gonzui

    1: /* @(#)e_pow.c 5.1 93/09/24 */
    2: /*
    3:  * ====================================================
    4:  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
    5:  *
    6:  * Developed at SunPro, a Sun Microsystems, Inc. business.
    7:  * Permission to use, copy, modify, and distribute this
    8:  * software is freely granted, provided that this notice
    9:  * is preserved.
   10:  * ====================================================
   11:  */
   12: 
   13: /* LINTLIBRARY */
   14: 
   15: /* pow(x,y) return x**y
   16:  *
   17:  *                    n
   18:  * Method:  Let x =  2   * (1+f)
   19:  *      1. Compute and return log2(x) in two pieces:
   20:  *              log2(x) = w1 + w2,
   21:  *         where w1 has 53-24 = 29 bit trailing zeros.
   22:  *      2. Perform y*log2(x) = n+y' by simulating multi-precision
   23:  *         arithmetic, where |y'|<=0.5.
   24:  *      3. Return x**y = 2**n*exp(y'*log2)
   25:  *
   26:  * Special cases:
   27:  *      1.  (anything) ** 0  is 1
   28:  *      2.  (anything) ** 1  is itself
   29:  *      3.  (anything) ** NAN is NAN
   30:  *      4.  NAN ** (anything except 0) is NAN
   31:  *      5.  +-(|x| > 1) **  +INF is +INF
   32:  *      6.  +-(|x| > 1) **  -INF is +0
   33:  *      7.  +-(|x| < 1) **  +INF is +0
   34:  *      8.  +-(|x| < 1) **  -INF is +INF
   35:  *      9.  +-1         ** +-INF is NAN
   36:  *      10. +0 ** (+anything except 0, NAN)               is +0
   37:  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
   38:  *      12. +0 ** (-anything except 0, NAN)               is +INF
   39:  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
   40:  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
   41:  *      15. +INF ** (+anything except 0,NAN) is +INF
   42:  *      16. +INF ** (-anything except 0,NAN) is +0
   43:  *      17. -INF ** (anything)  = -0 ** (-anything)
   44:  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
   45:  *      19. (-anything except 0 and inf) ** (non-integer) is NAN
   46:  *
   47:  * Accuracy:
   48:  *      pow(x,y) returns x**y nearly rounded. In particular
   49:  *                      pow(integer,integer)
   50:  *      always returns the correct integer provided it is
   51:  *      representable.
   52:  *
   53:  * Constants :
   54:  * The hexadecimal values are the intended ones for the following
   55:  * constants. The decimal values may be used, provided that the
   56:  * compiler will convert from decimal to binary accurately enough
   57:  * to produce the hexadecimal values shown.
   58:  */
   59: 
   60: #include <sys/cdefs.h>
   61: #include <float.h>
   62: #include <math.h>
   63: 
   64: #include "math_private.h"
   65: 
   66: static const double
   67: bp[] = {1.0, 1.5,},
   68: dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
   69: dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
   70: zero    =  0.0,
   71: one     =  1.0,
   72: two     =  2.0,
   73: two53   =  9007199254740992.0,    /* 0x43400000, 0x00000000 */
   74: huge    =  1.0e300,
   75: tiny    =  1.0e-300,
   76:         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
   77: L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
   78: L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
   79: L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
   80: L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
   81: L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
   82: L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
   83: P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
   84: P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
   85: P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
   86: P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
   87: P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
   88: lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
   89: lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
   90: lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
   91: ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
   92: cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
   93: cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
   94: cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
   95: ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
   96: ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
   97: ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
   98: 
   99: double
