libc/src/__support/math/pow_utils.h

//===----------------------------------------------------------------------===//
//
// 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
//
//===----------------------------------------------------------------------===//
///
/// \file
/// Common utilities and tables for double-precision pow(x, y).
///
//===----------------------------------------------------------------------===//

#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_POW_UTILS_H
#define LLVM_LIBC_SRC___SUPPORT_MATH_POW_UTILS_H

#include "hdr/errno_macros.h"
#include "hdr/fenv_macros.h"
#include "src/__support/CPP/bit.h"
#include "src/__support/CPP/optional.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/FPUtil/rounding_mode.h"
#include "src/__support/FPUtil/sqrt.h"
#include "src/__support/common.h"
#include "src/__support/macros/attributes.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h"
#include "src/__support/macros/properties/cpu_features.h"
#include "src/__support/math/common_constants.h"
#include "src/__support/math/exp10.h"
#include "src/__support/math/exp2.h"

namespace LIBC_NAMESPACE_DECL {
namespace math {
namespace pow_internal {

using fputil::DoubleDouble;
using namespace common_constants_internal;

// Constants for log2(x) range reduction, generated by Sollya with:
// > for i from 0 to 127 do {
//     r = 2^-8 * ceil( 2^8 * (1 - 2^(-8)) / (1 + i*2^-7) );
//     b = nearestint(log2(r) * 2^41) * 2^-41;
//     c = round(log2(r) - b, D, RN);
//     print("{ ", -c, ", ", -b, " }, ");
//   };
// This is the same as -log2(RD[i]), with the least significant bits of the
// high part set to be 2^-41, so that the sum of high parts + e_x is exact in
// double precision.
// We also replace the first and the last ones to be 0.
LIBC_INLINE_VAR constexpr DoubleDouble LOG2_R_DD[128] = {
    {0.0, 0.0},
    {-0x1.19b14945cf6bap-44, 0x1.72c7ba21p-7},
    {-0x1.95539356f93dcp-43, 0x1.743ee862p-6},
    {0x1.abe0a48f83604p-43, 0x1.184b8e4c5p-5},
    {0x1.635577970e04p-43, 0x1.77394c9d9p-5},
    {-0x1.401fbaaa67e3cp-45, 0x1.d6ebd1f2p-5},
    {-0x1.5b1799ceaeb51p-43, 0x1.1bb32a6008p-4},
    {0x1.7c407050799bfp-43, 0x1.4c560fe688p-4},
    {0x1.da6339da288fcp-43, 0x1.7d60496cf8p-4},
    {0x1.be4f6f22dbbadp-43, 0x1.960caf9ab8p-4},
    {-0x1.c760bc9b188c4p-45, 0x1.c7b528b71p-4},
    {0x1.164e932b2d51cp-44, 0x1.f9c95dc1dp-4},
    {0x1.924ae921f7ecap-45, 0x1.097e38ce6p-3},
    {-0x1.6d25a5b8a19b2p-44, 0x1.22dadc2ab4p-3},
    {0x1.e50a1644ac794p-43, 0x1.3c6fb650ccp-3},
    {0x1.f34baa74a7942p-43, 0x1.494f863b8cp-3},
    {-0x1.8f7aac147fdc1p-46, 0x1.633a8bf438p-3},
    {0x1.f84be19cb9578p-43, 0x1.7046031c78p-3},
