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