V FastLog2f(const DF df, V x) {

  // 2,2 rational polynomial approximation of std::log1p(x) / std::log(2).

  HWY_ALIGN const float p[4 * (2 + 1)] = {HWY_REP4(-1.8503833400518310E-06f),

                                          HWY_REP4(1.4287160470083755E+00f),

                                          HWY_REP4(7.4245873327820566E-01f)};

  HWY_ALIGN const float q[4 * (2 + 1)] = {HWY_REP4(9.9032814277590719E-01f),

                                          HWY_REP4(1.0096718572241148E+00f),

                                          HWY_REP4(1.7409343003366853E-01f)};

  const Rebind<int32_t, DF> di;

  const auto x_bits = BitCast(di, x);

  // Range reduction to [-1/3, 1/3] - 3 integer, 2 float ops

  const auto exp_bits = Sub(x_bits, Set(di, 0x3f2aaaab));  // = 2/3

  // Shifted exponent = log2; also used to clear mantissa.

  const auto exp_shifted = ShiftRight<23>(exp_bits);

  const auto mantissa = BitCast(df, Sub(x_bits, ShiftLeft<23>(exp_shifted)));

  const auto exp_val = ConvertTo(df, exp_shifted);

  return Add(EvalRationalPolynomial(df, Sub(mantissa, Set(df, 1.0f)), p, q),

             exp_val);

}

V FastLog2f(const DF df, V x) {

  // 2,2 rational polynomial approximation of std::log1p(x) / std::log(2).

  HWY_ALIGN const float p[4 * (2 + 1)] = {HWY_REP4(-1.8503833400518310E-06f),

                                          HWY_REP4(1.4287160470083755E+00f),

                                          HWY_REP4(7.4245873327820566E-01f)};

  HWY_ALIGN const float q[4 * (2 + 1)] = {HWY_REP4(9.9032814277590719E-01f),

                                          HWY_REP4(1.0096718572241148E+00f),

                                          HWY_REP4(1.7409343003366853E-01f)};

  const Rebind<int32_t, DF> di;

  const auto x_bits = BitCast(di, x);

  // Range reduction to [-1/3, 1/3] - 3 integer, 2 float ops

  const auto exp_bits = Sub(x_bits, Set(di, 0x3f2aaaab));  // = 2/3

  // Shifted exponent = log2; also used to clear mantissa.

  const auto exp_shifted = ShiftRight<23>(exp_bits);

  const auto mantissa = BitCast(df, Sub(x_bits, ShiftLeft<23>(exp_shifted)));

  const auto exp_val = ConvertTo(df, exp_shifted);

  return Add(EvalRationalPolynomial(df, Sub(mantissa, Set(df, 1.0f)), p, q),

             exp_val);

}

V FastLog2f(const DF df, V x) {

  // 2,2 rational polynomial approximation of std::log1p(x) / std::log(2).

  HWY_ALIGN const float p[4 * (2 + 1)] = {HWY_REP4(-1.8503833400518310E-06f),

                                          HWY_REP4(1.4287160470083755E+00f),

                                          HWY_REP4(7.4245873327820566E-01f)};

  HWY_ALIGN const float q[4 * (2 + 1)] = {HWY_REP4(9.9032814277590719E-01f),

                                          HWY_REP4(1.0096718572241148E+00f),

                                          HWY_REP4(1.7409343003366853E-01f)};

  const Rebind<int32_t, DF> di;

  const auto x_bits = BitCast(di, x);

  // Range reduction to [-1/3, 1/3] - 3 integer, 2 float ops

  const auto exp_bits = Sub(x_bits, Set(di, 0x3f2aaaab));  // = 2/3

  // Shifted exponent = log2; also used to clear mantissa.

  const auto exp_shifted = ShiftRight<23>(exp_bits);

  const auto mantissa = BitCast(df, Sub(x_bits, ShiftLeft<23>(exp_shifted)));

  const auto exp_val = ConvertTo(df, exp_shifted);

  return Add(EvalRationalPolynomial(df, Sub(mantissa, Set(df, 1.0f)), p, q),

             exp_val);

}

V FastLog2f(const DF df, V x) {

  // 2,2 rational polynomial approximation of std::log1p(x) / std::log(2).

