| t2ex/bsd_source/lib/libc/src_bsd/math/e_sqrt.c | bare source | permlink (0.01 seconds) |
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: /* sqrt(x) 14: * Return correctly rounded sqrt. 15: * ------------------------------------------ 16: * | Use the hardware sqrt if you have one | 17: * ------------------------------------------ 18: * Method: 19: * Bit by bit method using integer arithmetic. (Slow, but portable) 20: * 1. Normalization 21: * Scale x to y in [1,4) with even powers of 2: 22: * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then 23: * sqrt(x) = 2^k * sqrt(y) 24: * 2. Bit by bit computation 25: * Let q = sqrt(y) truncated to i bit after binary point (q = 1), 26: * i 0 27: * i+1 2 28: * s = 2*q , and y = 2 * ( y - q ). (1) 29: * i i i i 30: * 31: * To compute q from q , one checks whether 32: * i+1 i 33: * 34: * -(i+1) 2 35: * (q + 2 ) <= y. (2) 36: * i 37: * -(i+1) 38: * If (2) is false, then q = q ; otherwise q = q + 2 . 39: * i+1 i i+1 i 40: * 41: * With some algebric manipulation, it is not difficult to see 42: * that (2) is equivalent to 43: * -(i+1) 44: * s + 2 <= y (3) 45: * i i 46: * 47: * The advantage of (3) is that s and y can be computed by 48: * i i 49: * the following recurrence formula: 50: * if (3) is false 51: * 52: * s = s , y = y ; (4) 53: * i+1 i i+1 i 54: * 55: * otherwise, 56: * -i -(i+1) 57: * s = s + 2 , y = y - s - 2 (5) 58: * i+1 i i+1 i i 59: * 60: * One may easily use induction to prove (4) and (5). 61: * Note. Since the left hand side of (3) contain only i+2 bits, 62: * it does not necessary to do a full (53-bit) comparison 63: * in (3). 64: * 3. Final rounding 65: * After generating the 53 bits result, we compute one more bit. 66: * Together with the remainder, we can decide whether the 67: * result is exact, bigger than 1/2ulp, or less than 1/2ulp 68: * (it will never equal to 1/2ulp). 69: * The rounding mode can be detected by checking whether 70: * huge + tiny is equal to huge, and whether huge - tiny is 71: * equal to huge for some floating point number "huge" and "tiny". 72: * 73: * Special cases: 74: * sqrt(+-0) = +-0 ... exact 75: * sqrt(inf) = inf 76: * sqrt(-ve) = NaN ... with invalid signal 77: * sqrt(NaN) = NaN ... with invalid signal for signaling NaN 78: * 79: * Other methods : see the appended file at the end of the program below. 80: *--------------- 81: */ 82: 83: /* LINTLIBRARY */ 84: 85: #include <sys/cdefs.h> 86: #include <float.h> 87: #include <math.h> 88: 89: #include "math_private.h" 90: 91: static const double one = 1.0, tiny=1.0e-300; 92: 93: double 94: sqrt(double x) 95: { 96: double z; 97: int32_t sign = (int)0x80000000; 98: int32_t ix0,s0,q,m,t,i; 99: u_int32_t r,t1,s1,ix1,q1; 100: 101: EXTRACT_WORDS(ix0,ix1,x); 102: 103: /* take care of Inf and NaN */ 104: if((ix0&0x7ff00000)==0x7ff00000) { 105: return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf 106: sqrt(-inf)=sNaN */ 107: } 108: /* take care of zero */ 109: if(ix0<=0) { 110: if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */ 111: else if(ix0<0) 112: return (x-x)/(x-x); /* sqrt(-ve) = sNaN */ 113: } 114: /* normalize x */ 115: m = (ix0>>20); 116: if(m==0) { /* subnormal x */ 117: while(ix0==0) { 118: m -= 21; 119: ix0 |= (ix1>>11); ix1 <<= 21; 120: } 121: for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1; 122: m -= i-1; 123: ix0 |= (ix1>>(32-i)); 124: ix1 <<= i; 125: } 126: m -= 1023; /* unbias exponent */ 127: ix0 = (ix0&0x000fffff)|0x00100000; 128: if(m&1){ /* odd m, double x to make it even */ 129: ix0 += ix0 + ((ix1&sign)>>31); 130: ix1 += ix1; 131: } 132: m >>= 1; /* m = [m/2] */ 133: 134: /* generate sqrt(x) bit by bit */ 135: ix0 += ix0 + ((ix1&sign)>>31); 136: ix1 += ix1; 137: q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */ 138: r = 0x00200000; /* r = moving bit from right to left */ 139: 140: while(r!=0) { 141: t = s0+r; 142: if(t<=ix0) { 143: s0 = t+r; 144: ix0 -= t; 145: q += r; 146: } 147: ix0 += ix0 + ((ix1&sign)>>31); 148: ix1 += ix1; 149: r>>=1; 150: } 151: 152: r = sign; 153: while(r!=0) { 154: t1 = s1+r; 155: t = s0; 156: if((t<ix0)||((t==ix0)&&(t1<=ix1))) { 157: s1 = t1+r; 158: if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1; 159: ix0 -= t; 160: if (ix1 < t1) ix0 -= 1; 161: ix1 -= t1; 162: q1 += r; 163: } 164: ix0 += ix0 + ((ix1&sign)>>31); 165: ix1 += ix1; 166: r>>=1; 167: } 168: 169: /* use floating add to find out rounding direction */ 170: if((ix0|ix1)!=0) { 171: z = one-tiny; /* trigger inexact flag */ 172: if (z>=one) { 173: z = one+tiny; 174: if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;} 175: else if (z>one) { 176: if (q1==(u_int32_t)0xfffffffe) q+=1; 177: q1+=2; 178: } else 179: q1 += (q1&1); 180: } 181: } 182: ix0 = (q>>1)+0x3fe00000; 183: ix1 = q1>>1; 184: if ((q&1)==1) ix1 |= sign; 185: ix0 += (m <<20); 186: INSERT_WORDS(z,ix0,ix1); 187: return z; 188: } 189: 190: /* 191: Other methods (use floating-point arithmetic) 192: ------------- 193: (This is a copy of a drafted paper by Prof W. Kahan 194: and K.C. Ng, written in May, 1986) 195: 196: Two algorithms are given here to implement sqrt(x) 197: (IEEE double precision arithmetic) in software. 