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1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
2 *
3 * ***** BEGIN LICENSE BLOCK *****
4 * Version: MPL 1.1/GPL 2.0/LGPL 2.1
5 *
6 * The contents of this file are subject to the Mozilla Public License Version
7 * 1.1 (the "License"); you may not use this file except in compliance with
8 * the License. You may obtain a copy of the License at
9 * http://www.mozilla.org/MPL/
10 *
11 * Software distributed under the License is distributed on an "AS IS" basis,
12 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
13 * for the specific language governing rights and limitations under the
14 * License.
15 *
16 * The Original Code is Mozilla Communicator client code, released
17 * March 31, 1998.
18 *
19 * The Initial Developer of the Original Code is
20 * Sun Microsystems, Inc.
21 * Portions created by the Initial Developer are Copyright (C) 1998
22 * the Initial Developer. All Rights Reserved.
23 *
24 * Contributor(s):
25 *
26 * Alternatively, the contents of this file may be used under the terms of
27 * either of the GNU General Public License Version 2 or later (the "GPL"),
28 * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
29 * in which case the provisions of the GPL or the LGPL are applicable instead
30 * of those above. If you wish to allow use of your version of this file only
31 * under the terms of either the GPL or the LGPL, and not to allow others to
32 * use your version of this file under the terms of the MPL, indicate your
33 * decision by deleting the provisions above and replace them with the notice
34 * and other provisions required by the GPL or the LGPL. If you do not delete
35 * the provisions above, a recipient may use your version of this file under
36 * the terms of any one of the MPL, the GPL or the LGPL.
37 *
38 * ***** END LICENSE BLOCK ***** */
39 /* @(#)e_sqrt.c 1.3 95/01/18 */
40 /*
41 * ====================================================
42 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
43 *
44 * Developed at SunSoft, a Sun Microsystems, Inc. business.
45 * Permission to use, copy, modify, and distribute this
46 * software is freely granted, provided that this notice
47 * is preserved.
48 * ====================================================
49 */
50
51 /* __ieee754_sqrt(x)
52 * Return correctly rounded sqrt.
53 * ------------------------------------------
54 * | Use the hardware sqrt if you have one |
55 * ------------------------------------------
56 * Method:
57 * Bit by bit method using integer arithmetic. (Slow, but portable)
58 * 1. Normalization
59 * Scale x to y in [1,4) with even powers of 2:
60 * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
61 * sqrt(y) = 2^k * sqrt(x)
62 * 2. Bit by bit computation
63 * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
64 * i 0
65 * i+1 2
66 * s = 2*q , and y = 2 * ( y - q ). (1)
67 * i i i i
68 *
69 * To compute q from q , one checks whether
70 * i+1 i
71 *
72 * -(i+1) 2
73 * (q + 2 ) <= y. (2)
74 * i
75 * -(i+1)
76 * If (2) is false, then q = q ; otherwise q = q + 2 .
77 * i+1 i i+1 i
78 *
79 * With some algebric manipulation, it is not difficult to see
80 * that (2) is equivalent to
81 * -(i+1)
82 * s + 2 <= y (3)
83 * i i
84 *
85 * The advantage of (3) is that s and y can be computed by
86 * i i
87 * the following recurrence formula:
88 * if (3) is false
89 *
90 * s = s , y = y ; (4)
91 * i+1 i i+1 i
92 *
93 * otherwise,
94 * -i -(i+1)
95 * s = s + 2 , y = y - s - 2 (5)
96 * i+1 i i+1 i i
97 *
98 * One may easily use induction to prove (4) and (5).
99 * Note. Since the left hand side of (3) contain only i+2 bits,
100 * it does not necessary to do a full (53-bit) comparison
101 * in (3).
102 * 3. Final rounding
103 * After generating the 53 bits result, we compute one more bit.
104 * Together with the remainder, we can decide whether the
105 * result is exact, bigger than 1/2ulp, or less than 1/2ulp
106 * (it will never equal to 1/2ulp).
107 * The rounding mode can be detected by checking whether
108 * huge + tiny is equal to huge, and whether huge - tiny is
109 * equal to huge for some floating point number "huge" and "tiny".
