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f7eac6eb | 1 | /* |
e4d82761 | 2 | * IBM Accurate Mathematical Library |
aeb25823 | 3 | * written by International Business Machines Corp. |
0ac5ae23 | 4 | * Copyright (C) 2001, 2002, 2004, 2011 Free Software Foundation |
f7eac6eb | 5 | * |
e4d82761 UD |
6 | * This program is free software; you can redistribute it and/or modify |
7 | * it under the terms of the GNU Lesser General Public License as published by | |
cc7375ce | 8 | * the Free Software Foundation; either version 2.1 of the License, or |
e4d82761 | 9 | * (at your option) any later version. |
f7eac6eb | 10 | * |
e4d82761 UD |
11 | * This program is distributed in the hope that it will be useful, |
12 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
13 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
c6c6dd48 | 14 | * GNU Lesser General Public License for more details. |
f7eac6eb | 15 | * |
e4d82761 UD |
16 | * You should have received a copy of the GNU Lesser General Public License |
17 | * along with this program; if not, write to the Free Software | |
18 | * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. | |
f7eac6eb | 19 | */ |
e4d82761 UD |
20 | /***************************************************************************/ |
21 | /* MODULE_NAME: upow.c */ | |
22 | /* */ | |
23 | /* FUNCTIONS: upow */ | |
24 | /* power1 */ | |
0ac5ae23 | 25 | /* my_log2 */ |
e4d82761 UD |
26 | /* log1 */ |
27 | /* checkint */ | |
28 | /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h */ | |
29 | /* halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c */ | |
0ac5ae23 | 30 | /* uexp.c upow.c */ |
e4d82761 UD |
31 | /* root.tbl uexp.tbl upow.tbl */ |
32 | /* An ultimate power routine. Given two IEEE double machine numbers y,x */ | |
33 | /* it computes the correctly rounded (to nearest) value of x^y. */ | |
34 | /* Assumption: Machine arithmetic operations are performed in */ | |
35 | /* round to nearest mode of IEEE 754 standard. */ | |
36 | /* */ | |
37 | /***************************************************************************/ | |
38 | #include "endian.h" | |
39 | #include "upow.h" | |
c8b3296b | 40 | #include <dla.h> |
e4d82761 UD |
41 | #include "mydefs.h" |
42 | #include "MathLib.h" | |
43 | #include "upow.tbl" | |
e859d1d9 | 44 | #include "math_private.h" |
f7eac6eb | 45 | |
f7eac6eb | 46 | |
e4d82761 UD |
47 | double __exp1(double x, double xx, double error); |
48 | static double log1(double x, double *delta, double *error); | |
1f81acbc | 49 | static double my_log2(double x, double *delta, double *error); |
ca58f1db | 50 | double __slowpow(double x, double y,double z); |
e4d82761 UD |
51 | static double power1(double x, double y); |
52 | static int checkint(double x); | |
f7eac6eb | 53 | |
e4d82761 UD |
54 | /***************************************************************************/ |
55 | /* An ultimate power routine. Given two IEEE double machine numbers y,x */ | |
56 | /* it computes the correctly rounded (to nearest) value of X^y. */ | |
57 | /***************************************************************************/ | |
