1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25 /*
26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 #include "libm_synonyms.h"
31 #include "libm_inlines.h"
32
33 #ifdef __RESTRICT
34 #define restrict _Restrict
35 #else
36 #define restrict
37 #endif
38
39 /* float rsqrtf(float x)
40 *
41 * Method :
42 * 1. Special cases:
43 * for x = NaN => QNaN;
44 * for x = +Inf => 0;
45 * for x is negative, -Inf => QNaN + invalid;
46 * for x = +0 => +Inf + divide-by-zero;
47 * for x = -0 => -Inf + divide-by-zero.
48 * 2. Computes reciprocal square root from:
49 * x = m * 2**n
50 * Where:
51 * m = [0.5, 2),
52 * n = ((exponent + 1) & ~1).
53 * Then:
54 * rsqrtf(x) = 1/sqrt( m * 2**n ) = (2 ** (-n/2)) * (1/sqrt(m))
55 * 2. Computes 1/sqrt(m) from:
56 * 1/sqrt(m) = (1/sqrt(m0)) * (1/sqrt(1 + (1/m0)*dm))
57 * Where:
58 * m = m0 + dm,
59 * m0 = 0.5 * (1 + k/64) for m = [0.5, 0.5+127/256), k = [0, 63];
60 * m0 = 1.0 * (0 + k/64) for m = [0.5+127/256, 1.0+127/128), k = [64, 127];
61 * Then:
62 * 1/sqrt(m0), 1/m0 are looked up in a table,
63 * 1/sqrt(1 + (1/m0)*dm) is computed using approximation:
64 * 1/sqrt(1 + z) = ((a3 * z + a2) * z + a1) * z + a0
65 * where z = [-1/64, 1/64].
66 *
67 * Accuracy:
68 * The maximum relative error for the approximating
69 * polynomial is 2**(-27.87).
70 * Maximum error observed: less than 0.534 ulp for the
71 * whole float type range.
72 */
73
74 #define sqrtf __sqrtf
75
76 extern float sqrtf( float );
77
78 static const double __TBL_rsqrtf[] = {
79 /*
80 i = [0,63]
81 TBL[2*i ] = 1 / (*(double*)&(0x3fe0000000000000ULL + (i << 46))) * 2**-24;
82 TBL[2*i+1] = 1 / sqrtl(*(double*)&(0x3fe0000000000000ULL + (i << 46)));
83 i = [64,127]
84 TBL[2*i ] = 1 / (*(double*)&(0x3fe0000000000000ULL + (i << 46))) * 2**-23;
85 TBL[2*i+1] = 1 / sqrtl(*(double*)&(0x3fe0000000000000ULL + (i << 46)));
86 */
87 1.1920928955078125000e-07, 1.4142135623730951455e+00,
88 1.1737530048076923728e-07, 1.4032928308912466786e+00,
89 1.1559688683712121533e-07, 1.3926212476455828160e+00,
90 1.1387156016791044559e-07, 1.3821894809301762397e+00,
91 1.1219697840073529256e-07, 1.3719886811400707760e+00,
92 1.1057093523550724772e-07, 1.3620104492139977204e+00,
93 1.0899135044642856803e-07, 1.3522468075656264297e+00,
94 1.0745626100352112918e-07, 1.3426901732747025253e+00,
95 1.0596381293402777190e-07, 1.3333333333333332593e+00,
96 1.0451225385273972023e-07, 1.3241694217637887121e+00,
97 1.0309992609797297870e-07, 1.3151918984428583315e+00,
98 1.0172526041666667320e-07, 1.3063945294843617440e+00,
99 1.0038677014802631022e-07, 1.2977713690461003537e+00,
100 9.9083045860389616921e-08, 1.2893167424406084542e+00,
101 9.7812750400641022247e-08, 1.2810252304406970492e+00,
102 9.6574614319620251657e-08, 1.2728916546811681609e+00,
103 9.5367431640625005294e-08, 1.2649110640673517647e+00,
