00001 /* @(#)e_sqrt.c 5.1 93/09/24 */ 00002 /* 00003 * ==================================================== 00004 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 00005 * 00006 * Developed at SunPro, a Sun Microsystems, Inc. business. 00007 * Permission to use, copy, modify, and distribute this 00008 * software is freely granted, provided that this notice 00009 * is preserved. 00010 * ==================================================== 00011 */ 00012 00013 #if defined(LIBM_SCCS) && !defined(lint) 00014 static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $"; 00015 #endif 00016 00017 /* __ieee754_sqrt(x) 00018 * Return correctly rounded sqrt. 00019 * ------------------------------------------ 00020 * | Use the hardware sqrt if you have one | 00021 * ------------------------------------------ 00022 * Method: 00023 * Bit by bit method using integer arithmetic. (Slow, but portable) 00024 * 1. Normalization 00025 * Scale x to y in [1,4) with even powers of 2: 00026 * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then 00027 * sqrt(x) = 2^k * sqrt(y) 00028 * 2. Bit by bit computation 00029 * Let q = sqrt(y) truncated to i bit after binary point (q = 1), 00030 * i 0 00031 * i+1 2 00032 * s = 2*q , and y = 2 * ( y - q ). (1) 00033 * i i i i 00034 * 00035 * To compute q from q , one checks whether 00036 * i+1 i 00037 * 00038 * -(i+1) 2 00039 * (q + 2 ) <= y. (2) 00040 * i 00041 * -(i+1) 00042 * If (2) is false, then q = q ; otherwise q = q + 2 . 00043 * i+1 i i+1 i 00044 * 00045 * With some algebric manipulation, it is not difficult to see 00046 * that (2) is equivalent to 00047 * -(i+1) 00048 * s + 2 <= y (3) 00049 * i i 00050 * 00051 * The advantage of (3) is that s and y can be computed by 00052 * i i 00053 * the following recurrence formula: 00054 * if (3) is false 00055 * 00056 * s = s , y = y ; (4) 00057 * i+1 i i+1 i 00058 * 00059 * otherwise, 00060 * -i -(i+1) 00061 * s = s + 2 , y = y - s - 2 (5) 00062 * i+1 i i+1 i i 00063 * 00064 * One may easily use induction to prove (4) and (5). 00065 * Note. Since the left hand side of (3) contain only i+2 bits, 00066 * it does not necessary to do a full (53-bit) comparison 00067 * in (3). 00068 * 3. Final rounding 00069 * After generating the 53 bits result, we compute one more bit. 00070 * Together with the remainder, we can decide whether the 00071 * result is exact, bigger than 1/2ulp, or less than 1/2ulp 00072 * (it will never equal to 1/2ulp). 00073 * The rounding mode can be detected by checking whether 00074 * huge + tiny is equal to huge, and whether huge - tiny is 00075 * equal to huge for some floating point number "huge" and "tiny". 00076 * 00077 * Special cases: 00078 * sqrt(+-0) = +-0 ... exact 00079 * sqrt(inf) = inf 00080 * sqrt(-ve) = NaN ... with invalid signal 00081 * sqrt(NaN) = NaN ... with invalid signal for signaling NaN 00082 * 00083 * Other methods : see the appended file at the end of the program below. 00084 *--------------- 00085 */ 00086 00087 #include "math.h" 00088 #include "mathP.h" 00089 00090 #ifdef __STDC__ 00091 static const double one = 1.0, tiny=1.0e-300; 00092 #else 00093 static double one = 1.0, tiny=1.0e-300; 00094 #endif 00095 00096 #ifdef __STDC__ 00097 double __ieee754_sqrt(double x) 00098 #else 00099 double __ieee754_sqrt(x) 00100 double x; 00101 #endif 00102 { 00103 double z; 00104 int32_t sign = (int)0x80000000; 00105 int32_t ix0,s0,q,m,t,i; 00106 u_int32_t r,t1,s1,ix1,q1; 00107 00108 EXTRACT_WORDS(ix0,ix1,x); 00109 00110 /* take care of Inf and NaN */ 00111 if((ix0&0x7ff00000)==0x7ff00000) { 00112 return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf 00113 sqrt(-inf)=sNaN */ 00114 } 00115 /* take care of zero */ 00116 if(ix0<=0) { 00117 if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */ 00118 else if(ix0<0) 00119 return (x-x)/(x-x); /* sqrt(-ve) = sNaN */ 00120 } 00121 /* normalize x */ 00122 m = (ix0>>20); 00123 if(m==0) { /* subnormal