double FN_PROTOTYPE(atan)(double x)
{
/* Some constants and split constants. */
static double piby2 = 1.5707963267948966e+00; /* 0x3ff921fb54442d18 */
double chi, clo, v, s, q, z;
/* Find properties of argument x. */
unsigned long long ux, aux, xneg;
GET_BITS_DP64(x, ux);
aux = ux & ~SIGNBIT_DP64;
xneg = (ux != aux);
if (xneg) v = -x;
else v = x;
/* Argument reduction to range [-7/16,7/16] */
if (aux < 0x3e50000000000000) /* v < 2.0^(-26) */
{
/* x is a good approximation to atan(x) and avoids working on
intermediate denormal numbers */
if (aux == 0x0000000000000000)
return x;
else
return val_with_flags(x, AMD_F_INEXACT);
}
else if (aux > 0x4003800000000000) /* v > 39./16. */
{
if (aux > PINFBITPATT_DP64)
{
/* x is NaN */
#ifdef WINDOWS
return handle_error("atan", ux|0x0008000000000000, _DOMAIN, 0,
EDOM, x, 0.0);
#else
return x + x; /* Raise invalid if it's a signalling NaN */
#endif
}
else if (aux > 0x4370000000000000)
{ /* abs(x) > 2^56 => arctan(1/x) is
insignificant compared to piby2 */
if (xneg)
return val_with_flags(-piby2, AMD_F_INEXACT);
else
return val_with_flags(piby2, AMD_F_INEXACT);
}
x = -1.0/v;
/* (chi + clo) = arctan(infinity) */
chi = 1.57079632679489655800e+00; /* 0x3ff921fb54442d18 */
clo = 6.12323399573676480327e-17; /* 0x3c91a62633145c06 */
}
else if (aux > 0x3ff3000000000000) /* 39./16. > v > 19./16. */
{
x = (v-1.5)/(1.0+1.5*v);
/* (chi + clo) = arctan(1.5) */
chi = 9.82793723247329054082e-01; /* 0x3fef730bd281f69b */
clo = 1.39033110312309953701e-17; /* 0x3c7007887af0cbbc */
}
else if (aux > 0x3fe6000000000000) /* 19./16. > v > 11./16. */
{
x = (v-1.0)/(1.0+v);
/* (chi + clo) = arctan(1.) */
chi = 7.85398163397448278999e-01; /* 0x3fe921fb54442d18 */
clo = 3.06161699786838240164e-17; /* 0x3c81a62633145c06 */
}
else if (aux > 0x3fdc000000000000) /* 11./16. > v > 7./16. */
{
x = (2.0*v-1.0)/(2.0+v);
/* (chi + clo) = arctan(0.5) */
chi = 4.63647609000806093515e-01; /* 0x3fddac670561bb4f */
clo = 2.26987774529616809294e-17; /* 0x3c7a2b7f222f65e0 */
}
else /* v < 7./16. */
{
x = v;
chi = 0.0;
clo = 0.0;
}
/* Core approximation: Remez(4,4) on [-7/16,7/16] */
s = x*x;
q = x*s*
(0.268297920532545909e0 +
(0.447677206805497472e0 +
(0.220638780716667420e0 +
(0.304455919504853031e-1 +
0.142316903342317766e-3*s)*s)*s)*s)/
(0.804893761597637733e0 +
(0.182596787737507063e1 +
(0.141254259931958921e1 +
(0.424602594203847109e0 +
0.389525873944742195e-1*s)*s)*s)*s);
z = chi - ((q - clo) - x);
if (xneg) z = -z;
return z;
}
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(sinh)(double x)
{
/*
After dealing with special cases the computation is split into
regions as follows:
abs(x) >= max_sinh_arg:
sinh(x) = sign(x)*Inf
abs(x) >= small_threshold:
sinh(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)))
sinh(x) is then sign(x)*z. */
static const double
max_sinh_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;
/* (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 sinh(x) = x */
{
if (aux == 0)
/* with no inexact */
return x;
else
return val_with_flags(x, AMD_F_INEXACT);
}
else if (aux >= 0x7ff0000000000000) /* |x| is NaN or Inf */
{
return x + x;
}
xneg = (aux != ux);
y = x;
if (xneg) y = -x;
if (y >= max_sinh_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 sinh(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)
where sinh(y0) and cosh(y0) are tabulated above. */
int ind;
double dy, dy2, sdy, cdy, sdy1, sdy2;
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.