104 9.4190055941358019463e-08, 1.2570787221094177344e+00,
105 9.3041396722560978838e-08, 1.2493900951088485751e+00,
106 9.1920416039156631290e-08, 1.2418408411301324890e+00,
107 9.0826125372023804482e-08, 1.2344267996967352996e+00,
108 8.9757582720588234048e-08, 1.2271439821557927896e+00,
109 8.8713889898255812722e-08, 1.2199885626608373279e+00,
110 8.7694190014367814875e-08, 1.2129568697262453902e+00,
111 8.6697665127840911497e-08, 1.2060453783110545167e+00,
112 8.5723534058988761666e-08, 1.1992507023933782762e+00,
113 8.4771050347222225457e-08, 1.1925695879998878812e+00,
114 8.3839500343406599951e-08, 1.1859989066577618644e+00,
115 8.2928201426630432481e-08, 1.1795356492391770864e+00,
116 8.2036500336021511923e-08, 1.1731769201708264205e+00,
117 8.1163771609042551220e-08, 1.1669199319831564665e+00,
118 8.0309416118421050820e-08, 1.1607620001760186046e+00,
119 7.9472859700520828922e-08, 1.1547005383792514621e+00,
120 7.8653551868556699530e-08, 1.1487330537883810866e+00,
121 7.7850964604591830522e-08, 1.1428571428571427937e+00,
122 7.7064591224747481298e-08, 1.1370704872299222110e+00,
123 7.6293945312500001588e-08, 1.1313708498984760276e+00,
124 7.5538559715346535571e-08, 1.1257560715684669095e+00,
125 7.4797985600490195040e-08, 1.1202240672224077489e+00,
126 7.4071791565533974158e-08, 1.1147728228665882977e+00,
127 7.3359562800480773303e-08, 1.1094003924504582947e+00,
128 7.2660900297619054173e-08, 1.1041048949477667573e+00,
129 7.1975420106132072725e-08, 1.0988845115895122806e+00,
130 7.1302752628504667579e-08, 1.0937374832394612945e+00,
131 7.0642541956018514597e-08, 1.0886621079036347126e+00,
132 6.9994445240825691959e-08, 1.0836567383657542685e+00,
133 6.9358132102272723904e-08, 1.0787197799411873955e+00,
134 6.8733284065315314719e-08, 1.0738496883424388795e+00,
135 6.8119594029017853361e-08, 1.0690449676496975862e+00,
136 6.7516765763274335346e-08, 1.0643041683803828867e+00,
137 6.6924513432017540145e-08, 1.0596258856520350822e+00,
138 6.6342561141304348632e-08, 1.0550087574332591700e+00,
139 6.5770642510775861156e-08, 1.0504514628777803509e+00,
140 6.5208500267094023655e-08, 1.0459527207369814228e+00,
141 6.4655885858050847233e-08, 1.0415112878465908608e+00,
142 6.4112559086134451001e-08, 1.0371259576834630511e+00,
143 6.3578287760416665784e-08, 1.0327955589886446131e+00,
144 6.3052847365702481089e-08, 1.0285189544531601058e+00,
145 6.2536020747950822927e-08, 1.0242950394631678002e+00,
146 6.2027597815040656970e-08, 1.0201227409013413627e+00,
147 6.1527375252016127325e-08, 1.0160010160015240377e+00,
148 6.1035156250000001271e-08, 1.0119288512538813229e+00,
149 6.0550750248015869655e-08, 1.0079052613579393416e+00,
150 6.0073972687007873182e-08, 1.0039292882210537616e+00,
151 1.1920928955078125000e-07, 1.0000000000000000000e+00,
152 1.1737530048076923728e-07, 9.9227787671366762812e-01,