    {-0x1.66cccab240e9p-46, 0x1.8a8980abfcp-3},
    {-0x1.3f7a55cd2af4cp-47, 0x1.97c1cb13c8p-3},
    {0x1.3458cde69308cp-43, 0x1.b2602497d4p-3},
    {-0x1.667f21fa8423fp-44, 0x1.bfc67a8p-3},
    {0x1.d2fe4574e09b9p-47, 0x1.dac22d3e44p-3},
    {0x1.367bde40c5e6dp-43, 0x1.e857d3d36p-3},
    {0x1.d45da26510033p-46, 0x1.01d9bbcfa6p-2},
    {-0x1.7204f55bbf90dp-44, 0x1.08bce0d96p-2},
    {-0x1.d4f1b95e0ff45p-43, 0x1.169c05364p-2},
    {0x1.c20d74c0211bfp-44, 0x1.1d982c9d52p-2},
    {0x1.ad89a083e072ap-43, 0x1.249cd2b13cp-2},
    {0x1.cd0cb4492f1bcp-43, 0x1.32bfee370ep-2},
    {-0x1.2101a9685c779p-47, 0x1.39de8e155ap-2},
    {0x1.9451cd394fe8dp-43, 0x1.4106017c3ep-2},
    {0x1.661e393a16b95p-44, 0x1.4f6fbb2cecp-2},
    {-0x1.c6d8d86531d56p-44, 0x1.56b22e6b58p-2},
    {0x1.c1c885adb21d3p-43, 0x1.5dfdcf1eeap-2},
    {0x1.3bb5921006679p-45, 0x1.6552b49986p-2},
    {0x1.1d406db502403p-43, 0x1.6cb0f6865cp-2},
    {0x1.55a63e278bad5p-43, 0x1.7b89f02cf2p-2},
    {-0x1.66ae2a7ada553p-49, 0x1.8304d90c12p-2},
    {-0x1.66cccab240e9p-45, 0x1.8a8980abfcp-2},
    {-0x1.62404772a151dp-45, 0x1.921800924ep-2},
    {0x1.ac9bca36fd02ep-44, 0x1.99b072a96cp-2},
    {0x1.4bc302ffa76fbp-43, 0x1.a8ff97181p-2},
    {0x1.01fea1ec47c71p-43, 0x1.b0b67f4f46p-2},
    {-0x1.f20203b3186a6p-43, 0x1.b877c57b1cp-2},
    {-0x1.2642415d47384p-45, 0x1.c043859e3p-2},
    {-0x1.bc76a2753b99bp-50, 0x1.c819dc2d46p-2},
    {-0x1.da93ae3a5f451p-43, 0x1.cffae611aep-2},
    {-0x1.50e785694a8c6p-43, 0x1.d7e6c0abc4p-2},
    {0x1.c56138c894641p-43, 0x1.dfdd89d586p-2},
    {0x1.5669df6a2b592p-43, 0x1.e7df5fe538p-2},
    {-0x1.ea92d9e0e8ac2p-48, 0x1.efec61b012p-2},
    {0x1.a0331af2e6feap-43, 0x1.f804ae8d0cp-2},
    {0x1.9518ce032f41dp-48, 0x1.0014332bep-1},
    {-0x1.b3b3864c60011p-44, 0x1.042bd4b9a8p-1},
    {-0x1.103e8f00d41c8p-45, 0x1.08494c66b9p-1},
    {0x1.65be75cc3da17p-43, 0x1.0c6caaf0c5p-1},
    {0x1.3676289cd3dd4p-43, 0x1.1096015deep-1},
    {-0x1.41dfc7d7c3321p-43, 0x1.14c560fe69p-1},
    {0x1.e0cda8bd74461p-44, 0x1.18fadb6e2dp-1},
    {0x1.2a606046ad444p-44, 0x1.1d368296b5p-1},
    {0x1.f9ea977a639cp-43, 0x1.217868b0c3p-1},
    {-0x1.50520a377c7ecp-45, 0x1.25c0a0463cp-1},
    {0x1.6e3cb71b554e7p-47, 0x1.2a0f3c3407p-1},
    {-0x1.4275f1035e5e8p-48, 0x1.2e644fac05p-1},
    {-0x1.4275f1035e5e8p-48, 0x1.2e644fac05p-1},
    {-0x1.979a5db68721dp-45, 0x1.32bfee370fp-1},
    {0x1.1ee969a95f529p-43, 0x1.37222bb707p-1},
    {0x1.bb4b69336b66ep-43, 0x1.3b8b1c68fap-1},
    {0x1.d5e6a8a4fb059p-45, 0x1.3ffad4e74fp-1},
    {0x1.3106e404cabb7p-44, 0x1.44716a2c08p-1},
    {0x1.3106e404cabb7p-44, 0x1.44716a2c08p-1},
    {-0x1.9bcaf1aa4168ap-43, 0x1.48eef19318p-1},
