1 //===-- Double-precision atan2 function -----------------------------------===// 2 // 3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. 4 // See https://llvm.org/LICENSE.txt for license information. 5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception 6 // 7 //===----------------------------------------------------------------------===// 8 9 #include "src/math/atan2.h" 10 #include "inv_trigf_utils.h" 11 #include "src/__support/FPUtil/FPBits.h" 12 #include "src/__support/FPUtil/PolyEval.h" 13 #include "src/__support/FPUtil/double_double.h" 14 #include "src/__support/FPUtil/multiply_add.h" 15 #include "src/__support/FPUtil/nearest_integer.h" 16 #include "src/__support/FPUtil/rounding_mode.h" 17 #include "src/__support/macros/config.h" 18 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 19 20 namespace LIBC_NAMESPACE_DECL { 21 22 namespace { 23 24 using DoubleDouble = fputil::DoubleDouble; 25 26 // atan(i/64) with i = 0..64, generated by Sollya with: 27 // > for i from 0 to 64 do { 28 // a = round(atan(i/64), D, RN); 29 // b = round(atan(i/64) - a, D, RN); 30 // print("{", b, ",", a, "},"); 31 // }; 32 constexpr fputil::DoubleDouble ATAN_I[65] = { 33 {0.0, 0.0}, 34 {-0x1.220c39d4dff5p-61, 0x1.fff555bbb729bp-7}, 35 {-0x1.5ec431444912cp-60, 0x1.ffd55bba97625p-6}, 36 {-0x1.86ef8f794f105p-63, 0x1.7fb818430da2ap-5}, 37 {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5}, 38 {0x1.ac4ce285df847p-58, 0x1.3f59f0e7c559dp-4}, 39 {-0x1.cfb654c0c3d98p-58, 0x1.7ee182602f10fp-4}, 40 {0x1.f7b8f29a05987p-58, 0x1.be39ebe6f07c3p-4}, 41 {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4}, 42 {-0x1.b485914dacf8cp-59, 0x1.1e1fafb043727p-3}, 43 {0x1.61a3b0ce9281bp-57, 0x1.3d6eee8c6626cp-3}, 44 {-0x1.054ab2c010f3dp-58, 0x1.5c9811e3ec26ap-3}, 45 {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3}, 46 {0x1.cf601e7b4348ep-59, 0x1.9a6a8e96c8626p-3}, 47 {0x1.17b10d2e0e5abp-61, 0x1.b90d7529260a2p-3}, 48 {0x1.c648d1534597ep-57, 0x1.d77d5df205736p-3}, 49 {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3}, 50 {0x1.62e47390cb865p-56, 0x1.09dc597d86362p-2}, 51 {0x1.30ca4748b1bf9p-57, 0x1.18bf5a30bf178p-2}, 52 {-0x1.077cdd36dfc81p-56, 0x1.278372057ef46p-2}, 53 {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2}, 54 {-0x1.5d5e43c55b3bap-56, 0x1.44aa436c2af0ap-2}, 55 {-0x1.2566480884082p-57, 0x1.530ad9951cd4ap-2}, 56 {-0x1.a725715711fp-56, 0x1.614840309cfe2p-2}, 57 {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2}, 58 {0x1.69c885c2b249ap-56, 0x1.7d5604b63b3f7p-2}, 59 {0x1.b6d0ba3748fa8p-56, 0x1.8b24d394a1b25p-2}, 60 {0x1.9e6c988fd0a77p-56, 0x1.98cd5454d6b18p-2}, 61 {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2}, 62 {0x1.ae187b1ca504p-56, 0x1.b3a911da65c6cp-2}, 63 {-0x1.cc1ce70934c34p-56, 0x1.c0db4c94ec9fp-2}, 64 {-0x1.a2cfa4418f1adp-56, 0x1.cde53432c1351p-2}, 65 {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2}, 66 {0x1.0e53dc1bf3435p-56, 0x1.e77eb7f175a34p-2}, 67 {-0x1.a3992dc382a23p-57, 0x1.f40dd0b541418p-2}, 68 {-0x1.b32c949c9d593p-55, 0x1.0039c73c1a40cp-1}, 69 {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1}, 70 {0x1.974fa13b5404fp-58, 0x1.0c6145b5b43dap-1}, 71 {-0x1.2bdaee1c0ee35p-58, 0x1.1255d9bfbd2a9p-1}, 72 {0x1.c621cec00c301p-55, 0x1.1835a88be7c13p-1}, 73 {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1}, 74 {0x1.c421c9f38224ep-57, 0x1.23b71e2cc9e6ap-1}, 75 {-0x1.09e73b0c6c087p-56, 0x1.2958e59308e31p-1}, 