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      1 /* __gmpfr_isqrt && __gmpfr_cuberoot -- Integer square root and cube root
      2 
      3 Copyright 2004-2023 Free Software Foundation, Inc.
      4 Contributed by the AriC and Caramba projects, INRIA.
      5 
      6 This file is part of the GNU MPFR Library.
      7 
      8 The GNU MPFR Library is free software; you can redistribute it and/or modify
      9 it under the terms of the GNU Lesser General Public License as published by
     10 the Free Software Foundation; either version 3 of the License, or (at your
     11 option) any later version.
     12 
     13 The GNU MPFR Library is distributed in the hope that it will be useful, but
     14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
     15 or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
     16 License for more details.
     17 
     18 You should have received a copy of the GNU Lesser General Public License
     19 along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
     20 https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
     21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
     22 
     23 #include "mpfr-impl.h"
     24 
     25 /* returns floor(sqrt(n)) */
     26 unsigned long
     27 __gmpfr_isqrt (unsigned long n)
     28 {
     29   unsigned long i, s;
     30 
     31   /* First find an approximation to floor(sqrt(n)) of the form 2^k. */
     32   i = n;
     33   s = 1;
     34   while (i >= 2)
     35     {
     36       i >>= 2;
     37       s <<= 1;
     38     }
     39 
     40   do
     41     {
     42       s = (s + n / s) / 2;
     43     }
     44   while (!(s*s <= n && (s*s > s*(s+2) || n <= s*(s+2))));
     45   /* Short explanation: As mathematically s*(s+2) < 2*ULONG_MAX,
     46      the condition s*s > s*(s+2) is evaluated as true when s*(s+2)
     47      "overflows" but not s*s. This implies that mathematically, one
     48      has s*s <= n <= s*(s+2). If s*s "overflows", this means that n
     49      is "large" and the inequality n <= s*(s+2) cannot be satisfied. */
     50   return s;
     51 }
     52 
     53 /* returns floor(n^(1/3)) */
     54 unsigned long
     55 __gmpfr_cuberoot (unsigned long n)
     56 {
     57   unsigned long i, s;
     58 
     59   /* First find an approximation to floor(cbrt(n)) of the form 2^k. */
     60   i = n;
     61   s = 1;
     62   while (i >= 4)
     63     {
     64       i >>= 3;
     65       s <<= 1;
     66     }
     67 
     68   /* Improve the approximation (this is necessary if n is large, so that
     69      mathematically (s+1)*(s+1)*(s+1) isn't much larger than ULONG_MAX). */
     70   if (n >= 256)
     71     {
     72       s = (2 * s + n / (s * s)) / 3;
     73       s = (2 * s + n / (s * s)) / 3;
     74       s = (2 * s + n / (s * s)) / 3;
     75     }
     76 
     77   do
     78     {
     79       s = (2 * s + n / (s * s)) / 3;
     80     }
     81   while (!(s*s*s <= n && (s*s*s > (s+1)*(s+1)*(s+1) ||
     82                           n < (s+1)*(s+1)*(s+1))));
     83   return s;
     84 }
     85