ctaylor_math.hpp

/*
Copyright (c) 2009-2017 Ulf Ekstrom <uekstrom@gmail.com>

Permission is hereby granted, free of charge, to any person
obtaining a copy of this software and associated documentation
files (the "Software"), to deal in the Software without
restriction, including without limitation the rights to use,
copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the
Software is furnished to do so, subject to the following
conditions:

The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
OTHER DEALINGS IN THE SOFTWARE.
*/

#pragma once

// For inclusion in ctaylor.h only!
#include "tmath.hpp"

template <typename T, int Nvar, typename S>
static ctaylor<T, Nvar> operator/(const S & x, const ctaylor<T, Nvar> & t) {
  ctaylor<T, Nvar> res;
#ifdef CTAYLOR_SPARSE
  res.isscalar = t.isscalar;
  if (res.isscalar) {
    res.c[0] = x / t.c[0];
  } else {
    T tmp[Nvar + 1];
    inv_expand<T, Nvar>(tmp, t.c[0]);
    for (int i = 0; i <= Nvar; i++)
      tmp[i] *= x;
    ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  }
#else
  T tmp[Nvar + 1];
  inv_expand<T, Nvar>(tmp, t.c[0]);
  for (int i = 0; i <= Nvar; i++)
    tmp[i] *= x;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
#endif
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> operator/(const ctaylor<T, Nvar> & t1,
                                  const ctaylor<T, Nvar> & t2) {
  ctaylor<T, Nvar> res;
#ifdef CTAYLOR_SPARSE
  if (t1.isscalar)
    return t1.c[0] / t2;
  else if (t2.isscalar)
    return t1 / t2.c[0];
  res.isscalar = 0;
#endif
  T tmp[Nvar + 1];
  inv_expand<T, Nvar>(tmp, t2.c[0]);
  ctaylor_rec<T, Nvar>::compose(res.c, t2.c, tmp);
  res *= t1;
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> operator/(const ctaylor<T, Nvar> & t, const T & x) {
  ctaylor<T, Nvar> tmp = t;
  tmp *= 1 / x;
  return tmp;
}

template <typename T, int Nvar, typename S>
static ctaylor<T, Nvar> operator/(const ctaylor<T, Nvar> & t, const S & x) {
  ctaylor<T, Nvar> tmp = t;
  tmp *= 1 / T(x);
  return tmp;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> abs(const ctaylor<T, Nvar> & t) {
  if (t.c[0] < 0)
    return -t;
  else
    return t;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> exp(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(exp(t.c[0]));
#endif
  T tmp[Nvar + 1];
  exp_expand<T, Nvar>(tmp, t.c[0]);
  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

// exp(x)-1, but accurate for small x
template <typename T, int Nvar>
static ctaylor<T, Nvar> expm1(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(2 * exp(t.c[0] / 2) * sinh(t.c[0] / 2));
#endif
  T tmp[Nvar + 1];
  exp_expand<T, Nvar>(tmp, t.c[0]);

  // Only constant value is affected by the cancellation
  if (fabs(t.c[0]) > 1e-3)
    tmp[0] -= 1;
  else
    tmp[0] = 2 * exp(t.c[0] / 2) * sinh(t.c[0] / 2);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> log(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(log(t.c[0]));
#endif
  T tmp[Nvar + 1];
  log_expand<T, Nvar>(tmp, t.c[0]);
  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

// We need this version with double a argument to prevent truncation
// to int.
template <typename T, int Nvar>
static ctaylor<T, Nvar> pow(const ctaylor<T, Nvar> & t, const double & a) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(pow(t.c[0], a));
#endif
  T tmp[Nvar + 1];
  pow_expand<T, Nvar>(tmp, t.c[0], a);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> sqrt(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(sqrt(t.c[0]));
#endif
  T tmp[Nvar + 1];
  sqrt_expand<T, Nvar>(tmp, t.c[0]);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> cbrt(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(cbrt(t.c[0]));
#endif
  T tmp[Nvar + 1];
  cbrt_expand<T, Nvar>(tmp, t.c[0]);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