76 {0x1.c5d5e9ff0cf8dp-55, 0x1.2ee628406cbcap-1}, 77 {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1}, 78 {-0x1.2304331d8bf46p-55, 0x1.39c391cd4171ap-1}, 79 {0x1.ecf8b492644fp-56, 0x1.3f13fb89e96f4p-1}, 80 {-0x1.f76d0163f79c8p-56, 0x1.445065b795b56p-1}, 81 {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1}, 82 {0x1.4a33dbeb3796cp-55, 0x1.4e8de5bb6ec04p-1}, 83 {-0x1.1bb74abda520cp-55, 0x1.538f57b89061fp-1}, 84 {-0x1.5e5c9d8c5a95p-56, 0x1.587d81f732fbbp-1}, 85 {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1}, 86 {-0x1.2b785350ee8c1p-57, 0x1.6220d115d7b8ep-1}, 87 {-0x1.6ea6febe8bbbap-56, 0x1.66d663923e087p-1}, 88 {-0x1.a80386188c50ep-55, 0x1.6b798920b3d99p-1}, 89 {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1}, 90 {0x1.7b2a6165884a1p-59, 0x1.748978fba8e0fp-1}, 91 {0x1.406a08980374p-55, 0x1.78f6bbd5d315ep-1}, 92 {0x1.560821e2f3aa9p-55, 0x1.7d528289fa093p-1}, 93 {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1}, 94 {0x1.6b66e7fc8b8c3p-57, 0x1.85d69576cc2c5p-1}, 95 {-0x1.55b9a5e177a1bp-55, 0x1.89ff5ff57f1f8p-1}, 96 {-0x1.ec182ab042f61p-56, 0x1.8e17aa99cc05ep-1}, 97 {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1}, 98 }; 99 100 // Approximate atan(x) for |x| <= 2^-7. 101 // Using degree-9 Taylor polynomial: 102 // P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9; 103 // Then the absolute error is bounded by: 104 // |atan(x) - P(x)| < |x|^11/11 < 2^(-7*11) / 11 < 2^-80. 105 // And the relative error is bounded by: 106 // |(atan(x) - P(x))/atan(x)| < |x|^10 / 10 < 2^-73. 107 // For x = x_hi + x_lo, fully expand the polynomial and drop any terms less than 108 // ulp(x_hi^3 / 3) gives us: 109 // P(x) ~ x_hi - x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 + 110 // + x_lo * (1 - x_hi^2 + x_hi^4) 111 DoubleDouble atan_eval(const DoubleDouble &x) { 112 DoubleDouble p; 113 p.hi = x.hi; 114 double x_hi_sq = x.hi * x.hi; 115 // c0 ~ x_hi^2 * 1/5 - 1/3 116 double c0 = fputil::multiply_add(x_hi_sq, 0x1.999999999999ap-3, 117 -0x1.5555555555555p-2); 118 // c1 ~ x_hi^2 * 1/9 - 1/7 119 double c1 = fputil::multiply_add(x_hi_sq, 0x1.c71c71c71c71cp-4, 120 -0x1.2492492492492p-3); 121 // x_hi^3 122 double x_hi_3 = x_hi_sq * x.hi; 123 // x_hi^4 124 double x_hi_4 = x_hi_sq * x_hi_sq; 125 // d0 ~ 1/3 - x_hi^2 / 5 + x_hi^4 / 7 - x_hi^6 / 9 126 double d0 = fputil::multiply_add(x_hi_4, c1, c0); 127 // x_lo - x_lo * x_hi^2 + x_lo * x_hi^4 128 double d1 = fputil::multiply_add(x_hi_4 - x_hi_sq, x.lo, x.lo); 129 // p.lo ~ -x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 + 130 // + x_lo * (1 - x_hi^2 + x_hi^4) 131 p.lo = fputil::multiply_add(x_hi_3, d0, d1); 132 return p; 133 } 134 135 } // anonymous namespace 136 137 // There are several range reduction steps we can take for atan2(y, x) as 138 // follow: 139 140 // * Range reduction 1: signness 141 // atan2(y, x) will return a number between -PI and PI representing the angle 142 // forming by the 0x axis and the vector (x, y) on the 0xy-plane. 