]>
Commit | Line | Data |
---|---|---|
f7eac6eb | 1 | /* |
e4d82761 UD |
2 | * IBM Accurate Mathematical Library |
3 | * Copyright (c) International Business Machines Corp., 2001 | |
f7eac6eb | 4 | * |
e4d82761 UD |
5 | * This program is free software; you can redistribute it and/or modify |
6 | * it under the terms of the GNU Lesser General Public License as published by | |
7 | * the Free Software Foundation; either version 2 of the License, or | |
8 | * (at your option) any later version. | |
f7eac6eb | 9 | * |
e4d82761 UD |
10 | * This program is distributed in the hope that it will be useful, |
11 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
12 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
13 | * GNU General Public License for more details. | |
f7eac6eb | 14 | * |
e4d82761 UD |
15 | * You should have received a copy of the GNU Lesser General Public License |
16 | * along with this program; if not, write to the Free Software | |
17 | * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. | |
f7eac6eb | 18 | */ |
e4d82761 UD |
19 | /***************************************************************************/ |
20 | /* MODULE_NAME: upow.c */ | |
21 | /* */ | |
22 | /* FUNCTIONS: upow */ | |
23 | /* power1 */ | |
24 | /* log2 */ | |
25 | /* log1 */ | |
26 | /* checkint */ | |
27 | /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h */ | |
28 | /* halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c */ | |
29 | /* uexp.c upow.c */ | |
30 | /* root.tbl uexp.tbl upow.tbl */ | |
31 | /* An ultimate power routine. Given two IEEE double machine numbers y,x */ | |
32 | /* it computes the correctly rounded (to nearest) value of x^y. */ | |
33 | /* Assumption: Machine arithmetic operations are performed in */ | |
34 | /* round to nearest mode of IEEE 754 standard. */ | |
35 | /* */ | |
36 | /***************************************************************************/ | |
37 | #include "endian.h" | |
38 | #include "upow.h" | |
39 | #include "dla.h" | |
40 | #include "mydefs.h" | |
41 | #include "MathLib.h" | |
42 | #include "upow.tbl" | |
f7eac6eb | 43 | |
f7eac6eb | 44 | |
e4d82761 UD |
45 | double __exp1(double x, double xx, double error); |
46 | static double log1(double x, double *delta, double *error); | |
47 | static double log2(double x, double *delta, double *error); | |
48 | double slowpow(double x, double y,double z); | |
49 | static double power1(double x, double y); | |
50 | static int checkint(double x); | |
f7eac6eb | 51 | |
e4d82761 UD |
52 | /***************************************************************************/ |
53 | /* An ultimate power routine. Given two IEEE double machine numbers y,x */ | |
54 | /* it computes the correctly rounded (to nearest) value of X^y. */ | |
55 | /***************************************************************************/ | |
9a656848 | 56 | double __ieee754_pow(double x, double y) { |
50944bca UD |
57 | double z,a,aa,error, t,a1,a2,y1,y2; |
58 | #if 0 | |
59 | double gor=1.0; | |
60 | #endif | |
e4d82761 UD |
61 | mynumber u,v; |
62 | int k; | |
63 | int4 qx,qy; | |
64 | v.x=y; | |
65 | u.x=x; | |
66 | if (v.i[LOW_HALF] == 0) { /* of y */ | |
67 | qx = u.i[HIGH_HALF]&0x7fffffff; | |
68 | /* Checking if x is not too small to compute */ | |
69 | if (((qx==0x7ff00000)&&(u.i[LOW_HALF]!=0))||(qx>0x7ff00000)) return NaNQ.x; | |
70 | if (y == 1.0) return x; | |
71 | if (y == 2.0) return x*x; | |