9a656848 | 58 | double __ieee754_pow(double x, double y) { |
50944bca UD |
59 | double z,a,aa,error, t,a1,a2,y1,y2; |
60 | #if 0 | |
61 | double gor=1.0; | |
62 | #endif | |
e4d82761 UD |
63 | mynumber u,v; |
64 | int k; | |
65 | int4 qx,qy; | |
66 | v.x=y; | |
67 | u.x=x; | |
68 | if (v.i[LOW_HALF] == 0) { /* of y */ | |
69 | qx = u.i[HIGH_HALF]&0x7fffffff; | |
70 | /* Checking if x is not too small to compute */ | |
71 | if (((qx==0x7ff00000)&&(u.i[LOW_HALF]!=0))||(qx>0x7ff00000)) return NaNQ.x; | |
72 | if (y == 1.0) return x; | |
73 | if (y == 2.0) return x*x; | |
ca58f1db UD |
74 | if (y == -1.0) return 1.0/x; |
75 | if (y == 0) return 1.0; | |
e4d82761 UD |
76 | } |
77 | /* else */ | |
78 | if(((u.i[HIGH_HALF]>0 && u.i[HIGH_HALF]<0x7ff00000)|| /* x>0 and not x->0 */ | |
79 | (u.i[HIGH_HALF]==0 && u.i[LOW_HALF]!=0)) && | |
0ac5ae23 | 80 | /* 2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */ |
e4d82761 UD |
81 | (v.i[HIGH_HALF]&0x7fffffff) < 0x4ff00000) { /* if y<-1 or y>1 */ |
82 | z = log1(x,&aa,&error); /* x^y =e^(y log (X)) */ | |
83 | t = y*134217729.0; | |
84 | y1 = t - (t-y); | |
85 | y2 = y - y1; | |
86 | t = z*134217729.0; | |
87 | a1 = t - (t-z); | |
88 | a2 = (z - a1)+aa; | |
89 | a = y1*a1; | |
90 | aa = y2*a1 + y*a2; | |
91 | a1 = a+aa; | |
92 | a2 = (a-a1)+aa; | |
93 | error = error*ABS(y); | |
94 | t = __exp1(a1,a2,1.9e16*error); /* return -10 or 0 if wasn't computed exactly */ | |
95 | return (t>0)?t:power1(x,y); | |
96 | } | |
f7eac6eb | 97 | |
e4d82761 | 98 | if (x == 0) { |
ca58f1db UD |
99 | if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0) |
100 | || (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000) | |
101 | return y; | |
102 | if (ABS(y) > 1.0e20) return (y>0)?0:INF.x; | |
e4d82761 | 103 | k = checkint(y); |
ca58f1db UD |
104 | if (k == -1) |
105 | return y < 0 ? 1.0/x : x; | |
106 | else | |
107 | return y < 0 ? 1.0/ABS(x) : 0.0; /* return 0 */ | |
e4d82761 | 108 | } |
8f3edfee UD |
109 | |
110 | qx = u.i[HIGH_HALF]&0x7fffffff; /* no sign */ | |
111 | qy = v.i[HIGH_HALF]&0x7fffffff; /* no sign */ | |
112 | ||
113 | if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0)) return NaNQ.x; | |
114 | if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0)) | |
115 | return x == 1.0 ? 1.0 : NaNQ.x; | |
116 | ||
e4d82761 UD |
117 | /* if x<0 */ |
118 | if (u.i[HIGH_HALF] < 0) { | |
119 | k = checkint(y); | |
ca58f1db | 120 | if (k==0) { |
8f3edfee | 121 | if (qy == 0x7ff00000) { |
ca58f1db UD |
122 | if (x == -1.0) return 1.0; |
123 | else if (x > -1.0) return v.i[HIGH_HALF] < 0 ? INF.x : 0.0; | |
124 | else return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x; | |
125 | } | |
8f3edfee | 126 | else if (qx == 0x7ff00000) |
ca58f1db UD |
127 | return y < 0 ? 0.0 : INF.x; |
128 | return NaNQ.x; /* y not integer and x<0 */ | |
129 | } | |
8f3edfee | 130 | else if (qx == 0x7ff00000) |
ca58f1db UD |
131 | { |
132 | if (k < 0) | |
133 | return y < 0 ? nZERO.x : nINF.x; | |
134 | else | |
135 | return y < 0 ? 0.0 : INF.x; | |
136 | } | |
137 | return (k==1)?__ieee754_pow(-x,y):-__ieee754_pow(-x,y); /* if y even or odd */ | |
e4d82761 UD |
138 | } |