110 *
111 * Special cases:
112 * sqrt(+-0) = +-0 ... exact
113 * sqrt(inf) = inf
114 * sqrt(-ve) = NaN ... with invalid signal
115 * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
116 *
117 * Other methods : see the appended file at the end of the program below.
118 *---------------
119 */
120
121 #include "fdlibm.h"
122
123 #if defined(_MSC_VER)
124 /* Microsoft Compiler */
125 #pragma warning( disable : 4723 ) /* disables potential divide by 0 warning */
126 #endif
127
128 #ifdef __STDC__
129 static const double one = 1.0, tiny=1.0e-300;
130 #else
131 static double one = 1.0, tiny=1.0e-300;
132 #endif
133
134 #ifdef __STDC__
135 double __ieee754_sqrt(double x)
136 #else
137 double __ieee754_sqrt(x)
138 double x;
139 #endif
140 {
141 fd_twoints u;
142 double z;
143 int sign = (int)0x80000000;
144 unsigned r,t1,s1,ix1,q1;
145 int ix0,s0,q,m,t,i;
146
147 u.d = x;
148 ix0 = __HI(u); /* high word of x */
149 ix1 = __LO(u); /* low word of x */
150
151 /* take care of Inf and NaN */
152 if((ix0&0x7ff00000)==0x7ff00000) {
153 return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
154 sqrt(-inf)=sNaN */
155 }
156 /* take care of zero */
157 if(ix0<=0) {
158 if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
159 else if(ix0<0)
160 return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
161 }
162 /* normalize x */
163 m = (ix0>>20);
164 if(m==0) { /* subnormal x */
165 while(ix0==0) {
166 m -= 21;
167 ix0 |= (ix1>>11); ix1 <<= 21;
168 }
169 for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
170 m -= i-1;
171 ix0 |= (ix1>>(32-i));
172 ix1 <<= i;
173 }
174 m -= 1023; /* unbias exponent */
175 ix0 = (ix0&0x000fffff)|0x00100000;
176 if(m&1){ /* odd m, double x to make it even */
177 ix0 += ix0 + ((ix1&sign)>>31);
178 ix1 += ix1;
179 }
180 m >>= 1; /* m = [m/2] */
181
182 /* generate sqrt(x) bit by bit */
183 ix0 += ix0 + ((ix1&sign)>>31);
184 ix1 += ix1;
185 q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
186 r = 0x00200000; /* r = moving bit from right to left */
187
188 while(r!=0) {
189 t = s0+r;
190 if(t<=ix0) {
191 s0 = t+r;
192 ix0 -= t;
193 q += r;
194 }
195 ix0 += ix0 + ((ix1&sign)>>31);
196 ix1 += ix1;
197 r>>=1;
198 }
199
200 r = sign;
201 while(r!=0) {
202 t1 = s1+r;
203 t = s0;
204 if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
205 s1 = t1+r;
206 if(((int)(t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
207 ix0 -= t;
208 if (ix1 < t1) ix0 -= 1;
209 ix1 -= t1;
210 q1 += r;
211 }
212 ix0 += ix0 + ((ix1&sign)>>31);
213 ix1 += ix1;
214 r>>=1;
215 }
216
217 /* use floating add to find out rounding direction */
218 if((ix0|ix1)!=0) {
219 z = one-tiny; /* trigger inexact flag */
220 if (z>=one) {
221 z = one+tiny;
222 if (q1==(unsigned)0xffffffff) { q1=0; q += 1;}
223 else if (z>one) {
224 if (q1==(unsigned)0xfffffffe) q+=1;
225 q1+=2;
226 } else
227 q1 += (q1&1);
228 }
229 }
230 ix0 = (q>>1)+0x3fe00000;
231 ix1 = q1>>1;
232 if ((q&1)==1) ix1 |= sign;
233 ix0 += (m <<20);
234 u.d = z;
235 __HI(u) = ix0;
236 __LO(u) = ix1;
237 z = u.d;
238 return z;
239 }
240
241 /*
242 Other methods (use floating-point arithmetic)
243 -------------
244 (This is a copy of a drafted paper by Prof W. Kahan
245 and K.C. Ng, written in May, 1986)
246
247 Two algorithms are given here to implement sqrt(x)
248 (IEEE double precision arithmetic) in software.
