Annotation of src/lib/libm/src/e_sqrt.c, Revision 1.3
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:
! 13: #ifndef lint
! 14: static char rcsid[] = "$Id: $";
! 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:
1.2 jtc 87: #include <math.h>
1.1 jtc 88:
89: #ifdef __STDC__
90: static const double one = 1.0, tiny=1.0e-300;
91: #else
92: static double one = 1.0, tiny=1.0e-300;
93: #endif
94:
95: #ifdef __STDC__
96: double __ieee754_sqrt(double x)
97: #else
98: double __ieee754_sqrt(x)
99: double x;
100: #endif
101: {
102: int n0;
103:
104: double z;
105: int sign = (int)0x80000000;
106: unsigned r,t1,s1,ix1,q1;
107: int ix0,s0,q,m,t,i;
108:
109: n0 = ((*(int*)&one)>>29)^1; /* index of high word */
110: ix0 = *(n0+(int*)&x); /* high word of x */
111: ix1 = *((1-n0)+(int*)&x); /* low word of x */
112:
113: /* take care of Inf and NaN */
114: if((ix0&0x7ff00000)==0x7ff00000) {
115: return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
116: sqrt(-inf)=sNaN */
117: }
118: /* take care of zero */
119: if(ix0<=0) {
120: if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
121: else if(ix0<0)
122: return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
123: }
124: /* normalize x */
125: m = (ix0>>20);
126: if(m==0) { /* subnormal x */
127: while(ix0==0) {
128: m -= 21;
129: ix0 |= (ix1>>11); ix1 <<= 21;
130: }
131: for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
132: m -= i-1;
133: ix0 |= (ix1>>(32-i));
134: ix1 <<= i;
135: }
136: m -= 1023; /* unbias exponent */
137: ix0 = (ix0&0x000fffff)|0x00100000;
138: if(m&1){ /* odd m, double x to make it even */
139: ix0 += ix0 + ((ix1&sign)>>31);
140: ix1 += ix1;
141: }
142: m >>= 1; /* m = [m/2] */
143:
144: /* generate sqrt(x) bit by bit */
145: ix0 += ix0 + ((ix1&sign)>>31);
146: ix1 += ix1;
147: q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
148: r = 0x00200000; /* r = moving bit from right to left */
149:
150: while(r!=0) {
151: t = s0+r;
152: if(t<=ix0) {
153: s0 = t+r;
154: ix0 -= t;
155: q += r;
156: }
157: ix0 += ix0 + ((ix1&sign)>>31);
158: ix1 += ix1;
159: r>>=1;
160: }
161:
162: r = sign;
163: while(r!=0) {
164: t1 = s1+r;
165: t = s0;
166: if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
167: s1 = t1+r;
168: if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
169: ix0 -= t;
170: if (ix1 < t1) ix0 -= 1;
171: ix1 -= t1;
172: q1 += r;
173: }
174: ix0 += ix0 + ((ix1&sign)>>31);
175: ix1 += ix1;
176: r>>=1;
177: }
178:
179: /* use floating add to find out rounding direction */
180: if((ix0|ix1)!=0) {
181: z = one-tiny; /* trigger inexact flag */
182: if (z>=one) {
183: z = one+tiny;
184: if (q1==(unsigned)0xffffffff) { q1=0; q += 1;}
185: else if (z>one) {
186: if (q1==(unsigned)0xfffffffe) q+=1;
187: q1+=2;
188: } else
189: q1 += (q1&1);
190: }
191: }
192: ix0 = (q>>1)+0x3fe00000;
193: ix1 = q1>>1;
194: if ((q&1)==1) ix1 |= sign;
195: ix0 += (m <<20);
196: *(n0+(int*)&z) = ix0;
197: *((1-n0)+(int*)&z) = ix1;
198: return z;
199: }
200:
201: /*
202: Other methods (use floating-point arithmetic)
203: -------------
204: (This is a copy of a drafted paper by Prof W. Kahan
205: and K.C. Ng, written in May, 1986)
206:
207: Two algorithms are given here to implement sqrt(x)
208: (IEEE double precision arithmetic) in software.
