Annotation of src/lib/libm/src/e_sqrt.c, Revision 1.1.1.1
1.1 jtc 1:
2: /* @(#)e_sqrt.c 5.1 93/09/24 */
3: /*
4: * ====================================================
5: * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6: *
7: * Developed at SunPro, 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: /* __ieee754_sqrt(x)
15: * Return correctly rounded sqrt.
16: * ------------------------------------------
17: * | Use the hardware sqrt if you have one |
18: * ------------------------------------------
19: * Method:
20: * Bit by bit method using integer arithmetic. (Slow, but portable)
21: * 1. Normalization
22: * Scale x to y in [1,4) with even powers of 2:
23: * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
24: * sqrt(x) = 2^k * sqrt(y)
25: * 2. Bit by bit computation
26: * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
27: * i 0
28: * i+1 2
29: * s = 2*q , and y = 2 * ( y - q ). (1)
30: * i i i i
31: *
32: * To compute q from q , one checks whether
33: * i+1 i
34: *
35: * -(i+1) 2
36: * (q + 2 ) <= y. (2)
37: * i
38: * -(i+1)
39: * If (2) is false, then q = q ; otherwise q = q + 2 .
40: * i+1 i i+1 i
41: *
42: * With some algebric manipulation, it is not difficult to see
43: * that (2) is equivalent to
44: * -(i+1)
45: * s + 2 <= y (3)
46: * i i
47: *
48: * The advantage of (3) is that s and y can be computed by
49: * i i
50: * the following recurrence formula:
51: * if (3) is false
52: *
53: * s = s , y = y ; (4)
54: * i+1 i i+1 i
55: *
56: * otherwise,
57: * -i -(i+1)
58: * s = s + 2 , y = y - s - 2 (5)
59: * i+1 i i+1 i i
60: *
61: * One may easily use induction to prove (4) and (5).
62: * Note. Since the left hand side of (3) contain only i+2 bits,
63: * it does not necessary to do a full (53-bit) comparison
64: * in (3).
65: * 3. Final rounding
66: * After generating the 53 bits result, we compute one more bit.
67: * Together with the remainder, we can decide whether the
68: * result is exact, bigger than 1/2ulp, or less than 1/2ulp
69: * (it will never equal to 1/2ulp).
70: * The rounding mode can be detected by checking whether
71: * huge + tiny is equal to huge, and whether huge - tiny is
72: * equal to huge for some floating point number "huge" and "tiny".
73: *
74: * Special cases:
75: * sqrt(+-0) = +-0 ... exact
76: * sqrt(inf) = inf
77: * sqrt(-ve) = NaN ... with invalid signal
78: * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
79: *
80: * Other methods : see the appended file at the end of the program below.
81: *---------------
82: */
83:
84: #include "fdlibm.h"
85:
86: #ifdef __STDC__
87: static const double one = 1.0, tiny=1.0e-300;
88: #else
89: static double one = 1.0, tiny=1.0e-300;
90: #endif
91:
92: #ifdef __STDC__
93: double __ieee754_sqrt(double x)
94: #else
95: double __ieee754_sqrt(x)
96: double x;
97: #endif
98: {
99: int n0;
100:
101: double z;
102: int sign = (int)0x80000000;
103: unsigned r,t1,s1,ix1,q1;
104: int ix0,s0,q,m,t,i;
105:
106: n0 = ((*(int*)&one)>>29)^1; /* index of high word */
107: ix0 = *(n0+(int*)&x); /* high word of x */
108: ix1 = *((1-n0)+(int*)&x); /* low word of x */
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;
181: if (q1==(unsigned)0xffffffff) { q1=0; q += 1;}
182: else if (z>one) {
183: if (q1==(unsigned)0xfffffffe) q+=1;
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);
193: *(n0+(int*)&z) = ix0;
194: *((1-n0)+(int*)&z) = ix1;
195: return z;
196: }
197:
198: /*
199: Other methods (use floating-point arithmetic)
200: -------------
201: (This is a copy of a drafted paper by Prof W. Kahan
202: and K.C. Ng, written in May, 1986)
203:
204: Two algorithms are given here to implement sqrt(x)
205: (IEEE double precision arithmetic) in software.
