/* Deal with errno for out-of-range result */
static inline double retval_errno_erange(double x, int xneg)
{
struct exception exc;
exc.arg1 = x;
exc.arg2 = x;
exc.type = OVERFLOW;
exc.name = (char *)"sinh";
if (_LIB_VERSION == _SVID_)
{
if (xneg)
exc.retval = -HUGE;
else
exc.retval = HUGE;
}
else
{
if (xneg)
exc.retval = -infinity_with_flags(AMD_F_OVERFLOW);
else
exc.retval = infinity_with_flags(AMD_F_OVERFLOW);
}
if (_LIB_VERSION == _POSIX_)
__set_errno(ERANGE);
else if (!matherr(&exc))
__set_errno(ERANGE);
return exc.retval;
}
/* Deal with errno for out-of-range result */
static inline double retval_errno_erange_overflow(double x)
{
struct exception exc;
exc.arg1 = x;
exc.arg2 = x;
exc.type = OVERFLOW;
exc.name = (char *)"exp2";
if (PATH64_LIB_VERSION_SVID)
exc.retval = HUGE;
else
exc.retval = infinity_with_flags(AMD_F_OVERFLOW | AMD_F_INEXACT);
if (PATH64_LIB_VERSION_POSIX)
__set_errno(ERANGE);
else if (!matherr(&exc))
__set_errno(ERANGE);
return exc.retval;
}
double FN_PROTOTYPE(atanh)(double x)
{
unsigned long long ux, ax;
double r, absx, t, poly;
GET_BITS_DP64(x, ux);
ax = ux & ~SIGNBIT_DP64;
PUT_BITS_DP64(ax, absx);
if ((ux & EXPBITS_DP64) == EXPBITS_DP64)
{
/* x is either NaN or infinity */
if (ux & MANTBITS_DP64)
{
/* x is NaN */
#ifdef WINDOWS
return handle_error(_FUNCNAME, ux|0x0008000000000000, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0);
#else
return x + x; /* Raise invalid if it is a signalling NaN */
#endif
}
else
{
/* x is infinity; return a NaN */
#ifdef WINDOWS
return handle_error(_FUNCNAME, INDEFBITPATT_DP64, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0);
#else
return retval_errno_edom(x,nan_with_flags(AMD_F_INVALID));
#endif
}
}
else if (ax >= 0x3ff0000000000000)
{
if (ax > 0x3ff0000000000000)
{
/* abs(x) > 1.0; return NaN */
#ifdef WINDOWS
return handle_error(_FUNCNAME, INDEFBITPATT_DP64, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0);
#else
return retval_errno_edom(x,nan_with_flags(AMD_F_INVALID));
#endif
}
else if (ux == 0x3ff0000000000000)
{
/* x = +1.0; return infinity with the same sign as x
and set the divbyzero status flag */
#ifdef WINDOWS
return handle_error(_FUNCNAME, PINFBITPATT_DP64, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0);
#else
return retval_errno_edom(x,infinity_with_flags(AMD_F_DIVBYZERO));
#endif
}
else
{
/* x = -1.0; return infinity with the same sign as x */
#ifdef WINDOWS
return handle_error(_FUNCNAME, NINFBITPATT_DP64, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0);
#else
return retval_errno_edom(x,-infinity_with_flags(AMD_F_DIVBYZERO));
#endif
}
}
if (ax < 0x3e30000000000000)
{
if (ax == 0x0000000000000000)
{
/* x is +/-zero. Return the same zero. */
return x;
}
else
{
/* Arguments smaller than 2^(-28) in magnitude are
approximated by atanh(x) = x, raising inexact flag. */
return val_with_flags(x, AMD_F_INEXACT);
}
}
else
{
if (ax < 0x3fe0000000000000)
{
/* Arguments up to 0.5 in magnitude are
approximated by a [5,5] minimax polynomial */
t = x*x;
poly =
(0.47482573589747356373e0 +
(-0.11028356797846341457e1 +
(0.88468142536501647470e0 +
(-0.28180210961780814148e0 +
(0.28728638600548514553e-1 -
0.10468158892753136958e-3 * t) * t) * t) * t) * t) /
(0.14244772076924206909e1 +
(-0.41631933639693546274e1 +
(0.45414700626084508355e1 +
(-0.22608883748988489342e1 +
(0.49561196555503101989e0 -
0.35861554370169537512e-1 * t) * t) * t) * t) * t);
return x + x*t*poly;
}
else
{
/* abs(x) >= 0.5 */
/* Note that
atanh(x) = 0.5 * ln((1+x)/(1-x))
(see Abramowitz and Stegun 4.6.22).
