]>
Commit | Line | Data |
---|---|---|
f7eac6eb RM |
1 | /* @(#)e_pow.c 5.1 93/09/24 */ |
2 | /* | |
3 | * ==================================================== | |
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | |
5 | * | |
6 | * Developed at SunPro, a Sun Microsystems, Inc. business. | |
7 | * Permission to use, copy, modify, and distribute this | |
ba1ffaa1 | 8 | * software is freely granted, provided that this notice |
f7eac6eb RM |
9 | * is preserved. |
10 | * ==================================================== | |
11 | */ | |
923609d1 UD |
12 | /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, |
13 | for performance improvement on pipelined processors. | |
14 | */ | |
f7eac6eb RM |
15 | |
16 | #if defined(LIBM_SCCS) && !defined(lint) | |
17 | static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $"; | |
18 | #endif | |
19 | ||
20 | /* __ieee754_pow(x,y) return x**y | |
21 | * | |
22 | * n | |
23 | * Method: Let x = 2 * (1+f) | |
24 | * 1. Compute and return log2(x) in two pieces: | |
25 | * log2(x) = w1 + w2, | |
26 | * where w1 has 53-24 = 29 bit trailing zeros. | |
ba1ffaa1 | 27 | * 2. Perform y*log2(x) = n+y' by simulating muti-precision |
f7eac6eb RM |
28 | * arithmetic, where |y'|<=0.5. |
29 | * 3. Return x**y = 2**n*exp(y'*log2) | |
30 | * | |
31 | * Special cases: | |
32 | * 1. (anything) ** 0 is 1 | |
33 | * 2. (anything) ** 1 is itself | |
34 | * 3. (anything) ** NAN is NAN | |
35 | * 4. NAN ** (anything except 0) is NAN | |
36 | * 5. +-(|x| > 1) ** +INF is +INF | |
37 | * 6. +-(|x| > 1) ** -INF is +0 | |
38 | * 7. +-(|x| < 1) ** +INF is +0 | |
39 | * 8. +-(|x| < 1) ** -INF is +INF | |
40 | * 9. +-1 ** +-INF is NAN | |
41 | * 10. +0 ** (+anything except 0, NAN) is +0 | |
42 | * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 | |
43 | * 12. +0 ** (-anything except 0, NAN) is +INF | |
44 | * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF | |
45 | * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) | |
46 | * 15. +INF ** (+anything except 0,NAN) is +INF | |
47 | * 16. +INF ** (-anything except 0,NAN) is +0 | |
48 | * 17. -INF ** (anything) = -0 ** (-anything) | |
49 | * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) | |
50 | * 19. (-anything except 0 and inf) ** (non-integer) is NAN | |
51 | * | |
52 | * Accuracy: | |
53 | * pow(x,y) returns x**y nearly rounded. In particular | |
54 | * pow(integer,integer) | |
ba1ffaa1 | 55 | * always returns the correct integer provided it is |
f7eac6eb RM |
56 | * representable. |
57 | * | |
58 | * Constants : | |
ba1ffaa1 UD |
59 | * The hexadecimal values are the intended ones for the following |
60 | * constants. The decimal values may be used, provided that the | |
61 | * compiler will convert from decimal to binary accurately enough | |
f7eac6eb RM |
62 | * to produce the hexadecimal values shown. |
63 | */ | |
64 | ||
65 | #include "math.h" | |
66 | #include "math_private.h" | |
923609d1 UD |
67 | #define zero C[0] |
68 | #define one C[1] | |
69 | #define two C[2] | |
70 | #define two53 C[3] | |
71 | #define huge C[4] | |
72 | #define tiny C[5] | |
73 | #define L1 C[6] | |
74 | #define L2 C[7] | |
75 | #define L3 C[8] | |
76 | #define L4 C[9] | |
77 | #define L5 C[10] | |
78 | #define L6 C[11] | |
79 | #define P1 C[12] | |
80 | #define P2 C[13] | |
81 | #define P3 C[14] | |
82 | #define P4 C[15] | |
83 | #define P5 C[16] | |
84 | #define lg2 C[17] | |
85 | #define lg2_h C[18] | |
