e_powf.c revision 129955 
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#ifndef lint
17static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_powf.c 129955 2004-06-01 19:03:31Z bde $";
18#endif
19
20#include "math.h"
21#include "math_private.h"
22
23static const float
24bp[] = {1.0, 1.5,},
25dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
26dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
27zero    =  0.0,
28one	=  1.0,
29two	=  2.0,
30two24	=  16777216.0,	/* 0x4b800000 */
31huge	=  1.0e30,
32tiny    =  1.0e-30,
33	/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
34L1  =  6.0000002384e-01, /* 0x3f19999a */
35L2  =  4.2857143283e-01, /* 0x3edb6db7 */
36L3  =  3.3333334327e-01, /* 0x3eaaaaab */
37L4  =  2.7272811532e-01, /* 0x3e8ba305 */
38L5  =  2.3066075146e-01, /* 0x3e6c3255 */
39L6  =  2.0697501302e-01, /* 0x3e53f142 */
40P1   =  1.6666667163e-01, /* 0x3e2aaaab */
41P2   = -2.7777778450e-03, /* 0xbb360b61 */
42P3   =  6.6137559770e-05, /* 0x388ab355 */
43P4   = -1.6533901999e-06, /* 0xb5ddea0e */
44P5   =  4.1381369442e-08, /* 0x3331bb4c */
45lg2  =  6.9314718246e-01, /* 0x3f317218 */
46lg2_h  =  6.93145752e-01, /* 0x3f317200 */
47lg2_l  =  1.42860654e-06, /* 0x35bfbe8c */
48ovt =  4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
49cp    =  9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
50cp_h  =  9.6179199219e-01, /* 0x3f763800 =head of cp */
51cp_l  =  4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
52ivln2    =  1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
53ivln2_h  =  1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
54ivln2_l  =  7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
55
56float
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,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    /* (x<0)**(non-int) is NaN */
124	if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
125
126    /* |y| is huge */
127	if(iy>0x4d000000) { /* if |y| > 2**27 */
128	/* over/underflow if x is not close to one */
129	    if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny;
130	    if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny;
131	/* now |1-x| is tiny <= 2**-20, suffice to compute
132	   log(x) by x-x^2/2+x^3/3-x^4/4 */
133	    t = ax-1;		/* t has 20 trailing zeros */
134	    w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
135	    u = ivln2_h*t;	/* ivln2_h has 16 sig. bits */
136	    v = t*ivln2_l-w*ivln2;
137	    t1 = u+v;
138	    GET_FLOAT_WORD(is,t1);
139	    SET_FLOAT_WORD(t1,is&0xfffff000);
140	    t2 = v-(t1-u);
141	} else {
142	    float s2,s_h,s_l,t_h,t_l;
143	    n = 0;
144	/* take care subnormal number */
145	    if(ix<0x00800000)
146		{ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
147	    n  += ((ix)>>23)-0x7f;
148	    j  = ix&0x007fffff;
149	/* determine interval */
150	    ix = j|0x3f800000;		/* normalize ix */
151	    if(j<=0x1cc471) k=0;	/* |x|<sqrt(3/2) */
152	    else if(j<0x5db3d7) k=1;	/* |x|<sqrt(3)   */
153	    else {k=0;n+=1;ix -= 0x00800000;}
154	    SET_FLOAT_WORD(ax,ix);
155
156	/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
157	    u = ax-bp[k];		/* bp[0]=1.0, bp[1]=1.5 */
158	    v = one/(ax+bp[k]);
159	    s = u*v;
160	    s_h = s;
161	    GET_FLOAT_WORD(is,s_h);
162	    SET_FLOAT_WORD(s_h,is&0xfffff000);
163	/* t_h=ax+bp[k] High */
164	    is = ((ix>>1)&0xfffff000)|0x20000000;
165	    SET_FLOAT_WORD(t_h,is+0x00400000+(k<<21));
166	    t_l = ax - (t_h-bp[k]);
167	    s_l = v*((u-s_h*t_h)-s_h*t_l);
168	/* compute log(ax) */
169	    s2 = s*s;
170	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
171	    r += s_l*(s_h+s);
172	    s2  = s_h*s_h;
173	    t_h = (float)3.0+s2+r;
174	    GET_FLOAT_WORD(is,t_h);
175	    SET_FLOAT_WORD(t_h,is&0xfffff000);
176	    t_l = r-((t_h-(float)3.0)-s2);
177	/* u+v = s*(1+...) */
178	    u = s_h*t_h;
179	    v = s_l*t_h+t_l*s;
180	/* 2/(3log2)*(s+...) */
181	    p_h = u+v;
182	    GET_FLOAT_WORD(is,p_h);
183	    SET_FLOAT_WORD(p_h,is&0xfffff000);
184	    p_l = v-(p_h-u);
185	    z_h = cp_h*p_h;		/* cp_h+cp_l = 2/(3*log2) */
186	    z_l = cp_l*p_h+p_l*cp+dp_l[k];
187	/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
188	    t = (float)n;
189	    t1 = (((z_h+z_l)+dp_h[k])+t);
190	    GET_FLOAT_WORD(is,t1);
191	    SET_FLOAT_WORD(t1,is&0xfffff000);
192	    t2 = z_l-(((t1-t)-dp_h[k])-z_h);
193	}
194
195	s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
196	if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
197	    s = -one;	/* (-ve)**(odd int) */
198
199    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
200	GET_FLOAT_WORD(is,y);
201	SET_FLOAT_WORD(y1,is&0xfffff000);
202	p_l = (y-y1)*t1+y*t2;
203	p_h = y1*t1;
204	z = p_l+p_h;
205	GET_FLOAT_WORD(j,z);
206	if (j>0x43000000)				/* if z > 128 */
207	    return s*huge*huge;				/* overflow */
208	else if (j==0x43000000) {			/* if z == 128 */
209	    if(p_l+ovt>z-p_h) return s*huge*huge;	/* overflow */
210	}
211	else if ((j&0x7fffffff)>0x43160000)		/* z <= -150 */
212	    return s*tiny*tiny;				/* underflow */
213	else if (j==0xc3160000){			/* z == -150 */
214	    if(p_l<=z-p_h) return s*tiny*tiny;		/* underflow */
215	}
216    /*
217     * compute 2**(p_h+p_l)
218     */
219	i = j&0x7fffffff;
220	k = (i>>23)-0x7f;
221	n = 0;
222	if(i>0x3f000000) {		/* if |z| > 0.5, set n = [z+0.5] */
223	    n = j+(0x00800000>>(k+1));
224	    k = ((n&0x7fffffff)>>23)-0x7f;	/* new k for n */
225	    SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
226	    n = ((n&0x007fffff)|0x00800000)>>(23-k);
227	    if(j<0) n = -n;
228	    p_h -= t;
229	}
230	t = p_l+p_h;
231	GET_FLOAT_WORD(is,t);
232	SET_FLOAT_WORD(t,is&0xffff8000);
233	u = t*lg2_h;
234	v = (p_l-(t-p_h))*lg2+t*lg2_l;
235	z = u+v;
236	w = v-(z-u);
237	t  = z*z;
238	t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
239	r  = (z*t1)/(t1-two)-(w+z*w);
240	z  = one-(r-z);
241	GET_FLOAT_WORD(j,z);
242	j += (n<<23);
243	if((j>>23)<=0) z = scalbnf(z,n);	/* subnormal output */
244	else SET_FLOAT_WORD(z,j);
245	return s*z;
246}
247