For greater accuracy we use the variant formula
atanh(x) = log(1 + 2x/(1-x)) = log1p(2x/(1-x)).
*/
r = (2.0 * absx) / (1.0 - absx);
r = 0.5 * FN_PROTOTYPE(log1p)(r);
if (ux & SIGNBIT_DP64)
/* Argument x is negative */
return -r;
else
return r;
}
}
}
double FN_PROTOTYPE(cosh)(double x)
{
/*
Derived from sinh subroutine
After dealing with special cases the computation is split into
regions as follows:
abs(x) >= max_cosh_arg:
cosh(x) = sign(x)*Inf
abs(x) >= small_threshold:
cosh(x) = sign(x)*exp(abs(x))/2 computed using the
splitexp and scaleDouble functions as for exp_amd().
abs(x) < small_threshold:
compute p = exp(y) - 1 and then z = 0.5*(p+(p/(p+1.0)))
cosh(x) is then sign(x)*z. */
static const double
max_cosh_arg = 7.10475860073943977113e+02, /* 0x408633ce8fb9f87e */
thirtytwo_by_log2 = 4.61662413084468283841e+01, /* 0x40471547652b82fe */
log2_by_32_lead = 2.16608493356034159660e-02, /* 0x3f962e42fe000000 */
log2_by_32_tail = 5.68948749532545630390e-11, /* 0x3dcf473de6af278e */
// small_threshold = 8*BASEDIGITS_DP64*0.30102999566398119521373889;
small_threshold = 20.0;
/* (8*BASEDIGITS_DP64*log10of2) ' exp(-x) insignificant c.f. exp(x) */
/* Lead and tail tabulated values of sinh(i) and cosh(i)
for i = 0,...,36. The lead part has 26 leading bits. */
static const double sinh_lead[ 37] = {
0.00000000000000000000e+00, /* 0x0000000000000000 */
1.17520117759704589844e+00, /* 0x3ff2cd9fc0000000 */
3.62686038017272949219e+00, /* 0x400d03cf60000000 */
1.00178747177124023438e+01, /* 0x40240926e0000000 */
2.72899169921875000000e+01, /* 0x403b4a3800000000 */
7.42032089233398437500e+01, /* 0x40528d0160000000 */
2.01713153839111328125e+02, /* 0x406936d228000000 */
5.48316116333007812500e+02, /* 0x4081228768000000 */
1.49047882080078125000e+03, /* 0x409749ea50000000 */
4.05154187011718750000e+03, /* 0x40afa71570000000 */
1.10132326660156250000e+04, /* 0x40c5829dc8000000 */
2.99370708007812500000e+04, /* 0x40dd3c4488000000 */
8.13773945312500000000e+04, /* 0x40f3de1650000000 */
2.21206695312500000000e+05, /* 0x410b00b590000000 */
6.01302140625000000000e+05, /* 0x412259ac48000000 */
1.63450865625000000000e+06, /* 0x4138f0cca8000000 */
4.44305525000000000000e+06, /* 0x4150f2ebd0000000 */
1.20774762500000000000e+07, /* 0x4167093488000000 */
3.28299845000000000000e+07, /* 0x417f4f2208000000 */
8.92411500000000000000e+07, /* 0x419546d8f8000000 */
2.42582596000000000000e+08, /* 0x41aceb0888000000 */
6.59407856000000000000e+08, /* 0x41c3a6e1f8000000 */
1.79245641600000000000e+09, /* 0x41dab5adb8000000 */
4.87240166400000000000e+09, /* 0x41f226af30000000 */
1.32445608960000000000e+10, /* 0x4208ab7fb0000000 */
3.60024494080000000000e+10, /* 0x4220c3d390000000 */
9.78648043520000000000e+10, /* 0x4236c93268000000 */
2.66024116224000000000e+11, /* 0x424ef822f0000000 */
7.23128516608000000000e+11, /* 0x42650bba30000000 */
1.96566712320000000000e+12, /* 0x427c9aae40000000 */
5.34323724288000000000e+12, /* 0x4293704708000000 */
1.45244246507520000000e+13, /* 0x42aa6b7658000000 */
