Annotation of src/lib/libm/src/e_sqrt.c, Revision 1.8
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
8: * software is freely granted, provided that this notice
9: * is preserved.
10: * ====================================================
11: */
1.3 jtc 12:
1.7 jtc 13: #if defined(LIBM_SCCS) && !defined(lint)
1.8 ! jtc 14: static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.7 1994/09/22 16:40:13 jtc Exp $";
1.3 jtc 15: #endif
1.1 jtc 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:
1.5 jtc 87: #include "math.h"
88: #include "math_private.h"
1.1 jtc 89:
90: #ifdef __STDC__
91: static const double one = 1.0, tiny=1.0e-300;
92: #else
93: static double one = 1.0, tiny=1.0e-300;
94: #endif
95:
96: #ifdef __STDC__
97: double __ieee754_sqrt(double x)
98: #else
99: double __ieee754_sqrt(x)
100: double x;
101: #endif
102: {
103: double z;
1.6 jtc 104: int32_t sign = (int)0x80000000;
105: int32_t ix0,s0,q,m,t,i;
106: u_int32_t r,t1,s1,ix1,q1;
1.1 jtc 107:
1.5 jtc 108: EXTRACT_WORDS(ix0,ix1,x);
1.1 jtc 109:
110: /* take care of Inf and NaN */
111: if((ix0&0x7ff00000)==0x7ff00000) {
112: return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
113: sqrt(-inf)=sNaN */
114: }
115: /* take care of zero */
116: if(ix0<=0) {
117: if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
118: else if(ix0<0)
119: return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
120: }
121: /* normalize x */
122: m = (ix0>>20);
123: if(m==0) { /* subnormal x */
124: while(ix0==0) {
125: m -= 21;
126: ix0 |= (ix1>>11); ix1 <<= 21;
127: }
128: for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
129: m -= i-1;
130: ix0 |= (ix1>>(32-i));
131: ix1 <<= i;
132: }
133: m -= 1023; /* unbias exponent */
134: ix0 = (ix0&0x000fffff)|0x00100000;
135: if(m&1){ /* odd m, double x to make it even */
136: ix0 += ix0 + ((ix1&sign)>>31);
137: ix1 += ix1;
138: }
139: m >>= 1; /* m = [m/2] */
140:
141: /* generate sqrt(x) bit by bit */
142: ix0 += ix0 + ((ix1&sign)>>31);
143: ix1 += ix1;
144: q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
145: r = 0x00200000; /* r = moving bit from right to left */
146:
147: while(r!=0) {
148: t = s0+r;
149: if(t<=ix0) {
150: s0 = t+r;
151: ix0 -= t;
152: q += r;
153: }
154: ix0 += ix0 + ((ix1&sign)>>31);
155: ix1 += ix1;
156: r>>=1;
157: }
158:
159: r = sign;
160: while(r!=0) {
161: t1 = s1+r;
162: t = s0;
163: if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
164: s1 = t1+r;
165: if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
166: ix0 -= t;
167: if (ix1 < t1) ix0 -= 1;
168: ix1 -= t1;
169: q1 += r;
170: }
171: ix0 += ix0 + ((ix1&sign)>>31);
172: ix1 += ix1;
173: r>>=1;
174: }
175:
176: /* use floating add to find out rounding direction */
177: if((ix0|ix1)!=0) {
178: z = one-tiny; /* trigger inexact flag */
179: if (z>=one) {
180: z = one+tiny;
1.6 jtc 181: if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;}
1.1 jtc 182: else if (z>one) {
1.6 jtc 183: if (q1==(u_int32_t)0xfffffffe) q+=1;
1.1 jtc 184: q1+=2;
185: } else
186: q1 += (q1&1);
187: }
188: }
189: ix0 = (q>>1)+0x3fe00000;
190: ix1 = q1>>1;
191: if ((q&1)==1) ix1 |= sign;
192: ix0 += (m <<20);
1.5 jtc 193: INSERT_WORDS(z,ix0,ix1);
1.1 jtc 194: return z;
195: }
196:
197: /*
198: Other methods (use floating-point arithmetic)
199: -------------
200: (This is a copy of a drafted paper by Prof W. Kahan
201: and K.C. Ng, written in May, 1986)
202:
203: Two algorithms are given here to implement sqrt(x)
204: (IEEE double precision arithmetic) in software.
