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e_pow.c File Reference
#include "fdlibm.h"

Go to the source code of this file.

Functions

double __ieee754_pow (double x, double y)

Variables

static double bp [] = {1.0, 1.5,}
static double dp_h [] = { 0.0, 5.84962487220764160156e-01,}
static double dp_l [] = { 0.0, 1.35003920212974897128e-08,}
static double zero = 0.0
static double one = 1.0
static double two = 2.0
static double two53 = 9007199254740992.0
static double really_big = 1.0e300
static double tiny = 1.0e-300
static double L1 = 5.99999999999994648725e-01
static double L2 = 4.28571428578550184252e-01
static double L3 = 3.33333329818377432918e-01
static double L4 = 2.72728123808534006489e-01
static double L5 = 2.30660745775561754067e-01
static double L6 = 2.06975017800338417784e-01
static double P1 = 1.66666666666666019037e-01
static double P2 = -2.77777777770155933842e-03
static double P3 = 6.61375632143793436117e-05
static double P4 = -1.65339022054652515390e-06
static double P5 = 4.13813679705723846039e-08
static double lg2 = 6.93147180559945286227e-01
static double lg2_h = 6.93147182464599609375e-01
static double lg2_l = -1.90465429995776804525e-09
static double ovt = 8.0085662595372944372e-0017
static double cp = 9.61796693925975554329e-01
static double cp_h = 9.61796700954437255859e-01
static double cp_l = -7.02846165095275826516e-09
static double ivln2 = 1.44269504088896338700e+00
static double ivln2_h = 1.44269502162933349609e+00
static double ivln2_l = 1.92596299112661746887e-08

Function Documentation

double __ieee754_pow ( double  x,
double  y 
)

Definition at line 144 of file e_pow.c.

{
        fd_twoints ux, uy, uz;
        double y1,t1,p_h,t,z,ax;
       double z_h,z_l,p_l;
       double t2,r,s,u,v,w;
       int i,j,k,yisint,n;
       int hx,hy,ix,iy;
       unsigned lx,ly;

        ux.d = x; uy.d = y;
       hx = __HI(ux); lx = __LO(ux);
       hy = __HI(uy); ly = __LO(uy);
       ix = hx&0x7fffffff;  iy = hy&0x7fffffff;

    /* y==zero: x**0 = 1 */
       if((iy|ly)==0) return one;  

    /* +-NaN return x+y */
       if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
          iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=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>=0x43400000) yisint = 2; /* even integer y */
           else if(iy>=0x3ff00000) {
              k = (iy>>20)-0x3ff;     /* exponent */
              if(k>20) {
                  j = ly>>(52-k);
                  if((j<<(52-k))==(int)ly) yisint = 2-(j&1);
              } else if(ly==0) {
                  j = iy>>(20-k);
                  if((j<<(20-k))==iy) yisint = 2-(j&1);
              }
           }         
       } 

    /* special value of y */
       if(ly==0) {   
           if (iy==0x7ff00000) {   /* y is +-inf */
               if(((ix-0x3ff00000)|lx)==0)
#ifdef _WIN32
/* VC++ optimizer reduces y - y to 0 */ 
                    return y / y;
#else
                  return  y - y;   /* inf**+-1 is NaN */
#endif
               else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
                  return (hy>=0)? y: zero;
               else                /* (|x|<1)**-,+inf = inf,0 */
                  return (hy<0)?-y: zero;
           } 
           if(iy==0x3ff00000) {    /* y is  +-1 */
              if(hy<0) return one/x; else return x;
           }
           if(hy==0x40000000) return x*x; /* y is  2 */
           if(hy==0x3fe00000) {    /* y is  0.5 */
              if(hx>=0)     /* x >= +0 */
              return fd_sqrt(x);   
           }
       }

       ax   = fd_fabs(x);
    /* special value of x */
       if(lx==0) {
           if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
              z = ax;                     /*x is +-0,+-inf,+-1*/
              if(hy<0) z = one/z;  /* z = (1/|x|) */
              if(hx<0) {
                  if(((ix-0x3ff00000)|yisint)==0) {
                     z = (z-z)/(z-z); /* (-1)**non-int is NaN */
                  } else if(yisint==1) {
#ifdef HPUX
                        uz.d = z;
                     __HI(uz) ^= 1<<31; /* some HPUXes cannot negate 0.. */
                        z = uz.d;
#else
                     z = -z;              /* (x<0)**odd = -(|x|**odd) */
#endif
                     }
              }
              return z;
           }
       }
    
