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(exp2)(double x)
{
static const double
max_exp2_arg = 1024.0, /* 0x4090000000000000 */
min_exp2_arg = -1074.0, /* 0xc090c80000000000 */
log2 = 6.931471805599453094178e-01, /* 0x3fe62e42fefa39ef */
log2_lead = 6.93147167563438415527E-01, /* 0x3fe62e42f8000000 */
log2_tail = 1.29965068938898869640E-08, /* 0x3e4be8e7bcd5e4f1 */
one_by_32_lead = 0.03125;
double y, z1, z2, z, hx, tx, y1, y2;
int m;
unsigned long ux, ax;
/*
Computation of exp2(x).
We compute the values m, z1, and z2 such that
exp2(x) = 2**m * (z1 + z2), where exp2(x) is 2**x.
Computations needed in order to obtain m, z1, and z2
involve three steps.
First, we reduce the argument x to the form
x = n/32 + remainder,
where n has the value of an integer and |remainder| <= 1/64.
The value of n = x * 32 rounded to the nearest integer and
the remainder = x - n/32.
Second, we approximate exp2(r1 + r2) - 1 where r1 is the leading
part of the remainder and r2 is the trailing part of the remainder.
Third, we reconstruct exp2(x) so that
exp2(x) = 2**m * (z1 + z2).
*/
GET_BITS_DP64(x, ux);
ax = ux & (~SIGNBIT_DP64);
if (ax >= 0x4090000000000000) /* abs(x) >= 1024.0 */
{
if(ax >= 0x7ff0000000000000)
{
/* x is either NaN or infinity */
if (ux & MANTBITS_DP64)
/* x is NaN */
return x + x; /* Raise invalid if it is a signalling NaN */
else if (ux & SIGNBIT_DP64)
/* x is negative infinity; return 0.0 with no flags. */
return 0.0;
else
/* x is positive infinity */
return x;
}
if (x > max_exp2_arg)
/* Return +infinity with overflow flag */
return retval_errno_erange_overflow(x);
else if (x < min_exp2_arg)
/* x is negative. Return +zero with underflow and inexact flags */
return retval_errno_erange_underflow(x);
}
/* Handle small arguments separately */
if (ax < 0x3fb7154764ee6c2f) /* abs(x) < 1/(16*log2) */
{
if (ax < 0x3c00000000000000) /* abs(x) < 2^(-63) */
return 1.0 + x; /* Raises inexact if x is non-zero */
else
{
/* Split x into hx (head) and tx (tail). */
unsigned long u;
hx = x;
GET_BITS_DP64(hx, u);
u &= 0xfffffffff8000000;
PUT_BITS_DP64(u, hx);
tx = x - hx;
/* Carefully multiply x by log2. y1 is the most significant
part of the result, and y2 the least significant part */
y1 = x * log2_lead;
y2 = (((hx * log2_lead - y1) + hx * log2_tail) +
tx * log2_lead) + tx * log2_tail;
y = y1 + y2;
z = (9.99564649780173690e-1 +
(1.61251249355268050e-5 +
(2.37986978239838493e-2 +
2.68724774856111190e-7*y)*y)*y)/
(9.99564649780173692e-1 +
(-4.99766199765151309e-1 +
(1.070876894098586184e-1 +
(-1.189773642681502232e-2 +
5.9480622371960190616e-4*y)*y)*y)*y);
z = ((z * y1) + (z * y2)) + 1.0;
}
}
else
{
/* Find m, z1 and z2 such that exp2(x) = 2**m * (z1 + z2) */
splitexp(x, log2, 32.0, one_by_32_lead, 0.0, &m, &z1, &z2);
/* Scale (z1 + z2) by 2.0**m */
if (m > EMIN_DP64 && m < EMAX_DP64)
z = scaleDouble_1((z1+z2),m);
else
z = scaleDouble_2((z1+z2),m);
}
return z;
}
float FN_PROTOTYPE(coshf)(float fx)
{
/*
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
/* The max argument of coshf, but stored as a double */
max_cosh_arg = 8.94159862922329438106e+01, /* 0x40565a9f84f82e63 */
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) */
/* Tabulated values of sinh(i) and cosh(i) for i = 0,...,36. */
static const double sinh_lead[ 37] = {
0.00000000000000000000e+00, /* 0x0000000000000000 */
1.17520119364380137839e+00, /* 0x3ff2cd9fc44eb982 */
3.62686040784701857476e+00, /* 0x400d03cf63b6e19f */
1.00178749274099008204e+01, /* 0x40240926e70949ad */
2.72899171971277496596e+01, /* 0x403b4a3803703630 */
7.42032105777887522891e+01, /* 0x40528d0166f07374 */
