128 = idx / 64
Float128 k_f128(Sign::POS, -6, Float128::MantissaType(idx));
// Range reduction:
// atan(n/d) - atan(k) = atan((n/d - k/64) / (1 + (n/d) * (k/64)))
// = atan((n - d * k/64)) / (d + n * k/64))
// num_f128 = n - d * k/64
Float128 num_f128 = fputil::multiply_add(den, -k_f128, num);
// den_f128 = d + n * k/64
Float128 den_f128 = fputil::multiply_add(num, k_f128, den);
// q = (n - d * k) / (d + n * k)
Float128 q = fputil::quick_mul(num_f128, fputil::approx_reciprocal(den_f128));
// p ~ atan(q)
Float128 p = atan_eval(q);
Float128 r =
fputil::quick_add(const_term, fputil::quick_add(ATAN_I_F128[idx], p));
if (final_sign)
r.sign = r.sign.negate();
return static_cast<float128>(r);
}
} // namespace LIBC_NAMESPACE_DECL
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