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1//===-- Implementation header for atan2 -------------------------*- 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-exception6//7//===----------------------------------------------------------------------===//8 9#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2_H10#define LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2_H11 12#include "atan_utils.h"13#include "src/__support/FPUtil/FEnvImpl.h"14#include "src/__support/FPUtil/FPBits.h"15#include "src/__support/FPUtil/double_double.h"16#include "src/__support/FPUtil/multiply_add.h"17#include "src/__support/FPUtil/nearest_integer.h"18#include "src/__support/macros/config.h"19#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY20 21namespace LIBC_NAMESPACE_DECL {22 23namespace math {24 25// There are several range reduction steps we can take for atan2(y, x) as26// follow:27 28// * Range reduction 1: signness29// atan2(y, x) will return a number between -PI and PI representing the angle30// forming by the 0x axis and the vector (x, y) on the 0xy-plane.31// In particular, we have that:32// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)33// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)34// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)35// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)36// Since atan function is odd, we can use the formula:37// atan(-u) = -atan(u)38// to adjust the above conditions a bit further:39// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)40// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)41// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)42// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)43// Which can be simplified to:44// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 045// = sign(y) * (pi - atan( |y|/|x| )) if x < 046 47// * Range reduction 2: reciprocal48// Now that the argument inside atan is positive, we can use the formula:49// atan(1/x) = pi/2 - atan(x)50// to make the argument inside atan <= 1 as follow:51// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x52// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|53// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x54// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|55 56// * Range reduction 3: look up table.57// After the previous two range reduction steps, we reduce the problem to58// compute atan(u) with 0 <= u <= 1, or to be precise:59// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).60// An accurate polynomial approximation for the whole [0, 1] input range will61// require a very large degree. To make it more efficient, we reduce the input62// range further by finding an integer idx such that:63// | n/d - idx/64 | <= 1/128.64// In particular,65// idx := round(2^6 * n/d)66// Then for the fast pass, we find a polynomial approximation for:67// atan( n/d ) ~ atan( idx/64 ) + (n/d - idx/64) * Q(n/d - idx/64)68// For the accurate pass, we use the addition formula:69// atan( n/d ) - atan( idx/64 ) = atan( (n/d - idx/64)/(1 + (n*idx)/(64*d)) )70// = atan( (n - d*(idx/64))/(d + n*(idx/64)) )71// And for the fast pass, we use degree-9 Taylor polynomial to compute the RHS:72// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/973// with absolute errors bounded by:74// |atan(u) - P(u)| < |u|^11 / 11 < 2^-8075// and relative errors bounded by:76// |(atan(u) - P(u)) / P(u)| < u^10 / 11 < 2^-73.77 78LIBC_INLINE static constexpr double atan2(double y, double x) {79 using namespace atan_internal;80 using FPBits = fputil::FPBits<double>;81 82 constexpr double IS_NEG[2] = {1.0, -1.0};83 constexpr DoubleDouble ZERO = {0.0, 0.0};84 constexpr DoubleDouble MZERO = {-0.0, -0.0};85 constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p+1};86 constexpr DoubleDouble MPI = {-0x1.1a62633145c07p-53, -0x1.921fb54442d18p+1};87 constexpr DoubleDouble PI_OVER_2 = {0x1.1a62633145c07p-54,88 0x1.921fb54442d18p0};89 constexpr DoubleDouble MPI_OVER_2 = {-0x1.1a62633145c07p-54,90 -0x1.921fb54442d18p0};91 constexpr DoubleDouble PI_OVER_4 = {0x1.1a62633145c07p-55,92 0x1.921fb54442d18p-1};93 constexpr