    /* (x<0)**(non-int) is NaN */
       if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x);

    /* |y| is really_big */
       if(iy>0x41e00000) { /* if |y| > 2**31 */
           if(iy>0x43f00000){      /* if |y| > 2**64, must o/uflow */
              if(ix<=0x3fefffff) return (hy<0)? really_big*really_big:tiny*tiny;
              if(ix>=0x3ff00000) return (hy>0)? really_big*really_big:tiny*tiny;
           }
       /* over/underflow if x is not close to one */
           if(ix<0x3fefffff) return (hy<0)? really_big*really_big:tiny*tiny;
           if(ix>0x3ff00000) return (hy>0)? really_big*really_big:tiny*tiny;
       /* now |1-x| is tiny <= 2**-20, suffice to compute 
          log(x) by x-x^2/2+x^3/3-x^4/4 */
           t = x-1;         /* t has 20 trailing zeros */
           w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
           u = ivln2_h*t;   /* ivln2_h has 21 sig. bits */
           v = t*ivln2_l-w*ivln2;
           t1 = u+v;
            uz.d = t1;
           __LO(uz) = 0;
            t1 = uz.d;
           t2 = v-(t1-u);
       } else {
           double s_h,t_h;
           double s2,s_l,t_l;
           n = 0;
       /* take care subnormal number */
           if(ix<0x00100000) 
              {ax *= two53; n -= 53; uz.d = ax; ix = __HI(uz); }
           n  += ((ix)>>20)-0x3ff;
           j  = ix&0x000fffff;
       /* determine interval */
           ix = j|0x3ff00000;             /* normalize ix */
           if(j<=0x3988E) k=0;            /* |x|<sqrt(3/2) */
           else if(j<0xBB67A) k=1; /* |x|<sqrt(3)   */
           else {k=0;n+=1;ix -= 0x00100000;}
            uz.d = ax;
           __HI(uz) = ix;
            ax = uz.d;

       /* 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;
            uz.d = s_h;
           __LO(uz) = 0;
            s_h = uz.d;
       /* t_h=ax+bp[k] High */
           t_h = zero;
            uz.d = t_h;
           __HI(uz)=((ix>>1)|0x20000000)+0x00080000+(k<<18); 
            t_h = uz.d;
           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;
           r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
           r += s_l*(s_h+s);
           s2  = s_h*s_h;
           t_h = 3.0+s2+r;
            uz.d = t_h;
           __LO(uz) = 0;
            t_h = uz.d;
           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;
            uz.d = p_h;
           __LO(uz) = 0;
            p_h = uz.d;
           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 = (double)n;
           t1 = (((z_h+z_l)+dp_h[k])+t);
            uz.d = t1;
           __LO(uz) = 0;
            t1 = uz.d;
           t2 = z_l-(((t1-t)-dp_h[k])-z_h);
       }

       s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
       if((((hx>>31)+1)|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */

    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
       y1  = y;
        uy.d = y1;
       __LO(uy) = 0;
        y1 = uy.d;
       p_l = (y-y1)*t1+y*t2;
       p_h = y1*t1;
       z = p_l+p_h;
        uz.d = z;
       j = __HI(uz);
       i = __LO(uz);

