//===-- Implementation header for atan2f128 ---------------------*- 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_ATAN2F128_H #define LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2F128_H #include "include/llvm-libc-types/float128.h" #ifdef LIBC_TYPES_HAS_FLOAT128 #include "atan_utils.h" #include "src/__support/FPUtil/FPBits.h" #include "src/__support/FPUtil/dyadic_float.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/uint128.h" namespace LIBC_NAMESPACE_DECL { namespace math { // 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. LIBC_INLINE static constexpr float128 atan2f128(float128 y, float128 x) { using Float128 = fputil::DyadicFloat<128>; constexpr Float128 ZERO = {Sign::POS, 0, 0_u128}; constexpr Float128 MZERO = {Sign::NEG, 0, 0_u128}; constexpr Float128 PI = {Sign::POS, -126, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128}; constexpr Float128 MPI = {Sign::NEG, -126, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128}; constexpr Float128 PI_OVER_2 = {Sign::POS, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128}; constexpr Float128 MPI_OVER_2 = {Sign::NEG, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128}; constexpr Float128 PI_OVER_4 = {Sign::POS, -128, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128}; 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] constexpr Float128 CONST_ADJ[2][2][2] = { {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}}, {{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}}; using namespace atan_internal; using FPBits = fputil::FPBits; 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(min_abs >> FPBits::FRACTION_LEN); unsigned max_exp = static_cast(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(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(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(static_cast(num.mantissa >> 104) + (6U << fputil::FPBits::FRACTION_LEN)); float den_f = cpp::bit_cast( static_cast(den.mantissa >> 104) + (static_cast(exp_diff) << fputil::FPBits::FRACTION_LEN)); float k = fputil::nearest_integer(num_f / den_f); unsigned idx = static_cast(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(r); } } // namespace math } // namespace LIBC_NAMESPACE_DECL #endif // LIBC_TYPES_HAS_FLOAT128 #endif // LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2F128_H