GenericPacketMathTrig.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2007 Julien Pommier
// Copyright (C) 2009-2019 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2018-2025 Rasmus Munk Larsen <rmlarsen@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
// SPDX-License-Identifier: MPL-2.0
 
#ifndef EIGEN_ARCH_GENERIC_PACKET_MATH_TRIG_H
#define EIGEN_ARCH_GENERIC_PACKET_MATH_TRIG_H
 
// IWYU pragma: private
#include "../../InternalHeaderCheck.h"
 
namespace Eigen {
namespace internal {
 
//----------------------------------------------------------------------
// Trigonometric Functions
//----------------------------------------------------------------------
 
// Enum for selecting which function to compute. SinCos is intended to compute
// pairs of Sin and Cos of the even entries in the packet, e.g.
// SinCos([a, *, b, *]) = [sin(a), cos(a), sin(b), cos(b)].
enum class TrigFunction : uint8_t { Sin, Cos, Tan, SinCos };
 
// The following code is inspired by the following stack-overflow answer:
//   https://stackoverflow.com/questions/30463616/payne-hanek-algorithm-implementation-in-c/30465751#30465751
// It has been largely optimized:
//  - By-pass calls to frexp.
//  - Aligned loads of required 96 bits of 2/pi. This is accomplished by
//    (1) balancing the mantissa and exponent to the required bits of 2/pi are
//    aligned on 8-bits, and (2) replicating the storage of the bits of 2/pi.
//  - Avoid a branch in rounding and extraction of the remaining fractional part.
// Overall, I measured a speed up higher than x2 on x86-64.
inline float trig_reduce_huge(float xf, Eigen::numext::int32_t* quadrant) {
  using Eigen::numext::int32_t;
  using Eigen::numext::int64_t;
  using Eigen::numext::uint32_t;
  using Eigen::numext::uint64_t;
 
  const double pio2_62 = 3.4061215800865545e-19;     // pi/2 * 2^-62
  const uint64_t zero_dot_five = uint64_t(1) << 61;  // 0.5 in 2.62-bit fixed-point format
 
  // 192 bits of 2/pi for Payne-Hanek reduction
  // Bits are introduced by packet of 8 to enable aligned reads.
  static const uint32_t two_over_pi[] = {
      0x00000028, 0x000028be, 0x0028be60, 0x28be60db, 0xbe60db93, 0x60db9391, 0xdb939105, 0x9391054a, 0x91054a7f,
      0x054a7f09, 0x4a7f09d5, 0x7f09d5f4, 0x09d5f47d, 0xd5f47d4d, 0xf47d4d37, 0x7d4d3770, 0x4d377036, 0x377036d8,
      0x7036d8a5, 0x36d8a566, 0xd8a5664f, 0xa5664f10, 0x664f10e4, 0x4f10e410, 0x10e41000, 0xe4100000};
 
  uint32_t xi = numext::bit_cast<uint32_t>(xf);
  // Below, -118 = -126 + 8.
  //   -126 is to get the exponent,
  //   +8 is to enable alignment of 2/pi's bits on 8 bits.
  // This is possible because the fractional part of x has only 24 meaningful bits.
  uint32_t e = (xi >> 23) - 118;
  // Extract the mantissa and shift it to align it wrt the exponent
  xi = ((xi & 0x007fffffu) | 0x00800000u) << (e & 0x7);
 
  uint32_t i = e >> 3;
  uint32_t twoopi_1 = two_over_pi[i - 1];
  uint32_t twoopi_2 = two_over_pi[i + 3];
  uint32_t twoopi_3 = two_over_pi[i + 7];
 
  // Compute x * 2/pi in 2.62-bit fixed-point format.
  uint64_t p;
  p = uint64_t(xi) * twoopi_3;
  p = uint64_t(xi) * twoopi_2 + (p >> 32);
  p = (uint64_t(xi * twoopi_1) << 32) + p;
 
  // Round to nearest: add 0.5 and extract integral part.
  uint64_t q = (p + zero_dot_five) >> 62;
  *quadrant = int(q);
  // Now it remains to compute "r = x - q*pi/2" with high accuracy,
  // since we have p=x/(pi/2) with high accuracy, we can more efficiently compute r as:
  //   r = (p-q)*pi/2,
  // where the product can be carried out with sufficient accuracy using double precision.
  p -= q << 62;
  return float(double(int64_t(p)) * pio2_62);
}
 
template <TrigFunction Func, typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psincos_float(const Packet& _x) {
  using PacketI = typename unpacket_traits<Packet>::integer_packet;
 
  const Packet cst_2oPI = pset1<Packet>(0.636619746685028076171875f);  // 2/PI
  const Packet cst_rounding_magic = pset1<Packet>(12582912);           // 2^23 for rounding
  const PacketI csti_1 = pset1<PacketI>(1);
  const Packet cst_sign_mask = psignmask<Packet>();
 
  Packet x = pabs(_x);
 
  // Scale x by 2/Pi to find x's octant.
  Packet y = pmul(x, cst_2oPI);
 
  // Rounding trick to find nearest integer:
  Packet y_round = padd(y, cst_rounding_magic);
  EIGEN_OPTIMIZATION_BARRIER(y_round)
  PacketI y_int = preinterpret<PacketI>(y_round);  // last 23 digits represent integer (if abs(x)<2^24)
  y = psub(y_round, cst_rounding_magic);           // nearest integer to x * (2/pi)
 
// Subtract y * Pi/2 to reduce x to the interval -Pi/4 <= x <= +Pi/4
// using "Extended precision modular arithmetic"
#if defined(EIGEN_VECTORIZE_FMA)
  // This version requires true FMA for high accuracy.
  // It provides a max error of 1ULP up to (with absolute_error < 5.9605e-08):
  constexpr float huge_th = (Func == TrigFunction::Sin) ? 117435.992f : 71476.0625f;
  x = pmadd(y, pset1<Packet>(-1.57079601287841796875f), x);
  x = pmadd(y, pset1<Packet>(-3.1391647326017846353352069854736328125e-07f), x);
  x = pmadd(y, pset1<Packet>(-5.390302529957764765544681040410068817436695098876953125e-15f), x);
#else
  // Without true FMA, the previous set of coefficients maintain 1ULP accuracy
  // up to x<15.7 (for sin), but accuracy is immediately lost for x>15.7.
  // We thus use one more iteration to maintain 2ULPs up to reasonably large inputs.
 