// Integer exponent version is analytical at t[0] = 0
// This function gets priority over the normal pow
// when the exponent is an integer, but does not force
// conversion to integer.
template <typename T, int Nvar>
static ctaylor<T, Nvar> pow(const ctaylor<T, Nvar> & t, int n) {
  if (n > 0) {
    ctaylor<T, Nvar> res = t;
    while (n-- > 1)
      res *= t;
    return res;
  } else if (n < 0) {
    return pow(t, double(n));
  } else {
    ctaylor<T, Nvar> res(1);
    return res;
  }
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> atan(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(atan(t.c[0]));
#endif
  T tmp[Nvar + 1];
  atan_expand<T, Nvar>(tmp, t.c[0]);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> erf(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(erf(t.c[0]));
#endif
  T tmp[Nvar + 1];
  erf_expand<T, Nvar>(tmp, t.c[0]);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> sin(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(sin(t.c[0]));
#endif
  T tmp[Nvar + 1];
  sin_expand<T, Nvar>(tmp, t.c[0]);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> cos(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(cos(t.c[0]));
#endif
  T tmp[Nvar + 1];
  cos_expand<T, Nvar>(tmp, t.c[0]);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> asin(const ctaylor<T, Nvar> & t) {
  T tmp[Nvar + 1];
  asin_expand<T, Nvar>(tmp, t.c[0]);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> acos(const ctaylor<T, Nvar> & t) {
  T tmp[Nvar + 1];
  acos_expand<T, Nvar>(tmp, t.c[0]);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> asinh(const ctaylor<T, Nvar> & t) {
#ifdef CTAYLOR_SPARSE
  if (t.isscalar)
    return ctaylor<T, Nvar>(asinh(t.c[0]));
#endif
  T tmp[Nvar + 1];
  asinh_expand<T, Nvar>(tmp, t.c[0]);

  ctaylor<T, Nvar> res;
  ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
  return res;
}

/*
  The original function is unstable for small t[0] values similarly to
  the Boys function. Use an [8,8] Pade approximation when |t[0]| is
  small. This works less well but still ok in single precision.
 */
template <typename T, int Nvar>
static ctaylor<T, Nvar> sqrtx_asinh_sqrtx(const ctaylor<T, Nvar> & t) {
  assert(t.c[0] > -0.5);
#ifdef CTAYLOR_SPARSE
  if (t.isscalar) {
    T sqrtx = sqrt(t.c[0]);
    return ctaylor<T, Nvar>(sqrtx * asinh(sqrtx));
  }
#endif
    // Coefficients of an [8,8] Pade approximation at x = 0
#define ASINH_TABSIZE 9
  static const T P[ASINH_TABSIZE] = {0,
                                     3.510921856028398e3,
                                     1.23624388373212e4,
                                     1.734847003883674e4,
                                     1.235072285222234e4,
                                     4.691117148130619e3,
                                     9.119186273274577e2,
                                     7.815848629220836e1,
                                     1.96088643023654e0};
  static const T Q[ASINH_TABSIZE] = {3.510921856028398e3,
                                     1.29475924799926e4,
                                     1.924308297963337e4,
                                     1.474357149568687e4,
                                     6.176496729255528e3,
                                     1.379806958043824e3,
                                     1.471833349002349e2,
                                     5.666278232986776e0,
                                     2.865104054302032e-2};
  if (fabs(t.c[0]) < 0.5) {
    // Shift polys, divide and compose
    assert(Nvar < ASINH_TABSIZE);
    T tmp[Nvar + 1], pq[9];
    for (int i = 0; i < ASINH_TABSIZE; i++)
      pq[i] = Q[i];
    tfuns<T, ASINH_TABSIZE - 1>::shift(pq, t.c[0]);
    inv_expand<T, Nvar>(tmp, pq[0]);
    tfuns<T, Nvar>::compose(tmp, pq);
    for (int i = 0; i < ASINH_TABSIZE; i++)
      pq[i] = P[i];
    tfuns<T, ASINH_TABSIZE - 1>::shift(pq, t.c[0]);
    tfuns<T, Nvar>::multo(tmp, pq);
    ctaylor<T, Nvar> res;
    ctaylor_rec<T, Nvar>::compose(res.c, t.c, tmp);
    return res;
  } else {
    // This is the unstable form
    ctaylor<T, Nvar> s = sqrt(t);
    return s * asinh(s);
  }
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> min(const ctaylor<T, Nvar> & a, const ctaylor<T, Nvar> & b) {
  if (a <= b)
    return a;
  else
    return b;
}

template <typename T, int Nvar>
static ctaylor<T, Nvar> max(const ctaylor<T, Nvar> & a, const ctaylor<T, Nvar> & b) {
  if (a > b)
    return a;
  else
    return b;
}