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
__vrhypotf(int n,float * restrict px,int stridex,float * restrict py,int stridey,float * restrict pz,int stridez)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
__vrhypotf_n(int n,float * restrict px,int stridex,float * restrict py,int stridey,float * restrict pz,int stridez)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 }
462