    {0x1.1646b761c48dep-44, 0x1.4d7380dcc4p-1},
    {0x1.2f0c0bfe9dbecp-43, 0x1.51ff2e3021p-1},
    {0x1.29904613e33cp-43, 0x1.5692101d9bp-1},
    {0x1.1d406db502403p-44, 0x1.5b2c3da197p-1},
    {0x1.1d406db502403p-44, 0x1.5b2c3da197p-1},
    {-0x1.125d6cbcd1095p-44, 0x1.5fcdce2728p-1},
    {-0x1.bd9b32266d92cp-43, 0x1.6476d98adap-1},
    {0x1.54243b21709cep-44, 0x1.6927781d93p-1},
    {0x1.54243b21709cep-44, 0x1.6927781d93p-1},
    {-0x1.ce60916e52e91p-44, 0x1.6ddfc2a79p-1},
    {0x1.f1f5ae718f241p-43, 0x1.729fd26b7p-1},
    {-0x1.6eb9612e0b4f3p-43, 0x1.7767c12968p-1},
    {-0x1.6eb9612e0b4f3p-43, 0x1.7767c12968p-1},
    {0x1.fed21f9cb2cc5p-43, 0x1.7c37a9227ep-1},
    {0x1.7f5dc57266758p-43, 0x1.810fa51bf6p-1},
    {0x1.7f5dc57266758p-43, 0x1.810fa51bf6p-1},
    {0x1.5b338360c2ae2p-43, 0x1.85efd062c6p-1},
    {-0x1.96fc8f4b56502p-43, 0x1.8ad846cf37p-1},
    {-0x1.96fc8f4b56502p-43, 0x1.8ad846cf37p-1},
    {-0x1.bdc81c4db3134p-44, 0x1.8fc924c89bp-1},
    {0x1.36c101ee1344p-43, 0x1.94c287492cp-1},
    {0x1.36c101ee1344p-43, 0x1.94c287492cp-1},
    {0x1.e41fa0a62e6aep-44, 0x1.99c48be206p-1},
    {-0x1.d97ee9124773bp-46, 0x1.9ecf50bf44p-1},
    {-0x1.d97ee9124773bp-46, 0x1.9ecf50bf44p-1},
    {-0x1.3f94e00e7d6bcp-46, 0x1.a3e2f4ac44p-1},
    {-0x1.6879fa00b120ap-43, 0x1.a8ff971811p-1},
    {-0x1.6879fa00b120ap-43, 0x1.a8ff971811p-1},
    {0x1.1659d8e2d7d38p-44, 0x1.ae255819fp-1},
    {0x1.1e5e0ae0d3f8ap-43, 0x1.b35458761dp-1},
    {0x1.1e5e0ae0d3f8ap-43, 0x1.b35458761dp-1},
    {0x1.484a15babcf88p-43, 0x1.b88cb9a2abp-1},
    {0x1.484a15babcf88p-43, 0x1.b88cb9a2abp-1},
    {0x1.871a7610e40bdp-45, 0x1.bdce9dcc96p-1},
    {-0x1.2d90e5edaeceep-43, 0x1.c31a27dd01p-1},
    {-0x1.2d90e5edaeceep-43, 0x1.c31a27dd01p-1},
    {-0x1.5dd31d962d373p-43, 0x1.c86f7b7ea5p-1},
    {-0x1.5dd31d962d373p-43, 0x1.c86f7b7ea5p-1},
    {-0x1.9ad57391924a7p-43, 0x1.cdcebd2374p-1},
    {-0x1.3167ccc538261p-44, 0x1.d338120a6ep-1},
    {-0x1.3167ccc538261p-44, 0x1.d338120a6ep-1},
    {0x1.c7a4ff65ddbc9p-45, 0x1.d8aba045bp-1},
    {0x1.c7a4ff65ddbc9p-45, 0x1.d8aba045bp-1},
    {-0x1.f9ab3cf74babap-44, 0x1.de298ec0bbp-1},
    {-0x1.f9ab3cf74babap-44, 0x1.de298ec0bbp-1},
    {0x1.52842c1c1e586p-43, 0x1.e3b20546f5p-1},
    {0x1.52842c1c1e586p-43, 0x1.e3b20546f5p-1},
    {0x1.3c6764fc87b4ap-48, 0x1.e9452c8a71p-1},
    {0x1.3c6764fc87b4ap-48, 0x1.e9452c8a71p-1},
    {-0x1.a0976c0a2827dp-44, 0x1.eee32e2aedp-1},
    {-0x1.a0976c0a2827dp-44, 0x1.eee32e2aedp-1},
    {-0x1.a45314dc4fc42p-43, 0x1.f48c34bd1fp-1},
    {-0x1.a45314dc4fc42p-43, 0x1.f48c34bd1fp-1},
    {0x1.ef5d00e390ap-44, 0x1.fa406bd244p-1},
    {0.0, 1.0},
};