72 | if (y == -1.0) return (x!=0)?1.0/x:NaNQ.x; | |
73 | if (y == 0) return ((x>0)&&(qx<0x7ff00000))?1.0:NaNQ.x; | |
74 | } | |
75 | /* else */ | |
76 | if(((u.i[HIGH_HALF]>0 && u.i[HIGH_HALF]<0x7ff00000)|| /* x>0 and not x->0 */ | |
77 | (u.i[HIGH_HALF]==0 && u.i[LOW_HALF]!=0)) && | |
78 | /* 2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */ | |
79 | (v.i[HIGH_HALF]&0x7fffffff) < 0x4ff00000) { /* if y<-1 or y>1 */ | |
80 | z = log1(x,&aa,&error); /* x^y =e^(y log (X)) */ | |
81 | t = y*134217729.0; | |
82 | y1 = t - (t-y); | |
83 | y2 = y - y1; | |
84 | t = z*134217729.0; | |
85 | a1 = t - (t-z); | |
86 | a2 = (z - a1)+aa; | |
87 | a = y1*a1; | |
88 | aa = y2*a1 + y*a2; | |
89 | a1 = a+aa; | |
90 | a2 = (a-a1)+aa; | |
91 | error = error*ABS(y); | |
92 | t = __exp1(a1,a2,1.9e16*error); /* return -10 or 0 if wasn't computed exactly */ | |
93 | return (t>0)?t:power1(x,y); | |
94 | } | |
f7eac6eb | 95 | |
e4d82761 UD |
96 | if (x == 0) { |
97 | if (ABS(y) > 1.0e20) return (y>0)?0:NaNQ.x; | |
98 | k = checkint(y); | |
99 | if (k == 0 || y < 0) return NaNQ.x; | |
100 | else return (k==1)?0:x; /* return 0 */ | |
101 | } | |
102 | /* if x<0 */ | |
103 | if (u.i[HIGH_HALF] < 0) { | |
104 | k = checkint(y); | |
105 | if (k==0) return NaNQ.x; /* y not integer and x<0 */ | |
106 | return (k==1)?upow(-x,y):-upow(-x,y); /* if y even or odd */ | |
107 | } | |
108 | /* x>0 */ | |
109 | qx = u.i[HIGH_HALF]&0x7fffffff; /* no sign */ | |
110 | qy = v.i[HIGH_HALF]&0x7fffffff; /* no sign */ | |
f7eac6eb | 111 | |
e4d82761 UD |
112 | if (qx > 0x7ff00000 || (qx == 0x7ff00000 && u.i[LOW_HALF] != 0)) return NaNQ.x; |
113 | /* if 0<x<2^-0x7fe */ | |
114 | if (qy > 0x7ff00000 || (qy == 0x7ff00000 && v.i[LOW_HALF] != 0)) return NaNQ.x; | |
115 | /* if y<2^-0x7fe */ | |
f7eac6eb | 116 | |
e4d82761 UD |
117 | if (qx == 0x7ff00000) /* x= 2^-0x3ff */ |
118 | {if (y == 0) return NaNQ.x; | |
119 | return (y>0)?x:0; } | |
f14bd805 | 120 | |
e4d82761 UD |
121 | if (qy > 0x45f00000 && qy < 0x7ff00000) { |
122 | if (x == 1.0) return 1.0; | |
123 | if (y>0) return (x>1.0)?INF.x:0; | |
124 | if (y<0) return (x<1.0)?INF.x:0; | |
125 | } | |
f7eac6eb | 126 | |
e4d82761 UD |
127 | if (x == 1.0) return NaNQ.x; |
128 | if (y>0) return (x>1.0)?INF.x:0; | |
129 | if (y<0) return (x<1.0)?INF.x:0; | |
130 | return 0; /* unreachable, to make the compiler happy */ | |
131 | } | |
f7eac6eb | 132 | |
e4d82761 UD |
133 | /**************************************************************************/ |
134 | /* Computing x^y using more accurate but more slow log routine */ | |
135 | /**************************************************************************/ | |
136 | static double power1(double x, double y) { | |
137 | double z,a,aa,error, t,a1,a2,y1,y2; | |
138 | z = log2(x,&aa,&error); | |
139 | t = y*134217729.0; | |
140 | y1 = t - (t-y); | |
141 | y2 = y - y1; | |
142 | t = z*134217729.0; | |
143 | a1 = t - (t-z); | |
144 | a2 = z - a1; | |
145 | a = y*z; | |
146 | aa = ((y1*a1-a)+y1*a2+y2*a1)+y2*a2+aa*y; | |
147 | a1 = a+aa; | |
148 | a2 = (a-a1)+aa; | |
149 | error = error*ABS(y); | |
150 | t = __exp1(a1,a2,1.9e16*error); | |
151 | return (t >= 0)?t:slowpow(x,y,z); | |
152 | } | |
f7eac6eb | 153 | |
e4d82761 UD |
154 | /****************************************************************************/ |
155 | /* Computing log(x) (x is left argument). The result is the returned double */ | |
156 | /* + the parameter delta. */ | |
