| cc68a136 |
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 | */ |
| 12 | |
| 13 | #if defined(LIBM_SCCS) && !defined(lint) |
| 14 | static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $"; |
| 15 | #endif |
| 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 | |
| 87 | #include "math.h" |
| 88 | #include "math_private.h" |
| 89 | |
| 90 | #ifdef __STDC__ |
| 91 | static const double one = 1.0, tiny=1.0e-300; |
| 92 | #else |
| 93 | static double one = 1.0, tiny=1.0e-300; |
| 94 | #endif |
| 95 | |
| 96 | #ifdef __STDC__ |
| 97 | double __ieee754_sqrt(double x) |
| 98 | #else |
| 99 | double __ieee754_sqrt(x) |
| 100 | double x; |
| 101 | #endif |
| 102 | { |
| 103 | double z; |
| 104 | int32_t sign = (int)0x80000000; |
| 105 | int32_t ix0,s0,q,m,t,i; |
| 106 | u_int32_t r,t1,s1,ix1,q1; |
| 107 | |
| 108 | EXTRACT_WORDS(ix0,ix1,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==(u_int32_t)0xffffffff) { q1=0; q += 1;} |
| 182 | else if (z>one) { |
| 183 | if (q1==(u_int32_t)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 | INSERT_WORDS(z,ix0,ix1); |
| 194 | return z; |
| 195 | } |
| 196 | |
| 197 | /* |
| 198 | Other methods (use floating-point arithmetic) |
| 199 | ------------- |
| 200 | (This is a copy of a drafted paper by Prof W. Kahan |
| 201 | and K.C. Ng, written in May, 1986) |
| 202 | |
| 203 | Two algorithms are given here to implement sqrt(x) |
| 204 | (IEEE double precision arithmetic) in software. |
| 205 | Both supply sqrt(x) correctly rounded. The first algorithm (in |
| 206 | Section A) uses newton iterations and involves four divisions. |
| 207 | The second one uses reciproot iterations to avoid division, but |
| 208 | requires more multiplications. Both algorithms need the ability |
| 209 | to chop results of arithmetic operations instead of round them, |
| 210 | and the INEXACT flag to indicate when an arithmetic operation |
| 211 | is executed exactly with no roundoff error, all part of the |
| 212 | standard (IEEE 754-1985). The ability to perform shift, add, |
| 213 | subtract and logical AND operations upon 32-bit words is needed |
| 214 | too, though not part of the standard. |
| 215 | |
| 216 | A. sqrt(x) by Newton Iteration |
| 217 | |
| 218 | (1) Initial approximation |
| 219 | |
| 220 | Let x0 and x1 be the leading and the trailing 32-bit words of |
| 221 | a floating point number x (in IEEE double format) respectively |
| 222 | |
| 223 | 1 11 52 ...widths |
| 224 | ------------------------------------------------------ |
| 225 | x: |s| e | f | |
| 226 | ------------------------------------------------------ |
| 227 | msb lsb msb lsb ...order |
| 228 | |
| 229 | |
| 230 | ------------------------ ------------------------ |
| 231 | x0: |s| e | f1 | x1: | f2 | |
| 232 | ------------------------ ------------------------ |
| 233 | |
| 234 | By performing shifts and subtracts on x0 and x1 (both regarded |
| 235 | as integers), we obtain an 8-bit approximation of sqrt(x) as |
| 236 | follows. |
| 237 | |
| 238 | k := (x0>>1) + 0x1ff80000; |
| 239 | y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits |
| 240 | Here k is a 32-bit integer and T1[] is an integer array containing |
| 241 | correction terms. Now magically the floating value of y (y's |
| 242 | leading 32-bit word is y0, the value of its trailing word is 0) |
| 243 | approximates sqrt(x) to almost 8-bit. |
| 244 | |
| 245 | Value of T1: |
| 246 | static int T1[32]= { |
| 247 | 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592, |
| 248 | 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215, |
| 249 | 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581, |
| 250 | 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,}; |
| 251 | |
| 252 | (2) Iterative refinement |
| 253 | |
| 254 | Apply Heron's rule three times to y, we have y approximates |
| 255 | sqrt(x) to within 1 ulp (Unit in the Last Place): |
| 256 | |
| 257 | y := (y+x/y)/2 ... almost 17 sig. bits |
| 258 | y := (y+x/y)/2 ... almost 35 sig. bits |
| 259 | y := y-(y-x/y)/2 ... within 1 ulp |