153 1.1559688683712121533e-07, 9.8473192783466190203e-01,
154 1.1387156016791044559e-07, 9.7735555485044178781e-01,
155 1.1219697840073529256e-07, 9.7014250014533187638e-01,
156 1.1057093523550724772e-07, 9.6308682468615358641e-01,
157 1.0899135044642856803e-07, 9.5618288746751489704e-01,
158 1.0745626100352112918e-07, 9.4942532655508271588e-01,
159 1.0596381293402777190e-07, 9.4280904158206335630e-01,
160 1.0451225385273972023e-07, 9.3632917756904454620e-01,
161 1.0309992609797297870e-07, 9.2998110995055427441e-01,
162 1.0172526041666667320e-07, 9.2376043070340119190e-01,
163 1.0038677014802631022e-07, 9.1766293548224708854e-01,
164 9.9083045860389616921e-08, 9.1168461167710357351e-01,
165 9.7812750400641022247e-08, 9.0582162731567661407e-01,
166 9.6574614319620251657e-08, 9.0007032074081916306e-01,
167 9.5367431640625005294e-08, 8.9442719099991585541e-01,
168 9.4190055941358019463e-08, 8.8888888888888883955e-01,
169 9.3041396722560978838e-08, 8.8345220859877238162e-01,
170 9.1920416039156631290e-08, 8.7811407991752277180e-01,
171 9.0826125372023804482e-08, 8.7287156094396955996e-01,
172 8.9757582720588234048e-08, 8.6772183127462465535e-01,
173 8.8713889898255812722e-08, 8.6266218562750729415e-01,
174 8.7694190014367814875e-08, 8.5769002787023584933e-01,
175 8.6697665127840911497e-08, 8.5280286542244176928e-01,
176 8.5723534058988761666e-08, 8.4799830400508802164e-01,
177 8.4771050347222225457e-08, 8.4327404271156780613e-01,
178 8.3839500343406599951e-08, 8.3862786937753464045e-01,
179 8.2928201426630432481e-08, 8.3405765622829908246e-01,
180 8.2036500336021511923e-08, 8.2956135578434020417e-01,
181 8.1163771609042551220e-08, 8.2513699700703468931e-01,
182 8.0309416118421050820e-08, 8.2078268166812329287e-01,
183 7.9472859700520828922e-08, 8.1649658092772603446e-01,
184 7.8653551868556699530e-08, 8.1227693210689522196e-01,
185 7.7850964604591830522e-08, 8.0812203564176865456e-01,
186 7.7064591224747481298e-08, 8.0403025220736967782e-01,
187 7.6293945312500001588e-08, 8.0000000000000004441e-01,
188 7.5538559715346535571e-08, 7.9602975216799132241e-01,
189 7.4797985600490195040e-08, 7.9211803438133943089e-01,
190 7.4071791565533974158e-08, 7.8826342253143455441e-01,
191 7.3359562800480773303e-08, 7.8446454055273617811e-01,
192 7.2660900297619054173e-08, 7.8072005835882651859e-01,
193 7.1975420106132072725e-08, 7.7702868988581130782e-01,
194 7.1302752628504667579e-08, 7.7338919123653082632e-01,
195 7.0642541956018514597e-08, 7.6980035891950104876e-01,
196 6.9994445240825691959e-08, 7.6626102817692109959e-01,
197 6.9358132102272723904e-08, 7.6277007139647390321e-01,
198 6.8733284065315314719e-08, 7.5932639660199918730e-01,
199 6.8119594029017853361e-08, 7.5592894601845450619e-01,
200 6.7516765763274335346e-08, 7.5257669470687782454e-01,
