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 && !issignaling (x)) return one; |
66 | |
67 | /* x==+-1 */ |
68 | if(x == 1.0 && !issignaling (y)) 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 = ax-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, |
170 | ((((ix>>1)|0x20000000)+0x00400000+(k<<21)) |
171 | & 0xfffff000)); |
172 | t_l = ax - (t_h-bp[k]); |
173 | s_l = v*((u-s_h*t_h)-s_h*t_l); |
174 | /* compute log(ax) */ |
175 | s2 = s*s; |
176 | r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); |
177 | r += s_l*(s_h+s); |
178 | s2 = s_h*s_h; |
179 | t_h = (float)3.0+s2+r; |
180 | GET_FLOAT_WORD(is,t_h); |
181 | SET_FLOAT_WORD(t_h,is&0xfffff000); |
182 | t_l = r-((t_h-(float)3.0)-s2); |
183 | /* u+v = s*(1+...) */ |
184 | u = s_h*t_h; |
185 | v = s_l*t_h+t_l*s; |
186 | /* 2/(3log2)*(s+...) */ |
187 | p_h = u+v; |
188 | GET_FLOAT_WORD(is,p_h); |
189 | SET_FLOAT_WORD(p_h,is&0xfffff000); |
190 | p_l = v-(p_h-u); |
191 | z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ |
192 | z_l = cp_l*p_h+p_l*cp+dp_l[k]; |
193 | /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ |
194 | t = (float)n; |
195 | t1 = (((z_h+z_l)+dp_h[k])+t); |
196 | GET_FLOAT_WORD(is,t1); |
197 | SET_FLOAT_WORD(t1,is&0xfffff000); |
198 | t2 = z_l-(((t1-t)-dp_h[k])-z_h); |
199 | } |
200 | |
201 | s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ |
202 | if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) |
203 | s = -one; /* (-ve)**(odd int) */ |
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 (__builtin_expect(j>0x43000000, 0)) /* if z > 128 */ |
213 | return s*huge*huge; /* overflow */ |
214 | else if (__builtin_expect(j==0x43000000, 0)) { /* if z == 128 */ |
215 | if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ |
216 | } |
217 | else if (__builtin_expect((j&0x7fffffff)>0x43160000, 0))/* z <= -150 */ |
218 | return s*tiny*tiny; /* underflow */ |
219 | else if (__builtin_expect((u_int32_t) j==0xc3160000, 0)){/* z == -150*/ |
220 | if(p_l<=z-p_h) return s*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&0xfffff000); |
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) /* subnormal output */ |
250 | { |
251 | z = __scalbnf (z, n); |
252 | float force_underflow = z * z; |
253 | math_force_eval (force_underflow); |
254 | } |
255 | else SET_FLOAT_WORD(z,j); |
256 | return s*z; |
257 | } |
258 | strong_alias (__ieee754_powf, __powf_finite) |
259 | |