86 | #define lg2_l C[19] | |
87 | #define ovt C[20] | |
88 | #define cp C[21] | |
89 | #define cp_h C[22] | |
90 | #define cp_l C[23] | |
91 | #define ivln2 C[24] | |
92 | #define ivln2_h C[25] | |
93 | #define ivln2_l C[26] | |
f7eac6eb RM |
94 | |
95 | #ifdef __STDC__ | |
ba1ffaa1 | 96 | static const double |
f7eac6eb | 97 | #else |
ba1ffaa1 | 98 | static double |
f7eac6eb RM |
99 | #endif |
100 | bp[] = {1.0, 1.5,}, | |
101 | dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ | |
102 | dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ | |
923609d1 UD |
103 | C[] = { |
104 | 0.0, | |
105 | 1.0, | |
106 | 2.0, | |
107 | 9007199254740992.0 , | |
108 | 1.0e300, | |
109 | 1.0e-300, | |
110 | 5.99999999999994648725e-01 , | |
111 | 4.28571428578550184252e-01 , | |
112 | 3.33333329818377432918e-01 , | |
113 | 2.72728123808534006489e-01 , | |
114 | 2.30660745775561754067e-01 , | |
115 | 2.06975017800338417784e-01 , | |
116 | 1.66666666666666019037e-01 , | |
117 | -2.77777777770155933842e-03 , | |
118 | 6.61375632143793436117e-05 , | |
119 | -1.65339022054652515390e-06 , | |
120 | 4.13813679705723846039e-08 , | |
121 | 6.93147180559945286227e-01 , | |
122 | 6.93147182464599609375e-01 , | |
123 | -1.90465429995776804525e-09 , | |
124 | 8.0085662595372944372e-0017 , | |
125 | 9.61796693925975554329e-01 , | |
126 | 9.61796700954437255859e-01 , | |
127 | -7.02846165095275826516e-09 , | |
128 | 1.44269504088896338700e+00 , | |
129 | 1.44269502162933349609e+00 , | |
130 | 1.92596299112661746887e-08 }; | |
f7eac6eb RM |
131 | |
132 | #ifdef __STDC__ | |
133 | double __ieee754_pow(double x, double y) | |
134 | #else | |
135 | double __ieee754_pow(x,y) | |
136 | double x, y; | |
137 | #endif | |
138 | { | |
139 | double z,ax,z_h,z_l,p_h,p_l; | |
923609d1 | 140 | double y1,t1,t2,r,s,t,u,v,w, t12,t14,r_1,r_2,r_3; |
f7eac6eb RM |
141 | int32_t i,j,k,yisint,n; |
142 | int32_t hx,hy,ix,iy; | |
143 | u_int32_t lx,ly; | |
144 | ||
145 | EXTRACT_WORDS(hx,lx,x); | |
146 | EXTRACT_WORDS(hy,ly,y); | |
147 | ix = hx&0x7fffffff; iy = hy&0x7fffffff; | |
148 | ||
149 | /* y==zero: x**0 = 1 */ | |
923609d1 | 150 | if((iy|ly)==0) return C[1]; |
f7eac6eb | 151 | |
f14bd805 UD |
152 | /* x==+-1 */ |
153 | if(x == 1.0) return C[1]; | |
154 | if(x == -1.0 && isinf(y)) return C[1]; | |
155 | ||
f7eac6eb RM |
156 | /* +-NaN return x+y */ |
157 | if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || | |
ba1ffaa1 UD |
158 | iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) |
159 | return x+y; | |
f7eac6eb RM |
160 | |
161 | /* determine if y is an odd int when x < 0 | |
162 | * yisint = 0 ... y is not an integer | |
163 | * yisint = 1 ... y is an odd int | |
164 | * yisint = 2 ... y is an even int | |
165 | */ | |
166 | yisint = 0; | |
ba1ffaa1 | 167 | if(hx<0) { |
f7eac6eb RM |
168 | if(iy>=0x43400000) yisint = 2; /* even integer y */ |
169 | else if(iy>=0x3ff00000) { | |
170 | k = (iy>>20)-0x3ff; /* exponent */ | |
171 | if(k>20) { | |
172 | j = ly>>(52-k); | |
ba1ffaa1 | 173 | if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1); |
f7eac6eb RM |
174 | } else if(ly==0) { |
175 | j = iy>>(20-k); | |
ba1ffaa1 | 176 | if((int32_t)(j<<(20-k))==iy) yisint = 2-(j&1); |
f7eac6eb | 177 | } |
ba1ffaa1 UD |
178 | } |
179 | } | |
f7eac6eb RM |
180 | |
181 | /* special value of y */ | |
ba1ffaa1 | 182 | if(ly==0) { |
f7eac6eb RM |
183 | if (iy==0x7ff00000) { /* y is +-inf */ |
