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 #pragma weak __sqrtl = sqrtl
31
32 #include "libm.h"
33 #include "longdouble.h"
34
35 extern int __swapTE(int);
36 extern int __swapEX(int);
37 extern enum fp_direction_type __swapRD(enum fp_direction_type);
38
39 /*
40 * in struct longdouble, msw consists of
41 * unsigned short sgn:1;
42 * unsigned short exp:15;
43 * unsigned short frac1:16;
44 */
45
46 #ifdef __LITTLE_ENDIAN
47
48 /* array indices used to access words within a double */
49 #define HIWORD 1
50 #define LOWORD 0
51
52 /* structure used to access words within a quad */
53 union longdouble {
54 struct {
55 unsigned int frac4;
56 unsigned int frac3;
57 unsigned int frac2;
58 unsigned int msw;
59 } l;
60 long double d;
61 };
62
63 /* default NaN returned for sqrt(neg) */
64 static const union longdouble
65 qnan = { 0xffffffff, 0xffffffff, 0xffffffff, 0x7fffffff };
66
67 /* signalling NaN used to raise invalid */
68 static const union {
69 unsigned u[2];
70 double d;
71 } snan = { 0, 0x7ff00001 };
72
73 #else
74
75 /* array indices used to access words within a double */
76 #define HIWORD 0
77 #define LOWORD 1
78
79 /* structure used to access words within a quad */
80 union longdouble {
81 struct {
82 unsigned int msw;
83 unsigned int frac2;
84 unsigned int frac3;
85 unsigned int frac4;
86 } l;
87 long double d;
88 };
89
90 /* default NaN returned for sqrt(neg) */
91 static const union longdouble
92 qnan = { 0x7fffffff, 0xffffffff, 0xffffffff, 0xffffffff };
93
94 /* signalling NaN used to raise invalid */
95 static const union {
96 unsigned u[2];
97 double d;
98 } snan = { 0x7ff00001, 0 };
99
100 #endif /* __LITTLE_ENDIAN */
101
102
103 static const double
104 zero = 0.0,
105 half = 0.5,
106 one = 1.0,
107 huge = 1.0e300,
108 tiny = 1.0e-300,
109 two36 = 6.87194767360000000000e+10,
110 two30 = 1.07374182400000000000e+09,
111 two6 = 6.40000000000000000000e+01,
112 two4 = 1.60000000000000000000e+01,
113 twom18 = 3.81469726562500000000e-06,
114 twom28 = 3.72529029846191406250e-09,
115 twom42 = 2.27373675443232059479e-13,
116 twom60 = 8.67361737988403547206e-19,
117 twom62 = 2.16840434497100886801e-19,
118 twom66 = 1.35525271560688054251e-20,
119 twom90 = 8.07793566946316088742e-28,
120 twom113 = 9.62964972193617926528e-35,
121 twom124 = 4.70197740328915003187e-38;
122
123
124 /*
125 * Extract the exponent and normalized significand (represented as
126 * an array of five doubles) from a finite, nonzero quad.
127 */
128 static int
__q_unpack(const union longdouble * x,double * s)129 __q_unpack(const union longdouble *x, double *s)
130 {
131 union {
132 double d;
133 unsigned int l[2];
134 } u;
135 double b;
136 unsigned int lx, w[3];
137 int ex;
138
139 /* get the normalized significand and exponent */
140 ex = (int) ((x->l.msw & 0x7fffffff) >> 16);
141 lx = x->l.msw & 0xffff;
142 if (ex)
143 {
144 lx |= 0x10000;
145 w[0] = x->l.frac2;
146 w[1] = x->l.frac3;
147 w[2] = x->l.frac4;
148 }
149 else
150 {
151 if (lx | (x->l.frac2 & 0xfffe0000))
152 {
153 w[0] = x->l.frac2;
154 w[1] = x->l.frac3;
155 w[2] = x->l.frac4;
156 ex = 1;
157 }
158 else if (x->l.frac2 | (x->l.frac3 & 0xfffe0000))
159 {
160 lx = x->l.frac2;
161 w[0] = x->l.frac3;
162 w[1] = x->l.frac4;
163 w[2] = 0;
164 ex = -31;
165 }
166 else if (x->l.frac3 | (x->l.frac4 & 0xfffe0000))
167 {
168 lx = x->l.frac3;
169 w[0] = x->l.frac4;