  100: pow(double x, double y)
  101: {
  102:         double z,ax,z_h,z_l,p_h,p_l;
  103:         double yy1,t1,t2,r,s,t,u,v,w;
  104:         int32_t i,j,k,yisint,n;
  105:         int32_t hx,hy,ix,iy;
  106:         u_int32_t lx,ly;
  107: 
  108:         EXTRACT_WORDS(hx,lx,x);
  109:         EXTRACT_WORDS(hy,ly,y);
  110:         ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
  111: 
  112:     /* y==zero: x**0 = 1 */
  113:         if((iy|ly)==0) return one;
  114: 
  115:     /* +-NaN return x+y */
  116:         if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
  117:            iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
  118:                 return x+y;
  119: 
  120:     /* determine if y is an odd int when x < 0
  121:      * yisint = 0       ... y is not an integer
  122:      * yisint = 1       ... y is an odd int
  123:      * yisint = 2       ... y is an even int
  124:      */
  125:         yisint  = 0;
  126:         if(hx<0) {
  127:             if(iy>=0x43400000) yisint = 2; /* even integer y */
  128:             else if(iy>=0x3ff00000) {
  129:                 k = (iy>>20)-0x3ff;      /* exponent */
  130:                 if(k>20) {
  131:                     j = ly>>(52-k);
  132:                     if((j<<(52-k))==ly) yisint = 2-(j&1);
  133:                 } else if(ly==0) {
  134:                     j = iy>>(20-k);
  135:                     if((j<<(20-k))==iy) yisint = 2-(j&1);
  136:                 }
  137:             }
  138:         }
  139: 
  140:     /* special value of y */
  141:         if(ly==0) {
  142:             if (iy==0x7ff00000) {      /* y is +-inf */
  143:                 if(((ix-0x3ff00000)|lx)==0)
  144:                     return  y - y;    /* inf**+-1 is NaN */
  145:                 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
  146:                     return (hy>=0)? y: zero;
  147:                 else                   /* (|x|<1)**-,+inf = inf,0 */
  148:                     return (hy<0)?-y: zero;
  149:             }
  150:             if(iy==0x3ff00000) {       /* y is  +-1 */
  151:                 if(hy<0) return one/x; else return x;
  152:             }
  153:             if(hy==0x40000000) return x*x; /* y is  2 */
  154:             if(hy==0x3fe00000) {       /* y is  0.5 */
  155:                 if(hx>=0)     /* x >= +0 */
  156:                 return sqrt(x);
  157:             }
  158:         }
  159: 
  160:         ax   = fabs(x);
  161:     /* special value of x */
  162:         if(lx==0) {
  163:             if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
  164:                 z = ax;                       /*x is +-0,+-inf,+-1*/
  165:                 if(hy<0) z = one/z;   /* z = (1/|x|) */
  166:                 if(hx<0) {
  167:                     if(((ix-0x3ff00000)|yisint)==0) {
  168:                         z = (z-z)/(z-z); /* (-1)**non-int is NaN */
  169:                     } else if(yisint==1)
  170:                         z = -z;              /* (x<0)**odd = -(|x|**odd) */
  171:                 }
  172:                 return z;
  173:             }
  174:         }
  175: 
  176:         n = (hx>>31)+1;
  177: 
  178:     /* (x<0)**(non-int) is NaN */
  179:         if((n|yisint)==0) return (x-x)/(x-x);
  180: 
  181:         s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
  182:         if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
  183: 
  184:     /* |y| is huge */
  185:         if(iy>0x41e00000) { /* if |y| > 2**31 */
  186:             if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
  187:                 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
  188:                 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
  189:             }
  190:         /* over/underflow if x is not close to one */
  191:             if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
  192:             if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
  193:         /* now |1-x| is tiny <= 2**-20, suffice to compute
  194:            log(x) by x-x^2/2+x^3/3-x^4/4 */
  195:             t = ax-one;                /* t has 20 trailing zeros */
  196:             w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
  197:             u = ivln2_h*t;     /* ivln2_h has 21 sig. bits */
  198:             v = t*ivln2_l-w*ivln2;
  199:             t1 = u+v;
  200:             SET_LOW_WORD(t1,0);
  201:             t2 = v-(t1-u);
  202:         } else {
  203:             double ss,s2,s_h,s_l,t_h,t_l;