// Check if x is an odd integer: the lowest set bit must be at the unit
// position:
//   x_e + lsb == UNIT_EXPONENT.
LIBC_INLINE bool is_odd_integer(double x) {
  using FPBits = fputil::FPBits<double>;
  FPBits xbits(x);
  uint64_t x_u = xbits.uintval();
  unsigned x_e = static_cast<unsigned>(xbits.get_biased_exponent());
  unsigned lsb =
      static_cast<unsigned>(cpp::countr_zero(x_u | FPBits::EXP_MASK));
  constexpr unsigned UNIT_EXPONENT =
      static_cast<unsigned>(FPBits::EXP_BIAS + FPBits::FRACTION_LEN);
  return (x_e + lsb == UNIT_EXPONENT);
}

// Check if x is an integer: the lowest set bit must be at or above the unit
// position:
//   x_e + lsb >= UNIT_EXPONENT.
LIBC_INLINE bool is_integer(double x) {
  if (x == 0.0)
    return true;
  using FPBits = fputil::FPBits<double>;
  FPBits xbits(x);
  uint64_t x_u = xbits.uintval();
  unsigned x_e = static_cast<unsigned>(xbits.get_biased_exponent());
  unsigned lsb =
      static_cast<unsigned>(cpp::countr_zero(x_u | FPBits::EXP_MASK));
  constexpr unsigned UNIT_EXPONENT =
      static_cast<unsigned>(FPBits::EXP_BIAS + FPBits::FRACTION_LEN);
  return (x_e + lsb >= UNIT_EXPONENT);
}

LIBC_INLINE double set_overflow(bool is_neg) {
  fputil::set_errno_if_required(ERANGE);
  fputil::raise_overflow_except_if_required<double>();
  using FPBits = fputil::FPBits<double>;
#ifndef LIBC_MATH_HAS_ASSUME_ROUND_NEAREST_ONLY
  int rounding = fputil::quick_get_round();
  if (rounding == FE_TOWARDZERO)
    return is_neg ? -FPBits::max_normal().get_val()
                  : FPBits::max_normal().get_val();
  if (rounding == FE_DOWNWARD)
    return is_neg ? -FPBits::inf().get_val() : FPBits::max_normal().get_val();
  if (rounding == FE_UPWARD)
    return is_neg ? -FPBits::max_normal().get_val() : FPBits::inf().get_val();
#endif // LIBC_MATH_HAS_ASSUME_ROUND_NEAREST_ONLY
  return is_neg ? -FPBits::inf().get_val() : FPBits::inf().get_val();
}

LIBC_INLINE double set_underflow(bool is_neg) {
  fputil::set_errno_if_required(ERANGE);
  fputil::raise_underflow_except_if_required<double>();
#ifndef LIBC_MATH_HAS_ASSUME_ROUND_NEAREST_ONLY
  using FPBits = fputil::FPBits<double>;
  int rounding = fputil::quick_get_round();
  if (rounding == FE_UPWARD && !is_neg)
    return FPBits::min_subnormal().get_val();
  if (rounding == FE_DOWNWARD && is_neg)
    return -FPBits::min_subnormal().get_val();
#endif // LIBC_MATH_HAS_ASSUME_ROUND_NEAREST_ONLY
  return is_neg ? -0.0 : 0.0;
}