  // The following set of coefficients maintain 1ULP up to 9.43 and 14.16 for sin and cos respectively.
  // and 2 ULP up to:
  constexpr float huge_th = (Func == TrigFunction::Sin) ? 25966.f : 18838.f;
  x = pmadd(y, pset1<Packet>(-1.5703125), x);  // = 0xbfc90000
  EIGEN_OPTIMIZATION_BARRIER(x)
  x = pmadd(y, pset1<Packet>(-0.000483989715576171875), x);  // = 0xb9fdc000
  EIGEN_OPTIMIZATION_BARRIER(x)
  x = pmadd(y, pset1<Packet>(1.62865035235881805419921875e-07), x);  // = 0x342ee000
  EIGEN_OPTIMIZATION_BARRIER(x)
  x = pmadd(y, pset1<Packet>(5.5644315544167710640977020375430583953857421875e-11), x);  // = 0x2e74b9ee
 
// For the record, the following set of coefficients maintain 2ULP up
// to a slightly larger range:
// const float huge_th = ComputeSine ? 51981.f : 39086.125f;
// but it slightly fails to maintain 1ULP for two values of sin below pi.
// x = pmadd(y, pset1<Packet>(-3.140625/2.), x);
// x = pmadd(y, pset1<Packet>(-0.00048351287841796875), x);
// x = pmadd(y, pset1<Packet>(-3.13855707645416259765625e-07), x);
// x = pmadd(y, pset1<Packet>(-6.0771006282767103812147979624569416046142578125e-11), x);
 
// For the record, with only 3 iterations it is possible to maintain
// 1 ULP up to 3PI (maybe more) and 2ULP up to 255.
// The coefficients are: 0xbfc90f80, 0xb7354480, 0x2e74b9ee
#endif
 
  if (predux_any(pcmp_le(pset1<Packet>(huge_th), pabs(_x)))) {
    const int PacketSize = unpacket_traits<Packet>::size;
    EIGEN_ALIGN_TO_BOUNDARY(unpacket_traits<Packet>::alignment) float vals[PacketSize];
    EIGEN_ALIGN_TO_BOUNDARY(unpacket_traits<Packet>::alignment) float x_cpy[PacketSize];
    EIGEN_ALIGN_TO_BOUNDARY(unpacket_traits<Packet>::alignment) Eigen::numext::int32_t y_int2[PacketSize];
    pstoreu(vals, pabs(_x));
    pstoreu(x_cpy, x);
    pstoreu(y_int2, y_int);
    for (int k = 0; k < PacketSize; ++k) {
      float val = vals[k];
      if (val >= huge_th && (numext::isfinite)(val)) x_cpy[k] = trig_reduce_huge(val, &y_int2[k]);
    }
    x = ploadu<Packet>(x_cpy);
    y_int = ploadu<PacketI>(y_int2);
  }
 
  // Get the polynomial selection mask from the second bit of y_int
  // We'll calculate both (sin and cos) polynomials and then select from the two.
  Packet poly_mask = preinterpret<Packet>(pcmp_eq(pand(y_int, csti_1), pzero(y_int)));
 
  Packet x2 = pmul(x, x);
 
  // Evaluate the cos(x) polynomial. (-Pi/4 <= x <= Pi/4)
  Packet y1 = pset1<Packet>(2.4372266125283204019069671630859375e-05f);
  y1 = pmadd(y1, x2, pset1<Packet>(-0.00138865201734006404876708984375f));
  y1 = pmadd(y1, x2, pset1<Packet>(0.041666619479656219482421875f));
  y1 = pmadd(y1, x2, pset1<Packet>(-0.5f));
  y1 = pmadd(y1, x2, pset1<Packet>(1.f));
 
  // Evaluate the sin(x) polynomial. (-Pi/4 <= x <= Pi/4)
  // octave/matlab code to compute those coefficients:
  //    x = (0:0.0001:pi/4)';
  //    A = [x.^3 x.^5 x.^7];
  //    w = ((1.-(x/(pi/4)).^2).^5)*2000+1;         # weights trading relative accuracy
  //    c = (A'*diag(w)*A)\‍(A'*diag(w)*(sin(x)-x)); # weighted LS, linear coeff forced to 1
  //    printf('%.64f\n %.64f\n%.64f\n', c(3), c(2), c(1))
  //
  Packet y2 = pset1<Packet>(-0.0001959234114083702898469196984621021329076029360294342041015625f);
  y2 = pmadd(y2, x2, pset1<Packet>(0.0083326873655616851693794799871284340042620897293090820312500000f));
  y2 = pmadd(y2, x2, pset1<Packet>(-0.1666666203982298255503735617821803316473960876464843750000000000f));
  y2 = pmul(y2, x2);
  y2 = pmadd(y2, x, x);
 
  // Select the correct result from the two polynomials.
  // Compute the sign to apply to the polynomial.
  // sin: sign = second_bit(y_int) xor signbit(_x)
  // cos: sign = second_bit(y_int+1)
  Packet sign_bit = (Func == TrigFunction::Sin) ? pxor(_x, preinterpret<Packet>(plogical_shift_left<30>(y_int)))
                                                : preinterpret<Packet>(plogical_shift_left<30>(padd(y_int, csti_1)));
  sign_bit = pand(sign_bit, cst_sign_mask);  // clear all but left most bit
 
  if ((Func == TrigFunction::SinCos) || (Func == TrigFunction::Tan)) {
    Packet peven = peven_mask(x);
    Packet ysin = pselect(poly_mask, y2, y1);
    Packet ycos = pselect(poly_mask, y1, y2);
    Packet sign_bit_sin = pxor(_x, preinterpret<Packet>(plogical_shift_left<30>(y_int)));
    Packet sign_bit_cos = preinterpret<Packet>(plogical_shift_left<30>(padd(y_int, csti_1)));
    sign_bit_sin = pand(sign_bit_sin, cst_sign_mask);  // clear all but left most bit
    sign_bit_cos = pand(sign_bit_cos, cst_sign_mask);  // clear all but left most bit
    y = (Func == TrigFunction::SinCos) ? pselect(peven, pxor(ysin, sign_bit_sin), pxor(ycos, sign_bit_cos))
                                       : pdiv(pxor(ysin, sign_bit_sin), pxor(ycos, sign_bit_cos));
  } else {
    y = (Func == TrigFunction::Sin) ? pselect(poly_mask, y2, y1) : pselect(poly_mask, y1, y2);
    y = pxor(y, sign_bit);
  }
  return y;
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psin_float(const Packet& x) {
  return psincos_float<TrigFunction::Sin>(x);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcos_float(const Packet& x) {
  return psincos_float<TrigFunction::Cos>(x);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet ptan_float(const Packet& x) {
  return psincos_float<TrigFunction::Tan>(x);
}
 
// Pi/2 split into 3 double-precision parts (triple-double).
// c1 + c2 + c3 = pi/2 to ~159 bits. Computed by Sollya.
// c1 = RD(pi/2), c2 = RD(pi/2 - c1), c3 = RD(pi/2 - c1 - c2).
template <typename Packet>
Packet cst_pio2_1() {
  return pset1<Packet>(-1.5707963267948965579989817342720925807952880859375);  // -0x1.921fb54442d18p0
}
template <typename Packet>
Packet cst_pio2_2() {
  return pset1<Packet>(-6.12323399573676603586882014729198302312846062338790e-17);  // -0x1.1a62633145c07p-54
}
template <typename Packet>
Packet cst_pio2_3() {
  return pset1<Packet>(1.4973849048591698329435081771059920083527504761695190e-33);  //  0x1.f1976b7ed8fbcp-110
}
 
