Annotation of src/lib/libm/src/e_sqrt.c, Revision 1.13
1.1 jtc 1: /* @(#)e_sqrt.c 5.1 93/09/24 */
2: /*
3: * ====================================================
4: * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5: *
6: * Developed at SunPro, a Sun Microsystems, Inc. business.
7: * Permission to use, copy, modify, and distribute this
1.10 simonb 8: * software is freely granted, provided that this notice
1.1 jtc 9: * is preserved.
10: * ====================================================
11: */
1.3 jtc 12:
1.9 lukem 13: #include <sys/cdefs.h>
1.7 jtc 14: #if defined(LIBM_SCCS) && !defined(lint)
1.13 ! lukem 15: __RCSID("$NetBSD: e_sqrt.c,v 1.12 2002/05/26 22:01:52 wiz Exp $");
1.3 jtc 16: #endif
1.1 jtc 17:
18: /* __ieee754_sqrt(x)
19: * Return correctly rounded sqrt.
20: * ------------------------------------------
21: * | Use the hardware sqrt if you have one |
22: * ------------------------------------------
1.10 simonb 23: * Method:
24: * Bit by bit method using integer arithmetic. (Slow, but portable)
1.1 jtc 25: * 1. Normalization
1.10 simonb 26: * Scale x to y in [1,4) with even powers of 2:
1.1 jtc 27: * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
28: * sqrt(x) = 2^k * sqrt(y)
29: * 2. Bit by bit computation
30: * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
31: * i 0
32: * i+1 2
33: * s = 2*q , and y = 2 * ( y - q ). (1)
34: * i i i i
1.10 simonb 35: *
36: * To compute q from q , one checks whether
37: * i+1 i
1.1 jtc 38: *
39: * -(i+1) 2
40: * (q + 2 ) <= y. (2)
41: * i
42: * -(i+1)
43: * If (2) is false, then q = q ; otherwise q = q + 2 .
44: * i+1 i i+1 i
45: *
46: * With some algebric manipulation, it is not difficult to see
1.10 simonb 47: * that (2) is equivalent to
1.1 jtc 48: * -(i+1)
49: * s + 2 <= y (3)
50: * i i
51: *
1.10 simonb 52: * The advantage of (3) is that s and y can be computed by
1.1 jtc 53: * i i
54: * the following recurrence formula:
55: * if (3) is false
56: *
57: * s = s , y = y ; (4)
58: * i+1 i i+1 i
59: *
60: * otherwise,
61: * -i -(i+1)
62: * s = s + 2 , y = y - s - 2 (5)
63: * i+1 i i+1 i i
1.10 simonb 64: *
65: * One may easily use induction to prove (4) and (5).
1.1 jtc 66: * Note. Since the left hand side of (3) contain only i+2 bits,
1.10 simonb 67: * it does not necessary to do a full (53-bit) comparison
1.1 jtc 68: * in (3).
69: * 3. Final rounding
70: * After generating the 53 bits result, we compute one more bit.
71: * Together with the remainder, we can decide whether the
72: * result is exact, bigger than 1/2ulp, or less than 1/2ulp
73: * (it will never equal to 1/2ulp).
74: * The rounding mode can be detected by checking whether
75: * huge + tiny is equal to huge, and whether huge - tiny is
76: * equal to huge for some floating point number "huge" and "tiny".
1.10 simonb 77: *
1.1 jtc 78: * Special cases:
79: * sqrt(+-0) = +-0 ... exact
80: * sqrt(inf) = inf
81: * sqrt(-ve) = NaN ... with invalid signal
82: * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
83: *
84: * Other methods : see the appended file at the end of the program below.
