text: /*-*- mode:c;indent-tabs-mode:nil;c-basic-offset:4;tab-width:8;coding:utf-8 -*-│
  │vi: set net ft=c ts=4 sts=4 sw=4 fenc=utf-8                                :vi│
  ╞══════════════════════════════════════════════════════════════════════════════╡
  │ Python 3                                                                     │
  │ https://docs.python.org/3/license.html                                       │
  ╚─────────────────────────────────────────────────────────────────────────────*/
  #include "libc/errno.h"
  #include "third_party/python/Include/pymath.h"
  #include "third_party/python/Modules/_math.h"
  #include "third_party/python/pyconfig.h"
  /* clang-format off */
  
  /* Definitions of some C99 math library functions, for those platforms
     that don't implement these functions already. */
  
  /* The following copyright notice applies to the original
     implementations of acosh, asinh and atanh. */
  
  /*
   * ====================================================
   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   *
   * Developed at SunPro, a Sun Microsystems, Inc. business.
   * Permission to use, copy, modify, and distribute this
   * software is freely granted, provided that this notice
   * is preserved.
   * ====================================================
   */
  
  static const double ln2 = 6.93147180559945286227E-01;
  static const double two_pow_m28 = 3.7252902984619141E-09; /* 2**-28 */
  static const double two_pow_p28 = 268435456.0; /* 2**28 */
  #ifndef Py_NAN
  static const double zero = 0.0;
  #endif
  
  /* acosh(x)
   * Method :
   *      Based on
   *            acosh(x) = log [ x + sqrt(x*x-1) ]
   *      we have
   *            acosh(x) := log(x)+ln2, if x is large; else
   *            acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
   *            acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
   *
   * Special cases:
   *      acosh(x) is NaN with signal if x<1.
   *      acosh(NaN) is NaN without signal.
   */
  
  double
  _Py_acosh(double x)
  {
      if (Py_IS_NAN(x)) {
          return x+x;
      }
      if (x < 1.) {                       /* x < 1;  return a signaling NaN */
          errno = EDOM;
  #ifdef Py_NAN
          return Py_NAN;
  #else
          return (x-x)/(x-x);
  #endif
      }
      else if (x >= two_pow_p28) {        /* x > 2**28 */
          if (Py_IS_INFINITY(x)) {
              return x+x;
          }
          else {
              return log(x)+ln2;          /* acosh(huge)=log(2x) */
          }
      }
      else if (x == 1.) {
          return 0.0;                     /* acosh(1) = 0 */
      }
      else if (x > 2.) {                  /* 2 < x < 2**28 */
          double t = x*x;
          return log(2.0*x - 1.0 / (x + sqrt(t - 1.0)));
      }
      else {                              /* 1 < x <= 2 */
          double t = x - 1.0;
          return m_log1p(t + sqrt(2.0*t + t*t));
      }
  }
  
  
  /* asinh(x)
   * Method :
   *      Based on
   *              asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
   *      we have
   *      asinh(x) := x  if  1+x*x=1,
   *               := sign(x)*(log(x)+ln2)) for large |x|, else
   *               := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
   *               := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
   */
  
  double
  _Py_asinh(double x)
  {
      double w;
      double absx = fabs(x);
  
      if (Py_IS_NAN(x) || Py_IS_INFINITY(x)) {
          return x+x;
      }
      if (absx < two_pow_m28) {           /* |x| < 2**-28 */
          return x;                       /* return x inexact except 0 */
      }
      if (absx > two_pow_p28) {           /* |x| > 2**28 */
          w = log(absx)+ln2;
      }
      else if (absx > 2.0) {              /* 2 < |x| < 2**28 */
          w = log(2.0*absx + 1.0 / (sqrt(x*x + 1.0) + absx));
      }
      else {                              /* 2**-28 <= |x| < 2= */
          double t = x*x;
          w = m_log1p(absx + t / (1.0 + sqrt(1.0 + t)));
      }
      return copysign(w, x);
  