157 | /* The result is bounded by error (rightmost argument) */ | |
158 | /****************************************************************************/ | |
159 | static double log1(double x, double *delta, double *error) { | |
50944bca UD |
160 | int i,j,m; |
161 | #if 0 | |
162 | int n; | |
163 | #endif | |
164 | double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0; | |
165 | #if 0 | |
166 | double cor; | |
167 | #endif | |
e4d82761 | 168 | mynumber u,v; |
ba1ffaa1 | 169 | |
e4d82761 UD |
170 | u.x = x; |
171 | m = u.i[HIGH_HALF]; | |
172 | *error = 0; | |
173 | *delta = 0; | |
174 | if (m < 0x00100000) /* 1<x<2^-1007 */ | |
175 | { x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF];} | |
f7eac6eb | 176 | |
e4d82761 UD |
177 | if ((m&0x000fffff) < 0x0006a09e) |
178 | {u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); } | |
179 | else | |
180 | {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; } | |
f7eac6eb | 181 | |
e4d82761 UD |
182 | v.x = u.x + bigu.x; |
183 | uu = v.x - bigu.x; | |
184 | i = (v.i[LOW_HALF]&0x000003ff)<<2; | |
185 | if (two52.i[LOW_HALF] == 1023) /* nx = 0 */ | |
186 | { | |
187 | if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10 */ | |
188 | { | |
189 | t = x - 1.0; | |
190 | t1 = (t+5.0e6)-5.0e6; | |
191 | t2 = t-t1; | |
192 | e1 = t - 0.5*t1*t1; | |
193 | e2 = t*t*t*(r3+t*(r4+t*(r5+t*(r6+t*(r7+t*r8)))))-0.5*t2*(t+t1); | |
194 | res = e1+e2; | |
195 | *error = 1.0e-21*ABS(t); | |
196 | *delta = (e1-res)+e2; | |
197 | return res; | |
198 | } /* |x-1| < 1.5*2**-10 */ | |
199 | else | |
200 | { | |
201 | v.x = u.x*(ui.x[i]+ui.x[i+1])+bigv.x; | |
202 | vv = v.x-bigv.x; | |
203 | j = v.i[LOW_HALF]&0x0007ffff; | |
204 | j = j+j+j; | |
205 | eps = u.x - uu*vv; | |
206 | e1 = eps*ui.x[i]; | |
207 | e2 = eps*(ui.x[i+1]+vj.x[j]*(ui.x[i]+ui.x[i+1])); | |
208 | e = e1+e2; | |
209 | e2 = ((e1-e)+e2); | |
210 | t=ui.x[i+2]+vj.x[j+1]; | |
211 | t1 = t+e; | |
212 | t2 = (((t-t1)+e)+(ui.x[i+3]+vj.x[j+2]))+e2+e*e*(p2+e*(p3+e*p4)); | |
213 | res=t1+t2; | |
214 | *error = 1.0e-24; | |
215 | *delta = (t1-res)+t2; | |
216 | return res; | |
217 | } | |
218 | } /* nx = 0 */ | |
219 | else /* nx != 0 */ | |
220 | { | |
221 | eps = u.x - uu; | |
222 | nx = (two52.x - two52e.x)+add; | |
223 | e1 = eps*ui.x[i]; | |
224 | e2 = eps*ui.x[i+1]; | |
225 | e=e1+e2; | |
226 | e2 = (e1-e)+e2; | |
227 | t=nx*ln2a.x+ui.x[i+2]; | |
228 | t1=t+e; | |
229 | t2=(((t-t1)+e)+nx*ln2b.x+ui.x[i+3]+e2)+e*e*(q2+e*(q3+e*(q4+e*(q5+e*q6)))); | |
230 | res = t1+t2; | |
231 | *error = 1.0e-21; | |
232 | *delta = (t1-res)+t2; | |
233 | return res; | |
234 | } /* nx != 0 */ | |
235 | } | |
236 | ||
237 | /****************************************************************************/ | |
238 | /* More slow but more accurate routine of log */ | |
239 | /* Computing log(x)(x is left argument).The result is return double + delta.*/ | |
240 | /* The result is bounded by error (right argument) */ | |
241 | /****************************************************************************/ | |
242 | static double log2(double x, double *delta, double *error) { | |
50944bca UD |
243 | int i,j,m; |
244 | #if 0 | |
245 | int n; | |
246 | #endif | |
247 | double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0; | |
248 | #if 0 | |
249 | double cor; | |
250 | #endif | |
e4d82761 UD |
251 | double ou1,ou2,lu1,lu2,ov,lv1,lv2,a,a1,a2; |
252 | double y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8; | |
253 | mynumber u,v; | |
