449 lines
12 KiB
C
449 lines
12 KiB
C
/*
|
|
* ====================================================
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
|
* Permission to use, copy, modify, and distribute this
|
|
* software is freely granted, provided that this notice
|
|
* is preserved.
|
|
* ====================================================
|
|
*/
|
|
|
|
/* Expansions and modifications for 128-bit long double are
|
|
Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
|
|
and are incorporated herein by permission of the author. The author
|
|
reserves the right to distribute this material elsewhere under different
|
|
copying permissions. These modifications are distributed here under
|
|
the following terms:
|
|
|
|
This library is free software; you can redistribute it and/or
|
|
modify it under the terms of the GNU Lesser General Public
|
|
License as published by the Free Software Foundation; either
|
|
version 2.1 of the License, or (at your option) any later version.
|
|
|
|
This library is distributed in the hope that it will be useful,
|
|
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
|
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
|
|
Lesser General Public License for more details.
|
|
|
|
You should have received a copy of the GNU Lesser General Public
|
|
License along with this library; if not, write to the Free Software
|
|
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
|
|
|
|
/* powq(x,y) return x**y
|
|
*
|
|
* n
|
|
* Method: Let x = 2 * (1+f)
|
|
* 1. Compute and return log2(x) in two pieces:
|
|
* log2(x) = w1 + w2,
|
|
* where w1 has 113-53 = 60 bit trailing zeros.
|
|
* 2. Perform y*log2(x) = n+y' by simulating muti-precision
|
|
* arithmetic, where |y'|<=0.5.
|
|
* 3. Return x**y = 2**n*exp(y'*log2)
|
|
*
|
|
* Special cases:
|
|
* 1. (anything) ** 0 is 1
|
|
* 2. (anything) ** 1 is itself
|
|
* 3. (anything) ** NAN is NAN
|
|
* 4. NAN ** (anything except 0) is NAN
|
|
* 5. +-(|x| > 1) ** +INF is +INF
|
|
* 6. +-(|x| > 1) ** -INF is +0
|
|
* 7. +-(|x| < 1) ** +INF is +0
|
|
* 8. +-(|x| < 1) ** -INF is +INF
|
|
* 9. +-1 ** +-INF is NAN
|
|
* 10. +0 ** (+anything except 0, NAN) is +0
|
|
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
|
|
* 12. +0 ** (-anything except 0, NAN) is +INF
|
|
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
|
|
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
|
|
* 15. +INF ** (+anything except 0,NAN) is +INF
|
|
* 16. +INF ** (-anything except 0,NAN) is +0
|
|
* 17. -INF ** (anything) = -0 ** (-anything)
|
|
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
|
|
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
|
|
*
|
|
*/
|
|
|
|
#include "quadmath-imp.h"
|
|
|
|
static const __float128 bp[] = {
|
|
1.0Q,
|
|
1.5Q,
|
|
};
|
|
|
|
/* log_2(1.5) */
|
|
static const __float128 dp_h[] = {
|
|
0.0,
|
|
5.8496250072115607565592654282227158546448E-1Q
|
|
};
|
|
|
|
/* Low part of log_2(1.5) */
|
|
static const __float128 dp_l[] = {
|
|
0.0,
|
|
1.0579781240112554492329533686862998106046E-16Q
|
|
};
|
|
|
|
static const __float128 zero = 0.0Q,
|
|
one = 1.0Q,
|
|
two = 2.0Q,
|
|
two113 = 1.0384593717069655257060992658440192E34Q,
|
|
huge = 1.0e3000Q,
|