198: Both supply sqrt(x) correctly rounded. The first algorithm (in 199: Section A) uses newton iterations and involves four divisions. 200: The second one uses reciproot iterations to avoid division, but 201: requires more multiplications. Both algorithms need the ability 202: to chop results of arithmetic operations instead of round them, 203: and the INEXACT flag to indicate when an arithmetic operation 204: is executed exactly with no roundoff error, all part of the 205: standard (IEEE 754-1985). The ability to perform shift, add, 206: subtract and logical AND operations upon 32-bit words is needed 207: too, though not part of the standard. 208: 209: A. sqrt(x) by Newton Iteration 210: 211: (1) Initial approximation 212: 213: Let x0 and x1 be the leading and the trailing 32-bit words of 214: a floating point number x (in IEEE double format) respectively 215: 216: 1 11 52 ...widths 217: ------------------------------------------------------ 218: x: |s| e | f | 219: ------------------------------------------------------ 220: msb lsb msb lsb ...order 221: 222: 223: ------------------------ ------------------------ 224: x0: |s| e | f1 | x1: | f2 | 225: ------------------------ ------------------------ 226: 227: By performing shifts and subtracts on x0 and x1 (both regarded 228: as integers), we obtain an 8-bit approximation of sqrt(x) as 229: follows. 230: 231: k := (x0>>1) + 0x1ff80000; 232: y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits 233: Here k is a 32-bit integer and T1[] is an integer array containing 234: correction terms. Now magically the floating value of y (y's 235: leading 32-bit word is y0, the value of its trailing word is 0) 236: approximates sqrt(x) to almost 8-bit. 237: 238: Value of T1: 239: static int T1[32]= { 240: 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592, 241: 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215, 242: 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581, 243: 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,}; 244: 245: (2) Iterative refinement 246: 247: Apply Heron's rule three times to y, we have y approximates 248: sqrt(x) to within 1 ulp (Unit in the Last Place): 249: 250: y := (y+x/y)/2 ... almost 17 sig. bits 251: y := (y+x/y)/2 ... almost 35 sig. bits 252: y := y-(y-x/y)/2 ... within 1 ulp 253: 254: 255: Remark 1. 256: Another way to improve y to within 1 ulp is: 257: 258: y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x) 259: y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x) 260: 261: 2 262: (x-y )*y 263: y := y + 2* ---------- ...within 1 ulp 264: 2 265: 3y + x 266: 267: 268: This formula has one division fewer than the one above; however, 269: it requires more multiplications and additions. Also x must be 270: scaled in advance to avoid spurious overflow in evaluating the 271: expression 3y*y+x. Hence it is not recommended uless division 272: is slow. If division is very slow, then one should use the 273: reciproot algorithm given in section B. 274: 275: (3) Final adjustment 276: 277: By twiddling y's last bit it is possible to force y to be 278: correctly rounded according to the prevailing rounding mode 279: as follows. Let r and i be copies of the rounding mode and 280: inexact flag before entering the square root program. Also we 281: use the expression y+-ulp for the next representable floating 282: numbers (up and down) of y. Note that y+-ulp = either fixed 283: point y+-1, or multiply y by nextafter(1,+-inf) in chopped 284: mode. 285: 286: I := FALSE; ... reset INEXACT flag I 287: R := RZ; ... set rounding mode to round-toward-zero 288: z := x/y; ... chopped quotient, possibly inexact 289: If(not I) then { ... if the quotient is exact 290: if(z=y) { 291: I := i; ... restore inexact flag 292: R := r; ... restore rounded mode 293: return sqrt(x):=y. 294: } else { 295: z := z - ulp; ... special rounding 296: } 297: } 298: i := TRUE; ... sqrt(x) is inexact 299: If (r=RN) then z=z+ulp ... rounded-to-nearest 300: If (r=RP) then { ... round-toward-+inf 301: y = y+ulp; z=z+ulp; 302: } 303: y := y+z; ... chopped sum 304: y0:=y0-0x00100000; ... y := y/2 is correctly rounded. 305: I := i; ... restore inexact flag 306: R := r; ... restore rounded mode 307: return sqrt(x):=y. 308: 309: (4) Special cases 310: 311: Square root of +inf, +-0, or NaN is itself; 312: Square root of a negative number is NaN with invalid signal. 313: 314: 315: B. sqrt(x) by Reciproot Iteration 316: 317: (1) Initial approximation 318: 319: Let x0 and x1 be the leading and the trailing 32-bit words of 320: a floating point number x (in IEEE double format) respectively 321: (see section A). By performing shifs and subtracts on x0 and y0, 322: we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. 