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