Shift some significant bits from dy to sdy. */
GET_BITS_DP64(dy, ux);
ux &= 0xfffffffff8000000;
PUT_BITS_DP64(ux, sdy1);
sdy2 = sdy + (dy - sdy1);
z = ((((((cosh_tail[ind]*sdy2 + sinh_tail[ind]*cdy)
+ cosh_tail[ind]*sdy1) + sinh_tail[ind])
+ cosh_lead[ind]*sdy2) + sinh_lead[ind]*cdy)
+ cosh_lead[ind]*sdy1) + sinh_lead[ind];
}
if (xneg) z = - z;
return z;
}
double FN_PROTOTYPE(asinh)(double x)
{
unsigned long long ux, ax, xneg;
double absx, r, rarg, t, r1, r2, poly, s, v1, v2;
int xexp;
static const unsigned long long
rteps = 0x3e46a09e667f3bcd, /* sqrt(eps) = 1.05367121277235086670e-08 */
recrteps = 0x4196a09e667f3bcd; /* 1/rteps = 9.49062656242515593767e+07 */
/* log2_lead and log2_tail sum to an extra-precise version
of log(2) */
static const double
log2_lead = 6.93147122859954833984e-01, /* 0x3fe62e42e0000000 */
log2_tail = 5.76999904754328540596e-08; /* 0x3e6efa39ef35793c */
GET_BITS_DP64(x, ux);
ax = ux & ~SIGNBIT_DP64;
xneg = 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 the same infinity. */
#ifdef WINDOWS
if (ux & SIGNBIT_DP64)
return handle_error(_FUNCNAME, NINFBITPATT_DP64, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0);
else
return handle_error(_FUNCNAME, PINFBITPATT_DP64, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0);
#else
return x;
#endif
}
}
else if (ax < rteps) /* abs(x) < sqrt(epsilon) */
{
if (ax == 0x0000000000000000)
{
/* x is +/-zero. Return the same zero. */
return x;
}
else
{
/* Tiny arguments approximated by asinh(x) = x
- avoid slow operations on denormalized numbers */
return val_with_flags(x,AMD_F_INEXACT);
}
}
if (ax <= 0x3ff0000000000000) /* abs(x) <= 1.0 */
{
/* Arguments less than 1.0 in magnitude are
approximated by [4,4] or [5,4] minimax polynomials
fitted to asinh series 4.6.31 (x < 1) from Abramowitz and Stegun
*/
t = x*x;
if (ax < 0x3fd0000000000000)
{
/* [4,4] for 0 < abs(x) < 0.25 */
poly =
(-0.12845379283524906084997e0 +
(-0.21060688498409799700819e0 +
(-0.10188951822578188309186e0 +
(-0.13891765817243625541799e-1 -
0.10324604871728082428024e-3 * t) * t) * t) * t) /
(0.77072275701149440164511e0 +
(0.16104665505597338100747e1 +
(0.11296034614816689554875e1 +
(0.30079351943799465092429e0 +
0.235224464765951442265117e-1 * t) * t) * t) * t);
}
else if (ax < 0x3fe0000000000000)
{
/* [4,4] for 0.25 <= abs(x) < 0.5 */
poly =
(-0.12186605129448852495563e0 +
(-0.19777978436593069928318e0 +
(-0.94379072395062374824320e-1 +
(-0.12620141363821680162036e-1 -
0.903396794842691998748349e-4 * t) * t) * t) * t) /
(0.73119630776696495279434e0 +
(0.15157170446881616648338e1 +
(0.10524909506981282725413e1 +
(0.27663713103600182193817e0 +
0.21263492900663656707646e-1 * t) * t) * t) * t);
}
else if (ax < 0x3fe8000000000000)
{
/* [4,4] for 0.5 <= abs(x) < 0.75 */
poly =
(-0.81210026327726247622500e-1 +
(-0.12327355080668808750232e0 +
(-0.53704925162784720405664e-1 +
(-0.63106739048128554465450e-2 -
0.35326896180771371053534e-4 * t) * t) * t) * t) /
(0.48726015805581794231182e0 +
(0.95890837357081041150936e0 +
(0.62322223426940387752480e0 +
(0.15028684818508081155141e0 +
0.10302171620320141529445e-1 * t) * t) * t) * t);
}
else
{
/* [5,4] for 0.75 <= abs(x) <= 1.0 */
poly =
(-0.4638179204422665073e-1 +
(-0.7162729496035415183e-1 +
(-0.3247795155696775148e-1 +
(-0.4225785421291932164e-2 +
(-0.3808984717603160127e-4 +
0.8023464184964125826e-6 * t) * t) * t) * t) * t) /
(0.2782907534642231184e0 +
(0.5549945896829343308e0 +
(0.3700732511330698879e0 +
(0.9395783438240780722e-1 +
0.7200057974217143034e-2 * t) * t) * t) * t);
}
return x + x*t*poly;
}
else if (ax < 0x4040000000000000)
{
/* 1.0 <= abs(x) <= 32.0 */
/* Arguments in this region are approximated by various
minimax polynomials fitted to asinh series 4.6.31
in Abramowitz and Stegun.
