StdLib/LibC/Math/e_pow.c

   1 /** @file
   2   Compute the base 10 logrithm of x.
   3
   4   Copyright (c) 2010 - 2011, Intel Corporation. All rights reserved.<BR>
   5   This program and the accompanying materials are licensed and made available under
   6   the terms and conditions of the BSD License that accompanies this distribution.
   7   The full text of the license may be found at
   8   http://opensource.org/licenses/bsd-license.
   9
  10   THE PROGRAM IS DISTRIBUTED UNDER THE BSD LICENSE ON AN "AS IS" BASIS,
  11   WITHOUT WARRANTIES OR REPRESENTATIONS OF ANY KIND, EITHER EXPRESS OR IMPLIED.
  12
  13  * ====================================================
  14  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  15  *
  16  * Developed at SunPro, a Sun Microsystems, Inc. business.
  17  * Permission to use, copy, modify, and distribute this
  18  * software is freely granted, provided that this notice
  19  * is preserved.
  20  * ====================================================
  21
  22   e_pow.c 5.1 93/09/24
  23   NetBSD: e_pow.c,v 1.13 2004/06/30 18:43:15 drochner Exp
  24 **/
  25 #include  <LibConfig.h>
  26 #include  <sys/EfiCdefs.h>
  27
  28 #if defined(_MSC_VER)           /* Handle Microsoft VC++ compiler specifics. */
  29   // C4723: potential divide by zero.
  30   #pragma warning ( disable : 4723 )
  31   // C4756: overflow in constant arithmetic
  32   #pragma warning ( disable : 4756 )
  33 #endif
  34
  35 /* __ieee754_pow(x,y) return x**y
  36  *
  37  *          n
  38  * Method:  Let x =  2   * (1+f)
  39  *  1. Compute and return log2(x) in two pieces:
  40  *    log2(x) = w1 + w2,
  41  *     where w1 has 53-24 = 29 bit trailing zeros.
  42  *  2. Perform y*log2(x) = n+y' by simulating multi-precision
  43  *     arithmetic, where |y'|<=0.5.
  44  *  3. Return x**y = 2**n*exp(y'*log2)
  45  *
  46  * Special cases:
  47  *  1.  (anything) ** 0  is 1
  48  *  2.  (anything) ** 1  is itself
  49  *  3.  (anything) ** NAN is NAN
  50  *  4.  NAN ** (anything except 0) is NAN
  51  *  5.  +-(|x| > 1) **  +INF is +INF
  52  *  6.  +-(|x| > 1) **  -INF is +0
  53  *  7.  +-(|x| < 1) **  +INF is +0
  54  *  8.  +-(|x| < 1) **  -INF is +INF
  55  *  9.  +-1         ** +-INF is NAN
  56  *  10. +0 ** (+anything except 0, NAN)               is +0
  57  *  11. -0 ** (+anything except 0, NAN, odd integer)  is +0
  58  *  12. +0 ** (-anything except 0, NAN)               is +INF
  59  *  13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  60  *  14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  61  *  15. +INF ** (+anything except 0,NAN) is +INF
  62  *  16. +INF ** (-anything except 0,NAN) is +0
  63  *  17. -INF ** (anything)  = -0 ** (-anything)
  64  *  18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  65  *  19. (-anything except 0 and inf) ** (non-integer) is NAN
  66  *
  67  * Accuracy:
  68  *  pow(x,y) returns x**y nearly rounded. In particular
  69  *      pow(integer,integer)
  70  *  always returns the correct integer provided it is
  71  *  representable.
  72  *
  73  * Constants :
  74  * The hexadecimal values are the intended ones for the following
  75  * constants. The decimal values may be used, provided that the
  76  * compiler will convert from decimal to binary accurately enough
  77  * to produce the hexadecimal values shown.