       if (j>=0x40900000) {                      /* z >= 1024 */
           if(((j-0x40900000)|i)!=0)                    /* if z > 1024 */
              return s*really_big*really_big;                  /* overflow */
           else {
              if(p_l+ovt>z-p_h) return s*really_big*really_big;       /* overflow */
           }
       } else if((j&0x7fffffff)>=0x4090cc00 ) {  /* z <= -1075 */
           if(((j-0xc090cc00)|i)!=0)             /* z < -1075 */
              return s*tiny*tiny;         /* underflow */
           else {
              if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
           }
       }
    /*
     * compute 2**(p_h+p_l)
     */
       i = j&0x7fffffff;
       k = (i>>20)-0x3ff;
       n = 0;
       if(i>0x3fe00000) {          /* if |z| > 0.5, set n = [z+0.5] */
           n = j+(0x00100000>>(k+1));
           k = ((n&0x7fffffff)>>20)-0x3ff;       /* new k for n */
           t = zero;
            uz.d = t;
           __HI(uz) = (n&~(0x000fffff>>k));
            t = uz.d;
           n = ((n&0x000fffff)|0x00100000)>>(20-k);
           if(j<0) n = -n;
           p_h -= t;
       } 
       t = p_l+p_h;
        uz.d = t;
       __LO(uz) = 0;
        t = uz.d;
       u = t*lg2_h;
       v = (p_l-(t-p_h))*lg2+t*lg2_l;
       z = u+v;
       w = v-(z-u);
       t  = z*z;
       t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
       r  = (z*t1)/(t1-two)-(w+z*w);
       z  = one-(r-z);
        uz.d = z;
       j  = __HI(uz);
       j += (n<<20);
       if((j>>20)<=0) z = fd_scalbn(z,n); /* subnormal output */
       else { uz.d = z; __HI(uz) += (n<<20); z = uz.d; }
       return s*z;
}

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Variable Documentation

double bp[] = {1.0, 1.5,} [static]

Definition at line 109 of file e_pow.c.

double cp = 9.61796693925975554329e-01 [static]

Definition at line 134 of file e_pow.c.

double cp_h = 9.61796700954437255859e-01 [static]

Definition at line 135 of file e_pow.c.

double cp_l = -7.02846165095275826516e-09 [static]

Definition at line 136 of file e_pow.c.

double dp_h[] = { 0.0, 5.84962487220764160156e-01,} [static]

Definition at line 110 of file e_pow.c.

double dp_l[] = { 0.0, 1.35003920212974897128e-08,} [static]

Definition at line 111 of file e_pow.c.

double ivln2 = 1.44269504088896338700e+00 [static]

Definition at line 137 of file e_pow.c.

double ivln2_h = 1.44269502162933349609e+00 [static]

Definition at line 138 of file e_pow.c.

double ivln2_l = 1.92596299112661746887e-08 [static]

Definition at line 139 of file e_pow.c.

double L1 = 5.99999999999994648725e-01 [static]

Definition at line 119 of file e_pow.c.

double L2 = 4.28571428578550184252e-01 [static]

Definition at line 120 of file e_pow.c.

double L3 = 3.33333329818377432918e-01 [static]

Definition at line 121 of file e_pow.c.

double L4 = 2.72728123808534006489e-01 [static]

Definition at line 122 of file e_pow.c.

double L5 = 2.30660745775561754067e-01 [static]

Definition at line 123 of file e_pow.c.

double L6 = 2.06975017800338417784e-01 [static]

Definition at line 124 of file e_pow.c.

double lg2 = 6.93147180559945286227e-01 [static]

Definition at line 130 of file e_pow.c.

double lg2_h = 6.93147182464599609375e-01 [static]

Definition at line 131 of file e_pow.c.

double lg2_l = -1.90465429995776804525e-09 [static]

Definition at line 132 of file e_pow.c.

double one = 1.0 [static]

Definition at line 113 of file e_pow.c.

double ovt = 8.0085662595372944372e-0017 [static]

Definition at line 133 of file e_pow.c.

double P1 = 1.66666666666666019037e-01 [static]

Definition at line 125 of file e_pow.c.

double P2 = -2.77777777770155933842e-03 [static]

Definition at line 126 of file e_pow.c.

double P3 = 6.61375632143793436117e-05 [static]

Definition at line 127 of file e_pow.c.

double P4 = -1.65339022054652515390e-06 [static]

Definition at line 128 of file e_pow.c.

double P5 = 4.13813679705723846039e-08 [static]

Definition at line 129 of file e_pow.c.

double really_big = 1.0e300 [static]

Definition at line 116 of file e_pow.c.

double tiny = 1.0e-300 [static]

Definition at line 117 of file e_pow.c.

double two = 2.0 [static]

Definition at line 114 of file e_pow.c.

double two53 = 9007199254740992.0 [static]

Definition at line 115 of file e_pow.c.

double zero = 0.0 [static]

Definition at line 112 of file e_pow.c.