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 <sys/cdefs.h>
17 __FBSDID("$FreeBSD$");
18 
19 #include "math.h"
20 #include "math_private.h"
21 
22 static const float
23 bp[] = {1.0, 1.5,},
24 dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
25 dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
26 zero    =  0.0,
27 one	=  1.0,
28 two	=  2.0,
29 two24	=  16777216.0,	/* 0x4b800000 */
30 huge	=  1.0e30,
31 tiny    =  1.0e-30,
32 	/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
33 L1  =  6.0000002384e-01, /* 0x3f19999a */
34 L2  =  4.2857143283e-01, /* 0x3edb6db7 */
35 L3  =  3.3333334327e-01, /* 0x3eaaaaab */
36 L4  =  2.7272811532e-01, /* 0x3e8ba305 */
37 L5  =  2.3066075146e-01, /* 0x3e6c3255 */
38 L6  =  2.0697501302e-01, /* 0x3e53f142 */
39 P1   =  1.6666667163e-01, /* 0x3e2aaaab */
40 P2   = -2.7777778450e-03, /* 0xbb360b61 */
41 P3   =  6.6137559770e-05, /* 0x388ab355 */
42 P4   = -1.6533901999e-06, /* 0xb5ddea0e */
43 P5   =  4.1381369442e-08, /* 0x3331bb4c */
44 lg2  =  6.9314718246e-01, /* 0x3f317218 */
45 lg2_h  =  6.93145752e-01, /* 0x3f317200 */
46 lg2_l  =  1.42860654e-06, /* 0x35bfbe8c */
47 ovt =  4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
48 cp    =  9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
49 cp_h  =  9.6191406250e-01, /* 0x3f764000 =12b cp */
50 cp_l  = -1.1736857402e-04, /* 0xb8f623c6 =tail of cp_h */
51 ivln2    =  1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
52 ivln2_h  =  1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
53 ivln2_l  =  7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
54 
55 float
__ieee754_powf(float x,float y)56 __ieee754_powf(float x, float y)
57 {
58 	float z,ax,z_h,z_l,p_h,p_l;
59 	float y1,t1,t2,r,s,sn,t,u,v,w;
60 	int32_t i,j,k,yisint,n;
61 	int32_t hx,hy,ix,iy,is;
62 
63 	GET_FLOAT_WORD(hx,x);
64 	GET_FLOAT_WORD(hy,y);
65 	ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
66 
67     /* y==zero: x**0 = 1 */
68 	if(iy==0) return one;
69 
70     /* x==1: 1**y = 1, even if y is NaN */
71 	if (hx==0x3f800000) return one;
72 
73     /* y!=zero: result is NaN if either arg is NaN */
74 	if(ix > 0x7f800000 ||
75 	   iy > 0x7f800000)
76 		return (x+0.0F)+(y+0.0F);
77 
78     /* determine if y is an odd int when x < 0
79      * yisint = 0	... y is not an integer
80      * yisint = 1	... y is an odd int
81      * yisint = 2	... y is an even int
82      */
83 	yisint  = 0;
84 	if(hx<0) {
85 	    if(iy>=0x4b800000) yisint = 2; /* even integer y */
86 	    else if(iy>=0x3f800000) {
87 		k = (iy>>23)-0x7f;	   /* exponent */
88 		j = iy>>(23-k);
89 		if((j<<(23-k))==iy) yisint = 2-(j&1);
90 	    }
91 	}
92 
93     /* special value of y */
94 	if (iy==0x7f800000) {	/* y is +-inf */
95 	    if (ix==0x3f800000)
96 	        return  one;	/* (-1)**+-inf is NaN */
97 	    else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
98 	        return (hy>=0)? y: zero;
99 	    else			/* (|x|<1)**-,+inf = inf,0 */
100 	        return (hy<0)?-y: zero;
101 	}
102 	if(iy==0x3f800000) {	/* y is  +-1 */
103 	    if(hy<0) return one/x; else return x;
104 	}
105 	if(hy==0x40000000) return x*x; /* y is  2 */
106 	if(hy==0x3f000000) {	/* y is  0.5 */
107 	    if(hx>=0)	/* x >= +0 */
108 	    return __ieee754_sqrtf(x);
109 	}
110 
111 	ax   = fabsf(x);
112     /* special value of x */
113 	if(ix==0x7f800000||ix==0||ix==0x3f800000){
114 	    z = ax;			/*x is +-0,+-inf,+-1*/
115 	    if(hy<0) z = one/z;	/* z = (1/|x|) */
116 	    if(hx<0) {
117 		if(((ix-0x3f800000)|yisint)==0) {
118 		    z = (z-z)/(z-z); /* (-1)**non-int is NaN */
119 		} else if(yisint==1)
120 		    z = -z;		/* (x<0)**odd = -(|x|**odd) */
121 	    }
122 	    return z;
123 	}
124 
125 	n = ((u_int32_t)hx>>31)-1;
126 
127     /* (x<0)**(non-int) is NaN */
128 	if((n|yisint)==0) return (x-x)/(x-x);
129 
130 	sn = one; /* s (sign of result -ve**odd) = -1 else = 1 */
131 	if((n|(yisint-1))==0) sn = -one;/* (-ve)**(odd int) */
132 
133     /* |y| is huge */
134 	if(iy>0x4d000000) { /* if |y| > 2**27 */
