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      1 /*-
      2  * ====================================================
      3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
      4  *
      5  * Developed at SunPro, a Sun Microsystems, Inc. business.
      6  * Permission to use, copy, modify, and distribute this
      7  * software is freely granted, provided that this notice
      8  * is preserved.
      9  * ====================================================
     10  */
     11 
     12 /*
     13  * Copyright (c) 2008 Stephen L. Moshier <steve (at) moshier.net>
     14  *
     15  * Permission to use, copy, modify, and distribute this software for any
     16  * purpose with or without fee is hereby granted, provided that the above
     17  * copyright notice and this permission notice appear in all copies.
     18  *
     19  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
     20  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
     21  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
     22  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
     23  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
     24  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
     25  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
     26  */
     27 
     28 /* powl(x,y) return x**y
     29  *
     30  *		      n
     31  * Method:  Let x =  2   * (1+f)
     32  *	1. Compute and return log2(x) in two pieces:
     33  *		log2(x) = w1 + w2,
     34  *	   where w1 has 113-53 = 60 bit trailing zeros.
     35  *	2. Perform y*log2(x) = n+y' by simulating multi-precision
     36  *	   arithmetic, where |y'|<=0.5.
     37  *	3. Return x**y = 2**n*exp(y'*log2)
     38  *
     39  * Special cases:
     40  *	1.  (anything) ** 0  is 1
     41  *	2.  (anything) ** 1  is itself
     42  *	3.  (anything) ** NAN is NAN
     43  *	4.  NAN ** (anything except 0) is NAN
     44  *	5.  +-(|x| > 1) **  +INF is +INF
     45  *	6.  +-(|x| > 1) **  -INF is +0
     46  *	7.  +-(|x| < 1) **  +INF is +0
     47  *	8.  +-(|x| < 1) **  -INF is +INF
     48  *	9.  +-1         ** +-INF is NAN
     49  *	10. +0 ** (+anything except 0, NAN)               is +0
     50  *	11. -0 ** (+anything except 0, NAN, odd integer)  is +0
     51  *	12. +0 ** (-anything except 0, NAN)               is +INF
     52  *	13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
     53  *	14. -0 ** (odd integer) = -( +0 ** (odd integer) )
     54  *	15. +INF ** (+anything except 0,NAN) is +INF
     55  *	16. +INF ** (-anything except 0,NAN) is +0
     56  *	17. -INF ** (anything)  = -0 ** (-anything)
     57  *	18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
     58  *	19. (-anything except 0 and inf) ** (non-integer) is NAN
     59  *
     60  */
     61 
     62 #include <sys/cdefs.h>
     63 __FBSDID("$FreeBSD: head/lib/msun/ld128/e_powl.c 342651 2018-12-31 15:43:06Z pfg $");
     64 
     65 #include <float.h>
     66 #include <math.h>
     67 
     68 #include "math_private.h"
     69 
     70 static const long double bp[] = {
     71   1.0L,
     72   1.5L,
     73 };
     74 
     75 /* log_2(1.5) */
     76 static const long double dp_h[] = {
     77   0.0,
     78   5.8496250072115607565592654282227158546448E-1L
     79 };
     80 
     81 /* Low part of log_2(1.5) */
     82 static const long double dp_l[] = {
     83   0.0,
     84   1.0579781240112554492329533686862998106046E-16L
     85 };
     86 
     87 static const long double zero = 0.0L,
     88   one = 1.0L,
     89   two = 2.0L,
     90   two113 = 1.0384593717069655257060992658440192E34L,
     91   huge = 1.0e3000L,
     92   tiny = 1.0e-3000L;
     93 
     94 /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
     95    z = (x-1)/(x+1)
     96    1 <= x <= 1.25
     97    Peak relative error 2.3e-37 */
     98 static const long double LN[] =