143 // In particular, we have that: 144 // atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant) 145 // = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant) 146 // = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant) 147 // = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant) 148 // Since atan function is odd, we can use the formula: 149 // atan(-u) = -atan(u) 150 // to adjust the above conditions a bit further: 151 // atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant) 152 // = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant) 153 // = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant) 154 // = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant) 155 // Which can be simplified to: 156 // atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0 157 // = sign(y) * (pi - atan( |y|/|x| )) if x < 0 158 159 // * Range reduction 2: reciprocal 160 // Now that the argument inside atan is positive, we can use the formula: 161 // atan(1/x) = pi/2 - atan(x) 162 // to make the argument inside atan <= 1 as follow: 163 // atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x 164 // = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y| 165 // = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x 166 // = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y| 167 168 // * Range reduction 3: look up table. 169 // After the previous two range reduction steps, we reduce the problem to 170 // compute atan(u) with 0 <= u <= 1, or to be precise: 171 // atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|). 172 // An accurate polynomial approximation for the whole [0, 1] input range will 173 // require a very large degree. To make it more efficient, we reduce the input 174 // range further by finding an integer idx such that: 175 // | n/d - idx/64 | <= 1/128. 176 // In particular, 177 // idx := round(2^6 * n/d) 178 // Then for the fast pass, we find a polynomial approximation for: 179 // atan( n/d ) ~ atan( idx/64 ) + (n/d - idx/64) * Q(n/d - idx/64) 180 // For the accurate pass, we use the addition formula: 181 // atan( n/d ) - atan( idx/64 ) = atan( (n/d - idx/64)/(1 + (n*idx)/(64*d)) ) 182 // = atan( (n - d*(idx/64))/(d + n*(idx/64)) ) 183 // And for the fast pass, we use degree-9 Taylor polynomial to compute the RHS: 184 // atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 185 // with absolute errors bounded by: 186 // |atan(u) - P(u)| < |u|^11 / 11 < 2^-80 187 // and relative errors bounded by: 188 // |(atan(u) - P(u)) / P(u)| < u^10 / 11 < 2^-73. 189 190 LLVM_LIBC_FUNCTION(double, atan2, (double y, double x)) { 191 using FPBits = fputil::FPBits<double>; 192 193 constexpr double IS_NEG[2] = {1.0, -1.0}; 194 constexpr DoubleDouble ZERO = {0.0, 0.0}; 195 constexpr DoubleDouble MZERO = {-0.0, -0.0}; 196 constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p+1}; 197 constexpr DoubleDouble MPI = {-0x1.1a62633145c07p-53, -0x1.921fb54442d18p+1}; 198 constexpr DoubleDouble PI_OVER_2 = {0x1.1a62633145c07p-54, 199 0x1.921fb54442d18p0}; 200 constexpr DoubleDouble MPI_OVER_2 = {-0x1.1a62633145c07p-54, 201 -0x1.921fb54442d18p0}; 202 constexpr DoubleDouble PI_OVER_4 = {0x1.1a62633145c07p-55, 203 0x1.921fb54442d18p-1}; 204 constexpr DoubleDouble THREE_PI_OVER_4 = {0x1.a79394c9e8a0ap-54, 205 0x1.2d97c7f3321d2p+1}; 206 // Adjustment for constant term: 207 // CONST_ADJ[x_sign][y_sign][recip] 208 constexpr DoubleDouble CONST_ADJ[2][2][2] = { 209 {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}}, 210 {{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}}; 211 212 FPBits x_bits(x), y_bits(y); 213 bool x_sign = x_bits.sign().is_neg(); 214 bool y_sign = y_bits.sign().is_neg(); 215 x_bits = x_bits.abs(); 216 y_bits = y_bits.abs(); 217 uint64_t x_abs = x_bits.uintval(); 218 uint64_t y_abs = y_bits.uintval(); 219 bool recip = x_abs < y_abs; 220 uint64_t min_abs = recip ? x_abs : y_abs; 221 uint64_t max_abs = !recip ? x_abs : y_abs; 222 unsigned min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN); 223 unsigned max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN); 224 225 double num = FPBits(min_abs).get_val(); 226 double den = FPBits(max_abs).get_val(); 227 228 // Check for exceptional cases, whether inputs are 0, inf, nan, or close to 229 // overflow, or close to underflow. 