  }
  
  /* atanh(x)
   * Method :
   *    1.Reduced x to positive by atanh(-x) = -atanh(x)
   *    2.For x>=0.5
   *                  1              2x                          x
   *      atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * -------)
   *                  2             1 - x                      1 - x
   *
   *      For x<0.5
   *      atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
   *
   * Special cases:
   *      atanh(x) is NaN if |x| >= 1 with signal;
   *      atanh(NaN) is that NaN with no signal;
   *
   */
  
  double
  _Py_atanh(double x)
  {
      double absx;
      double t;
  
      if (Py_IS_NAN(x)) {
          return x+x;
      }
      absx = fabs(x);
      if (absx >= 1.) {                   /* |x| >= 1 */
          errno = EDOM;
  #ifdef Py_NAN
          return Py_NAN;
  #else
          return x/zero;
  #endif
      }
      if (absx < two_pow_m28) {           /* |x| < 2**-28 */
          return x;
      }
      if (absx < 0.5) {                   /* |x| < 0.5 */
          t = absx+absx;
          t = 0.5 * m_log1p(t + t*absx / (1.0 - absx));
      }
      else {                              /* 0.5 <= |x| <= 1.0 */
          t = 0.5 * m_log1p((absx + absx) / (1.0 - absx));
      }
      return copysign(t, x);
  }
  
  /* Mathematically, expm1(x) = exp(x) - 1.  The expm1 function is designed
     to avoid the significant loss of precision that arises from direct
     evaluation of the expression exp(x) - 1, for x near 0. */
  
  double
  _Py_expm1(double x)
  {
      /* For abs(x) >= log(2), it's safe to evaluate exp(x) - 1 directly; this
         also works fine for infinities and nans.
  
         For smaller x, we can use a method due to Kahan that achieves close to
         full accuracy.
      */
  
      if (fabs(x) < 0.7) {
          double u;
          u = exp(x);
          if (u == 1.0)
              return x;
          else
              return (u - 1.0) * x / log(u);
      }
      else
          return exp(x) - 1.0;
  }
  
  /* log1p(x) = log(1+x).  The log1p function is designed to avoid the
     significant loss of precision that arises from direct evaluation when x is
     small. */
  
  #ifdef HAVE_LOG1P
  
  double
  _Py_log1p(double x)
  {
      /* Some platforms supply a log1p function but don't respect the sign of
         zero:  log1p(-0.0) gives 0.0 instead of the correct result of -0.0.
  
         To save fiddling with configure tests and platform checks, we handle the
         special case of zero input directly on all platforms.
      */
      if (x == 0.0) {
          return x;
      }
      else {
          return log1p(x);
      }
  }
  
  #else
  
  double
  _Py_log1p(double x)
  {
      /* For x small, we use the following approach.  Let y be the nearest float
         to 1+x, then
  
           1+x = y * (1 - (y-1-x)/y)
  
         so log(1+x) = log(y) + log(1-(y-1-x)/y).  Since (y-1-x)/y is tiny, the
         second term is well approximated by (y-1-x)/y.  If abs(x) >=
         DBL_EPSILON/2 or the rounding-mode is some form of round-to-nearest
         then y-1-x will be exactly representable, and is computed exactly by
         (y-1)-x.
  
         If abs(x) < DBL_EPSILON/2 and the rounding mode is not known to be
         round-to-nearest then this method is slightly dangerous: 1+x could be
         rounded up to 1+DBL_EPSILON instead of down to 1, and in that case
         y-1-x will not be exactly representable any more and the result can be
         off by many ulps.  But this is easily fixed: for a floating-point
         number |x| < DBL_EPSILON/2., the closest floating-point number to
         log(1+x) is exactly x.
      */
  
      double y;
      if (fabs(x) < DBL_EPSILON/2.) {
          return x;
      }
      else if (-0.5 <= x && x <= 1.) {
          /* WARNING: it's possible that an overeager compiler
             will incorrectly optimize the following two lines
             to the equivalent of "return log(1.+x)". If this
             happens, then results from log1p will be inaccurate
             for small x. */
          y = 1.+x;
          return log(y)-((y-1.)-x)/y;
      }
      else {
          /* NaNs and infinities should end up here */
          return log(1.+x);
      }
  }
  
  #endif /* ifdef HAVE_LOG1P */
  