src/math/sqrt.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_sqrt.c */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   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
   9  * is preserved.
  10  * ====================================================
  11  */
  12 /* sqrt(x)
  13  * Return correctly rounded sqrt.
  14  *           ------------------------------------------
  15  *           |  Use the hardware sqrt if you have one |
  16  *           ------------------------------------------
  17  * Method:
  18  *   Bit by bit method using integer arithmetic. (Slow, but portable)
  19  *   1. Normalization
  20  *      Scale x to y in [1,4) with even powers of 2:
  21  *      find an integer k such that  1 <= (y=x*2^(2k)) < 4, then
  22  *              sqrt(x) = 2^k * sqrt(y)
  23  *   2. Bit by bit computation
  24  *      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
  25  *           i                                                   0
  26  *                                     i+1         2
  27  *          s  = 2*q , and      y  =  2   * ( y - q  ).         (1)
  28  *           i      i            i                 i
  29  *
  30  *      To compute q    from q , one checks whether
  31  *                  i+1       i
  32  *
  33  *                            -(i+1) 2
  34  *                      (q + 2      ) <= y.                     (2)
  35  *                        i
  36  *                                                            -(i+1)
  37  *      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
  38  *                             i+1   i             i+1   i
  39  *
  40  *      With some algebric manipulation, it is not difficult to see
  41  *      that (2) is equivalent to
  42  *                             -(i+1)
  43  *                      s  +  2       <= y                      (3)
  44  *                       i                i
  45  *
  46  *      The advantage of (3) is that s  and y  can be computed by
  47  *                                    i      i
  48  *      the following recurrence formula:
  49  *          if (3) is false
  50  *
  51  *          s     =  s  ,       y    = y   ;                    (4)
  52  *           i+1      i          i+1    i
  53  *
  54  *          otherwise,
  55  *                         -i                     -(i+1)
  56  *          s     =  s  + 2  ,  y    = y  -  s  - 2             (5)
  57  *           i+1      i          i+1    i     i
  58  *
  59  *      One may easily use induction to prove (4) and (5).
  60  *      Note. Since the left hand side of (3) contain only i+2 bits,
  61  *            it does not necessary to do a full (53-bit) comparison
  62  *            in (3).
  63  *   3. Final rounding
  64  *      After generating the 53 bits result, we compute one more bit.
  65  *      Together with the remainder, we can decide whether the
  66  *      result is exact, bigger than 1/2ulp, or less than 1/2ulp
  67  *      (it will never equal to 1/2ulp).
  68  *      The rounding mode can be detected by checking whether
  69  *      huge + tiny is equal to huge, and whether huge - tiny is
  70  *      equal to huge for some floating point number "huge" and "tiny".
  71  *
  72  * Special cases:
  73  *      sqrt(+-0) = +-0         ... exact
  74  *      sqrt(inf) = inf
  75  *      sqrt(-ve) = NaN         ... with invalid signal
  76  *      sqrt(NaN) = NaN         ... with invalid signal for signaling NaN
  77  */
  78
  79 #include "libm.h"
  80
  81 static const double tiny = 1.0e-300;
  82
  83 double sqrt(double x)
  84 {
  85         double z;
  86         int32_t sign = (int)0x80000000;
  87         int32_t ix0,s0,q,m,t,i;
  88         uint32_t r,t1,s1,ix1,q1;
  89
  90         EXTRACT_WORDS(ix0, ix1, x);
  91
  92         /* take care of Inf and NaN */
  93         if ((ix0&0x7ff00000) == 0x7ff00000) {
  94                 return x*x + x;  /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */
  95         }
  96         /* take care of zero */
  97         if (ix0 <= 0) {
  98                 if (((ix0&~sign)|ix1) == 0)
  99                         return x;  /* sqrt(+-0) = +-0 */
 100                 if (ix0 < 0)
 101                         return (x-x)/(x-x);  /* sqrt(-ve) = sNaN */
 102         }
 103         /* normalize x */
 104         m = ix0>>20;
 105         if (m == 0) {  /* subnormal x */
 106                 while (ix0 == 0) {
 107                         m -= 21;
 108                         ix0 |= (ix1>>11);
 109                         ix1 <<= 21;
 110                 }
 111                 for (i=0; (ix0&0x00100000) == 0; i++)
 112                         ix0<<=1;
 113                 m -= i - 1;
 114                 ix0 |= ix1>>(32-i);
 115                 ix1 <<= i;
 116         }
 117         m -= 1023;    /* unbias exponent */
 118         ix0 = (ix0&0x000fffff)|0x00100000;
 119         if (m & 1) {  /* odd m, double x to make it even */
 120                 ix0 += ix0 + ((ix1&sign)>>31);
 121                 ix1 += ix1;
 122         }
 123         m >>= 1;      /* m = [m/2] */
 124
 125         /* generate sqrt(x) bit by bit */
 126         ix0 += ix0 + ((ix1&sign)>>31);
 127         ix1 += ix1;
 128         q = q1 = s0 = s1 = 0;  /* [q,q1] = sqrt(x) */
 129         r = 0x00200000;        /* r = moving bit from right to left */
 130
 131         while (r != 0) {
 132                 t = s0 + r;
 133                 if (t <= ix0) {
 134                         s0   = t + r;
 135                         ix0 -= t;
 136                         q   += r;
 137                 }
 138                 ix0 += ix0 + ((ix1&sign)>>31);
 139                 ix1 += ix1;
 140                 r >>= 1;
 141         }
 142
 143         r = sign;
 144         while (r != 0) {
 145                 t1 = s1 + r;
 146                 t  = s0;
 147                 if (t < ix0 || (t == ix0 && t1 <= ix1)) {
 148                         s1 = t1 + r;
 149                         if ((t1&sign) == sign && (s1&sign) == 0)
 150                                 s0++;
 151                         ix0 -= t;
 152                         if (ix1 < t1)
 153                                 ix0--;
 154                         ix1 -= t1;
 155                         q1 += r;
 156                 }
 157                 ix0 += ix0 + ((ix1&sign)>>31);
 158                 ix1 += ix1;
 159                 r >>= 1;
 160         }
 161
 162         /* use floating add to find out rounding direction */
 163         if ((ix0|ix1) != 0) {
 164                 z = 1.0 - tiny; /* raise inexact flag */
 165                 if (z >= 1.0) {
 166                         z = 1.0 + tiny;
 167                         if (q1 == (uint32_t)0xffffffff) {
 168                                 q1 = 0;
 169                                 q++;
 170                         } else if (z > 1.0) {
 171                                 if (q1 == (uint32_t)0xfffffffe)
 172                                         q++;
 173                                 q1 += 2;
 174                         } else
 175                                 q1 += q1 & 1;
 176                 }
 177         }
 178         ix0 = (q>>1) + 0x3fe00000;
 179         ix1 = q1>>1;
 180         if (q&1)
 181                 ix1 |= sign;
 182         ix0 += m << 20;
 183         INSERT_WORDS(z, ix0, ix1);
 184         return z;
 185 }