170 w[1] = w[2] = 0;
171 ex = -63;
172 }
173 else
174 {
175 lx = x->l.frac4;
176 w[0] = w[1] = w[2] = 0;
177 ex = -95;
178 }
179 while ((lx & 0x10000) == 0)
180 {
181 lx = (lx << 1) | (w[0] >> 31);
182 w[0] = (w[0] << 1) | (w[1] >> 31);
183 w[1] = (w[1] << 1) | (w[2] >> 31);
184 w[2] <<= 1;
185 ex--;
186 }
187 }
188
189 /* extract the significand into five doubles */
190 u.l[HIWORD] = 0x42300000;
191 u.l[LOWORD] = 0;
192 b = u.d;
193 u.l[LOWORD] = lx;
194 s[0] = u.d - b;
195
196 u.l[HIWORD] = 0x40300000;
197 u.l[LOWORD] = 0;
198 b = u.d;
199 u.l[LOWORD] = w[0] & 0xffffff00;
200 s[1] = u.d - b;
201
202 u.l[HIWORD] = 0x3e300000;
203 u.l[LOWORD] = 0;
204 b = u.d;
205 u.l[HIWORD] |= w[0] & 0xff;
206 u.l[LOWORD] = w[1] & 0xffff0000;
207 s[2] = u.d - b;
208
209 u.l[HIWORD] = 0x3c300000;
210 u.l[LOWORD] = 0;
211 b = u.d;
212 u.l[HIWORD] |= w[1] & 0xffff;
213 u.l[LOWORD] = w[2] & 0xff000000;
214 s[3] = u.d - b;
215
216 u.l[HIWORD] = 0x3c300000;
217 u.l[LOWORD] = 0;
218 b = u.d;
219 u.l[LOWORD] = w[2] & 0xffffff;
220 s[4] = u.d - b;
221
222 return ex - 0x3fff;
223 }
224
225
226 /*
227 * Pack an exponent and array of three doubles representing a finite,
228 * nonzero number into a quad. Assume the sign is already there and
229 * the rounding mode has been fudged accordingly.
230 */
231 static void
__q_pack(const double * z,int exp,enum fp_direction_type rm,union longdouble * x,int * inexact)232 __q_pack(const double *z, int exp, enum fp_direction_type rm,
233 union longdouble *x, int *inexact)
234 {
235 union {
236 double d;
237 unsigned int l[2];
238 } u;
239 double s[3], t, t2;
240 unsigned int msw, frac2, frac3, frac4;
241
242 /* bias exponent and strip off integer bit */
243 exp += 0x3fff;
244 s[0] = z[0] - one;
245 s[1] = z[1];
246 s[2] = z[2];
247
248 /*
249 * chop the significand to obtain the fraction;
250 * use round-to-minus-infinity to ensure chopping
251 */
252 (void) __swapRD(fp_negative);
253
254 /* extract the first eighty bits of fraction */
255 t = s[1] + s[2];
256 u.d = two36 + (s[0] + t);
257 msw = u.l[LOWORD];
258 s[0] -= (u.d - two36);
259
260 u.d = two4 + (s[0] + t);
261 frac2 = u.l[LOWORD];
262 s[0] -= (u.d - two4);
263
264 u.d = twom28 + (s[0] + t);
265 frac3 = u.l[LOWORD];
266 s[0] -= (u.d - twom28);
267
268 /* condense the remaining fraction; errors here won't matter */
269 t = s[0] + s[1];
270 s[1] = ((s[0] - t) + s[1]) + s[2];
271 s[0] = t;
272
273 /* get the last word of fraction */
274 u.d = twom60 + (s[0] + s[1]);
275 frac4 = u.l[LOWORD];
276 s[0] -= (u.d - twom60);
277
278 /*
279 * keep track of what's left for rounding; note that
280 * t2 will be non-negative due to rounding mode
281 */
282 t = s[0] + s[1];
283 t2 = (s[0] - t) + s[1];
284
285 if (t != zero)
286 {
287 *inexact = 1;
288
289 /* decide whether to round the fraction up */
290 if (rm == fp_positive || (rm == fp_nearest && (t > twom113 ||
291 (t == twom113 && (t2 != zero || frac4 & 1)))))
292 {
293 /* round up and renormalize if necessary */
294 if (++frac4 == 0)
295 if (++frac3 == 0)
296 if (++frac2 == 0)
297 if (++msw == 0x10000)
298 {
299 msw = 0;
300 exp++;
301 }
302 }
303 }
304
305 /* assemble the result */
306 x->l.msw |= msw | (exp << 16);
307 x->l.frac2 = frac2;
308 x->l.frac3 = frac3;
309 x->l.frac4 = frac4;
310 }
311
312
313 /*
314 * Compute the square root of x and place the TP result in s.