  204:             n = 0;
  205:         /* take care subnormal number */
  206:             if(ix<0x00100000)
  207:                 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
  208:             n  += ((ix)>>20)-0x3ff;
  209:             j  = ix&0x000fffff;
  210:         /* determine interval */
  211:             ix = j|0x3ff00000;         /* normalize ix */
  212:             if(j<=0x3988E) k=0;                /* |x|<sqrt(3/2) */
  213:             else if(j<0xBB67A) k=1;    /* |x|<sqrt(3)   */
  214:             else {k=0;n+=1;ix -= 0x00100000;}
  215:             SET_HIGH_WORD(ax,ix);
  216: 
  217:         /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
  218:             u = ax-bp[k];              /* bp[0]=1.0, bp[1]=1.5 */
  219:             v = one/(ax+bp[k]);
  220:             ss = u*v;
  221:             s_h = ss;
  222:             SET_LOW_WORD(s_h,0);
  223:         /* t_h=ax+bp[k] High */
  224:             t_h = zero;
  225:             SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
  226:             t_l = ax - (t_h-bp[k]);
  227:             s_l = v*((u-s_h*t_h)-s_h*t_l);
  228:         /* compute log(ax) */
  229:             s2 = ss*ss;
  230:             r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
  231:             r += s_l*(s_h+ss);
  232:             s2  = s_h*s_h;
  233:             t_h = 3.0+s2+r;
  234:             SET_LOW_WORD(t_h,0);
  235:             t_l = r-((t_h-3.0)-s2);
  236:         /* u+v = ss*(1+...) */
  237:             u = s_h*t_h;
  238:             v = s_l*t_h+t_l*ss;
  239:         /* 2/(3log2)*(ss+...) */
  240:             p_h = u+v;
  241:             SET_LOW_WORD(p_h,0);
  242:             p_l = v-(p_h-u);
  243:             z_h = cp_h*p_h;            /* cp_h+cp_l = 2/(3*log2) */
  244:             z_l = cp_l*p_h+p_l*cp+dp_l[k];
  245:         /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
  246:             t = (double)n;
  247:             t1 = (((z_h+z_l)+dp_h[k])+t);
  248:             SET_LOW_WORD(t1,0);
  249:             t2 = z_l-(((t1-t)-dp_h[k])-z_h);
  250:         }
  251: 
  252:     /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
  253:         yy1  = y;
  254:         SET_LOW_WORD(yy1,0);
  255:         p_l = (y-yy1)*t1+y*t2;
  256:         p_h = yy1*t1;
  257:         z = p_l+p_h;
  258:         EXTRACT_WORDS(j,i,z);
  259:         if (j>=0x40900000) {                           /* z >= 1024 */
  260:             if(((j-0x40900000)|i)!=0)                  /* if z > 1024 */
  261:                 return s*huge*huge;                   /* overflow */
  262:             else {
  263:                 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
  264:             }
  265:         } else if((j&0x7fffffff)>=0x4090cc00 ) {       /* z <= -1075 */
  266:             if(((j-0xc090cc00)|i)!=0)          /* z < -1075 */
  267:                 return s*tiny*tiny;           /* underflow */
  268:             else {
  269:                 if(p_l<=z-p_h) return s*tiny*tiny;    /* underflow */
  270:             }
  271:         }
  272:     /*
  273:      * compute 2**(p_h+p_l)
  274:      */
  275:         i = j&0x7fffffff;
  276:         k = (i>>20)-0x3ff;
  277:         n = 0;
  278:         if(i>0x3fe00000) {             /* if |z| > 0.5, set n = [z+0.5] */
  279:             n = j+(0x00100000>>(k+1));
  280:             k = ((n&0x7fffffff)>>20)-0x3ff;    /* new k for n */
  281:             t = zero;
  282:             SET_HIGH_WORD(t,n&~(0x000fffff>>k));
  283:             n = ((n&0x000fffff)|0x00100000)>>(20-k);
  284:             if(j<0) n = -n;
  285:             p_h -= t;
  286:         }
  287:         t = p_l+p_h;
  288:         SET_LOW_WORD(t,0);
  289:         u = t*lg2_h;
  290:         v = (p_l-(t-p_h))*lg2+t*lg2_l;
  291:         z = u+v;
  292:         w = v-(z-u);
  293:         t  = z*z;
  294:         t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
  295:         r  = (z*t1)/(t1-two)-(w+z*w);
  296:         z  = one-(r-z);
  297:         GET_HIGH_WORD(j,z);
  298:         j += (n<<20);
  299:         if((j>>20)<=0) z = scalbn(z,n);        /* subnormal output */
  300:         else SET_HIGH_WORD(z,j);
  301:         return s*z;
  302: }
  303: 
  304: #if     LDBL_MANT_DIG == 53
  305: #ifdef  lint
  306: /* PROTOLIB1 */
  307: long double powl(long double, long double);
  308: #else   /* lint */
  309: __weak_alias(powl, pow);
  310: #endif  /* lint */
  311: #endif  /* LDBL_MANT_DIG == 53 */