1 //===-- Single-precision atan2f 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/atan2f.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 #ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS 25 26 // Look up tables for accurate pass: 27 28 // atan(i/16) with i = 0..16, generated by Sollya with: 29 // > for i from 0 to 16 do { 30 // a = round(atan(i/16), D, RN); 31 // b = round(atan(i/16) - a, D, RN); 32 // print("{", b, ",", a, "},"); 33 // }; 34 constexpr fputil::DoubleDouble ATAN_I[17] = { 35 {0.0, 0.0}, 36 {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5}, 37 {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4}, 38 {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3}, 39 {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3}, 40 {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2}, 41 {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2}, 42 {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2}, 43 {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2}, 44 {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1}, 45 {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1}, 46 {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1}, 47 {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1}, 48 {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1}, 49 {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1}, 50 {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1}, 51 {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1}, 52 }; 53 54 // Taylor polynomial, generated by Sollya with: 55 // > for i from 0 to 8 do { 56 // j = (-1)^(i + 1)/(2*i + 1); 57 // a = round(j, D, RN); 58 // b = round(j - a, D, RN); 59 // print("{", b, ",", a, "},"); 60 // }; 61 constexpr fputil::DoubleDouble COEFFS[9] = { 62 {0.0, 1.0}, // 1 63 {-0x1.5555555555555p-56, -0x1.5555555555555p-2}, // -1/3 64 {-0x1.999999999999ap-57, 0x1.999999999999ap-3}, // 1/5 65 {-0x1.2492492492492p-57, -0x1.2492492492492p-3}, // -1/7 66 {0x1.c71c71c71c71cp-58, 0x1.c71c71c71c71cp-4}, // 1/9 67 {0x1.745d1745d1746p-59, -0x1.745d1745d1746p-4}, // -1/11 68 {-0x1.3b13b13b13b14p-58, 0x1.3b13b13b13b14p-4}, // 1/13 69 {-0x1.1111111111111p-60, -0x1.1111111111111p-4}, // -1/15 70 {0x1.e1e1e1e1e1e1ep-61, 0x1.e1e1e1e1e1e1ep-5}, // 1/17 71 }; 72 73 // Veltkamp's splitting of a double precision into hi + lo, where the hi part is 74 // slightly smaller than an even split, so that the product of 75 // hi * (s1 * k + s2) is exact, 76 // where: 77 // s1, s2 are single precsion, 78 // 1/16 <= s1/s2 <= 1 79 // 1/16 <= k <= 1 is an integer. 80 // So the maximal precision of (s1 * k + s2) is: 81 // prec(s1 * k + s2) = 2 + log2(msb(s2)) - log2(lsb(k_d * s1)) 82 // = 2 + log2(msb(s1)) + 4 - log2(lsb(k_d)) - log2(lsb(s1)) 83 // = 2 + log2(lsb(s1)) + 23 + 4 - (-4) - log2(lsb(s1)) 84 // = 33. 85 // Thus, the Veltkamp splitting constant is C = 2^33 + 1. 86 // This is used when FMA instruction is not available. 87 [[maybe_unused]] constexpr fputil::DoubleDouble split_d(double a) { 88 fputil::DoubleDouble r{0.0, 0.0}; 89 constexpr double C = 0x1.0p33 + 1.0; 90 double t1 = C * a; 91 double t2 = a - t1; 92 r.hi = t1 + t2; 93 r.lo = a - r.hi; 94 return r; 95 } 96 97 // Compute atan( num_d / den_d ) in double-double precision. 98 // num_d = min(|x|, |y|) 99 // den_d = max(|x|, |y|) 100 // q_d = num_d / den_d 101 // idx, k_d = round( 2^4 * num_d / den_d ) 102 // final_sign = sign of the final result 103 // const_term = the constant term in the final expression. 104 float atan2f_double_double(double num_d, double den_d, double q_d, int idx, 105 double k_d, double final_sign, 106 const fputil::DoubleDouble &const_term) { 107 fputil::DoubleDouble q; 108 double num_r, den_r; 109 110 if (idx != 0) { 111 // The following range reduction is accurate even without fma for 112 // 1/16 <= n/d <= 1. 113 // atan(n/d) - atan(idx/16) = atan((n/d - idx/16) / (1 + (n/d) * (idx/16))) 114 // = atan((n - d*(idx/16)) / (d + n*idx/16)) 115 k_d *= 0x1.0p-4; 116 num_r = fputil::multiply_add(k_d, -den_d, num_d); // Exact 117 den_r = fputil::multiply_add(k_d, num_d, den_d); // Exact 118 q.hi = num_r / den_r; 119 } else { 120 // For 0 < n/d < 1/16, we just need to calculate the lower part of their 121 // quotient. 