     99 {
    100  -3.0779177200290054398792536829702930623200E1L,
    101   6.5135778082209159921251824580292116201640E1L,
    102  -4.6312921812152436921591152809994014413540E1L,
    103   1.2510208195629420304615674658258363295208E1L,
    104  -9.9266909031921425609179910128531667336670E-1L
    105 };
    106 static const long double LD[] =
    107 {
    108  -5.129862866715009066465422805058933131960E1L,
    109   1.452015077564081884387441590064272782044E2L,
    110  -1.524043275549860505277434040464085593165E2L,
    111   7.236063513651544224319663428634139768808E1L,
    112  -1.494198912340228235853027849917095580053E1L
    113   /* 1.0E0 */
    114 };
    115 
    116 /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
    117    0 <= x <= 0.5
    118    Peak relative error 5.7e-38  */
    119 static const long double PN[] =
    120 {
    121   5.081801691915377692446852383385968225675E8L,
    122   9.360895299872484512023336636427675327355E6L,
    123   4.213701282274196030811629773097579432957E4L,
    124   5.201006511142748908655720086041570288182E1L,
    125   9.088368420359444263703202925095675982530E-3L,
    126 };
    127 static const long double PD[] =
    128 {
    129   3.049081015149226615468111430031590411682E9L,
    130   1.069833887183886839966085436512368982758E8L,
    131   8.259257717868875207333991924545445705394E5L,
    132   1.872583833284143212651746812884298360922E3L,
    133   /* 1.0E0 */
    134 };
    135 
    136 static const long double
    137   /* ln 2 */
    138   lg2 = 6.9314718055994530941723212145817656807550E-1L,
    139   lg2_h = 6.9314718055994528622676398299518041312695E-1L,
    140   lg2_l = 2.3190468138462996154948554638754786504121E-17L,
    141   ovt = 8.0085662595372944372e-0017L,
    142   /* 2/(3*log(2)) */
    143   cp = 9.6179669392597560490661645400126142495110E-1L,
    144   cp_h = 9.6179669392597555432899980587535537779331E-1L,
    145   cp_l = 5.0577616648125906047157785230014751039424E-17L;
    146 
    147 long double
    148 powl(long double x, long double y)
    149 {
    150   long double z, ax, z_h, z_l, p_h, p_l;
    151   long double yy1, t1, t2, r, s, t, u, v, w;
    152   long double s2, s_h, s_l, t_h, t_l;
    153   int32_t i, j, k, yisint, n;
    154   u_int32_t ix, iy;
    155   int32_t hx, hy;
    156   ieee_quad_shape_type o, p, q;
    157 
    158   p.value = x;
    159   hx = p.parts32.mswhi;
    160   ix = hx & 0x7fffffff;
    161 
    162   q.value = y;
    163   hy = q.parts32.mswhi;
    164   iy = hy & 0x7fffffff;
    165 
    166 
    167   /* y==zero: x**0 = 1 */
    168   if ((iy | q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
    169     return one;
    170 
    171   /* 1.0**y = 1; -1.0**+-Inf = 1 */
    172   if (x == one)
    173     return one;
    174   if (x == -1.0L && iy == 0x7fff0000
    175       && (q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
    176     return one;
    177 
    178   /* +-NaN return x+y */
    179   if ((ix > 0x7fff0000)
    180       || ((ix == 0x7fff0000)
    181 	  && ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) != 0))
    182       || (iy > 0x7fff0000)
    183       || ((iy == 0x7fff0000)
    184 	  && ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) != 0)))
    185     return nan_mix(x, y);
    186 
    187   /* determine if y is an odd int when x < 0
    188    * yisint = 0       ... y is not an integer
    189    * yisint = 1       ... y is an odd int
    190    * yisint = 2       ... y is an even int
    191    */
    192   yisint = 0;
    193   if (hx < 0)
    194     {
    195       if (iy >= 0x40700000)	/* 2^113 */
    196 	yisint = 2;		/* even integer y */
    197       else if (iy >= 0x3fff0000)	/* 1.0 */
    198 	{
    199 	  if (floorl (y) == y)
    200 	    {
    201 	      z = 0.5 * y;
    202 	      if (floorl (z) == z)