x */ 00124 while(ix0==0) { 00125 m -= 21; 00126 ix0 |= (ix1>>11); ix1 <<= 21; 00127 } 00128 for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1; 00129 m -= i-1; 00130 ix0 |= (ix1>>(32-i)); 00131 ix1 <<= i; 00132 } 00133 m -= 1023; /* unbias exponent */ 00134 ix0 = (ix0&0x000fffff)|0x00100000; 00135 if(m&1){ /* odd m, double x to make it even */ 00136 ix0 += ix0 + ((ix1&sign)>>31); 00137 ix1 += ix1; 00138 } 00139 m >>= 1; /* m = [m/2] */ 00140 00141 /* generate sqrt(x) bit by bit */ 00142 ix0 += ix0 + ((ix1&sign)>>31); 00143 ix1 += ix1; 00144 q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */ 00145 r = 0x00200000; /* r = moving bit from right to left */ 00146 00147 while(r!=0) { 00148 t = s0+r; 00149 if(t<=ix0) { 00150 s0 = t+r; 00151 ix0 -= t; 00152 q += r; 00153 } 00154 ix0 += ix0 + ((ix1&sign)>>31); 00155 ix1 += ix1; 00156 r>>=1; 00157 } 00158 00159 r = sign; 00160 while(r!=0) { 00161 t1 = s1+r; 00162 t = s0; 00163 if((t<ix0)||((t==ix0)&&(t1<=ix1))) { 00164 s1 = t1+r; 00165 if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1; 00166 ix0 -= t; 00167 if (ix1 < t1) ix0 -= 1; 00168 ix1 -= t1; 00169 q1 += r; 00170 } 00171 ix0 += ix0 + ((ix1&sign)>>31); 00172 ix1 += ix1; 00173 r>>=1; 00174 } 00175 00176 /* use floating add to find out rounding direction */ 00177 if((ix0|ix1)!=0) { 00178 z = one-tiny; /* trigger inexact flag */ 00179 if (z>=one) { 00180 z = one+tiny; 00181 if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;} 00182 else if (z>one) { 00183 if (q1==(u_int32_t)0xfffffffe) q+=1; 00184 q1+=2; 00185 } else 00186 q1 += (q1&1); 00187 } 00188 } 00189 ix0 = (q>>1)+0x3fe00000; 00190 ix1 = q1>>1; 00191 if ((q&1)==1) ix1 |= sign; 00192 ix0 += (m <<20); 00193 INSERT_WORDS(z,ix0,ix1); 00194 return z; 00195 } 00196 00197 /* 00198 Other methods (use floating-point arithmetic) 00199 ------------- 00200 (This is a copy of a drafted paper by Prof W. Kahan 00201 and K.C. Ng, written in May, 1986) 00202 00203 Two algorithms are given here to implement sqrt(x) 00204 (IEEE double precision arithmetic) in software. 00205 Both supply sqrt(x) correctly rounded. The first algorithm (in 00206 Section A) uses newton iterations and involves four divisions. 00207 The second one uses reciproot iterations to avoid division, but 00208 requires more multiplications. Both algorithms need the ability 00209 to chop results of arithmetic operations instead of round them, 00210 and the INEXACT flag to indicate when an arithmetic operation 00211 is executed exactly with no roundoff error, all part of the 00212 standard (IEEE 754-1985). The ability to perform shift, add, 00213 subtract and logical AND operations upon 32-bit words is needed 00214 too, though not part of the standard. 00215 00216 A. sqrt(x) by Newton Iteration 00217 00218 (1) Initial approximation 00219 00220 Let x0 and x1 be the leading and the trailing 32-bit words of 00221 a floating point number x (in IEEE double format) respectively 00222 00223 1 11 52 ...widths 00224 ------------------------------------------------------ 00225 x: |s| e | f | 00226 ------------------------------------------------------ 00227 msb lsb msb lsb ...order 00228 00229 00230 ------------------------ ------------------------ 00231 x0: |s| e | f1 | x1: | f2 | 00232 ------------------------ ------------------------ 00233 00234 By performing shifts and subtracts on x0 and x1 (both regarded 00235 as integers), we obtain an 8-bit approximation of sqrt(x) as 00236 follows. 