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 <sys/isa_defs.h>
31 #include "libm_synonyms.h"
32 #include "libm_inlines.h"
33
34 #ifdef _LITTLE_ENDIAN
35 #define HI(x) *(1+(int*)x)
36 #define LO(x) *(unsigned*)x
37 #else
38 #define HI(x) *(int*)x
39 #define LO(x) *(1+(unsigned*)x)
40 #endif
41
42 #ifdef __RESTRICT
43 #define restrict _Restrict
44 #else
45 #define restrict
46 #endif
47
48 /* float rhypotf(float x, float y)
49 *
50 * Method :
51 * 1. Special cases:
52 * for x or y = Inf => 0;
53 * for x or y = NaN => QNaN;
54 * for x and y = 0 => +Inf + divide-by-zero;
55 * 2. Computes d = x * x + y * y;
56 * 3. Computes reciprocal square root from:
57 * d = m * 2**n
58 * Where:
59 * m = [0.5, 2),
60 * n = ((exponent + 1) & ~1).
61 * Then:
62 * rsqrtf(d) = 1/sqrt( m * 2**n ) = (2 ** (-n/2)) * (1/sqrt(m))
63 * 4. Computes 1/sqrt(m) from:
64 * 1/sqrt(m) = (1/sqrt(m0)) * (1/sqrt(1 + (1/m0)*dm))
65 * Where:
66 * m = m0 + dm,
67 * m0 = 0.5 * (1 + k/64) for m = [0.5, 0.5+127/256), k = [0, 63];
68 * m0 = 1.0 * (0 + k/64) for m = [0.5+127/256, 1.0+127/128), k = [64, 127];
69 * Then:
70 * 1/sqrt(m0), 1/m0 are looked up in a table,
71 * 1/sqrt(1 + (1/m0)*dm) is computed using approximation:
72 * 1/sqrt(1 + z) = ((a3 * z + a2) * z + a1) * z + a0
73 * where z = [-1/64, 1/64].
74 *
75 * Accuracy:
76 * The maximum relative error for the approximating
77 * polynomial is 2**(-27.87).
78 * Maximum error observed: less than 0.535 ulp after 3.000.000.000
79 * results.
80 */
81
82 #pragma align 32 (__vlibm_TBL_rhypotf)
83
84 static const double __vlibm_TBL_rhypotf[] = {
85 /*
86 i = [0,63]
87 TBL[2*i+0] = 1.0 / (*(double*)&(0x3ff0000000000000LL + (i << 46)));
88 TBL[2*i+1] = (double)(0.5/sqrtl(2) / sqrtl(*(double*)&(0x3ff0000000000000LL + (i << 46))));
89 TBL[128+2*i+0] = 1.0 / (*(double*)&(0x3ff0000000000000LL + (i << 46)));
90 TBL[128+2*i+1] = (double)(0.25 / sqrtl(*(double*)&(0x3ff0000000000000LL + (i << 46))));
91 */
92 1.0000000000000000000e+00, 3.5355339059327378637e-01,
93 9.8461538461538467004e-01, 3.5082320772281166965e-01,
94 9.6969696969696972388e-01, 3.4815531191139570399e-01,
95 9.5522388059701490715e-01, 3.4554737023254405992e-01,
96 9.4117647058823528106e-01, 3.4299717028501769400e-01,
97 9.2753623188405798228e-01, 3.4050261230349943009e-01,
98 9.1428571428571425717e-01, 3.3806170189140660742e-01,
99 9.0140845070422537244e-01, 3.3567254331867563133e-01,
100 8.8888888888888883955e-01, 3.3333333333333331483e-01,
101 8.7671232876712323900e-01, 3.3104235544094717802e-01,
102 8.6486486486486491287e-01, 3.2879797461071458287e-01,
103 8.5333333333333338810e-01, 3.2659863237109043599e-01,
104 8.4210526315789469010e-01, 3.2444284226152508843e-01,
105 8.3116883116883122362e-01, 3.2232918561015211356e-01,
106 8.2051282051282048435e-01, 3.2025630761017426229e-01,
107 8.1012658227848100001e-01, 3.1822291367029204023e-01,
108 8.0000000000000004441e-01, 3.1622776601683794118e-01,
109 7.9012345679012341293e-01, 3.1426968052735443360e-01,
110 7.8048780487804880757e-01, 3.1234752377721214378e-01,
111 7.7108433734939763049e-01, 3.1046021028253312224e-01,