3.94814795284480000000e+13, /* 0x42c1f43fc8000000 */
1.07321789251584000000e+14, /* 0x42d866f348000000 */
2.91730863685632000000e+14, /* 0x42f0953e28000000 */
7.93006722514944000000e+14, /* 0x430689e220000000 */
2.15561576592179200000e+15}; /* 0x431ea215a0000000 */
static const double sinh_tail[ 37] = {
0.00000000000000000000e+00, /* 0x0000000000000000 */
1.60467555584448807892e-08, /* 0x3e513ae6096a0092 */
2.76742892754807136947e-08, /* 0x3e5db70cfb79a640 */
2.09697499555224576530e-07, /* 0x3e8c2526b66dc067 */
2.04940252448908240062e-07, /* 0x3e8b81b18647f380 */
1.65444891522700935932e-06, /* 0x3ebbc1cdd1e1eb08 */
3.53116789999998198721e-06, /* 0x3ecd9f201534fb09 */
6.94023870987375490695e-06, /* 0x3edd1c064a4e9954 */
4.98876893611587449271e-06, /* 0x3ed4eca65d06ea74 */
3.19656024605152215752e-05, /* 0x3f00c259bcc0ecc5 */
2.08687768377236501204e-04, /* 0x3f2b5a6647cf9016 */
4.84668088325403796299e-05, /* 0x3f09691adefb0870 */
1.17517985422733832468e-03, /* 0x3f53410fc29cde38 */
6.90830086959560562415e-04, /* 0x3f46a31a50b6fb3c */
1.45697262451506548420e-03, /* 0x3f57defc71805c40 */
2.99859023684906737806e-02, /* 0x3f9eb49fd80e0bab */
1.02538800507941396667e-02, /* 0x3f84fffc7bcd5920 */
1.26787628407699110022e-01, /* 0x3fc03a93b6c63435 */
6.86652479544033744752e-02, /* 0x3fb1940bb255fd1c */
4.81593627621056619148e-01, /* 0x3fded26e14260b50 */
1.70489513795397629181e+00, /* 0x3ffb47401fc9f2a2 */
1.12416073482258713767e+01, /* 0x40267bb3f55634f1 */
7.06579578070110514432e+00, /* 0x401c435ff8194ddc */
5.91244512999659974639e+01, /* 0x404d8fee052ba63a */
1.68921736147050694399e+02, /* 0x40651d7edccde3f6 */
2.60692936262073658327e+02, /* 0x40704b1644557d1a */
3.62419382134885609048e+02, /* 0x4076a6b5ca0a9dc4 */
4.07689930834187271103e+03, /* 0x40afd9cc72249aba */
1.55377375868385224749e+04, /* 0x40ce58de693edab5 */
2.53720210371943067003e+04, /* 0x40d8c70158ac6363 */
4.78822310734952334315e+04, /* 0x40e7614764f43e20 */
1.81871712615542812273e+05, /* 0x4106337db36fc718 */
5.62892347580489004031e+05, /* 0x41212d98b1f611e2 */
6.41374032312148716301e+05, /* 0x412392bc108b37cc */
7.57809544070145115256e+06, /* 0x415ce87bdc3473dc */
3.64177136406482197344e+06, /* 0x414bc8d5ae99ad14 */
7.63580561355670914054e+06}; /* 0x415d20d76744835c */
static const double cosh_lead[ 37] = {
1.00000000000000000000e+00, /* 0x3ff0000000000000 */
1.54308062791824340820e+00, /* 0x3ff8b07550000000 */
3.76219564676284790039e+00, /* 0x400e18fa08000000 */
1.00676617622375488281e+01, /* 0x402422a490000000 */
2.73082327842712402344e+01, /* 0x403b4ee858000000 */
7.42099475860595703125e+01, /* 0x40528d6fc8000000 */
2.01715633392333984375e+02, /* 0x406936e678000000 */
5.48317031860351562500e+02, /* 0x4081228948000000 */
1.49047915649414062500e+03, /* 0x409749eaa8000000 */
4.05154199218750000000e+03, /* 0x40afa71580000000 */
1.10132329101562500000e+04, /* 0x40c5829dd0000000 */
2.99370708007812500000e+04, /* 0x40dd3c4488000000 */
8.13773945312500000000e+04, /* 0x40f3de1650000000 */
