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Annotation of src/lib/libm/src/e_sqrt.c, Revision 1.1.1.1

1.1       jtc         1:
                      2: /* @(#)e_sqrt.c 5.1 93/09/24 */
                      3: /*
                      4:  * ====================================================
                      5:  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
                      6:  *
                      7:  * Developed at SunPro, a Sun Microsystems, Inc. business.
                      8:  * Permission to use, copy, modify, and distribute this
                      9:  * software is freely granted, provided that this notice
                     10:  * is preserved.
                     11:  * ====================================================
                     12:  */
                     13:
                     14: /* __ieee754_sqrt(x)
                     15:  * Return correctly rounded sqrt.
                     16:  *           ------------------------------------------
                     17:  *          |  Use the hardware sqrt if you have one |
                     18:  *           ------------------------------------------
                     19:  * Method:
                     20:  *   Bit by bit method using integer arithmetic. (Slow, but portable)
                     21:  *   1. Normalization
                     22:  *     Scale x to y in [1,4) with even powers of 2:
                     23:  *     find an integer k such that  1 <= (y=x*2^(2k)) < 4, then
                     24:  *             sqrt(x) = 2^k * sqrt(y)
                     25:  *   2. Bit by bit computation
                     26:  *     Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
                     27:  *          i                                                   0
                     28:  *                                     i+1         2
                     29:  *         s  = 2*q , and      y  =  2   * ( y - q  ).         (1)
                     30:  *          i      i            i                 i
                     31:  *
                     32:  *     To compute q    from q , one checks whether
                     33:  *                 i+1       i
                     34:  *
                     35:  *                           -(i+1) 2
                     36:  *                     (q + 2      ) <= y.                     (2)
                     37:  *                               i
                     38:  *                                                           -(i+1)
                     39:  *     If (2) is false, then q   = q ; otherwise q   = q  + 2      .
                     40:  *                            i+1   i             i+1   i
                     41:  *
                     42:  *     With some algebric manipulation, it is not difficult to see
                     43:  *     that (2) is equivalent to
                     44:  *                             -(i+1)
                     45:  *                     s  +  2       <= y                      (3)
                     46:  *                      i                i
                     47:  *
                     48:  *     The advantage of (3) is that s  and y  can be computed by
                     49:  *                                   i      i
                     50:  *     the following recurrence formula:
                     51:  *         if (3) is false
                     52:  *
                     53:  *         s     =  s  ,       y    = y   ;                    (4)
                     54:  *          i+1      i          i+1    i
                     55:  *
                     56:  *         otherwise,
                     57:  *                         -i                     -(i+1)
                     58:  *         s     =  s  + 2  ,  y    = y  -  s  - 2             (5)
                     59:  *           i+1      i          i+1    i     i
                     60:  *
                     61:  *     One may easily use induction to prove (4) and (5).
                     62:  *     Note. Since the left hand side of (3) contain only i+2 bits,
                     63:  *           it does not necessary to do a full (53-bit) comparison
                     64:  *           in (3).
                     65:  *   3. Final rounding
                     66:  *     After generating the 53 bits result, we compute one more bit.
                     67:  *     Together with the remainder, we can decide whether the
                     68:  *     result is exact, bigger than 1/2ulp, or less than 1/2ulp
                     69:  *     (it will never equal to 1/2ulp).
                     70:  *     The rounding mode can be detected by checking whether
                     71:  *     huge + tiny is equal to huge, and whether huge - tiny is
                     72:  *     equal to huge for some floating point number "huge" and "tiny".
                     73:  *
                     74:  * Special cases:
                     75:  *     sqrt(+-0) = +-0         ... exact
                     76:  *     sqrt(inf) = inf
                     77:  *     sqrt(-ve) = NaN         ... with invalid signal
                     78:  *     sqrt(NaN) = NaN         ... with invalid signal for signaling NaN
                     79:  *
                     80:  * Other methods : see the appended file at the end of the program below.