// Rounding tests for 2^hi * (mid + lo) when the output might be denormal.  We
// assume further that
//   1 <= mid < 2,
//   mid + lo < 2,
//   and |lo| << mid.
//
// Notice that, if 0 < x < 2^-1022:
//   double( 2^-1022 + x ) -  2^-1022 = double(x),
// and if -2^-1022 < x < 0:
//   double(-2^-1022 + x ) - -2^-1022 = double(x).
//
// So if we scale |x| up by 2^1022, we can use the "+1 trick" (or "-1 trick"
// when x < 0):
//   double( 1.0 + 2^1022 * x) -  1.0, for x > 0, and
//   double(-1.0 + 2^1022 * x) - -1.0, for x < 0,
// to test how x is rounded in the denormal range.  By adding +-1.0, the
// results will have similar rounding points as denormal outputs, aligning the
// least significant bit of the mantissa with 2^-52 * 2^-1022 = 2^-1074.
//
// Finally, subtracting 1023 in the exponent field converts the rounded normal
// value into the subnormal value.
template <bool SKIP_ZIV_TEST = false>
LIBC_INLINE constexpr cpp::optional<double>
ziv_test_denorm(int hi, double mid, double lo, double err, bool is_neg) {
  using FPBits = typename fputil::FPBits<double>;

  uint64_t scale_u = static_cast<uint64_t>(hi + 2045) << FPBits::FRACTION_LEN;
  double exp_scale = cpp::bit_cast<double>(scale_u);

  double mid_hi = mid * exp_scale;
  double lo_scaled = lo * exp_scale;

  double extra_factor = 0.0;
  uint64_t scale_down = 0x3FE0'0000'0000'0000ULL; // 1022 in the exponent field.

  // Result is denormal if |mid_hi + lo_scaled| < 1.0.
  // In that case, add +-1.0 to mimic denormal rounding points.
  if (is_neg) {
    if ((-1.0 - mid_hi) < lo_scaled) {
      // Extra rounding step is needed, which adds more rounding errors.
      extra_factor = -1.0;
      scale_down = 0x3FF0'0000'0000'0000ULL; // 1023 in the exponent field.
      err += 0x1.0p-52;
    }
  } else {
    if ((1.0 - mid_hi) > lo_scaled) {
      // Extra rounding step is needed, which adds more rounding errors.
      extra_factor = 1.0;
      scale_down = 0x3FF0'0000'0000'0000ULL; // 1023 in the exponent field.
      err += 0x1.0p-52;
    }
  }

  // By adding +-1.0, the hardware adder performs rounding at the exact same
  // bit position as denormal rounding.
  if constexpr (SKIP_ZIV_TEST) {
    double r = extra_factor + (mid_hi + lo_scaled);
    return cpp::bit_cast<double>(cpp::bit_cast<uint64_t>(r) - scale_down);
  } else {
    double err_scaled = err * exp_scale;

    double lo_u = lo_scaled + err_scaled;
    double lo_l = lo_scaled - err_scaled;

    double upper = extra_factor + (mid_hi + lo_u);
    double lower = extra_factor + (mid_hi + lo_l);

    if (LIBC_LIKELY(upper == lower)) {
      return cpp::bit_cast<double>(cpp::bit_cast<uint64_t>(upper) - scale_down);
    }

    return cpp::nullopt;
  }
}

// Upper bound for y = 1075 / |log2(1 - 2^(-53)|, generated by Sollya:
// > y = round(-1075 / log2(1 - 2^(-53)), D, RU);
// > y;
// 0x1.74910d52d3052p62
// > printdouble(y);
// 0x43d74910d52d3052
constexpr uint64_t Y_UPPER_BOUND = 0x43d7'4910'd52d'3052;
// Lower bound for y = 2^(-54) / 1074), generated by Sollya:
// > y = round(2^(-54) / 1074), D, RD);
// > y;
// 0x1.e829f39aef509p-65
// > printdouble(y);
// 0x3bee829f39ae'f509
constexpr uint64_t Y_LOWER_BOUND = 0x3bee'829f'39ae'f509;

