2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
9 * ====================================================
13 * Return correctly rounded sqrt.
14 * ------------------------------------------
15 * | Use the hardware sqrt if you have one |
16 * ------------------------------------------
18 * Bit by bit method using integer arithmetic. (Slow, but portable)
20 * Scale x to y in [1,4) with even powers of 2:
21 * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
22 * sqrt(x) = 2^k * sqrt(y)
23 * 2. Bit by bit computation
24 * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
27 * s = 2*q , and y = 2 * ( y - q ). (1)
30 * To compute q from q , one checks whether
37 * If (2) is false, then q = q ; otherwise q = q + 2 .
40 * With some algebric manipulation, it is not difficult to see
41 * that (2) is equivalent to
46 * The advantage of (3) is that s and y can be computed by
48 * the following recurrence formula:
56 * s = s + 2 , y = y - s - 2 (5)
59 * One may easily use induction to prove (4) and (5).
60 * Note. Since the left hand side of (3) contain only i+2 bits,
61 * it does not necessary to do a full (53-bit) comparison
64 * After generating the 53 bits result, we compute one more bit.
65 * Together with the remainder, we can decide whether the
66 * result is exact, bigger than 1/2ulp, or less than 1/2ulp
67 * (it will never equal to 1/2ulp).
68 * The rounding mode can be detected by checking whether
69 * huge + tiny is equal to huge, and whether huge - tiny is
70 * equal to huge for some floating point number "huge" and "tiny".
73 * sqrt(+-0) = +-0 ... exact
75 * sqrt(-ve) = NaN ... with invalid signal
76 * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
78 * Other methods : see the appended file at the end of the program below.
83 #include "math_private.h"
85 static const double one = 1.0, tiny = 1.0e-300;
87 double __ieee754_sqrt(double x)
90 int32_t sign = (int)0x80000000;
91 int32_t ix0,s0,q,m,t,i;
92 u_int32_t r,t1,s1,ix1,q1;
94 EXTRACT_WORDS(ix0,ix1,x);
96 /* take care of Inf and NaN */
97 if((ix0&0x7ff00000)==0x7ff00000) {
98 return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
101 /* take care of zero */
103 if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
105 return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
109 if(m==0) { /* subnormal x */
112 ix0 |= (ix1>>11); ix1 <<= 21;
114 for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
116 ix0 |= (ix1>>(32-i));
119 m -= 1023; /* unbias exponent */
120 ix0 = (ix0&0x000fffff)|0x00100000;
121 if(m&1){ /* odd m, double x to make it even */
122 ix0 += ix0 + ((ix1&sign)>>31);
125 m >>= 1; /* m = [m/2] */
127 /* generate sqrt(x) bit by bit */
128 ix0 += ix0 + ((ix1&sign)>>31);
130 q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
131 r = 0x00200000; /* r = moving bit from right to left */
140 ix0 += ix0 + ((ix1&sign)>>31);
149 if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
151 if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
153 if (ix1 < t1) ix0 -= 1;
157 ix0 += ix0 + ((ix1&sign)>>31);
162 /* use floating add to find out rounding direction */
164 z = one-tiny; /* trigger inexact flag */
167 if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;}
169 if (q1==(u_int32_t)0xfffffffe) q+=1;
175 ix0 = (q>>1)+0x3fe00000;
177 if ((q&1)==1) ix1 |= sign;
179 INSERT_WORDS(z,ix0,ix1);
187 double sqrt(double x)
189 double z = __ieee754_sqrt(x);
190 if (_LIB_VERSION == _IEEE_ || isnan(x))
193 return __kernel_standard(x, x, 26); /* sqrt(negative) */
197 strong_alias(__ieee754_sqrt, sqrt)
199 libm_hidden_def(sqrt)
203 Other methods (use floating-point arithmetic)
205 (This is a copy of a drafted paper by Prof W. Kahan
206 and K.C. Ng, written in May, 1986)
208 Two algorithms are given here to implement sqrt(x)
209 (IEEE double precision arithmetic) in software.
210 Both supply sqrt(x) correctly rounded. The first algorithm (in
211 Section A) uses newton iterations and involves four divisions.
212 The second one uses reciproot iterations to avoid division, but
213 requires more multiplications. Both algorithms need the ability
214 to chop results of arithmetic operations instead of round them,
215 and the INEXACT flag to indicate when an arithmetic operation
216 is executed exactly with no roundoff error, all part of the
217 standard (IEEE 754-1985). The ability to perform shift, add,
218 subtract and logical AND operations upon 32-bit words is needed
219 too, though not part of the standard.
221 A. sqrt(x) by Newton Iteration
223 (1) Initial approximation
225 Let x0 and x1 be the leading and the trailing 32-bit words of
226 a floating point number x (in IEEE double format) respectively
229 ------------------------------------------------------
231 ------------------------------------------------------
232 msb lsb msb lsb ...order
235 ------------------------ ------------------------
236 x0: |s| e | f1 | x1: | f2 |
237 ------------------------ ------------------------
239 By performing shifts and subtracts on x0 and x1 (both regarded
240 as integers), we obtain an 8-bit approximation of sqrt(x) as
243 k := (x0>>1) + 0x1ff80000;
244 y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
245 Here k is a 32-bit integer and T1[] is an integer array containing
246 correction terms. Now magically the floating value of y (y's
247 leading 32-bit word is y0, the value of its trailing word is 0)
248 approximates sqrt(x) to almost 8-bit.