|
tiny = 1.0e-3000Q;
|
|
|
|
/* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
|
|
z = (x-1)/(x+1)
|
|
1 <= x <= 1.25
|
|
Peak relative error 2.3e-37 */
|
|
static const __float128 LN[] =
|
|
{
|
|
-3.0779177200290054398792536829702930623200E1Q,
|
|
6.5135778082209159921251824580292116201640E1Q,
|
|
-4.6312921812152436921591152809994014413540E1Q,
|
|
1.2510208195629420304615674658258363295208E1Q,
|
|
-9.9266909031921425609179910128531667336670E-1Q
|
|
};
|
|
static const __float128 LD[] =
|
|
{
|
|
-5.129862866715009066465422805058933131960E1Q,
|
|
1.452015077564081884387441590064272782044E2Q,
|
|
-1.524043275549860505277434040464085593165E2Q,
|
|
7.236063513651544224319663428634139768808E1Q,
|
|
-1.494198912340228235853027849917095580053E1Q
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
/* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
|
|
0 <= x <= 0.5
|
|
Peak relative error 5.7e-38 */
|
|
static const __float128 PN[] =
|
|
{
|
|
5.081801691915377692446852383385968225675E8Q,
|
|
9.360895299872484512023336636427675327355E6Q,
|
|
4.213701282274196030811629773097579432957E4Q,
|
|
5.201006511142748908655720086041570288182E1Q,
|
|
9.088368420359444263703202925095675982530E-3Q,
|
|
};
|
|
static const __float128 PD[] =
|
|
{
|
|
3.049081015149226615468111430031590411682E9Q,
|
|
1.069833887183886839966085436512368982758E8Q,
|
|
8.259257717868875207333991924545445705394E5Q,
|
|
1.872583833284143212651746812884298360922E3Q,
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
static const __float128
|
|
/* ln 2 */
|
|
lg2 = 6.9314718055994530941723212145817656807550E-1Q,
|
|
lg2_h = 6.9314718055994528622676398299518041312695E-1Q,
|
|
lg2_l = 2.3190468138462996154948554638754786504121E-17Q,
|
|
ovt = 8.0085662595372944372e-0017Q,
|
|
/* 2/(3*log(2)) */
|
|
cp = 9.6179669392597560490661645400126142495110E-1Q,
|
|
cp_h = 9.6179669392597555432899980587535537779331E-1Q,
|
|
cp_l = 5.0577616648125906047157785230014751039424E-17Q;
|
|
|
|
__float128
|
|
powq (__float128 x, __float128 y)
|
|
{
|
|
__float128 z, ax, z_h, z_l, p_h, p_l;
|
|
__float128 y1, t1, t2, r, s, sgn, t, u, v, w;
|
|
__float128 s2, s_h, s_l, t_h, t_l, ay;
|
|
int32_t i, j, k, yisint, n;
|
|
uint32_t ix, iy;
|
|
int32_t hx, hy;
|
|
ieee854_float128 o, p, q;
|
|
|
|
p.value = x;
|
|
hx = p.words32.w0;
|
|
ix = hx & 0x7fffffff;
|
|
|
|
q.value = y;
|
|
hy = q.words32.w0;
|
|
iy = hy & 0x7fffffff;
|
|
|
|
|
|
/* y==zero: x**0 = 1 */
|
|
if ((iy | q.words32.w1 | q.words32.w2 | q.words32.w3) == 0)
|
|
return one;
|
|
|
|
/* 1.0**y = 1; -1.0**+-Inf = 1 */
|
|
if (x == one)
|
|
return one;
|
|
if (x == -1.0Q && iy == 0x7fff0000
|
|
&& (q.words32.w1 | q.words32.w2 | q.words32.w3) == 0)
|
|
return one;
|
|
|
|
/* +-NaN return x+y */
|
|
if ((ix > 0x7fff0000)
|
|
|| ((ix == 0x7fff0000)
|
|
&& ((p.words32.w1 | p.words32.w2 | p.words32.w3) != 0))
|
|
|| (iy > 0x7fff0000)
|
|
|| ((iy == 0x7fff0000)
|
|
&& ((q.words32.w1 | q.words32.w2 | q.words32.w3) != 0)))
|
|
return x + y;
|
|
|
|
/* determine if y is an odd int when x < 0
|
|
* yisint = 0 ... y is not an integer
|
|
* yisint = 1 ... y is an odd int
|
|
* yisint = 2 ... y is an even int
|
|
*/
|
|
yisint = 0;
|
|
if (hx < 0)
|
|
{
|
|