112 7.6190476190476186247e-01, 3.0860669992418382490e-01,
113 7.5294117647058822484e-01, 3.0678599553894819740e-01,
114 7.4418604651162789665e-01, 3.0499714066520933198e-01,
115 7.3563218390804596680e-01, 3.0323921743156134756e-01,
116 7.2727272727272729291e-01, 3.0151134457776362918e-01,
117 7.1910112359550559802e-01, 2.9981267559834456904e-01,
118 7.1111111111111113825e-01, 2.9814239699997197031e-01,
119 7.0329670329670335160e-01, 2.9649972666444046610e-01,
120 6.9565217391304345895e-01, 2.9488391230979427160e-01,
121 6.8817204301075274309e-01, 2.9329423004270660513e-01,
122 6.8085106382978721751e-01, 2.9172998299578911663e-01,
123 6.7368421052631577428e-01, 2.9019050004400465115e-01,
124 6.6666666666666662966e-01, 2.8867513459481286553e-01,
125 6.5979381443298967813e-01, 2.8718326344709527165e-01,
126 6.5306122448979586625e-01, 2.8571428571428569843e-01,
127 6.4646464646464651960e-01, 2.8426762180748055275e-01,
128 6.4000000000000001332e-01, 2.8284271247461900689e-01,
129 6.3366336633663367106e-01, 2.8143901789211672737e-01,
130 6.2745098039215685404e-01, 2.8005601680560193723e-01,
131 6.2135922330097081989e-01, 2.7869320571664707442e-01,
132 6.1538461538461541878e-01, 2.7735009811261457369e-01,
133 6.0952380952380957879e-01, 2.7602622373694168934e-01,
134 6.0377358490566035432e-01, 2.7472112789737807015e-01,
135 5.9813084112149528249e-01, 2.7343437080986532361e-01,
136 5.9259259259259255970e-01, 2.7216552697590867815e-01,
137 5.8715596330275232617e-01, 2.7091418459143856712e-01,
138 5.8181818181818178992e-01, 2.6967994498529684888e-01,
139 5.7657657657657657158e-01, 2.6846242208560971987e-01,
140 5.7142857142857139685e-01, 2.6726124191242439654e-01,
141 5.6637168141592919568e-01, 2.6607604209509572168e-01,
142 5.6140350877192979340e-01, 2.6490647141300877054e-01,
143 5.5652173913043478937e-01, 2.6375218935831479250e-01,
144 5.5172413793103447510e-01, 2.6261286571944508772e-01,
145 5.4700854700854706358e-01, 2.6148818018424535570e-01,
146 5.4237288135593220151e-01, 2.6037782196164771520e-01,
147 5.3781512605042014474e-01, 2.5928148942086576278e-01,
148 5.3333333333333332593e-01, 2.5819888974716115326e-01,
149 5.2892561983471075848e-01, 2.5712973861329002645e-01,
150 5.2459016393442625681e-01, 2.5607375986579195004e-01,
151 5.2032520325203257539e-01, 2.5503068522533534068e-01,
152 5.1612903225806450180e-01, 2.5400025400038100942e-01,
153 5.1200000000000001066e-01, 2.5298221281347033074e-01,
154 5.0793650793650790831e-01, 2.5197631533948483540e-01,
155 5.0393700787401574104e-01, 2.5098232205526344041e-01,
156 1.0000000000000000000e+00, 2.5000000000000000000e-01,
157 9.8461538461538467004e-01, 2.4806946917841690703e-01,
158 9.6969696969696972388e-01, 2.4618298195866547551e-01,
159 9.5522388059701490715e-01, 2.4433888871261044695e-01,
160 9.4117647058823528106e-01, 2.4253562503633296910e-01,
161 9.2753623188405798228e-01, 2.4077170617153839660e-01,
162 9.1428571428571425717e-01, 2.3904572186687872426e-01,
163 9.0140845070422537244e-01, 2.3735633163877067897e-01,
164 8.8888888888888883955e-01, 2.3570226039551583908e-01,
