sysdeps/ieee754/dbl-64/mpsqrt.c

   1
   2 /*
   3  * IBM Accurate Mathematical Library
   4  * Copyright (c) International Business Machines Corp., 2001
   5  *
   6  * This program is free software; you can redistribute it and/or modify
   7  * it under the terms of the GNU Lesser General Public License as published by
   8  * the Free Software Foundation; either version 2 of the License, or
   9  * (at your option) any later version.
  10  *
  11  * This program is distributed in the hope that it will be useful,
  12  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  13  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  14  * GNU General Public License for more details.
  15  *
  16  * You should have received a copy of the GNU Lesser General Public License
  17  * along with this program; if not, write to the Free Software
  18  * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
  19  */
  20 /****************************************************************************/
  21 /*  MODULE_NAME:mpsqrt.c                                                    */
  22 /*                                                                          */
  23 /*  FUNCTION:mpsqrt                                                         */
  24 /*           fastiroot                                                      */
  25 /*                                                                          */
  26 /* FILES NEEDED:endian.h mpa.h mpsqrt.h                                     */
  27 /*              mpa.c                                                       */
  28 /* Multi-Precision square root function subroutine for precision p >= 4.    */
  29 /* The relative error is bounded by 3.501*r**(1-p), where r=2**24.          */
  30 /*                                                                          */
  31 /****************************************************************************/
  32 #include "endian.h"
  33 #include "mpa.h"
  34
  35 /****************************************************************************/
  36 /* Multi-Precision square root function subroutine for precision p >= 4.    */
  37 /* The relative error is bounded by 3.501*r**(1-p), where r=2**24.          */
  38 /* Routine receives two pointers to  Multi Precision numbers:               */
  39 /* x (left argument) and y (next argument). Routine also receives precision */
  40 /* p as integer. Routine computes sqrt(*x) and stores result in *y          */
  41 /****************************************************************************/
  42
  43 double fastiroot(double);
  44
  45 void mpsqrt(mp_no *x, mp_no *y, int p) {
  46 #include "mpsqrt.h"
  47
  48   int i,m,ex,ey;
  49   double dx,dy;
  50   mp_no
  51     mphalf   = {0,{0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
  52                    0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
  53                    0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}},
  54     mp3halfs = {0,{0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
  55                    0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
  56                    0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
  57   mp_no mpxn,mpz,mpu,mpt1,mpt2;
  58
  59   /* Prepare multi-precision 1/2 and 3/2 */
  60   mphalf.e  =0;  mphalf.d[0]  =ONE;  mphalf.d[1]  =HALFRAD;
  61   mp3halfs.e=1;  mp3halfs.d[0]=ONE;  mp3halfs.d[1]=ONE;  mp3halfs.d[2]=HALFRAD;
  62
  63   ex=EX;      ey=EX/2;     cpy(x,&mpxn,p);    mpxn.e -= (ey+ey);
  64   __mp_dbl(&mpxn,&dx,p);   dy=fastiroot(dx);    dbl_mp(dy,&mpu,p);
  65   mul(&mpxn,&mphalf,&mpz,p);
  66
  67   m=mp[p];
  68   for (i=0; i<m; i++) {
  69     mul(&mpu,&mpu,&mpt1,p);
  70     mul(&mpt1,&mpz,&mpt2,p);
  71     sub(&mp3halfs,&mpt2,&mpt1,p);
  72     mul(&mpu,&mpt1,&mpt2,p);
  73     cpy(&mpt2,&mpu,p);
  74   }
  75   mul(&mpxn,&mpu,y,p);  EY += ey;
  76
  77   return;
  78 }
  79
  80 /***********************************************************/
  81 /* Compute a double precision approximation for 1/sqrt(x)  */
  82 /* with the relative error bounded by 2**-51.              */
  83 /***********************************************************/
  84 double fastiroot(double x) {
  85   union {long i[2]; double d;} p,q;
  86   double y,z, t;
  87   long n;
  88   static const double c0 = 0.99674, c1 = -0.53380, c2 = 0.45472, c3 = -0.21553;
  89
  90   p.d = x;
  91   p.i[HIGH_HALF] = (p.i[HIGH_HALF] & 0x3FFFFFFF ) | 0x3FE00000 ;
  92   q.d = x;
  93   y = p.d;
  94   z = y -1.0;
  95   n = (q.i[HIGH_HALF] - p.i[HIGH_HALF])>>1;
  96   z = ((c3*z + c2)*z + c1)*z + c0;            /* 2**-7         */
  97   z = z*(1.5 - 0.5*y*z*z);                    /* 2**-14        */
  98   p.d = z*(1.5 - 0.5*y*z*z);                  /* 2**-28        */
  99   p.i[HIGH_HALF] -= n;
 100   t = x*p.d;
 101   return p.d*(1.5 - 0.5*p.d*t);
 102 }