254 | ||
255 | u.x = x; | |
256 | m = u.i[HIGH_HALF]; | |
257 | *error = 0; | |
258 | *delta = 0; | |
259 | add=0; | |
260 | if (m<0x00100000) { /* x < 2^-1022 */ | |
261 | x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF]; } | |
f7eac6eb | 262 | |
e4d82761 UD |
263 | if ((m&0x000fffff) < 0x0006a09e) |
264 | {u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); } | |
265 | else | |
266 | {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; } | |
267 | ||
268 | v.x = u.x + bigu.x; | |
269 | uu = v.x - bigu.x; | |
270 | i = (v.i[LOW_HALF]&0x000003ff)<<2; | |
271 | /*------------------------------------- |x-1| < 2**-11------------------------------- */ | |
272 | if ((two52.i[LOW_HALF] == 1023) && (i == 1200)) | |
273 | { | |
274 | t = x - 1.0; | |
275 | EMULV(t,s3,y,yy,j1,j2,j3,j4,j5); | |
276 | ADD2(-0.5,0,y,yy,z,zz,j1,j2); | |
277 | MUL2(t,0,z,zz,y,yy,j1,j2,j3,j4,j5,j6,j7,j8); | |
278 | MUL2(t,0,y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8); | |
279 | ||
280 | e1 = t+z; | |
281 | e2 = (((t-e1)+z)+zz)+t*t*t*(ss3+t*(s4+t*(s5+t*(s6+t*(s7+t*s8))))); | |
282 | res = e1+e2; | |
283 | *error = 1.0e-25*ABS(t); | |
284 | *delta = (e1-res)+e2; | |
285 | return res; | |
286 | } | |
287 | /*----------------------------- |x-1| > 2**-11 -------------------------- */ | |
288 | else | |
289 | { /*Computing log(x) according to log table */ | |
290 | nx = (two52.x - two52e.x)+add; | |
291 | ou1 = ui.x[i]; | |
292 | ou2 = ui.x[i+1]; | |
293 | lu1 = ui.x[i+2]; | |
294 | lu2 = ui.x[i+3]; | |
295 | v.x = u.x*(ou1+ou2)+bigv.x; | |
296 | vv = v.x-bigv.x; | |
297 | j = v.i[LOW_HALF]&0x0007ffff; | |
298 | j = j+j+j; | |
299 | eps = u.x - uu*vv; | |
300 | ov = vj.x[j]; | |
301 | lv1 = vj.x[j+1]; | |
302 | lv2 = vj.x[j+2]; | |
303 | a = (ou1+ou2)*(1.0+ov); | |
304 | a1 = (a+1.0e10)-1.0e10; | |
305 | a2 = a*(1.0-a1*uu*vv); | |
306 | e1 = eps*a1; | |
307 | e2 = eps*a2; | |
308 | e = e1+e2; | |
309 | e2 = (e1-e)+e2; | |
310 | t=nx*ln2a.x+lu1+lv1; | |
311 | t1 = t+e; | |
312 | t2 = (((t-t1)+e)+(lu2+lv2+nx*ln2b.x+e2))+e*e*(p2+e*(p3+e*p4)); | |
313 | res=t1+t2; | |
314 | *error = 1.0e-27; | |
315 | *delta = (t1-res)+t2; | |
316 | return res; | |
317 | } | |
318 | } | |
f7eac6eb | 319 | |
e4d82761 UD |
320 | /**********************************************************************/ |
321 | /* Routine receives a double x and checks if it is an integer. If not */ | |
322 | /* it returns 0, else it returns 1 if even or -1 if odd. */ | |
323 | /**********************************************************************/ | |
324 | static int checkint(double x) { | |
325 | union {int4 i[2]; double x;} u; | |
50944bca UD |
326 | int k,m,n; |
327 | #if 0 | |
328 | int l; | |
329 | #endif | |
e4d82761 UD |
330 | u.x = x; |
331 | m = u.i[HIGH_HALF]&0x7fffffff; /* no sign */ | |
332 | if (m >= 0x7ff00000) return 0; /* x is +/-inf or NaN */ | |
333 | if (m >= 0x43400000) return 1; /* |x| >= 2**53 */ | |
334 | if (m < 0x40000000) return 0; /* |x| < 2, can not be 0 or 1 */ | |
335 | n = u.i[LOW_HALF]; | |
336 | k = (m>>20)-1023; /* 1 <= k <= 52 */ | |
337 | if (k == 52) return (n&1)? -1:1; /* odd or even*/ | |
338 | if (k>20) { | |
339 | if (n<<(k-20)) return 0; /* if not integer */ | |
340 | return (n<<(k-21))?-1:1; | |
341 | } | |
342 | if (n) return 0; /*if not integer*/ | |
343 | if (k == 20) return (m&1)? -1:1; | |
344 | if (m<<(k+12)) return 0; | |
345 | return (m<<(k+11))?-1:1; | |
f7eac6eb | 346 | } |