165 8.7671232876712323900e-01, 2.3408229439226113655e-01,
166 8.6486486486486491287e-01, 2.3249527748763856860e-01,
167 8.5333333333333338810e-01, 2.3094010767585029797e-01,
168 8.4210526315789469010e-01, 2.2941573387056177213e-01,
169 8.3116883116883122362e-01, 2.2792115291927589338e-01,
170 8.2051282051282048435e-01, 2.2645540682891915352e-01,
171 8.1012658227848100001e-01, 2.2501758018520479077e-01,
172 8.0000000000000004441e-01, 2.2360679774997896385e-01,
173 7.9012345679012341293e-01, 2.2222222222222220989e-01,
174 7.8048780487804880757e-01, 2.2086305214969309541e-01,
175 7.7108433734939763049e-01, 2.1952851997938069295e-01,
176 7.6190476190476186247e-01, 2.1821789023599238999e-01,
177 7.5294117647058822484e-01, 2.1693045781865616384e-01,
178 7.4418604651162789665e-01, 2.1566554640687682354e-01,
179 7.3563218390804596680e-01, 2.1442250696755896233e-01,
180 7.2727272727272729291e-01, 2.1320071635561044232e-01,
181 7.1910112359550559802e-01, 2.1199957600127200541e-01,
182 7.1111111111111113825e-01, 2.1081851067789195153e-01,
183 7.0329670329670335160e-01, 2.0965696734438366011e-01,
184 6.9565217391304345895e-01, 2.0851441405707477061e-01,
185 6.8817204301075274309e-01, 2.0739033894608505104e-01,
186 6.8085106382978721751e-01, 2.0628424925175867233e-01,
187 6.7368421052631577428e-01, 2.0519567041703082322e-01,
188 6.6666666666666662966e-01, 2.0412414523193150862e-01,
189 6.5979381443298967813e-01, 2.0306923302672380549e-01,
190 6.5306122448979586625e-01, 2.0203050891044216364e-01,
191 6.4646464646464651960e-01, 2.0100756305184241945e-01,
192 6.4000000000000001332e-01, 2.0000000000000001110e-01,
193 6.3366336633663367106e-01, 1.9900743804199783060e-01,
194 6.2745098039215685404e-01, 1.9802950859533485772e-01,
195 6.2135922330097081989e-01, 1.9706585563285863860e-01,
196 6.1538461538461541878e-01, 1.9611613513818404453e-01,
197 6.0952380952380957879e-01, 1.9518001458970662965e-01,
198 6.0377358490566035432e-01, 1.9425717247145282696e-01,
199 5.9813084112149528249e-01, 1.9334729780913270658e-01,
200 5.9259259259259255970e-01, 1.9245008972987526219e-01,
201 5.8715596330275232617e-01, 1.9156525704423027490e-01,
202 5.8181818181818178992e-01, 1.9069251784911847580e-01,
203 5.7657657657657657158e-01, 1.8983159915049979682e-01,
204 5.7142857142857139685e-01, 1.8898223650461362655e-01,
205 5.6637168141592919568e-01, 1.8814417367671945613e-01,
206 5.6140350877192979340e-01, 1.8731716231633879777e-01,
207 5.5652173913043478937e-01, 1.8650096164806276300e-01,
208 5.5172413793103447510e-01, 1.8569533817705186074e-01,
209 5.4700854700854706358e-01, 1.8490006540840969729e-01,
210 5.4237288135593220151e-01, 1.8411492357966466327e-01,
211 5.3781512605042014474e-01, 1.8333969940564226464e-01,
212 5.3333333333333332593e-01, 1.8257418583505535814e-01,
213 5.2892561983471075848e-01, 1.8181818181818182323e-01,
214 5.2459016393442625681e-01, 1.8107149208503706128e-01,
215 5.2032520325203257539e-01, 1.8033392693348646030e-01,
216 5.1612903225806450180e-01, 1.7960530202677491007e-01,
217 5.1200000000000001066e-01, 1.7888543819998317663e-01,