206: Both supply sqrt(x) correctly rounded. The first algorithm (in
207: Section A) uses newton iterations and involves four divisions.
208: The second one uses reciproot iterations to avoid division, but
209: requires more multiplications. Both algorithms need the ability
210: to chop results of arithmetic operations instead of round them,
211: and the INEXACT flag to indicate when an arithmetic operation
212: is executed exactly with no roundoff error, all part of the
213: standard (IEEE 754-1985). The ability to perform shift, add,
214: subtract and logical AND operations upon 32-bit words is needed
215: too, though not part of the standard.
216:
217: A. sqrt(x) by Newton Iteration
218:
219: (1) Initial approximation
220:
221: Let x0 and x1 be the leading and the trailing 32-bit words of
222: a floating point number x (in IEEE double format) respectively
223:
224: 1 11 52 ...widths
225: ------------------------------------------------------
226: x: |s| e | f |
227: ------------------------------------------------------
228: msb lsb msb lsb ...order
229:
230:
231: ------------------------ ------------------------
232: x0: |s| e | f1 | x1: | f2 |
233: ------------------------ ------------------------
234:
235: By performing shifts and subtracts on x0 and x1 (both regarded
236: as integers), we obtain an 8-bit approximation of sqrt(x) as
237: follows.
238:
239: k := (x0>>1) + 0x1ff80000;
240: y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
241: Here k is a 32-bit integer and T1[] is an integer array containing
242: correction terms. Now magically the floating value of y (y's
243: leading 32-bit word is y0, the value of its trailing word is 0)
244: approximates sqrt(x) to almost 8-bit.
245:
246: Value of T1:
247: static int T1[32]= {
248: 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
249: 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
250: 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
251: 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
252:
253: (2) Iterative refinement
254:
255: Apply Heron's rule three times to y, we have y approximates
256: sqrt(x) to within 1 ulp (Unit in the Last Place):
257:
258: y := (y+x/y)/2 ... almost 17 sig. bits
259: y := (y+x/y)/2 ... almost 35 sig. bits
260: y := y-(y-x/y)/2 ... within 1 ulp
261:
262:
263: Remark 1.
264: Another way to improve y to within 1 ulp is:
265:
266: y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
267: y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
268:
269: 2
270: (x-y )*y
271: y := y + 2* ---------- ...within 1 ulp
272: 2
273: 3y + x
274:
275:
276: This formula has one division fewer than the one above; however,
277: it requires more multiplications and additions. Also x must be
278: scaled in advance to avoid spurious overflow in evaluating the
279: expression 3y*y+x. Hence it is not recommended uless division
280: is slow. If division is very slow, then one should use the
281: reciproot algorithm given in section B.
282:
283: (3) Final adjustment
284:
285: By twiddling y's last bit it is possible to force y to be
286: correctly rounded according to the prevailing rounding mode
287: as follows. Let r and i be copies of the rounding mode and
288: inexact flag before entering the square root program. Also we
289: use the expression y+-ulp for the next representable floating
290: numbers (up and down) of y. Note that y+-ulp = either fixed
291: point y+-1, or multiply y by nextafter(1,+-inf) in chopped
292: mode.
293:
294: I := FALSE; ... reset INEXACT flag I
295: R := RZ; ... set rounding mode to round-toward-zero
296: z := x/y; ... chopped quotient, possibly inexact
297: If(not I) then { ... if the quotient is exact
298: if(z=y) {
299: I := i; ... restore inexact flag
300: R := r; ... restore rounded mode
301: return sqrt(x):=y.
302: } else {
303: z := z - ulp; ... special rounding
304: }
305: }
306: i := TRUE; ... sqrt(x) is inexact
307: If (r=RN) then z=z+ulp ... rounded-to-nearest
308: If (r=RP) then { ... round-toward-+inf
309: y = y+ulp; z=z+ulp;
310: }
311: y := y+z; ... chopped sum
312: y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
313: I := i; ... restore inexact flag
314: R := r; ... restore rounded mode
315: return sqrt(x):=y.
316:
317: (4) Special cases
318:
319: Square root of +inf, +-0, or NaN is itself;
320: Square root of a negative number is NaN with invalid signal.
