diff options
Diffstat (limited to 'libc/src/math/generic/atan2f.cpp')
| -rw-r--r-- | libc/src/math/generic/atan2f.cpp | 328 |
1 files changed, 2 insertions, 326 deletions
diff --git a/libc/src/math/generic/atan2f.cpp b/libc/src/math/generic/atan2f.cpp index 32b977f..7c56788 100644 --- a/libc/src/math/generic/atan2f.cpp +++ b/libc/src/math/generic/atan2f.cpp @@ -7,336 +7,12 @@ //===----------------------------------------------------------------------===// #include "src/math/atan2f.h" -#include "hdr/fenv_macros.h" -#include "src/__support/FPUtil/FEnvImpl.h" -#include "src/__support/FPUtil/FPBits.h" -#include "src/__support/FPUtil/PolyEval.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/FPUtil/rounding_mode.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY -#include "src/__support/math/inv_trigf_utils.h" - -#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \ - defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT) - -// We use float-float implementation to reduce size. -#include "src/math/generic/atan2f_float.h" - -#else +#include "src/__support/math/atan2f.h" namespace LIBC_NAMESPACE_DECL { -namespace { - -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -// Look up tables for accurate pass: - -// atan(i/16) with i = 0..16, generated by Sollya with: -// > for i from 0 to 16 do { -// a = round(atan(i/16), D, RN); -// b = round(atan(i/16) - a, D, RN); -// print("{", b, ",", a, "},"); -// }; -constexpr fputil::DoubleDouble ATAN_I[17] = { - {0.0, 0.0}, - {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5}, - {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4}, - {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3}, - {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3}, - {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2}, - {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2}, - {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2}, - {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2}, - {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1}, - {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1}, - {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1}, - {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1}, - {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1}, - {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1}, - {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1}, - {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1}, -}; - -// Taylor polynomial, generated by Sollya with: -// > for i from 0 to 8 do { -// j = (-1)^(i + 1)/(2*i + 1); -// a = round(j, D, RN); -// b = round(j - a, D, RN); -// print("{", b, ",", a, "},"); -// }; -constexpr fputil::DoubleDouble COEFFS[9] = { - {0.0, 1.0}, // 1 - {-0x1.5555555555555p-56, -0x1.5555555555555p-2}, // -1/3 - {-0x1.999999999999ap-57, 0x1.999999999999ap-3}, // 1/5 - {-0x1.2492492492492p-57, -0x1.2492492492492p-3}, // -1/7 - {0x1.c71c71c71c71cp-58, 0x1.c71c71c71c71cp-4}, // 1/9 - {0x1.745d1745d1746p-59, -0x1.745d1745d1746p-4}, // -1/11 - {-0x1.3b13b13b13b14p-58, 0x1.3b13b13b13b14p-4}, // 1/13 - {-0x1.1111111111111p-60, -0x1.1111111111111p-4}, // -1/15 - {0x1.e1e1e1e1e1e1ep-61, 0x1.e1e1e1e1e1e1ep-5}, // 1/17 -}; - -// Veltkamp's splitting of a double precision into hi + lo, where the hi part is -// slightly smaller than an even split, so that the product of -// hi * (s1 * k + s2) is exact, -// where: -// s1, s2 are single precsion, -// 1/16 <= s1/s2 <= 1 -// 1/16 <= k <= 1 is an integer. -// So the maximal precision of (s1 * k + s2) is: -// prec(s1 * k + s2) = 2 + log2(msb(s2)) - log2(lsb(k_d * s1)) -// = 2 + log2(msb(s1)) + 4 - log2(lsb(k_d)) - log2(lsb(s1)) -// = 2 + log2(lsb(s1)) + 23 + 4 - (-4) - log2(lsb(s1)) -// = 33. -// Thus, the Veltkamp splitting constant is C = 2^33 + 1. -// This is used when FMA instruction is not available. -[[maybe_unused]] constexpr fputil::DoubleDouble split_d(double a) { - fputil::DoubleDouble r{0.0, 0.0}; - constexpr double C = 0x1.0p33 + 1.0; - double t1 = C * a; - double t2 = a - t1; - r.hi = t1 + t2; - r.lo = a - r.hi; - return r; -} - -// Compute atan( num_d / den_d ) in double-double precision. -// num_d = min(|x|, |y|) -// den_d = max(|x|, |y|) -// q_d = num_d / den_d -// idx, k_d = round( 2^4 * num_d / den_d ) -// final_sign = sign of the final result -// const_term = the constant term in the final expression. -float atan2f_double_double(double num_d, double den_d, double q_d, int idx, - double k_d, double final_sign, - const fputil::DoubleDouble &const_term) { - fputil::DoubleDouble q; - double num_r, den_r; - - if (idx != 0) { - // The following range reduction is accurate even without fma for - // 1/16 <= n/d <= 1. - // atan(n/d) - atan(idx/16) = atan((n/d - idx/16) / (1 + (n/d) * (idx/16))) - // = atan((n - d*(idx/16)) / (d + n*idx/16)) - k_d *= 0x1.0p-4; - num_r = fputil::multiply_add(k_d, -den_d, num_d); // Exact - den_r = fputil::multiply_add(k_d, num_d, den_d); // Exact - q.hi = num_r / den_r; - } else { - // For 0 < n/d < 1/16, we just need to calculate the lower part of their - // quotient. - q.hi = q_d; - num_r = num_d; - den_r = den_d; - } -#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE - q.lo = fputil::multiply_add(q.hi, -den_r, num_r) / den_r; -#else - // Compute `(num_r - q.hi * den_r) / den_r` accurately without FMA - // instructions. - fputil::DoubleDouble q_hi_dd = split_d(q.hi); - double t1 = fputil::multiply_add(q_hi_dd.hi, -den_r, num_r); // Exact - double t2 = fputil::multiply_add(q_hi_dd.lo, -den_r, t1); - q.lo = t2 / den_r; -#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE - - // Taylor polynomial, evaluating using Horner's scheme: - // P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15 - // + x^17/17 - // = x*(1 + x^2*(-1/3 + x^2*(1/5 + x^2*(-1/7 + x^2*(1/9 + x^2* - // *(-1/11 + x^2*(1/13 + x^2*(-1/15 + x^2 * 1/17)))))))) - fputil::DoubleDouble q2 = fputil::quick_mult(q, q); - fputil::DoubleDouble p_dd = - fputil::polyeval(q2, COEFFS[0], COEFFS[1], COEFFS[2], COEFFS[3], - COEFFS[4], COEFFS[5], COEFFS[6], COEFFS[7], COEFFS[8]); - fputil::DoubleDouble r_dd = - fputil::add(const_term, fputil::multiply_add(q, p_dd, ATAN_I[idx])); - r_dd.hi *= final_sign; - r_dd.lo *= final_sign; - - // Make sure the sum is normalized: - fputil::DoubleDouble rr = fputil::exact_add(r_dd.hi, r_dd.lo); - // Round to odd. - uint64_t rr_bits = cpp::bit_cast<uint64_t>(rr.hi); - if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) { - Sign hi_sign = fputil::FPBits<double>(rr.hi).sign(); - Sign lo_sign = fputil::FPBits<double>(rr.lo).sign(); - if (hi_sign == lo_sign) { - ++rr_bits; - } else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) { - --rr_bits; - } - } - - return static_cast<float>(cpp::bit_cast<double>(rr_bits)); -} - -#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -} // 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/16 | <= 1/32. -// In particular, -// idx := 2^-4 * round(2^4 * n/d) -// Then for the fast pass, we find a polynomial approximation for: -// atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16) -// For the accurate pass, we use the addition formula: -// atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (n*idx)/(16*d)) ) -// = atan( (n - d * idx/16)/(d + n * idx/16) ) -// And finally we use Taylor polynomial to compute the RHS in the accurate pass: -// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 - -// - u^15/15 + u^17/17 -// It's error in double-double precision is estimated in Sollya to be: -// > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15 -// + x^17/17; -// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]); -// 0x1.aec6f...p-100 -// which is about rounding errors of double-double (2^-104). - LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) { - using namespace inv_trigf_utils_internal; - using FPBits = typename fputil::FPBits<float>; - constexpr double IS_NEG[2] = {1.0, -1.0}; - constexpr double PI = 0x1.921fb54442d18p1; - constexpr double PI_LO = 0x1.1a62633145c07p-53; - constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1; - constexpr double PI_OVER_2 = 0x1.921fb54442d18p0; - constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1; - // Adjustment for constant term: - // CONST_ADJ[x_sign][y_sign][recip] - constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = { - {{{0.0, 0.0}, {-PI_LO / 2, -PI_OVER_2}}, - {{-0.0, -0.0}, {-PI_LO / 2, -PI_OVER_2}}}, - {{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}}, - {{-PI_LO, -PI}, {PI_LO / 2, 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.set_sign(Sign::POS); - y_bits.set_sign(Sign::POS); - uint32_t x_abs = x_bits.uintval(); - uint32_t y_abs = y_bits.uintval(); - uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs; - uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs; - float num_f = FPBits(min_abs).get_val(); - float den_f = FPBits(max_abs).get_val(); - double num_d = static_cast<double>(num_f); - double den_d = static_cast<double>(den_f); - - if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || num_d == 0.0)) { - 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(); - } - double x_d = static_cast<double>(x); - double y_d = static_cast<double>(y); - size_t x_except = (x_d == 0.0) ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1); - size_t y_except = (y_d == 0.0) ? 0 : (y_abs == 0x7f80'0000 ? 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 double EXCEPTS[3][3][2] = { - {{0.0, PI}, {0.0, PI}, {0.0, PI}}, - {{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}}, - {{PI_OVER_2, PI_OVER_2}, - {PI_OVER_2, PI_OVER_2}, - {PI_OVER_4, THREE_PI_OVER_4}}, - }; - - double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign]; - - return static_cast<float>(r); - } - - bool recip = x_abs < y_abs; - double final_sign = IS_NEG[(x_sign != y_sign) != recip]; - fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip]; - double q_d = num_d / den_d; - - double k_d = fputil::nearest_integer(q_d * 0x1.0p4); - int idx = static_cast<int>(k_d); - double r; - -#ifdef LIBC_MATH_HAS_SMALL_TABLES - double p = atan_eval_no_table(num_d, den_d, k_d * 0x1.0p-4); - r = final_sign * (p + (const_term.hi + ATAN_K_OVER_16[idx])); -#else - q_d = fputil::multiply_add(k_d, -0x1.0p-4, q_d); - - double p = atan_eval(q_d, idx); - r = final_sign * - fputil::multiply_add(q_d, p, const_term.hi + ATAN_COEFFS[idx][0]); -#endif // LIBC_MATH_HAS_SMALL_TABLES - -#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - return static_cast<float>(r); -#else - constexpr uint32_t LOWER_ERR = 4; - // Mask sticky bits in double precision before rounding to single precision. - constexpr uint32_t MASK = - mask_trailing_ones<uint32_t, fputil::FPBits<double>::SIG_LEN - - FPBits::SIG_LEN - 1>(); - constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR; - - uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(r)) & MASK; - - // Ziv's rounding test. - if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR)) - return static_cast<float>(r); - - return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign, - const_term); -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS + return math::atan2f(y, x); } } // namespace LIBC_NAMESPACE_DECL - -#endif |