| 260 | |
| 261 | |
| 262 | Remark 1. |
| 263 | Another way to improve y to within 1 ulp is: |
| 264 | |
| 265 | y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x) |
| 266 | y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x) |
| 267 | |
| 268 | 2 |
| 269 | (x-y )*y |
| 270 | y := y + 2* ---------- ...within 1 ulp |
| 271 | 2 |
| 272 | 3y + x |
| 273 | |
| 274 | |
| 275 | This formula has one division fewer than the one above; however, |
| 276 | it requires more multiplications and additions. Also x must be |
| 277 | scaled in advance to avoid spurious overflow in evaluating the |
| 278 | expression 3y*y+x. Hence it is not recommended uless division |
| 279 | is slow. If division is very slow, then one should use the |
| 280 | reciproot algorithm given in section B. |
| 281 | |
| 282 | (3) Final adjustment |
| 283 | |
| 284 | By twiddling y's last bit it is possible to force y to be |
| 285 | correctly rounded according to the prevailing rounding mode |
| 286 | as follows. Let r and i be copies of the rounding mode and |
| 287 | inexact flag before entering the square root program. Also we |
| 288 | use the expression y+-ulp for the next representable floating |
| 289 | numbers (up and down) of y. Note that y+-ulp = either fixed |
| 290 | point y+-1, or multiply y by nextafter(1,+-inf) in chopped |
| 291 | mode. |
| 292 | |
| 293 | I := FALSE; ... reset INEXACT flag I |
| 294 | R := RZ; ... set rounding mode to round-toward-zero |
| 295 | z := x/y; ... chopped quotient, possibly inexact |
| 296 | If(not I) then { ... if the quotient is exact |
| 297 | if(z=y) { |
| 298 | I := i; ... restore inexact flag |
| 299 | R := r; ... restore rounded mode |
| 300 | return sqrt(x):=y. |
| 301 | } else { |
| 302 | z := z - ulp; ... special rounding |
| 303 | } |
| 304 | } |
| 305 | i := TRUE; ... sqrt(x) is inexact |
| 306 | If (r=RN) then z=z+ulp ... rounded-to-nearest |
| 307 | If (r=RP) then { ... round-toward-+inf |
| 308 | y = y+ulp; z=z+ulp; |
| 309 | } |
| 310 | y := y+z; ... chopped sum |
| 311 | y0:=y0-0x00100000; ... y := y/2 is correctly rounded. |
| 312 | I := i; ... restore inexact flag |
| 313 | R := r; ... restore rounded mode |
| 314 | return sqrt(x):=y. |
| 315 | |
| 316 | (4) Special cases |
| 317 | |
| 318 | Square root of +inf, +-0, or NaN is itself; |
| 319 | Square root of a negative number is NaN with invalid signal. |
| 320 | |
| 321 | |
| 322 | B. sqrt(x) by Reciproot Iteration |
| 323 | |
| 324 | (1) Initial approximation |
| 325 | |
| 326 | Let x0 and x1 be the leading and the trailing 32-bit words of |
| 327 | a floating point number x (in IEEE double format) respectively |
| 328 | (see section A). By performing shifs and subtracts on x0 and y0, |
| 329 | we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. |
| 330 | |
| 331 | k := 0x5fe80000 - (x0>>1); |
| 332 | y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits |
| 333 | |
| 334 | Here k is a 32-bit integer and T2[] is an integer array |
| 335 | containing correction terms. Now magically the floating |
| 336 | value of y (y's leading 32-bit word is y0, the value of |
| 337 | its trailing word y1 is set to zero) approximates 1/sqrt(x) |
| 338 | to almost 7.8-bit. |
| 339 | |
| 340 | Value of T2: |
| 341 | static int T2[64]= { |
| 342 | 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, |
| 343 | 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, |
| 344 | 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, |
| 345 | 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, |
| 346 | 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, |
| 347 | 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, |
| 348 | 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, |
| 349 | 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; |
| 350 | |
| 351 | (2) Iterative refinement |
| 352 | |
| 353 | Apply Reciproot iteration three times to y and multiply the |
| 354 | result by x to get an approximation z that matches sqrt(x) |
| 355 | to about 1 ulp. To be exact, we will have |