if (iy >= 0x40700000) /* 2^113 */
|
|
yisint = 2; /* even integer y */
|
|
else if (iy >= 0x3fff0000) /* 1.0 */
|
|
{
|
|
if (floorq (y) == y)
|
|
{
|
|
z = 0.5 * y;
|
|
if (floorq (z) == z)
|
|
yisint = 2;
|
|
else
|
|
yisint = 1;
|
|
}
|
|
}
|
|
}
|
|
|
|
/* special value of y */
|
|
if ((q.words32.w1 | q.words32.w2 | q.words32.w3) == 0)
|
|
{
|
|
if (iy == 0x7fff0000) /* y is +-inf */
|
|
{
|
|
if (((ix - 0x3fff0000) | p.words32.w1 | p.words32.w2 | p.words32.w3)
|
|
== 0)
|
|
return y - y; /* +-1**inf is NaN */
|
|
else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */
|
|
return (hy >= 0) ? y : zero;
|
|
else /* (|x|<1)**-,+inf = inf,0 */
|
|
return (hy < 0) ? -y : zero;
|
|
}
|
|
if (iy == 0x3fff0000)
|
|
{ /* y is +-1 */
|
|
if (hy < 0)
|
|
return one / x;
|
|
else
|
|
return x;
|
|
}
|
|
if (hy == 0x40000000)
|
|
return x * x; /* y is 2 */
|
|
if (hy == 0x3ffe0000)
|
|
{ /* y is 0.5 */
|
|
if (hx >= 0) /* x >= +0 */
|
|
return sqrtq (x);
|
|
}
|
|
}
|
|
|
|
ax = fabsq (x);
|
|
/* special value of x */
|
|
if ((p.words32.w1 | p.words32.w2 | p.words32.w3) == 0)
|
|
{
|
|
if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000)
|
|
{
|
|
z = ax; /*x is +-0,+-inf,+-1 */
|
|
if (hy < 0)
|
|
z = one / z; /* z = (1/|x|) */
|
|
if (hx < 0)
|
|
{
|
|
if (((ix - 0x3fff0000) | yisint) == 0)
|
|
{
|
|
z = (z - z) / (z - z); /* (-1)**non-int is NaN */
|
|
}
|
|
else if (yisint == 1)
|
|
z = -z; /* (x<0)**odd = -(|x|**odd) */
|
|
}
|
|
return z;
|
|
}
|
|
}
|
|
|
|
/* (x<0)**(non-int) is NaN */
|
|
if (((((uint32_t) hx >> 31) - 1) | yisint) == 0)
|
|
return (x - x) / (x - x);
|
|
|
|
/* sgn (sign of result -ve**odd) = -1 else = 1 */
|
|
sgn = one;
|
|
if (((((uint32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
|
|
sgn = -one; /* (-ve)**(odd int) */
|
|
|
|
/* |y| is huge.
|
|
2^-16495 = 1/2 of smallest representable value.
|
|
If (1 - 1/131072)^y underflows, y > 1.4986e9 */
|
|
if (iy > 0x401d654b)
|
|
{
|
|
/* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
|
|
if (iy > 0x407d654b)
|
|
{
|
|
if (ix <= 0x3ffeffff)
|
|
return (hy < 0) ? huge * huge : tiny * tiny;
|
|
if (ix >= 0x3fff0000)
|
|
return (hy > 0) ? huge * huge : tiny * tiny;
|
|
}
|
|
/* over/underflow if x is not close to one */
|
|
if (ix < 0x3ffeffff)
|
|
return (hy < 0) ? sgn * huge * huge : sgn * tiny * tiny;
|
|
if (ix > 0x3fff0000)
|
|
return (hy > 0) ? sgn * huge * huge : sgn * tiny * tiny;
|
|
}
|
|
|
|
ay = y > 0 ? y : -y;
|
|
if (ay < 0x1p-128)
|
|
y = y < 0 ? -0x1p-128 : 0x1p-128;
|
|
|
|
n = 0;
|
|
/* take care subnormal number */
|
|
if (ix < 0x00010000)
|
|
{
|
|
ax *= two113;
|
|
n -= 113;
|
|
o.value = ax;
|
|
ix = o.words32.w0;
|
|
}
|
|
n += ((ix) >> 16) - 0x3fff;
|
|
j = ix & 0x0000ffff;
|
|
/* determine interval */
|
|
ix = j | 0x3fff0000; /* normalize ix */
|
|
if (j <= 0x3988)
|
|
k = 0; /* |x|<sqrt(3/2) */
|
|
else if (j < 0xbb67)
|
|
k = 1; /* |x|<sqrt(3) */
|
|
else
|
|
{
|
|
k = 0;
|
|
n += 1;
|
|
ix -= 0x00010000;
|
|
}
|
|
|
|
o.value = ax;
|
|
o.words32.w0 = ix;
|
|
ax = o.value;
|
|
|
|
/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
|
|
u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
|
|
v = one / (ax + bp[k]);
|
|
s = u * v;
|
|
