xref: /llvm-project/libc/src/math/generic/atan2.cpp (revision 0f4b3c409fbd756d826c89d5539d9ea22bcc56aa)

//===-- 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