// Fast checks for special inputs:
//   x = 0, +-1, 2^k, 10, +- inf
//   y = 0, +-1, 2, 0.5
LIBC_ALWAYS_INLINE cpp::optional<double> check_special_inputs(double x,
                                                              double y) {
  using FPBits = fputil::FPBits<double>;
  FPBits xbits(x), ybits(y);

  bool x_sign = xbits.sign() == Sign::NEG;
  bool y_sign = ybits.sign() == Sign::NEG;

  FPBits x_abs = xbits.abs();
  FPBits y_abs = ybits.abs();

  uint64_t x_u = xbits.uintval();
  uint64_t y_u = ybits.uintval();
  uint64_t y_a = y_abs.uintval();

  if (LIBC_UNLIKELY((x_u & 0x0003'FFFF'FFFF'FFFF) == 0) ||
      ((y_u & 0x000F'FFFF'FFFF'FFFF) == 0)) {
    // If x or y is signaling NaN
    if (x_abs.is_signaling_nan() || y_abs.is_signaling_nan()) {
      fputil::raise_except_if_required(FE_INVALID);
      return FPBits::quiet_nan().get_val();
    }

    if (x == 1.0 || y == 0.0)
      return 1.0;

    if (x == 0.0) {
      if (y_abs.is_nan())
        return y;
      if (y_abs.is_inf())
        return y_sign ? FPBits::inf().get_val() : 0.0;
      bool out_is_neg = x_sign && is_odd_integer(y);
      if (y_sign) {
        // pow(0, negative number) = inf
        fputil::set_errno_if_required(EDOM);
        fputil::raise_except_if_required(FE_DIVBYZERO);
        return FPBits::inf(out_is_neg ? Sign::NEG : Sign::POS).get_val();
      }
      // pow(0, positive number) = 0
      return out_is_neg ? -0.0 : 0.0;
    }

    if (y == 1.0)
      return x;

    if (y == 2.0)
      return x * x;

    if (y == 0.5 && !x_sign)
      return fputil::sqrt<double>(x);

    // TODO: Add special case y = -0.5 when rsqrt is available.

    if (x == 2.0)
      return math::exp2(y);

    if (x == 10.0)
      return math::exp10(y);

    // For x = 2^(+- 2^n):
    //   x^y = 2^(+- 2^n * y).
    if (!x_sign && x_abs.is_normal() && xbits.get_mantissa() == 0 &&
        y_a > Y_LOWER_BOUND && y_a < Y_UPPER_BOUND) {
      int e_x = xbits.get_exponent();
      uint32_t abs_e = static_cast<uint32_t>(e_x > 0 ? e_x : -e_x);
      if (cpp::has_single_bit(abs_e)) {
        double hi = static_cast<double>(e_x) * y;
        if (hi >= 1024.0)
          return set_overflow(false);
        if (hi < -1075.0)
          return set_underflow(false);
        if (hi >= -1022.0)
          return math::exp2(hi);
      }
    }
  }

  return cpp::nullopt;
}

// Filters out extreme input ranges, infinities, NaNs, normalizes denormal
// inputs, and handles negative bases. Returns a value if the result is
// determined, or nullopt if regular evaluation should proceed (in which case x,
// y, e_x, x_mant, is_neg, sign_d may be updated).
LIBC_ALWAYS_INLINE cpp::optional<double>
check_exceptional_cases(double &x, double &y, double &e_x, uint64_t &x_mant,
                        bool &is_neg, double &sign_d) {
  using FPBits = fputil::FPBits<double>;

  FPBits xbits(x), ybits(y);
  bool x_sign = xbits.sign() == Sign::NEG;
  bool y_sign = ybits.sign() == Sign::NEG;