s_h = s;
|
|
|
|
o.value = s_h;
|
|
o.words32.w3 = 0;
|
|
o.words32.w2 &= 0xf8000000;
|
|
s_h = o.value;
|
|
/* t_h=ax+bp[k] High */
|
|
t_h = ax + bp[k];
|
|
o.value = t_h;
|
|
o.words32.w3 = 0;
|
|
o.words32.w2 &= 0xf8000000;
|
|
t_h = o.value;
|
|
t_l = ax - (t_h - bp[k]);
|
|
s_l = v * ((u - s_h * t_h) - s_h * t_l);
|
|
/* compute log(ax) */
|
|
s2 = s * s;
|
|
u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4])));
|
|
v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2))));
|
|
r = s2 * s2 * u / v;
|
|
r += s_l * (s_h + s);
|
|
s2 = s_h * s_h;
|
|
t_h = 3.0 + s2 + r;
|
|
o.value = t_h;
|
|
o.words32.w3 = 0;
|
|
o.words32.w2 &= 0xf8000000;
|
|
t_h = o.value;
|
|
t_l = r - ((t_h - 3.0) - s2);
|
|
/* u+v = s*(1+...) */
|
|
u = s_h * t_h;
|
|
v = s_l * t_h + t_l * s;
|
|
/* 2/(3log2)*(s+...) */
|
|
p_h = u + v;
|
|
o.value = p_h;
|
|
o.words32.w3 = 0;
|
|
o.words32.w2 &= 0xf8000000;
|
|
p_h = o.value;
|
|
p_l = v - (p_h - u);
|
|
z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
|
|
z_l = cp_l * p_h + p_l * cp + dp_l[k];
|
|
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
|
|
t = (__float128) n;
|
|
t1 = (((z_h + z_l) + dp_h[k]) + t);
|
|
o.value = t1;
|
|
o.words32.w3 = 0;
|
|
o.words32.w2 &= 0xf8000000;
|
|
t1 = o.value;
|
|
t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
|
|
|
|
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
|
|
y1 = y;
|
|
o.value = y1;
|
|
o.words32.w3 = 0;
|
|
o.words32.w2 &= 0xf8000000;
|
|
y1 = o.value;
|
|
p_l = (y - y1) * t1 + y * t2;
|
|
p_h = y1 * t1;
|
|
z = p_l + p_h;
|
|
o.value = z;
|
|
j = o.words32.w0;
|
|
if (j >= 0x400d0000) /* z >= 16384 */
|
|
{
|
|
/* if z > 16384 */
|
|
if (((j - 0x400d0000) | o.words32.w1 | o.words32.w2 | o.words32.w3) != 0)
|
|
return sgn * huge * huge; /* overflow */
|
|
else
|
|
{
|
|
if (p_l + ovt > z - p_h)
|
|
return sgn * huge * huge; /* overflow */
|
|
}
|
|
}
|
|
else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */
|
|
{
|
|
/* z < -16495 */
|
|
if (((j - 0xc00d01bc) | o.words32.w1 | o.words32.w2 | o.words32.w3)
|
|
!= 0)
|
|
return sgn * tiny * tiny; /* underflow */
|
|
else
|
|
{
|
|
if (p_l <= z - p_h)
|
|
return sgn * tiny * tiny; /* underflow */
|
|
}
|
|
}
|
|
/* compute 2**(p_h+p_l) */
|
|
i = j & 0x7fffffff;
|
|
k = (i >> 16) - 0x3fff;
|
|
n = 0;
|
|
if (i > 0x3ffe0000)
|
|
{ /* if |z| > 0.5, set n = [z+0.5] */
|
|
n = floorq (z + 0.5Q);
|
|
t = n;
|
|
p_h -= t;
|
|
}
|
|
t = p_l + p_h;
|
|
o.value = t;
|
|
o.words32.w3 = 0;
|
|
o.words32.w2 &= 0xf8000000;
|
|
t = o.value;
|
|
u = t * lg2_h;
|
|
v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
|
|
z = u + v;
|
|
w = v - (z - u);
|
|
/* exp(z) */
|
|
t = z * z;
|
|
u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4])));
|
|
v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t)));
|
|
t1 = z - t * u / v;
|
|
r = (z * t1) / (t1 - two) - (w + z * w);
|
|
z = one - (r - z);
|
|
o.value = z;
|
|
j = o.words32.w0;
|
|
j += (n << 16);
|
|
if ((j >> 16) <= 0)
|
|
{
|
|
z = scalbnq (z, n); /* subnormal output */
|
|
__float128 force_underflow = z * z;
|
|
math_force_eval (force_underflow);
|
|
}
|
|
else
|
|
{
|
|
o.words32.w0 = j;
|
|
z = o.value;
|
|
}
|
|
return sgn * z;
|
|
}
|