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      1 /* e_powf.c -- float version of e_pow.c.
      2  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian (at) 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 #ifndef lint
     17 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_powf.c,v 1.12 2004/06/01 19:33:30 bde Exp $";
     18 #endif
     19 
     20 #include "math.h"
     21 #include "math_private.h"
     22 
     23 static const float
     24 bp[] = {1.0, 1.5,},
     25 dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
     26 dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
     27 zero    =  0.0,
     28 one	=  1.0,
     29 two	=  2.0,
     30 two24	=  16777216.0,	/* 0x4b800000 */
     31 huge	=  1.0e30,
     32 tiny    =  1.0e-30,
     33 	/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
     34 L1  =  6.0000002384e-01, /* 0x3f19999a */
     35 L2  =  4.2857143283e-01, /* 0x3edb6db7 */
     36 L3  =  3.3333334327e-01, /* 0x3eaaaaab */
     37 L4  =  2.7272811532e-01, /* 0x3e8ba305 */
     38 L5  =  2.3066075146e-01, /* 0x3e6c3255 */
     39 L6  =  2.0697501302e-01, /* 0x3e53f142 */
     40 P1   =  1.6666667163e-01, /* 0x3e2aaaab */
     41 P2   = -2.7777778450e-03, /* 0xbb360b61 */
     42 P3   =  6.6137559770e-05, /* 0x388ab355 */
     43 P4   = -1.6533901999e-06, /* 0xb5ddea0e */
     44 P5   =  4.1381369442e-08, /* 0x3331bb4c */
     45 lg2  =  6.9314718246e-01, /* 0x3f317218 */
     46 lg2_h  =  6.93145752e-01, /* 0x3f317200 */
     47 lg2_l  =  1.42860654e-06, /* 0x35bfbe8c */
     48 ovt =  4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
     49 cp    =  9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
     50 cp_h  =  9.6179199219e-01, /* 0x3f763800 =head of cp */
     51 cp_l  =  4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
     52 ivln2    =  1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
     53 ivln2_h  =  1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
     54 ivln2_l  =  7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
     55 
     56 float
     57 __ieee754_powf(float x, float y)
     58 {
     59 	float z,ax,z_h,z_l,p_h,p_l;
     60 	float y1,t1,t2,r,s,sn,t,u,v,w;
     61 	int32_t i,j,k,yisint,n;
     62 	int32_t hx,hy,ix,iy,is;
     63 
     64 	GET_FLOAT_WORD(hx,x);
     65 	GET_FLOAT_WORD(hy,y);
     66 	ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
     67 
     68     /* y==zero: x**0 = 1 */
     69 	if(iy==0) return one;
     70 
     71     /* +-NaN return x+y */
     72 	if(ix > 0x7f800000 ||
     73 	   iy > 0x7f800000)
     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 (iy==0x7f800000) {	/* 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(hx>=0)	/* x >= +0 */
    106 	    return __ieee754_sqrtf(x);
    107 	}
    108 
    109 	ax   = fabsf(x);
    110     /* special value of x */
    111 	if(ix==0x7f800000||ix==0||ix==0x3f800000){
    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 	n = ((u_int32_t)hx>>31)-1;
    124 
    125     /* (x<0)**(non-int) is NaN */
    126 	if((n|yisint)==0) return (x-x)/(x-x);
    127 
    128 	sn = one; /* s (sign of result -ve**odd) = -1 else = 1 */
    129 	if((n|(yisint-1))==0) sn = -one;/* (-ve)**(odd int) */
    130 
    131     /* |y| is huge */
    132 	if(iy>0x4d000000) { /* if |y| > 2**27 */
    133 	/* over/underflow if x is not close to one */
    134 	    if(ix<0x3f7ffff8) return (hy<0)? sn*huge*huge:sn*tiny*tiny;
    135 	    if(ix>0x3f800007) return (hy>0)? sn*huge*huge:sn*tiny*tiny;
