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I used these shell commands: ../glibc/scripts/update-copyrights $PWD/../gnulib/build-aux/update-copyright (cd ../glibc && git commit -am"[this commit message]") and then ignored the output, which consisted lines saying "FOO: warning: copyright statement not found" for each of 6694 files FOO. I then removed trailing white space from benchtests/bench-pthread-locks.c and iconvdata/tst-iconv-big5-hkscs-to-2ucs4.c, to work around this diagnostic from Savannah: remote: *** pre-commit check failed ... remote: *** error: lines with trailing whitespace found remote: error: hook declined to update refs/heads/master
914 lines
19 KiB
C
914 lines
19 KiB
C
/*
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* IBM Accurate Mathematical Library
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* written by International Business Machines Corp.
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* Copyright (C) 2001-2021 Free Software Foundation, Inc.
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU Lesser General Public License as published by
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* the Free Software Foundation; either version 2.1 of the License, or
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* (at your option) any later version.
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*
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* This program is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU Lesser General Public License for more details.
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*
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* You should have received a copy of the GNU Lesser General Public License
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* along with this program; if not, see <https://www.gnu.org/licenses/>.
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*/
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/************************************************************************/
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/* MODULE_NAME: mpa.c */
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/* */
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/* FUNCTIONS: */
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/* mcr */
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/* acr */
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/* cpy */
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/* norm */
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/* denorm */
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/* mp_dbl */
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/* dbl_mp */
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/* add_magnitudes */
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/* sub_magnitudes */
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/* add */
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/* sub */
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/* mul */
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/* inv */
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/* dvd */
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/* */
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/* Arithmetic functions for multiple precision numbers. */
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/* Relative errors are bounded */
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/************************************************************************/
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#include "endian.h"
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#include "mpa.h"
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#include <sys/param.h>
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#include <alloca.h>
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#ifndef SECTION
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# define SECTION
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#endif
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#ifndef NO__CONST
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const mp_no __mpone = { 1, { 1.0, 1.0 } };
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const mp_no __mptwo = { 1, { 1.0, 2.0 } };
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#endif
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#ifndef NO___ACR
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/* Compare mantissa of two multiple precision numbers regardless of the sign
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and exponent of the numbers. */
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static int
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mcr (const mp_no *x, const mp_no *y, int p)
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{
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long i;
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long p2 = p;
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for (i = 1; i <= p2; i++)
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{
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if (X[i] == Y[i])
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continue;
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else if (X[i] > Y[i])
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return 1;
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else
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return -1;
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}
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return 0;
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}
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/* Compare the absolute values of two multiple precision numbers. */
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int
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__acr (const mp_no *x, const mp_no *y, int p)
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{
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long i;
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if (X[0] == 0)
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{
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if (Y[0] == 0)
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i = 0;
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else
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i = -1;
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}
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else if (Y[0] == 0)
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i = 1;
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else
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{
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if (EX > EY)
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i = 1;
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else if (EX < EY)
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i = -1;
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else
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i = mcr (x, y, p);
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}
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return i;
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}
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#endif
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#ifndef NO___CPY
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/* Copy multiple precision number X into Y. They could be the same
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number. */
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void
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__cpy (const mp_no *x, mp_no *y, int p)
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{
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long i;
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EY = EX;
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for (i = 0; i <= p; i++)
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Y[i] = X[i];
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}
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#endif
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#ifndef NO___MP_DBL
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/* Convert a multiple precision number *X into a double precision
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number *Y, normalized case (|x| >= 2**(-1022))). X has precision
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P, which is positive. */
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static void
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norm (const mp_no *x, double *y, int p)
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{
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# define R RADIXI
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long i;
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double c;
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mantissa_t a, u, v, z[5];
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if (p < 5)
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{
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if (p == 1)
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c = X[1];
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else if (p == 2)
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c = X[1] + R * X[2];
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else if (p == 3)
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c = X[1] + R * (X[2] + R * X[3]);
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else /* p == 4. */
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c = (X[1] + R * X[2]) + R * R * (X[3] + R * X[4]);
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}
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else
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{
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for (a = 1, z[1] = X[1]; z[1] < TWO23; )
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{
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a *= 2;
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z[1] *= 2;
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}
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for (i = 2; i < 5; i++)
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{
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mantissa_store_t d, r;
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d = X[i] * (mantissa_store_t) a;
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DIV_RADIX (d, r);
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z[i] = r;
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z[i - 1] += d;
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}
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u = ALIGN_DOWN_TO (z[3], TWO19);
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v = z[3] - u;
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if (v == TWO18)
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{
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if (z[4] == 0)
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{
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for (i = 5; i <= p; i++)
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{
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if (X[i] == 0)
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continue;
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else
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{
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z[3] += 1;
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break;
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}
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}
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}
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else
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z[3] += 1;
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}
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c = (z[1] + R * (z[2] + R * z[3])) / a;
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}
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c *= X[0];
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for (i = 1; i < EX; i++)
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c *= RADIX;
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for (i = 1; i > EX; i--)
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c *= RADIXI;
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*y = c;
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# undef R
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}
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/* Convert a multiple precision number *X into a double precision
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number *Y, Denormal case (|x| < 2**(-1022))). */
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static void
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denorm (const mp_no *x, double *y, int p)
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{
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long i, k;
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long p2 = p;
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double c;
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mantissa_t u, z[5];
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# define R RADIXI