321:
322:
323: B. sqrt(x) by Reciproot Iteration
324:
325: (1) Initial approximation
326:
327: Let x0 and x1 be the leading and the trailing 32-bit words of
328: a floating point number x (in IEEE double format) respectively
329: (see section A). By performing shifs and subtracts on x0 and y0,
330: we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
331:
332: k := 0x5fe80000 - (x0>>1);
333: y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
334:
335: Here k is a 32-bit integer and T2[] is an integer array
336: containing correction terms. Now magically the floating
337: value of y (y's leading 32-bit word is y0, the value of
338: its trailing word y1 is set to zero) approximates 1/sqrt(x)
339: to almost 7.8-bit.
340:
341: Value of T2:
342: static int T2[64]= {
343: 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
344: 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
345: 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
346: 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
347: 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
348: 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
349: 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
350: 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
351:
352: (2) Iterative refinement
353:
354: Apply Reciproot iteration three times to y and multiply the
355: result by x to get an approximation z that matches sqrt(x)
356: to about 1 ulp. To be exact, we will have
357: -1ulp < sqrt(x)-z<1.0625ulp.
358:
359: ... set rounding mode to Round-to-nearest
360: y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
361: y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
362: ... special arrangement for better accuracy
363: z := x*y ... 29 bits to sqrt(x), with z*y<1
364: z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
365:
366: Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
367: (a) the term z*y in the final iteration is always less than 1;
368: (b) the error in the final result is biased upward so that
369: -1 ulp < sqrt(x) - z < 1.0625 ulp
370: instead of |sqrt(x)-z|<1.03125ulp.
371:
372: (3) Final adjustment
373:
374: By twiddling y's last bit it is possible to force y to be
375: correctly rounded according to the prevailing rounding mode
376: as follows. Let r and i be copies of the rounding mode and
377: inexact flag before entering the square root program. Also we
378: use the expression y+-ulp for the next representable floating
379: numbers (up and down) of y. Note that y+-ulp = either fixed
380: point y+-1, or multiply y by nextafter(1,+-inf) in chopped
381: mode.
382:
383: R := RZ; ... set rounding mode to round-toward-zero
384: switch(r) {
385: case RN: ... round-to-nearest
386: if(x<= z*(z-ulp)...chopped) z = z - ulp; else
387: if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
388: break;
389: case RZ:case RM: ... round-to-zero or round-to--inf
390: R:=RP; ... reset rounding mod to round-to-+inf
391: if(x<z*z ... rounded up) z = z - ulp; else
392: if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
393: break;
394: case RP: ... round-to-+inf
395: if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
396: if(x>z*z ...chopped) z = z+ulp;
397: break;
398: }
399:
400: Remark 3. The above comparisons can be done in fixed point. For
401: example, to compare x and w=z*z chopped, it suffices to compare
402: x1 and w1 (the trailing parts of x and w), regarding them as
403: two's complement integers.
404:
405: ...Is z an exact square root?
406: To determine whether z is an exact square root of x, let z1 be the
407: trailing part of z, and also let x0 and x1 be the leading and
408: trailing parts of x.
409:
410: If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
411: I := 1; ... Raise Inexact flag: z is not exact
412: else {
413: j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
414: k := z1 >> 26; ... get z's 25-th and 26-th
415: fraction bits
416: I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
417: }
418: R:= r ... restore rounded mode
419: return sqrt(x):=z.
420:
421: If multiplication is cheaper then the foregoing red tape, the
422: Inexact flag can be evaluated by
423:
424: I := i;
425: I := (z*z!=x) or I.
426:
427: Note that z*z can overwrite I; this value must be sensed if it is
428: True.
429:
430: Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
431: zero.
432:
433: --------------------
434: z1: | f2 |
435: --------------------
436: bit 31 bit 0
437:
438: Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
439: or even of logb(x) have the following relations:
440:
441: -------------------------------------------------
442: bit 27,26 of z1 bit 1,0 of x1 logb(x)
443: -------------------------------------------------
444: 00 00 odd and even
445: 01 01 even
446: 10 10 odd
447: 10 00 even
448: 11 01 even
449: -------------------------------------------------
450:
451: (4) Special cases (see (4) of Section A).
452:
453: */
454:
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