  HWY_ALIGN const float p[4 * (2 + 1)] = {HWY_REP4(-1.8503833400518310E-06f),

                                          HWY_REP4(1.4287160470083755E+00f),

                                          HWY_REP4(7.4245873327820566E-01f)};

  HWY_ALIGN const float q[4 * (2 + 1)] = {HWY_REP4(9.9032814277590719E-01f),

                                          HWY_REP4(1.0096718572241148E+00f),

                                          HWY_REP4(1.7409343003366853E-01f)};

  const Rebind<int32_t, DF> di;

  const auto x_bits = BitCast(di, x);

  // Range reduction to [-1/3, 1/3] - 3 integer, 2 float ops

  const auto exp_bits = Sub(x_bits, Set(di, 0x3f2aaaab));  // = 2/3

  // Shifted exponent = log2; also used to clear mantissa.

  const auto exp_shifted = ShiftRight<23>(exp_bits);

  const auto mantissa = BitCast(df, Sub(x_bits, ShiftLeft<23>(exp_shifted)));

  const auto exp_val = ConvertTo(df, exp_shifted);

  return Add(EvalRationalPolynomial(df, Sub(mantissa, Set(df, 1.0f)), p, q),

             exp_val);

}

V FastPow2f(const DF df, V x) {

  const Rebind<int32_t, DF> di;

  auto floorx = Floor(x);

  auto exp =

      BitCast(df, ShiftLeft<23>(Add(ConvertTo(di, floorx), Set(di, 127))));

  auto frac = Sub(x, floorx);

  auto num = Add(frac, Set(df, 1.01749063e+01));

  num = MulAdd(num, frac, Set(df, 4.88687798e+01));

  num = MulAdd(num, frac, Set(df, 9.85506591e+01));

  num = Mul(num, exp);

  auto den = MulAdd(frac, Set(df, 2.10242958e-01), Set(df, -2.22328856e-02));

  den = MulAdd(den, frac, Set(df, -1.94414990e+01));

  den = MulAdd(den, frac, Set(df, 9.85506633e+01));

  return Div(num, den);

}

V FastPow2f(const DF df, V x) {

  const Rebind<int32_t, DF> di;

  auto floorx = Floor(x);

  auto exp =

      BitCast(df, ShiftLeft<23>(Add(ConvertTo(di, floorx), Set(di, 127))));

  auto frac = Sub(x, floorx);

  auto num = Add(frac, Set(df, 1.01749063e+01));

  num = MulAdd(num, frac, Set(df, 4.88687798e+01));

  num = MulAdd(num, frac, Set(df, 9.85506591e+01));

  num = Mul(num, exp);

  auto den = MulAdd(frac, Set(df, 2.10242958e-01), Set(df, -2.22328856e-02));

  den = MulAdd(den, frac, Set(df, -1.94414990e+01));

  den = MulAdd(den, frac, Set(df, 9.85506633e+01));

  return Div(num, den);

}

V FastPow2f(const DF df, V x) {

  const Rebind<int32_t, DF> di;

  auto floorx = Floor(x);

  auto exp =

      BitCast(df, ShiftLeft<23>(Add(ConvertTo(di, floorx), Set(di, 127))));

  auto frac = Sub(x, floorx);

  auto num = Add(frac, Set(df, 1.01749063e+01));

  num = MulAdd(num, frac, Set(df, 4.88687798e+01));

  num = MulAdd(num, frac, Set(df, 9.85506591e+01));

  num = Mul(num, exp);

  auto den = MulAdd(frac, Set(df, 2.10242958e-01), Set(df, -2.22328856e-02));

  den = MulAdd(den, frac, Set(df, -1.94414990e+01));

  den = MulAdd(den, frac, Set(df, 9.85506633e+01));

  return Div(num, den);

}

V FastPowf(const DF df, V base, V exponent) {

  return FastPow2f(df, Mul(FastLog2f(df, base), exponent));

}

V FastPowf(const DF df, V base, V exponent) {

  return FastPow2f(df, Mul(FastLog2f(df, base), exponent));

}

V FastPowf(const DF df, V base, V exponent) {

  return FastPow2f(df, Mul(FastLog2f(df, base), exponent));

}

V FastCosf(const DF df, V x) {

  // Step 1: range reduction to [0, 2pi)

  const auto pi2 = Set(df, kPi * 2.0f);

  const auto pi2_inv = Set(df, 0.5f / kPi);

  const auto npi2 = Mul(Floor(Mul(x, pi2_inv)), pi2);

  const auto xmodpi2 = Sub(x, npi2);

  // Step 2: range reduction to [0, pi]

  const auto x_pi = Min(xmodpi2, Sub(pi2, xmodpi2));

  // Step 3: range reduction to [0, pi/2]

  const auto above_pihalf = Ge(x_pi, Set(df, kPi / 2.0f));

  const auto x_pihalf = IfThenElse(above_pihalf, Sub(Set(df, kPi), x_pi), x_pi);

  // Step 4: Taylor-like approximation, scaled by 2**0.75 to make angle

  // duplication steps faster, on x/4.

  const auto xs = Mul(x_pihalf, Set(df, 0.25f));

  const auto x2 = Mul(xs, xs);

  const auto x4 = Mul(x2, x2);

  const auto cosx_prescaling =

      MulAdd(x4, Set(df, 0.06960438),

             MulAdd(x2, Set(df, -0.84087373), Set(df, 1.68179268)));

  // Step 5: angle duplication.

  const auto cosx_scale1 =

      MulAdd(cosx_prescaling, cosx_prescaling, Set(df, -1.414213562));

  const auto cosx_scale2 = MulAdd(cosx_scale1, cosx_scale1, Set(df, -1));

