Eigen/src/Core/MathFunctionsImpl.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2014 Pedro Gonnet (pedro.gonnet@gmail.com)
// Copyright (C) 2016 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// 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/.

#ifndef EIGEN_MATHFUNCTIONSIMPL_H
#define EIGEN_MATHFUNCTIONSIMPL_H

#include "./InternalHeaderCheck.h"

namespace Eigen {

namespace internal {

/** \internal Fast reciprocal using Newton-Raphson's method.

 Preconditions:
   1. The starting guess provided in approx_a_recip must have at least half
      the leading mantissa bits in the correct result, such that a single
      Newton-Raphson step is sufficient to get within 1-2 ulps of the currect
      result.
   2. If a is zero, approx_a_recip must be infinite with the same sign as a.
   3. If a is infinite, approx_a_recip must be zero with the same sign as a.

   If the preconditions are satisfied, which they are for for the _*_rcp_ps
   instructions on x86, the result has a maximum relative error of 2 ulps,
   and correctly handles reciprocals of zero, infinity, and NaN.
*/
template <typename Packet, int Steps>
struct generic_reciprocal_newton_step {
  static_assert(Steps > 0, "Steps must be at least 1.");
  EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE  Packet
  run(const Packet& a, const Packet& approx_a_recip) {
    using Scalar = typename unpacket_traits<Packet>::type;
    const Packet two = pset1<Packet>(Scalar(2));
    // Refine the approximation using one Newton-Raphson step:
    //   x_{i} = x_{i-1} * (2 - a * x_{i-1})
     const Packet x =
         generic_reciprocal_newton_step<Packet,Steps - 1>::run(a, approx_a_recip);
     const Packet tmp = pnmadd(a, x, two);
     // If tmp is NaN, it means that a is either +/-0 or +/-Inf.
     // In this case return the approximation directly.
     const Packet is_not_nan = pcmp_eq(tmp, tmp);
     return pselect(is_not_nan, pmul(x, tmp), x);
  }
};

template<typename Packet>
struct generic_reciprocal_newton_step<Packet, 0> {
   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Packet
   run(const Packet& /*unused*/, const Packet& approx_rsqrt) {
    return approx_rsqrt;
  }
};


/** \internal Fast reciprocal sqrt using Newton-Raphson's method.

 Preconditions:
   1. The starting guess provided in approx_a_recip must have at least half
      the leading mantissa bits in the correct result, such that a single
      Newton-Raphson step is sufficient to get within 1-2 ulps of the currect
      result.
   2. If a is zero, approx_a_recip must be infinite with the same sign as a.
   3. If a is infinite, approx_a_recip must be zero with the same sign as a.

   If the preconditions are satisfied, which they are for for the _*_rcp_ps
   instructions on x86, the result has a maximum relative error of 2 ulps,
   and correctly handles zero, infinity, and NaN. Positive denormals are
   treated as zero.
*/
template <typename Packet, int Steps>
struct generic_rsqrt_newton_step {
  static_assert(Steps > 0, "Steps must be at least 1.");

  EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE  Packet
  run(const Packet& a, const Packet& approx_rsqrt) {
    using Scalar = typename unpacket_traits<Packet>::type;
    const Packet one_point_five = pset1<Packet>(Scalar(1.5));
    const Packet minus_half = pset1<Packet>(Scalar(-0.5));
    
    // Refine the approximation using one Newton-Raphson step:
    //   x_{n+1} = x_n * (1.5 + (-0.5 * x_n) * (a * x_n)).
    // The approximation is expressed this way to avoid over/under-flows.  
    Packet x_newton  = pmul(approx_rsqrt, pmadd(pmul(minus_half, approx_rsqrt), pmul(a, approx_rsqrt), one_point_five));
    for (int step = 1; step < Steps; ++step) {
      x_newton  = pmul(x_newton, pmadd(pmul(minus_half, x_newton), pmul(a, x_newton), one_point_five));
    }
    