2.21206695312500000000e+05, /* 0x410b00b590000000 */
6.01302140625000000000e+05, /* 0x412259ac48000000 */
1.63450865625000000000e+06, /* 0x4138f0cca8000000 */
4.44305525000000000000e+06, /* 0x4150f2ebd0000000 */
1.20774762500000000000e+07, /* 0x4167093488000000 */
3.28299845000000000000e+07, /* 0x417f4f2208000000 */
8.92411500000000000000e+07, /* 0x419546d8f8000000 */
2.42582596000000000000e+08, /* 0x41aceb0888000000 */
6.59407856000000000000e+08, /* 0x41c3a6e1f8000000 */
1.79245641600000000000e+09, /* 0x41dab5adb8000000 */
4.87240166400000000000e+09, /* 0x41f226af30000000 */
1.32445608960000000000e+10, /* 0x4208ab7fb0000000 */
3.60024494080000000000e+10, /* 0x4220c3d390000000 */
9.78648043520000000000e+10, /* 0x4236c93268000000 */
2.66024116224000000000e+11, /* 0x424ef822f0000000 */
7.23128516608000000000e+11, /* 0x42650bba30000000 */
1.96566712320000000000e+12, /* 0x427c9aae40000000 */
5.34323724288000000000e+12, /* 0x4293704708000000 */
1.45244246507520000000e+13, /* 0x42aa6b7658000000 */
3.94814795284480000000e+13, /* 0x42c1f43fc8000000 */
1.07321789251584000000e+14, /* 0x42d866f348000000 */
2.91730863685632000000e+14, /* 0x42f0953e28000000 */
7.93006722514944000000e+14, /* 0x430689e220000000 */
2.15561576592179200000e+15}; /* 0x431ea215a0000000 */
static const double cosh_tail[ 37] = {
0.00000000000000000000e+00, /* 0x0000000000000000 */
6.89700037027478056904e-09, /* 0x3e3d9f5504c2bd28 */
4.43207835591715833630e-08, /* 0x3e67cb66f0a4c9fd */
2.33540217013828929694e-07, /* 0x3e8f58617928e588 */
5.17452463948269748331e-08, /* 0x3e6bc7d000c38d48 */
9.38728274131605919153e-07, /* 0x3eaf7f9d4e329998 */
2.73012191010840495544e-06, /* 0x3ec6e6e464885269 */
3.29486051438996307950e-06, /* 0x3ecba3a8b946c154 */
4.75803746362771416375e-06, /* 0x3ed3f4e76110d5a4 */
3.33050940471947692369e-05, /* 0x3f017622515a3e2b */
9.94707313972136215365e-06, /* 0x3ee4dc4b528af3d0 */
6.51685096227860253398e-05, /* 0x3f11156278615e10 */
1.18132406658066663359e-03, /* 0x3f535ad50ed821f5 */
6.93090416366541877541e-04, /* 0x3f46b61055f2935c */
1.45780415323416845386e-03, /* 0x3f57e2794a601240 */
2.99862082708111758744e-02, /* 0x3f9eb4b45f6aadd3 */
1.02539925859688602072e-02, /* 0x3f85000b967b3698 */
1.26787669807076286421e-01, /* 0x3fc03a940fadc092 */
6.86652631843830962843e-02, /* 0x3fb1940bf3bf874c */
4.81593633223853068159e-01, /* 0x3fded26e1a2a2110 */
1.70489514001513020602e+00, /* 0x3ffb4740205796d6 */
1.12416073489841270572e+01, /* 0x40267bb3f55cb85d */
7.06579578098005001152e+00, /* 0x401c435ff81e18ac */
5.91244513000686140458e+01, /* 0x404d8fee052bdea4 */
1.68921736147088438429e+02, /* 0x40651d7edccde926 */
2.60692936262087528121e+02, /* 0x40704b1644557e0e */
3.62419382134890611269e+02, /* 0x4076a6b5ca0a9e1c */
4.07689930834187453002e+03, /* 0x40afd9cc72249abe */
1.55377375868385224749e+04, /* 0x40ce58de693edab5 */
2.53720210371943103382e+04, /* 0x40d8c70158ac6364 */
4.78822310734952334315e+04, /* 0x40e7614764f43e20 */
1.81871712615542812273e+05, /* 0x4106337db36fc718 */
