Line data Source code
1 :
2 : /* @(#)e_sqrt.c 1.3 95/01/18 */
3 : /*
4 : * ====================================================
5 : * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 : *
7 : * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 : * Permission to use, copy, modify, and distribute this
9 : * software is freely granted, provided that this notice
10 : * is preserved.
11 : * ====================================================
12 : */
13 :
14 : //#include <sys/cdefs.h>
15 : //__FBSDID("$FreeBSD$");
16 :
17 : /* __ieee754_sqrt(x)
18 : * Return correctly rounded sqrt.
19 : * ------------------------------------------
20 : * | Use the hardware sqrt if you have one |
21 : * ------------------------------------------
22 : * Method:
23 : * Bit by bit method using integer arithmetic. (Slow, but portable)
24 : * 1. Normalization
25 : * Scale x to y in [1,4) with even powers of 2:
26 : * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
27 : * sqrt(x) = 2^k * sqrt(y)
28 : * 2. Bit by bit computation
29 : * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
30 : * i 0
31 : * i+1 2
32 : * s = 2*q , and y = 2 * ( y - q ). (1)
33 : * i i i i
34 : *
35 : * To compute q from q , one checks whether
36 : * i+1 i
37 : *
38 : * -(i+1) 2
39 : * (q + 2 ) <= y. (2)
40 : * i
41 : * -(i+1)
42 : * If (2) is false, then q = q ; otherwise q = q + 2 .
43 : * i+1 i i+1 i
44 : *
45 : * With some algebric manipulation, it is not difficult to see
46 : * that (2) is equivalent to
47 : * -(i+1)
48 : * s + 2 <= y (3)
49 : * i i
50 : *
51 : * The advantage of (3) is that s and y can be computed by
52 : * i i
53 : * the following recurrence formula:
54 : * if (3) is false
55 : *
56 : * s = s , y = y ; (4)
57 : * i+1 i i+1 i
58 : *
59 : * otherwise,
60 : * -i -(i+1)
61 : * s = s + 2 , y = y - s - 2 (5)
62 : * i+1 i i+1 i i
63 : *
64 : * One may easily use induction to prove (4) and (5).
65 : * Note. Since the left hand side of (3) contain only i+2 bits,
66 : * it does not necessary to do a full (53-bit) comparison
67 : * in (3).
68 : * 3. Final rounding
69 : * After generating the 53 bits result, we compute one more bit.
70 : * Together with the remainder, we can decide whether the
71 : * result is exact, bigger than 1/2ulp, or less than 1/2ulp
72 : * (it will never equal to 1/2ulp).
73 : * The rounding mode can be detected by checking whether
74 : * huge + tiny is equal to huge, and whether huge - tiny is
75 : * equal to huge for some floating point number "huge" and "tiny".
76 : *
77 : * Special cases:
78 : * sqrt(+-0) = +-0 ... exact
79 : * sqrt(inf) = inf
80 : * sqrt(-ve) = NaN ... with invalid signal
81 : * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
82 : *
83 : * Other methods : see the appended file at the end of the program below.