201 6.6924513432017540145e-08, 7.4926864926535519107e-01,
202 6.6342561141304348632e-08, 7.4600384659225105199e-01,
203 6.5770642510775861156e-08, 7.4278135270820744296e-01,
204 6.5208500267094023655e-08, 7.3960026163363878915e-01,
205 6.4655885858050847233e-08, 7.3645969431865865307e-01,
206 6.4112559086134451001e-08, 7.3335879762256905856e-01,
207 6.3578287760416665784e-08, 7.3029674334022143256e-01,
208 6.3052847365702481089e-08, 7.2727272727272729291e-01,
209 6.2536020747950822927e-08, 7.2428596834014824513e-01,
210 6.2027597815040656970e-08, 7.2133570773394584119e-01,
211 6.1527375252016127325e-08, 7.1842120810709964029e-01,
212 6.1035156250000001271e-08, 7.1554175279993270653e-01,
213 6.0550750248015869655e-08, 7.1269664509979835376e-01,
214 6.0073972687007873182e-08, 7.0988520753289097165e-01,
215 };
216
217 static const unsigned long long LCONST[] = {
218 0x3feffffffee7f18fULL, /* A0 = 9.99999997962321453275e-01 */
219 0xbfdffffffe07e52fULL, /* A1 =-4.99999998166077580600e-01 */
220 0x3fd801180ca296d9ULL, /* A2 = 3.75066768969515586277e-01 */
221 0xbfd400fc0bbb8e78ULL, /* A3 =-3.12560092408808548438e-01 */
222 };
223
224 static void
225 __vrsqrtf_n( int n, float * restrict px, int stridex, float * restrict py, int stridey );
226
227 #pragma no_inline(__vrsqrtf_n)
228
229 #define RETURN(ret) \
230 { \
231 *py = (ret); \
232 py += stridey; \
233 if ( n_n == 0 ) \
234 { \
235 spx = px; spy = py; \
236 ax0 = *(int*)px; \
237 continue; \
238 } \
239 n--; \
240 break; \
241 }
242
243 void
244 __vrsqrtf( int n, float * restrict px, int stridex, float * restrict py, int stridey )
245 {
246 float *spx, *spy;
247 int ax0, n_n;
248 float res;
249 float FONE = 1.0f, FTWO = 2.0f;
250
251 while ( n > 1 )
252 {
253 n_n = 0;
254 spx = px;
255 spy = py;
256 ax0 = *(int*)px;
257 for ( ; n > 1 ; n-- )
258 {
259 px += stridex;
260 if ( ax0 >= 0x7f800000 ) /* X = NaN or Inf */
261 {
262 res = *(px - stridex);
263 RETURN ( FONE / res )
264 }
265
266 py += stridey;
267
268 if ( ax0 < 0x00800000 ) /* X = denormal, zero or negative */
269 {
270 py -= stridey;
271 res = *(px - stridex);
272
273 if ( (ax0 & 0x7fffffff) == 0 ) /* |X| = zero */
274 {
275 RETURN ( FONE / res )
276 }
277 else if ( ax0 >= 0 ) /* X = denormal */
278 {
279 double A0 = ((double*)LCONST)[0]; /* 9.99999997962321453275e-01 */
280 double A1 = ((double*)LCONST)[1]; /* -4.99999998166077580600e-01 */
281 double A2 = ((double*)LCONST)[2]; /* 3.75066768969515586277e-01 */
282 double A3 = ((double*)LCONST)[3]; /* -3.12560092408808548438e-01 */
283
284 double res0, xx0, tbl_div0, tbl_sqrt0;
285 float fres0;
286 int iax0, si0, iexp0;
287
288 res = *(int*)&res;
289 res *= FTWO;
290 ax0 = *(int*)&res;
291 iexp0 = ax0 >> 24;
292 iexp0 = 0x3f + 0x4b - iexp0;
293 iexp0 = iexp0 << 23;
294
295 si0 = (ax0 >> 13) & 0x7f0;
296
