sysdeps/ieee754/flt-32/e_powf.c

   1 /* e_powf.c -- float version of e_pow.c.
   2  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
   3  */
   4
   5 /*
   6  * ====================================================
   7  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   8  *
   9  * Developed at SunPro, a Sun Microsystems, Inc. business.
  10  * Permission to use, copy, modify, and distribute this
  11  * software is freely granted, provided that this notice
  12  * is preserved.
  13  * ====================================================
  14  */
  15
  16 #include <math.h>
  17 #include <math_private.h>
  18
  19 static const float huge = 1.0e+30, tiny = 1.0e-30;
  20
  21 static const float
  22 bp[] = {1.0, 1.5,},
  23 dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
  24 dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
  25 zero    =  0.0,
  26 one     =  1.0,
  27 two     =  2.0,
  28 two24   =  16777216.0,  /* 0x4b800000 */
  29         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  30 L1  =  6.0000002384e-01, /* 0x3f19999a */
  31 L2  =  4.2857143283e-01, /* 0x3edb6db7 */
  32 L3  =  3.3333334327e-01, /* 0x3eaaaaab */
  33 L4  =  2.7272811532e-01, /* 0x3e8ba305 */
  34 L5  =  2.3066075146e-01, /* 0x3e6c3255 */
  35 L6  =  2.0697501302e-01, /* 0x3e53f142 */
  36 P1   =  1.6666667163e-01, /* 0x3e2aaaab */
  37 P2   = -2.7777778450e-03, /* 0xbb360b61 */
  38 P3   =  6.6137559770e-05, /* 0x388ab355 */
  39 P4   = -1.6533901999e-06, /* 0xb5ddea0e */
  40 P5   =  4.1381369442e-08, /* 0x3331bb4c */
  41 lg2  =  6.9314718246e-01, /* 0x3f317218 */
  42 lg2_h  =  6.93145752e-01, /* 0x3f317200 */
  43 lg2_l  =  1.42860654e-06, /* 0x35bfbe8c */
  44 ovt =  4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
  45 cp    =  9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
  46 cp_h  =  9.6179199219e-01, /* 0x3f763800 =head of cp */
  47 cp_l  =  4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
  48 ivln2    =  1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
  49 ivln2_h  =  1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
  50 ivln2_l  =  7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
  51
  52 float
  53 __ieee754_powf(float x, float y)
  54 {
  55         float z,ax,z_h,z_l,p_h,p_l;
  56         float y1,t1,t2,r,s,t,u,v,w;
  57         int32_t i,j,k,yisint,n;
  58         int32_t hx,hy,ix,iy,is;
  59
  60         GET_FLOAT_WORD(hx,x);
  61         GET_FLOAT_WORD(hy,y);
  62         ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
  63
  64     /* y==zero: x**0 = 1 */
  65         if(iy==0) return one;
  66
  67     /* x==+-1 */
  68         if(x == 1.0) return one;
  69         if(x == -1.0 && isinf(y)) return one;
  70
  71     /* +-NaN return x+y */
  72         if(__builtin_expect(ix > 0x7f800000 ||
  73                             iy > 0x7f800000, 0))
  74                 return x+y;
  75
  76     /* determine if y is an odd int when x < 0
  77      * yisint = 0       ... y is not an integer
  78      * yisint = 1       ... y is an odd int
  79      * yisint = 2       ... y is an even int
  80      */
  81         yisint  = 0;
  82         if(hx<0) {
  83             if(iy>=0x4b800000) yisint = 2; /* even integer y */
  84             else if(iy>=0x3f800000) {
  85                 k = (iy>>23)-0x7f;         /* exponent */
  86                 j = iy>>(23-k);
  87                 if((j<<(23-k))==iy) yisint = 2-(j&1);
  88             }
  89         }
  90
  91     /* special value of y */
  92         if (__builtin_expect(iy==0x7f800000, 0)) {      /* y is +-inf */
