3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
16 #include "math_private.h"
18 #ifdef USE_BUILTIN_SQRT
22 return (__builtin_sqrt(x));
26 * Return correctly rounded sqrt.
27 * ------------------------------------------
28 * | Use the hardware sqrt if you have one |
29 * ------------------------------------------
31 * Bit by bit method using integer arithmetic. (Slow, but portable)
33 * Scale x to y in [1,4) with even powers of 2:
34 * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
35 * sqrt(x) = 2^k * sqrt(y)
36 * 2. Bit by bit computation
37 * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
40 * s = 2*q , and y = 2 * ( y - q ). (1)
43 * To compute q from q , one checks whether
50 * If (2) is false, then q = q ; otherwise q = q + 2 .
53 * With some algebric manipulation, it is not difficult to see
54 * that (2) is equivalent to
59 * The advantage of (3) is that s and y can be computed by
61 * the following recurrence formula:
69 * s = s + 2 , y = y - s - 2 (5)
72 * One may easily use induction to prove (4) and (5).
73 * Note. Since the left hand side of (3) contain only i+2 bits,
74 * it does not necessary to do a full (53-bit) comparison
77 * After generating the 53 bits result, we compute one more bit.
78 * Together with the remainder, we can decide whether the
79 * result is exact, bigger than 1/2ulp, or less than 1/2ulp
80 * (it will never equal to 1/2ulp).
81 * The rounding mode can be detected by checking whether
82 * huge + tiny is equal to huge, and whether huge - tiny is
83 * equal to huge for some floating point number "huge" and "tiny".
86 * sqrt(+-0) = +-0 ... exact
88 * sqrt(-ve) = NaN ... with invalid signal
89 * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
91 * Other methods : see the appended file at the end of the program below.
95 static const double one = 1.0, tiny=1.0e-300;
101 int32_t sign = (int)0x80000000;
102 int32_t ix0,s0,q,m,t,i;
103 u_int32_t r,t1,s1,ix1,q1;
105 EXTRACT_WORDS(ix0,ix1,x);
107 /* take care of Inf and NaN */
108 if((ix0&0x7ff00000)==0x7ff00000) {
109 return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
112 /* take care of zero */
114 if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
116 return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
120 if(m==0) { /* subnormal x */
123 ix0 |= (ix1>>11); ix1 <<= 21;
125 for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
127 ix0 |= (ix1>>(32-i));
130 m -= 1023; /* unbias exponent */
131 ix0 = (ix0&0x000fffff)|0x00100000;
132 if(m&1){ /* odd m, double x to make it even */
133 ix0 += ix0 + ((ix1&sign)>>31);
136 m >>= 1; /* m = [m/2] */
138 /* generate sqrt(x) bit by bit */
139 ix0 += ix0 + ((ix1&sign)>>31);
141 q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
142 r = 0x00200000; /* r = moving bit from right to left */
151 ix0 += ix0 + ((ix1&sign)>>31);
160 if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
162 if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
164 if (ix1 < t1) ix0 -= 1;
168 ix0 += ix0 + ((ix1&sign)>>31);
173 /* use floating add to find out rounding direction */
175 z = one-tiny; /* trigger inexact flag */
178 if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;}
180 if (q1==(u_int32_t)0xfffffffe) q+=1;
186 ix0 = (q>>1)+0x3fe00000;
188 if ((q&1)==1) ix1 |= sign;
190 INSERT_WORDS(z,ix0,ix1);
195 #if (LDBL_MANT_DIG == 53)
196 __weak_reference(sqrt, sqrtl);
200 Other methods (use floating-point arithmetic)
202 (This is a copy of a drafted paper by Prof W. Kahan
203 and K.C. Ng, written in May, 1986)
205 Two algorithms are given here to implement sqrt(x)
206 (IEEE double precision arithmetic) in software.
207 Both supply sqrt(x) correctly rounded. The first algorithm (in
208 Section A) uses newton iterations and involves four divisions.
209 The second one uses reciproot iterations to avoid division, but
210 requires more multiplications. Both algorithms need the ability
211 to chop results of arithmetic operations instead of round them,
212 and the INEXACT flag to indicate when an arithmetic operation
213 is executed exactly with no roundoff error, all part of the
214 standard (IEEE 754-1985). The ability to perform shift, add,
215 subtract and logical AND operations upon 32-bit words is needed
216 too, though not part of the standard.
218 A. sqrt(x) by Newton Iteration
220 (1) Initial approximation
222 Let x0 and x1 be the leading and the trailing 32-bit words of
223 a floating point number x (in IEEE double format) respectively
226 ------------------------------------------------------
228 ------------------------------------------------------
229 msb lsb msb lsb ...order
232 ------------------------ ------------------------
233 x0: |s| e | f1 | x1: | f2 |
234 ------------------------ ------------------------
236 By performing shifts and subtracts on x0 and x1 (both regarded
237 as integers), we obtain an 8-bit approximation of sqrt(x) as
240 k := (x0>>1) + 0x1ff80000;
241 y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
242 Here k is a 32-bit integer and T1[] is an integer array containing
243 correction terms. Now magically the floating value of y (y's
244 leading 32-bit word is y0, the value of its trailing word is 0)
245 approximates sqrt(x) to almost 8-bit.