85: *---------------
86: */
87:
1.5 jtc 88: #include "math.h"
89: #include "math_private.h"
1.1 jtc 90:
91: static const double one = 1.0, tiny=1.0e-300;
92:
1.12 wiz 93: double
94: __ieee754_sqrt(double x)
1.1 jtc 95: {
96: double z;
1.10 simonb 97: int32_t sign = (int)0x80000000;
1.6 jtc 98: int32_t ix0,s0,q,m,t,i;
99: u_int32_t r,t1,s1,ix1,q1;
1.1 jtc 100:
1.5 jtc 101: EXTRACT_WORDS(ix0,ix1,x);
1.1 jtc 102:
103: /* take care of Inf and NaN */
1.10 simonb 104: if((ix0&0x7ff00000)==0x7ff00000) {
1.1 jtc 105: return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
106: sqrt(-inf)=sNaN */
1.10 simonb 107: }
1.1 jtc 108: /* take care of zero */
109: if(ix0<=0) {
110: if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
111: else if(ix0<0)
112: return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
113: }
114: /* normalize x */
115: m = (ix0>>20);
116: if(m==0) { /* subnormal x */
117: while(ix0==0) {
118: m -= 21;
119: ix0 |= (ix1>>11); ix1 <<= 21;
120: }
121: for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
122: m -= i-1;
123: ix0 |= (ix1>>(32-i));
124: ix1 <<= i;
125: }
126: m -= 1023; /* unbias exponent */
127: ix0 = (ix0&0x000fffff)|0x00100000;
128: if(m&1){ /* odd m, double x to make it even */
129: ix0 += ix0 + ((ix1&sign)>>31);
130: ix1 += ix1;
131: }
132: m >>= 1; /* m = [m/2] */
133:
134: /* generate sqrt(x) bit by bit */
135: ix0 += ix0 + ((ix1&sign)>>31);
136: ix1 += ix1;
137: q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
138: r = 0x00200000; /* r = moving bit from right to left */
139:
140: while(r!=0) {
1.10 simonb 141: t = s0+r;
142: if(t<=ix0) {
143: s0 = t+r;
144: ix0 -= t;
145: q += r;
146: }
1.1 jtc 147: ix0 += ix0 + ((ix1&sign)>>31);
148: ix1 += ix1;
149: r>>=1;
150: }
151:
152: r = sign;
153: while(r!=0) {
1.10 simonb 154: t1 = s1+r;
1.1 jtc 155: t = s0;
1.10 simonb 156: if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
1.1 jtc 157: s1 = t1+r;
1.13 ! lukem 158: if(((t1&sign)==(u_int32_t)sign)&&(s1&sign)==0) s0 += 1;
1.1 jtc 159: ix0 -= t;
160: if (ix1 < t1) ix0 -= 1;
161: ix1 -= t1;
162: q1 += r;
163: }
164: ix0 += ix0 + ((ix1&sign)>>31);
165: ix1 += ix1;
166: r>>=1;
167: }
168:
169: /* use floating add to find out rounding direction */
170: if((ix0|ix1)!=0) {
171: z = one-tiny; /* trigger inexact flag */
172: if (z>=one) {
173: z = one+tiny;
1.6 jtc 174: if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;}
1.1 jtc 175: else if (z>one) {
1.6 jtc 176: if (q1==(u_int32_t)0xfffffffe) q+=1;
1.10 simonb 177: q1+=2;
1.1 jtc 178: } else
179: q1 += (q1&1);
180: }
181: }
182: ix0 = (q>>1)+0x3fe00000;
183: ix1 = q1>>1;
184: if ((q&1)==1) ix1 |= sign;
185: ix0 += (m <<20);
1.5 jtc 186: INSERT_WORDS(z,ix0,ix1);
1.1 jtc 187: return z;
188: }
189:
190: /*
191: Other methods (use floating-point arithmetic)
192: -------------
1.10 simonb 193: (This is a copy of a drafted paper by Prof W. Kahan
1.1 jtc 194: and K.C. Ng, written in May, 1986)
195:
1.10 simonb 196: Two algorithms are given here to implement sqrt(x)
1.1 jtc 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
1.10 simonb 202: to chop results of arithmetic operations instead of round them,
1.1 jtc 203: and the INEXACT flag to indicate when an arithmetic operation
1.10 simonb 204: is executed exactly with no roundoff error, all part of the
1.1 jtc 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
1.10 simonb 214: a floating point number x (in IEEE double format) respectively
1.1 jtc 215:
216: 1 11 52 ...widths
217: ------------------------------------------------------
218: x: |s| e | f |
219: ------------------------------------------------------
220: msb lsb msb lsb ...order
221:
1.10 simonb 222:
1.1 jtc 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:
1.10 simonb 247: Apply Heron's rule three times to y, we have y approximates
1.1 jtc 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
1.10 simonb 272: is slow. If division is very slow, then one should use the
1.1 jtc 273: reciproot algorithm given in section B.
274:
275: (3) Final adjustment
276:
1.10 simonb 277: By twiddling y's last bit it is possible to force y to be
1.1 jtc 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.
1.10 simonb 308:
1.1 jtc 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:
1.10 simonb 327: Here k is a 32-bit integer and T2[] is an integer array
1.1 jtc 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)
1.10 simonb 348: to about 1 ulp. To be exact, we will have
1.1 jtc 349: -1ulp < sqrt(x)-z<1.0625ulp.
1.10 simonb 350:
1.1 jtc 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
1.10 simonb 359: (a) the term z*y in the final iteration is always less than 1;
1.1 jtc 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:
1.10 simonb 366: By twiddling y's last bit it is possible to force y to be
1.1 jtc 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
1.10 simonb 406: k := z1 >> 26; ... get z's 25-th and 26-th
1.1 jtc 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:
1.11 wiz 413: If multiplication is cheaper than the foregoing red tape, the
1.1 jtc 414: Inexact flag can be evaluated by
415:
416: I := i;
417: I := (z*z!=x) or I.
418:
1.10 simonb 419: Note that z*z can overwrite I; this value must be sensed if it is
1.1 jtc 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: --------------------
1.10 simonb 426: z1: | f2 |
1.1 jtc 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:
1.10 simonb 443: (4) Special cases (see (4) of Section A).
444:
1.1 jtc 445: */
1.10 simonb 446:
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