  93             if (ix==0x3f800000)
  94                 return  y - y;  /* inf**+-1 is NaN */
  95             else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
  96                 return (hy>=0)? y: zero;
  97             else                        /* (|x|<1)**-,+inf = inf,0 */
  98                 return (hy<0)?-y: zero;
  99         }
 100         if(iy==0x3f800000) {    /* y is  +-1 */
 101             if(hy<0) return one/x; else return x;
 102         }
 103         if(hy==0x40000000) return x*x; /* y is  2 */
 104         if(hy==0x3f000000) {    /* y is  0.5 */
 105             if(__builtin_expect(hx>=0, 1))      /* x >= +0 */
 106             return __ieee754_sqrtf(x);
 107         }
 108
 109         ax   = fabsf(x);
 110     /* special value of x */
 111         if(__builtin_expect(ix==0x7f800000||ix==0||ix==0x3f800000, 0)){
 112             z = ax;                     /*x is +-0,+-inf,+-1*/
 113             if(hy<0) z = one/z; /* z = (1/|x|) */
 114             if(hx<0) {
 115                 if(((ix-0x3f800000)|yisint)==0) {
 116                     z = (z-z)/(z-z); /* (-1)**non-int is NaN */
 117                 } else if(yisint==1)
 118                     z = -z;             /* (x<0)**odd = -(|x|**odd) */
 119             }
 120             return z;
 121         }
 122
 123     /* (x<0)**(non-int) is NaN */
 124         if(__builtin_expect(((((u_int32_t)hx>>31)-1)|yisint)==0, 0))
 125             return (x-x)/(x-x);
 126
 127     /* |y| is huge */
 128         if(__builtin_expect(iy>0x4d000000, 0)) { /* if |y| > 2**27 */
 129         /* over/underflow if x is not close to one */
 130             if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny;
 131             if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny;
 132         /* now |1-x| is tiny <= 2**-20, suffice to compute
 133            log(x) by x-x^2/2+x^3/3-x^4/4 */
 134             t = x-1;            /* t has 20 trailing zeros */
 135             w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
 136             u = ivln2_h*t;      /* ivln2_h has 16 sig. bits */
 137             v = t*ivln2_l-w*ivln2;
 138             t1 = u+v;
 139             GET_FLOAT_WORD(is,t1);
 140             SET_FLOAT_WORD(t1,is&0xfffff000);
 141             t2 = v-(t1-u);
 142         } else {
 143             float s2,s_h,s_l,t_h,t_l;
 144             /* Avoid internal underflow for tiny y.  The exact value
 145                of y does not matter if |y| <= 2**-32.  */
 146             if (iy < 0x2f800000)
 147               SET_FLOAT_WORD (y, (hy & 0x80000000) | 0x2f800000);
 148             n = 0;
 149         /* take care subnormal number */
 150             if(ix<0x00800000)
 151                 {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
 152             n  += ((ix)>>23)-0x7f;
 153             j  = ix&0x007fffff;
 154         /* determine interval */
 155             ix = j|0x3f800000;          /* normalize ix */
 156             if(j<=0x1cc471) k=0;        /* |x|<sqrt(3/2) */
 157             else if(j<0x5db3d7) k=1;    /* |x|<sqrt(3)   */
 158             else {k=0;n+=1;ix -= 0x00800000;}
 159             SET_FLOAT_WORD(ax,ix);
 160
 161         /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 162             u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
 163             v = one/(ax+bp[k]);
 164             s = u*v;
 165             s_h = s;
 166             GET_FLOAT_WORD(is,s_h);
 167             SET_FLOAT_WORD(s_h,is&0xfffff000);
 168         /* t_h=ax+bp[k] High */
