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: head/lib/msun/src/e_powf.c 336362 2018-07-17 07:42:14Z bde $");
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 half = 0.5,
28 qrtr = 0.25,
29 thrd = 3.33333343e-01, /* 0x3eaaaaab */
30 one = 1.0,
31 two = 2.0,
32 two24 = 16777216.0, /* 0x4b800000 */
33 huge = 1.0e30,
34 tiny = 1.0e-30,
35 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
36 L1 = 6.0000002384e-01, /* 0x3f19999a */
37 L2 = 4.2857143283e-01, /* 0x3edb6db7 */
38 L3 = 3.3333334327e-01, /* 0x3eaaaaab */
39 L4 = 2.7272811532e-01, /* 0x3e8ba305 */
40 L5 = 2.3066075146e-01, /* 0x3e6c3255 */
41 L6 = 2.0697501302e-01, /* 0x3e53f142 */
42 P1 = 1.6666667163e-01, /* 0x3e2aaaab */
43 P2 = -2.7777778450e-03, /* 0xbb360b61 */
44 P3 = 6.6137559770e-05, /* 0x388ab355 */
45 P4 = -1.6533901999e-06, /* 0xb5ddea0e */
46 P5 = 4.1381369442e-08, /* 0x3331bb4c */
47 lg2 = 6.9314718246e-01, /* 0x3f317218 */
48 lg2_h = 6.93145752e-01, /* 0x3f317200 */
49 lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
50 ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
51 cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
52 cp_h = 9.6191406250e-01, /* 0x3f764000 =12b cp */
53 cp_l = -1.1736857402e-04, /* 0xb8f623c6 =tail of cp_h */
54 ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
55 ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
56 ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
57
58 float
__ieee754_powf(float x,float y)59 __ieee754_powf(float x, float y)
60 {
61 float z,ax,z_h,z_l,p_h,p_l;
62 float y1,t1,t2,r,s,sn,t,u,v,w;
63 int32_t i,j,k,yisint,n;
64 int32_t hx,hy,ix,iy,is;
65
66 GET_FLOAT_WORD(hx,x);
67 GET_FLOAT_WORD(hy,y);
68 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
69
70 /* y==zero: x**0 = 1 */
71 if(iy==0) return one;
72
73 /* x==1: 1**y = 1, even if y is NaN */
74 if (hx==0x3f800000) return one;
75
76 /* y!=zero: result is NaN if either arg is NaN */
77 if(ix > 0x7f800000 ||
78 iy > 0x7f800000)
79 return nan_mix(x, y);
80
81 /* determine if y is an odd int when x < 0
82 * yisint = 0 ... y is not an integer
83 * yisint = 1 ... y is an odd int
84 * yisint = 2 ... y is an even int
85 */
86 yisint = 0;
87 if(hx<0) {
88 if(iy>=0x4b800000) yisint = 2; /* even integer y */
89 else if(iy>=0x3f800000) {
90 k = (iy>>23)-0x7f; /* exponent */
91 j = iy>>(23-k);
92 if((j<<(23-k))==iy) yisint = 2-(j&1);
93 }
94 }
95
96 /* special value of y */
97 if (iy==0x7f800000) { /* y is +-inf */
98 if (ix==0x3f800000)
99 return one; /* (-1)**+-inf is NaN */
100 else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
101 return (hy>=0)? y: zero;
102 else /* (|x|<1)**-,+inf = inf,0 */
103 return (hy<0)?-y: zero;
104 }
105 if(iy==0x3f800000) { /* y is +-1 */
106 if(hy<0) return one/x; else return x;
107 }
108 if(hy==0x40000000) return x*x; /* y is 2 */
109 if(hy==0x3f000000) { /* y is 0.5 */
110 if(hx>=0) /* x >= +0 */
111 return __ieee754_sqrtf(x);
112 }
113
114 ax = fabsf(x);
115 /* special value of x */
116 if(ix==0x7f800000||ix==0||ix==0x3f800000){
117 z = ax; /*x is +-0,+-inf,+-1*/
118 if(hy<0) z = one/z; /* z = (1/|x|) */
119 if(hx<0) {
120 if(((ix-0x3f800000)|yisint)==0) {
121 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
122 } else if(yisint==1)
123 z = -z; /* (x<0)**odd = -(|x|**odd) */
124 }
125 return z;
126 }
127
128 n = ((u_int32_t)hx>>31)-1;
129
130 /* (x<0)**(non-int) is NaN */
131 if((n|yisint)==0) return (x-x)/(x-x);
132
133 sn = one; /* s (sign of result -ve**odd) = -1 else = 1 */
134 if((n|(yisint-1))==0) sn = -one;/* (-ve)**(odd int) */
135
136 /* |y| is huge */
137 if(iy>0x4d000000) { /* if |y| > 2**27 */