205: Both supply sqrt(x) correctly rounded. The first algorithm (in
206: Section A) uses newton iterations and involves four divisions.
207: The second one uses reciproot iterations to avoid division, but
208: requires more multiplications. Both algorithms need the ability
209: to chop results of arithmetic operations instead of round them,
210: and the INEXACT flag to indicate when an arithmetic operation
211: is executed exactly with no roundoff error, all part of the
212: standard (IEEE 754-1985). The ability to perform shift, add,
213: subtract and logical AND operations upon 32-bit words is needed
214: too, though not part of the standard.
215:
216: A. sqrt(x) by Newton Iteration
217:
218: (1) Initial approximation
219:
220: Let x0 and x1 be the leading and the trailing 32-bit words of
221: a floating point number x (in IEEE double format) respectively
222:
223: 1 11 52 ...widths
224: ------------------------------------------------------
225: x: |s| e | f |
226: ------------------------------------------------------
227: msb lsb msb lsb ...order
228:
229:
230: ------------------------ ------------------------
231: x0: |s| e | f1 | x1: | f2 |
232: ------------------------ ------------------------
233:
234: By performing shifts and subtracts on x0 and x1 (both regarded
235: as integers), we obtain an 8-bit approximation of sqrt(x) as
236: follows.
237:
238: k := (x0>>1) + 0x1ff80000;
239: y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
240: Here k is a 32-bit integer and T1[] is an integer array containing
241: correction terms. Now magically the floating value of y (y's
242: leading 32-bit word is y0, the value of its trailing word is 0)
243: approximates sqrt(x) to almost 8-bit.
244:
245: Value of T1:
246: static int T1[32]= {
247: 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
248: 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
249: 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
250: 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
251:
252: (2) Iterative refinement
253:
254: Apply Heron's rule three times to y, we have y approximates
255: sqrt(x) to within 1 ulp (Unit in the Last Place):
256:
257: y := (y+x/y)/2 ... almost 17 sig. bits
258: y := (y+x/y)/2 ... almost 35 sig. bits
259: y := y-(y-x/y)/2 ... within 1 ulp
260:
261:
262: Remark 1.
263: Another way to improve y to within 1 ulp is:
264:
265: y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
266: y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
267:
268: 2
269: (x-y )*y
270: y := y + 2* ---------- ...within 1 ulp
271: 2
272: 3y + x
273:
274:
275: This formula has one division fewer than the one above; however,
276: it requires more multiplications and additions. Also x must be
277: scaled in advance to avoid spurious overflow in evaluating the
278: expression 3y*y+x. Hence it is not recommended uless division
279: is slow. If division is very slow, then one should use the
280: reciproot algorithm given in section B.
281:
282: (3) Final adjustment
283:
284: By twiddling y's last bit it is possible to force y to be
285: correctly rounded according to the prevailing rounding mode
286: as follows. Let r and i be copies of the rounding mode and
287: inexact flag before entering the square root program. Also we
288: use the expression y+-ulp for the next representable floating
289: numbers (up and down) of y. Note that y+-ulp = either fixed
290: point y+-1, or multiply y by nextafter(1,+-inf) in chopped
291: mode.
292:
293: I := FALSE; ... reset INEXACT flag I
294: R := RZ; ... set rounding mode to round-toward-zero
295: z := x/y; ... chopped quotient, possibly inexact
296: If(not I) then { ... if the quotient is exact
297: if(z=y) {
298: I := i; ... restore inexact flag
299: R := r; ... restore rounded mode
300: return sqrt(x):=y.
301: } else {
302: z := z - ulp; ... special rounding
303: }
304: }
305: i := TRUE; ... sqrt(x) is inexact
306: If (r=RN) then z=z+ulp ... rounded-to-nearest
307: If (r=RP) then { ... round-toward-+inf
308: y = y+ulp; z=z+ulp;
309: }
310: y := y+z; ... chopped sum
311: y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
312: I := i; ... restore inexact flag
313: R := r; ... restore rounded mode
314: return sqrt(x):=y.
315:
316: (4) Special cases
317:
318: Square root of +inf, +-0, or NaN is itself;
319: Square root of a negative number is NaN with invalid signal.