218 5.0793650793650790831e-01, 1.7817416127494958844e-01,
219 5.0393700787401574104e-01, 1.7747130188322274291e-01,
220 };
221
222 #define fabsf __fabsf
223
224 extern float fabsf( float );
225
226 static const double
227 A0 = 9.99999997962321453275e-01,
228 A1 =-4.99999998166077580600e-01,
229 A2 = 3.75066768969515586277e-01,
230 A3 =-3.12560092408808548438e-01;
231
232 static void
233 __vrhypotf_n( int n, float * restrict px, int stridex, float * restrict py,
234 int stridey, float * restrict pz, int stridez );
235
236 #pragma no_inline(__vrhypotf_n)
237
238 #define RETURN(ret) \
239 { \
240 *pz = (ret); \
241 pz += stridez; \
242 if ( n_n == 0 ) \
243 { \
244 spx = px; spy = py; spz = pz; \
245 ay0 = *(int*)py; \
246 continue; \
247 } \
248 n--; \
249 break; \
250 }
251
252
253 void
254 __vrhypotf( int n, float * restrict px, int stridex, float * restrict py,
255 int stridey, float * restrict pz, int stridez )
256 {
257 float *spx, *spy, *spz;
258 int ax0, ay0, n_n;
259 float res, x0, y0;
260
261 while ( n > 1 )
262 {
263 n_n = 0;
264 spx = px;
265 spy = py;
266 spz = pz;
267 ax0 = *(int*)px;
268 ay0 = *(int*)py;
269 for ( ; n > 1 ; n-- )
270 {
271 ax0 &= 0x7fffffff;
272 ay0 &= 0x7fffffff;
273
274 px += stridex;
275
276 if ( ax0 >= 0x7f800000 || ay0 >= 0x7f800000 ) /* X or Y = NaN or Inf */
277 {
278 x0 = *(px - stridex);
279 y0 = *py;
280 res = fabsf(x0) + fabsf(y0);
281 if( ax0 == 0x7f800000 ) res = 0.0f;
282 else if( ay0 == 0x7f800000 ) res = 0.0f;
283 ax0 = *(int*)px;
284 py += stridey;
285 RETURN ( res )
286 }
287 ax0 = *(int*)px;
288 py += stridey;
289 if ( ay0 == 0 ) /* Y = 0 */
290 {
291 int tx = *(int*)(px - stridex) & 0x7fffffff;
292 if ( tx == 0 ) /* X = 0 */
293 {
294 RETURN ( 1.0f / 0.0f )
295 }
296 }
297 pz += stridez;
298 n_n++;
299 ay0 = *(int*)py;
300 }
301 if ( n_n > 0 )
302 __vrhypotf_n( n_n, spx, stridex, spy, stridey, spz, stridez );
303 }
304 if ( n > 0 )
305 {
306 ax0 = *(int*)px;
307 ay0 = *(int*)py;
308 x0 = *px;
309 y0 = *py;
310
311 ax0 &= 0x7fffffff;
312 ay0 &= 0x7fffffff;
313
314 if ( ax0 >= 0x7f800000 || ay0 >= 0x7f800000 ) /* X or Y = NaN or Inf */
315 {
316 res = fabsf(x0) + fabsf(y0);
317 if( ax0 == 0x7f800000 ) res = 0.0f;
318 else if( ay0 == 0x7f800000 ) res = 0.0f;
319 *pz = res;
320 }
321 else if ( ax0 == 0 && ay0 == 0 ) /* X and Y = 0 */
322 {
323 *pz = 1.0f / 0.0f;
324 }
325 else
326 {
327 double xx0, res0, hyp0, h_hi0 = 0, dbase0 = 0;
328 int ibase0, si0, hyp0h;
329
330 hyp0 = x0 * (double)x0 + y0 * (double)y0;
331
332 ibase0 = HI(&hyp0);
333
334 HI(&dbase0) = (0x60000000 - ((ibase0 & 0x7fe00000) >> 1));
335
336 hyp0h = (ibase0 & 0x000fffff) | 0x3ff00000;
337 HI(&hyp0) = hyp0h;
338 HI(&h_hi0) = hyp0h & 0x7fffc000;
339
340 ibase0 >>= 10;
341 si0 = ibase0 & 0x7f0;
342 xx0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[0];
343
344 xx0 = (hyp0 - h_hi0) * xx0;
345 res0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[1];
346 res0 *= (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0);
347 res0 *= dbase0;
348 *pz = res0;
349 }
350 }
351 }
352
353 static void
