compiler/stdc/math/e_pow.c

   1 /* @(#)e_pow.c 1.5 04/04/22 SMI */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
   5  *
   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 #ifndef lint
  13 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_pow.c,v 1.14 2011/10/21 06:26:07 das Exp $";
  14 #endif
  15
  16 /* __ieee754_pow(x,y) return x**y
  17  *
  18  *                    n
  19  * Method:  Let x =  2   * (1+f)
  20  *      1. Compute and return log2(x) in two pieces:
  21  *              log2(x) = w1 + w2,
  22  *         where w1 has 53-24 = 29 bit trailing zeros.
  23  *      2. Perform y*log2(x) = n+y' by simulating multi-precision
  24  *         arithmetic, where |y'|<=0.5.
  25  *      3. Return x**y = 2**n*exp(y'*log2)
  26  *
  27  * Special cases:
  28  *      1.  (anything) ** 0  is 1
  29  *      2.  (anything) ** 1  is itself
  30  *      3.  (anything) ** NAN is NAN except 1 ** NAN = 1
  31  *      4.  NAN ** (anything except 0) is NAN
  32  *      5.  +-(|x| > 1) **  +INF is +INF
  33  *      6.  +-(|x| > 1) **  -INF is +0
  34  *      7.  +-(|x| < 1) **  +INF is +0
  35  *      8.  +-(|x| < 1) **  -INF is +INF
  36  *      9.  +-1         ** +-INF is 1
  37  *      10. +0 ** (+anything except 0, NAN)               is +0
  38  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
  39  *      12. +0 ** (-anything except 0, NAN)               is +INF
  40  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  41  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  42  *      15. +INF ** (+anything except 0,NAN) is +INF
  43  *      16. +INF ** (-anything except 0,NAN) is +0
  44  *      17. -INF ** (anything)  = -0 ** (-anything)
  45  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  46  *      19. (-anything except 0 and inf) ** (non-integer) is NAN
  47  *
  48  * Accuracy:
  49  *      pow(x,y) returns x**y nearly rounded. In particular
  50  *                      pow(integer,integer)
  51  *      always returns the correct integer provided it is
  52  *      representable.
  53  *
  54  * Constants :
  55  * The hexadecimal values are the intended ones for the following
  56  * constants. The decimal values may be used, provided that the
  57  * compiler will convert from decimal to binary accurately enough
  58  * to produce the hexadecimal values shown.
  59  */
  60
  61 #include <float.h>
  62 #include "math.h"
  63 #include "math_private.h"
  64
  65 static const double
  66 bp[] = {1.0, 1.5,},
  67 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
  68 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
  69 zero    =  0.0,
  70 one     =  1.0,
  71 two     =  2.0,
  72 two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
  73 huge    =  1.0e300,
  74 tiny    =  1.0e-300,
  75         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  76 L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
  77 L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
  78 L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
  79 L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
  80 L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
  81 L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
  82 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  83 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  84 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  85 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  86 P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
  87 lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
  88 lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
  89 lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
  90 ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
  91 cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
  92 cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
  93 cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
  94 ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
  95 ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
  96 ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
  97
  98 double
  99 __ieee754_pow(double x, double y)
 100 {
 101         double z,ax,z_h,z_l,p_h,p_l;
 102         double y1,t1,t2,r,s,t,u,v,w;
 103         int32_t i,j,k,yisint,n;
 104         int32_t hx,hy,ix,iy;
 105         uint32_t lx,ly;
 106
 107         EXTRACT_WORDS(hx,lx,x);
 108         EXTRACT_WORDS(hy,ly,y);
 109         ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
 110
 111     /* y==zero: x**0 = 1 */
 112         if((iy|ly)==0) return one;
 113
 114     /* x==1: 1**y = 1, even if y is NaN */
 115         if (hx==0x3ff00000 && lx == 0) return one;
 116
 117     /* y!=zero: result is NaN if either arg is NaN */
 118         if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
 119            iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
 120                 return (x+0.0)+(y+0.0);
 121
 122     /* determine if y is an odd int when x < 0
 123      * yisint = 0       ... y is not an integer
 124      * yisint = 1       ... y is an odd int
 125      * yisint = 2       ... y is an even int
 126      */
 127         yisint  = 0;
 128         if(hx<0) {
 129             if(iy>=0x43400000) yisint = 2; /* even integer y */
 130             else if(iy>=0x3ff00000) {
 131                 k = (iy>>20)-0x3ff;        /* exponent */
 132                 if(k>20) {
 133                     j = ly>>(52-k);
 134                     if((j<<(52-k))==ly) yisint = 2-(j&1);
 135                 } else if(ly==0) {
 136                     j = iy>>(20-k);
 137                     if((j<<(20-k))==iy) yisint = 2-(j&1);
 138                 }
 139             }
 140         }
 141
 142     /* special value of y */
 143         if(ly==0) {
 144             if (iy==0x7ff00000) {       /* y is +-inf */
 145                 if(((ix-0x3ff00000)|lx)==0)
 146                     return  one;        /* (-1)**+-inf is 1 */
 147                 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
 148                     return (hy>=0)? y: zero;
 149                 else                    /* (|x|<1)**-,+inf = inf,0 */
 150                     return (hy<0)?-y: zero;
 151             }
 152             if(iy==0x3ff00000) {        /* y is  +-1 */
 153                 if(hy<0) return one/x; else return x;
 154             }
 155             if(hy==0x40000000) return x*x;   /* y is  2 */
 156             if(hy==0x3fe00000) {             /* y is  0.5 */
 157                 if(hx>=0)       /* x >= +0 */