    // If approx_rsqrt is 0 or +/-inf, we should return it as is.  Note:
    // on intel, approx_rsqrt can be inf for small denormal values.
    const Packet return_approx = por(pcmp_eq(approx_rsqrt, pzero(a)),
                                     pcmp_eq(pabs(approx_rsqrt), pset1<Packet>(NumTraits<Scalar>::infinity())));
    return pselect(return_approx, approx_rsqrt, x_newton);
  }
};

template<typename Packet>
struct generic_rsqrt_newton_step<Packet, 0> {
   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Packet
   run(const Packet& /*unused*/, const Packet& approx_rsqrt) {
    return approx_rsqrt;
  }
};


/** \internal Fast sqrt using Newton-Raphson's method.

 Preconditions:
   1. The starting guess for the reciprocal sqrt provided in approx_rsqrt must
      have at least half the leading mantissa bits in the correct result, such
      that a single Newton-Raphson step is sufficient to get within 1-2 ulps of
      the currect result.
   2. If a is zero, approx_rsqrt must be infinite.
   3. If a is infinite, approx_rsqrt must be zero.

   If the preconditions are satisfied, which they are for for the _*_rsqrt_ps
   instructions on x86, the result has a maximum relative error of 2 ulps,
   and correctly handles zero and infinity, and NaN. Positive denormal inputs
   are treated as zero.
*/
template <typename Packet, int Steps=1>
struct generic_sqrt_newton_step {
  static_assert(Steps > 0, "Steps must be at least 1.");

  EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE  Packet
  run(const Packet& a, const Packet& approx_rsqrt) {
    using Scalar = typename unpacket_traits<Packet>::type;
    const Packet one_point_five = pset1<Packet>(Scalar(1.5));
    const Packet minus_half = pset1<Packet>(Scalar(-0.5));
    // If a is inf or zero, return a directly.
    const Packet inf_mask = pcmp_eq(a, pset1<Packet>(NumTraits<Scalar>::infinity()));
    const Packet return_a = por(pcmp_eq(a, pzero(a)), inf_mask);
    // Do a single step of Newton's iteration for reciprocal square root:
    //   x_{n+1} = x_n * (1.5 + (-0.5 * x_n) * (a * x_n))).
    // The Newton's step is computed this way to avoid over/under-flows.
    Packet rsqrt = pmul(approx_rsqrt, pmadd(pmul(minus_half, approx_rsqrt), pmul(a, approx_rsqrt), one_point_five));
    for (int step = 1; step < Steps; ++step) {
      rsqrt = pmul(rsqrt, pmadd(pmul(minus_half, rsqrt), pmul(a, rsqrt), one_point_five));
    }

    // Return sqrt(x) = x * rsqrt(x) for non-zero finite positive arguments.
    // Return a itself for 0 or +inf, NaN for negative arguments.
    return pselect(return_a, a, pmul(a, rsqrt));
  }
};

/** \internal \returns the hyperbolic tan of \a a (coeff-wise)
    Doesn't do anything fancy, just a 13/6-degree rational interpolant which
    is accurate up to a couple of ulps in the (approximate) range [-8, 8],
    outside of which tanh(x) = +/-1 in single precision. The input is clamped
    to the range [-c, c]. The value c is chosen as the smallest value where
    the approximation evaluates to exactly 1. In the reange [-0.0004, 0.0004]
    the approximation tanh(x) ~= x is used for better accuracy as x tends to zero.