354 __vrhypotf_n( int n, float * restrict px, int stridex, float * restrict py,
355 int stridey, float * restrict pz, int stridez )
356 {
357 double xx0, res0, hyp0, h_hi0 = 0, dbase0 = 0;
358 double xx1, res1, hyp1, h_hi1 = 0, dbase1 = 0;
359 double xx2, res2, hyp2, h_hi2 = 0, dbase2 = 0;
360 float x0, y0;
361 float x1, y1;
362 float x2, y2;
363 int ibase0, si0, hyp0h;
364 int ibase1, si1, hyp1h;
365 int ibase2, si2, hyp2h;
366
367 for ( ; n > 2 ; n -= 3 )
368 {
369 x0 = *px;
370 px += stridex;
371 x1 = *px;
372 px += stridex;
373 x2 = *px;
374 px += stridex;
375
376 y0 = *py;
377 py += stridey;
378 y1 = *py;
379 py += stridey;
380 y2 = *py;
381 py += stridey;
382
383 hyp0 = x0 * (double)x0 + y0 * (double)y0;
384 hyp1 = x1 * (double)x1 + y1 * (double)y1;
385 hyp2 = x2 * (double)x2 + y2 * (double)y2;
386
387 ibase0 = HI(&hyp0);
388 ibase1 = HI(&hyp1);
389 ibase2 = HI(&hyp2);
390
391 HI(&dbase0) = (0x60000000 - ((ibase0 & 0x7fe00000) >> 1));
392 HI(&dbase1) = (0x60000000 - ((ibase1 & 0x7fe00000) >> 1));
393 HI(&dbase2) = (0x60000000 - ((ibase2 & 0x7fe00000) >> 1));
394
395 hyp0h = (ibase0 & 0x000fffff) | 0x3ff00000;
396 hyp1h = (ibase1 & 0x000fffff) | 0x3ff00000;
397 hyp2h = (ibase2 & 0x000fffff) | 0x3ff00000;
398 HI(&hyp0) = hyp0h;
399 HI(&hyp1) = hyp1h;
400 HI(&hyp2) = hyp2h;
401 HI(&h_hi0) = hyp0h & 0x7fffc000;
402 HI(&h_hi1) = hyp1h & 0x7fffc000;
403 HI(&h_hi2) = hyp2h & 0x7fffc000;
404
405 ibase0 >>= 10;
406 ibase1 >>= 10;
407 ibase2 >>= 10;
408 si0 = ibase0 & 0x7f0;
409 si1 = ibase1 & 0x7f0;
410 si2 = ibase2 & 0x7f0;
411 xx0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[0];
412 xx1 = ((double*)((char*)__vlibm_TBL_rhypotf + si1))[0];
413 xx2 = ((double*)((char*)__vlibm_TBL_rhypotf + si2))[0];
414
415 xx0 = (hyp0 - h_hi0) * xx0;
416 xx1 = (hyp1 - h_hi1) * xx1;
417 xx2 = (hyp2 - h_hi2) * xx2;
418 res0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[1];
419 res1 = ((double*)((char*)__vlibm_TBL_rhypotf + si1))[1];
420 res2 = ((double*)((char*)__vlibm_TBL_rhypotf + si2))[1];
421 res0 *= (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0);
422 res1 *= (((A3 * xx1 + A2) * xx1 + A1) * xx1 + A0);
423 res2 *= (((A3 * xx2 + A2) * xx2 + A1) * xx2 + A0);
424 res0 *= dbase0;
425 res1 *= dbase1;
426 res2 *= dbase2;
427 *pz = res0;
428 pz += stridez;
429 *pz = res1;
430 pz += stridez;
431 *pz = res2;
432 pz += stridez;
433 }
434
435 for ( ; n > 0 ; n-- )
436 {
437 x0 = *px;
438 px += stridex;
439
440 y0 = *py;
441 py += stridey;
442
443 hyp0 = x0 * (double)x0 + y0 * (double)y0;
444
445 ibase0 = HI(&hyp0);
446
447 HI(&dbase0) = (0x60000000 - ((ibase0 & 0x7fe00000) >> 1));
448
449 hyp0h = (ibase0 & 0x000fffff) | 0x3ff00000;
450 HI(&hyp0) = hyp0h;
451 HI(&h_hi0) = hyp0h & 0x7fffc000;
452
453 ibase0 >>= 10;
454 si0 = ibase0 & 0x7f0;
455 xx0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[0];
456
457 xx0 = (hyp0 - h_hi0) * xx0;
458 res0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[1];
459 res0 *= (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0);
460 res0 *= dbase0;
461 *pz = res0;
462 pz += stridez;
463 }
464 }
465