249 Both supply sqrt(x) correctly rounded. The first algorithm (in
250 Section A) uses newton iterations and involves four divisions.
251 The second one uses reciproot iterations to avoid division, but
252 requires more multiplications. Both algorithms need the ability
253 to chop results of arithmetic operations instead of round them,
254 and the INEXACT flag to indicate when an arithmetic operation
255 is executed exactly with no roundoff error, all part of the
256 standard (IEEE 754-1985). The ability to perform shift, add,
257 subtract and logical AND operations upon 32-bit words is needed
258 too, though not part of the standard.
259
260 A. sqrt(x) by Newton Iteration
261
262 (1) Initial approximation
263
264 Let x0 and x1 be the leading and the trailing 32-bit words of
265 a floating point number x (in IEEE double format) respectively
266
267 1 11 52 ...widths
268 ------------------------------------------------------
269 x: |s| e | f |
270 ------------------------------------------------------
271 msb lsb msb lsb ...order
272
273
274 ------------------------ ------------------------
275 x0: |s| e | f1 | x1: | f2 |
276 ------------------------ ------------------------
277
278 By performing shifts and subtracts on x0 and x1 (both regarded
279 as integers), we obtain an 8-bit approximation of sqrt(x) as
280 follows.
281
282 k := (x0>>1) + 0x1ff80000;
283 y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
284 Here k is a 32-bit integer and T1[] is an integer array containing
285 correction terms. Now magically the floating value of y (y's
286 leading 32-bit word is y0, the value of its trailing word is 0)
287 approximates sqrt(x) to almost 8-bit.
288
289 Value of T1:
290 static int T1[32]= {
291 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
292 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
293 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
294 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
295
296 (2) Iterative refinement
297
298 Apply Heron's rule three times to y, we have y approximates
299 sqrt(x) to within 1 ulp (Unit in the Last Place):
300
301 y := (y+x/y)/2 ... almost 17 sig. bits
302 y := (y+x/y)/2 ... almost 35 sig. bits
303 y := y-(y-x/y)/2 ... within 1 ulp
304
305
306 Remark 1.
307 Another way to improve y to within 1 ulp is:
308
309 y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
310 y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
311
312 2
313 (x-y )*y
314 y := y + 2* ---------- ...within 1 ulp
315 2
316 3y + x
317
318
319 This formula has one division fewer than the one above; however,
320 it requires more multiplications and additions. Also x must be
321 scaled in advance to avoid spurious overflow in evaluating the
322 expression 3y*y+x. Hence it is not recommended uless division
323 is slow. If division is very slow, then one should use the
324 reciproot algorithm given in section B.
325
326 (3) Final adjustment
327
328 By twiddling y's last bit it is possible to force y to be
329 correctly rounded according to the prevailing rounding mode
330 as follows. Let r and i be copies of the rounding mode and
331 inexact flag before entering the square root program. Also we
332 use the expression y+-ulp for the next representable floating
333 numbers (up and down) of y. Note that y+-ulp = either fixed
334 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
335 mode.
336
337 I := FALSE; ... reset INEXACT flag I
338 R := RZ; ... set rounding mode to round-toward-zero
339 z := x/y; ... chopped quotient, possibly inexact
340 If(not I) then { ... if the quotient is exact
341 if(z=y) {
342 I := i; ... restore inexact flag
343 R := r; ... restore rounded mode
344 return sqrt(x):=y.
345 } else {
346 z := z - ulp; ... special rounding
347 }
348 }
349 i := TRUE; ... sqrt(x) is inexact
350 If (r=RN) then z=z+ulp ... rounded-to-nearest
351 If (r=RP) then { ... round-toward-+inf
352 y = y+ulp; z=z+ulp;
353 }
354 y := y+z; ... chopped sum
355 y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
356 I := i; ... restore inexact flag
357 R := r; ... restore rounded mode
358 return sqrt(x):=y.