135 	/* over/underflow if x is not close to one */
136 	    if(ix<0x3f7ffff8) return (hy<0)? sn*huge*huge:sn*tiny*tiny;
137 	    if(ix>0x3f800007) return (hy>0)? sn*huge*huge:sn*tiny*tiny;
138 	/* now |1-x| is tiny <= 2**-20, suffice to compute
139 	   log(x) by x-x^2/2+x^3/3-x^4/4 */
140 	    t = ax-1;		/* t has 20 trailing zeros */
141 	    w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
142 	    u = ivln2_h*t;	/* ivln2_h has 16 sig. bits */
143 	    v = t*ivln2_l-w*ivln2;
144 	    t1 = u+v;
145 	    GET_FLOAT_WORD(is,t1);
146 	    SET_FLOAT_WORD(t1,is&0xfffff000);
147 	    t2 = v-(t1-u);
148 	} else {
149 	    float s2,s_h,s_l,t_h,t_l;
150 	    n = 0;
151 	/* take care subnormal number */
152 	    if(ix<0x00800000)
153 		{ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
154 	    n  += ((ix)>>23)-0x7f;
155 	    j  = ix&0x007fffff;
156 	/* determine interval */
157 	    ix = j|0x3f800000;		/* normalize ix */
158 	    if(j<=0x1cc471) k=0;	/* |x|<sqrt(3/2) */
159 	    else if(j<0x5db3d7) k=1;	/* |x|<sqrt(3)   */
160 	    else {k=0;n+=1;ix -= 0x00800000;}
161 	    SET_FLOAT_WORD(ax,ix);
162 
163 	/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
164 	    u = ax-bp[k];		/* bp[0]=1.0, bp[1]=1.5 */
165 	    v = one/(ax+bp[k]);
166 	    s = u*v;
167 	    s_h = s;
168 	    GET_FLOAT_WORD(is,s_h);
169 	    SET_FLOAT_WORD(s_h,is&0xfffff000);
170 	/* t_h=ax+bp[k] High */
171 	    is = ((ix>>1)&0xfffff000)|0x20000000;
172 	    SET_FLOAT_WORD(t_h,is+0x00400000+(k<<21));
173 	    t_l = ax - (t_h-bp[k]);
174 	    s_l = v*((u-s_h*t_h)-s_h*t_l);
175 	/* compute log(ax) */
176 	    s2 = s*s;
177 	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
178 	    r += s_l*(s_h+s);
179 	    s2  = s_h*s_h;
180 	    t_h = (float)3.0+s2+r;
181 	    GET_FLOAT_WORD(is,t_h);
182 	    SET_FLOAT_WORD(t_h,is&0xfffff000);
183 	    t_l = r-((t_h-(float)3.0)-s2);
184 	/* u+v = s*(1+...) */
185 	    u = s_h*t_h;
186 	    v = s_l*t_h+t_l*s;
187 	/* 2/(3log2)*(s+...) */
188 	    p_h = u+v;
189 	    GET_FLOAT_WORD(is,p_h);
190 	    SET_FLOAT_WORD(p_h,is&0xfffff000);
191 	    p_l = v-(p_h-u);
192 	    z_h = cp_h*p_h;		/* cp_h+cp_l = 2/(3*log2) */
193 	    z_l = cp_l*p_h+p_l*cp+dp_l[k];
194 	/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
195 	    t = (float)n;
196 	    t1 = (((z_h+z_l)+dp_h[k])+t);
197 	    GET_FLOAT_WORD(is,t1);
198 	    SET_FLOAT_WORD(t1,is&0xfffff000);
199 	    t2 = z_l-(((t1-t)-dp_h[k])-z_h);
200 	}
201 
202     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
203 	GET_FLOAT_WORD(is,y);
204 	SET_FLOAT_WORD(y1,is&0xfffff000);
205 	p_l = (y-y1)*t1+y*t2;
206 	p_h = y1*t1;
207 	z = p_l+p_h;
208 	GET_FLOAT_WORD(j,z);
209 	if (j>0x43000000)				/* if z > 128 */
210 	    return sn*huge*huge;			/* overflow */
211 	else if (j==0x43000000) {			/* if z == 128 */
212 	    if(p_l+ovt>z-p_h) return sn*huge*huge;	/* overflow */
213 	}
214 	else if ((j&0x7fffffff)>0x43160000)		/* z <= -150 */
215 	    return sn*tiny*tiny;			/* underflow */
216 	else if (j==0xc3160000){			/* z == -150 */
217 	    if(p_l<=z-p_h) return sn*tiny*tiny;		/* underflow */
218 	}
219     /*
220      * compute 2**(p_h+p_l)
221      */
222 	i = j&0x7fffffff;
223 	k = (i>>23)-0x7f;
224 	n = 0;
225 	if(i>0x3f000000) {		/* if |z| > 0.5, set n = [z+0.5] */
226 	    n = j+(0x00800000>>(k+1));
227 	    k = ((n&0x7fffffff)>>23)-0x7f;	/* new k for n */
228 	    SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
229 	    n = ((n&0x007fffff)|0x00800000)>>(23-k);
230 	    if(j<0) n = -n;
231 	    p_h -= t;
232 	}
233 	t = p_l+p_h;
234 	GET_FLOAT_WORD(is,t);
235 	SET_FLOAT_WORD(t,is&0xffff8000);
236 	u = t*lg2_h;
237 	v = (p_l-(t-p_h))*lg2+t*lg2_l;
238 	z = u+v;
239 	w = v-(z-u);
240 	t  = z*z;
241 	t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
242 	r  = (z*t1)/(t1-two)-(w+z*w);
243 	z  = one-(r-z);
244 	GET_FLOAT_WORD(j,z);
245 	j += (n<<23);
246 	if((j>>23)<=0) z = scalbnf(z,n);	/* subnormal output */
247 	else SET_FLOAT_WORD(z,j);
248 	return sn*z;
249 }
250