  78  */
  79
  80 #include "math.h"
  81 #include "math_private.h"
  82 #include  <errno.h>
  83
  84 static const double
  85 bp[] = {1.0, 1.5,},
  86 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
  87 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
  88 zero    =  0.0,
  89 one =  1.0,
  90 two =  2.0,
  91 two53 =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
  92 huge  =  1.0e300,
  93 tiny    =  1.0e-300,
  94   /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  95 L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
  96 L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
  97 L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
  98 L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
  99 L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
 100 L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
 101 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
 102 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
 103 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
 104 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
 105 P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
 106 lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
 107 lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
 108 lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
 109 ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
 110 cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
 111 cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
 112 cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
 113 ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
 114 ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
 115 ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
 116
 117 double
 118 __ieee754_pow(double x, double y)
 119 {
 120   double z,ax,z_h,z_l,p_h,p_l;
 121   double y1,t1,t2,r,s,t,u,v,w;
 122   int32_t i,j,k,yisint,n;
 123   int32_t hx,hy,ix,iy;
 124   u_int32_t lx,ly;
 125
 126   EXTRACT_WORDS(hx,lx,x);
 127   EXTRACT_WORDS(hy,ly,y);
 128   ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
 129
 130     /* y==zero: x**0 = 1 */
 131   if((iy|ly)==0) return one;
 132
 133     /* +-NaN return x+y */
 134   if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
 135      iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
 136     return x+y;
 137
 138     /* determine if y is an odd int when x < 0
 139      * yisint = 0 ... y is not an integer
 140      * yisint = 1 ... y is an odd int
 141      * yisint = 2 ... y is an even int
 142      */
 143   yisint  = 0;
 144   if(hx<0) {
 145       if(iy>=0x43400000) yisint = 2; /* even integer y */
 146       else if(iy>=0x3ff00000) {
 147     k = (iy>>20)-0x3ff;    /* exponent */
 148     if(k>20) {
 149         j = ly>>(52-k);
 150         if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
 151     } else if(ly==0) {
 152         j = iy>>(20-k);
 153         if((j<<(20-k))==iy) yisint = 2-(j&1);
 154     }
 155       }
 156   }
 157
 158     /* special value of y */
 159   if(ly==0) {
 160       if (iy==0x7ff00000) { /* y is +-inf */
 161           if(((ix-0x3ff00000)|lx)==0)
 162         return  y - y;  /* inf**+-1 is NaN */
 163           else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
 164         return (hy>=0)? y: zero;
 165           else      /* (|x|<1)**-,+inf = inf,0 */
 166         return (hy<0)?-y: zero;
 167       }
 168       if(iy==0x3ff00000) {  /* y is  +-1 */
 169     if(hy<0) return one/x; else return x;
 170       }
 171       if(hy==0x40000000) return x*x; /* y is  2 */
 172       if(hy==0x3fe00000) {  /* y is  0.5 */
 173     if(hx>=0) /* x >= +0 */
 174     return __ieee754_sqrt(x);
 175       }
 176   }
 177
 178   ax   = fabs(x);
 179     /* special value of x */
 180   if(lx==0) {
 181       if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
 182     z = ax;     /*x is +-0,+-inf,+-1*/
 183     if(hy<0) z = one/z; /* z = (1/|x|) */
 184     if(hx<0) {
 185         if(((ix-0x3ff00000)|yisint)==0) {
 186       z = (z-z)/(z-z); /* (-1)**non-int is NaN */
 187         } else if(yisint==1)
 188       z = -z;   /* (x<0)**odd = -(|x|**odd) */
 189     }
 190     return z;
 191       }
 192   }
 193
 194   n = (hx>>31)+1;
 195
 196     /* (x<0)**(non-int) is NaN */
 197     if((n|yisint)==0) {
 198       errno = EDOM;
 199       return (x-x)/(x-x);
 200     }