359
360 (4) Special cases
361
362 Square root of +inf, +-0, or NaN is itself;
363 Square root of a negative number is NaN with invalid signal.
364
365
366 B. sqrt(x) by Reciproot Iteration
367
368 (1) Initial approximation
369
370 Let x0 and x1 be the leading and the trailing 32-bit words of
371 a floating point number x (in IEEE double format) respectively
372 (see section A). By performing shifs and subtracts on x0 and y0,
373 we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
374
375 k := 0x5fe80000 - (x0>>1);
376 y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
377
378 Here k is a 32-bit integer and T2[] is an integer array
379 containing correction terms. Now magically the floating
380 value of y (y's leading 32-bit word is y0, the value of
381 its trailing word y1 is set to zero) approximates 1/sqrt(x)
382 to almost 7.8-bit.
383
384 Value of T2:
385 static int T2[64]= {
386 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
387 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
388 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
389 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
390 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
391 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
392 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
393 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
394
395 (2) Iterative refinement
396
397 Apply Reciproot iteration three times to y and multiply the
398 result by x to get an approximation z that matches sqrt(x)
399 to about 1 ulp. To be exact, we will have
400 -1ulp < sqrt(x)-z<1.0625ulp.
401
402 ... set rounding mode to Round-to-nearest
403 y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
404 y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
405 ... special arrangement for better accuracy
406 z := x*y ... 29 bits to sqrt(x), with z*y<1
407 z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
408
409 Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
410 (a) the term z*y in the final iteration is always less than 1;
411 (b) the error in the final result is biased upward so that
412 -1 ulp < sqrt(x) - z < 1.0625 ulp
413 instead of |sqrt(x)-z|<1.03125ulp.
414
415 (3) Final adjustment
416
417 By twiddling y's last bit it is possible to force y to be
418 correctly rounded according to the prevailing rounding mode
419 as follows. Let r and i be copies of the rounding mode and
420 inexact flag before entering the square root program. Also we
421 use the expression y+-ulp for the next representable floating
422 numbers (up and down) of y. Note that y+-ulp = either fixed
423 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
424 mode.
425
426 R := RZ; ... set rounding mode to round-toward-zero
427 switch(r) {
428 case RN: ... round-to-nearest
429 if(x<= z*(z-ulp)...chopped) z = z - ulp; else
430 if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
431 break;
432 case RZ:case RM: ... round-to-zero or round-to--inf
433 R:=RP; ... reset rounding mod to round-to-+inf
434 if(x<z*z ... rounded up) z = z - ulp; else
435 if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
436 break;
437 case RP: ... round-to-+inf
438 if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
439 if(x>z*z ...chopped) z = z+ulp;
440 break;
441 }
442
443 Remark 3. The above comparisons can be done in fixed point. For
444 example, to compare x and w=z*z chopped, it suffices to compare
445 x1 and w1 (the trailing parts of x and w), regarding them as
446 two's complement integers.
447
448 ...Is z an exact square root?
449 To determine whether z is an exact square root of x, let z1 be the
450 trailing part of z, and also let x0 and x1 be the leading and
451 trailing parts of x.
452
453 If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
454 I := 1; ... Raise Inexact flag: z is not exact
455 else {
456 j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
457 k := z1 >> 26; ... get z's 25-th and 26-th
458 fraction bits
459 I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
460 }
461 R:= r ... restore rounded mode
462 return sqrt(x):=z.
463
464 If multiplication is cheaper then the foregoing red tape, the
465 Inexact flag can be evaluated by
466
467 I := i;
468 I := (z*z!=x) or I.
469
470 Note that z*z can overwrite I; this value must be sensed if it is
471 True.
472
473 Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
474 zero.
475
476 --------------------
477 z1: | f2 |
478 --------------------
479 bit 31 bit 0
480
481 Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
482 or even of logb(x) have the following relations:
483
484 -------------------------------------------------
485 bit 27,26 of z1 bit 1,0 of x1 logb(x)
486 -------------------------------------------------
487 00 00 odd and even
488 01 01 even
489 10 10 odd
490 10 00 even
491 11 01 even
492 -------------------------------------------------
493
494 (4) Special cases (see (4) of Section A).
495
496 */
497

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