                     81:  *---------------
                     82:  */
                     83:
                     84: #include "fdlibm.h"
                     85:
                     86: #ifdef __STDC__
                     87: static const double    one     = 1.0, tiny=1.0e-300;
                     88: #else
                     89: static double  one     = 1.0, tiny=1.0e-300;
                     90: #endif
                     91:
                     92: #ifdef __STDC__
                     93:        double __ieee754_sqrt(double x)
                     94: #else
                     95:        double __ieee754_sqrt(x)
                     96:        double x;
                     97: #endif
                     98: {
                     99:        int     n0;
                    100:
                    101:        double z;
                    102:        int     sign = (int)0x80000000;
                    103:        unsigned r,t1,s1,ix1,q1;
                    104:        int ix0,s0,q,m,t,i;
                    105:
                    106:        n0 = ((*(int*)&one)>>29)^1;             /* index of high word */
                    107:        ix0 = *(n0+(int*)&x);                   /* high word of x */
                    108:        ix1 = *((1-n0)+(int*)&x);               /* low word of 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==(unsigned)0xffffffff) { q1=0; q += 1;}
                    182:                else if (z>one) {
                    183:                    if (q1==(unsigned)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:        *(n0+(int*)&z) = ix0;
                    194:        *((1-n0)+(int*)&z) = ix1;
                    195:        return z;
                    196: }
                    197:
                    198: /*
                    199: Other methods  (use floating-point arithmetic)
                    200: -------------
                    201: (This is a copy of a drafted paper by Prof W. Kahan
                    202: and K.C. Ng, written in May, 1986)
                    203:
                    204:        Two algorithms are given here to implement sqrt(x)
                    205:        (IEEE double precision arithmetic) in software.
                    206:        Both supply sqrt(x) correctly rounded. The first algorithm (in
                    207:        Section A) uses newton iterations and involves four divisions.
                    208:        The second one uses reciproot iterations to avoid division, but
                    209:        requires more multiplications. Both algorithms need the ability
                    210:        to chop results of arithmetic operations instead of round them,
                    211:        and the INEXACT flag to indicate when an arithmetic operation
                    212:        is executed exactly with no roundoff error, all part of the
                    213:        standard (IEEE 754-1985). The ability to perform shift, add,
                    214:        subtract and logical AND operations upon 32-bit words is needed
                    215:        too, though not part of the standard.
                    216:
                    217: A.  sqrt(x) by Newton Iteration
                    218:
                    219:    (1) Initial approximation
                    220:
                    221:        Let x0 and x1 be the leading and the trailing 32-bit words of
                    222:        a floating point number x (in IEEE double format) respectively
                    223:
                    224:            1    11                  52                           ...widths
                    225:           ------------------------------------------------------
                    226:        x: |s|    e     |             f                         |
                    227:           ------------------------------------------------------
                    228:              msb    lsb  msb                                 lsb ...order
                    229:
                    230:
                    231:             ------------------------        ------------------------
                    232:        x0:  |s|   e    |    f1     |    x1: |          f2           |
                    233:             ------------------------        ------------------------
                    234:
                    235:        By performing shifts and subtracts on x0 and x1 (both regarded
                    236:        as integers), we obtain an 8-bit approximation of sqrt(x) as
                    237:        follows.
                    238:
                    239:                k  := (x0>>1) + 0x1ff80000;
                    240:                y0 := k - T1[31&(k>>15)].       ... y ~ sqrt(x) to 8 bits
                    241:        Here k is a 32-bit integer and T1[] is an integer array containing
                    242:        correction terms. Now magically the floating value of y (y's
                    243:        leading 32-bit word is y0, the value of its trailing word is 0)
                    244:        approximates sqrt(x) to almost 8-bit.
                    245:
                    246:        Value of T1:
                    247:        static int T1[32]= {
                    248:        0,      1024,   3062,   5746,   9193,   13348,  18162,  23592,
                    249:        29598,  36145,  43202,  50740,  58733,  67158,  75992,  85215,
                    250:        83599,  71378,  60428,  50647,  41945,  34246,  27478,  21581,
                    251:        16499,  12183,  8588,   5674,   3403,   1742,   661,    130,};
                    252:
                    253:     (2)        Iterative refinement
                    254:
                    255:        Apply Heron's rule three times to y, we have y approximates
                    256:        sqrt(x) to within 1 ulp (Unit in the Last Place):
                    257:
                    258:                y := (y+x/y)/2          ... almost 17 sig. bits
                    259:                y := (y+x/y)/2          ... almost 35 sig. bits
                    260:                y := y-(y-x/y)/2        ... within 1 ulp
                    261:
                    262:
                    263:        Remark 1.