  // Step 6: change sign if needed.

  const Rebind<uint32_t, DF> du;

  auto signbit = ShiftLeft<31>(BitCast(du, VecFromMask(df, above_pihalf)));

  return BitCast(df, Xor(signbit, BitCast(du, cosx_scale2)));

}

V FastErff(const DF df, V x) {

  // Formula from

  // https://en.wikipedia.org/wiki/Error_function#Numerical_approximations

  // but constants have been recomputed.

  const auto xle0 = Le(x, Zero(df));

  const auto absx = Abs(x);

  // Compute 1 - 1 / ((((x * a + b) * x + c) * x + d) * x + 1)**4

  const auto denom1 =

      MulAdd(absx, Set(df, 7.77394369e-02), Set(df, 2.05260015e-04));

  const auto denom2 = MulAdd(denom1, absx, Set(df, 2.32120216e-01));

  const auto denom3 = MulAdd(denom2, absx, Set(df, 2.77820801e-01));

  const auto denom4 = MulAdd(denom3, absx, Set(df, 1.0f));

  const auto denom5 = Mul(denom4, denom4);

  const auto inv_denom5 = Div(Set(df, 1.0f), denom5);

  const auto result = NegMulAdd(inv_denom5, inv_denom5, Set(df, 1.0f));

  // Change sign if needed.

  const Rebind<uint32_t, DF> du;

  auto signbit = ShiftLeft<31>(BitCast(du, VecFromMask(df, xle0)));

  return BitCast(df, Xor(signbit, BitCast(du, result)));

}

V FastErff(const DF df, V x) {

  // Formula from

  // https://en.wikipedia.org/wiki/Error_function#Numerical_approximations

  // but constants have been recomputed.

  const auto xle0 = Le(x, Zero(df));

  const auto absx = Abs(x);

  // Compute 1 - 1 / ((((x * a + b) * x + c) * x + d) * x + 1)**4

  const auto denom1 =

      MulAdd(absx, Set(df, 7.77394369e-02), Set(df, 2.05260015e-04));

  const auto denom2 = MulAdd(denom1, absx, Set(df, 2.32120216e-01));

  const auto denom3 = MulAdd(denom2, absx, Set(df, 2.77820801e-01));

  const auto denom4 = MulAdd(denom3, absx, Set(df, 1.0f));

  const auto denom5 = Mul(denom4, denom4);

  const auto inv_denom5 = Div(Set(df, 1.0f), denom5);

  const auto result = NegMulAdd(inv_denom5, inv_denom5, Set(df, 1.0f));

  // Change sign if needed.

  const Rebind<uint32_t, DF> du;

  auto signbit = ShiftLeft<31>(BitCast(du, VecFromMask(df, xle0)));

  return BitCast(df, Xor(signbit, BitCast(du, result)));

}

inline float FastLog2f(float f) {

  HWY_CAPPED(float, 1) D;

  return GetLane(FastLog2f(D, Set(D, f)));

}

inline float FastPow2f(float f) {

  HWY_CAPPED(float, 1) D;

  return GetLane(FastPow2f(D, Set(D, f)));

}

inline float FastPowf(float b, float e) {

  HWY_CAPPED(float, 1) D;

  return GetLane(FastPowf(D, Set(D, b), Set(D, e)));

}

V CubeRootAndAdd(const V x, const V add) {

  const HWY_FULL(float) df;

  const HWY_FULL(int32_t) di;

  const auto kExpBias = Set(di, 0x54800000);  // cast(1.) + cast(1.) / 3

  const auto kExpMul = Set(di, 0x002AAAAA);   // shifted 1/3

  const auto k1_3 = Set(df, 1.0f / 3);

  const auto k4_3 = Set(df, 4.0f / 3);

  const auto xa = x;  // assume inputs never negative

  const auto xa_3 = Mul(k1_3, xa);

  // Multiply exponent by -1/3

  const auto m1 = BitCast(di, xa);

  // Special case for 0. 0 is represented with an exponent of 0, so the

  // "kExpBias - 1/3 * exp" below gives the wrong result. The IfThenZeroElse()

  // sets those values as 0, which prevents having NaNs in the computations

  // below.

  // TODO(eustas): use fused op

  const auto m2 = IfThenZeroElse(

      Eq(m1, Zero(di)), Sub(kExpBias, Mul((ShiftRight<23>(m1)), kExpMul)));

  auto r = BitCast(df, m2);

  // Newton-Raphson iterations

  for (int i = 0; i < 3; i++) {

    const auto r2 = Mul(r, r);

    r = NegMulAdd(xa_3, Mul(r2, r2), Mul(k4_3, r));

  }

  // Final iteration

  auto r2 = Mul(r, r);

  r = MulAdd(k1_3, NegMulAdd(xa, Mul(r2, r2), r), r);

  r2 = Mul(r, r);

  r = MulAdd(r2, x, add);

  return r;

}