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;
}
float FN_PROTOTYPE(atanhf)(float x)
{
double dx;
unsigned int ux, ax;
double r, t, poly;
GET_BITS_SP32(x, ux);
ax = ux & ~SIGNBIT_SP32;
if ((ux & EXPBITS_SP32) == EXPBITS_SP32)
{
/* x is either NaN or infinity */
if (ux & MANTBITS_SP32)
{
/* x is NaN */
#ifdef WINDOWS
return handle_errorf(_FUNCNAME, ux|0x00400000, _DOMAIN,
0, EDOM, x, 0.0F);
#else
return x + x; /* Raise invalid if it is a signalling NaN */
#endif
}
else
{
/* x is infinity; return a NaN */
#ifdef WINDOWS
return handle_errorf(_FUNCNAME, INDEFBITPATT_SP32, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0F);
#else
return retval_errno_edom(x,nanf_with_flags(AMD_F_INVALID));
#endif
}
}
else if (ax >= 0x3f800000)
{
if (ax > 0x3f800000)
{
/* abs(x) > 1.0; return NaN */
#ifdef WINDOWS
return handle_errorf(_FUNCNAME, INDEFBITPATT_SP32, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0F);
#else
return retval_errno_edom(x,nanf_with_flags(AMD_F_INVALID));
#endif
}
else if (ux == 0x3f800000)
{
/* x = +1.0; return infinity with the same sign as x
and set the divbyzero status flag */
#ifdef WINDOWS
return handle_errorf(_FUNCNAME, PINFBITPATT_SP32, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0F);
#else
return retval_errno_edom(x,infinityf_with_flags(AMD_F_DIVBYZERO));
#endif
}
else
{
/* x = -1.0; return infinity with the same sign as x */
#ifdef WINDOWS
return handle_errorf(_FUNCNAME, NINFBITPATT_SP32, _DOMAIN,
AMD_F_INVALID, EDOM, x, 0.0F);
#else
return retval_errno_edom(x,-infinityf_with_flags(AMD_F_DIVBYZERO));
#endif
}
}
if (ax < 0x39000000)
{
if (ax == 0x00000000)
{
/* x is +/-zero. Return the same zero. */
return x;
}
else
{
/* Arguments smaller than 2^(-13) in magnitude are
approximated by atanhf(x) = x, raising inexact flag. */
return valf_with_flags(x, AMD_F_INEXACT);
}
}
else
{
dx = x;
if (ax < 0x3f000000)
{
/* Arguments up to 0.5 in magnitude are
approximated by a [2,2] minimax polynomial */
t = dx*dx;
poly =
(0.39453629046e0 +
(-0.28120347286e0 +
0.92834212715e-2 * t) * t) /
(0.11836088638e1 +
(-0.15537744551e1 +
0.45281890445e0 * t) * t);
return (float)(dx + dx*t*poly);
}
else
{
/* abs(x) >= 0.5 */
/* Note that
atanhf(x) = 0.5 * ln((1+x)/(1-x))
(see Abramowitz and Stegun 4.6.22).
For greater accuracy we use the variant formula
atanhf(x) = log(1 + 2x/(1-x)) = log1p(2x/(1-x)).
*/
if (ux & SIGNBIT_SP32)
{
/* Argument x is negative */
r = (-2.0 * dx) / (1.0 + dx);
r = 0.5 * FN_PROTOTYPE(log1p)(r);
return (float)-r;
}
else
{
r = (2.0 * dx) / (1.0 - dx);
r = 0.5 * FN_PROTOTYPE(log1p)(r);
return (float)r;
}
}
}
}
