| //===-- Double-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/atan2.h" |
| #include "atan_utils.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/macros/config.h" |
| #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY |
| |
| namespace LIBC_NAMESPACE_DECL { |
| |
| // 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-9 Taylor polynomial to compute the RHS: |
| // atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 |
| // with absolute errors bounded by: |
| // |atan(u) - P(u)| < |u|^11 / 11 < 2^-80 |
| // and relative errors bounded by: |
| // |(atan(u) - P(u)) / P(u)| < u^10 / 11 < 2^-73. |
| |
| LLVM_LIBC_FUNCTION(double, atan2, (double y, double x)) { |
| using FPBits = fputil::FPBits<double>; |
| |
| constexpr double IS_NEG[2] = {1.0, -1.0}; |
| constexpr DoubleDouble ZERO = {0.0, 0.0}; |
| constexpr DoubleDouble MZERO = {-0.0, -0.0}; |
| constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p+1}; |
| constexpr DoubleDouble MPI = {-0x1.1a62633145c07p-53, -0x1.921fb54442d18p+1}; |
| constexpr DoubleDouble PI_OVER_2 = {0x1.1a62633145c07p-54, |
| 0x1.921fb54442d18p0}; |
| constexpr DoubleDouble MPI_OVER_2 = {-0x1.1a62633145c07p-54, |
| -0x1.921fb54442d18p0}; |
| constexpr DoubleDouble PI_OVER_4 = {0x1.1a62633145c07p-55, |
| 0x1.921fb54442d18p-1}; |
| constexpr DoubleDouble THREE_PI_OVER_4 = {0x1.a79394c9e8a0ap-54, |
| 0x1.2d97c7f3321d2p+1}; |
| // Adjustment for constant term: |
| // CONST_ADJ[x_sign][y_sign][recip] |
| constexpr DoubleDouble CONST_ADJ[2][2][2] = { |
| {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}}, |
| {{MPI, PI_OVER_2}, {MPI, 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 = x_bits.abs(); |
| y_bits = y_bits.abs(); |
| uint64_t x_abs = x_bits.uintval(); |
| uint64_t y_abs = y_bits.uintval(); |
| bool recip = x_abs < y_abs; |
| uint64_t min_abs = recip ? x_abs : y_abs; |
| uint64_t 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); |
| |
| double num = FPBits(min_abs).get_val(); |
| double 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 > 0x7ffU - 128U || min_exp < 128U)) { |
| if (x_bits.is_nan() || y_bits.is_nan()) |
| return FPBits::quiet_nan().get_val(); |
| unsigned x_except = x == 0.0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1); |
| unsigned y_except = y == 0.0 ? 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 DoubleDouble 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)) { |
| DoubleDouble r = EXCEPTS[y_except][x_except][x_sign]; |
| return fputil::multiply_add(IS_NEG[y_sign], r.hi, IS_NEG[y_sign] * r.lo); |
| } |
| bool scale_up = min_exp < 128U; |
| bool scale_down = max_exp > 0x7ffU - 128U; |
| // At least one input is denormal, multiply both numerator and denominator |
| // by some large enough power of 2 to normalize denormal inputs. |
| if (scale_up) { |
| num *= 0x1.0p64; |
| if (!scale_down) |
| den *= 0x1.0p64; |
| } else if (scale_down) { |
| den *= 0x1.0p-64; |
| if (!scale_up) |
| num *= 0x1.0p-64; |
| } |
| |
| min_abs = FPBits(num).uintval(); |
| max_abs = FPBits(den).uintval(); |
| min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN); |
| max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN); |
| } |
| |
| double final_sign = IS_NEG[(x_sign != y_sign) != recip]; |
| DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip]; |
| unsigned exp_diff = max_exp - min_exp; |
| // 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 > 54)) { |
| return fputil::multiply_add(final_sign, const_term.hi, |
| final_sign * (const_term.lo + num / den)); |
| } |
| |
| double k = fputil::nearest_integer(64.0 * num / den); |
| unsigned idx = static_cast<unsigned>(k); |
| // k = idx / 64 |
| k *= 0x1.0p-6; |
| |
| // Range reduction: |
| // atan(n/d) - atan(k/64) = atan((n/d - k/64) / (1 + (n/d) * (k/64))) |
| // = atan((n - d * k/64)) / (d + n * k/64)) |
| DoubleDouble num_k = fputil::exact_mult(num, k); |
| DoubleDouble den_k = fputil::exact_mult(den, k); |
| |
| // num_dd = n - d * k |
| DoubleDouble num_dd = fputil::exact_add(num - den_k.hi, -den_k.lo); |
| // den_dd = d + n * k |
| DoubleDouble den_dd = fputil::exact_add(den, num_k.hi); |
| den_dd.lo += num_k.lo; |
| |
| // q = (n - d * k) / (d + n * k) |
| DoubleDouble q = fputil::div(num_dd, den_dd); |
| // p ~ atan(q) |
| DoubleDouble p = atan_eval(q); |
| |
| DoubleDouble r = fputil::add(const_term, fputil::add(ATAN_I[idx], p)); |
| r.hi *= final_sign; |
| r.lo *= final_sign; |
| |
| return r.hi + r.lo; |
| } |
| |
| } // namespace LIBC_NAMESPACE_DECL |