// Trigonometric argument reduction for double.
// Reduces x to t such that x + q * pi/2 = t, where |t| <= pi/4.
// Uses a triple-double split of pi/2 (cst_pio2_{1,2,3}).
template <typename Packet>
Packet trig_reduce_small_double(const Packet& x, const Packet& q) {
#ifdef EIGEN_HAS_SINGLE_INSTRUCTION_MADD
  // With FMA, pmadd(a, b, c) = fl(a*b + c) in a single rounding,
  // so Cody-Waite reduction is accurate even under catastrophic cancellation.
  Packet t;
  t = pmadd(cst_pio2_1<Packet>(), q, x);
  t = pmadd(cst_pio2_2<Packet>(), q, t);
  t = pmadd(cst_pio2_3<Packet>(), q, t);
  return t;
#else
  // Without FMA, pmadd is mul + add (two roundings). For large q,
  // pmul(pio2_1, q) rounds before the cancellation with x, losing
  // catastrophic amounts of precision (observed: ~10 digits lost).
  // Use error-free transformations to preserve accuracy.
 
  // Compute q * pio2_1 exactly as a double-word using Dekker's algorithm.
  Packet qp_hi, qp_lo;
  twoprod(cst_pio2_1<Packet>(), q, qp_hi, qp_lo);
 
  // Error-free addition of x and qp_hi using Knuth's 2sum.
  // Returns t_hi + t_lo = x + qp_hi exactly, with t_hi = fl(x + qp_hi).
  Packet t_hi = padd(x, qp_hi);
  Packet v = psub(t_hi, x);
  Packet t_lo = padd(psub(x, psub(t_hi, v)), psub(qp_hi, v));
 
  // Accumulate the low part of the product and the remaining pi/2 terms.
  t_lo = padd(t_lo, qp_lo);
  t_lo = pmadd(cst_pio2_2<Packet>(), q, t_lo);
  t_lo = pmadd(cst_pio2_3<Packet>(), q, t_lo);
 
  return padd(t_hi, t_lo);
#endif
}
 
template <TrigFunction Func, typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psincos_double(const Packet& x) {
  using PacketI = typename unpacket_traits<Packet>::integer_packet;
  using ScalarI = typename unpacket_traits<PacketI>::type;
 
  const Packet cst_sign_mask = psignmask<Packet>();
 
  // If the argument is smaller than this value, use a simpler argument reduction
  const double small_th = 15;
  // If the argument is bigger than this value, use the non-vectorized std version
  const double huge_th = 1e14;
 
  // 2/PI as a double-word: hi + lo = 2/pi to ~107 bits. Computed by Sollya.
  const Packet cst_2oPI_hi =
      pset1<Packet>(0.63661977236758138243288840385503135621547698974609375);  // 0x1.45f306dc9c883p-1
  const Packet cst_2oPI_lo =
      pset1<Packet>(-3.9357353350364971763790381828183628368294820823718866e-17);  // -0x1.6b01ec5417056p-55
  // Integer Packet constants
  const PacketI cst_one = pset1<PacketI>(ScalarI(1));
 
  Packet x_abs = pabs(x);
 
  // Scale x by 2/Pi
  PacketI q_int;
  Packet s;
 
  if (EIGEN_PREDICT_FALSE(predux_any(pcmp_le(pset1<Packet>(small_th), x_abs)))) {
    // Medium path: use double-word product x * (2/pi) for precise quadrant computation.
    Packet prod_hi, prod_lo;
    twoprod(x_abs, cst_2oPI_hi, prod_hi, prod_lo);
    // Correction for 2/pi truncation: add x * lo(2/pi)
    prod_lo = pmadd(x_abs, cst_2oPI_lo, prod_lo);
 
    // Round the double-word (prod_hi, prod_lo) to the nearest integer.
    Packet q = pround(prod_hi);
    // Compute exact fractional part to check if rounding was correct.
    Packet frac = padd(psub(prod_hi, q), prod_lo);
    // Correct if fractional part crossed +-0.5 boundary.
    q = padd(q, pand(pcmp_lt(pset1<Packet>(0.5), frac), pset1<Packet>(1.0)));
    q = padd(q, pand(pcmp_lt(frac, pset1<Packet>(-0.5)), pset1<Packet>(-1.0)));
 
    q_int = pcast<Packet, PacketI>(q);
    s = trig_reduce_small_double(x_abs, q);
  } else {
    // Small path: simple reduction with triple-double pi/2 split.
    Packet qval_noround = pmul(x_abs, cst_2oPI_hi);
    q_int = pcast<Packet, PacketI>(padd(qval_noround, pset1<Packet>(0.5)));
    Packet q = pcast<PacketI, Packet>(q_int);
    s = trig_reduce_small_double(x_abs, q);
  }
 
  Packet ss = pmul(s, s);
 
  // Minimax polynomial approximation of cos(x) on [-pi/4, pi/4].
  // cos(x) = 1 + u * P(u), where u = x^2 and P is degree 6 (7 FMAs total).
  // Coefficients computed by Sollya fpminimax. Max polynomial error ~1.3e-19.
  Packet scos = pset1<Packet>(-1.1368926065317776472832699312119132152576472805094454088248312473297119140625e-11);
  scos = pmadd(scos, ss, pset1<Packet>(2.0875905481768720039634091158002593413556269297259859740734100341796875e-09));
  scos = pmadd(scos, ss, pset1<Packet>(-2.7557315712466412785356544880299711763882442028261721134185791015625e-07));
  scos = pmadd(scos, ss, pset1<Packet>(2.480158729424286522739599714082459058772656135261058807373046875e-05));
  scos = pmadd(scos, ss, pset1<Packet>(-1.388888888888178789471350427220386336557567119598388671875e-03));
  scos = pmadd(scos, ss, pset1<Packet>(4.166666666666664353702032030923874117434024810791015625e-02));
  scos = pmadd(scos, ss, pset1<Packet>(-0.5));
  scos = pmadd(scos, ss, pset1<Packet>(1.0));
 