84 : *---------------
85 : */
86 :
87 : #include <float.h>
88 :
89 : #include "math_private.h"
90 :
91 : static const double one = 1.0, tiny=1.0e-300;
92 :
93 : double
94 0 : __ieee754_sqrt(double x)
95 : {
96 : double z;
97 0 : int32_t sign = (int)0x80000000;
98 : int32_t ix0,s0,q,m,t,i;
99 : u_int32_t r,t1,s1,ix1,q1;
100 :
101 0 : EXTRACT_WORDS(ix0,ix1,x);
102 :
103 : /* take care of Inf and NaN */
104 0 : if((ix0&0x7ff00000)==0x7ff00000) {
105 0 : return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
106 : sqrt(-inf)=sNaN */
107 : }
108 : /* take care of zero */
109 0 : if(ix0<=0) {
110 0 : if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
111 0 : else if(ix0<0)
112 0 : return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
113 : }
114 : /* normalize x */
115 0 : m = (ix0>>20);
116 0 : if(m==0) { /* subnormal x */
117 0 : while(ix0==0) {
118 0 : m -= 21;
119 0 : ix0 |= (ix1>>11); ix1 <<= 21;
120 : }
121 0 : for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
122 0 : m -= i-1;
123 0 : ix0 |= (ix1>>(32-i));
124 0 : ix1 <<= i;
125 : }
126 0 : m -= 1023; /* unbias exponent */
127 0 : ix0 = (ix0&0x000fffff)|0x00100000;
128 0 : if(m&1){ /* odd m, double x to make it even */
129 0 : ix0 += ix0 + ((ix1&sign)>>31);
130 0 : ix1 += ix1;
131 : }
132 0 : m >>= 1; /* m = [m/2] */
133 :
134 : /* generate sqrt(x) bit by bit */
135 0 : ix0 += ix0 + ((ix1&sign)>>31);
136 0 : ix1 += ix1;
137 0 : q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
138 0 : r = 0x00200000; /* r = moving bit from right to left */
139 :
140 0 : while(r!=0) {
141 0 : t = s0+r;
142 0 : if(t<=ix0) {
143 0 : s0 = t+r;
144 0 : ix0 -= t;
145 0 : q += r;
146 : }
147 0 : ix0 += ix0 + ((ix1&sign)>>31);
148 0 : ix1 += ix1;
149 0 : r>>=1;
150 : }
151 :
152 0 : r = sign;
153 0 : while(r!=0) {
154 0 : t1 = s1+r;
155 0 : t = s0;
156 0 : if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
157 0 : s1 = t1+r;
158 0 : if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
159 0 : ix0 -= t;
160 0 : if (ix1 < t1) ix0 -= 1;
161 0 : ix1 -= t1;
162 0 : q1 += r;
163 : }
164 0 : ix0 += ix0 + ((ix1&sign)>>31);
165 0 : ix1 += ix1;
166 0 : r>>=1;
167 : }
168 :
169 : /* use floating add to find out rounding direction */
170 0 : if((ix0|ix1)!=0) {
171 0 : z = one-tiny; /* trigger inexact flag */
172 0 : if (z>=one) {
173 0 : z = one+tiny;
174 0 : if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;}
175 0 : else if (z>one) {
176 0 : if (q1==(u_int32_t)0xfffffffe) q+=1;
177 0 : q1+=2;
178 : } else
179 0 : q1 += (q1&1);
180 : }
181 : }
182 0 : ix0 = (q>>1)+0x3fe00000;
183 0 : ix1 = q1>>1;
184 0 : if ((q&1)==1) ix1 |= sign;
185 0 : ix0 += (m <<20);
186 0 : INSERT_WORDS(z,ix0,ix1);
187 0 : return z;
188 : }
189 :
190 : /*
191 : Other methods (use floating-point arithmetic)
192 : -------------
193 : (This is a copy of a drafted paper by Prof W. Kahan
194 : and K.C. Ng, written in May, 1986)
195 :
196 : Two algorithms are given here to implement sqrt(x)
197 : (IEEE double precision arithmetic) in software.
198 : Both supply sqrt(x) correctly rounded. The first algorithm (in
199 : Section A) uses newton iterations and involves four divisions.
200 : The second one uses reciproot iterations to avoid division, but
201 : requires more multiplications. Both algorithms need the ability
202 : to chop results of arithmetic operations instead of round them,
203 : and the INEXACT flag to indicate when an arithmetic operation
204 : is executed exactly with no roundoff error, all part of the
205 : standard (IEEE 754-1985). The ability to perform shift, add,
206 : subtract and logical AND operations upon 32-bit words is needed
207 : too, though not part of the standard.
