diff options
Diffstat (limited to 'libc/src/math/generic/atan2.cpp')
| -rw-r--r-- | libc/src/math/generic/atan2.cpp | 186 |
1 files changed, 2 insertions, 184 deletions
diff --git a/libc/src/math/generic/atan2.cpp b/libc/src/math/generic/atan2.cpp index aa770de..4aaa63d 100644 --- a/libc/src/math/generic/atan2.cpp +++ b/libc/src/math/generic/atan2.cpp @@ -7,194 +7,12 @@ //===----------------------------------------------------------------------===// #include "src/math/atan2.h" -#include "atan_utils.h" -#include "src/__support/FPUtil/FEnvImpl.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 +#include "src/__support/math/atan2.h" 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()) { - if (x_bits.is_signaling_nan() || y_bits.is_signaling_nan()) - fputil::raise_except_if_required(FE_INVALID); - 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; + return math::atan2(y, x); } } // namespace LIBC_NAMESPACE_DECL |