DoubleDouble THREE_PI_OVER_4 = {0x1.a79394c9e8a0ap-54,94 0x1.2d97c7f3321d2p+1};95 // Adjustment for constant term:96 // CONST_ADJ[x_sign][y_sign][recip]97 constexpr DoubleDouble CONST_ADJ[2][2][2] = {98 {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}},99 {{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}};100 101 FPBits x_bits(x), y_bits(y);102 bool x_sign = x_bits.sign().is_neg();103 bool y_sign = y_bits.sign().is_neg();104 x_bits = x_bits.abs();105 y_bits = y_bits.abs();106 uint64_t x_abs = x_bits.uintval();107 uint64_t y_abs = y_bits.uintval();108 bool recip = x_abs < y_abs;109 uint64_t min_abs = recip ? x_abs : y_abs;110 uint64_t max_abs = !recip ? x_abs : y_abs;111 unsigned min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);112 unsigned max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);113 114 double num = FPBits(min_abs).get_val();115 double den = FPBits(max_abs).get_val();116 117 // Check for exceptional cases, whether inputs are 0, inf, nan, or close to118 // overflow, or close to underflow.119 if (LIBC_UNLIKELY(max_exp > 0x7ffU - 128U || min_exp < 128U)) {120 if (x_bits.is_nan() || y_bits.is_nan()) {121 if (x_bits.is_signaling_nan() || y_bits.is_signaling_nan())122 fputil::raise_except_if_required(FE_INVALID);123 return FPBits::quiet_nan().get_val();124 }125 unsigned x_except = x == 0.0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);126 unsigned y_except = y == 0.0 ? 0 : (FPBits(y_abs).is_inf() ? 2 : 1);127 128 // Exceptional cases:129 // EXCEPT[y_except][x_except][x_is_neg]130 // with x_except & y_except:131 // 0: zero132 // 1: finite, non-zero133 // 2: infinity134 constexpr DoubleDouble EXCEPTS[3][3][2] = {135 {{ZERO, PI}, {ZERO, PI}, {ZERO, PI}},136 {{PI_OVER_2, PI_OVER_2}, {ZERO, ZERO}, {ZERO, PI}},137 {{PI_OVER_2, PI_OVER_2},138 {PI_OVER_2, PI_OVER_2},139 {PI_OVER_4, THREE_PI_OVER_4}},140 };141 142 if ((x_except != 1) || (y_except != 1)) {143 DoubleDouble r = EXCEPTS[y_except][x_except][x_sign];144 return fputil::multiply_add(IS_NEG[y_sign], r.hi, IS_NEG[y_sign] * r.lo);145 }146 bool scale_up = min_exp < 128U;147 bool scale_down = max_exp > 0x7ffU - 128U;148 // At least one input is denormal, multiply both numerator and denominator149 // by some large enough power of 2 to normalize denormal inputs.150 if (scale_up) {151 num *= 0x1.0p64;152 if (!scale_down)153 den *= 0x1.0p64;154 } else if (scale_down) {155 den *= 0x1.0p-64;156 if (!scale_up)157 num *= 0x1.0p-64;158 }159 160 min_abs = FPBits(num).uintval();161 max_abs = FPBits(den).uintval();162 min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);163 max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);164 }165 166 double final_sign = IS_NEG[(x_sign != y_sign) != recip];167 DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];168 unsigned exp_diff = max_exp - min_exp;169 // We have the following bound for normalized n and d:170 // 2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1).171 if (LIBC_UNLIKELY(exp_diff > 54)) {172 return fputil::multiply_add(final_sign, const_term.hi,173 final_sign * (const_term.lo + num / den));174 }175 176 double k = fputil::nearest_integer(64.0 * num / den);177 unsigned idx = static_cast<unsigned>(k);178 // k = idx / 64179 k *= 0x1.0p-6;180 181 // Range reduction:182 // atan(n/d) - atan(k/64) = atan((n/d - k/64) / (1 + (n/d) * (k/64)))183 // = atan((n - d * k/64)) / (d + n * k/64))184 DoubleDouble num_k = fputil::exact_mult(num, k);185 DoubleDouble den_k = fputil::exact_mult(den, k);186 187 // num_dd = n - d * k188 DoubleDouble num_dd = fputil::exact_add(num - den_k.hi, -den_k.lo);189 // den_dd = d + n * k190 DoubleDouble den_dd = fputil::exact_add(den, num_k.hi);191 den_dd.lo += num_k.lo;192 193 // q = (n - d * k) / (d + n * k)194 DoubleDouble q = fputil::div(num_dd, den_dd);195 // p ~ atan(q)196 DoubleDouble p = atan_eval(q);197 198 DoubleDouble r = fputil::add(const_term, fputil::add(ATAN_I[idx], p));199 r.hi *= final_sign;200 r.lo *= final_sign;201 202 return r.hi + r.lo;203}204 205} // namespace math206 207} // namespace LIBC_NAMESPACE_DECL208 209#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2_H210