*/
t = x*x;
if (ax >= 0x4020000000000000)
{
/* [3,3] for 8.0 <= abs(x) <= 32.0 */
poly =
(-0.538003743384069117e-10 +
(-0.273698654196756169e-9 +
(-0.268129826956403568e-9 -
0.804163374628432850e-29 * t) * t) * t) /
(0.238083376363471960e-9 +
(0.203579344621125934e-8 +
(0.450836980450693209e-8 +
0.286005148753497156e-8 * t) * t) * t);
}
else if (ax >= 0x4010000000000000)
{
/* [4,3] for 4.0 <= abs(x) <= 8.0 */
poly =
(-0.178284193496441400e-6 +
(-0.928734186616614974e-6 +
(-0.923318925566302615e-6 +
(-0.776417026702577552e-19 +
0.290845644810826014e-21 * t) * t) * t) * t) /
(0.786694697277890964e-6 +
(0.685435665630965488e-5 +
(0.153780175436788329e-4 +
0.984873520613417917e-5 * t) * t) * t);
}
else if (ax >= 0x4000000000000000)
{
/* [5,4] for 2.0 <= abs(x) <= 4.0 */
poly =
(-0.209689451648100728e-6 +
(-0.219252358028695992e-5 +
(-0.551641756327550939e-5 +
(-0.382300259826830258e-5 +
(-0.421182121910667329e-17 +
0.492236019998237684e-19 * t) * t) * t) * t) * t) /
(0.889178444424237735e-6 +
(0.131152171690011152e-4 +
(0.537955850185616847e-4 +
(0.814966175170941864e-4 +
0.407786943832260752e-4 * t) * t) * t) * t);
}
else if (ax >= 0x3ff8000000000000)
{
/* [5,4] for 1.5 <= abs(x) <= 2.0 */
poly =
(-0.195436610112717345e-4 +
(-0.233315515113382977e-3 +
(-0.645380957611087587e-3 +
(-0.478948863920281252e-3 +
(-0.805234112224091742e-12 +
0.246428598194879283e-13 * t) * t) * t) * t) * t) /
(0.822166621698664729e-4 +
(0.135346265620413852e-2 +
(0.602739242861830658e-2 +
(0.972227795510722956e-2 +
0.510878800983771167e-2 * t) * t) * t) * t);
}
else
{
/* [5,5] for 1.0 <= abs(x) <= 1.5 */
poly =
(-0.121224194072430701e-4 +
(-0.273145455834305218e-3 +
(-0.152866982560895737e-2 +
(-0.292231744584913045e-2 +
(-0.174670900236060220e-2 -
0.891754209521081538e-12 * t) * t) * t) * t) * t) /
(0.499426632161317606e-4 +
(0.139591210395547054e-2 +
(0.107665231109108629e-1 +
(0.325809818749873406e-1 +
(0.415222526655158363e-1 +
0.186315628774716763e-1 * t) * t) * t) * t) * t);
}
log_kernel_amd64(absx, ax, &xexp, &r1, &r2);
r1 = ((xexp+1) * log2_lead + r1);
r2 = ((xexp+1) * log2_tail + r2);
/* Now (r1,r2) sum to log(2x). Add the term
1/(2.2.x^2) = 0.25/t, and add poly/t, carefully
to maintain precision. (Note that we add poly/t
rather than poly because of the *x factor used
when generating the minimax polynomial) */
v2 = (poly+0.25)/t;
r = v2 + r1;
s = ((r1 - r) + v2) + r2;
v1 = r + s;
v2 = (r - v1) + s;
r = v1 + v2;
if (xneg)
return -r;
else
return r;
}
else
{
/* abs(x) > 32.0 */
if (ax > recrteps)
{
/* Arguments greater than 1/sqrt(epsilon) in magnitude are
approximated by asinh(x) = ln(2) + ln(abs(x)), with sign of x */
/* log_kernel_amd(x) returns xexp, r1, r2 such that
log(x) = xexp*log(2) + r1 + r2 */
log_kernel_amd64(absx, ax, &xexp, &r1, &r2);
/* Add (xexp+1) * log(2) to z1,z2 to get the result asinh(x).
The computed r1 is not subject to rounding error because
(xexp+1) has at most 10 significant bits, log(2) has 24 significant
bits, and r1 has up to 24 bits; and the exponents of r1
and r2 differ by at most 6. */
r1 = ((xexp+1) * log2_lead + r1);
r2 = ((xexp+1) * log2_tail + r2);
if (xneg)
return -(r1 + r2);
else
return r1 + r2;
}
else
{
rarg = absx*absx+1.0;
/* Arguments such that 32.0 <= abs(x) <= 1/sqrt(epsilon) are
approximated by
asinh(x) = ln(abs(x) + sqrt(x*x+1))
with the sign of x (see Abramowitz and Stegun 4.6.20) */
/* Use assembly instruction to compute r = sqrt(rarg); */
ASMSQRT(rarg,r);
r += absx;
GET_BITS_DP64(r, ax);
log_kernel_amd64(r, ax, &xexp, &r1, &r2);
r1 = (xexp * log2_lead + r1);
r2 = (xexp * log2_tail + r2);
if (xneg)
return -(r1 + r2);
else
return r1 + r2;
}
}
}