00237 00238 k := (x0>>1) + 0x1ff80000; 00239 y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits 00240 Here k is a 32-bit integer and T1[] is an integer array containing 00241 correction terms. Now magically the floating value of y (y's 00242 leading 32-bit word is y0, the value of its trailing word is 0) 00243 approximates sqrt(x) to almost 8-bit. 00244 00245 Value of T1: 00246 static int T1[32]= { 00247 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592, 00248 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215, 00249 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581, 00250 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,}; 00251 00252 (2) Iterative refinement 00253 00254 Apply Heron's rule three times to y, we have y approximates 00255 sqrt(x) to within 1 ulp (Unit in the Last Place): 00256 00257 y := (y+x/y)/2 ... almost 17 sig. bits 00258 y := (y+x/y)/2 ... almost 35 sig. bits 00259 y := y-(y-x/y)/2 ... within 1 ulp 00260 00261 00262 Remark 1. 00263 Another way to improve y to within 1 ulp is: 00264 00265 y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x) 00266 y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x) 00267 00268 2 00269 (x-y )*y 00270 y := y + 2* ---------- ...within 1 ulp 00271 2 00272 3y + x 00273 00274 00275 This formula has one division fewer than the one above; however, 00276 it requires more multiplications and additions. Also x must be 00277 scaled in advance to avoid spurious overflow in evaluating the 00278 expression 3y*y+x. Hence it is not recommended uless division 00279 is slow. If division is very slow, then one should use the 00280 reciproot algorithm given in section B. 00281 00282 (3) Final adjustment 00283 00284 By twiddling y's last bit it is possible to force y to be 00285 correctly rounded according to the prevailing rounding mode 00286 as follows. Let r and i be copies of the rounding mode and 00287 inexact flag before entering the square root program. Also we 00288 use the expression y+-ulp for the next representable floating 00289 numbers (up and down) of y. Note that y+-ulp = either fixed 00290 point y+-1, or multiply y by nextafter(1,+-inf) in chopped 00291 mode. 00292 00293 I := FALSE; ... reset INEXACT flag I 00294 R := RZ; ... set rounding mode to round-toward-zero 00295 z := x/y; ... chopped quotient, possibly inexact 00296 If(not I) then { ... if the quotient is exact 00297 if(z=y) { 00298 I := i; ... restore inexact flag 00299 R := r; ... restore rounded mode 00300 return sqrt(x):=y. 00301 } else { 00302 z := z - ulp; ... special rounding 00303 } 00304 } 00305 i := TRUE; ... sqrt(x) is inexact 00306 If (r=RN) then z=z+ulp ... rounded-to-nearest 00307 If (r=RP) then { ... round-toward-+inf 00308 y = y+ulp; z=z+ulp; 00309 } 00310 y := y+z; ... chopped sum 00311 y0:=y0-0x00100000; ... y := y/2 is correctly rounded. 00312 I := i; ... restore inexact flag 00313 R := r; ... restore rounded mode 00314 return sqrt(x):=y. 00315 00316 (4) Special cases 00317 00318 Square root of +inf, +-0, or NaN is itself; 00319 Square root of a negative number is NaN with invalid signal. 00320 00321 00322 B. sqrt(x) by Reciproot Iteration 00323 00324 (1) Initial approximation 00325 00326 Let x0 and x1 be the leading and the trailing 32-bit words of 00327 a floating point number x (in IEEE double format) respectively 00328 (see section A). By performing shifs and subtracts on x0 and y0, 00329 we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. 00330 00331 k := 0x5fe80000 - (x0>>1); 00332 y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits 00333 00334 Here k is a 32-bit integer and T2[] is an integer array 00335 containing correction terms. Now magically the floating 00336 value of y (y's leading 32-bit word is y0, the value of 00337 its trailing word y1 is set to zero) approximates 1/sqrt(x) 00338 to almost 7.8-bit. 00339 00340 Value of T2: 00341 static int T2[64]= { 00342 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, 00343 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, 00344 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, 00345 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, 00346 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, 00347 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, 00348 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, 00349 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; 00350 00351 (2) Iterative refinement 00352 00353 Apply Reciproot iteration three times to y and multiply the 00354 result by x to get an approximation z that matches sqrt(x) 00355 to about 1 ulp. To be exact, we will have 00356 -1ulp < sqrt(x)-z<1.0625ulp. 00357 00358 ... set rounding mode to Round-to-nearest 00359 y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) 00360 y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) 00361 ... special arrangement for better accuracy 00362 z := x*y ... 