320:
321:
322: B. sqrt(x) by Reciproot Iteration
323:
324: (1) Initial approximation
325:
326: Let x0 and x1 be the leading and the trailing 32-bit words of
327: a floating point number x (in IEEE double format) respectively
328: (see section A). By performing shifs and subtracts on x0 and y0,
329: we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
330:
331: k := 0x5fe80000 - (x0>>1);
332: y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
333:
334: Here k is a 32-bit integer and T2[] is an integer array
335: containing correction terms. Now magically the floating
336: value of y (y's leading 32-bit word is y0, the value of
337: its trailing word y1 is set to zero) approximates 1/sqrt(x)
338: to almost 7.8-bit.
339:
340: Value of T2:
341: static int T2[64]= {
342: 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
343: 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
344: 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
345: 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
346: 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
347: 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
348: 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
349: 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
350:
351: (2) Iterative refinement
352:
353: Apply Reciproot iteration three times to y and multiply the
354: result by x to get an approximation z that matches sqrt(x)
355: to about 1 ulp. To be exact, we will have
356: -1ulp < sqrt(x)-z<1.0625ulp.
357:
358: ... set rounding mode to Round-to-nearest
359: y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
360: y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
361: ... special arrangement for better accuracy
362: z := x*y ... 29 bits to sqrt(x), with z*y<1
363: z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
364:
365: Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
366: (a) the term z*y in the final iteration is always less than 1;
367: (b) the error in the final result is biased upward so that
368: -1 ulp < sqrt(x) - z < 1.0625 ulp
369: instead of |sqrt(x)-z|<1.03125ulp.
370:
371: (3) Final adjustment
372:
373: By twiddling y's last bit it is possible to force y to be
374: correctly rounded according to the prevailing rounding mode
375: as follows. Let r and i be copies of the rounding mode and
376: inexact flag before entering the square root program. Also we
377: use the expression y+-ulp for the next representable floating
378: numbers (up and down) of y. Note that y+-ulp = either fixed
379: point y+-1, or multiply y by nextafter(1,+-inf) in chopped
380: mode.
381:
382: R := RZ; ... set rounding mode to round-toward-zero
383: switch(r) {
384: case RN: ... round-to-nearest
385: if(x<= z*(z-ulp)...chopped) z = z - ulp; else
386: if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
387: break;
388: case RZ:case RM: ... round-to-zero or round-to--inf
389: R:=RP; ... reset rounding mod to round-to-+inf
390: if(x<z*z ... rounded up) z = z - ulp; else
391: if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
392: break;
393: case RP: ... round-to-+inf
394: if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
395: if(x>z*z ...chopped) z = z+ulp;
396: break;
397: }
398:
399: Remark 3. The above comparisons can be done in fixed point. For
400: example, to compare x and w=z*z chopped, it suffices to compare
401: x1 and w1 (the trailing parts of x and w), regarding them as
402: two's complement integers.
403:
404: ...Is z an exact square root?
405: To determine whether z is an exact square root of x, let z1 be the
406: trailing part of z, and also let x0 and x1 be the leading and
407: trailing parts of x.
408:
409: If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
410: I := 1; ... Raise Inexact flag: z is not exact
411: else {
412: j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
413: k := z1 >> 26; ... get z's 25-th and 26-th
414: fraction bits
415: I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
416: }
417: R:= r ... restore rounded mode
418: return sqrt(x):=z.
419:
420: If multiplication is cheaper then the foregoing red tape, the
421: Inexact flag can be evaluated by
422:
423: I := i;
424: I := (z*z!=x) or I.
425:
426: Note that z*z can overwrite I; this value must be sensed if it is
427: True.
428:
429: Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
430: zero.
431:
432: --------------------
433: z1: | f2 |
434: --------------------
435: bit 31 bit 0
436:
437: Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
438: or even of logb(x) have the following relations:
439:
440: -------------------------------------------------
441: bit 27,26 of z1 bit 1,0 of x1 logb(x)
442: -------------------------------------------------
443: 00 00 odd and even
444: 01 01 even
445: 10 10 odd
446: 10 00 even
447: 11 01 even
448: -------------------------------------------------
449:
450: (4) Special cases (see (4) of Section A).
451:
452: */
453:
CVSweb <webmaster@jp.NetBSD.org>
![[BACK]](/icons/back.gif){kind=icon}
Return to e_sqrt.c CVS log ![[TXT]](/icons/text.gif){kind=icon}
![[DIR]](/icons/dir.gif){kind=icon}
Up to [cvs.NetBSD.org] / src / lib / libm / src