297 tbl_div0 = ((double*)((char*)__TBL_rsqrtf + si0))[0];
298 tbl_sqrt0 = ((double*)((char*)__TBL_rsqrtf + si0))[1];
299 iax0 = ax0 & 0x7ffe0000;
300 iax0 = ax0 - iax0;
301 xx0 = iax0 * tbl_div0;
302 res0 = tbl_sqrt0 * (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0);
303
304 fres0 = res0;
305 iexp0 += *(int*)&fres0;
306 RETURN(*(float*)&iexp0)
307 }
308 else /* X = negative */
309 {
310 RETURN ( sqrtf(res) )
311 }
312 }
313 n_n++;
314 ax0 = *(int*)px;
315 }
316 if ( n_n > 0 )
317 __vrsqrtf_n( n_n, spx, stridex, spy, stridey );
318 }
319
320 if ( n > 0 )
321 {
322 ax0 = *(int*)px;
323
324 if ( ax0 >= 0x7f800000 ) /* X = NaN or Inf */
325 {
326 res = *px;
327 *py = FONE / res;
328 }
329 else if ( ax0 < 0x00800000 ) /* X = denormal, zero or negative */
330 {
331 res = *px;
332
333 if ( (ax0 & 0x7fffffff) == 0 ) /* |X| = zero */
334 {
335 *py = FONE / res;
336 }
337 else if ( ax0 >= 0 ) /* X = denormal */
338 {
339 double A0 = ((double*)LCONST)[0]; /* 9.99999997962321453275e-01 */
340 double A1 = ((double*)LCONST)[1]; /* -4.99999998166077580600e-01 */
341 double A2 = ((double*)LCONST)[2]; /* 3.75066768969515586277e-01 */
342 double A3 = ((double*)LCONST)[3]; /* -3.12560092408808548438e-01 */
343 double res0, xx0, tbl_div0, tbl_sqrt0;
344 float fres0;
345 int iax0, si0, iexp0;
346
347 res = *(int*)&res;
348 res *= FTWO;
349 ax0 = *(int*)&res;
350 iexp0 = ax0 >> 24;
351 iexp0 = 0x3f + 0x4b - iexp0;
352 iexp0 = iexp0 << 23;
353
354 si0 = (ax0 >> 13) & 0x7f0;
355
356 tbl_div0 = ((double*)((char*)__TBL_rsqrtf + si0))[0];
357 tbl_sqrt0 = ((double*)((char*)__TBL_rsqrtf + si0))[1];
358 iax0 = ax0 & 0x7ffe0000;
359 iax0 = ax0 - iax0;
360 xx0 = iax0 * tbl_div0;
361 res0 = tbl_sqrt0 * (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0);
362
363 fres0 = res0;
364 iexp0 += *(int*)&fres0;
365
366 *(int*)py = iexp0;
367 }
368 else /* X = negative */
369 {
370 *py = sqrtf(res);
371 }
372 }
373 else
374 {
375 double A0 = ((double*)LCONST)[0]; /* 9.99999997962321453275e-01 */
376 double A1 = ((double*)LCONST)[1]; /* -4.99999998166077580600e-01 */
377 double A2 = ((double*)LCONST)[2]; /* 3.75066768969515586277e-01 */
378 double A3 = ((double*)LCONST)[3]; /* -3.12560092408808548438e-01 */
379 double res0, xx0, tbl_div0, tbl_sqrt0;
380 float fres0;
381 int iax0, si0, iexp0;
382
383 iexp0 = ax0 >> 24;
384 iexp0 = 0x3f - iexp0;
385 iexp0 = iexp0 << 23;
386
387 si0 = (ax0 >> 13) & 0x7f0;
388
389 tbl_div0 = ((double*)((char*)__TBL_rsqrtf + si0))[0];
390 tbl_sqrt0 = ((double*)((char*)__TBL_rsqrtf + si0))[1];
391 iax0 = ax0 & 0x7ffe0000;
392 iax0 = ax0 - iax0;
393 xx0 = iax0 * tbl_div0;
394 res0 = tbl_sqrt0 * (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0);
395
396 fres0 = res0;
397 iexp0 += *(int*)&fres0;
398
399 *(int*)py = iexp0;
400 }
401 }
402 }
403
404 void