315 */
316 static void
__q_tp_sqrt(const double * x,double * s)317 __q_tp_sqrt(const double *x, double *s)
318 {
319 double c, rr, r[3], tt[3], t[5];
320
321 /* approximate the divisor for the Newton iteration */
322 c = sqrt((x[0] + x[1]) + x[2]);
323 rr = half / c;
324
325 /* compute the first five "digits" of the square root */
326 t[0] = (c + two30) - two30;
327 tt[0] = t[0] + t[0];
328 r[0] = ((x[0] - t[0] * t[0]) + x[1]) + x[2];
329
330 t[1] = (rr * (r[0] + x[3]) + two6) - two6;
331 tt[1] = t[1] + t[1];
332 r[0] -= tt[0] * t[1];
333 r[1] = x[3] - t[1] * t[1];
334 c = (r[1] + twom18) - twom18;
335 r[0] += c;
336 r[1] = (r[1] - c) + x[4];
337
338 t[2] = (rr * (r[0] + r[1]) + twom18) - twom18;
339 tt[2] = t[2] + t[2];
340 r[0] -= tt[0] * t[2];
341 r[1] -= tt[1] * t[2];
342 c = (r[1] + twom42) - twom42;
343 r[0] += c;
344 r[1] = (r[1] - c) - t[2] * t[2];
345
346 t[3] = (rr * (r[0] + r[1]) + twom42) - twom42;
347 r[0] = ((r[0] - tt[0] * t[3]) + r[1]) - tt[1] * t[3];
348 r[1] = -tt[2] * t[3];
349 c = (r[1] + twom90) - twom90;
350 r[0] += c;
351 r[1] = (r[1] - c) - t[3] * t[3];
352
353 t[4] = (rr * (r[0] + r[1]) + twom66) - twom66;
354
355 /* here we just need to get the sign of the remainder */
356 c = (((((r[0] - tt[0] * t[4]) - tt[1] * t[4]) + r[1])
357 - tt[2] * t[4]) - (t[3] + t[3]) * t[4]) - t[4] * t[4];
358
359 /* reduce to three doubles */
360 t[0] += t[1];
361 t[1] = t[2] + t[3];
362 t[2] = t[4];
363
364 /* if the third term might lie on a rounding boundary, perturb it */
365 if (c != zero && t[2] == (twom62 + t[2]) - twom62)
366 {
367 if (c < zero)
368 t[2] -= twom124;
369 else
370 t[2] += twom124;
371 }
372
373 /* condense the square root */
374 c = t[1] + t[2];
375 t[2] += (t[1] - c);
376 t[1] = c;
377 c = t[0] + t[1];
378 s[1] = t[1] + (t[0] - c);
379 s[0] = c;
380 if (s[1] == zero)
381 {
382 c = s[0] + t[2];
383 s[1] = t[2] + (s[0] - c);
384 s[0] = c;
385 s[2] = zero;
386 }
387 else
388 {
389 c = s[1] + t[2];
390 s[2] = t[2] + (s[1] - c);
391 s[1] = c;
392 }
393 }
394
395
396 long double
sqrtl(long double ldx)397 sqrtl(long double ldx)
398 {
399 union longdouble x;
400 volatile double t;
401 double xx[5], zz[3];
402 enum fp_direction_type rm;
403 int ex, inexact, exc, traps;
404
405 /* clear cexc */
406 t = zero;
407 t -= zero;
408
409 /* check for zero operand */
410 x.d = ldx;
411 if (!((x.l.msw & 0x7fffffff) | x.l.frac2 | x.l.frac3 | x.l.frac4))
412 return ldx;
413
414 /* handle nan and inf cases */
415 if ((x.l.msw & 0x7fffffff) >= 0x7fff0000)
416 {
417 if ((x.l.msw & 0xffff) | x.l.frac2 | x.l.frac3 | x.l.frac4)
418 {
419 if (!(x.l.msw & 0x8000))
420 {
421 /* snan, signal invalid */
422 t += snan.d;
423 }
424 x.l.msw |= 0x8000;
425 return x.d;
426 }
427 if (x.l.msw & 0x80000000)
428 {
429 /* sqrt(-inf), signal invalid */
430 t = -one;
431 t = sqrt(t);
432 return qnan.d;
433 }
434 /* sqrt(inf), return inf */
435 return x.d;
436 }
437
438 /* handle negative numbers */
439 if (x.l.msw & 0x80000000)
440 {
441 t = -one;
442 t = sqrt(t);
443 return qnan.d;
444 }
445
446 /* now x is finite, positive */
447
448 traps = __swapTE(0);
449 exc = __swapEX(0);
450 rm = __swapRD(fp_nearest);
451
452 ex = __q_unpack(&x, xx);
453 if (ex & 1)
454 {
455 /* make exponent even */
456 xx[0] += xx[0];
457 xx[1] += xx[1];
458 xx[2] += xx[2];
459 xx[3] += xx[3];
460 xx[4] += xx[4];
461 ex--;
462 }
463 __q_tp_sqrt(xx, zz);
464
465 /* put everything together */
466 x.l.msw = 0;
467 inexact = 0;
468 __q_pack(zz, ex >> 1, rm, &x, &inexact);
469
470 (void) __swapRD(rm);
471 (void) __swapEX(exc);
472 (void) __swapTE(traps);
473 if (inexact)
474 {
475 t = huge;
476 t += tiny;
477 }
478 return x.d;
479 }
480