| /* |
| * IBM Accurate Mathematical Library |
| * written by International Business Machines Corp. |
| * Copyright (C) 2001-2018 Free Software Foundation, Inc. |
| * |
| * This program is free software; you can redistribute it and/or modify |
| * it under the terms of the GNU Lesser General Public License as published by |
| * the Free Software Foundation; either version 2.1 of the License, or |
| * (at your option) any later version. |
| * |
| * This program is distributed in the hope that it will be useful, |
| * but WITHOUT ANY WARRANTY; without even the implied warranty of |
| * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| * GNU Lesser General Public License for more details. |
| * |
| * You should have received a copy of the GNU Lesser General Public License |
| * along with this program; if not, see <http://www.gnu.org/licenses/>. |
| */ |
| /****************************************************************************/ |
| /* MODULE_NAME:mpsqrt.c */ |
| /* */ |
| /* FUNCTION:mpsqrt */ |
| /* fastiroot */ |
| /* */ |
| /* FILES NEEDED:endian.h mpa.h mpsqrt.h */ |
| /* mpa.c */ |
| /* Multi-Precision square root function subroutine for precision p >= 4. */ |
| /* The relative error is bounded by 3.501*r**(1-p), where r=2**24. */ |
| /* */ |
| /****************************************************************************/ |
| #include "endian.h" |
| #include "mpa.h" |
| |
| #ifndef SECTION |
| # define SECTION |
| #endif |
| |
| #include "mpsqrt.h" |
| |
| /****************************************************************************/ |
| /* Multi-Precision square root function subroutine for precision p >= 4. */ |
| /* The relative error is bounded by 3.501*r**(1-p), where r=2**24. */ |
| /* Routine receives two pointers to Multi Precision numbers: */ |
| /* x (left argument) and y (next argument). Routine also receives precision */ |
| /* p as integer. Routine computes sqrt(*x) and stores result in *y */ |
| /****************************************************************************/ |
| |
| static double fastiroot (double); |
| |
| void |
| SECTION |
| __mpsqrt (mp_no *x, mp_no *y, int p) |
| { |
| int i, m, ey; |
| double dx, dy; |
| static const mp_no mphalf = {0, {1.0, HALFRAD}}; |
| static const mp_no mp3halfs = {1, {1.0, 1.0, HALFRAD}}; |
| mp_no mpxn, mpz, mpu, mpt1, mpt2; |
| |
| ey = EX / 2; |
| __cpy (x, &mpxn, p); |
| mpxn.e -= (ey + ey); |
| __mp_dbl (&mpxn, &dx, p); |
| dy = fastiroot (dx); |
| __dbl_mp (dy, &mpu, p); |
| __mul (&mpxn, &mphalf, &mpz, p); |
| |
| m = __mpsqrt_mp[p]; |
| for (i = 0; i < m; i++) |
| { |
| __sqr (&mpu, &mpt1, p); |
| __mul (&mpt1, &mpz, &mpt2, p); |
| __sub (&mp3halfs, &mpt2, &mpt1, p); |
| __mul (&mpu, &mpt1, &mpt2, p); |
| __cpy (&mpt2, &mpu, p); |
| } |
| __mul (&mpxn, &mpu, y, p); |
| EY += ey; |
| } |
| |
| /***********************************************************/ |
| /* Compute a double precision approximation for 1/sqrt(x) */ |
| /* with the relative error bounded by 2**-51. */ |
| /***********************************************************/ |
| static double |
| SECTION |
| fastiroot (double x) |
| { |
| union |
| { |
| int i[2]; |
| double d; |
| } p, q; |
| double y, z, t; |
| int n; |
| static const double c0 = 0.99674, c1 = -0.53380; |
| static const double c2 = 0.45472, c3 = -0.21553; |
| |
| p.d = x; |
| p.i[HIGH_HALF] = (p.i[HIGH_HALF] & 0x3FFFFFFF) | 0x3FE00000; |
| q.d = x; |
| y = p.d; |
| z = y - 1.0; |
| n = (q.i[HIGH_HALF] - p.i[HIGH_HALF]) >> 1; |
| z = ((c3 * z + c2) * z + c1) * z + c0; /* 2**-7 */ |
| z = z * (1.5 - 0.5 * y * z * z); /* 2**-14 */ |
| p.d = z * (1.5 - 0.5 * y * z * z); /* 2**-28 */ |
| p.i[HIGH_HALF] -= n; |
| t = x * p.d; |
| return p.d * (1.5 - 0.5 * p.d * t); |
| } |