122 q.hi = q_d; 123 num_r = num_d; 124 den_r = den_d; 125 } 126 #ifdef LIBC_TARGET_CPU_HAS_FMA 127 q.lo = fputil::multiply_add(q.hi, -den_r, num_r) / den_r; 128 #else 129 // Compute `(num_r - q.hi * den_r) / den_r` accurately without FMA 130 // instructions. 131 fputil::DoubleDouble q_hi_dd = split_d(q.hi); 132 double t1 = fputil::multiply_add(q_hi_dd.hi, -den_r, num_r); // Exact 133 double t2 = fputil::multiply_add(q_hi_dd.lo, -den_r, t1); 134 q.lo = t2 / den_r; 135 #endif // LIBC_TARGET_CPU_HAS_FMA 136 137 // Taylor polynomial, evaluating using Horner's scheme: 138 // P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15 139 // + x^17/17 140 // = x*(1 + x^2*(-1/3 + x^2*(1/5 + x^2*(-1/7 + x^2*(1/9 + x^2* 141 // *(-1/11 + x^2*(1/13 + x^2*(-1/15 + x^2 * 1/17)))))))) 142 fputil::DoubleDouble q2 = fputil::quick_mult(q, q); 143 fputil::DoubleDouble p_dd = 144 fputil::polyeval(q2, COEFFS[0], COEFFS[1], COEFFS[2], COEFFS[3], 145 COEFFS[4], COEFFS[5], COEFFS[6], COEFFS[7], COEFFS[8]); 146 fputil::DoubleDouble r_dd = 147 fputil::add(const_term, fputil::multiply_add(q, p_dd, ATAN_I[idx])); 148 r_dd.hi *= final_sign; 149 r_dd.lo *= final_sign; 150 151 // Make sure the sum is normalized: 152 fputil::DoubleDouble rr = fputil::exact_add(r_dd.hi, r_dd.lo); 153 // Round to odd. 154 uint64_t rr_bits = cpp::bit_cast<uint64_t>(rr.hi); 155 if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) { 156 Sign hi_sign = fputil::FPBits<double>(rr.hi).sign(); 157 Sign lo_sign = fputil::FPBits<double>(rr.lo).sign(); 158 if (hi_sign == lo_sign) { 159 ++rr_bits; 160 } else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) { 161 --rr_bits; 162 } 163 } 164 165 return static_cast<float>(cpp::bit_cast<double>(rr_bits)); 166 } 167 168 #endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS 169 170 } // anonymous namespace 171 172 // There are several range reduction steps we can take for atan2(y, x) as 173 // follow: 174 175 // * Range reduction 1: signness 176 // atan2(y, x) will return a number between -PI and PI representing the angle 177 // forming by the 0x axis and the vector (x, y) on the 0xy-plane. 178 // In particular, we have that: 179 // atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant) 180 // = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant) 181 // = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant) 182 // = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant) 183 // Since atan function is odd, we can use the formula: 184 // atan(-u) = -atan(u) 185 // to adjust the above conditions a bit further: 186 // atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant) 187 // = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant) 188 // = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant) 189 // = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant) 190 // Which can be simplified to: 191 // atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0 192 // = sign(y) * (pi - atan( |y|/|x| )) if x < 0 193 194 // * Range reduction 2: reciprocal 195 // Now that the argument inside atan is positive, we can use the formula: 196 // atan(1/x) = pi/2 - atan(x) 197 // to make the argument inside atan <= 1 as follow: 198 // atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x 199 // = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y| 200 // = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x 201 // = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y| 202 203 // * Range reduction 3: look up table. 204 // After the previous two range reduction steps, we reduce the problem to 205 // compute atan(u) with 0 <= u <= 1, or to be precise: 206 // atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|). 207 // An accurate polynomial approximation for the whole [0, 1] input range will 208 // require a very large degree. To make it more efficient, we reduce the input 209 // range further by finding an integer idx such that: 210 // | n/d - idx/16 | <= 1/32. 