 201
 202   s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
 203   if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
 204
 205     /* |y| is huge */
 206   if(iy>0x41e00000) { /* if |y| > 2**31 */
 207       if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
 208     if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
 209     if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
 210       }
 211   /* over/underflow if x is not close to one */
 212       if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
 213       if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
 214   /* now |1-x| is tiny <= 2**-20, suffice to compute
 215      log(x) by x-x^2/2+x^3/3-x^4/4 */
 216       t = ax-one;   /* t has 20 trailing zeros */
 217       w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
 218       u = ivln2_h*t;  /* ivln2_h has 21 sig. bits */
 219       v = t*ivln2_l-w*ivln2;
 220       t1 = u+v;
 221       SET_LOW_WORD(t1,0);
 222       t2 = v-(t1-u);
 223   } else {
 224       double ss,s2,s_h,s_l,t_h,t_l;
 225       n = 0;
 226   /* take care subnormal number */
 227       if(ix<0x00100000)
 228     {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
 229       n  += ((ix)>>20)-0x3ff;
 230       j  = ix&0x000fffff;
 231   /* determine interval */
 232       ix = j|0x3ff00000;    /* normalize ix */
 233       if(j<=0x3988E) k=0;   /* |x|<sqrt(3/2) */
 234       else if(j<0xBB67A) k=1; /* |x|<sqrt(3)   */
 235       else {k=0;n+=1;ix -= 0x00100000;}
 236       SET_HIGH_WORD(ax,ix);
 237
 238   /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 239       u = ax-bp[k];   /* bp[0]=1.0, bp[1]=1.5 */
 240       v = one/(ax+bp[k]);
 241       ss = u*v;
 242       s_h = ss;
 243       SET_LOW_WORD(s_h,0);
 244   /* t_h=ax+bp[k] High */
 245       t_h = zero;
 246       SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
 247       t_l = ax - (t_h-bp[k]);
 248       s_l = v*((u-s_h*t_h)-s_h*t_l);
 249   /* compute log(ax) */
 250       s2 = ss*ss;
 251       r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
 252       r += s_l*(s_h+ss);
 253       s2  = s_h*s_h;
 254       t_h = 3.0+s2+r;
 255       SET_LOW_WORD(t_h,0);
 256       t_l = r-((t_h-3.0)-s2);
 257   /* u+v = ss*(1+...) */
 258       u = s_h*t_h;
 259       v = s_l*t_h+t_l*ss;
 260   /* 2/(3log2)*(ss+...) */
 261       p_h = u+v;
 262       SET_LOW_WORD(p_h,0);
 263       p_l = v-(p_h-u);
 264       z_h = cp_h*p_h;   /* cp_h+cp_l = 2/(3*log2) */
 265       z_l = cp_l*p_h+p_l*cp+dp_l[k];
 266   /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 267       t = (double)n;
 268       t1 = (((z_h+z_l)+dp_h[k])+t);
 269       SET_LOW_WORD(t1,0);
 270       t2 = z_l-(((t1-t)-dp_h[k])-z_h);
 271   }
 272
 273     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
 274   y1  = y;
 275   SET_LOW_WORD(y1,0);
 276   p_l = (y-y1)*t1+y*t2;
 277   p_h = y1*t1;
 278   z = p_l+p_h;
 279   EXTRACT_WORDS(j,i,z);
 280   if (j>=0x40900000) {        /* z >= 1024 */
 281       if(((j-0x40900000)|i)!=0)     /* if z > 1024 */
 282     return s*huge*huge;     /* overflow */
 283       else {
 284     if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
 285       }
 286   } else if((j&0x7fffffff)>=0x4090cc00 ) {  /* z <= -1075 */
 287       if(((j-0xc090cc00)|i)!=0)     /* z < -1075 */
 288     return s*tiny*tiny;   /* underflow */
 289       else {
 290     if(p_l<=z-p_h) return s*tiny*tiny;  /* underflow */
 291       }
 292   }
 293     /*
 294      * compute 2**(p_h+p_l)
 295      */
 296   i = j&0x7fffffff;
 297   k = (i>>20)-0x3ff;
 298   n = 0;
 299   if(i>0x3fe00000) {    /* if |z| > 0.5, set n = [z+0.5] */
 300       n = j+(0x00100000>>(k+1));
 301       k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
 302       t = zero;
 303       SET_HIGH_WORD(t,n&~(0x000fffff>>k));
 304       n = ((n&0x000fffff)|0x00100000)>>(20-k);
 305       if(j<0) n = -n;
 306       p_h -= t;
 307   }
 308   t = p_l+p_h;
 309   SET_LOW_WORD(t,0);
 310   u = t*lg2_h;
 311   v = (p_l-(t-p_h))*lg2+t*lg2_l;
 312   z = u+v;
 313   w = v-(z-u);
 314   t  = z*z;
 315   t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 316   r  = (z*t1)/(t1-two)-(w+z*w);
 317   z  = one-(r-z);
 318   GET_HIGH_WORD(j,z);
 319   j += (n<<20);
 320   if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
 321   else SET_HIGH_WORD(z,j);
 322   return s*z;
 323 }