| //===-- 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<double>; |
| |
| 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<unsigned>(min_abs >> FPBits::FRACTION_LEN); |
| unsigned max_exp = static_cast<unsigned>(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<unsigned>(min_abs >> FPBits::FRACTION_LEN); |
| max_exp = static_cast<unsigned>(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<unsigned>(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 |