  // Minimax polynomial approximation of sin(x) on [-pi/4, pi/4].
  // sin(x) = x * (1 + u * R(u)), where u = x^2 and R is degree 5.
  // Computed as: x + x * u * R(u) (6 FMAs + 1 mul).
  // Coefficients computed by Sollya fpminimax. Max polynomial error ~1.0e-17.
  Packet ssin = pset1<Packet>(1.59193066075142890698150587293845624470289834562208852730691432952880859375e-10);
  ssin = pmadd(ssin, ss, pset1<Packet>(-2.50511517945670206974594627392927126408039839589037001132965087890625e-08));
  ssin = pmadd(ssin, ss, pset1<Packet>(2.755731622544328228235042954619160582296899519860744476318359375e-06));
  ssin = pmadd(ssin, ss, pset1<Packet>(-1.9841269837089632013978068858506276228581555187702178955078125e-04));
  ssin = pmadd(ssin, ss, pset1<Packet>(8.333333333331312264835588621281203813850879669189453125e-03));
  ssin = pmadd(ssin, ss, pset1<Packet>(-0.1666666666666666574148081281236954964697360992431640625));
  ssin = pmul(ssin, ss);
  ssin = pmadd(ssin, s, s);
 
  Packet poly_mask = preinterpret<Packet>(pcmp_eq(pand(q_int, cst_one), pzero(q_int)));
 
  Packet sign_sin = pxor(x, preinterpret<Packet>(plogical_shift_left<62>(q_int)));
  Packet sign_cos = preinterpret<Packet>(plogical_shift_left<62>(padd(q_int, cst_one)));
  Packet sign_bit, sFinalRes;
  if (Func == TrigFunction::Sin) {
    sign_bit = sign_sin;
    sFinalRes = pselect(poly_mask, ssin, scos);
  } else if (Func == TrigFunction::Cos) {
    sign_bit = sign_cos;
    sFinalRes = pselect(poly_mask, scos, ssin);
  } else if (Func == TrigFunction::Tan) {
    sign_bit = pxor(sign_sin, sign_cos);
    sFinalRes = pdiv(pselect(poly_mask, ssin, scos), pselect(poly_mask, scos, ssin));
  } else if (Func == TrigFunction::SinCos) {
    Packet peven = peven_mask(x);
    sign_bit = pselect(peven, sign_sin, sign_cos);
    sFinalRes = pselect(pxor(peven, poly_mask), scos, ssin);
  }
  sign_bit = pand(sign_bit, cst_sign_mask);  // clear all but left most bit
  sFinalRes = pxor(sFinalRes, sign_bit);
 
  // For inputs above huge_th the medium-path reduction loses too much precision. A vectorized
  // Payne-Hanek reduction was investigated and judged not worthwhile (high implementation cost
  // for what is in practice a rare path), so these inputs fall back to the scalar libm.
  if (EIGEN_PREDICT_FALSE(predux_any(pcmp_le(pset1<Packet>(huge_th), x_abs)))) {
    const int PacketSize = unpacket_traits<Packet>::size;
    EIGEN_ALIGN_TO_BOUNDARY(unpacket_traits<Packet>::alignment) double sincos_vals[PacketSize];
    EIGEN_ALIGN_TO_BOUNDARY(unpacket_traits<Packet>::alignment) double x_cpy[PacketSize];
    pstoreu(x_cpy, x);
    pstoreu(sincos_vals, sFinalRes);
    for (int k = 0; k < PacketSize; ++k) {
      double val = x_cpy[k];
      if (numext::abs(val) > huge_th && (numext::isfinite)(val)) {
        if (Func == TrigFunction::Sin) {
          sincos_vals[k] = numext::sin(val);
        } else if (Func == TrigFunction::Cos) {
          sincos_vals[k] = numext::cos(val);
        } else if (Func == TrigFunction::Tan) {
          sincos_vals[k] = numext::tan(val);
        } else if (Func == TrigFunction::SinCos) {
          sincos_vals[k] = k % 2 == 0 ? numext::sin(val) : numext::cos(val);
        }
      }
    }
    sFinalRes = ploadu<Packet>(sincos_vals);
  }
  return sFinalRes;
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psin_double(const Packet& x) {
  return psincos_double<TrigFunction::Sin>(x);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcos_double(const Packet& x) {
  return psincos_double<TrigFunction::Cos>(x);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet ptan_double(const Packet& x) {
  return psincos_double<TrigFunction::Tan>(x);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
    std::enable_if_t<std::is_same<typename unpacket_traits<Packet>::type, float>::value, Packet>
    psincos_selector(const Packet& x) {
  return psincos_float<TrigFunction::SinCos, Packet>(x);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
    std::enable_if_t<std::is_same<typename unpacket_traits<Packet>::type, double>::value, Packet>
    psincos_selector(const Packet& x) {
  return psincos_double<TrigFunction::SinCos, Packet>(x);
}
 
//----------------------------------------------------------------------
// Inverse Trigonometric Functions
//----------------------------------------------------------------------
 
// Generic implementation of acos(x).
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pacos_float(const Packet& x_in) {
  using Scalar = typename unpacket_traits<Packet>::type;
  static_assert(std::is_same<Scalar, float>::value, "Scalar type must be float");
 
  const Packet cst_one = pset1<Packet>(Scalar(1));
  const Packet cst_pi = pset1<Packet>(Scalar(EIGEN_PI));
  const Packet p6 = pset1<Packet>(Scalar(2.36423197202384471893310546875e-3));
  const Packet p5 = pset1<Packet>(Scalar(-1.1368644423782825469970703125e-2));
  const Packet p4 = pset1<Packet>(Scalar(2.717843465507030487060546875e-2));
  const Packet p3 = pset1<Packet>(Scalar(-4.8969544470310211181640625e-2));
  const Packet p2 = pset1<Packet>(Scalar(8.8804088532924652099609375e-2));
  const Packet p1 = pset1<Packet>(Scalar(-0.214591205120086669921875));
  const Packet p0 = pset1<Packet>(Scalar(1.57079637050628662109375));
 
  // For x in [0:1], we approximate acos(x)/sqrt(1-x), which is a smooth
  // function, by a 6'th order polynomial.
  // For x in [-1:0) we use that acos(-x) = pi - acos(x).
  const Packet neg_mask = psignbit(x_in);
  const Packet abs_x = pabs(x_in);
 
  // Evaluate the polynomial using Horner's rule:
  //   P(x) = p0 + x * (p1 +  x * (p2 + ... (p5 + x * p6)) ... ) .
  // We evaluate even and odd terms independently to increase
  // instruction level parallelism.
  Packet x2 = pmul(x_in, x_in);
  Packet p_even = pmadd(p6, x2, p4);
  Packet p_odd = pmadd(p5, x2, p3);
  p_even = pmadd(p_even, x2, p2);
  p_odd = pmadd(p_odd, x2, p1);
  p_even = pmadd(p_even, x2, p0);
  Packet p = pmadd(p_odd, abs_x, p_even);
 
  // The polynomial approximates acos(x)/sqrt(1-x), so
  // multiply by sqrt(1-x) to get acos(x).
  // Conveniently returns NaN for arguments outside [-1:1].
  Packet denom = psqrt(psub(cst_one, abs_x));
  Packet result = pmul(denom, p);
  // Undo mapping for negative arguments.
  return pselect(neg_mask, psub(cst_pi, result), result);
}
 
