mingw-w64-crt/complex/casinh.def.h

/*
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 Copyright (c) 2009, 2010 by the mingw-w64 project

 See the AUTHORS file for the list of contributors to the mingw-w64 project.

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 as GPL compatible by the Free Software Foundation (FSF).

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      in the documentation and/or other materials provided with the
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      products derived from this software without prior written permission
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      with the copyright holders.
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 OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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*/

__FLT_TYPE __complex__ __cdecl
__FLT_ABI(casinh) (__FLT_TYPE __complex__ z)
{
  __complex__ __FLT_TYPE ret;
  __complex__ __FLT_TYPE x;
  __FLT_TYPE arz, aiz;
  int r_class = fpclassify (__real__ z);
  int i_class = fpclassify (__imag__ z);

  if (i_class == FP_INFINITE)
  {
    __real__ ret = __FLT_ABI(copysign) (__FLT_HUGE_VAL, __real__ z);
    __imag__ ret = (r_class == FP_NAN
      ? __FLT_NAN
      : (__FLT_ABI(copysign) ((r_class != FP_NAN && r_class != FP_INFINITE) ? __FLT_PI_2 : __FLT_PI_4, __imag__ z)));
    return ret;
  }

  if (r_class == FP_INFINITE)
  {
    __real__ ret = __real__ z;
    __imag__ ret = (i_class != FP_NAN
      ? __FLT_ABI(copysign) (__FLT_CST(0.0), __imag__ z)
      : __FLT_NAN);
    return ret;
  }

  if (r_class == FP_NAN)
  {
    __real__ ret = __real__ z;
    __imag__ ret = (i_class == FP_ZERO
      ? __FLT_ABI(copysign) (__FLT_CST(0.0), __imag__ z)
      : __FLT_NAN);
    return ret;
  }

  if (i_class == FP_NAN)
  {
    __real__ ret = __FLT_NAN;
    __imag__ ret = __FLT_NAN;
    return ret;
  }

  if (r_class == FP_ZERO && i_class == FP_ZERO)
    return z;

  /* casinh(z) = log(z + sqrt(z*z + 1)) */

  /* Use symmetries to perform the calculation in the first quadrant. */
  arz = __FLT_ABI(fabs) (__real__ z);
  aiz = __FLT_ABI(fabs) (__imag__ z);

  if (arz >= __FLT_CST(1.0)/__FLT_EPSILON
      || aiz >= __FLT_CST(1.0)/__FLT_EPSILON)
  {
    /* For large z, z + sqrt(z*z + 1) is approximately 2*z.
    Use that approximation to avoid overflow when squaring. */
    __real__ x = arz;
    __imag__ x = aiz;
    ret = __FLT_ABI(clog) (x);
    __real__ ret += M_LN2;
  }
  else if (aiz < __FLT_CST(1.0) && arz <= __FLT_EPSILON)
  {
    /* Taylor series expansion around arz=0 for z + sqrt(z*z + 1):
    c = arz + sqrt(1-aiz^2) + i*(aiz + arz*aiz / sqrt(1-aiz^2)) + O(arz^2)
    Identity: clog(c) = log(|c|) + i*arg(c)
    For real part of result:
    |c| = 1 + arz / sqrt(1-aiz^2) + O(arz^2)  (Taylor series expansion)
    For imaginary part of result:
    c = (arz + sqrt(1-aiz^2))/sqrt(1-aiz^2) * (sqrt(1-aiz^2) + i*aiz) + O(arz^6)
    */
    __FLT_TYPE s1maiz2 = __FLT_ABI(sqrt) ((__FLT_CST(1.0)+aiz)*(__FLT_CST(1.0)-aiz));
    __real__ ret = __FLT_ABI(log1p) (arz / s1maiz2);
    __imag__ ret = __FLT_ABI(atan2) (aiz, s1maiz2);
  }
  else if (aiz < __FLT_CST(1.0) && arz*arz <= __FLT_EPSILON)
  {
    /* Taylor series expansion around arz=0 for z + sqrt(z*z + 1):
    c = arz + sqrt(1-aiz^2) + arz^2 / (2*(1-aiz^2)^(3/2)) + i*(aiz + arz*aiz / sqrt(1-aiz^2)) + O(arz^4)
    Identity: clog(c) = log(|c|) + i*arg(c)
    For real part of result:
    |c| = 1 + arz / sqrt(1-aiz^2) + arz^2/(2*(1-aiz^2)) + O(arz^3)  (Taylor series expansion)
    For imaginary part of result:
    c = 1/sqrt(1-aiz^2) * ((1-aiz^2) + arz*sqrt(1-aiz^2) + arz^2/(2*(1-aiz^2)) + i*aiz*(sqrt(1-aiz^2)+arz)) + O(arz^3)
    */
    __FLT_TYPE onemaiz2 = (__FLT_CST(1.0)+aiz)*(__FLT_CST(1.0)-aiz);
    __FLT_TYPE s1maiz2 = __FLT_ABI(sqrt) (onemaiz2);
    __FLT_TYPE arz2red = arz * arz / __FLT_CST(2.0) / s1maiz2;
    __real__ ret = __FLT_ABI(log1p) ((arz + arz2red) / s1maiz2);
    __imag__ ret = __FLT_ABI(atan2) (aiz * (s1maiz2 + arz),
                                     onemaiz2 + arz*s1maiz2 + arz2red);
  }
  else
  {
    __real__ x = (arz - aiz) * (arz + aiz) + __FLT_CST(1.0);
    __imag__ x = __FLT_CST(2.0) * arz * aiz;

    x = __FLT_ABI(csqrt) (x);

    __real__ x += arz;
    __imag__ x += aiz;

    ret = __FLT_ABI(clog) (x);
  }

  /* adjust signs for input quadrant */
  __real__ ret = __FLT_ABI(copysign) (__real__ ret, __real__ z);
  __imag__ ret = __FLT_ABI(copysign) (__imag__ ret, __imag__ z);

  return ret;
}