| //===-- 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 |