// Generic implementation of asin(x).
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pasin_float(const Packet& x_in) {
  using Scalar = typename unpacket_traits<Packet>::type;
  static_assert(std::is_same<Scalar, float>::value, "Scalar type must be float");
 
  constexpr float kPiOverTwo = static_cast<float>(EIGEN_PI / 2);
 
  const Packet cst_half = pset1<Packet>(0.5f);
  const Packet cst_one = pset1<Packet>(1.0f);
  const Packet cst_two = pset1<Packet>(2.0f);
  const Packet cst_pi_over_two = pset1<Packet>(kPiOverTwo);
 
  const Packet abs_x = pabs(x_in);
  const Packet sign_mask = pandnot(x_in, abs_x);
  const Packet invalid_mask = pcmp_lt(cst_one, abs_x);
 
  // For arguments |x| > 0.5, we map x back to [0:0.5] using
  // the transformation x_large = sqrt(0.5*(1-x)), and use the
  // identity
  //   asin(x) = pi/2 - 2 * asin( sqrt( 0.5 * (1 - x)))
 
  const Packet x_large = psqrt(pnmadd(cst_half, abs_x, cst_half));
  const Packet large_mask = pcmp_lt(cst_half, abs_x);
  const Packet x = pselect(large_mask, x_large, abs_x);
  const Packet x2 = pmul(x, x);
 
  // For |x| < 0.5 approximate asin(x)/x by an 8th order polynomial with
  // even terms only.
  constexpr float alpha[] = {5.08838854730129241943359375e-2f, 3.95139865577220916748046875e-2f,
                             7.550220191478729248046875e-2f, 0.16664917767047882080078125f, 1.00000011920928955078125f};
  Packet p = ppolevl<Packet, 4>::run(x2, alpha);
  p = pmul(p, x);
 
  const Packet p_large = pnmadd(cst_two, p, cst_pi_over_two);
  p = pselect(large_mask, p_large, p);
  // Flip the sign for negative arguments.
  p = pxor(p, sign_mask);
  // Return NaN for arguments outside [-1:1].
  return por(invalid_mask, p);
}
 
template <typename Scalar>
struct patan_reduced {
  template <typename Packet>
  static EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet run(const Packet& x);
};
 
template <>
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patan_reduced<double>::run(const Packet& x) {
  constexpr double alpha[] = {2.6667153866462208e-05, 3.0917513112462781e-03, 5.2574296781008604e-02,
                              3.0409318473444424e-01, 7.5365702534987022e-01, 8.2704055405494614e-01,
                              3.3004361289279920e-01};
 
  constexpr double beta[] = {
      2.7311202462436667e-04, 1.0899150928962708e-02, 1.1548932646420353e-01, 4.9716458728465573e-01, 1.0,
      9.3705509168587852e-01, 3.3004361289279920e-01};
 
  Packet x2 = pmul(x, x);
  Packet p = ppolevl<Packet, 6>::run(x2, alpha);
  Packet q = ppolevl<Packet, 6>::run(x2, beta);
  return pmul(x, pdiv(p, q));
}
 
// Computes elementwise atan(x) for x in [-1:1] with 2 ulp accuracy.
template <>
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patan_reduced<float>::run(const Packet& x) {
  constexpr float alpha[] = {1.12026982009410858154296875e-01f, 7.296695709228515625e-01f, 8.109951019287109375e-01f};
 
  constexpr float beta[] = {1.00917108356952667236328125e-02f, 2.8318560123443603515625e-01f, 1.0f,
                            8.109951019287109375e-01f};
 
  Packet x2 = pmul(x, x);
  Packet p = ppolevl<Packet, 2>::run(x2, alpha);
  Packet q = ppolevl<Packet, 3>::run(x2, beta);
  return pmul(x, pdiv(p, q));
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet generic_atan(const Packet& x_in) {
  using Scalar = typename unpacket_traits<Packet>::type;
 
  constexpr Scalar kPiOverTwo = static_cast<Scalar>(EIGEN_PI / 2);
 
  const Packet cst_signmask = psignmask<Packet>();
  const Packet cst_one = pset1<Packet>(Scalar(1));
  const Packet cst_pi_over_two = pset1<Packet>(kPiOverTwo);
 
  //   "Large": For |x| > 1, use atan(1/x) = sign(x)*pi/2 - atan(x).
  //   "Small": For |x| <= 1, approximate atan(x) directly by a polynomial
  //            calculated using Rminimax.
 
  const Packet abs_x = pabs(x_in);
  const Packet x_signmask = pand(x_in, cst_signmask);
  const Packet large_mask = pcmp_lt(cst_one, abs_x);
  const Packet x = pselect(large_mask, preciprocal(abs_x), abs_x);
  const Packet p = patan_reduced<Scalar>::run(x);
  // Apply transformations according to the range reduction masks.
  Packet result = pselect(large_mask, psub(cst_pi_over_two, p), p);
  // Return correct sign
  return pxor(result, x_signmask);
}
 
//----------------------------------------------------------------------
// Hyperbolic Functions
//----------------------------------------------------------------------
 
#ifdef EIGEN_FAST_MATH

template <typename T>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS T ptanh_float(const T& a_x) {
  // Clamp the inputs to the range [-c, c] and set everything
  // outside that range to 1.0. The value c is chosen as the smallest
  // floating point argument such that the approximation is exactly 1.
  // This saves clamping the value at the end.
#ifdef EIGEN_VECTORIZE_FMA
  const T plus_clamp = pset1<T>(8.01773357391357422f);
  const T minus_clamp = pset1<T>(-8.01773357391357422f);
#else
  const T plus_clamp = pset1<T>(7.90738964080810547f);
  const T minus_clamp = pset1<T>(-7.90738964080810547f);
#endif
  const T x = pmax(pmin(a_x, plus_clamp), minus_clamp);
 
  // The following rational approximation was generated by rminimax
  // (https://gitlab.inria.fr/sfilip/rminimax) using the following
  // command:
  // $ ratapprox --function="tanh(x)" --dom='[-8.67,8.67]' --num="odd"
  //   --den="even" --type="[9,8]" --numF="[SG]" --denF="[SG]" --log
  //   --output=tanhf.sollya --dispCoeff="dec"
 
  // The monomial coefficients of the numerator polynomial (odd).
  constexpr float alpha[] = {1.394553628e-8f, 2.102733560e-5f, 3.520756727e-3f, 1.340216100e-1f};
 
  // The monomial coefficients of the denominator polynomial (even).
  constexpr float beta[] = {8.015776984e-7f, 3.326951409e-4f, 2.597254514e-2f, 4.673548340e-1f, 1.0f};
 
  // Since the polynomials are odd/even, we need x^2.
  const T x2 = pmul(x, x);
  const T x3 = pmul(x2, x);
 