2.01713157370279219549e+02, /* 0x406936d22f67c805 */
5.48316123273246489589e+02, /* 0x408122876ba380c9 */
1.49047882578955000099e+03, /* 0x409749ea514eca65 */
4.05154190208278987484e+03, /* 0x40afa7157430966f */
1.10132328747033916443e+04, /* 0x40c5829dced69991 */
2.99370708492480553105e+04, /* 0x40dd3c4488cb48d6 */
8.13773957064298447222e+04, /* 0x40f3de1654d043f0 */
2.21206696003330085659e+05, /* 0x410b00b5916a31a5 */
6.01302142081972560845e+05, /* 0x412259ac48bef7e3 */
1.63450868623590236530e+06, /* 0x4138f0ccafad27f6 */
4.44305526025387924165e+06, /* 0x4150f2ebd0a7ffe3 */
1.20774763767876271158e+07, /* 0x416709348c0ea4ed */
3.28299845686652474105e+07, /* 0x417f4f22091940bb */
8.92411504815936237574e+07, /* 0x419546d8f9ed26e1 */
2.42582597704895108938e+08, /* 0x41aceb088b68e803 */
6.59407867241607308388e+08, /* 0x41c3a6e1fd9eecfd */
1.79245642306579566002e+09, /* 0x41dab5adb9c435ff */
4.87240172312445068359e+09, /* 0x41f226af33b1fdc0 */
1.32445610649217357635e+10, /* 0x4208ab7fb5475fb7 */
3.60024496686929321289e+10, /* 0x4220c3d3920962c8 */
9.78648047144193725586e+10, /* 0x4236c932696a6b5c */
2.66024120300899291992e+11, /* 0x424ef822f7f6731c */
7.23128532145737548828e+11, /* 0x42650bba3796379a */
1.96566714857202099609e+12, /* 0x427c9aae4631c056 */
5.34323729076223046875e+12, /* 0x429370470aec28ec */
1.45244248326237109375e+13, /* 0x42aa6b765d8cdf6c */
3.94814800913403437500e+13, /* 0x42c1f43fcc4b662c */
1.07321789892958031250e+14, /* 0x42d866f34a725782 */
2.91730871263727437500e+14, /* 0x42f0953e2f3a1ef7 */
7.93006726156715250000e+14, /* 0x430689e221bc8d5a */
2.15561577355759750000e+15}; /* 0x431ea215a1d20d76 */
static const double cosh_lead[ 37] = {
1.00000000000000000000e+00, /* 0x3ff0000000000000 */
1.54308063481524371241e+00, /* 0x3ff8b07551d9f550 */
3.76219569108363138810e+00, /* 0x400e18fa0df2d9bc */
1.00676619957777653269e+01, /* 0x402422a497d6185e */
2.73082328360164865444e+01, /* 0x403b4ee858de3e80 */
7.42099485247878334349e+01, /* 0x40528d6fcbeff3a9 */
2.01715636122455890700e+02, /* 0x406936e67db9b919 */
5.48317035155212010977e+02, /* 0x4081228949ba3a8b */
1.49047916125217807348e+03, /* 0x409749eaa93f4e76 */
4.05154202549259389343e+03, /* 0x40afa715845d8894 */
1.10132329201033226127e+04, /* 0x40c5829dd053712d */
2.99370708659497577173e+04, /* 0x40dd3c4489115627 */
8.13773957125740562333e+04, /* 0x40f3de1654d6b543 */
2.21206696005590405548e+05, /* 0x410b00b5916b6105 */
6.01302142082804115489e+05, /* 0x412259ac48bf13ca */
1.63450868623620807193e+06, /* 0x4138f0ccafad2d17 */
4.44305526025399193168e+06, /* 0x4150f2ebd0a8005c */
1.20774763767876680940e+07, /* 0x416709348c0ea503 */
3.28299845686652623117e+07, /* 0x417f4f22091940bf */
8.92411504815936237574e+07, /* 0x419546d8f9ed26e1 */
2.42582597704895138741e+08, /* 0x41aceb088b68e804 */
6.59407867241607308388e+08, /* 0x41c3a6e1fd9eecfd */
1.79245642306579566002e+09, /* 0x41dab5adb9c435ff */
4.87240172312445068359e+09, /* 0x41f226af33b1fdc0 */
1.32445610649217357635e+10, /* 0x4208ab7fb5475fb7 */
3.60024496686929321289e+10, /* 0x4220c3d3920962c8 */
9.78648047144193725586e+10, /* 0x4236c932696a6b5c */
2.66024120300899291992e+11, /* 0x424ef822f7f6731c */
7.23128532145737548828e+11, /* 0x42650bba3796379a */
1.96566714857202099609e+12, /* 0x427c9aae4631c056 */
5.34323729076223046875e+12, /* 0x429370470aec28ec */
1.45244248326237109375e+13, /* 0x42aa6b765d8cdf6c */
3.94814800913403437500e+13, /* 0x42c1f43fcc4b662c */
1.07321789892958031250e+14, /* 0x42d866f34a725782 */
2.91730871263727437500e+14, /* 0x42f0953e2f3a1ef7 */
