//===-- 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 "inv_trigf_utils.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 namespace LIBC_NAMESPACE_DECL { namespace { using DoubleDouble = fputil::DoubleDouble; // atan(i/64) with i = 0..64, generated by Sollya with: // > for i from 0 to 64 do { // a = round(atan(i/64), D, RN); // b = round(atan(i/64) - a, D, RN); // print("{", b, ",", a, "},"); // }; constexpr fputil::DoubleDouble ATAN_I[65] = { {0.0, 0.0}, {-0x1.220c39d4dff5p-61, 0x1.fff555bbb729bp-7}, {-0x1.5ec431444912cp-60, 0x1.ffd55bba97625p-6}, {-0x1.86ef8f794f105p-63, 0x1.7fb818430da2ap-5}, {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5}, {0x1.ac4ce285df847p-58, 0x1.3f59f0e7c559dp-4}, {-0x1.cfb654c0c3d98p-58, 0x1.7ee182602f10fp-4}, {0x1.f7b8f29a05987p-58, 0x1.be39ebe6f07c3p-4}, {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4}, {-0x1.b485914dacf8cp-59, 0x1.1e1fafb043727p-3}, {0x1.61a3b0ce9281bp-57, 0x1.3d6eee8c6626cp-3}, {-0x1.054ab2c010f3dp-58, 0x1.5c9811e3ec26ap-3}, {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3}, {0x1.cf601e7b4348ep-59, 0x1.9a6a8e96c8626p-3}, {0x1.17b10d2e0e5abp-61, 0x1.b90d7529260a2p-3}, {0x1.c648d1534597ep-57, 0x1.d77d5df205736p-3}, {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3}, {0x1.62e47390cb865p-56, 0x1.09dc597d86362p-2}, {0x1.30ca4748b1bf9p-57, 0x1.18bf5a30bf178p-2}, {-0x1.077cdd36dfc81p-56, 0x1.278372057ef46p-2}, {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2}, {-0x1.5d5e43c55b3bap-56, 0x1.44aa436c2af0ap-2}, {-0x1.2566480884082p-57, 0x1.530ad9951cd4ap-2}, {-0x1.a725715711fp-56, 0x1.614840309cfe2p-2}, {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2}, {0x1.69c885c2b249ap-56, 0x1.7d5604b63b3f7p-2}, {0x1.b6d0ba3748fa8p-56, 0x1.8b24d394a1b25p-2}, {0x1.9e6c988fd0a77p-56, 0x1.98cd5454d6b18p-2}, {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2}, {0x1.ae187b1ca504p-56, 0x1.b3a911da65c6cp-2}, {-0x1.cc1ce70934c34p-56, 0x1.c0db4c94ec9fp-2}, {-0x1.a2cfa4418f1adp-56, 0x1.cde53432c1351p-2}, {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2}, {0x1.0e53dc1bf3435p-56, 0x1.e77eb7f175a34p-2}, {-0x1.a3992dc382a23p-57, 0x1.f40dd0b541418p-2}, {-0x1.b32c949c9d593p-55, 0x1.0039c73c1a40cp-1}, {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1}, {0x1.974fa13b5404fp-58, 0x1.0c6145b5b43dap-1}, {-0x1.2bdaee1c0ee35p-58, 0x1.1255d9bfbd2a9p-1}, {0x1.c621cec00c301p-55, 0x1.1835a88be7c13p-1}, {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1}, {0x1.c421c9f38224ep-57, 0x1.23b71e2cc9e6ap-1}, {-0x1.09e73b0c6c087p-56, 0x1.2958e59308e31p-1}, {0x1.c5d5e9ff0cf8dp-55, 0x1.2ee628406cbcap-1}, {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1}, {-0x1.2304331d8bf46p-55, 0x1.39c391cd4171ap-1}, {0x1.ecf8b492644fp-56, 0x1.3f13fb89e96f4p-1}, {-0x1.f76d0163f79c8p-56, 0x1.445065b795b56p-1}, {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1}, {0x1.4a33dbeb3796cp-55, 0x1.4e8de5bb6ec04p-1}, {-0x1.1bb74abda520cp-55, 0x1.538f57b89061fp-1}, {-0x1.5e5c9d8c5a95p-56, 0x1.587d81f732fbbp-1}, {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1}, {-0x1.2b785350ee8c1p-57, 0x1.6220d115d7b8ep-1}, {-0x1.6ea6febe8bbbap-56, 0x1.66d663923e087p-1}, {-0x1.a80386188c50ep-55, 0x1.6b798920b3d99p-1}, {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1}, {0x1.7b2a6165884a1p-59, 0x1.748978fba8e0fp-1}, {0x1.406a08980374p-55, 0x1.78f6bbd5d315ep-1}, {0x1.560821e2f3aa9p-55, 0x1.7d528289fa093p-1}, {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1}, {0x1.6b66e7fc8b8c3p-57, 0x1.85d69576cc2c5p-1}, {-0x1.55b9a5e177a1bp-55, 0x1.89ff5ff57f1f8p-1}, {-0x1.ec182ab042f61p-56, 0x1.8e17aa99cc05ep-1}, {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1}, }; // Approximate atan(x) for |x| <= 2^-7. // Using degree-9 Taylor polynomial: // P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9; // Then the absolute error is bounded by: // |atan(x) - P(x)| < |x|^11/11 < 2^(-7*11) / 11 < 2^-80. // And the relative error is bounded by: // |(atan(x) - P(x))/atan(x)| < |x|^10 / 10 < 2^-73. // For x = x_hi + x_lo, fully expand the polynomial and drop any terms less than // ulp(x_hi^3 / 3) gives us: // P(x) ~ x_hi - x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 + // + x_lo * (1 - x_hi^2 + x_hi^4) DoubleDouble atan_eval(const DoubleDouble &x) { DoubleDouble p; p.hi = x.hi; double x_hi_sq = x.hi * x.hi; // c0 ~ x_hi^2 * 1/5 - 1/3 double c0 = fputil::multiply_add(x_hi_sq, 0x1.999999999999ap-3, -0x1.5555555555555p-2); // c1 ~ x_hi^2 * 1/9 - 1/7 double c1 = fputil::multiply_add(x_hi_sq, 0x1.c71c71c71c71cp-4, -0x1.2492492492492p-3); // x_hi^3 double x_hi_3 = x_hi_sq * x.hi; // x_hi^4 double x_hi_4 = x_hi_sq * x_hi_sq; // d0 ~ 1/3 - x_hi^2 / 5 + x_hi^4 / 7 - x_hi^6 / 9 double d0 = fputil::multiply_add(x_hi_4, c1, c0); // x_lo - x_lo * x_hi^2 + x_lo * x_hi^4 double d1 = fputil::multiply_add(x_hi_4 - x_hi_sq, x.lo, x.lo); // p.lo ~ -x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 + // + x_lo * (1 - x_hi^2 + x_hi^4) p.lo = fputil::multiply_add(x_hi_3, d0, d1); return p; } } // 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-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; 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(min_abs >> FPBits::FRACTION_LEN); unsigned max_exp = static_cast(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(min_abs >> FPBits::FRACTION_LEN); max_exp = static_cast(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(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