  T p = ppolevl<T, 3>::run(x2, alpha);
  T q = ppolevl<T, 4>::run(x2, beta);
  // Take advantage of the fact that the constant term in p is 1 to compute
  // x*(x^2*p + 1) = x^3 * p + x.
  p = pmadd(x3, p, x);
 
  // Divide the numerator by the denominator.
  return pdiv(p, q);
}
 
#else

template <typename T>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS T ptanh_float(const T& x) {
  // The polynomial coefficients were computed using Rminimax:
  // % ./ratapprox --function="tanh(x)-x" --dom='[-1.25,1.25]' --num="[x^3,x^5]" --den="even"
  //     --type="[3,4]" --numF="[SG]" --denF="[SG]" --log --dispCoeff="dec" --output=tanhf.solly
  constexpr float alpha[] = {-1.46725140511989593505859375e-02f, -3.333333432674407958984375e-01f};
  constexpr float beta[] = {1.570280082523822784423828125e-02, 4.4401752948760986328125e-01, 1.0f};
  const T x2 = pmul(x, x);
  const T x3 = pmul(x2, x);
  const T p = ppolevl<T, 1>::run(x2, alpha);
  const T q = ppolevl<T, 2>::run(x2, beta);
  const T small_tanh = pmadd(x3, pdiv(p, q), x);
 
  const T sign_mask = psignmask<T>();
  const T abs_x = pandnot(x, sign_mask);
  constexpr float kSmallThreshold = 1.25f;
  const T large_mask = pcmp_lt(pset1<T>(kSmallThreshold), abs_x);
  // Fast exit if all elements are small.
  if (!predux_any(large_mask)) {
    return small_tanh;
  }
 
  //  Compute as 1 - (2 / (1 + exp(2*x)))
  const T one = pset1<T>(1.0f);
  const T two = pset1<T>(2.0f);
  const T s = pexp_float<T, true>(pmul(two, abs_x));
  const T abs_tanh = psub(one, pdiv(two, padd(s, one)));
 
  // Handle infinite inputs and set sign bit.
  constexpr float kHugeThreshold = 16.0f;
  const T huge_mask = pcmp_lt(pset1<T>(kHugeThreshold), abs_x);
  const T x_sign = pand(sign_mask, x);
  const T large_tanh = por(x_sign, pselect(huge_mask, one, abs_tanh));
  return pselect(large_mask, large_tanh, small_tanh);
}
 
#endif  // EIGEN_FAST_MATH

template <typename T>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS T ptanh_double(const T& a_x) {
  // Clamp the inputs to the range [-c, c] and set everything
  // outside that range to 1.0. The value c is chosen as the smallest
  // floating point argument such that the approximation is exactly 1.
  // This saves clamping the value at the end.
#ifdef EIGEN_VECTORIZE_FMA
  const T plus_clamp = pset1<T>(17.6610191624600077);
  const T minus_clamp = pset1<T>(-17.6610191624600077);
#else
  const T plus_clamp = pset1<T>(17.714196154005176);
  const T minus_clamp = pset1<T>(-17.714196154005176);
#endif
  const T x = pmax(pmin(a_x, plus_clamp), minus_clamp);
  // The following rational approximation was generated by rminimax
  // (https://gitlab.inria.fr/sfilip/rminimax) using the following
  // command:
  // $ ./ratapprox --function="tanh(x)" --dom='[-18.72,18.72]'
  //   --num="odd" --den="even" --type="[19,18]" --numF="[D]"
  //   --denF="[D]" --log --output=tanh.sollya --dispCoeff="dec"
 
  // The monomial coefficients of the numerator polynomial (odd).
  constexpr double alpha[] = {2.6158007860482230e-23, 7.6534862268749319e-19, 3.1309488231386680e-15,
                              4.2303918148209176e-12, 2.4618379131293676e-09, 6.8644367682497074e-07,
                              9.3839087674268880e-05, 5.9809711724441161e-03, 1.5184719640284322e-01};
 
  // The monomial coefficients of the denominator polynomial (even).
  constexpr double beta[] = {6.463747022670968018e-21, 5.782506856739003571e-17,
                             1.293019623712687916e-13, 1.123643448069621992e-10,
                             4.492975677839633985e-08, 8.785185266237658698e-06,
                             8.295161192716231542e-04, 3.437448108450402717e-02,
                             4.851805297361760360e-01, 1.0};
 
  // Since the polynomials are odd/even, we need x^2.
  const T x2 = pmul(x, x);
  const T x3 = pmul(x2, x);
 
  // Interleave the evaluation of the numerator polynomial p and
  // denominator polynomial q.
  T p = ppolevl<T, 8>::run(x2, alpha);
  T q = ppolevl<T, 9>::run(x2, beta);
  // Take advantage of the fact that the constant term in p is 1 to compute
  // x*(x^2*p + 1) = x^3 * p + x.
  p = pmadd(x3, p, x);
 
  // Divide the numerator by the denominator.
  return pdiv(p, q);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patanh_float(const Packet& x) {
  using Scalar = typename unpacket_traits<Packet>::type;
  static_assert(std::is_same<Scalar, float>::value, "Scalar type must be float");
 
  // For |x| in [0:0.5] we use a polynomial approximation of the form
  // P(x) = x + x^3*(alpha[4] + x^2 * (alpha[3] + x^2 * (... x^2 * alpha[0]) ... )).
  constexpr float alpha[] = {0.1819281280040740966796875f, 8.2311116158962249755859375e-2f,
                             0.14672131836414337158203125f, 0.1997792422771453857421875f, 0.3333373963832855224609375f};
  const Packet x2 = pmul(x, x);
  const Packet x3 = pmul(x, x2);
  Packet p = ppolevl<Packet, 4>::run(x2, alpha);
  p = pmadd(x3, p, x);
 
  const Packet half = pset1<Packet>(0.5f);
  const Packet one = pset1<Packet>(1.0f);
  const Packet x_gt_half = pcmp_le(half, pabs(x));
  // Fast exit: if all |x| <= 0.5, skip the expensive plog/pdiv branch.
  if (!predux_any(x_gt_half)) {
    return p;
  }
 