    203 		yisint = 2;
    204 	      else
    205 		yisint = 1;
    206 	    }
    207 	}
    208     }
    209 
    210   /* special value of y */
    211   if ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
    212     {
    213       if (iy == 0x7fff0000)	/* y is +-inf */
    214 	{
    215 	  if (((ix - 0x3fff0000) | p.parts32.mswlo | p.parts32.lswhi |
    216 	    p.parts32.lswlo) == 0)
    217 	    return y - y;	/* +-1**inf is NaN */
    218 	  else if (ix >= 0x3fff0000)	/* (|x|>1)**+-inf = inf,0 */
    219 	    return (hy >= 0) ? y : zero;
    220 	  else			/* (|x|<1)**-,+inf = inf,0 */
    221 	    return (hy < 0) ? -y : zero;
    222 	}
    223       if (iy == 0x3fff0000)
    224 	{			/* y is  +-1 */
    225 	  if (hy < 0)
    226 	    return one / x;
    227 	  else
    228 	    return x;
    229 	}
    230       if (hy == 0x40000000)
    231 	return x * x;		/* y is  2 */
    232       if (hy == 0x3ffe0000)
    233 	{			/* y is  0.5 */
    234 	  if (hx >= 0)		/* x >= +0 */
    235 	    return sqrtl (x);
    236 	}
    237     }
    238 
    239   ax = fabsl (x);
    240   /* special value of x */
    241   if ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0)
    242     {
    243       if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000)
    244 	{
    245 	  z = ax;		/*x is +-0,+-inf,+-1 */
    246 	  if (hy < 0)
    247 	    z = one / z;	/* z = (1/|x|) */
    248 	  if (hx < 0)
    249 	    {
    250 	      if (((ix - 0x3fff0000) | yisint) == 0)
    251 		{
    252 		  z = (z - z) / (z - z);	/* (-1)**non-int is NaN */
    253 		}
    254 	      else if (yisint == 1)
    255 		z = -z;		/* (x<0)**odd = -(|x|**odd) */
    256 	    }
    257 	  return z;
    258 	}
    259     }
    260 
    261   /* (x<0)**(non-int) is NaN */
    262   if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
    263     return (x - x) / (x - x);
    264 
    265   /* |y| is huge.
    266      2^-16495 = 1/2 of smallest representable value.
    267      If (1 - 1/131072)^y underflows, y > 1.4986e9 */
    268   if (iy > 0x401d654b)
    269     {
    270       /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
    271       if (iy > 0x407d654b)
    272 	{
    273 	  if (ix <= 0x3ffeffff)
    274 	    return (hy < 0) ? huge * huge : tiny * tiny;
    275 	  if (ix >= 0x3fff0000)
    276 	    return (hy > 0) ? huge * huge : tiny * tiny;
    277 	}
    278       /* over/underflow if x is not close to one */
    279       if (ix < 0x3ffeffff)
    280 	return (hy < 0) ? huge * huge : tiny * tiny;
    281       if (ix > 0x3fff0000)
    282 	return (hy > 0) ? huge * huge : tiny * tiny;
    283     }
    284 
    285   n = 0;
    286   /* take care subnormal number */
    287   if (ix < 0x00010000)
    288     {
    289       ax *= two113;
    290       n -= 113;
    291       o.value = ax;
    292       ix = o.parts32.mswhi;
    293     }
    294   n += ((ix) >> 16) - 0x3fff;
    295   j = ix & 0x0000ffff;
    296   /* determine interval */
    297   ix = j | 0x3fff0000;		/* normalize ix */
    298   if (j <= 0x3988)
    299     k = 0;			/* |x|<sqrt(3/2) */
    300   else if (j < 0xbb67)
    301     k = 1;			/* |x|<sqrt(3)   */
    302   else
    303     {
    304       k = 0;
    305       n += 1;
    306       ix -= 0x00010000;
    307     }
    308 
    309   o.value = ax;
    310   o.parts32.mswhi = ix;
    311   ax = o.value;
    312 
    313   /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
    314   u = ax - bp[k];		/* bp[0]=1.0, bp[1]=1.5 */
    315   v = one / (ax + bp[k]);
    316   s = u * v;
    317   s_h = s;
    318 
    319   o.value = s_h;
    320   o.parts32.lswlo = 0;
    321   o.parts32.lswhi &= 0xf8000000;
    322   s_h = o.value;
    323   /* t_h=ax+bp[k] High */
    324   t_h = ax + bp[k];
    325   o.value = t_h;
    326   o.parts32.lswlo = 0;
    327   o.parts32.lswhi &= 0xf8000000;