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if (EX < -44 || (EX == -44 && X[1] < TWO5))
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{
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*y = 0;
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return;
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}
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if (p2 == 1)
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{
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if (EX == -42)
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{
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z[1] = X[1] + TWO10;
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z[2] = 0;
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z[3] = 0;
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k = 3;
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}
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else if (EX == -43)
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{
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z[1] = TWO10;
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z[2] = X[1];
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z[3] = 0;
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k = 2;
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}
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else
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{
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z[1] = TWO10;
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z[2] = 0;
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z[3] = X[1];
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k = 1;
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}
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}
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else if (p2 == 2)
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{
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if (EX == -42)
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{
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z[1] = X[1] + TWO10;
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z[2] = X[2];
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z[3] = 0;
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k = 3;
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}
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else if (EX == -43)
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{
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z[1] = TWO10;
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z[2] = X[1];
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z[3] = X[2];
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k = 2;
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}
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else
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{
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z[1] = TWO10;
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z[2] = 0;
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z[3] = X[1];
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k = 1;
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}
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}
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else
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{
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if (EX == -42)
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{
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z[1] = X[1] + TWO10;
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z[2] = X[2];
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k = 3;
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}
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else if (EX == -43)
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{
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z[1] = TWO10;
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z[2] = X[1];
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k = 2;
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}
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else
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{
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z[1] = TWO10;
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z[2] = 0;
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k = 1;
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}
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z[3] = X[k];
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}
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u = ALIGN_DOWN_TO (z[3], TWO5);
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if (u == z[3])
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{
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for (i = k + 1; i <= p2; i++)
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{
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if (X[i] == 0)
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continue;
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else
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{
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z[3] += 1;
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break;
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}
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}
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}
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c = X[0] * ((z[1] + R * (z[2] + R * z[3])) - TWO10);
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*y = c * TWOM1032;
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# undef R
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}
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/* Convert multiple precision number *X into double precision number *Y. The
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result is correctly rounded to the nearest/even. */
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void
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__mp_dbl (const mp_no *x, double *y, int p)
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{
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if (X[0] == 0)
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{
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*y = 0;
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return;
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}
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if (__glibc_likely (EX > -42 || (EX == -42 && X[1] >= TWO10)))
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norm (x, y, p);
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else
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denorm (x, y, p);
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}
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#endif
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/* Get the multiple precision equivalent of X into *Y. If the precision is too
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small, the result is truncated. */
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void
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SECTION
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__dbl_mp (double x, mp_no *y, int p)
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{
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long i, n;
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long p2 = p;
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/* Sign. */
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if (x == 0)
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{
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Y[0] = 0;
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return;
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}
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else if (x > 0)
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Y[0] = 1;
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else
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{
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Y[0] = -1;
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x = -x;
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}
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/* Exponent. */
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for (EY = 1; x >= RADIX; EY += 1)
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x *= RADIXI;
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for (; x < 1; EY -= 1)
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x *= RADIX;
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/* Digits. */
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n = MIN (p2, 4);
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for (i = 1; i <= n; i++)
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{
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INTEGER_OF (x, Y[i]);
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x *= RADIX;
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}
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for (; i <= p2; i++)
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Y[i] = 0;
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}
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/* Add magnitudes of *X and *Y assuming that abs (*X) >= abs (*Y) > 0. The
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sign of the sum *Z is not changed. X and Y may overlap but not X and Z or
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Y and Z. No guard digit is used. The result equals the exact sum,
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truncated. */
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static void
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SECTION
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add_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p)
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{
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long i, j, k;
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long p2 = p;
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mantissa_t zk;
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EZ = EX;
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i = p2;
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j = p2 + EY - EX;
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k = p2 + 1;
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if (__glibc_unlikely (j < 1))
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{
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__cpy (x, z, p);
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return;
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}
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zk = 0;
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for (; j > 0; i--, j--)
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{
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zk += X[i] + Y[j];
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if (zk >= RADIX)
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{
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Z[k--] = zk - RADIX;
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zk = 1;
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}
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else
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{
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Z[k--] = zk;
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zk = 0;
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}
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}
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for (; i > 0; i--)
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{
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zk += X[i];
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if (zk >= RADIX)
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{
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Z[k--] = zk - RADIX;
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zk = 1;
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}
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else
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{
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Z[k--] = zk;
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zk = 0;
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}
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}
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if (zk == 0)
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{
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for (i = 1; i <= p2; i++)
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Z[i] = Z[i + 1];
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}
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else
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{
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Z[1] = zk;
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EZ += 1;
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}
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}
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/* Subtract the magnitudes of *X and *Y assuming that abs (*x) > abs (*y) > 0.
