dtanh.c

/* ============================================================
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============================================================ */
#include "libm_amd.h"
#include "libm_util_amd.h"

#define USE_SPLITEXP
#define USE_SCALEDOUBLE_2
#define USE_VAL_WITH_FLAGS
#include "libm_inlines_amd.h"
#undef USE_SPLITEXP
#undef USE_SCALEDOUBLE_2
#undef USE_VAL_WITH_FLAGS

#include "libm_errno_amd.h"

double
FN_PROTOTYPE(mth_i_dtanh)(double x)
{
  /*
    The definition of tanh(x) is sinh(x)/cosh(x), which is also equivalent
    to the following three formulae:
    1.  (exp(x) - exp(-x))/(exp(x) + exp(-x))
    2.  (1 - (2/(exp(2*x) + 1 )))
    3.  (exp(2*x) - 1)/(exp(2*x) + 1)
    but computationally, some formulae are better on some ranges.
  */
  static const double thirtytwo_by_log2 =
                          4.61662413084468283841e+01, /* 0x40471547652b82fe */
      log2_by_32_lead = 2.16608493356034159660e-02,   /* 0x3f962e42fe000000 */
      log2_by_32_tail = 5.68948749532545630390e-11,   /* 0x3dcf473de6af278e */
      large_threshold = 20.0;                         /* 0x4034000000000000 */

  __UINT8_T ux, aux, xneg;
  double y, z, p, z1, z2;
  int m;

  /* Special cases */

  GET_BITS_DP64(x, ux);
  aux = ux & ~SIGNBIT_DP64;
  if (aux < 0x3e30000000000000) /* |x| small enough that tanh(x) = x */
  {
    if (aux == 0)
      return x; /* with no inexact */
    else
      return val_with_flags(x, AMD_F_INEXACT);
  } else if (aux > 0x7ff0000000000000) /* |x| is NaN */
    return x + x;

  xneg = (aux != ux);

  y = x;
  if (xneg)
    y = -x;

  if (y > large_threshold) {
    /* If x is large then exp(-x) is negligible and
       formula 1 reduces to plus or minus 1.0 */
    z = 1.0;
  } else if (y <= 1.0) {
    double y2;
    y2 = y * y;
    if (y < 0.9) {
      /* Use a [3,3] Remez approximation on [0,0.9]. */
      z = y +
          y * y2 * (-0.274030424656179760118928e0 +
                    (-0.176016349003044679402273e-1 +
                     (-0.200047621071909498730453e-3 -
                      0.142077926378834722618091e-7 * y2) *
                         y2) *
                        y2) /
              (0.822091273968539282568011e0 +
               (0.381641414288328849317962e0 +
                (0.201562166026937652780575e-1 +
                 0.2091140262529164482568557e-3 * y2) *
                    y2) *
                   y2);
    } else {
      /* Use a [3,3] Remez approximation on [0.9,1]. */
      z = y +
          y * y2 * (-0.227793870659088295252442e0 +
                    (-0.146173047288731678404066e-1 +
                     (-0.165597043903549960486816e-3 -
                      0.115475878996143396378318e-7 * y2) *
                         y2) *
                        y2) /
              (0.683381611977295894959554e0 +
               (0.317204558977294374244770e0 +
                (0.167358775461896562588695e-1 +
                 0.173076050126225961768710e-3 * y2) *
                    y2) *
                   y2);
    }
  } else {
    /* Compute p = exp(2*y) + 1. The code is basically inlined
       from exp_amd. */

    splitexp(2 * y, 1.0, thirtytwo_by_log2, log2_by_32_lead, log2_by_32_tail,
             &m, &z1, &z2);
    p = scaleDouble_2(z1 + z2, m) + 1.0;

    /* Now reconstruct tanh from p. */
    z = (1.0 - 2.0 / p);
  }

  if (xneg)
    z = -z;
  return z;
}