libc/src/__support/math/atanf_float.h

//===-- Single-precision atanf float 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
//
//===----------------------------------------------------------------------===//

#ifndef LIBC_SRC___SUPPORT_MATH_ATANF_FLOAT_H
#define LIBC_SRC___SUPPORT_MATH_ATANF_FLOAT_H

#include "src/__support/FPUtil/FEnvImpl.h"
#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/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

namespace LIBC_NAMESPACE_DECL {

namespace math {

namespace atanf_internal {

using 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, "},");
//   };
static 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},
};

// 1 / (1 + (i/16)^2)  with i = 0..16, generated by Sollya with:
// > for i from 0 to 16 do {
//     a = round(1 / (1 + (i/16)^2), SG, RN);
//     print(a, ",");
//   };
static constexpr float ATANF_REDUCED_ARG[17] = {
    0x1.0p0f,       0x1.fe01fep-1f, 0x1.f81f82p-1f, 0x1.ee9c8p-1f,
    0x1.e1e1e2p-1f, 0x1.d272cap-1f, 0x1.c0e07p-1f,  0x1.adbe88p-1f,
    0x1.99999ap-1f, 0x1.84f00cp-1f, 0x1.702e06p-1f, 0x1.5babccp-1f,
    0x1.47ae14p-1f, 0x1.34679ap-1f, 0x1.21fb78p-1f, 0x1.107fbcp-1f,
    0x1p-1f,
};

// Approximating atan( u / (1 + u * k/16) )
//   atan( u / (1 + u * k/16) ) / u ~ 1 - k/16 * u + (k^2/256 - 1/3) * u^2 +
//                                    + (k/16 - (k/16)^3) * u^3 + O(u^4)
LIBC_INLINE static float atanf_eval(float u, float k_over_16) {
  // (k/16)^2
  float c2 = k_over_16 * k_over_16;
  // -(k/16)^3
  float c3 = fputil::multiply_add(-k_over_16, c2, k_over_16);
  float u2 = u * u;
  // (k^2/256 - 1/3) + u * (k/16 - (k/16)^3)
  float a0 = fputil::multiply_add(c3, u, c2 - 0x1.555556p-2f);
  // -k/16 + u*(k^2/256 - 1/3) + u^2 * (k/16 - (k/16)^3)
  float a1 = fputil::multiply_add(u, a0, -k_over_16);
  // u - u^2 * k/16 + u^3 * ((k^2/256 - 1/3) + u^4 * (k/16 - (k/16)^3))
  return fputil::multiply_add(u2, a1, u);
}

} // namespace atanf_internal

// There are several range reduction steps we can take for atan2(y, x) as
// follow:

LIBC_INLINE static float atanf(float x) {
  using namespace atanf_internal;
  using FPBits = typename fputil::FPBits<float>;
  using FPBits = typename fputil::FPBits<float>;

  constexpr float SIGN[2] = {1.0f, -1.0f};
  constexpr FloatFloat PI_OVER_2 = {-0x1.777a5cp-25f, 0x1.921fb6p0f};

  FPBits x_bits(x);
  Sign s = x_bits.sign();
  float sign = SIGN[s.is_neg()];
  uint32_t x_abs = x_bits.uintval() & 0x7fff'ffffU;

  // x is inf or nan, |x| <= 2^-11 or |x|= > 2^11.
  if (LIBC_UNLIKELY(x_abs <= 0x3a00'0000U || x_abs >= 0x4500'0000U)) {
    if (LIBC_UNLIKELY(x_bits.is_inf()))
      return sign * PI_OVER_2.hi;
    // atan(NaN) = NaN
    if (LIBC_UNLIKELY(x_bits.is_nan())) {
      if (x_bits.is_signaling_nan()) {
        fputil::raise_except_if_required(FE_INVALID);
        return FPBits::quiet_nan().get_val();
      }

      return x;
    }
    // |x| >= 2^11:
    // atan(x) = sign(x) * pi/2 - atan(1/x)
    //         ~ sign(x) * pi/2 - 1/x
    if (LIBC_UNLIKELY(x_abs >= 0x4200'0000))
      return fputil::multiply_add(sign, PI_OVER_2.hi, -1.0f / x);
    // x <= 2^-11:
    // atan(x) ~ x
    return x;
  }

  // Range reduction steps:
  // 1)  atan(x) = sign(x) * atan(|x|)
  // 2)  If |x| > 1, atan(|x|) = pi/2 - atan(1/|x|)
  // 3)  For 1/16 < |x| < 1 + 1/32, we find k such that: | |x| - k/16 | <= 1/32.
  //     Let y = |x| - k/16, then using the angle summation formula, we have:
  //   atan(|x|) = atan(k/16) + atan( (|x| - k/16) / (1 + |x| * k/16) )
  //             = atan(k/16) + atan( y / (1 + (y + k/16) * k/16 )
  //             = atan(k/16) + atan( y / ((1 + k^2/256) + y * k/16) )
  // 4)  Let u = y / (1 + k^2/256), then we can rewritten the above as:
  //   atan(|x|) = atan(k/16) + atan( u / (1 + u * k/16) )
  //             ~ atan(k/16) + (u - k/16 * u^2 + (k^2/256 - 1/3) * u^3 +
  //                             + (k/16 - (k/16)^3) * u^4) + O(u^5)
  float x_a = cpp::bit_cast<float>(x_abs);
  // |x| > 1 + 1/32, we need to invert x, so we will perform the division in
  // float-float.
  if (x_abs > 0x3f84'0000U)
    x_a = 1.0f / x_a;
  // Perform range reduction.
  //   k = nearestint(x * 16)
  float k_f = fputil::nearest_integer(x_a * 0x1.0p4f);
  unsigned idx = static_cast<unsigned>(k_f);
  float k_over_16 = k_f * 0x1.0p-4f;
  float y = x_a - k_over_16;
  // u = (x - k/16) / (1 + (k/16)^2)
  float u = y * ATANF_REDUCED_ARG[idx];

  // atan(x) = sign(x) * atan(|x|)
  //         = sign(x) * (atan(k/16) + atan(|))
  // p ~ atan(u)
  float p = atanf_eval(u, k_over_16);
  // |x|  > 1 + 1/32: q ~ (pi/2 - atan(1/|x|))
  // |x| <= 1 + 1/32: q ~ atan(|x|)
  float q = (p + ATAN_I[idx].lo) + ATAN_I[idx].hi;
  if (x_abs > 0x3f84'0000U)
    q = PI_OVER_2.hi + (PI_OVER_2.lo - q);
  // |x|  > 1 + 1/32: sign(x) * (pi/2 - atan(1/|x|))
  // |x| <= 1 + 1/32: sign(x) * atan(|x|)
  return sign * q;
}

} // namespace math

} // namespace LIBC_NAMESPACE_DECL

#endif // LIBC_SRC___SUPPORT_MATH_ATANF_FLOAT_H