138 /* over/underflow if x is not close to one */
139 if(ix<0x3f7ffff8) return (hy<0)? sn*huge*huge:sn*tiny*tiny;
140 if(ix>0x3f800007) return (hy>0)? sn*huge*huge:sn*tiny*tiny;
141 /* now |1-x| is tiny <= 2**-20, suffice to compute
142 log(x) by x-x^2/2+x^3/3-x^4/4 */
143 t = ax-1; /* t has 20 trailing zeros */
144 w = (t*t)*(half-t*(thrd-t*qrtr));
145 u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
146 v = t*ivln2_l-w*ivln2;
147 t1 = u+v;
148 GET_FLOAT_WORD(is,t1);
149 SET_FLOAT_WORD(t1,is&0xfffff000);
150 t2 = v-(t1-u);
151 } else {
152 float s2,s_h,s_l,t_h,t_l;
153 n = 0;
154 /* take care subnormal number */
155 if(ix<0x00800000)
156 {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
157 n += ((ix)>>23)-0x7f;
158 j = ix&0x007fffff;
159 /* determine interval */
160 ix = j|0x3f800000; /* normalize ix */
161 if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */
162 else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */
163 else {k=0;n+=1;ix -= 0x00800000;}
164 SET_FLOAT_WORD(ax,ix);
165
166 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
167 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
168 v = one/(ax+bp[k]);
169 s = u*v;
170 s_h = s;
171 GET_FLOAT_WORD(is,s_h);
172 SET_FLOAT_WORD(s_h,is&0xfffff000);
173 /* t_h=ax+bp[k] High */
174 is = ((ix>>1)&0xfffff000)|0x20000000;
175 SET_FLOAT_WORD(t_h,is+0x00400000+(k<<21));
176 t_l = ax - (t_h-bp[k]);
177 s_l = v*((u-s_h*t_h)-s_h*t_l);
178 /* compute log(ax) */
179 s2 = s*s;
180 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
181 r += s_l*(s_h+s);
182 s2 = s_h*s_h;
183 t_h = 3+s2+r;
184 GET_FLOAT_WORD(is,t_h);
185 SET_FLOAT_WORD(t_h,is&0xfffff000);
186 t_l = r-((t_h-3)-s2);
187 /* u+v = s*(1+...) */
188 u = s_h*t_h;
189 v = s_l*t_h+t_l*s;
190 /* 2/(3log2)*(s+...) */
191 p_h = u+v;
192 GET_FLOAT_WORD(is,p_h);
193 SET_FLOAT_WORD(p_h,is&0xfffff000);
194 p_l = v-(p_h-u);
195 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
196 z_l = cp_l*p_h+p_l*cp+dp_l[k];
197 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
198 t = n;
199 t1 = (((z_h+z_l)+dp_h[k])+t);
200 GET_FLOAT_WORD(is,t1);
201 SET_FLOAT_WORD(t1,is&0xfffff000);
202 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
203 }
204
205 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
206 GET_FLOAT_WORD(is,y);
207 SET_FLOAT_WORD(y1,is&0xfffff000);
208 p_l = (y-y1)*t1+y*t2;
209 p_h = y1*t1;
210 z = p_l+p_h;
211 GET_FLOAT_WORD(j,z);
212 if (j>0x43000000) /* if z > 128 */
213 return sn*huge*huge; /* overflow */
214 else if (j==0x43000000) { /* if z == 128 */
215 if(p_l+ovt>z-p_h) return sn*huge*huge; /* overflow */
216 }
217 else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */
218 return sn*tiny*tiny; /* underflow */
219 else if (j==0xc3160000){ /* z == -150 */
220 if(p_l<=z-p_h) return sn*tiny*tiny; /* underflow */
221 }
222 /*
223 * compute 2**(p_h+p_l)
224 */
225 i = j&0x7fffffff;
226 k = (i>>23)-0x7f;
227 n = 0;
228 if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
229 n = j+(0x00800000>>(k+1));
230 k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */
231 SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
232 n = ((n&0x007fffff)|0x00800000)>>(23-k);
233 if(j<0) n = -n;
234 p_h -= t;
235 }
236 t = p_l+p_h;
237 GET_FLOAT_WORD(is,t);
238 SET_FLOAT_WORD(t,is&0xffff8000);
239 u = t*lg2_h;
240 v = (p_l-(t-p_h))*lg2+t*lg2_l;
241 z = u+v;
242 w = v-(z-u);
243 t = z*z;
244 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
245 r = (z*t1)/(t1-two)-(w+z*w);
246 z = one-(r-z);
247 GET_FLOAT_WORD(j,z);
248 j += (n<<23);
249 if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */
250 else SET_FLOAT_WORD(z,j);
251 return sn*z;
252 }
253