139 | /* x>0 */ | |
f7eac6eb | 140 | |
e4d82761 UD |
141 | if (qx == 0x7ff00000) /* x= 2^-0x3ff */ |
142 | {if (y == 0) return NaNQ.x; | |
143 | return (y>0)?x:0; } | |
f14bd805 | 144 | |
e4d82761 UD |
145 | if (qy > 0x45f00000 && qy < 0x7ff00000) { |
146 | if (x == 1.0) return 1.0; | |
147 | if (y>0) return (x>1.0)?INF.x:0; | |
148 | if (y<0) return (x<1.0)?INF.x:0; | |
149 | } | |
f7eac6eb | 150 | |
ca58f1db | 151 | if (x == 1.0) return 1.0; |
e4d82761 UD |
152 | if (y>0) return (x>1.0)?INF.x:0; |
153 | if (y<0) return (x<1.0)?INF.x:0; | |
154 | return 0; /* unreachable, to make the compiler happy */ | |
155 | } | |
0ac5ae23 | 156 | strong_alias (__ieee754_pow, __pow_finite) |
f7eac6eb | 157 | |
e4d82761 UD |
158 | /**************************************************************************/ |
159 | /* Computing x^y using more accurate but more slow log routine */ | |
160 | /**************************************************************************/ | |
161 | static double power1(double x, double y) { | |
162 | double z,a,aa,error, t,a1,a2,y1,y2; | |
1f81acbc | 163 | z = my_log2(x,&aa,&error); |
e4d82761 UD |
164 | t = y*134217729.0; |
165 | y1 = t - (t-y); | |
166 | y2 = y - y1; | |
167 | t = z*134217729.0; | |
168 | a1 = t - (t-z); | |
169 | a2 = z - a1; | |
170 | a = y*z; | |
171 | aa = ((y1*a1-a)+y1*a2+y2*a1)+y2*a2+aa*y; | |
172 | a1 = a+aa; | |
173 | a2 = (a-a1)+aa; | |
174 | error = error*ABS(y); | |
175 | t = __exp1(a1,a2,1.9e16*error); | |
ca58f1db | 176 | return (t >= 0)?t:__slowpow(x,y,z); |
e4d82761 | 177 | } |
f7eac6eb | 178 | |
e4d82761 UD |
179 | /****************************************************************************/ |
180 | /* Computing log(x) (x is left argument). The result is the returned double */ | |
181 | /* + the parameter delta. */ | |
182 | /* The result is bounded by error (rightmost argument) */ | |
183 | /****************************************************************************/ | |
184 | static double log1(double x, double *delta, double *error) { | |
50944bca UD |
185 | int i,j,m; |
186 | #if 0 | |
187 | int n; | |
188 | #endif | |
189 | double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0; | |
190 | #if 0 | |
191 | double cor; | |
192 | #endif | |
e4d82761 | 193 | mynumber u,v; |
27692f89 UD |
194 | #ifdef BIG_ENDI |
195 | mynumber | |
196 | /**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */ | |
197 | #else | |
198 | #ifdef LITTLE_ENDI | |
199 | mynumber | |
200 | /**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */ | |
201 | #endif | |
202 | #endif | |
ba1ffaa1 | 203 | |
e4d82761 UD |
204 | u.x = x; |
205 | m = u.i[HIGH_HALF]; | |
206 | *error = 0; | |
207 | *delta = 0; | |
208 | if (m < 0x00100000) /* 1<x<2^-1007 */ | |
209 | { x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF];} | |
f7eac6eb | 210 | |
e4d82761 UD |
211 | if ((m&0x000fffff) < 0x0006a09e) |
212 | {u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); } | |
213 | else | |
214 | {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; } | |
f7eac6eb | 215 | |
e4d82761 UD |
216 | v.x = u.x + bigu.x; |
217 | uu = v.x - bigu.x; | |