252 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
253 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
254 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
255 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
257 (2) Iterative refinement
259 Apply Heron's rule three times to y, we have y approximates
260 sqrt(x) to within 1 ulp (Unit in the Last Place):
262 y := (y+x/y)/2 ... almost 17 sig. bits
263 y := (y+x/y)/2 ... almost 35 sig. bits
264 y := y-(y-x/y)/2 ... within 1 ulp
268 Another way to improve y to within 1 ulp is:
270 y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
271 y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
275 y := y + 2* ---------- ...within 1 ulp
280 This formula has one division fewer than the one above; however,
281 it requires more multiplications and additions. Also x must be
282 scaled in advance to avoid spurious overflow in evaluating the
283 expression 3y*y+x. Hence it is not recommended uless division
284 is slow. If division is very slow, then one should use the
285 reciproot algorithm given in section B.
289 By twiddling y's last bit it is possible to force y to be
290 correctly rounded according to the prevailing rounding mode
291 as follows. Let r and i be copies of the rounding mode and
292 inexact flag before entering the square root program. Also we
293 use the expression y+-ulp for the next representable floating
294 numbers (up and down) of y. Note that y+-ulp = either fixed
295 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
298 I := FALSE; ... reset INEXACT flag I
299 R := RZ; ... set rounding mode to round-toward-zero
300 z := x/y; ... chopped quotient, possibly inexact
301 If(not I) then { ... if the quotient is exact
303 I := i; ... restore inexact flag
304 R := r; ... restore rounded mode
307 z := z - ulp; ... special rounding
310 i := TRUE; ... sqrt(x) is inexact
311 If (r=RN) then z=z+ulp ... rounded-to-nearest
312 If (r=RP) then { ... round-toward-+inf
315 y := y+z; ... chopped sum
316 y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
317 I := i; ... restore inexact flag
318 R := r; ... restore rounded mode
323 Square root of +inf, +-0, or NaN is itself;
324 Square root of a negative number is NaN with invalid signal.
327 B. sqrt(x) by Reciproot Iteration
329 (1) Initial approximation
331 Let x0 and x1 be the leading and the trailing 32-bit words of
332 a floating point number x (in IEEE double format) respectively
333 (see section A). By performing shifs and subtracts on x0 and y0,
334 we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
336 k := 0x5fe80000 - (x0>>1);
337 y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
339 Here k is a 32-bit integer and T2[] is an integer array
340 containing correction terms. Now magically the floating
341 value of y (y's leading 32-bit word is y0, the value of
342 its trailing word y1 is set to zero) approximates 1/sqrt(x)
347 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
348 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
349 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
350 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
351 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
352 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
353 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
354 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
356 (2) Iterative refinement
358 Apply Reciproot iteration three times to y and multiply the
359 result by x to get an approximation z that matches sqrt(x)
360 to about 1 ulp. To be exact, we will have
361 -1ulp < sqrt(x)-z<1.0625ulp.
363 ... set rounding mode to Round-to-nearest
364 y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
365 y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
366 ... special arrangement for better accuracy
367 z := x*y ... 29 bits to sqrt(x), with z*y<1
368 z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
370 Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
371 (a) the term z*y in the final iteration is always less than 1;
372 (b) the error in the final result is biased upward so that
373 -1 ulp < sqrt(x) - z < 1.0625 ulp
374 instead of |sqrt(x)-z|<1.03125ulp.
378 By twiddling y's last bit it is possible to force y to be
379 correctly rounded according to the prevailing rounding mode
380 as follows. Let r and i be copies of the rounding mode and
381 inexact flag before entering the square root program. Also we
382 use the expression y+-ulp for the next representable floating
383 numbers (up and down) of y. Note that y+-ulp = either fixed
384 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
387 R := RZ; ... set rounding mode to round-toward-zero
389 case RN: ... round-to-nearest
390 if(x<= z*(z-ulp)...chopped) z = z - ulp; else
391 if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
393 case RZ:case RM: ... round-to-zero or round-to--inf
394 R:=RP; ... reset rounding mod to round-to-+inf
395 if(x<z*z ... rounded up) z = z - ulp; else
396 if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
398 case RP: ... round-to-+inf
399 if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
400 if(x>z*z ...chopped) z = z+ulp;
404 Remark 3. The above comparisons can be done in fixed point. For
405 example, to compare x and w=z*z chopped, it suffices to compare
406 x1 and w1 (the trailing parts of x and w), regarding them as
407 two's complement integers.
409 ...Is z an exact square root?
410 To determine whether z is an exact square root of x, let z1 be the
411 trailing part of z, and also let x0 and x1 be the leading and
414 If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
415 I := 1; ... Raise Inexact flag: z is not exact
417 j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
418 k := z1 >> 26; ... get z's 25-th and 26-th
420 I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
422 R:= r ... restore rounded mode
425 If multiplication is cheaper then the foregoing red tape, the
426 Inexact flag can be evaluated by
431 Note that z*z can overwrite I; this value must be sensed if it is
434 Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
442 Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
443 or even of logb(x) have the following relations:
445 -------------------------------------------------
446 bit 27,26 of z1 bit 1,0 of x1 logb(x)
447 -------------------------------------------------
453 -------------------------------------------------
455 (4) Special cases (see (4) of Section A).