    This implementation works on both scalars and packets.
*/
template<typename T>
T generic_fast_tanh_float(const T& a_x)
{
  // Clamp the inputs to the range [-c, c]
#ifdef EIGEN_VECTORIZE_FMA
  const T plus_clamp = pset1<T>(7.99881172180175781f);
  const T minus_clamp = pset1<T>(-7.99881172180175781f);
#else
  const T plus_clamp = pset1<T>(7.90531110763549805f);
  const T minus_clamp = pset1<T>(-7.90531110763549805f);
#endif
  const T tiny = pset1<T>(0.0004f);
  const T x = pmax(pmin(a_x, plus_clamp), minus_clamp);
  const T tiny_mask = pcmp_lt(pabs(a_x), tiny);
  // The monomial coefficients of the numerator polynomial (odd).
  const T alpha_1 = pset1<T>(4.89352455891786e-03f);
  const T alpha_3 = pset1<T>(6.37261928875436e-04f);
  const T alpha_5 = pset1<T>(1.48572235717979e-05f);
  const T alpha_7 = pset1<T>(5.12229709037114e-08f);
  const T alpha_9 = pset1<T>(-8.60467152213735e-11f);
  const T alpha_11 = pset1<T>(2.00018790482477e-13f);
  const T alpha_13 = pset1<T>(-2.76076847742355e-16f);

  // The monomial coefficients of the denominator polynomial (even).
  const T beta_0 = pset1<T>(4.89352518554385e-03f);
  const T beta_2 = pset1<T>(2.26843463243900e-03f);
  const T beta_4 = pset1<T>(1.18534705686654e-04f);
  const T beta_6 = pset1<T>(1.19825839466702e-06f);

  // Since the polynomials are odd/even, we need x^2.
  const T x2 = pmul(x, x);

  // Evaluate the numerator polynomial p.
  T p = pmadd(x2, alpha_13, alpha_11);
  p = pmadd(x2, p, alpha_9);
  p = pmadd(x2, p, alpha_7);
  p = pmadd(x2, p, alpha_5);
  p = pmadd(x2, p, alpha_3);
  p = pmadd(x2, p, alpha_1);
  p = pmul(x, p);

  // Evaluate the denominator polynomial q.
  T q = pmadd(x2, beta_6, beta_4);
  q = pmadd(x2, q, beta_2);
  q = pmadd(x2, q, beta_0);

  // Divide the numerator by the denominator.
  return pselect(tiny_mask, x, pdiv(p, q));
}

template<typename RealScalar>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
RealScalar positive_real_hypot(const RealScalar& x, const RealScalar& y)
{
  // IEEE IEC 6059 special cases.
  if ((numext::isinf)(x) || (numext::isinf)(y))
    return NumTraits<RealScalar>::infinity();
  if ((numext::isnan)(x) || (numext::isnan)(y))
    return NumTraits<RealScalar>::quiet_NaN();
    
  EIGEN_USING_STD(sqrt);
  RealScalar p, qp;
  p = numext::maxi(x,y);
  if(numext::is_exactly_zero(p)) return RealScalar(0);
  qp = numext::mini(y,x) / p;
  return p * sqrt(RealScalar(1) + qp*qp);
}

template<typename Scalar>
struct hypot_impl
{
  typedef typename NumTraits<Scalar>::Real RealScalar;
  static EIGEN_DEVICE_FUNC
  inline RealScalar run(const Scalar& x, const Scalar& y)
  {
    EIGEN_USING_STD(abs);
    return positive_real_hypot<RealScalar>(abs(x), abs(y));
  }
};

// Generic complex sqrt implementation that correctly handles corner cases
// according to https://en.cppreference.com/w/cpp/numeric/complex/sqrt
template<typename T>
EIGEN_DEVICE_FUNC std::complex<T> complex_sqrt(const std::complex<T>& z) {
  // Computes the principal sqrt of the input.
  //
  // For a complex square root of the number x + i*y. We want to find real
  // numbers u and v such that
  //    (u + i*v)^2 = x + i*y  <=>
  //    u^2 - v^2 + i*2*u*v = x + i*v.
  // By equating the real and imaginary parts we get:
  //    u^2 - v^2 = x
  //    2*u*v = y.
  //
  // For x >= 0, this has the numerically stable solution
  //    u = sqrt(0.5 * (x + sqrt(x^2 + y^2)))
  //    v = y / (2 * u)
  // and for x < 0,
  //    v = sign(y) * sqrt(0.5 * (-x + sqrt(x^2 + y^2)))
  //    u = y / (2 * v)
  //
  // Letting w = sqrt(0.5 * (|x| + |z|)),
  //   if x == 0: u = w, v = sign(y) * w
  //   if x > 0:  u = w, v = y / (2 * w)
  //   if x < 0:  u = |y| / (2 * w), v = sign(y) * w