 169             SET_FLOAT_WORD(t_h,((ix>>1)|0x20000000)+0x0040000+(k<<21));
 170             t_l = ax - (t_h-bp[k]);
 171             s_l = v*((u-s_h*t_h)-s_h*t_l);
 172         /* compute log(ax) */
 173             s2 = s*s;
 174             r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
 175             r += s_l*(s_h+s);
 176             s2  = s_h*s_h;
 177             t_h = (float)3.0+s2+r;
 178             GET_FLOAT_WORD(is,t_h);
 179             SET_FLOAT_WORD(t_h,is&0xfffff000);
 180             t_l = r-((t_h-(float)3.0)-s2);
 181         /* u+v = s*(1+...) */
 182             u = s_h*t_h;
 183             v = s_l*t_h+t_l*s;
 184         /* 2/(3log2)*(s+...) */
 185             p_h = u+v;
 186             GET_FLOAT_WORD(is,p_h);
 187             SET_FLOAT_WORD(p_h,is&0xfffff000);
 188             p_l = v-(p_h-u);
 189             z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
 190             z_l = cp_l*p_h+p_l*cp+dp_l[k];
 191         /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 192             t = (float)n;
 193             t1 = (((z_h+z_l)+dp_h[k])+t);
 194             GET_FLOAT_WORD(is,t1);
 195             SET_FLOAT_WORD(t1,is&0xfffff000);
 196             t2 = z_l-(((t1-t)-dp_h[k])-z_h);
 197         }
 198
 199         s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
 200         if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
 201             s = -one;   /* (-ve)**(odd int) */
 202
 203     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
 204         GET_FLOAT_WORD(is,y);
 205         SET_FLOAT_WORD(y1,is&0xfffff000);
 206         p_l = (y-y1)*t1+y*t2;
 207         p_h = y1*t1;
 208         z = p_l+p_h;
 209         GET_FLOAT_WORD(j,z);
 210         if (__builtin_expect(j>0x43000000, 0))          /* if z > 128 */
 211             return s*huge*huge;                         /* overflow */
 212         else if (__builtin_expect(j==0x43000000, 0)) {  /* if z == 128 */
 213             if(p_l+ovt>z-p_h) return s*huge*huge;       /* overflow */
 214         }
 215         else if (__builtin_expect((j&0x7fffffff)>0x43160000, 0))/* z <= -150 */
 216             return s*tiny*tiny;                         /* underflow */
 217         else if (__builtin_expect((u_int32_t) j==0xc3160000, 0)){/* z == -150*/
 218             if(p_l<=z-p_h) return s*tiny*tiny;          /* underflow */
 219         }
 220     /*
 221      * compute 2**(p_h+p_l)
 222      */
 223         i = j&0x7fffffff;
 224         k = (i>>23)-0x7f;
 225         n = 0;
 226         if(i>0x3f000000) {              /* if |z| > 0.5, set n = [z+0.5] */
 227             n = j+(0x00800000>>(k+1));
 228             k = ((n&0x7fffffff)>>23)-0x7f;      /* new k for n */
 229             SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
 230             n = ((n&0x007fffff)|0x00800000)>>(23-k);
 231             if(j<0) n = -n;
 232             p_h -= t;
 233         }
 234         t = p_l+p_h;
 235         GET_FLOAT_WORD(is,t);
 236         SET_FLOAT_WORD(t,is&0xfffff000);
 237         u = t*lg2_h;
 238         v = (p_l-(t-p_h))*lg2+t*lg2_l;
 239         z = u+v;
 240         w = v-(z-u);
 241         t  = z*z;
 242         t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 243         r  = (z*t1)/(t1-two)-(w+z*w);
 244         z  = one-(r-z);
 245         GET_FLOAT_WORD(j,z);
 246         j += (n<<23);
 247         if((j>>23)<=0) z = __scalbnf(z,n);      /* subnormal output */
 248         else SET_FLOAT_WORD(z,j);
 249         return s*z;
 250 }
 251 strong_alias (__ieee754_powf, __powf_finite)