29 bits to sqrt(x), with z*y<1 00363 z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) 00364 00365 Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that 00366 (a) the term z*y in the final iteration is always less than 1; 00367 (b) the error in the final result is biased upward so that 00368 -1 ulp < sqrt(x) - z < 1.0625 ulp 00369 instead of |sqrt(x)-z|<1.03125ulp. 00370 00371 (3) Final adjustment 00372 00373 By twiddling y's last bit it is possible to force y to be 00374 correctly rounded according to the prevailing rounding mode 00375 as follows. Let r and i be copies of the rounding mode and 00376 inexact flag before entering the square root program. Also we 00377 use the expression y+-ulp for the next representable floating 00378 numbers (up and down) of y. Note that y+-ulp = either fixed 00379 point y+-1, or multiply y by nextafter(1,+-inf) in chopped 00380 mode. 00381 00382 R := RZ; ... set rounding mode to round-toward-zero 00383 switch(r) { 00384 case RN: ... round-to-nearest 00385 if(x<= z*(z-ulp)...chopped) z = z - ulp; else 00386 if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; 00387 break; 00388 case RZ:case RM: ... round-to-zero or round-to--inf 00389 R:=RP; ... reset rounding mod to round-to-+inf 00390 if(x<z*z ... rounded up) z = z - ulp; else 00391 if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; 00392 break; 00393 case RP: ... round-to-+inf 00394 if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else 00395 if(x>z*z ...chopped) z = z+ulp; 00396 break; 00397 } 00398 00399 Remark 3. The above comparisons can be done in fixed point. For 00400 example, to compare x and w=z*z chopped, it suffices to compare 00401 x1 and w1 (the trailing parts of x and w), regarding them as 00402 two's complement integers. 00403 00404 ...Is z an exact square root? 00405 To determine whether z is an exact square root of x, let z1 be the 00406 trailing part of z, and also let x0 and x1 be the leading and 00407 trailing parts of x. 00408 00409 If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 00410 I := 1; ... Raise Inexact flag: z is not exact 00411 else { 00412 j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 00413 k := z1 >> 26; ... get z's 25-th and 26-th 00414 fraction bits 00415 I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); 00416 } 00417 R:= r ... restore rounded mode 00418 return sqrt(x):=z. 00419 00420 If multiplication is cheaper then the foregoing red tape, the 00421 Inexact flag can be evaluated by 00422 00423 I := i; 00424 I := (z*z!=x) or I. 00425 00426 Note that z*z can overwrite I; this value must be sensed if it is 00427 True. 00428 00429 Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be 00430 zero. 00431 00432 -------------------- 00433 z1: | f2 | 00434 -------------------- 00435 bit 31 bit 0 00436 00437 Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd 00438 or even of logb(x) have the following relations: 00439 00440 ------------------------------------------------- 00441 bit 27,26 of z1 bit 1,0 of x1 logb(x) 00442 ------------------------------------------------- 00443 00 00 odd and even 00444 01 01 even 00445 10 10 odd 00446 10 00 even 00447 11 01 even 00448 ------------------------------------------------- 00449 00450 (4) Special cases (see (4) of Section A). 00451 00452 */ 00453
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