405 __vrsqrtf_n( int n, float * restrict px, int stridex, float * restrict py, int stridey )
406 {
407 double A0 = ((double*)LCONST)[0]; /* 9.99999997962321453275e-01 */
408 double A1 = ((double*)LCONST)[1]; /* -4.99999998166077580600e-01 */
409 double A2 = ((double*)LCONST)[2]; /* 3.75066768969515586277e-01 */
410 double A3 = ((double*)LCONST)[3]; /* -3.12560092408808548438e-01 */
411 double res0, xx0, tbl_div0, tbl_sqrt0;
412 float fres0;
413 int iax0, ax0, si0, iexp0;
414
415 #if defined(ARCH_v7) || defined(ARCH_v8)
416 double res1, xx1, tbl_div1, tbl_sqrt1;
417 double res2, xx2, tbl_div2, tbl_sqrt2;
418 float fres1, fres2;
419 int iax1, ax1, si1, iexp1;
420 int iax2, ax2, si2, iexp2;
421
422 for( ; n > 2 ; n -= 3 )
423 {
424 ax0 = *(int*)px;
425 px += stridex;
426
427 ax1 = *(int*)px;
428 px += stridex;
429
430 ax2 = *(int*)px;
431 px += stridex;
432
433 iexp0 = ax0 >> 24;
434 iexp1 = ax1 >> 24;
435 iexp2 = ax2 >> 24;
436 iexp0 = 0x3f - iexp0;
437 iexp1 = 0x3f - iexp1;
438 iexp2 = 0x3f - iexp2;
439
440 iexp0 = iexp0 << 23;
441 iexp1 = iexp1 << 23;
442 iexp2 = iexp2 << 23;
443
444 si0 = (ax0 >> 13) & 0x7f0;
445 si1 = (ax1 >> 13) & 0x7f0;
446 si2 = (ax2 >> 13) & 0x7f0;
447
448 tbl_div0 = ((double*)((char*)__TBL_rsqrtf + si0))[0];
449 tbl_div1 = ((double*)((char*)__TBL_rsqrtf + si1))[0];
450 tbl_div2 = ((double*)((char*)__TBL_rsqrtf + si2))[0];
451 tbl_sqrt0 = ((double*)((char*)__TBL_rsqrtf + si0))[1];
452 tbl_sqrt1 = ((double*)((char*)__TBL_rsqrtf + si1))[1];
453 tbl_sqrt2 = ((double*)((char*)__TBL_rsqrtf + si2))[1];
454 iax0 = ax0 & 0x7ffe0000;
455 iax1 = ax1 & 0x7ffe0000;
456 iax2 = ax2 & 0x7ffe0000;
457 iax0 = ax0 - iax0;
458 iax1 = ax1 - iax1;
459 iax2 = ax2 - iax2;
460 xx0 = iax0 * tbl_div0;
461 xx1 = iax1 * tbl_div1;
462 xx2 = iax2 * tbl_div2;
463 res0 = tbl_sqrt0 * (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0);
464 res1 = tbl_sqrt1 * (((A3 * xx1 + A2) * xx1 + A1) * xx1 + A0);
465 res2 = tbl_sqrt2 * (((A3 * xx2 + A2) * xx2 + A1) * xx2 + A0);
466
467 fres0 = res0;
468 fres1 = res1;
469 fres2 = res2;
470
471 iexp0 += *(int*)&fres0;
472 iexp1 += *(int*)&fres1;
473 iexp2 += *(int*)&fres2;
474 *(int*)py = iexp0;
475 py += stridey;
476 *(int*)py = iexp1;
477 py += stridey;
478 *(int*)py = iexp2;
479 py += stridey;
480 }
481 #endif
482 for( ; n > 0 ; n-- )
483 {
484 ax0 = *(int*)px;
485 px += stridex;
486
487 iexp0 = ax0 >> 24;
488 iexp0 = 0x3f - iexp0;
489 iexp0 = iexp0 << 23;
490
491 si0 = (ax0 >> 13) & 0x7f0;
492
493 tbl_div0 = ((double*)((char*)__TBL_rsqrtf + si0))[0];
494 tbl_sqrt0 = ((double*)((char*)__TBL_rsqrtf + si0))[1];
495 iax0 = ax0 & 0x7ffe0000;
496 iax0 = ax0 - iax0;
497 xx0 = iax0 * tbl_div0;
498 res0 = tbl_sqrt0 * (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0);
499
500 fres0 = res0;
501 iexp0 += *(int*)&fres0;
502 *(int*)py = iexp0;
503 py += stridey;
504 }
505 }
506