1  /* mpf_sqrt -- Compute the square root of a float.
       2  
       3  Copyright 1993, 1994, 1996, 2000, 2001, 2004, 2005, 2012 Free Software
       4  Foundation, Inc.
       5  
       6  This file is part of the GNU MP Library.
       7  
       8  The GNU MP Library is free software; you can redistribute it and/or modify
       9  it under the terms of either:
      10  
      11    * the GNU Lesser General Public License as published by the Free
      12      Software Foundation; either version 3 of the License, or (at your
      13      option) any later version.
      14  
      15  or
      16  
      17    * the GNU General Public License as published by the Free Software
      18      Foundation; either version 2 of the License, or (at your option) any
      19      later version.
      20  
      21  or both in parallel, as here.
      22  
      23  The GNU MP Library is distributed in the hope that it will be useful, but
      24  WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
      25  or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
      26  for more details.
      27  
      28  You should have received copies of the GNU General Public License and the
      29  GNU Lesser General Public License along with the GNU MP Library.  If not,
      30  see https://www.gnu.org/licenses/.  */
      31  
      32  #include <stdio.h> /* for NULL */
      33  #include "gmp-impl.h"
      34  
      35  
      36  /* As usual, the aim is to produce PREC(r) limbs of result, with the high
      37     limb non-zero.  This is accomplished by applying mpn_sqrtrem to either
      38     2*prec or 2*prec-1 limbs, both such sizes resulting in prec limbs.
      39  
      40     The choice between 2*prec or 2*prec-1 limbs is based on the input
      41     exponent.  With b=2^GMP_NUMB_BITS the limb base then we can think of
      42     effectively taking out a factor b^(2k), for suitable k, to get to an
      43     integer input of the desired size ready for mpn_sqrtrem.  It must be an
      44     even power taken out, ie. an even number of limbs, so the square root
      45     gives factor b^k and the radix point is still on a limb boundary.  So if
      46     EXP(r) is even we'll get an even number of input limbs 2*prec, or if
      47     EXP(r) is odd we get an odd number 2*prec-1.
      48  
      49     Further limbs below the 2*prec or 2*prec-1 used don't affect the result
      50     and are simply truncated.  This can be seen by considering an integer x,
      51     with s=floor(sqrt(x)).  s is the unique integer satisfying s^2 <= x <
      52     (s+1)^2.  Notice that adding a fraction part to x (ie. some further bits)
      53     doesn't change the inequality, s remains the unique solution.  Working
      54     suitable factors of 2 into this argument lets it apply to an intended
      55     precision at any position for any x, not just the integer binary point.
      56  
      57     If the input is smaller than 2*prec or 2*prec-1, then we just pad with
      58     zeros, that of course being our usual interpretation of short inputs.
      59     The effect is to extend the root beyond the size of the input (for
      60     instance into fractional limbs if u is an integer).  */
      61  
      62  void
      63  mpf_sqrt (mpf_ptr r, mpf_srcptr u)
      64  {
      65    mp_size_t usize;
      66    mp_ptr up, tp;
      67    mp_size_t prec, tsize;
      68    mp_exp_t uexp, expodd;
      69    TMP_DECL;
      70  
      71    usize = u->_mp_size;
      72    if (UNLIKELY (usize <= 0))
      73      {
      74        if (usize < 0)
      75          SQRT_OF_NEGATIVE;
      76        r->_mp_size = 0;
      77        r->_mp_exp = 0;
      78        return;
      79      }
      80  
      81    TMP_MARK;
      82  
      83    uexp = u->_mp_exp;
      84    prec = r->_mp_prec;
      85    up = u->_mp_d;
      86  
      87    expodd = (uexp & 1);
      88    tsize = 2 * prec - expodd;
      89    r->_mp_size = prec;
      90    r->_mp_exp = (uexp + expodd) / 2;    /* ceil(uexp/2) */
      91  
      92    /* root size is ceil(tsize/2), this will be our desired "prec" limbs */
      93    ASSERT ((tsize + 1) / 2 == prec);
      94  
      95    tp = TMP_ALLOC_LIMBS (tsize);
      96  
      97    if (usize > tsize)
      98      {
      99        up += usize - tsize;
     100        usize = tsize;
     101        MPN_COPY (tp, up, tsize);
     102      }
     103    else
     104      {
     105        MPN_ZERO (tp, tsize - usize);
     106        MPN_COPY (tp + (tsize - usize), up, usize);
     107      }
     108  
     109    mpn_sqrtrem (r->_mp_d, NULL, tp, tsize);
     110  
     111    TMP_FREE;
     112  }