math/generic/atan2f.cpp

//===-- 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/optimization.h" // LIBC_UNLIKELY

namespace LIBC_NAMESPACE {

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<uint64_t>(rr.hi);
  if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) {
    Sign hi_sign = fputil::FPBits<double>(rr.hi).sign();
    Sign lo_sign = fputil::FPBits<double>(rr.lo).sign();
    if (hi_sign == lo_sign) {
      ++rr_bits;
    } else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) {
      --rr_bits;
    }
  }

  return static_cast<float>(cpp::bit_cast<double>(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<float>;
  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;

  if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || min_abs == 0U)) {
    if (x_bits.is_nan() || y_bits.is_nan())
      return FPBits::quiet_nan().get_val();
    size_t x_except = x_abs == 0 ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1);
    size_t y_except = y_abs == 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<float>(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 num_d = static_cast<double>(FPBits(min_abs).get_val());
  double den_d = static_cast<double>(FPBits(max_abs).get_val());
  double q_d = num_d / den_d;

  double k_d = fputil::nearest_integer(q_d * 0x1.0p4f);
  int idx = static_cast<int>(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<uint32_t, fputil::FPBits<double>::SIG_LEN -
                                       FPBits::SIG_LEN - 1>();
  constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR;

  uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(r)) & MASK;

  // Ziv's rounding test.
  if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR))
    return static_cast<float>(r);

  return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign,
                              const_term);
}

} // namespace LIBC_NAMESPACE