path: root/libc/src/math/generic/pow.cpp

//===-- Double-precision x^y 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/pow.h"
#include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
#include "hdr/errno_macros.h"
#include "hdr/fenv_macros.h"
#include "src/__support/CPP/bit.h"
#include "src/__support/FPUtil/FEnvImpl.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/sqrt.h" // Speedup for pow(x, 1/2) = sqrt(x)
#include "src/__support/common.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

namespace LIBC_NAMESPACE_DECL {

using fputil::DoubleDouble;

namespace {

// 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.
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},
};

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);
}

bool is_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);
}

} // namespace

LLVM_LIBC_FUNCTION(double, pow, (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_mant = xbits.get_mantissa();
  uint64_t y_mant = ybits.get_mantissa();
  uint64_t x_u = xbits.uintval();
  uint64_t x_a = x_abs.uintval();
  uint64_t y_a = y_abs.uintval();

  double e_x = static_cast<double>(xbits.get_exponent());
  uint64_t sign = 0;

  ///////// BEGIN - Check exceptional cases ////////////////////////////////////
  // 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();
  }

  // The double precision number that is closest to 1 is (1 - 2^-53), which has
  //   log2(1 - 2^-53) ~ -1.715...p-53.
  // So if |y| > |1075 / log2(1 - 2^-53)|, and x is finite:
  //   |y * log2(x)| = 0 or > 1075.
  // Hence x^y will either overflow or underflow if x is not zero.
  if (LIBC_UNLIKELY(y_mant == 0 || y_a > 0x43d7'4910'd52d'3052 ||
                    x_u == FPBits::one().uintval() ||
                    x_u >= FPBits::inf().uintval() ||
                    x_u < FPBits::min_normal().uintval())) {
    // Exceptional exponents.
    if (y == 0.0)
      return 1.0;

    switch (y_a) {
    case 0x3fe0'0000'0000'0000: { // y = +-0.5
      // TODO: speed up x^(-1/2) with rsqrt(x) when available.
      if (LIBC_UNLIKELY(
              (x == 0.0 || x_u == FPBits::inf(Sign::NEG).uintval()))) {
        // pow(-0, 1/2) = +0
        // pow(-inf, 1/2) = +inf
        // Make sure it works correctly for FTZ/DAZ.
        return y_sign ? 1.0 / (x * x) : (x * x);
      }
      return y_sign ? (1.0 / fputil::sqrt<double>(x)) : fputil::sqrt<double>(x);
    }
    case 0x3ff0'0000'0000'0000: // y = +-1.0
      return y_sign ? (1.0 / x) : x;
    case 0x4000'0000'0000'0000: // y = +-2.0;
      return y_sign ? (1.0 / (x * x)) : (x * x);
    }

    // |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
          // pow(x, NaN) = NaN
          return (x_u == FPBits::one().uintval()) ? 1.0 : 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;
        }

        if (x == 0.0 && y_sign) {
          // pow(+-0, -Inf) = +inf and raise FE_DIVBYZERO
          fputil::set_errno_if_required(EDOM);
          fputil::raise_except_if_required(FE_DIVBYZERO);
          return FPBits::inf().get_val();
        }
        // 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_u == FPBits::one().uintval()) {
      // pow(1, y) = 1
      return 1.0;
    }

    // TODO: Speed things up with pow(2, y) = exp2(y) and pow(10, y) = exp10(y).

    if (x == 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 (x_a == FPBits::inf().uintval()) {
      bool out_is_neg = x_sign && is_odd_integer(y);
      if (y_sign)
        return out_is_neg ? -0.0 : 0.0;
      return FPBits::inf(out_is_neg ? Sign::NEG : Sign::POS).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()) {
      e_x -= 64.0;
      x_mant = FPBits(x * 0x1.0p64).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))
          // sign = -1.0;
          sign = 0x8000'0000'0000'0000;
      } 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();
      }
    }
  }

  ///////// END - Check exceptional cases //////////////////////////////////////

  // x^y = 2^( y * log2(x) )
  //     = 2^( y * ( e_x + log2(m_x) ) )
  // First we compute log2(x) = e_x + log2(m_x)

  // Extract exponent field of x.

  // Use the highest 7 fractional bits of m_x as the index for look up tables.
  unsigned idx_x = static_cast<unsigned>(x_mant >> (FPBits::FRACTION_LEN - 7));
  // Add the hidden bit to the mantissa.
  // 1 <= m_x < 2
  FPBits m_x = FPBits(x_mant | 0x3ff0'0000'0000'0000);

  // Reduced argument for log2(m_x):
  //   dx = r * m_x - 1.
  // The computation is exact, and -2^-8 <= dx < 2^-7.
  // Then m_x = (1 + dx) / r, and
  //   log2(m_x) = log2( (1 + dx) / r )
  //             = log2(1 + dx) - log2(r).

