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