184 | if(((ix-0x3ff00000)|lx)==0) | |
185 | return y - y; /* inf**+-1 is NaN */ | |
186 | else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ | |
923609d1 | 187 | return (hy>=0)? y: C[0]; |
f7eac6eb | 188 | else /* (|x|<1)**-,+inf = inf,0 */ |
923609d1 | 189 | return (hy<0)?-y: C[0]; |
ba1ffaa1 | 190 | } |
f7eac6eb | 191 | if(iy==0x3ff00000) { /* y is +-1 */ |
923609d1 | 192 | if(hy<0) return C[1]/x; else return x; |
f7eac6eb RM |
193 | } |
194 | if(hy==0x40000000) return x*x; /* y is 2 */ | |
195 | if(hy==0x3fe00000) { /* y is 0.5 */ | |
196 | if(hx>=0) /* x >= +0 */ | |
ba1ffaa1 | 197 | return __ieee754_sqrt(x); |
f7eac6eb RM |
198 | } |
199 | } | |
200 | ||
201 | ax = fabs(x); | |
202 | /* special value of x */ | |
203 | if(lx==0) { | |
204 | if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ | |
205 | z = ax; /*x is +-0,+-inf,+-1*/ | |
923609d1 | 206 | if(hy<0) z = C[1]/z; /* z = (1/|x|) */ |
f7eac6eb RM |
207 | if(hx<0) { |
208 | if(((ix-0x3ff00000)|yisint)==0) { | |
209 | z = (z-z)/(z-z); /* (-1)**non-int is NaN */ | |
ba1ffaa1 | 210 | } else if(yisint==1) |
f7eac6eb RM |
211 | z = -z; /* (x<0)**odd = -(|x|**odd) */ |
212 | } | |
213 | return z; | |
214 | } | |
215 | } | |
ba1ffaa1 | 216 | |
f7eac6eb RM |
217 | /* (x<0)**(non-int) is NaN */ |
218 | if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); | |
219 | ||
220 | /* |y| is huge */ | |
221 | if(iy>0x41e00000) { /* if |y| > 2**31 */ | |
222 | if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ | |
923609d1 UD |
223 | if(ix<=0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5]; |
224 | if(ix>=0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5]; | |
f7eac6eb RM |
225 | } |
226 | /* over/underflow if x is not close to one */ | |
923609d1 UD |
227 | if(ix<0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5]; |
228 | if(ix>0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5]; | |
ba1ffaa1 | 229 | /* now |1-x| is tiny <= 2**-20, suffice to compute |
f7eac6eb RM |
230 | log(x) by x-x^2/2+x^3/3-x^4/4 */ |
231 | t = x-1; /* t has 20 trailing zeros */ | |
232 | w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); | |
923609d1 UD |
233 | u = C[25]*t; /* ivln2_h has 21 sig. bits */ |
234 | v = t*C[26]-w*C[24]; | |
f7eac6eb RM |
235 | t1 = u+v; |
236 | SET_LOW_WORD(t1,0); | |
237 | t2 = v-(t1-u); | |
238 | } else { | |
923609d1 | 239 | double s2,s_h,s_l,t_h,t_l,s22,s24,s26,r1,r2,r3; |
f7eac6eb RM |
240 | n = 0; |
241 | /* take care subnormal number */ | |
242 | if(ix<0x00100000) | |
923609d1 | 243 | {ax *= C[3]; n -= 53; GET_HIGH_WORD(ix,ax); } |
f7eac6eb RM |
244 | n += ((ix)>>20)-0x3ff; |
245 | j = ix&0x000fffff; | |
246 | /* determine interval */ | |
247 | ix = j|0x3ff00000; /* normalize ix */ | |
248 | if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ | |
249 | else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ | |
250 | else {k=0;n+=1;ix -= 0x00100000;} | |
251 | SET_HIGH_WORD(ax,ix); | |
252 | ||
253 | /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ | |
254 | u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ | |
923609d1 | 255 | v = C[1]/(ax+bp[k]); |
f7eac6eb RM |
256 | s = u*v; |
257 | s_h = s; | |
258 | SET_LOW_WORD(s_h,0); | |
259 | /* t_h=ax+bp[k] High */ | |
923609d1 | 260 | t_h = C[0]; |
f7eac6eb RM |
261 | SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); |
262 | t_l = ax - (t_h-bp[k]); | |
263 | s_l = v*((u-s_h*t_h)-s_h*t_l); | |
