sysdeps/ieee754/dbl-64/mpsqrt.c

   1
   2 /*
   3  * IBM Accurate Mathematical Library
   4  * written by International Business Machines Corp.
   5  * Copyright (C) 2001 Free Software Foundation
   6  *
   7  * This program is free software; you can redistribute it and/or modify
   8  * it under the terms of the GNU Lesser General Public License as published by
   9  * the Free Software Foundation; either version 2.1 of the License, or
  10  * (at your option) any later version.
  11  *
  12  * This program is distributed in the hope that it will be useful,
  13  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  14  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  15  * GNU Lesser General Public License for more details.
  16  *
  17  * You should have received a copy of the GNU Lesser General Public License
  18  * along with this program; if not, write to the Free Software
  19  * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
  20  */
  21 /****************************************************************************/
  22 /*  MODULE_NAME:mpsqrt.c                                                    */
  23 /*                                                                          */
  24 /*  FUNCTION:mpsqrt                                                         */
  25 /*           fastiroot                                                      */
  26 /*                                                                          */
  27 /* FILES NEEDED:endian.h mpa.h mpsqrt.h                                     */
  28 /*              mpa.c                                                       */
  29 /* Multi-Precision square root function subroutine for precision p >= 4.    */
  30 /* The relative error is bounded by 3.501*r**(1-p), where r=2**24.          */
  31 /*                                                                          */
  32 /****************************************************************************/
  33 #include "endian.h"
  34 #include "mpa.h"
  35
  36 /****************************************************************************/
  37 /* Multi-Precision square root function subroutine for precision p >= 4.    */
  38 /* The relative error is bounded by 3.501*r**(1-p), where r=2**24.          */
  39 /* Routine receives two pointers to  Multi Precision numbers:               */
  40 /* x (left argument) and y (next argument). Routine also receives precision */
  41 /* p as integer. Routine computes sqrt(*x) and stores result in *y          */
  42 /****************************************************************************/
  43
  44 double fastiroot(double);
  45
  46 void __mpsqrt(mp_no *x, mp_no *y, int p) {
  47 #include "mpsqrt.h"
  48
  49   int i,m,ex,ey;
  50   double dx,dy;
  51   mp_no
  52     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,
  53                    0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
  54                    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     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,
  56                    0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
  57                    0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
  58   mp_no mpxn,mpz,mpu,mpt1,mpt2;
  59
  60   /* Prepare multi-precision 1/2 and 3/2 */
  61   mphalf.e  =0;  mphalf.d[0]  =ONE;  mphalf.d[1]  =HALFRAD;
  62   mp3halfs.e=1;  mp3halfs.d[0]=ONE;  mp3halfs.d[1]=ONE;  mp3halfs.d[2]=HALFRAD;
  63
  64   ex=EX;      ey=EX/2;     __cpy(x,&mpxn,p);    mpxn.e -= (ey+ey);
  65   __mp_dbl(&mpxn,&dx,p);   dy=fastiroot(dx);    __dbl_mp(dy,&mpu,p);
  66   __mul(&mpxn,&mphalf,&mpz,p);
  67
  68   m=mp[p];
  69   for (i=0; i<m; i++) {
  70     __mul(&mpu,&mpu,&mpt1,p);
  71     __mul(&mpt1,&mpz,&mpt2,p);
  72     __sub(&mp3halfs,&mpt2,&mpt1,p);
  73     __mul(&mpu,&mpt1,&mpt2,p);
  74     __cpy(&mpt2,&mpu,p);
  75   }
  76   __mul(&mpxn,&mpu,y,p);  EY += ey;
  77
  78   return;
  79 }
  80
  81 /***********************************************************/
  82 /* Compute a double precision approximation for 1/sqrt(x)  */
  83 /* with the relative error bounded by 2**-51.              */
  84 /***********************************************************/
  85 double fastiroot(double x) {
  86   union {int i[2]; double d;} p,q;
  87   double y,z, t;
  88   int n;
  89   static const double c0 = 0.99674, c1 = -0.53380, c2 = 0.45472, c3 = -0.21553;
  90
  91   p.d = x;
  92   p.i[HIGH_HALF] = (p.i[HIGH_HALF] & 0x3FFFFFFF ) | 0x3FE00000 ;
  93   q.d = x;
  94   y = p.d;
  95   z = y -1.0;
  96   n = (q.i[HIGH_HALF] - p.i[HIGH_HALF])>>1;
  97   z = ((c3*z + c2)*z + c1)*z + c0;            /* 2**-7         */
  98   z = z*(1.5 - 0.5*y*z*z);                    /* 2**-14        */
  99   p.d = z*(1.5 - 0.5*y*z*z);                  /* 2**-28        */
 100   p.i[HIGH_HALF] -= n;
 101   t = x*p.d;
 102   return p.d*(1.5 - 0.5*p.d*t);
 103 }