                    264:            Another way to improve y to within 1 ulp is:
                    265:
                    266:                y := (y+x/y)            ... almost 17 sig. bits to 2*sqrt(x)
                    267:                y := y - 0x00100006     ... almost 18 sig. bits to sqrt(x)
                    268:
                    269:                                2
                    270:                            (x-y )*y
                    271:                y := y + 2* ----------  ...within 1 ulp
                    272:                               2
                    273:                             3y  + x
                    274:
                    275:
                    276:        This formula has one division fewer than the one above; however,
                    277:        it requires more multiplications and additions. Also x must be
                    278:        scaled in advance to avoid spurious overflow in evaluating the
                    279:        expression 3y*y+x. Hence it is not recommended uless division
                    280:        is slow. If division is very slow, then one should use the
                    281:        reciproot algorithm given in section B.
                    282:
                    283:     (3) Final adjustment
                    284:
                    285:        By twiddling y's last bit it is possible to force y to be
                    286:        correctly rounded according to the prevailing rounding mode
                    287:        as follows. Let r and i be copies of the rounding mode and
                    288:        inexact flag before entering the square root program. Also we
                    289:        use the expression y+-ulp for the next representable floating
                    290:        numbers (up and down) of y. Note that y+-ulp = either fixed
                    291:        point y+-1, or multiply y by nextafter(1,+-inf) in chopped
                    292:        mode.
                    293:
                    294:                I := FALSE;     ... reset INEXACT flag I
                    295:                R := RZ;        ... set rounding mode to round-toward-zero
                    296:                z := x/y;       ... chopped quotient, possibly inexact
                    297:                If(not I) then {        ... if the quotient is exact
                    298:                    if(z=y) {
                    299:                        I := i;  ... restore inexact flag
                    300:                        R := r;  ... restore rounded mode
                    301:                        return sqrt(x):=y.
                    302:                    } else {
                    303:                        z := z - ulp;   ... special rounding
                    304:                    }
                    305:                }
                    306:                i := TRUE;              ... sqrt(x) is inexact
                    307:                If (r=RN) then z=z+ulp  ... rounded-to-nearest
                    308:                If (r=RP) then {        ... round-toward-+inf
                    309:                    y = y+ulp; z=z+ulp;
                    310:                }
                    311:                y := y+z;               ... chopped sum
                    312:                y0:=y0-0x00100000;      ... y := y/2 is correctly rounded.
                    313:                I := i;                 ... restore inexact flag
                    314:                R := r;                 ... restore rounded mode
                    315:                return sqrt(x):=y.
                    316:
                    317:     (4)        Special cases
                    318:
                    319:        Square root of +inf, +-0, or NaN is itself;
                    320:        Square root of a negative number is NaN with invalid signal.
                    321:
                    322:
                    323: B.  sqrt(x) by Reciproot Iteration
                    324:
                    325:    (1) Initial approximation
                    326:
                    327:        Let x0 and x1 be the leading and the trailing 32-bit words of
                    328:        a floating point number x (in IEEE double format) respectively
                    329:        (see section A). By performing shifs and subtracts on x0 and y0,
                    330:        we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
                    331:
                    332:            k := 0x5fe80000 - (x0>>1);
                    333:            y0:= k - T2[63&(k>>14)].    ... y ~ 1/sqrt(x) to 7.8 bits
                    334:
                    335:        Here k is a 32-bit integer and T2[] is an integer array
                    336:        containing correction terms. Now magically the floating
                    337:        value of y (y's leading 32-bit word is y0, the value of
                    338:        its trailing word y1 is set to zero) approximates 1/sqrt(x)
                    339:        to almost 7.8-bit.
                    340:
                    341:        Value of T2:
                    342:        static int T2[64]= {
                    343:        0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
                    344:        0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
                    345:        0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
                    346:        0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
                    347:        0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
                    348:        0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
                    349:        0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
                    350:        0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
                    351:
                    352:     (2)        Iterative refinement
                    353:
                    354:        Apply Reciproot iteration three times to y and multiply the
                    355:        result by x to get an approximation z that matches sqrt(x)
                    356:        to about 1 ulp. To be exact, we will have
                    357:                -1ulp < sqrt(x)-z<1.0625ulp.