    136 	/* now |1-x| is tiny <= 2**-20, suffice to compute
    137 	   log(x) by x-x^2/2+x^3/3-x^4/4 */
    138 	    t = ax-1;		/* t has 20 trailing zeros */
    139 	    w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
    140 	    u = ivln2_h*t;	/* ivln2_h has 16 sig. bits */
    141 	    v = t*ivln2_l-w*ivln2;
    142 	    t1 = u+v;
    143 	    GET_FLOAT_WORD(is,t1);
    144 	    SET_FLOAT_WORD(t1,is&0xfffff000);
    145 	    t2 = v-(t1-u);
    146 	} else {
    147 	    float s2,s_h,s_l,t_h,t_l;
    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 	    is = ((ix>>1)&0xfffff000)|0x20000000;
    170 	    SET_FLOAT_WORD(t_h,is+0x00400000+(k<<21));
    171 	    t_l = ax - (t_h-bp[k]);
    172 	    s_l = v*((u-s_h*t_h)-s_h*t_l);
    173 	/* compute log(ax) */
    174 	    s2 = s*s;
    175 	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
    176 	    r += s_l*(s_h+s);
    177 	    s2  = s_h*s_h;
    178 	    t_h = (float)3.0+s2+r;
    179 	    GET_FLOAT_WORD(is,t_h);
    180 	    SET_FLOAT_WORD(t_h,is&0xfffff000);
    181 	    t_l = r-((t_h-(float)3.0)-s2);
    182 	/* u+v = s*(1+...) */
    183 	    u = s_h*t_h;
    184 	    v = s_l*t_h+t_l*s;
    185 	/* 2/(3log2)*(s+...) */
    186 	    p_h = u+v;
    187 	    GET_FLOAT_WORD(is,p_h);
    188 	    SET_FLOAT_WORD(p_h,is&0xfffff000);
    189 	    p_l = v-(p_h-u);
    190 	    z_h = cp_h*p_h;		/* cp_h+cp_l = 2/(3*log2) */
    191 	    z_l = cp_l*p_h+p_l*cp+dp_l[k];
    192 	/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
    193 	    t = (float)n;
    194 	    t1 = (((z_h+z_l)+dp_h[k])+t);
    195 	    GET_FLOAT_WORD(is,t1);
    196 	    SET_FLOAT_WORD(t1,is&0xfffff000);
    197 	    t2 = z_l-(((t1-t)-dp_h[k])-z_h);
    198 	}
    199 
    200     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
    201 	GET_FLOAT_WORD(is,y);
    202 	SET_FLOAT_WORD(y1,is&0xfffff000);
    203 	p_l = (y-y1)*t1+y*t2;
    204 	p_h = y1*t1;
    205 	z = p_l+p_h;
    206 	GET_FLOAT_WORD(j,z);
    207 	if (j>0x43000000)				/* if z > 128 */
    208 	    return sn*huge*huge;			/* overflow */
    209 	else if (j==0x43000000) {			/* if z == 128 */
    210 	    if(p_l+ovt>z-p_h) return sn*huge*huge;	/* overflow */
    211 	}
    212 	else if ((j&0x7fffffff)>0x43160000)		/* z <= -150 */
    213 	    return sn*tiny*tiny;			/* underflow */
    214 	else if (j==0xc3160000){			/* z == -150 */
    215 	    if(p_l<=z-p_h) return sn*tiny*tiny;		/* underflow */
    216 	}
    217     /*
    218      * compute 2**(p_h+p_l)
    219      */
    220 	i = j&0x7fffffff;
    221 	k = (i>>23)-0x7f;
    222 	n = 0;
    223 	if(i>0x3f000000) {		/* if |z| > 0.5, set n = [z+0.5] */
    224 	    n = j+(0x00800000>>(k+1));
    225 	    k = ((n&0x7fffffff)>>23)-0x7f;	/* new k for n */
    226 	    SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
    227 	    n = ((n&0x007fffff)|0x00800000)>>(23-k);
    228 	    if(j<0) n = -n;
    229 	    p_h -= t;
    230 	}
    231 	t = p_l+p_h;
    232 	GET_FLOAT_WORD(is,t);
    233 	SET_FLOAT_WORD(t,is&0xffff8000);
    234 	u = t*lg2_h;
    235 	v = (p_l-(t-p_h))*lg2+t*lg2_l;
    236 	z = u+v;
    237 	w = v-(z-u);
    238 	t  = z*z;
    239 	t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
    240 	r  = (z*t1)/(t1-two)-(w+z*w);
    241 	z  = one-(r-z);
    242 	GET_FLOAT_WORD(j,z);
    243 	j += (n<<23);
    244 	if((j>>23)<=0) z = scalbnf(z,n);	/* subnormal output */
    245 	else SET_FLOAT_WORD(z,j);
    246 	return sn*z;
    247 }
    248