7.93006726156715250000e+14, /* 0x430689e221bc8d5a */
2.15561577355759750000e+15}; /* 0x431ea215a1d20d76 */
unsigned long ux, aux, xneg;
double x = fx, 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) return (float)1.0; /* with no inexact */
if (LAMBDA_DP64 + x > 1.0) return valf_with_flags((float)1.0, AMD_F_INEXACT); /* with inexact */
}
else if (aux >= PINFBITPATT_DP64) /* |x| is NaN or Inf */
{
if (aux > PINFBITPATT_DP64) /* |x| is a NaN? */
return fx + fx;
else /* x is infinity */
return infinityf_with_flags(0);
}
xneg = (aux != ux);
y = x;
if (xneg) y = -x;
if (y >= max_cosh_arg)
{
/* Return infinity with overflow flag. */
#if 0
/* This way handles non-POSIX behaviour but weirdly causes
sinhf to run half as fast for all arguments on Hammer */
return retval_errno_erange(fx, xneg);
#else
/* This handles POSIX behaviour */
__set_errno(ERANGE);
z = infinityf_with_flags(AMD_F_OVERFLOW);
#endif
}
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 + 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 = 1 + 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);
z = cosh_lead[ind]*cdy + sinh_lead[ind]*sdy;
}
// if (xneg) z = - z;
return (float)z;
}
double FN_PROTOTYPE(exp)(double x)
{
static const double
max_exp_arg = 7.09782712893383973096e+02, /* 0x40862e42fefa39ef */
min_exp_arg = -7.45133219101941108420e+02, /* 0xc0874910d52d3051 */
thirtytwo_by_log2 = 4.61662413084468283841e+01, /* 0x40471547652b82fe */
log2_by_32_lead = 2.16608493356034159660e-02, /* 0x3f962e42fe000000 */
log2_by_32_trail = 5.68948749532545630390e-11; /* 0x3dcf473de6af278e */
double z1, z2, z;
int m;
unsigned long ux, ax;
/*
Computation of exp(x).
We compute the values m, z1, and z2 such that
exp(x) = 2**m * (z1 + z2), where
exp(x) is the natural exponential of x.
Computations needed in order to obtain m, z1, and z2
involve three steps.
First, we reduce the argument x to the form
x = n * log2/32 + remainder,
where n has the value of an integer and |remainder| <= log2/64.
The value of n = x * 32/log2 rounded to the nearest integer and
the remainder = x - n*log2/32.
Second, we approximate exp(r1 + r2) - 1 where r1 is the leading
part of the remainder and r2 is the trailing part of the remainder.
Third, we reconstruct the exponential of x so that
exp(x) = 2**m * (z1 + z2).
*/
GET_BITS_DP64(x, ux);
ax = ux & (~SIGNBIT_DP64);
if (ax >= 0x40862e42fefa39ef) /* abs(x) >= 709.78... */
{
if(ax >= 0x7ff0000000000000)
{
/* x is either NaN or infinity */
if (ux & MANTBITS_DP64)
/* x is NaN */
return x + x; /* Raise invalid if it is a signalling NaN */
else if (ux & SIGNBIT_DP64)
/* x is negative infinity; return 0.0 with no flags. */
return 0.0;
else
/* x is positive infinity */
return x;
}
if (x > max_exp_arg)
/* Return +infinity with overflow flag */
return retval_errno_erange_overflow(x);
else if (x < min_exp_arg)
/* x is negative. Return +zero with underflow and inexact flags */
return retval_errno_erange_underflow(x);
}
/* Handle small arguments separately */
if (ax < 0x3fb0000000000000) /* abs(x) < 1/16 */
{
if (ax < 0x3c00000000000000) /* abs(x) < 2^(-63) */
z = 1.0 + x; /* Raises inexact if x is non-zero */
else
z = ((((((((((
1.0/3628800)*x+
1.0/362880)*x+
1.0/40320)*x+
1.0/5040)*x+
1.0/720)*x+
1.0/120)*x+
1.0/24)*x+
1.0/6)*x+
1.0/2)*x+
1.0)*x + 1.0;
}
else
{
/* Find m, z1 and z2 such that exp(x) = 2**m * (z1 + z2) */
splitexp(x, 1.0, thirtytwo_by_log2, log2_by_32_lead, log2_by_32_trail,
&m, &z1, &z2);
/* Scale (z1 + z2) by 2.0**m */
if (m >= EMIN_DP64 && m <= EMAX_DP64)
z = scaleDouble_1((z1+z2),m);
else
z = scaleDouble_2((z1+z2),m);
}
return z;
}