218 | i = (v.i[LOW_HALF]&0x000003ff)<<2; | |
219 | if (two52.i[LOW_HALF] == 1023) /* nx = 0 */ | |
220 | { | |
221 | if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10 */ | |
222 | { | |
223 | t = x - 1.0; | |
224 | t1 = (t+5.0e6)-5.0e6; | |
225 | t2 = t-t1; | |
226 | e1 = t - 0.5*t1*t1; | |
227 | e2 = t*t*t*(r3+t*(r4+t*(r5+t*(r6+t*(r7+t*r8)))))-0.5*t2*(t+t1); | |
228 | res = e1+e2; | |
229 | *error = 1.0e-21*ABS(t); | |
230 | *delta = (e1-res)+e2; | |
231 | return res; | |
232 | } /* |x-1| < 1.5*2**-10 */ | |
233 | else | |
234 | { | |
235 | v.x = u.x*(ui.x[i]+ui.x[i+1])+bigv.x; | |
236 | vv = v.x-bigv.x; | |
237 | j = v.i[LOW_HALF]&0x0007ffff; | |
238 | j = j+j+j; | |
239 | eps = u.x - uu*vv; | |
240 | e1 = eps*ui.x[i]; | |
241 | e2 = eps*(ui.x[i+1]+vj.x[j]*(ui.x[i]+ui.x[i+1])); | |
242 | e = e1+e2; | |
243 | e2 = ((e1-e)+e2); | |
244 | t=ui.x[i+2]+vj.x[j+1]; | |
245 | t1 = t+e; | |
246 | t2 = (((t-t1)+e)+(ui.x[i+3]+vj.x[j+2]))+e2+e*e*(p2+e*(p3+e*p4)); | |
247 | res=t1+t2; | |
248 | *error = 1.0e-24; | |
249 | *delta = (t1-res)+t2; | |
250 | return res; | |
251 | } | |
252 | } /* nx = 0 */ | |
253 | else /* nx != 0 */ | |
254 | { | |
255 | eps = u.x - uu; | |
256 | nx = (two52.x - two52e.x)+add; | |
257 | e1 = eps*ui.x[i]; | |
258 | e2 = eps*ui.x[i+1]; | |
259 | e=e1+e2; | |
260 | e2 = (e1-e)+e2; | |
261 | t=nx*ln2a.x+ui.x[i+2]; | |
262 | t1=t+e; | |
263 | t2=(((t-t1)+e)+nx*ln2b.x+ui.x[i+3]+e2)+e*e*(q2+e*(q3+e*(q4+e*(q5+e*q6)))); | |
264 | res = t1+t2; | |
265 | *error = 1.0e-21; | |
266 | *delta = (t1-res)+t2; | |
267 | return res; | |
268 | } /* nx != 0 */ | |
269 | } | |
270 | ||
271 | /****************************************************************************/ | |
272 | /* More slow but more accurate routine of log */ | |
273 | /* Computing log(x)(x is left argument).The result is return double + delta.*/ | |
274 | /* The result is bounded by error (right argument) */ | |
275 | /****************************************************************************/ | |
1f81acbc | 276 | static double my_log2(double x, double *delta, double *error) { |
50944bca UD |
277 | int i,j,m; |
278 | #if 0 | |
279 | int n; | |
280 | #endif | |
281 | double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0; | |
282 | #if 0 | |
283 | double cor; | |
284 | #endif | |
e4d82761 | 285 | double ou1,ou2,lu1,lu2,ov,lv1,lv2,a,a1,a2; |
a1a87169 | 286 | double y,yy,z,zz,j1,j2,j7,j8; |
58985aa9 | 287 | #ifndef DLA_FMS |
a1a87169 UD |
288 | double j3,j4,j5,j6; |
289 | #endif | |
e4d82761 | 290 | mynumber u,v; |
27692f89 UD |
291 | #ifdef BIG_ENDI |
292 | mynumber | |
293 | /**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */ | |
294 | #else | |
295 | #ifdef LITTLE_ENDI | |
296 | mynumber | |
297 | /**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */ | |
298 | #endif | |
299 | #endif | |
e4d82761 UD |
300 | |
301 | u.x = x; | |
302 | m = u.i[HIGH_HALF]; | |
303 | *error = 0; | |
304 | *delta = 0; | |
305 | add=0; | |
306 | if (m<0x00100000) { /* x < 2^-1022 */ | |
307 | x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF]; } | |
f7eac6eb | 308 | |
e4d82761 UD |
309 | if ((m&0x000fffff) < 0x0006a09e) |