  // In order for the overall computations x^y = 2^(y * log2(x)) to have the
  // relative errors < 2^-52 (1ULP), we will need to evaluate the exponent part
  // y * log2(x) with absolute errors < 2^-52 (or better, 2^-53).  Since the
  // whole exponent range for double precision is bounded by
  // |y * log2(x)| < 1076 ~ 2^10, we need to evaluate log2(x) with absolute
  // errors < 2^-53 * 2^-10 = 2^-63.

  // With that requirement, we use the following degree-6 polynomial
  // approximation:
  //   P(dx) ~ log2(1 + dx) / dx
  // Generated by Sollya with:
  // > P = fpminimax(log2(1 + x)/x, 6, [|D...|], [-2^-8, 2^-7]); P;
  // > dirtyinfnorm(log2(1 + x) - x*P, [-2^-8, 2^-7]);
  //   0x1.d03cc...p-66
  constexpr double COEFFS[] = {0x1.71547652b82fep0,  -0x1.71547652b82e7p-1,
                               0x1.ec709dc3b1fd5p-2, -0x1.7154766124215p-2,
                               0x1.2776bd90259d8p-2, -0x1.ec586c6f3d311p-3,
                               0x1.9c4775eccf524p-3};
  // Error: ulp(dx^2) <= (2^-7)^2 * 2^-52 = 2^-66
  // Extra errors from various computations and rounding directions, the overall
  // errors we can be bounded by 2^-65.

  double dx;
  DoubleDouble dx_c0;

  // Perform exact range reduction and exact product dx * c0.
#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
  dx = fputil::multiply_add(RD[idx_x], m_x.get_val(), -1.0); // Exact
  dx_c0 = fputil::exact_mult(COEFFS[0], dx);
#else
  double c = FPBits(m_x.uintval() & 0x3fff'e000'0000'0000).get_val();
  dx = fputil::multiply_add(RD[idx_x], m_x.get_val() - c, CD[idx_x]); // Exact
  dx_c0 = fputil::exact_mult<double, 28>(dx, COEFFS[0]);              // Exact
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE

  double dx2 = dx * dx;
  double c0 = fputil::multiply_add(dx, COEFFS[2], COEFFS[1]);
  double c1 = fputil::multiply_add(dx, COEFFS[4], COEFFS[3]);
  double c2 = fputil::multiply_add(dx, COEFFS[6], COEFFS[5]);

  double p = fputil::polyeval(dx2, c0, c1, c2);

  // s = e_x - log2(r) + dx * P(dx)
  // Absolute error bound:
  //   |log2(x) - log2_x.hi - log2_x.lo| < 2^-65.

  // Notice that e_x - log2(r).hi is exact, so we perform an exact sum of
  // e_x - log2(r).hi and the high part of the product dx * c0:
  //   log2_x_hi.hi + log2_x_hi.lo = e_x - log2(r).hi + (dx * c0).hi
  DoubleDouble log2_x_hi =
      fputil::exact_add(e_x + LOG2_R_DD[idx_x].hi, dx_c0.hi);
  // The low part is dx^2 * p + low part of (dx * c0) + low part of -log2(r).
  double log2_x_lo =
      fputil::multiply_add(dx2, p, dx_c0.lo + LOG2_R_DD[idx_x].lo);
  // Perform accurate sums.
  DoubleDouble log2_x = fputil::exact_add(log2_x_hi.hi, log2_x_lo);
  log2_x.lo += log2_x_hi.lo;

  // To compute 2^(y * log2(x)), we break the exponent into 3 parts:
  //   y * log(2) = hi + mid + lo, where
  //   hi is an integer
  //   mid * 2^6 is an integer
  //   |lo| <= 2^-7
  // Then:
  //   x^y = 2^(y * log2(x)) = 2^hi * 2^mid * 2^lo,
  // In which 2^mid is obtained from a look-up table of size 2^6 = 64 elements,
  // and 2^lo ~ 1 + lo * P(lo).
  // Thus, we have:
  //   hi + mid = 2^-6 * round( 2^6 * y * log2(x) )
  // If we restrict the output such that |hi| < 150, (hi + mid) uses (8 + 6)
  // bits, hence, if we use double precision to perform
  //   round( 2^6 * y * log2(x))
  // the lo part is bounded by 2^-7 + 2^(-(52 - 14)) = 2^-7 + 2^-38

  // In the following computations:
  //   y6  = 2^6 * y
  //   hm  = 2^6 * (hi + mid) = round(2^6 * y * log2(x)) ~ round(y6 * s)
  //   lo6 = 2^6 * lo = 2^6 * (y - (hi + mid)) = y6 * log2(x) - hm.
  double y6 = y * 0x1.0p6; // Exact.