  // For |x| in ]0.5:1.0] we use atanh = 0.5*ln((1+x)/(1-x));
  Packet r = pdiv(padd(one, x), psub(one, x));
  r = pmul(half, plog(r));
 
  const Packet x_eq_one = pcmp_eq(one, pabs(x));
  const Packet x_gt_one = pcmp_lt(one, pabs(x));
  const Packet sign_mask = psignmask<Packet>();
  const Packet x_sign = pand(sign_mask, x);
  const Packet inf = pinf<Packet>();
  return por(x_gt_one, pselect(x_eq_one, por(x_sign, inf), pselect(x_gt_half, r, p)));
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patanh_double(const Packet& x) {
  using Scalar = typename unpacket_traits<Packet>::type;
  static_assert(std::is_same<Scalar, double>::value, "Scalar type must be double");
  // For x in [-0.5:0.5] we use a rational approximation of the form
  // R(x) = x + x^3*P(x^2)/Q(x^2), where P is or order 4 and Q is of order 5.
  constexpr double alpha[] = {3.3071338469301391e-03, -4.7129526768798737e-02, 1.8185306179826699e-01,
                              -2.5949536095445679e-01, 1.2306328729812676e-01};
 
  constexpr double beta[] = {-3.8679974580640881e-03, 7.6391885763341910e-02,  -4.2828141436397615e-01,
                             9.8733495886883648e-01,  -1.0000000000000000e+00, 3.6918986189438030e-01};
 
  const Packet x2 = pmul(x, x);
  const Packet x3 = pmul(x, x2);
  Packet p = ppolevl<Packet, 4>::run(x2, alpha);
  Packet q = ppolevl<Packet, 5>::run(x2, beta);
  Packet y_small = pmadd(x3, pdiv(p, q), x);
 
  const Packet half = pset1<Packet>(0.5);
  const Packet one = pset1<Packet>(1.0);
  const Packet x_gt_half = pcmp_le(half, pabs(x));
  // Fast exit: if all |x| <= 0.5, skip the expensive plog/pdiv branch.
  if (!predux_any(x_gt_half)) {
    return y_small;
  }
 
  // For |x| in ]0.5:1.0] we use atanh = 0.5*ln((1+x)/(1-x));
  Packet y_large = pdiv(padd(one, x), psub(one, x));
  y_large = pmul(half, plog(y_large));
 
  const Packet x_eq_one = pcmp_eq(one, pabs(x));
  const Packet x_gt_one = pcmp_lt(one, pabs(x));
  const Packet sign_mask = psignmask<Packet>();
  const Packet x_sign = pand(sign_mask, x);
  const Packet inf = pinf<Packet>();
  return por(x_gt_one, pselect(x_eq_one, por(x_sign, inf), pselect(x_gt_half, y_large, y_small)));
}
 
//----------------------------------------------------------------------
// sinh / cosh
//----------------------------------------------------------------------

template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psinh_float(const Packet& x) {
  using Scalar = typename unpacket_traits<Packet>::type;
  static_assert(std::is_same<Scalar, float>::value, "Scalar type must be float");
 
  const Packet sign_mask = psignmask<Packet>();
  const Packet abs_x = pandnot(x, sign_mask);
  const Packet x_sign = pand(x, sign_mask);
 
  // For |x| < 1, use a polynomial approximation to avoid
  // cancellation in exp(x) - exp(-x).
  constexpr float alpha[] = {2.7557314045e-06f, 1.9841270114e-04f, 8.3333335817e-03f, 1.6666666716e-01f};
  const Packet x2 = pmul(x, x);
  Packet p_small = ppolevl<Packet, 3>::run(x2, alpha);
  p_small = pmadd(pmul(x2, x), p_small, x);
 
  // Compute e = exp(|x|) / 2 = exp(|x| - 1) * (e/2), where e is Euler's number.
  // Using a single exp avoids a second expensive call, and subtracting 1 (exactly
  // representable) instead of ln2 avoids rounding error in the argument to exp,
  // which would be amplified into large relative output error.
  const Packet half_e = pset1<Packet>(1.3591409142295225f);  // e/2
  const Packet one = pset1<Packet>(1.0f);
  const Packet e = pmul(pexp(psub(abs_x, one)), half_e);
 
  // Medium path (1 <= |x| <= 20):
  //   sinh(x) = (exp(|x|) - exp(-|x|)) / 2
  //           = (2*e - 1/(2*e)) / 2 = e - 1/(4*e)
  const Packet quarter = pset1<Packet>(0.25f);
  Packet p_medium = psub(e, pdiv(quarter, e));
 
  // Large path (|x| > 20): exp(-|x|) is negligible, sinh(x) ~ exp(|x|)/2 = e.
  const Packet large_threshold = pset1<Packet>(20.0f);
  const Packet large_mask = pcmp_lt(large_threshold, abs_x);
  Packet p_large = pselect(large_mask, e, p_medium);
  p_large = por(x_sign, p_large);
 
  const Packet small_mask = pcmp_lt(abs_x, one);
  return pselect(small_mask, p_small, p_large);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psinh_double(const Packet& x) {
  using Scalar = typename unpacket_traits<Packet>::type;
  static_assert(std::is_same<Scalar, double>::value, "Scalar type must be double");
 
  const Packet sign_mask = psignmask<Packet>();
  const Packet abs_x = pandnot(x, sign_mask);
  const Packet x_sign = pand(x, sign_mask);
 
  // Taylor series: sinh(x) = x + x^3/3! + x^5/5! + ... + x^19/19!
  // Polynomial form: sinh(x) = x + x^3 * P(x^2) where P(t) = sum_{k=0}^{8} t^k/(2k+3)!
  // ppolevl stores highest-degree coefficient first.
  constexpr double alpha[] = {
      8.2206352466243297e-18,  // t^8: 1/19!
      2.8114572543455206e-15,  // t^7: 1/17!
      7.6471637318198164e-13,  // t^6: 1/15!
      1.6059043836821613e-10,  // t^5: 1/13!
      2.5052108385441718e-08,  // t^4: 1/11!
      2.7557319223985893e-06,  // t^3: 1/9!
      1.9841269841269841e-04,  // t^2: 1/7!
      8.3333333333333332e-03,  // t^1: 1/5!
      1.6666666666666666e-01,  // t^0: 1/3!
  };
  const Packet x2 = pmul(x, x);
  Packet p_small = ppolevl<Packet, 8>::run(x2, alpha);
  p_small = pmadd(pmul(x2, x), p_small, x);
 
  // Compute e = exp(|x|) / 2 = exp(|x| - 1) * (e/2), where e is Euler's number.
  // Subtracting 1 (exactly representable) instead of ln2 avoids rounding error
  // in the argument to exp, which would be amplified into large relative error.
  const Packet half_e = pset1<Packet>(1.3591409142295225);  // e/2
  const Packet one = pset1<Packet>(1.0);
  const Packet e = pmul(pexp(psub(abs_x, one)), half_e);
 
  // Medium path (1 <= |x| <= 20):
  //   sinh(x) = (exp(|x|) - exp(-|x|)) / 2 = e - 1/(4*e)
  const Packet quarter = pset1<Packet>(0.25);
  Packet p_medium = psub(e, pdiv(quarter, e));
 
  // Large path (|x| > 20): exp(-|x|) is negligible, sinh(x) ~ exp(|x|)/2 = e.
  const Packet large_threshold = pset1<Packet>(20.0);
  const Packet large_mask = pcmp_lt(large_threshold, abs_x);
  Packet p_large = pselect(large_mask, e, p_medium);
  p_large = por(x_sign, p_large);
  const Packet small_mask = pcmp_lt(abs_x, one);
  return pselect(small_mask, p_small, p_large);
}

template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcosh_float(const Packet& x) {
  const Packet abs_x = pabs(x);
 