                    358:
                    359:        ... set rounding mode to Round-to-nearest
                    360:           y := y*(1.5-0.5*x*y*y)       ... almost 15 sig. bits to 1/sqrt(x)
                    361:           y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
                    362:        ... special arrangement for better accuracy
                    363:           z := x*y                     ... 29 bits to sqrt(x), with z*y<1
                    364:           z := z + 0.5*z*(1-z*y)       ... about 1 ulp to sqrt(x)
                    365:
                    366:        Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
                    367:        (a) the term z*y in the final iteration is always less than 1;
                    368:        (b) the error in the final result is biased upward so that
                    369:                -1 ulp < sqrt(x) - z < 1.0625 ulp
                    370:            instead of |sqrt(x)-z|<1.03125ulp.
                    371:
                    372:     (3)        Final adjustment
                    373:
                    374:        By twiddling y's last bit it is possible to force y to be
                    375:        correctly rounded according to the prevailing rounding mode
                    376:        as follows. Let r and i be copies of the rounding mode and
                    377:        inexact flag before entering the square root program. Also we
                    378:        use the expression y+-ulp for the next representable floating
                    379:        numbers (up and down) of y. Note that y+-ulp = either fixed
                    380:        point y+-1, or multiply y by nextafter(1,+-inf) in chopped
                    381:        mode.
                    382:
                    383:        R := RZ;                ... set rounding mode to round-toward-zero
                    384:        switch(r) {
                    385:            case RN:            ... round-to-nearest
                    386:               if(x<= z*(z-ulp)...chopped) z = z - ulp; else
                    387:               if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
                    388:               break;
                    389:            case RZ:case RM:    ... round-to-zero or round-to--inf
                    390:               R:=RP;           ... reset rounding mod to round-to-+inf
                    391:               if(x<z*z ... rounded up) z = z - ulp; else
                    392:               if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
                    393:               break;
                    394:            case RP:            ... round-to-+inf
                    395:               if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
                    396:               if(x>z*z ...chopped) z = z+ulp;
                    397:               break;
                    398:        }
                    399:
                    400:        Remark 3. The above comparisons can be done in fixed point. For
                    401:        example, to compare x and w=z*z chopped, it suffices to compare
                    402:        x1 and w1 (the trailing parts of x and w), regarding them as
                    403:        two's complement integers.
                    404:
                    405:        ...Is z an exact square root?
                    406:        To determine whether z is an exact square root of x, let z1 be the
                    407:        trailing part of z, and also let x0 and x1 be the leading and
                    408:        trailing parts of x.
                    409:
                    410:        If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
                    411:            I := 1;             ... Raise Inexact flag: z is not exact
                    412:        else {
                    413:            j := 1 - [(x0>>20)&1]       ... j = logb(x) mod 2
                    414:            k := z1 >> 26;              ... get z's 25-th and 26-th
                    415:                                            fraction bits
                    416:            I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
                    417:        }
                    418:        R:= r           ... restore rounded mode
                    419:        return sqrt(x):=z.
                    420:
                    421:        If multiplication is cheaper then the foregoing red tape, the
                    422:        Inexact flag can be evaluated by
                    423:
                    424:            I := i;
                    425:            I := (z*z!=x) or I.
                    426:
                    427:        Note that z*z can overwrite I; this value must be sensed if it is
                    428:        True.
                    429:
                    430:        Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
                    431:        zero.
                    432:
                    433:                    --------------------
                    434:                z1: |        f2        |
                    435:                    --------------------
                    436:                bit 31             bit 0
                    437:
                    438:        Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
                    439:        or even of logb(x) have the following relations:
                    440:
                    441:        -------------------------------------------------
                    442:        bit 27,26 of z1         bit 1,0 of x1   logb(x)
                    443:        -------------------------------------------------
                    444:        00                      00              odd and even
                    445:        01                      01              even
                    446:        10                      10              odd
                    447:        10                      00              even
                    448:        11                      01              even
                    449:        -------------------------------------------------
                    450:
                    451:     (4)        Special cases (see (4) of Section A).
                    452:
                    453:  */
                    454:

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