209: Both supply sqrt(x) correctly rounded. The first algorithm (in
210: Section A) uses newton iterations and involves four divisions.
211: The second one uses reciproot iterations to avoid division, but
212: requires more multiplications. Both algorithms need the ability
213: to chop results of arithmetic operations instead of round them,
214: and the INEXACT flag to indicate when an arithmetic operation
215: is executed exactly with no roundoff error, all part of the
216: standard (IEEE 754-1985). The ability to perform shift, add,
217: subtract and logical AND operations upon 32-bit words is needed
218: too, though not part of the standard.
219:
220: A. sqrt(x) by Newton Iteration
221:
222: (1) Initial approximation
223:
224: Let x0 and x1 be the leading and the trailing 32-bit words of
225: a floating point number x (in IEEE double format) respectively
226:
227: 1 11 52 ...widths
228: ------------------------------------------------------
229: x: |s| e | f |
230: ------------------------------------------------------
231: msb lsb msb lsb ...order
232:
233:
234: ------------------------ ------------------------
235: x0: |s| e | f1 | x1: | f2 |
236: ------------------------ ------------------------
237:
238: By performing shifts and subtracts on x0 and x1 (both regarded
239: as integers), we obtain an 8-bit approximation of sqrt(x) as
240: follows.
241:
242: k := (x0>>1) + 0x1ff80000;
243: y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
244: Here k is a 32-bit integer and T1[] is an integer array containing
245: correction terms. Now magically the floating value of y (y's
246: leading 32-bit word is y0, the value of its trailing word is 0)
247: approximates sqrt(x) to almost 8-bit.
248:
249: Value of T1:
250: static int T1[32]= {
251: 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
252: 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
253: 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
254: 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
255:
256: (2) Iterative refinement
257:
258: Apply Heron's rule three times to y, we have y approximates
259: sqrt(x) to within 1 ulp (Unit in the Last Place):
260:
261: y := (y+x/y)/2 ... almost 17 sig. bits
262: y := (y+x/y)/2 ... almost 35 sig. bits
263: y := y-(y-x/y)/2 ... within 1 ulp
264:
265:
266: Remark 1.
267: Another way to improve y to within 1 ulp is:
268:
269: y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
270: y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
271:
272: 2
273: (x-y )*y
274: y := y + 2* ---------- ...within 1 ulp
275: 2
276: 3y + x
277:
278:
279: This formula has one division fewer than the one above; however,
280: it requires more multiplications and additions. Also x must be
281: scaled in advance to avoid spurious overflow in evaluating the
282: expression 3y*y+x. Hence it is not recommended uless division
283: is slow. If division is very slow, then one should use the
284: reciproot algorithm given in section B.
285:
286: (3) Final adjustment
287:
288: By twiddling y's last bit it is possible to force y to be
289: correctly rounded according to the prevailing rounding mode
290: as follows. Let r and i be copies of the rounding mode and
291: inexact flag before entering the square root program. Also we
292: use the expression y+-ulp for the next representable floating
293: numbers (up and down) of y. Note that y+-ulp = either fixed
294: point y+-1, or multiply y by nextafter(1,+-inf) in chopped
295: mode.
296:
297: I := FALSE; ... reset INEXACT flag I
298: R := RZ; ... set rounding mode to round-toward-zero
299: z := x/y; ... chopped quotient, possibly inexact
300: If(not I) then { ... if the quotient is exact
301: if(z=y) {
302: I := i; ... restore inexact flag
303: R := r; ... restore rounded mode
304: return sqrt(x):=y.
305: } else {
306: z := z - ulp; ... special rounding
307: }
308: }
309: i := TRUE; ... sqrt(x) is inexact
310: If (r=RN) then z=z+ulp ... rounded-to-nearest
311: If (r=RP) then { ... round-toward-+inf
312: y = y+ulp; z=z+ulp;
313: }
314: y := y+z; ... chopped sum
315: y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
316: I := i; ... restore inexact flag
317: R := r; ... restore rounded mode
318: return sqrt(x):=y.
319:
320: (4) Special cases
321:
322: Square root of +inf, +-0, or NaN is itself;
323: Square root of a negative number is NaN with invalid signal.