  DoubleDouble y6_log2_x = fputil::exact_mult(y6, log2_x.hi);
  y6_log2_x.lo = fputil::multiply_add(y6, log2_x.lo, y6_log2_x.lo);

  // Check overflow/underflow.
  double scale = 1.0;

  // |2^(hi + mid) - exp2_hi_mid| <= ulp(exp2_hi_mid) / 2
  // Clamp the exponent part into smaller range that fits double precision.
  // For those exponents that are out of range, the final conversion will round
  // them correctly to inf/max float or 0/min float accordingly.
  constexpr double UPPER_EXP_BOUND = 512.0 * 0x1.0p6;
  if (LIBC_UNLIKELY(FPBits(y6_log2_x.hi).abs().get_val() >= UPPER_EXP_BOUND)) {
    if (FPBits(y6_log2_x.hi).sign() == Sign::POS) {
      scale = 0x1.0p512;
      y6_log2_x.hi -= 512.0 * 64.0;
      if (y6_log2_x.hi > 513.0 * 64.0)
        y6_log2_x.hi = 513.0 * 64.0;
    } else {
      scale = 0x1.0p-512;
      y6_log2_x.hi += 512.0 * 64.0;
      if (y6_log2_x.hi < (-1076.0 + 512.0) * 64.0)
        y6_log2_x.hi = -564.0 * 64.0;
    }
  }

  double hm = fputil::nearest_integer(y6_log2_x.hi);

  // lo6 = 2^6 * lo.
  double lo6_hi = y6_log2_x.hi - hm;
  double lo6 = lo6_hi + y6_log2_x.lo;

  int hm_i = static_cast<int>(hm);
  unsigned idx_y = static_cast<unsigned>(hm_i) & 0x3f;

  // 2^hi
  int64_t exp2_hi_i = static_cast<int64_t>(
      static_cast<uint64_t>(static_cast<int64_t>(hm_i >> 6))
      << FPBits::FRACTION_LEN);
  // 2^mid
  int64_t exp2_mid_hi_i =
      static_cast<int64_t>(FPBits(EXP2_MID1[idx_y].hi).uintval());
  int64_t exp2_mid_lo_i =
      static_cast<int64_t>(FPBits(EXP2_MID1[idx_y].mid).uintval());
  // (-1)^sign * 2^hi * 2^mid
  // Error <= 2^hi * 2^-53
  uint64_t exp2_hm_hi_i =
      static_cast<uint64_t>(exp2_hi_i + exp2_mid_hi_i) + sign;
  // The low part could be 0.
  uint64_t exp2_hm_lo_i =
      idx_y != 0 ? static_cast<uint64_t>(exp2_hi_i + exp2_mid_lo_i) + sign
                 : sign;
  double exp2_hm_hi = FPBits(exp2_hm_hi_i).get_val();
  double exp2_hm_lo = FPBits(exp2_hm_lo_i).get_val();

  // Degree-5 polynomial approximation P(lo6) ~ 2^(lo6 / 2^6) = 2^(lo).
  // Generated by Sollya with:
  // > P = fpminimax(2^(x/64), 5, [|1, D...|], [-2^-1, 2^-1]);
  // > dirtyinfnorm(2^(x/64) - P, [-0.5, 0.5]);
  // 0x1.a2b77e618f5c4c176fd11b7659016cde5de83cb72p-60
  constexpr double EXP2_COEFFS[] = {0x1p0,
                                    0x1.62e42fefa39efp-7,
                                    0x1.ebfbdff82a23ap-15,
                                    0x1.c6b08d7076268p-23,
                                    0x1.3b2ad33f8b48bp-31,
                                    0x1.5d870c4d84445p-40};

  double lo6_sqr = lo6 * lo6;

  double d0 = fputil::multiply_add(lo6, EXP2_COEFFS[2], EXP2_COEFFS[1]);
  double d1 = fputil::multiply_add(lo6, EXP2_COEFFS[4], EXP2_COEFFS[3]);
  double pp = fputil::polyeval(lo6_sqr, d0, d1, EXP2_COEFFS[5]);

  double r = fputil::multiply_add(exp2_hm_hi * lo6, pp, exp2_hm_lo);
  r += exp2_hm_hi;

  return r * scale;
}

} // namespace LIBC_NAMESPACE_DECL