310 | {u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); } | |
311 | else | |
312 | {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; } | |
313 | ||
314 | v.x = u.x + bigu.x; | |
315 | uu = v.x - bigu.x; | |
316 | i = (v.i[LOW_HALF]&0x000003ff)<<2; | |
317 | /*------------------------------------- |x-1| < 2**-11------------------------------- */ | |
318 | if ((two52.i[LOW_HALF] == 1023) && (i == 1200)) | |
319 | { | |
320 | t = x - 1.0; | |
321 | EMULV(t,s3,y,yy,j1,j2,j3,j4,j5); | |
322 | ADD2(-0.5,0,y,yy,z,zz,j1,j2); | |
323 | MUL2(t,0,z,zz,y,yy,j1,j2,j3,j4,j5,j6,j7,j8); | |
324 | MUL2(t,0,y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8); | |
325 | ||
326 | e1 = t+z; | |
327 | e2 = (((t-e1)+z)+zz)+t*t*t*(ss3+t*(s4+t*(s5+t*(s6+t*(s7+t*s8))))); | |
328 | res = e1+e2; | |
329 | *error = 1.0e-25*ABS(t); | |
330 | *delta = (e1-res)+e2; | |
331 | return res; | |
332 | } | |
333 | /*----------------------------- |x-1| > 2**-11 -------------------------- */ | |
334 | else | |
335 | { /*Computing log(x) according to log table */ | |
336 | nx = (two52.x - two52e.x)+add; | |
337 | ou1 = ui.x[i]; | |
338 | ou2 = ui.x[i+1]; | |
339 | lu1 = ui.x[i+2]; | |
340 | lu2 = ui.x[i+3]; | |
341 | v.x = u.x*(ou1+ou2)+bigv.x; | |
342 | vv = v.x-bigv.x; | |
343 | j = v.i[LOW_HALF]&0x0007ffff; | |
344 | j = j+j+j; | |
345 | eps = u.x - uu*vv; | |
346 | ov = vj.x[j]; | |
347 | lv1 = vj.x[j+1]; | |
348 | lv2 = vj.x[j+2]; | |
349 | a = (ou1+ou2)*(1.0+ov); | |
350 | a1 = (a+1.0e10)-1.0e10; | |
351 | a2 = a*(1.0-a1*uu*vv); | |
352 | e1 = eps*a1; | |
353 | e2 = eps*a2; | |
354 | e = e1+e2; | |
355 | e2 = (e1-e)+e2; | |
356 | t=nx*ln2a.x+lu1+lv1; | |
357 | t1 = t+e; | |
358 | t2 = (((t-t1)+e)+(lu2+lv2+nx*ln2b.x+e2))+e*e*(p2+e*(p3+e*p4)); | |
359 | res=t1+t2; | |
360 | *error = 1.0e-27; | |
361 | *delta = (t1-res)+t2; | |
362 | return res; | |
363 | } | |
364 | } | |
f7eac6eb | 365 | |
e4d82761 UD |
366 | /**********************************************************************/ |
367 | /* Routine receives a double x and checks if it is an integer. If not */ | |
368 | /* it returns 0, else it returns 1 if even or -1 if odd. */ | |
369 | /**********************************************************************/ | |
370 | static int checkint(double x) { | |
371 | union {int4 i[2]; double x;} u; | |
50944bca UD |
372 | int k,m,n; |
373 | #if 0 | |
374 | int l; | |
375 | #endif | |
e4d82761 UD |
376 | u.x = x; |
377 | m = u.i[HIGH_HALF]&0x7fffffff; /* no sign */ | |
378 | if (m >= 0x7ff00000) return 0; /* x is +/-inf or NaN */ | |
379 | if (m >= 0x43400000) return 1; /* |x| >= 2**53 */ | |
380 | if (m < 0x40000000) return 0; /* |x| < 2, can not be 0 or 1 */ | |
381 | n = u.i[LOW_HALF]; | |
382 | k = (m>>20)-1023; /* 1 <= k <= 52 */ | |
383 | if (k == 52) return (n&1)? -1:1; /* odd or even*/ | |
384 | if (k>20) { | |
385 | if (n<<(k-20)) return 0; /* if not integer */ | |
386 | return (n<<(k-21))?-1:1; | |
387 | } | |
388 | if (n) return 0; /*if not integer*/ | |
389 | if (k == 20) return (m&1)? -1:1; | |
390 | if (m<<(k+12)) return 0; | |
391 | return (m<<(k+11))?-1:1; | |
f7eac6eb | 392 | } |