5.62892347580489004031e+05, /* 0x41212d98b1f611e2 */
6.41374032312148716301e+05, /* 0x412392bc108b37cc */
7.57809544070145115256e+06, /* 0x415ce87bdc3473dc */
3.64177136406482197344e+06, /* 0x414bc8d5ae99ad14 */
7.63580561355670914054e+06}; /* 0x415d20d76744835c */
unsigned long ux, aux, xneg;
double y, z, z1, z2;
int m;
/* Special cases */
GET_BITS_DP64(x, ux);
aux = ux & ~SIGNBIT_DP64;
if (aux < 0x3e30000000000000) /* |x| small enough that cosh(x) = 1 */
{
if (aux == 0)
/* with no inexact */
return 1.0;
else
return val_with_flags(1.0, AMD_F_INEXACT);
}
else if (aux >= PINFBITPATT_DP64) /* |x| is NaN or Inf */
{
if (aux > PINFBITPATT_DP64) /* |x| is a NaN? */
return x + x;
else /* x is infinity */
return infinity_with_flags(0);
}
xneg = (aux != ux);
y = x;
if (xneg) y = -x;
if (y >= max_cosh_arg)
{
/* Return +/-infinity with overflow flag */
return retval_errno_erange(x, xneg);
}
else if (y >= small_threshold)
{
/* In this range y is large enough so that
the negative exponential is negligible,
so cosh(y) is approximated by sign(x)*exp(y)/2. The
code below is an inlined version of that from
exp() with two changes (it operates on
y instead of x, and the division by 2 is
done by reducing m by 1). */
splitexp(y, 1.0, thirtytwo_by_log2, log2_by_32_lead,
log2_by_32_tail, &m, &z1, &z2);
m -= 1;
if (m >= EMIN_DP64 && m <= EMAX_DP64)
z = scaleDouble_1((z1+z2),m);
else
z = scaleDouble_2((z1+z2),m);
}
else
{
/* In this range we find the integer part y0 of y
and the increment dy = y - y0. We then compute
z = sinh(y) = sinh(y0)cosh(dy) + cosh(y0)sinh(dy)
z = cosh(y) = cosh(y0)cosh(dy) + sinh(y0)sinh(dy)
where sinh(y0) and cosh(y0) are tabulated above. */
int ind;
double dy, dy2, sdy, cdy;
ind = (int)y;
dy = y - ind;
dy2 = dy*dy;
sdy = dy*dy2*(0.166666666666666667013899e0 +
(0.833333333333329931873097e-2 +
(0.198412698413242405162014e-3 +
(0.275573191913636406057211e-5 +
(0.250521176994133472333666e-7 +
(0.160576793121939886190847e-9 +
0.7746188980094184251527126e-12*dy2)*dy2)*dy2)*dy2)*dy2)*dy2);
cdy = dy2*(0.500000000000000005911074e0 +
(0.416666666666660876512776e-1 +
(0.138888888889814854814536e-2 +
(0.248015872460622433115785e-4 +
(0.275573350756016588011357e-6 +
(0.208744349831471353536305e-8 +
0.1163921388172173692062032e-10*dy2)*dy2)*dy2)*dy2)*dy2)*dy2);
/* At this point sinh(dy) is approximated by dy + sdy, and cosh(dy) is approximated by 1 + cdy.
Shift some significant bits from dy to cdy. */
#if 0
double sdy1,sdy2;
GET_BITS_DP64(dy, ux);
ux &= 0xfffffffff8000000;
PUT_BITS_DP64(ux, sdy1); // sdy1 is upper 53-27=26 significant bits of dy.
sdy2 = sdy + (dy - sdy1); // sdy2 is sdy + lower bits of dy
z = ((((((cosh_tail[ind]*cdy + sinh_tail[ind]*sdy2)
+ sinh_tail[ind]*sdy1) + cosh_tail[ind])
+ cosh_lead[ind]*cdy) + sinh_lead[ind]*sdy2)
+ sinh_lead[ind]*sdy1) + cosh_lead[ind];
#else
z = ((((((cosh_tail[ind]*cdy + sinh_tail[ind]*sdy)
+ sinh_tail[ind]*dy) + cosh_tail[ind])
+ cosh_lead[ind]*cdy) + sinh_lead[ind]*sdy)
+ sinh_lead[ind]*dy) + cosh_lead[ind];
#endif
}
return z;
}
