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