  const T x = numext::real(z);
  const T y = numext::imag(z);
  const T zero = T(0);
  const T w = numext::sqrt(T(0.5) * (numext::abs(x) + numext::hypot(x, y)));

  return
    (numext::isinf)(y) ? std::complex<T>(NumTraits<T>::infinity(), y)
      : numext::is_exactly_zero(x) ? std::complex<T>(w, y < zero ? -w : w)
                                   : x > zero ? std::complex<T>(w, y / (2 * w))
      : std::complex<T>(numext::abs(y) / (2 * w), y < zero ? -w : w );
}

// Generic complex rsqrt implementation.
template<typename T>
EIGEN_DEVICE_FUNC std::complex<T> complex_rsqrt(const std::complex<T>& z) {
  // Computes the principal reciprocal sqrt of the input.
  //
  // For a complex reciprocal square root of the number z = x + i*y. We want to
  // find real numbers u and v such that
  //    (u + i*v)^2 = 1 / (x + i*y)  <=>
  //    u^2 - v^2 + i*2*u*v = x/|z|^2 - i*v/|z|^2.
  // By equating the real and imaginary parts we get:
  //    u^2 - v^2 = x/|z|^2
  //    2*u*v = y/|z|^2.
  //
  // For x >= 0, this has the numerically stable solution
  //    u = sqrt(0.5 * (x + |z|)) / |z|
  //    v = -y / (2 * u * |z|)
  // and for x < 0,
  //    v = -sign(y) * sqrt(0.5 * (-x + |z|)) / |z|
  //    u = -y / (2 * v * |z|)
  //
  // Letting w = sqrt(0.5 * (|x| + |z|)),
  //   if x == 0: u = w / |z|, v = -sign(y) * w / |z|
  //   if x > 0:  u = w / |z|, v = -y / (2 * w * |z|)
  //   if x < 0:  u = |y| / (2 * w * |z|), v = -sign(y) * w / |z|

  const T x = numext::real(z);
  const T y = numext::imag(z);
  const T zero = T(0);

  const T abs_z = numext::hypot(x, y);
  const T w = numext::sqrt(T(0.5) * (numext::abs(x) + abs_z));
  const T woz = w / abs_z;
  // Corner cases consistent with 1/sqrt(z) on gcc/clang.
  return
          numext::is_exactly_zero(abs_z) ? std::complex<T>(NumTraits<T>::infinity(), NumTraits<T>::quiet_NaN())
                                         : ((numext::isinf)(x) || (numext::isinf)(y)) ? std::complex<T>(zero, zero)
      : numext::is_exactly_zero(x) ? std::complex<T>(woz, y < zero ? woz : -woz)
                                   : x > zero ? std::complex<T>(woz, -y / (2 * w * abs_z))
      : std::complex<T>(numext::abs(y) / (2 * w * abs_z), y < zero ? woz : -woz );
}

template<typename T>
EIGEN_DEVICE_FUNC std::complex<T> complex_log(const std::complex<T>& z) {
  // Computes complex log.
  T a = numext::abs(z);
  EIGEN_USING_STD(atan2);
  T b = atan2(z.imag(), z.real());
  return std::complex<T>(numext::log(a), b);
}

} // end namespace internal

} // end namespace Eigen

#endif // EIGEN_MATHFUNCTIONSIMPL_H