    328   t_h = o.value;
    329   t_l = ax - (t_h - bp[k]);
    330   s_l = v * ((u - s_h * t_h) - s_h * t_l);
    331   /* compute log(ax) */
    332   s2 = s * s;
    333   u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4])));
    334   v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2))));
    335   r = s2 * s2 * u / v;
    336   r += s_l * (s_h + s);
    337   s2 = s_h * s_h;
    338   t_h = 3.0 + s2 + r;
    339   o.value = t_h;
    340   o.parts32.lswlo = 0;
    341   o.parts32.lswhi &= 0xf8000000;
    342   t_h = o.value;
    343   t_l = r - ((t_h - 3.0) - s2);
    344   /* u+v = s*(1+...) */
    345   u = s_h * t_h;
    346   v = s_l * t_h + t_l * s;
    347   /* 2/(3log2)*(s+...) */
    348   p_h = u + v;
    349   o.value = p_h;
    350   o.parts32.lswlo = 0;
    351   o.parts32.lswhi &= 0xf8000000;
    352   p_h = o.value;
    353   p_l = v - (p_h - u);
    354   z_h = cp_h * p_h;		/* cp_h+cp_l = 2/(3*log2) */
    355   z_l = cp_l * p_h + p_l * cp + dp_l[k];
    356   /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
    357   t = (long double) n;
    358   t1 = (((z_h + z_l) + dp_h[k]) + t);
    359   o.value = t1;
    360   o.parts32.lswlo = 0;
    361   o.parts32.lswhi &= 0xf8000000;
    362   t1 = o.value;
    363   t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
    364 
    365   /* s (sign of result -ve**odd) = -1 else = 1 */
    366   s = one;
    367   if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
    368     s = -one;			/* (-ve)**(odd int) */
    369 
    370   /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
    371   yy1 = y;
    372   o.value = yy1;
    373   o.parts32.lswlo = 0;
    374   o.parts32.lswhi &= 0xf8000000;
    375   yy1 = o.value;
    376   p_l = (y - yy1) * t1 + y * t2;
    377   p_h = yy1 * t1;
    378   z = p_l + p_h;
    379   o.value = z;
    380   j = o.parts32.mswhi;
    381   if (j >= 0x400d0000) /* z >= 16384 */
    382     {
    383       /* if z > 16384 */
    384       if (((j - 0x400d0000) | o.parts32.mswlo | o.parts32.lswhi |
    385 	o.parts32.lswlo) != 0)
    386 	return s * huge * huge;	/* overflow */
    387       else
    388 	{
    389 	  if (p_l + ovt > z - p_h)
    390 	    return s * huge * huge;	/* overflow */
    391 	}
    392     }
    393   else if ((j & 0x7fffffff) >= 0x400d01b9)	/* z <= -16495 */
    394     {
    395       /* z < -16495 */
    396       if (((j - 0xc00d01bc) | o.parts32.mswlo | o.parts32.lswhi |
    397 	o.parts32.lswlo)
    398 	  != 0)
    399 	return s * tiny * tiny;	/* underflow */
    400       else
    401 	{
    402 	  if (p_l <= z - p_h)
    403 	    return s * tiny * tiny;	/* underflow */
    404 	}
    405     }
    406   /* compute 2**(p_h+p_l) */
    407   i = j & 0x7fffffff;
    408   k = (i >> 16) - 0x3fff;
    409   n = 0;
    410   if (i > 0x3ffe0000)
    411     {				/* if |z| > 0.5, set n = [z+0.5] */
    412       n = floorl (z + 0.5L);
    413       t = n;
    414       p_h -= t;
    415     }
    416   t = p_l + p_h;
    417   o.value = t;
    418   o.parts32.lswlo = 0;
    419   o.parts32.lswhi &= 0xf8000000;
    420   t = o.value;
    421   u = t * lg2_h;
    422   v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
    423   z = u + v;
    424   w = v - (z - u);
    425   /*  exp(z) */
    426   t = z * z;
    427   u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4])));
    428   v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t)));
    429   t1 = z - t * u / v;
    430   r = (z * t1) / (t1 - two) - (w + z * w);
    431   z = one - (r - z);
    432   o.value = z;
    433   j = o.parts32.mswhi;
    434   j += (n << 16);
    435   if ((j >> 16) <= 0)
    436     z = scalbnl (z, n);	/* subnormal output */
    437   else
    438     {
    439       o.parts32.mswhi = j;
    440       z = o.value;
    441     }
    442   return s * z;
    443 }
    444