249 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
250 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
251 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
252 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
254 (2) Iterative refinement
256 Apply Heron's rule three times to y, we have y approximates
257 sqrt(x) to within 1 ulp (Unit in the Last Place):
259 y := (y+x/y)/2 ... almost 17 sig. bits
260 y := (y+x/y)/2 ... almost 35 sig. bits
261 y := y-(y-x/y)/2 ... within 1 ulp
265 Another way to improve y to within 1 ulp is:
267 y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
268 y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
272 y := y + 2* ---------- ...within 1 ulp
277 This formula has one division fewer than the one above; however,
278 it requires more multiplications and additions. Also x must be
279 scaled in advance to avoid spurious overflow in evaluating the
280 expression 3y*y+x. Hence it is not recommended uless division
281 is slow. If division is very slow, then one should use the
282 reciproot algorithm given in section B.
286 By twiddling y's last bit it is possible to force y to be
287 correctly rounded according to the prevailing rounding mode
288 as follows. Let r and i be copies of the rounding mode and
289 inexact flag before entering the square root program. Also we
290 use the expression y+-ulp for the next representable floating
291 numbers (up and down) of y. Note that y+-ulp = either fixed
292 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
295 I := FALSE; ... reset INEXACT flag I
296 R := RZ; ... set rounding mode to round-toward-zero
297 z := x/y; ... chopped quotient, possibly inexact
298 If(not I) then { ... if the quotient is exact
300 I := i; ... restore inexact flag
301 R := r; ... restore rounded mode
304 z := z - ulp; ... special rounding
307 i := TRUE; ... sqrt(x) is inexact
308 If (r=RN) then z=z+ulp ... rounded-to-nearest
309 If (r=RP) then { ... round-toward-+inf
312 y := y+z; ... chopped sum
313 y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
314 I := i; ... restore inexact flag
315 R := r; ... restore rounded mode
320 Square root of +inf, +-0, or NaN is itself;
321 Square root of a negative number is NaN with invalid signal.
324 B. sqrt(x) by Reciproot Iteration
326 (1) Initial approximation
328 Let x0 and x1 be the leading and the trailing 32-bit words of
329 a floating point number x (in IEEE double format) respectively
330 (see section A). By performing shifs and subtracts on x0 and y0,
331 we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
333 k := 0x5fe80000 - (x0>>1);
334 y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
336 Here k is a 32-bit integer and T2[] is an integer array
337 containing correction terms. Now magically the floating
338 value of y (y's leading 32-bit word is y0, the value of
339 its trailing word y1 is set to zero) approximates 1/sqrt(x)
344 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
345 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
346 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
347 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
348 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
349 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
350 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
351 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
353 (2) Iterative refinement
355 Apply Reciproot iteration three times to y and multiply the
356 result by x to get an approximation z that matches sqrt(x)
357 to about 1 ulp. To be exact, we will have
358 -1ulp < sqrt(x)-z<1.0625ulp.
360 ... set rounding mode to Round-to-nearest
361 y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
362 y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
363 ... special arrangement for better accuracy
364 z := x*y ... 29 bits to sqrt(x), with z*y<1
365 z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
367 Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
368 (a) the term z*y in the final iteration is always less than 1;
369 (b) the error in the final result is biased upward so that
370 -1 ulp < sqrt(x) - z < 1.0625 ulp
371 instead of |sqrt(x)-z|<1.03125ulp.
375 By twiddling y's last bit it is possible to force y to be
376 correctly rounded according to the prevailing rounding mode
377 as follows. Let r and i be copies of the rounding mode and
378 inexact flag before entering the square root program. Also we
379 use the expression y+-ulp for the next representable floating
380 numbers (up and down) of y. Note that y+-ulp = either fixed
381 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
384 R := RZ; ... set rounding mode to round-toward-zero
386 case RN: ... round-to-nearest
387 if(x<= z*(z-ulp)...chopped) z = z - ulp; else
388 if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
390 case RZ:case RM: ... round-to-zero or round-to--inf
391 R:=RP; ... reset rounding mod to round-to-+inf
392 if(x<z*z ... rounded up) z = z - ulp; else
393 if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
395 case RP: ... round-to-+inf
396 if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
397 if(x>z*z ...chopped) z = z+ulp;
401 Remark 3. The above comparisons can be done in fixed point. For
402 example, to compare x and w=z*z chopped, it suffices to compare
403 x1 and w1 (the trailing parts of x and w), regarding them as
404 two's complement integers.
406 ...Is z an exact square root?
407 To determine whether z is an exact square root of x, let z1 be the
408 trailing part of z, and also let x0 and x1 be the leading and
411 If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
412 I := 1; ... Raise Inexact flag: z is not exact
414 j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
415 k := z1 >> 26; ... get z's 25-th and 26-th
417 I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
419 R:= r ... restore rounded mode
422 If multiplication is cheaper then the foregoing red tape, the
423 Inexact flag can be evaluated by
428 Note that z*z can overwrite I; this value must be sensed if it is
431 Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
439 Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
440 or even of logb(x) have the following relations:
442 -------------------------------------------------
443 bit 27,26 of z1 bit 1,0 of x1 logb(x)
444 -------------------------------------------------
450 -------------------------------------------------
452 (4) Special cases (see (4) of Section A).