264 | /* compute log(ax) */ | |
265 | s2 = s*s; | |
923609d1 | 266 | #ifdef DO_NOT_USE_THIS |
f7eac6eb | 267 | r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); |
923609d1 UD |
268 | #else |
269 | r1 = C[10]+s2*C[11]; s22=s2*s2; | |
270 | r2 = C[8]+s2*C[9]; s24=s22*s22; | |
271 | r3 = C[6]+s2*C[7]; s26=s24*s22; | |
272 | r = r3*s22 + r2*s24 + r1*s26; | |
b9337b6a | 273 | #endif |
f7eac6eb RM |
274 | r += s_l*(s_h+s); |
275 | s2 = s_h*s_h; | |
276 | t_h = 3.0+s2+r; | |
277 | SET_LOW_WORD(t_h,0); | |
278 | t_l = r-((t_h-3.0)-s2); | |
279 | /* u+v = s*(1+...) */ | |
280 | u = s_h*t_h; | |
281 | v = s_l*t_h+t_l*s; | |
282 | /* 2/(3log2)*(s+...) */ | |
283 | p_h = u+v; | |
284 | SET_LOW_WORD(p_h,0); | |
285 | p_l = v-(p_h-u); | |
923609d1 UD |
286 | z_h = C[22]*p_h; /* cp_h+cp_l = 2/(3*log2) */ |
287 | z_l = C[23]*p_h+p_l*C[21]+dp_l[k]; | |
f7eac6eb RM |
288 | /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ |
289 | t = (double)n; | |
290 | t1 = (((z_h+z_l)+dp_h[k])+t); | |
291 | SET_LOW_WORD(t1,0); | |
292 | t2 = z_l-(((t1-t)-dp_h[k])-z_h); | |
293 | } | |
294 | ||
923609d1 | 295 | s = C[1]; /* s (sign of result -ve**odd) = -1 else = 1 */ |
f7eac6eb | 296 | if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) |
923609d1 | 297 | s = -C[1];/* (-ve)**(odd int) */ |
f7eac6eb RM |
298 | |
299 | /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ | |
300 | y1 = y; | |
301 | SET_LOW_WORD(y1,0); | |
302 | p_l = (y-y1)*t1+y*t2; | |
303 | p_h = y1*t1; | |
304 | z = p_l+p_h; | |
305 | EXTRACT_WORDS(j,i,z); | |
306 | if (j>=0x40900000) { /* z >= 1024 */ | |
307 | if(((j-0x40900000)|i)!=0) /* if z > 1024 */ | |
923609d1 | 308 | return s*C[4]*C[4]; /* overflow */ |
f7eac6eb | 309 | else { |
923609d1 | 310 | if(p_l+C[20]>z-p_h) return s*C[4]*C[4]; /* overflow */ |
f7eac6eb RM |
311 | } |
312 | } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ | |
313 | if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ | |
923609d1 | 314 | return s*C[5]*C[5]; /* underflow */ |
f7eac6eb | 315 | else { |
923609d1 | 316 | if(p_l<=z-p_h) return s*C[5]*C[5]; /* underflow */ |
f7eac6eb RM |
317 | } |
318 | } | |
319 | /* | |
320 | * compute 2**(p_h+p_l) | |
321 | */ | |
322 | i = j&0x7fffffff; | |
323 | k = (i>>20)-0x3ff; | |
324 | n = 0; | |
325 | if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ | |
326 | n = j+(0x00100000>>(k+1)); | |
327 | k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ | |
923609d1 | 328 | t = C[0]; |
f7eac6eb RM |
329 | SET_HIGH_WORD(t,n&~(0x000fffff>>k)); |
330 | n = ((n&0x000fffff)|0x00100000)>>(20-k); | |
331 | if(j<0) n = -n; | |
332 | p_h -= t; | |
ba1ffaa1 | 333 | } |
f7eac6eb RM |
334 | t = p_l+p_h; |
335 | SET_LOW_WORD(t,0); | |
923609d1 UD |
336 | u = t*C[18]; |
337 | v = (p_l-(t-p_h))*C[17]+t*C[19]; | |
f7eac6eb RM |
338 | z = u+v; |
339 | w = v-(z-u); | |
340 | t = z*z; | |
923609d1 UD |
341 | #ifdef DO_NOT_USE_THIS |
342 | t1 = z - t*(C[12]+t*(C[13]+t*(C[14]+t*(C[15]+t*C[16])))); | |
343 | #else | |
344 | r_1 = C[15]+t*C[16]; t12 = t*t; | |
345 | r_2 = C[13]+t*C[14]; t14 = t12*t12; | |
346 | r_3 = t*C[12]; | |
347 | t1 = z - r_3 - t12*r_2 - t14*r_1; | |
348 | #endif | |
349 | r = (z*t1)/(t1-C[2])-(w+z*w); | |
350 | z = C[1]-(r-z); | |
f7eac6eb RM |
351 | GET_HIGH_WORD(j,z); |
352 | j += (n<<20); | |
353 | if((j>>20)<=0) z = __scalbn(z,n); /* subnormal output */ | |
354 | else SET_HIGH_WORD(z,j); | |
355 | return s*z; | |
356 | } |