 158                     return sqrt(x);
 159             }
 160         }
 161
 162         ax   = fabs(x);
 163     /* special value of x */
 164         if(lx==0) {
 165             if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
 166                 z = ax;                 /*x is +-0,+-inf,+-1*/
 167                 if(hy<0) z = one/z;     /* z = (1/|x|) */
 168                 if(hx<0) {
 169                     if(((ix-0x3ff00000)|yisint)==0) {
 170                         z = (z-z)/(z-z); /* (-1)**non-int is NaN */
 171                     } else if(yisint==1)
 172                         z = -z;         /* (x<0)**odd = -(|x|**odd) */
 173                 }
 174                 return z;
 175             }
 176         }
 177
 178     /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
 179         n = (hx>>31)+1;
 180        but ANSI C says a right shift of a signed negative quantity is
 181        implementation defined.  */
 182         n = ((uint32_t)hx>>31)-1;
 183
 184     /* (x<0)**(non-int) is NaN */
 185         if((n|yisint)==0) return (x-x)/(x-x);
 186
 187         s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
 188         if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
 189
 190     /* |y| is huge */
 191         if(iy>0x41e00000) { /* if |y| > 2**31 */
 192             if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
 193                 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
 194                 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
 195             }
 196         /* over/underflow if x is not close to one */
 197             if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
 198             if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
 199         /* now |1-x| is tiny <= 2**-20, suffice to compute
 200            log(x) by x-x^2/2+x^3/3-x^4/4 */
 201             t = ax-one;         /* t has 20 trailing zeros */
 202             w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
 203             u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
 204             v = t*ivln2_l-w*ivln2;
 205             t1 = u+v;
 206             SET_LOW_WORD(t1,0);
 207             t2 = v-(t1-u);
 208         } else {
 209             double ss,s2,s_h,s_l,t_h,t_l;
 210             n = 0;
 211         /* take care subnormal number */
 212             if(ix<0x00100000)
 213                 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
 214             n  += ((ix)>>20)-0x3ff;
 215             j  = ix&0x000fffff;
 216         /* determine interval */
 217             ix = j|0x3ff00000;          /* normalize ix */
 218             if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
 219             else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
 220             else {k=0;n+=1;ix -= 0x00100000;}
 221             SET_HIGH_WORD(ax,ix);
 222
 223         /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 224             u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
 225             v = one/(ax+bp[k]);
 226             ss = u*v;
 227             s_h = ss;
 228             SET_LOW_WORD(s_h,0);
 229         /* t_h=ax+bp[k] High */
 230             t_h = zero;
 231             SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
 232             t_l = ax - (t_h-bp[k]);
 233             s_l = v*((u-s_h*t_h)-s_h*t_l);
 234         /* compute log(ax) */
 235             s2 = ss*ss;
 236             r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
 237             r += s_l*(s_h+ss);
 238             s2  = s_h*s_h;
 239             t_h = 3.0+s2+r;
 240             SET_LOW_WORD(t_h,0);
 241             t_l = r-((t_h-3.0)-s2);
 242         /* u+v = ss*(1+...) */
 243             u = s_h*t_h;
 244             v = s_l*t_h+t_l*ss;
 245         /* 2/(3log2)*(ss+...) */
 246             p_h = u+v;
 247             SET_LOW_WORD(p_h,0);
 248             p_l = v-(p_h-u);
 249             z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
 250             z_l = cp_l*p_h+p_l*cp+dp_l[k];
 251         /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 252             t = (double)n;
 253             t1 = (((z_h+z_l)+dp_h[k])+t);
 254             SET_LOW_WORD(t1,0);
 255             t2 = z_l-(((t1-t)-dp_h[k])-z_h);
 256         }
 257
 258     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
 259         y1  = y;
 260         SET_LOW_WORD(y1,0);
 261         p_l = (y-y1)*t1+y*t2;
 262         p_h = y1*t1;
 263         z = p_l+p_h;
 264         EXTRACT_WORDS(j,i,z);
 265         if (j>=0x40900000) {                            /* z >= 1024 */
 266             if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
 267                 return s*huge*huge;                     /* overflow */
 268             else {
 269                 if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
 270             }
 271         } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */
 272             if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
 273                 return s*tiny*tiny;             /* underflow */
 274             else {
 275                 if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
 276             }
 277         }
 278     /*
 279      * compute 2**(p_h+p_l)
 280      */
 281         i = j&0x7fffffff;
 282         k = (i>>20)-0x3ff;
 283         n = 0;
 284         if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */
 285             n = j+(0x00100000>>(k+1));
 286             k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
 287             t = zero;
 288             SET_HIGH_WORD(t,n&~(0x000fffff>>k));
 289             n = ((n&0x000fffff)|0x00100000)>>(20-k);
 290             if(j<0) n = -n;
 291             p_h -= t;
 292         }
 293         t = p_l+p_h;
 294         SET_LOW_WORD(t,0);
 295         u = t*lg2_h;
 296         v = (p_l-(t-p_h))*lg2+t*lg2_l;
 297         z = u+v;
 298         w = v-(z-u);
 299         t  = z*z;
 300         t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 301         r  = (z*t1)/(t1-two)-(w+z*w);
 302         z  = one-(r-z);
 303         GET_HIGH_WORD(j,z);
 304         j += (n<<20);
 305         if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
 306         else SET_HIGH_WORD(z,j);
 307         return s*z;
 308 }
 309
 310 #if     (LDBL_MANT_DIG == DBL_MANT_DIG)
 311 AROS_MAKE_ASM_SYM(typeof(powl), powl, AROS_CSYM_FROM_ASM_NAME(powl), AROS_CSYM_FROM_ASM_NAME(pow));
 312 AROS_EXPORT_ASM_SYM(AROS_CSYM_FROM_ASM_NAME(powl));
 313 #endif