  FPBits x_abs = xbits.abs();
  FPBits y_abs = ybits.abs();

  uint64_t x_u = xbits.uintval();
  uint64_t y_mant = ybits.get_mantissa();
  uint64_t x_a = x_abs.uintval();
  uint64_t y_a = y_abs.uintval();

  if (LIBC_UNLIKELY(y_a <= Y_LOWER_BOUND || y_a >= Y_UPPER_BOUND ||
                    x_u >= FPBits::inf().uintval() ||
                    x_u < FPBits::min_normal().uintval())) {
    // If x or y is signaling NaN
    if (x_abs.is_signaling_nan() || y_abs.is_signaling_nan()) {
      fputil::raise_except_if_required(FE_INVALID);
      return FPBits::quiet_nan().get_val();
    }

    // |y| > |1075 / log2(1 - 2^-53)|.
    if (y_a > 0x43d7'4910'd52d'3052) {
      if (y_a >= 0x7ff0'0000'0000'0000) {
        // y is inf or nan
        if (y_mant != 0) {
          // y is NaN
          // pow(1, NaN) = 1 is already dealt with.
          // pow(x, NaN) = NaN
          return y;
        }

        // Now y is +-Inf
        if (x_abs.is_nan()) {
          // pow(NaN, +-Inf) = NaN
          return x;
        }

        if (x_a == 0x3ff0'0000'0000'0000) {
          // pow(+-1, +-Inf) = 1.0
          return 1.0;
        }

        // pow (|x| < 1, -inf) = +inf
        // pow (|x| < 1, +inf) = 0.0
        // pow (|x| > 1, -inf) = 0.0
        // pow (|x| > 1, +inf) = +inf
        return ((x_a < FPBits::one().uintval()) == y_sign)
                   ? FPBits::inf().get_val()
                   : 0.0;
      }
      // x^y will overflow / underflow in double precision.  Set y to a
      // large enough exponent but not too large, so that the computations
      // won't overflow in double precision.
      y = y_sign ? -0x1.0p100 : 0x1.0p100;
    }

    // y is finite and non-zero.

    if (x_a == FPBits::inf().uintval()) {
      bool out_is_neg = x_sign && is_odd_integer(y);
      Sign out_sign = out_is_neg ? Sign::NEG : Sign::POS;
      return y_sign ? FPBits::zero(out_sign).get_val()
                    : FPBits::inf(out_sign).get_val();
    }

    if (x_a > FPBits::inf().uintval()) {
      // x is NaN.
      // pow (aNaN, 0) is already taken care above.
      return x;
    }

    // Normalize denormal inputs.
    if (x_a < FPBits::min_normal().uintval()) {
      FPBits x_norm(x * 0x1.0p64);
      e_x = static_cast<double>(x_norm.get_exponent()) - 64.0;
      x_mant = x_norm.get_mantissa();
    }

    // x is finite and negative, and y is a finite integer.
    if (x_sign) {
      if (is_integer(y)) {
        x = -x;
        if (is_odd_integer(y)) {
          is_neg = true;
          sign_d = -1.0;
        }
      } else {
        // pow( negative, non-integer ) = NaN
        fputil::set_errno_if_required(EDOM);
        fputil::raise_except_if_required(FE_INVALID);
        return FPBits::quiet_nan().get_val();
      }
    }

    if (y_a <= Y_LOWER_BOUND) {
      volatile double one = 1.0;
      volatile double eps = ((x_a < FPBits::one().uintval()) == !y_sign)
                                ? -0x1.0p-100
                                : 0x1.0p-100;
      return one + eps;
    }
  }