  // Compute e = exp(|x|) / 2 = exp(|x| - 1) * (e/2), where e is Euler's number.
  // Using a single exp avoids a second expensive call, and subtracting 1 (exactly
  // representable) instead of ln2 avoids rounding error in the argument to exp,
  // which would be amplified into large relative output error.
  const Packet half_e = pset1<Packet>(1.3591409142295225f);  // e/2
  const Packet one = pset1<Packet>(1.0f);
  const Packet e = pmul(pexp(psub(abs_x, one)), half_e);
 
  // Medium path: cosh(x) = (exp(|x|) + exp(-|x|)) / 2
  //            = (2*e + 1/(2*e)) / 2 = e + 1/(4*e)
  const Packet quarter = pset1<Packet>(0.25f);
  Packet p_medium = padd(e, pdiv(quarter, e));
 
  // Large path (|x| > 20): exp(-|x|) is negligible, cosh(x) ~ exp(|x|)/2 = e.
  const Packet large_threshold = pset1<Packet>(20.0f);
  const Packet large_mask = pcmp_lt(large_threshold, abs_x);
  return pselect(large_mask, e, p_medium);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcosh_double(const Packet& x) {
  const Packet abs_x = pabs(x);
 
  // Compute e = exp(|x|) / 2 = exp(|x| - 1) * (e/2), where e is Euler's number.
  // Subtracting 1 (exactly representable) instead of ln2 avoids rounding error
  // in the argument to exp, which would be amplified into large relative error.
  const Packet half_e = pset1<Packet>(1.3591409142295225);  // e/2
  const Packet one = pset1<Packet>(1.0);
  const Packet e = pmul(pexp(psub(abs_x, one)), half_e);
 
  // Medium path: cosh(x) = (exp(|x|) + exp(-|x|)) / 2 = e + 1/(4*e)
  const Packet quarter = pset1<Packet>(0.25);
  Packet p_medium = padd(e, pdiv(quarter, e));
 
  // Large path (|x| > 20): exp(-|x|) is negligible, cosh(x) ~ exp(|x|)/2 = e.
  const Packet large_threshold = pset1<Packet>(20.0);
  const Packet large_mask = pcmp_lt(large_threshold, abs_x);
  return pselect(large_mask, e, p_medium);
}
 
//----------------------------------------------------------------------
// asinh / acosh
//----------------------------------------------------------------------

template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pasinh_float(const Packet& x) {
  const Packet sign_mask = psignmask<Packet>();
  const Packet abs_x = pandnot(x, sign_mask);
  const Packet x_sign = pand(x, sign_mask);
  const Packet one = pset1<Packet>(1.0f);
 
  // For |x| >= 1e10, use log(2|x|) = log1p(|x| - 1) + ln2 to avoid x^2 overflow.
  const Packet large_mask = pcmp_lt(pset1<Packet>(1e10f), abs_x);
  // Guard x^2 against overflow in the large case.
  const Packet x2 = pmul(abs_x, pselect(large_mask, pzero(abs_x), abs_x));
  // For |x| < 1e10: log1p(|x| + x^2 / (1 + sqrt(1 + x^2))).
  // Algebraically equivalent to log(|x| + sqrt(x^2 + 1))
  // but avoids cancellation for small |x|.
  Packet normal_arg = padd(abs_x, pdiv(x2, padd(one, psqrt(padd(one, x2)))));
  // For |x| >= 1e10: log1p(|x| - 1), then add ln2 after.
  Packet large_arg = psub(abs_x, one);
  // Select argument, then call log1p once.
  Packet result = generic_log1p(pselect(large_mask, large_arg, normal_arg));
  // Add ln2 for the large path: log(2|x|) = log(|x|) + ln2 = log1p(|x|-1) + ln2.
  const Packet ln2 = pset1<Packet>(0.6931471805599453f);
  result = pselect(large_mask, padd(result, ln2), result);
  return por(x_sign, result);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pasinh_double(const Packet& x) {
  const Packet sign_mask = psignmask<Packet>();
  const Packet abs_x = pandnot(x, sign_mask);
  const Packet x_sign = pand(x, sign_mask);
  const Packet one = pset1<Packet>(1.0);
 
  const Packet large_mask = pcmp_lt(pset1<Packet>(1e150), abs_x);
  const Packet x2 = pmul(abs_x, pselect(large_mask, pzero(abs_x), abs_x));
  Packet normal_arg = padd(abs_x, pdiv(x2, padd(one, psqrt(padd(one, x2)))));
  Packet large_arg = psub(abs_x, one);
  Packet result = generic_log1p(pselect(large_mask, large_arg, normal_arg));
  const Packet ln2 = pset1<Packet>(0.6931471805599453);
  result = pselect(large_mask, padd(result, ln2), result);
  return por(x_sign, result);
}

template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pacosh_float(const Packet& x) {
  const Packet one = pset1<Packet>(1.0f);
  const Packet two = pset1<Packet>(2.0f);
  const Packet t = psub(x, one);
  const Packet huge_mask = pcmp_lt(pset1<Packet>(1e10f), x);
  // Guard t*(t+2) against overflow in the huge case.
  const Packet t_tp2 = pmul(pselect(huge_mask, pzero(t), t), padd(t, two));
  Packet normal_arg = padd(t, psqrt(t_tp2));
  // For huge x: acosh(x) = log(2x) = log1p(x - 1) + ln2.
  Packet huge_arg = t;
  // Select argument, then call log1p once.
  Packet result = generic_log1p(pselect(huge_mask, huge_arg, normal_arg));
  const Packet ln2 = pset1<Packet>(0.6931471805599453f);
  result = pselect(huge_mask, padd(result, ln2), result);
  // Return NaN for x < 1.
  const Packet invalid_mask = pcmp_lt(x, one);
  return por(invalid_mask, result);
}
 
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pacosh_double(const Packet& x) {
  const Packet one = pset1<Packet>(1.0);
  const Packet two = pset1<Packet>(2.0);
  const Packet t = psub(x, one);
  const Packet huge_mask = pcmp_lt(pset1<Packet>(1e150), x);
  const Packet t_tp2 = pmul(pselect(huge_mask, pzero(t), t), padd(t, two));
  Packet normal_arg = padd(t, psqrt(t_tp2));
  Packet huge_arg = t;
  Packet result = generic_log1p(pselect(huge_mask, huge_arg, normal_arg));
  const Packet ln2 = pset1<Packet>(0.6931471805599453);
  result = pselect(huge_mask, padd(result, ln2), result);
  const Packet invalid_mask = pcmp_lt(x, one);
  return por(invalid_mask, result);
}
 
}  // end namespace internal
}  // end namespace Eigen
 
#endif  // EIGEN_ARCH_GENERIC_PACKET_MATH_TRIG_H