| 356 | -1ulp < sqrt(x)-z<1.0625ulp. |
| 357 | |
| 358 | ... set rounding mode to Round-to-nearest |
| 359 | y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) |
| 360 | y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) |
| 361 | ... special arrangement for better accuracy |
| 362 | z := x*y ... 29 bits to sqrt(x), with z*y<1 |
| 363 | z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) |
| 364 | |
| 365 | Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that |
| 366 | (a) the term z*y in the final iteration is always less than 1; |
| 367 | (b) the error in the final result is biased upward so that |
| 368 | -1 ulp < sqrt(x) - z < 1.0625 ulp |
| 369 | instead of |sqrt(x)-z|<1.03125ulp. |
| 370 | |
| 371 | (3) Final adjustment |
| 372 | |
| 373 | By twiddling y's last bit it is possible to force y to be |
| 374 | correctly rounded according to the prevailing rounding mode |
| 375 | as follows. Let r and i be copies of the rounding mode and |
| 376 | inexact flag before entering the square root program. Also we |
| 377 | use the expression y+-ulp for the next representable floating |
| 378 | numbers (up and down) of y. Note that y+-ulp = either fixed |
| 379 | point y+-1, or multiply y by nextafter(1,+-inf) in chopped |
| 380 | mode. |
| 381 | |
| 382 | R := RZ; ... set rounding mode to round-toward-zero |
| 383 | switch(r) { |
| 384 | case RN: ... round-to-nearest |
| 385 | if(x<= z*(z-ulp)...chopped) z = z - ulp; else |
| 386 | if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; |
| 387 | break; |
| 388 | case RZ:case RM: ... round-to-zero or round-to--inf |
| 389 | R:=RP; ... reset rounding mod to round-to-+inf |
| 390 | if(x<z*z ... rounded up) z = z - ulp; else |
| 391 | if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; |
| 392 | break; |
| 393 | case RP: ... round-to-+inf |
| 394 | if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else |
| 395 | if(x>z*z ...chopped) z = z+ulp; |
| 396 | break; |
| 397 | } |
| 398 | |
| 399 | Remark 3. The above comparisons can be done in fixed point. For |
| 400 | example, to compare x and w=z*z chopped, it suffices to compare |
| 401 | x1 and w1 (the trailing parts of x and w), regarding them as |
| 402 | two's complement integers. |
| 403 | |
| 404 | ...Is z an exact square root? |
| 405 | To determine whether z is an exact square root of x, let z1 be the |
| 406 | trailing part of z, and also let x0 and x1 be the leading and |
| 407 | trailing parts of x. |
| 408 | |
| 409 | If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 |
| 410 | I := 1; ... Raise Inexact flag: z is not exact |
| 411 | else { |
| 412 | j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 |
| 413 | k := z1 >> 26; ... get z's 25-th and 26-th |
| 414 | fraction bits |
| 415 | I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); |
| 416 | } |
| 417 | R:= r ... restore rounded mode |
| 418 | return sqrt(x):=z. |
| 419 | |
| 420 | If multiplication is cheaper then the foregoing red tape, the |
| 421 | Inexact flag can be evaluated by |
| 422 | |
| 423 | I := i; |
| 424 | I := (z*z!=x) or I. |
| 425 | |
| 426 | Note that z*z can overwrite I; this value must be sensed if it is |
| 427 | True. |
| 428 | |
| 429 | Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be |
| 430 | zero. |
| 431 | |
| 432 | -------------------- |
| 433 | z1: | f2 | |
| 434 | -------------------- |
| 435 | bit 31 bit 0 |
| 436 | |
| 437 | Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd |
| 438 | or even of logb(x) have the following relations: |
| 439 | |
| 440 | ------------------------------------------------- |
| 441 | bit 27,26 of z1 bit 1,0 of x1 logb(x) |
| 442 | ------------------------------------------------- |
| 443 | 00 00 odd and even |
| 444 | 01 01 even |
| 445 | 10 10 odd |
| 446 | 10 00 even |
| 447 | 11 01 even |
| 448 | ------------------------------------------------- |
| 449 | |
| 450 | (4) Special cases (see (4) of Section A). |
| 451 | |
| 452 | */ |
| 453 | |
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