libc/src/math/generic/atan2f_float.h

//===-- 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/__support/FPUtil/FPBits.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
#include "src/math/atan2f.h"

namespace LIBC_NAMESPACE_DECL {

namespace {

using FloatFloat = fputil::FloatFloat;

// atan(i/64) with i = 0..16, generated by Sollya with:
// > for i from 0 to 16 do {
//     a = round(atan(i/16), SG, RN);
//     b = round(atan(i/16) - a, SG, RN);
//     print("{", b, ",", a, "},");
//   };
constexpr FloatFloat ATAN_I[17] = {
    {0.0f, 0.0f},
    {-0x1.1a6042p-30f, 0x1.ff55bcp-5f},
    {-0x1.54f424p-30f, 0x1.fd5baap-4f},
    {0x1.79cb6p-28f, 0x1.7b97b4p-3f},
    {-0x1.b4dfc8p-29f, 0x1.f5b76p-3f},
    {-0x1.1f0286p-27f, 0x1.362774p-2f},
    {0x1.e4defp-30f, 0x1.6f6194p-2f},
    {0x1.e611fep-29f, 0x1.a64eecp-2f},
    {0x1.586ed4p-28f, 0x1.dac67p-2f},
    {-0x1.6499e6p-26f, 0x1.0657eap-1f},
    {0x1.7bdfd6p-26f, 0x1.1e00bap-1f},
    {-0x1.98e422p-28f, 0x1.345f02p-1f},
    {0x1.934f7p-28f, 0x1.4978fap-1f},
    {0x1.c5a6c6p-27f, 0x1.5d5898p-1f},
    {0x1.5e118cp-27f, 0x1.700a7cp-1f},
    {-0x1.1d4eb6p-26f, 0x1.819d0cp-1f},
    {-0x1.777a5cp-26f, 0x1.921fb6p-1f},
};

// Approximate atan(x) for |x| <= 2^-5.
// Using degree-3 Taylor polynomial:
//  P = x - x^3/3
// Then the absolute error is bounded by:
//   |atan(x) - P(x)| < |x|^5/5 < 2^(-5*5) / 5 < 2^-27.
// And the relative error is bounded by:
//   |(atan(x) - P(x))/atan(x)| < |x|^4 / 4 < 2^-22.
// 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_lo * (1 - x_hi^2)
FloatFloat atan_eval(const FloatFloat &x) {
  FloatFloat p;
  p.hi = x.hi;
  float x_hi_sq = x.hi * x.hi;
  // c0 ~ - x_hi^2 / 3
  float c0 = -0x1.555556p-2f * x_hi_sq;
  // c1 ~ x_lo * (1 - x_hi^2)
  float c1 = fputil::multiply_add(x_hi_sq, -x.lo, x.lo);
  // p.lo ~ - x_hi^3 / 3 + x_lo * (1 - x_hi*2)
  p.lo = fputil::multiply_add(x.hi, c0, c1);
  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/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)
// with Q(x) = x - x^3/3 be the cubic Taylor polynomial of atan(x).
// It's error in float-float precision is estimated in Sollya to be:
// > P = x - x^3/3;
// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
// 0x1.995...p-28.

LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
  using FPBits = typename fputil::FPBits<float>;
  constexpr float IS_NEG[2] = {1.0f, -1.0f};
  constexpr FloatFloat ZERO = {0.0f, 0.0f};
  constexpr FloatFloat MZERO = {-0.0f, -0.0f};
  constexpr FloatFloat PI = {-0x1.777a5cp-24f, 0x1.921fb6p1f};
  constexpr FloatFloat MPI = {0x1.777a5cp-24f, -0x1.921fb6p1f};
  constexpr FloatFloat PI_OVER_4 = {-0x1.777a5cp-26f, 0x1.921fb6p-1f};
  constexpr FloatFloat PI_OVER_2 = {-0x1.777a5cp-25f, 0x1.921fb6p0f};
  constexpr FloatFloat MPI_OVER_2 = {-0x1.777a5cp-25f, 0x1.921fb6p0f};
  constexpr FloatFloat THREE_PI_OVER_4 = {-0x1.99bc5cp-28f, 0x1.2d97c8p1f};
  // Adjustment for constant term:
  //   CONST_ADJ[x_sign][y_sign][recip]
  constexpr FloatFloat 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();
  uint32_t x_abs = x_bits.uintval();
  uint32_t y_abs = y_bits.uintval();
  bool recip = x_abs < y_abs;
  uint32_t min_abs = recip ? x_abs : y_abs;
  uint32_t max_abs = !recip ? x_abs : y_abs;
  auto min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
  auto max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);

  float num = FPBits(min_abs).get_val();
  float 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 > 0xffU - 64U || min_exp < 64U)) {
    if (x_bits.is_nan() || y_bits.is_nan())
      return FPBits::quiet_nan().get_val();
    unsigned x_except = x == 0.0f ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);
    unsigned y_except = y == 0.0f ? 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 FloatFloat 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)) {
      FloatFloat 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 < 64U;
    bool scale_down = max_exp > 0xffU - 64U;
    // 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 *= 0x1.0p32f;
      if (!scale_down)
        den *= 0x1.0p32f;
    } else if (scale_down) {
      den *= 0x1.0p-32f;
      num *= 0x1.0p-32f;
    }

    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);
  }

  float final_sign = IS_NEG[(x_sign != y_sign) != recip];
  FloatFloat 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 > 25))
    return fputil::multiply_add(final_sign, const_term.hi,
                                final_sign * (const_term.lo + num / den));

  float k = fputil::nearest_integer(16.0f * num / den);
  unsigned idx = static_cast<unsigned>(k);
  // k = idx / 16
  k *= 0x1.0p-4f;

  // Range reduction:
  // atan(n/d) - atan(k/64) = atan((n/d - k/16) / (1 + (n/d) * (k/16)))
  //                        = atan((n - d * k/16)) / (d + n * k/16))
  FloatFloat num_k = fputil::exact_mult(num, k);
  FloatFloat den_k = fputil::exact_mult(den, k);

  // num_dd = n - d * k
  FloatFloat num_ff = fputil::exact_add(num - den_k.hi, -den_k.lo);
  // den_dd = d + n * k
  FloatFloat den_ff = fputil::exact_add(den, num_k.hi);
  den_ff.lo += num_k.lo;

  // q = (n - d * k) / (d + n * k)
  FloatFloat q = fputil::div(num_ff, den_ff);
  // p ~ atan(q)
  FloatFloat p = atan_eval(q);

  FloatFloat r = fputil::add(const_term, fputil::add(ATAN_I[idx], p));
  return final_sign * r.hi;
}

} // namespace LIBC_NAMESPACE_DECL