230 if (LIBC_UNLIKELY(max_exp > 0x7ffU - 128U || min_exp < 128U)) { 231 if (x_bits.is_nan() || y_bits.is_nan()) 232 return FPBits::quiet_nan().get_val(); 233 unsigned x_except = x == 0.0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1); 234 unsigned y_except = y == 0.0 ? 0 : (FPBits(y_abs).is_inf() ? 2 : 1); 235 236 // Exceptional cases: 237 // EXCEPT[y_except][x_except][x_is_neg] 238 // with x_except & y_except: 239 // 0: zero 240 // 1: finite, non-zero 241 // 2: infinity 242 constexpr DoubleDouble EXCEPTS[3][3][2] = { 243 {{ZERO, PI}, {ZERO, PI}, {ZERO, PI}}, 244 {{PI_OVER_2, PI_OVER_2}, {ZERO, ZERO}, {ZERO, PI}}, 245 {{PI_OVER_2, PI_OVER_2}, 246 {PI_OVER_2, PI_OVER_2}, 247 {PI_OVER_4, THREE_PI_OVER_4}}, 248 }; 249 250 if ((x_except != 1) || (y_except != 1)) { 251 DoubleDouble r = EXCEPTS[y_except][x_except][x_sign]; 252 return fputil::multiply_add(IS_NEG[y_sign], r.hi, IS_NEG[y_sign] * r.lo); 253 } 254 bool scale_up = min_exp < 128U; 255 bool scale_down = max_exp > 0x7ffU - 128U; 256 // At least one input is denormal, multiply both numerator and denominator 257 // by some large enough power of 2 to normalize denormal inputs. 258 if (scale_up) { 259 num *= 0x1.0p64; 260 if (!scale_down) 261 den *= 0x1.0p64; 262 } else if (scale_down) { 263 den *= 0x1.0p-64; 264 if (!scale_up) 265 num *= 0x1.0p-64; 266 } 267 268 min_abs = FPBits(num).uintval(); 269 max_abs = FPBits(den).uintval(); 270 min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN); 271 max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN); 272 } 273 274 double final_sign = IS_NEG[(x_sign != y_sign) != recip]; 275 DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip]; 276 unsigned exp_diff = max_exp - min_exp; 277 // We have the following bound for normalized n and d: 278 // 2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1). 279 if (LIBC_UNLIKELY(exp_diff > 54)) { 280 return fputil::multiply_add(final_sign, const_term.hi, 281 final_sign * (const_term.lo + num / den)); 282 } 283 284 double k = fputil::nearest_integer(64.0 * num / den); 285 unsigned idx = static_cast<unsigned>(k); 286 // k = idx / 64 287 k *= 0x1.0p-6; 288 289 // Range reduction: 290 // atan(n/d) - atan(k/64) = atan((n/d - k/64) / (1 + (n/d) * (k/64))) 291 // = atan((n - d * k/64)) / (d + n * k/64)) 292 DoubleDouble num_k = fputil::exact_mult(num, k); 293 DoubleDouble den_k = fputil::exact_mult(den, k); 294 295 // num_dd = n - d * k 296 DoubleDouble num_dd = fputil::exact_add(num - den_k.hi, -den_k.lo); 297 // den_dd = d + n * k 298 DoubleDouble den_dd = fputil::exact_add(den, num_k.hi); 299 den_dd.lo += num_k.lo; 300 301 // q = (n - d * k) / (d + n * k) 302 DoubleDouble q = fputil::div(num_dd, den_dd); 303 // p ~ atan(q) 304 DoubleDouble p = atan_eval(q); 305 306 DoubleDouble r = fputil::add(const_term, fputil::add(ATAN_I[idx], p)); 307 r.hi *= final_sign; 308 r.lo *= final_sign; 309 310 return r.hi + r.lo; 311 } 312 313 } // namespace LIBC_NAMESPACE_DECL 314