323: 324: k := 0x5fe80000 - (x0>>1); 325: y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits 326: 327: Here k is a 32-bit integer and T2[] is an integer array 328: containing correction terms. Now magically the floating 329: value of y (y's leading 32-bit word is y0, the value of 330: its trailing word y1 is set to zero) approximates 1/sqrt(x) 331: to almost 7.8-bit. 332: 333: Value of T2: 334: static int T2[64]= { 335: 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, 336: 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, 337: 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, 338: 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, 339: 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, 340: 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, 341: 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, 342: 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; 343: 344: (2) Iterative refinement 345: 346: Apply Reciproot iteration three times to y and multiply the 347: result by x to get an approximation z that matches sqrt(x) 348: to about 1 ulp. To be exact, we will have 349: -1ulp < sqrt(x)-z<1.0625ulp. 350: 351: ... set rounding mode to Round-to-nearest 352: y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) 353: y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) 354: ... special arrangement for better accuracy 355: z := x*y ... 29 bits to sqrt(x), with z*y<1 356: z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) 357: 358: Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that 359: (a) the term z*y in the final iteration is always less than 1; 360: (b) the error in the final result is biased upward so that 361: -1 ulp < sqrt(x) - z < 1.0625 ulp 362: instead of |sqrt(x)-z|<1.03125ulp. 363: 364: (3) Final adjustment 365: 366: By twiddling y's last bit it is possible to force y to be 367: correctly rounded according to the prevailing rounding mode 368: as follows. Let r and i be copies of the rounding mode and 369: inexact flag before entering the square root program. Also we 370: use the expression y+-ulp for the next representable floating 371: numbers (up and down) of y. Note that y+-ulp = either fixed 372: point y+-1, or multiply y by nextafter(1,+-inf) in chopped 373: mode. 374: 375: R := RZ; ... set rounding mode to round-toward-zero 376: switch(r) { 377: case RN: ... round-to-nearest 378: if(x<= z*(z-ulp)...chopped) z = z - ulp; else 379: if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; 380: break; 381: case RZ:case RM: ... round-to-zero or round-to--inf 382: R:=RP; ... reset rounding mod to round-to-+inf 383: if(x<z*z ... rounded up) z = z - ulp; else 384: if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; 385: break; 386: case RP: ... round-to-+inf 387: if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else 388: if(x>z*z ...chopped) z = z+ulp; 389: break; 390: } 391: 392: Remark 3. The above comparisons can be done in fixed point. For 393: example, to compare x and w=z*z chopped, it suffices to compare 394: x1 and w1 (the trailing parts of x and w), regarding them as 395: two's complement integers. 396: 397: ...Is z an exact square root? 398: To determine whether z is an exact square root of x, let z1 be the 399: trailing part of z, and also let x0 and x1 be the leading and 400: trailing parts of x. 401: 402: If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 403: I := 1; ... Raise Inexact flag: z is not exact 404: else { 405: j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 406: k := z1 >> 26; ... get z's 25-th and 26-th 407: fraction bits 408: I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); 409: } 410: R:= r ... restore rounded mode 411: return sqrt(x):=z. 412: 413: If multiplication is cheaper then the foregoing red tape, the 414: Inexact flag can be evaluated by 415: 416: I := i; 417: I := (z*z!=x) or I. 418: 419: Note that z*z can overwrite I; this value must be sensed if it is 420: True. 421: 422: Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be 423: zero. 424: 425: -------------------- 426: z1: | f2 | 427: -------------------- 428: bit 31 bit 0 429: 430: Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd 431: or even of logb(x) have the following relations: 432: 433: ------------------------------------------------- 434: bit 27,26 of z1 bit 1,0 of x1 logb(x) 435: ------------------------------------------------- 436: 00 00 odd and even 437: 01 01 even 438: 10 10 odd 439: 10 00 even 440: 11 01 even 441: ------------------------------------------------- 442: 443: (4) Special cases (see (4) of Section A). 444: 445: */ 446: 447: #if LDBL_MANT_DIG == 53 448: #ifdef lint 449: /* PROTOLIB1 */ 450: long double sqrtl(long double); 451: #else /* lint */ 452: __weak_alias(sqrtl, sqrt); 453: #endif /* lint */ 454: #endif /* LDBL_MANT_DIG == 53 */