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
__mpsqrt(mp_no * x,mp_no * y,int p)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 /***********************************************************/
fastiroot(double x)85 double fastiroot(double x) {
86 union {long i[2]; double d;} p,q;
87 double y,z, t;
88 long 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 }
104