  return cpp::nullopt;
}

// Check if x^y is an exact rounding boundary case (a 54-bit dyadic float,
// which is either an exact 53-bit float or an exact halfway midpoint).
//
// Reference:
//   Lauter, C. and Lefevre, V., "Rounding Boundary Cases for Values of the
//   Exponential and Power Functions," IEEE Trans. Comput. 58(8):1063-1074.
//
// For a normalized 128-bit dyadic float with MSB at bit 127, the top 54 bits
// occupy bits [127:74]. For any 54-bit number, bits [73:0] are zero.
//
// If x^y is not in F_54:
//   |x^y - o_54(x^y)| / x^y >= 2^-114.
// With an evaluation error < 2^-115, the distance from bits [73:0] to the
// nearest 54-bit boundary must satisfy:
//   dist >= (2^-114 - 2^-115) * 2^127 = 2^12.
// We use a threshold of 2^-114 (dist < 2^13) to capture all boundary
// candidates:
//
// 1. x = 2^e: x^y = 2^(e * y) is in F_54 iff e * y is an integer.
// 2. x = m * 2^e, y = n * 2^f with m, n odd integers:
//    x^y in F_54 implies:
//      - 0 <= y <= 35, n <= 35, and f >= -5.
//      - e * y is an integer.
//      - When f < 0 (i.e. y = n / 2^(-f)): m^(2^f) is an integer, verified
//        by taking repeated square roots -f times.
//      - The resulting significand (m^(2^f))^n <= 2^54.
//
// Returns true and sets exact_m and exact_exp if x^y is an exact boundary.
LIBC_INLINE bool is_exact_rounding_boundary(double x, double y,
                                            uint64_t &exact_m, int &exact_exp) {
  using FPBits = fputil::FPBits<double>;

  FPBits xbits(x);
  int x_e = 0;
  uint64_t x_mant = xbits.get_mantissa();
  if (LIBC_UNLIKELY(xbits.get_biased_exponent() == 0)) {
    if (x_mant != 0) {
      int shift = cpp::countl_zero(x_mant) - 11;
      x_mant = (x_mant << shift) & FPBits::FRACTION_MASK;
      x_e = -1022 - shift;
    }
  } else {
    x_e = xbits.get_exponent();
  }

  // Case 1: x = 2^e.
  if (x_mant == 0) {
    double e = static_cast<double>(x_e);
    double ey = e * y;
    if (is_integer(ey)) {
      exact_m = 1;
      exact_exp = static_cast<int>(ey);
      return true;
    }
    return false;
  }

  // Case 2: x is not a power of 2.
  if (y < 0.0 || y > 35.0)
    return false;

  // Decompose y = n * 2^f with n odd.
  FPBits ybits(y);
  uint64_t y_mant = ybits.get_mantissa() | (1ULL << FPBits::FRACTION_LEN);
  int y_exp = ybits.get_exponent() - static_cast<int>(FPBits::FRACTION_LEN);
  int tz_y = cpp::countr_zero(y_mant);
  uint64_t n = y_mant >> tz_y;
  int f = y_exp + tz_y;

  if (n > 35 || f < -5)
    return false;

  // Decompose x = m * 2^e with m odd.
  uint64_t full_x_mant = x_mant | (1ULL << FPBits::FRACTION_LEN);
  int tz_x = cpp::countr_zero(full_x_mant);
  uint64_t m = full_x_mant >> tz_x;
  int e = x_e - static_cast<int>(FPBits::FRACTION_LEN) + tz_x;

  if (f < 0) {
    // Non-integer exponent: y = n / 2^(-f) with -f in {1, ..., 5}.
    double ey = static_cast<double>(e) * y;
    if (!is_integer(ey))
      return false;

    // Check if m^(2^f) is an integer by taking repeated square roots -f times.
    int count = -f;
    uint64_t cur = m;
    for (int i = 0; i < count; ++i) {
      uint64_t s =
          static_cast<uint64_t>(fputil::sqrt<double>(static_cast<double>(cur)));
      if (s * s != cur)
        return false;
      cur = s;
    }

    // Compute res = cur^n and ensure res <= 2^54.
    uint64_t res = 1;
    for (uint64_t i = 0; i < n; ++i) {
      if (res > (1ULL << 54) / cur)
        return false;
      res *= cur;
    }
    exact_m = res;
    exact_exp = static_cast<int>(ey);
    return true;
  } else {
    // Integer exponent: f >= 0, so y is an integer.
    int y_int = static_cast<int>(y);
    uint64_t res = 1;
    for (int i = 0; i < y_int; ++i) {
      if (res > (1ULL << 54) / m)
        return false;
      res *= m;
    }
    exact_m = res;
    exact_exp = e * y_int;
    return true;
  }
}

} // namespace pow_internal
} // namespace math
} // namespace LIBC_NAMESPACE_DECL

#endif // LLVM_LIBC_SRC___SUPPORT_MATH_POW_UTILS_H