324:
325:
326: B. sqrt(x) by Reciproot Iteration
327:
328: (1) Initial approximation
329:
330: Let x0 and x1 be the leading and the trailing 32-bit words of
331: a floating point number x (in IEEE double format) respectively
332: (see section A). By performing shifs and subtracts on x0 and y0,
333: we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
334:
335: k := 0x5fe80000 - (x0>>1);
336: y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
337:
338: Here k is a 32-bit integer and T2[] is an integer array
339: containing correction terms. Now magically the floating
340: value of y (y's leading 32-bit word is y0, the value of
341: its trailing word y1 is set to zero) approximates 1/sqrt(x)
342: to almost 7.8-bit.
343:
344: Value of T2:
345: static int T2[64]= {
346: 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
347: 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
348: 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
349: 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
350: 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
351: 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
352: 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
353: 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
354:
355: (2) Iterative refinement
356:
357: Apply Reciproot iteration three times to y and multiply the
358: result by x to get an approximation z that matches sqrt(x)
359: to about 1 ulp. To be exact, we will have
360: -1ulp < sqrt(x)-z<1.0625ulp.
361:
362: ... set rounding mode to Round-to-nearest
363: y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
364: y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
365: ... special arrangement for better accuracy
366: z := x*y ... 29 bits to sqrt(x), with z*y<1
367: z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
368:
369: Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
370: (a) the term z*y in the final iteration is always less than 1;
371: (b) the error in the final result is biased upward so that
372: -1 ulp < sqrt(x) - z < 1.0625 ulp
373: instead of |sqrt(x)-z|<1.03125ulp.
374:
375: (3) Final adjustment
376:
377: By twiddling y's last bit it is possible to force y to be
378: correctly rounded according to the prevailing rounding mode
379: as follows. Let r and i be copies of the rounding mode and
380: inexact flag before entering the square root program. Also we
381: use the expression y+-ulp for the next representable floating
382: numbers (up and down) of y. Note that y+-ulp = either fixed
383: point y+-1, or multiply y by nextafter(1,+-inf) in chopped
384: mode.
385:
386: R := RZ; ... set rounding mode to round-toward-zero
387: switch(r) {
388: case RN: ... round-to-nearest
389: if(x<= z*(z-ulp)...chopped) z = z - ulp; else
390: if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
391: break;
392: case RZ:case RM: ... round-to-zero or round-to--inf
393: R:=RP; ... reset rounding mod to round-to-+inf
394: if(x<z*z ... rounded up) z = z - ulp; else
395: if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
396: break;
397: case RP: ... round-to-+inf
398: if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
399: if(x>z*z ...chopped) z = z+ulp;
400: break;
401: }
402:
403: Remark 3. The above comparisons can be done in fixed point. For
404: example, to compare x and w=z*z chopped, it suffices to compare
405: x1 and w1 (the trailing parts of x and w), regarding them as
406: two's complement integers.
407:
408: ...Is z an exact square root?
409: To determine whether z is an exact square root of x, let z1 be the
410: trailing part of z, and also let x0 and x1 be the leading and
411: trailing parts of x.
412:
413: If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
414: I := 1; ... Raise Inexact flag: z is not exact
415: else {
416: j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
417: k := z1 >> 26; ... get z's 25-th and 26-th
418: fraction bits
419: I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
420: }
421: R:= r ... restore rounded mode
422: return sqrt(x):=z.
423:
424: If multiplication is cheaper then the foregoing red tape, the
425: Inexact flag can be evaluated by
426:
427: I := i;
428: I := (z*z!=x) or I.
429:
430: Note that z*z can overwrite I; this value must be sensed if it is
431: True.
432:
433: Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
434: zero.
435:
436: --------------------
437: z1: | f2 |
438: --------------------
439: bit 31 bit 0
440:
441: Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
442: or even of logb(x) have the following relations:
443:
444: -------------------------------------------------
445: bit 27,26 of z1 bit 1,0 of x1 logb(x)
446: -------------------------------------------------
447: 00 00 odd and even
448: 01 01 even
449: 10 10 odd
450: 10 00 even
451: 11 01 even
452: -------------------------------------------------
453:
454: (4) Special cases (see (4) of Section A).
455:
456: */
457:
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