208 :
209 : A. sqrt(x) by Newton Iteration
210 :
211 : (1) Initial approximation
212 :
213 : Let x0 and x1 be the leading and the trailing 32-bit words of
214 : a floating point number x (in IEEE double format) respectively
215 :
216 : 1 11 52 ...widths
217 : ------------------------------------------------------
218 : x: |s| e | f |
219 : ------------------------------------------------------
220 : msb lsb msb lsb ...order
221 :
222 :
223 : ------------------------ ------------------------
224 : x0: |s| e | f1 | x1: | f2 |
225 : ------------------------ ------------------------
226 :
227 : By performing shifts and subtracts on x0 and x1 (both regarded
228 : as integers), we obtain an 8-bit approximation of sqrt(x) as
229 : follows.
230 :
231 : k := (x0>>1) + 0x1ff80000;
232 : y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
233 : Here k is a 32-bit integer and T1[] is an integer array containing
234 : correction terms. Now magically the floating value of y (y's
235 : leading 32-bit word is y0, the value of its trailing word is 0)
236 : approximates sqrt(x) to almost 8-bit.
237 :
238 : Value of T1:
239 : static int T1[32]= {
240 : 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
241 : 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
242 : 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
243 : 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
244 :
245 : (2) Iterative refinement
246 :
247 : Apply Heron's rule three times to y, we have y approximates
248 : sqrt(x) to within 1 ulp (Unit in the Last Place):
249 :
250 : y := (y+x/y)/2 ... almost 17 sig. bits
251 : y := (y+x/y)/2 ... almost 35 sig. bits
252 : y := y-(y-x/y)/2 ... within 1 ulp
253 :
254 :
255 : Remark 1.
256 : Another way to improve y to within 1 ulp is:
257 :
258 : y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
259 : y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
260 :
261 : 2
262 : (x-y )*y
263 : y := y + 2* ---------- ...within 1 ulp
264 : 2
265 : 3y + x
266 :
267 :
268 : This formula has one division fewer than the one above; however,
269 : it requires more multiplications and additions. Also x must be
270 : scaled in advance to avoid spurious overflow in evaluating the
271 : expression 3y*y+x. Hence it is not recommended uless division
272 : is slow. If division is very slow, then one should use the
273 : reciproot algorithm given in section B.
274 :
275 : (3) Final adjustment
276 :
277 : By twiddling y's last bit it is possible to force y to be
278 : correctly rounded according to the prevailing rounding mode
279 : as follows. Let r and i be copies of the rounding mode and
280 : inexact flag before entering the square root program. Also we
281 : use the expression y+-ulp for the next representable floating
282 : numbers (up and down) of y. Note that y+-ulp = either fixed
283 : point y+-1, or multiply y by nextafter(1,+-inf) in chopped
284 : mode.
285 :
286 : I := FALSE; ... reset INEXACT flag I
287 : R := RZ; ... set rounding mode to round-toward-zero
288 : z := x/y; ... chopped quotient, possibly inexact
289 : If(not I) then { ... if the quotient is exact
290 : if(z=y) {
291 : I := i; ... restore inexact flag
292 : R := r; ... restore rounded mode
293 : return sqrt(x):=y.
294 : } else {
295 : z := z - ulp; ... special rounding
296 : }
297 : }
298 : i := TRUE; ... sqrt(x) is inexact
299 : If (r=RN) then z=z+ulp ... rounded-to-nearest
300 : If (r=RP) then { ... round-toward-+inf
301 : y = y+ulp; z=z+ulp;
302 : }
303 : y := y+z; ... chopped sum
304 : y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
305 : I := i; ... restore inexact flag
306 : R := r; ... restore rounded mode
307 : return sqrt(x):=y.
308 :
309 : (4) Special cases
310 :
311 : Square root of +inf, +-0, or NaN is itself;
312 : Square root of a negative number is NaN with invalid signal.
313 :
314 :
315 : B. sqrt(x) by Reciproot Iteration
316 :
317 : (1) Initial approximation
318 :
319 : Let x0 and x1 be the leading and the trailing 32-bit words of
320 : a floating point number x (in IEEE double format) respectively
321 : (see section A). By performing shifs and subtracts on x0 and y0,
322 : we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
323 :
324 : k := 0x5fe80000 - (x0>>1);
325 : y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
326 :
327 : Here k is a 32-bit integer and T2[] is an integer array
328 : containing correction terms. Now magically the floating
329 : value of y (y's leading 32-bit word is y0, the value of
330 : its trailing word y1 is set to zero) approximates 1/sqrt(x)
331 : to almost 7.8-bit.
