libc/src/math/generic/cbrtf.cpp

//===-- Implementation of cbrtf 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/cbrtf.h"
#include "hdr/fenv_macros.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/common.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

namespace LIBC_NAMESPACE_DECL {

namespace {

// Look up table for 2^(i/3) for i = 0, 1, 2.
constexpr double CBRT2[3] = {1.0, 0x1.428a2f98d728bp0, 0x1.965fea53d6e3dp0};

// Degree-7 polynomials approximation of ((1 + x)^(1/3) - 1)/x for 0 <= x <= 1
// generated by Sollya with:
// > for i from 0 to 15 do {
//     P = fpminimax((1 + x)^(1/3) - 1)/x, 6, [|D...|], [i/16, (i + 1)/16]);
//     print("{", coeff(P, 0), ",", coeff(P, 1), ",", coeff(P, 2), ",",
//           coeff(P, 3), ",", coeff(P, 4), ",", coeff(P, 5), ",",
//           coeff(P, 6), "},");
// };
// Then (1 + x)^(1/3) ~ 1 + x * P(x).
constexpr double COEFFS[16][7] = {
    {0x1.55555555554ebp-2, -0x1.c71c71c678c0cp-4, 0x1.f9add2776de81p-5,
     -0x1.511e10aa964a7p-5, 0x1.ee44165937fa2p-6, -0x1.7c5c9e059345dp-6,
     0x1.047f75e0aff14p-6},
    {0x1.5555554d1149ap-2, -0x1.c71c676fcb5bp-4, 0x1.f9ab127dc57ebp-5,
     -0x1.50ea8fd1d4c15p-5, 0x1.e9d68f28ced43p-6, -0x1.60e0e1e661311p-6,
     0x1.716eca1d6e3bcp-7},
    {0x1.5555546377d45p-2, -0x1.c71bc1c6d49d2p-4, 0x1.f9924cc0ed24dp-5,
     -0x1.4fea3beb53b3bp-5, 0x1.de028a9a07b1bp-6, -0x1.3b090d2233524p-6,
     0x1.0aeca34893785p-7},
    {0x1.55554dce9f649p-2, -0x1.c7188b34b98f8p-4, 0x1.f93e1af34af49p-5,
     -0x1.4d9a06be75c63p-5, 0x1.cb943f4f68992p-6, -0x1.139a685a5e3c4p-6,
     0x1.88410674c6a5dp-8},
    {0x1.5555347d211c3p-2, -0x1.c70f2a4b1a5fap-4, 0x1.f88420e8602c3p-5,
     -0x1.49becfa4ed3ep-5, 0x1.b475cd9013162p-6, -0x1.dcfee1dd2f8efp-7,
     0x1.249bb51a1c498p-8},
    {0x1.5554f01b33dbap-2, -0x1.c6facb929dbf1p-4, 0x1.f73fb7861252ep-5,
     -0x1.4459a4a0071fap-5, 0x1.9a8df2b504fc2p-6, -0x1.9a7ce3006d06ep-7,
     0x1.ba9230918fa2ep-9},
    {0x1.55545c695db5fp-2, -0x1.c6d6089f20275p-4, 0x1.f556e0ea80efp-5,
     -0x1.3d91372d083f4p-5, 0x1.7f66cff331f4p-6, -0x1.606a562491737p-7,
     0x1.52e3e17c71069p-9},
    {0x1.55534a879232ap-2, -0x1.c69b836998b84p-4, 0x1.f2bb26dac0e4cp-5,
     -0x1.359eed43716d7p-5, 0x1.64218cd824fbcp-6, -0x1.2e703e2e091e8p-7,
     0x1.0677d9af6aad4p-9},
    {0x1.5551836bb5494p-2, -0x1.c64658c15353bp-4, 0x1.ef68517451a6ep-5,
     -0x1.2cc20a980dceep-5, 0x1.49843e0fad93ap-6, -0x1.03c59ccb68e54p-7,
     0x1.9ad325dc7adcbp-10},
    {0x1.554ecacb0d035p-2, -0x1.c5d2664026ffcp-4, 0x1.eb624796ba809p-5,
     -0x1.233803d19a535p-5, 0x1.300decb1c3c28p-6, -0x1.befe18031ec3dp-8,
     0x1.449f5ee175c69p-10},
    {0x1.554ae1f5ae815p-2, -0x1.c53c6b14ff6b2p-4, 0x1.e6b2d5127bb5bp-5,
     -0x1.19387336788a3p-5, 0x1.180955a6ab255p-6, -0x1.81696703ba369p-8,
     0x1.02cb36389bd79p-10},
    {0x1.55458a59f356ep-2, -0x1.c4820dd631ae9p-4, 0x1.e167af818bd15p-5,
     -0x1.0ef35f6f72e52p-5, 0x1.019c33b65e4ebp-6, -0x1.4d25bdd52d3a5p-8,
     0x1.a008ae91f5936p-11},
    {0x1.553e878eafee1p-2, -0x1.c3a1d0b2a3db2p-4, 0x1.db90d8ed9f89bp-5,
     -0x1.0490e20f1ae91p-5, 0x1.d9a5d1fc42fe3p-7, -0x1.20bf8227c2abfp-8,
     0x1.50f8174cdb6e9p-11},
    {0x1.5535a0dedf1b1p-2, -0x1.c29afb8bd01a1p-4, 0x1.d53f6371c1e27p-5,
     -0x1.f463209b433e2p-6, 0x1.b35222a17e44p-7, -0x1.f5efbf505e133p-9,
     0x1.12e0e94e8586dp-11},
    {0x1.552aa25e57bfdp-2, -0x1.c16d811e4acadp-4, 0x1.ce8489b47aa51p-5,
     -0x1.dfde7ff758ea8p-6, 0x1.901f43aac38c8p-7, -0x1.b581d07df5ad5p-9,
     0x1.c3726535f1fc6p-12},
    {0x1.551d5d9b204d3p-2, -0x1.c019e328f8db1p-4, 0x1.c7710f44fc3cep-5,
     -0x1.cbbbe25ea8ba4p-6, 0x1.6fe270088623dp-7, -0x1.7e6fc79733761p-9,
     0x1.75077abf18d84p-12},
};