211 // In particular, 212 // idx := 2^-4 * round(2^4 * n/d) 213 // Then for the fast pass, we find a polynomial approximation for: 214 // atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16) 215 // For the accurate pass, we use the addition formula: 216 // atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (n*idx)/(16*d)) ) 217 // = atan( (n - d * idx/16)/(d + n * idx/16) ) 218 // And finally we use Taylor polynomial to compute the RHS in the accurate pass: 219 // atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 - 220 // - u^15/15 + u^17/17 221 // It's error in double-double precision is estimated in Sollya to be: 222 // > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15 223 // + x^17/17; 224 // > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]); 225 // 0x1.aec6f...p-100 226 // which is about rounding errors of double-double (2^-104). 227 228 LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) { 229 using FPBits = typename fputil::FPBits<float>; 230 constexpr double IS_NEG[2] = {1.0, -1.0}; 231 constexpr double PI = 0x1.921fb54442d18p1; 232 constexpr double PI_LO = 0x1.1a62633145c07p-53; 233 constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1; 234 constexpr double PI_OVER_2 = 0x1.921fb54442d18p0; 235 constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1; 236 // Adjustment for constant term: 237 // CONST_ADJ[x_sign][y_sign][recip] 238 constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = { 239 {{{0.0, 0.0}, {-PI_LO / 2, -PI_OVER_2}}, 240 {{-0.0, -0.0}, {-PI_LO / 2, -PI_OVER_2}}}, 241 {{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}}, 242 {{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}}}}; 243 244 FPBits x_bits(x), y_bits(y); 245 bool x_sign = x_bits.sign().is_neg(); 246 bool y_sign = y_bits.sign().is_neg(); 247 x_bits.set_sign(Sign::POS); 248 y_bits.set_sign(Sign::POS); 249 uint32_t x_abs = x_bits.uintval(); 250 uint32_t y_abs = y_bits.uintval(); 251 uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs; 252 uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs; 253 float num_f = FPBits(min_abs).get_val(); 254 float den_f = FPBits(max_abs).get_val(); 255 double num_d = static_cast<double>(num_f); 256 double den_d = static_cast<double>(den_f); 257 258 if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || num_d == 0.0)) { 259 if (x_bits.is_nan() || y_bits.is_nan()) 260 return FPBits::quiet_nan().get_val(); 261 double x_d = static_cast<double>(x); 262 double y_d = static_cast<double>(y); 263 size_t x_except = (x_d == 0.0) ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1); 264 size_t y_except = (y_d == 0.0) ? 0 : (y_abs == 0x7f80'0000 ? 2 : 1); 265 266 // Exceptional cases: 267 // EXCEPT[y_except][x_except][x_is_neg] 268 // with x_except & y_except: 269 // 0: zero 270 // 1: finite, non-zero 271 // 2: infinity 272 constexpr double EXCEPTS[3][3][2] = { 273 {{0.0, PI}, {0.0, PI}, {0.0, PI}}, 274 {{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}}, 275 {{PI_OVER_2, PI_OVER_2}, 276 {PI_OVER_2, PI_OVER_2}, 277 {PI_OVER_4, THREE_PI_OVER_4}}, 278 }; 279 280 double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign]; 281 282 return static_cast<float>(r); 283 } 284 285 bool recip = x_abs < y_abs; 286 double final_sign = IS_NEG[(x_sign != y_sign) != recip]; 287 fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip]; 288 double q_d = num_d / den_d; 289 290 double k_d = fputil::nearest_integer(q_d * 0x1.0p4); 291 int idx = static_cast<int>(k_d); 292 double r; 293 294 #ifdef LIBC_MATH_HAS_SMALL_TABLES 295 double p = atan_eval_no_table(num_d, den_d, k_d * 0x1.0p-4); 296 r = final_sign * (p + (const_term.hi + ATAN_K_OVER_16[idx])); 297 #else 298 q_d = fputil::multiply_add(k_d, -0x1.0p-4, q_d); 299 300 double p = atan_eval(q_d, idx); 301 r = final_sign * 302 fputil::multiply_add(q_d, p, const_term.hi + ATAN_COEFFS[idx][0]); 303 #endif // LIBC_MATH_HAS_SMALL_TABLES 304 305 #ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS 306 return static_cast<float>(r); 307 #else 308 constexpr uint32_t LOWER_ERR = 4; 309 // Mask sticky bits in double precision before rounding to single precision. 310 constexpr uint32_t MASK = 311 mask_trailing_ones<uint32_t, fputil::FPBits<double>::SIG_LEN - 312 FPBits::SIG_LEN - 1>(); 313 constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR; 314 315 uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(r)) & MASK; 316 317 // Ziv's rounding test. 318 if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR)) 319 return static_cast<float>(r); 320 321 return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign, 322 const_term); 323 #endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS 324 } 325 326 } // namespace LIBC_NAMESPACE_DECL 327