332 :
333 : Value of T2:
334 : static int T2[64]= {
335 : 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
336 : 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
337 : 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
338 : 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
339 : 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
340 : 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
341 : 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
342 : 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
343 :
344 : (2) Iterative refinement
345 :
346 : Apply Reciproot iteration three times to y and multiply the
347 : result by x to get an approximation z that matches sqrt(x)
348 : to about 1 ulp. To be exact, we will have
349 : -1ulp < sqrt(x)-z<1.0625ulp.
350 :
351 : ... set rounding mode to Round-to-nearest
352 : y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
353 : y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
354 : ... special arrangement for better accuracy
355 : z := x*y ... 29 bits to sqrt(x), with z*y<1
356 : z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
357 :
358 : Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
359 : (a) the term z*y in the final iteration is always less than 1;
360 : (b) the error in the final result is biased upward so that
361 : -1 ulp < sqrt(x) - z < 1.0625 ulp
362 : instead of |sqrt(x)-z|<1.03125ulp.
363 :
364 : (3) Final adjustment
365 :
366 : By twiddling y's last bit it is possible to force y to be
367 : correctly rounded according to the prevailing rounding mode
368 : as follows. Let r and i be copies of the rounding mode and
369 : inexact flag before entering the square root program. Also we
370 : use the expression y+-ulp for the next representable floating
371 : numbers (up and down) of y. Note that y+-ulp = either fixed
372 : point y+-1, or multiply y by nextafter(1,+-inf) in chopped
373 : mode.
374 :
375 : R := RZ; ... set rounding mode to round-toward-zero
376 : switch(r) {
377 : case RN: ... round-to-nearest
378 : if(x<= z*(z-ulp)...chopped) z = z - ulp; else
379 : if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
380 : break;
381 : case RZ:case RM: ... round-to-zero or round-to--inf
382 : R:=RP; ... reset rounding mod to round-to-+inf
383 : if(x<z*z ... rounded up) z = z - ulp; else
384 : if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
385 : break;
386 : case RP: ... round-to-+inf
387 : if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
388 : if(x>z*z ...chopped) z = z+ulp;
389 : break;
390 : }
391 :
392 : Remark 3. The above comparisons can be done in fixed point. For
393 : example, to compare x and w=z*z chopped, it suffices to compare
394 : x1 and w1 (the trailing parts of x and w), regarding them as
395 : two's complement integers.
396 :
397 : ...Is z an exact square root?
398 : To determine whether z is an exact square root of x, let z1 be the
399 : trailing part of z, and also let x0 and x1 be the leading and
400 : trailing parts of x.
401 :
402 : If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
403 : I := 1; ... Raise Inexact flag: z is not exact
404 : else {
405 : j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
406 : k := z1 >> 26; ... get z's 25-th and 26-th
407 : fraction bits
408 : I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
409 : }
410 : R:= r ... restore rounded mode
411 : return sqrt(x):=z.
412 :
413 : If multiplication is cheaper then the foregoing red tape, the
414 : Inexact flag can be evaluated by
415 :
416 : I := i;
417 : I := (z*z!=x) or I.
418 :
419 : Note that z*z can overwrite I; this value must be sensed if it is
420 : True.
421 :
422 : Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
423 : zero.
424 :
425 : --------------------
426 : z1: | f2 |
427 : --------------------
428 : bit 31 bit 0
429 :
430 : Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
431 : or even of logb(x) have the following relations:
432 :
433 : -------------------------------------------------
434 : bit 27,26 of z1 bit 1,0 of x1 logb(x)
435 : -------------------------------------------------
436 : 00 00 odd and even
437 : 01 01 even
438 : 10 10 odd
439 : 10 00 even
440 : 11 01 even
441 : -------------------------------------------------
442 :
443 : (4) Special cases (see (4) of Section A).
444 :
445 : */
446 :
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