//===-- Single-precision atan2f 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/atan2f.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 { // 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 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 // 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(rr.hi); if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) { Sign hi_sign = fputil::FPBits(rr.hi).sign(); Sign lo_sign = fputil::FPBits(rr.lo).sign(); if (hi_sign == lo_sign) { ++rr_bits; } else if ((rr_bits & fputil::FPBits::FRACTION_MASK) > 0) { --rr_bits; } } return static_cast(cpp::bit_cast(rr_bits)); } } // 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 FPBits = typename fputil::FPBits; 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(num_f); double den_d = static_cast(den_f); if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || num_d == 0.0)) { if (x_bits.is_nan() || y_bits.is_nan()) return FPBits::quiet_nan().get_val(); double x_d = static_cast(x); double y_d = static_cast(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(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 * 0x1.0p4f); int idx = static_cast(k_d); q_d = fputil::multiply_add(k_d, -0x1.0p-4, q_d); double p = atan_eval(q_d, idx); double r = final_sign * fputil::multiply_add(q_d, p, const_term.hi + ATAN_COEFFS[idx][0]); constexpr uint32_t LOWER_ERR = 4; // Mask sticky bits in double precision before rounding to single precision. constexpr uint32_t MASK = mask_trailing_ones::SIG_LEN - FPBits::SIG_LEN - 1>(); constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR; uint32_t r_bits = static_cast(cpp::bit_cast(r)) & MASK; // Ziv's rounding test. if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR)) return static_cast(r); return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign, const_term); } } // namespace LIBC_NAMESPACE_DECL