} // anonymous namespace

LLVM_LIBC_FUNCTION(float, cbrtf, (float x)) {
  using FloatBits = typename fputil::FPBits<float>;
  using DoubleBits = typename fputil::FPBits<double>;

  FloatBits x_bits(x);

  uint32_t x_abs = x_bits.uintval() & 0x7fff'ffff;
  uint32_t sign_bit = (x_bits.uintval() >> 31) << DoubleBits::EXP_LEN;

  if (LIBC_UNLIKELY(x == 0.0f || x_abs >= 0x7f80'0000)) {
    // x is 0, Inf, or NaN.
    // Make sure it works for FTZ/DAZ modes.
    return x + x;
  }

  double xd = static_cast<double>(x);
  DoubleBits xd_bits(xd);

  // When using biased exponent of x in double precision,
  //   x_e = real_exponent_of_x + 1023
  // Then:
  //   x_e / 3 = real_exponent_of_x / 3 + 1023/3
  //           = real_exponent_of_x / 3 + 341
  // So to make it the correct biased exponent of x^(1/3), we add
  //   1023 - 341 = 682
  // to the quotient x_e / 3.
  unsigned x_e = static_cast<unsigned>(xd_bits.get_biased_exponent());
  unsigned out_e = (x_e / 3 + 682) | sign_bit;
  unsigned shift_e = x_e % 3;

  // Set x_m = 2^(x_e % 3) * (1.mantissa)
  uint64_t x_m = xd_bits.get_mantissa();
  // Use the leading 4 bits for look up table
  unsigned idx = static_cast<unsigned>(x_m >> (DoubleBits::FRACTION_LEN - 4));

  x_m |= static_cast<uint64_t>(DoubleBits::EXP_BIAS)
         << DoubleBits::FRACTION_LEN;

  double x_reduced = DoubleBits(x_m).get_val();
  double dx = x_reduced - 1.0;

  double dx_sq = dx * dx;
  double c0 = fputil::multiply_add(dx, COEFFS[idx][0], 1.0);
  double c1 = fputil::multiply_add(dx, COEFFS[idx][2], COEFFS[idx][1]);
  double c2 = fputil::multiply_add(dx, COEFFS[idx][4], COEFFS[idx][3]);
  double c3 = fputil::multiply_add(dx, COEFFS[idx][6], COEFFS[idx][5]);

  double dx_4 = dx_sq * dx_sq;
  double p0 = fputil::multiply_add(dx_sq, c1, c0);
  double p1 = fputil::multiply_add(dx_sq, c3, c2);

  double r = fputil::multiply_add(dx_4, p1, p0) * CBRT2[shift_e];

  uint64_t r_m = DoubleBits(r).get_mantissa();
  // Check if the output is exact.  To be exact, the smallest 1-bit of the
  // output has to be at least 2^-7 or higher.  So we check the lowest 44 bits
  // to see if they are within 2^(-52 + 3) errors from all zeros, then the
  // result cube root is exact.
  if (LIBC_UNLIKELY(((r_m + 8) & 0xfffffffffff) <= 16)) {
    if ((r_m & 0xfffffffffff) <= 8)
      r_m &= 0xffff'ffff'ffff'ffe0;
    else
      r_m = (r_m & 0xffff'ffff'ffff'ffe0) + 0x20;
    fputil::clear_except_if_required(FE_INEXACT);
  }
  // Adjust exponent and sign.
  uint64_t r_bits =
      r_m | (static_cast<uint64_t>(out_e) << DoubleBits::FRACTION_LEN);

  return static_cast<float>(DoubleBits(r_bits).get_val());
}

} // namespace LIBC_NAMESPACE_DECL