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The sign of the difference *Z is not changed. X and Y may overlap but not X
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and Z or Y and Z. One guard digit is used. The error is less than one
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ULP. */
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static void
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SECTION
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sub_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p)
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{
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long i, j, k;
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long p2 = p;
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mantissa_t zk;
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EZ = EX;
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i = p2;
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j = p2 + EY - EX;
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k = p2;
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/* Y is too small compared to X, copy X over to the result. */
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if (__glibc_unlikely (j < 1))
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{
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__cpy (x, z, p);
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return;
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}
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|
|
/* The relevant least significant digit in Y is non-zero, so we factor it in
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|
to enhance accuracy. */
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if (j < p2 && Y[j + 1] > 0)
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{
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Z[k + 1] = RADIX - Y[j + 1];
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zk = -1;
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}
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else
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zk = Z[k + 1] = 0;
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/* Subtract and borrow. */
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for (; j > 0; i--, j--)
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{
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zk += (X[i] - Y[j]);
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if (zk < 0)
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{
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Z[k--] = zk + RADIX;
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zk = -1;
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}
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else
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{
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Z[k--] = zk;
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zk = 0;
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}
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}
|
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/* We're done with digits from Y, so it's just digits in X. */
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for (; i > 0; i--)
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{
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zk += X[i];
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if (zk < 0)
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{
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Z[k--] = zk + RADIX;
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zk = -1;
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}
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else
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{
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Z[k--] = zk;
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zk = 0;
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}
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}
|
|
|
|
/* Normalize. */
|
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for (i = 1; Z[i] == 0; i++)
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;
|
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EZ = EZ - i + 1;
|
|
for (k = 1; i <= p2 + 1; )
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Z[k++] = Z[i++];
|
|
for (; k <= p2; )
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|
Z[k++] = 0;
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}
|
|
|
|
/* Add *X and *Y and store the result in *Z. X and Y may overlap, but not X
|
|
and Z or Y and Z. One guard digit is used. The error is less than one
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|
ULP. */
|
|
void
|
|
SECTION
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|
__add (const mp_no *x, const mp_no *y, mp_no *z, int p)
|
|
{
|
|
int n;
|
|
|
|
if (X[0] == 0)
|
|
{
|
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__cpy (y, z, p);
|
|
return;
|
|
}
|
|
else if (Y[0] == 0)
|
|
{
|
|
__cpy (x, z, p);
|
|
return;
|
|
}
|
|
|
|
if (X[0] == Y[0])
|
|
{
|
|
if (__acr (x, y, p) > 0)
|
|
{
|
|
add_magnitudes (x, y, z, p);
|
|
Z[0] = X[0];
|
|
}
|
|
else
|
|
{
|
|
add_magnitudes (y, x, z, p);
|
|
Z[0] = Y[0];
|
|
}
|
|
}
|
|
else
|
|
{
|
|
if ((n = __acr (x, y, p)) == 1)
|
|
{
|
|
sub_magnitudes (x, y, z, p);
|
|
Z[0] = X[0];
|
|
}
|
|
else if (n == -1)
|
|
{
|
|
sub_magnitudes (y, x, z, p);
|
|
Z[0] = Y[0];
|
|
}
|
|
else
|
|
Z[0] = 0;
|
|
}
|
|
}
|
|
|
|
/* Subtract *Y from *X and return the result in *Z. X and Y may overlap but
|
|
not X and Z or Y and Z. One guard digit is used. The error is less than
|
|
one ULP. */
|
|
void
|
|
SECTION
|
|
__sub (const mp_no *x, const mp_no *y, mp_no *z, int p)
|
|
{
|
|
int n;
|
|
|
|
if (X[0] == 0)
|
|
{
|
|
__cpy (y, z, p);
|
|
Z[0] = -Z[0];
|
|
return;
|
|
}
|
|
else if (Y[0] == 0)
|
|
{
|
|
__cpy (x, z, p);
|
|
return;
|
|
}
|
|
|
|
if (X[0] != Y[0])
|
|
{
|
|
if (__acr (x, y, p) > 0)
|
|
{
|
|
add_magnitudes (x, y, z, p);
|
|
Z[0] = X[0];
|
|
}
|
|
else
|
|
{
|
|
add_magnitudes (y, x, z, p);
|
|
Z[0] = -Y[0];
|
|
}
|
|
}
|
|
else
|
|
{
|
|
if ((n = __acr (x, y, p)) == 1)
|
|
{
|
|
sub_magnitudes (x, y, z, p);
|
|
Z[0] = X[0];
|
|
}
|
|
else if (n == -1)
|
|
{
|
|
sub_magnitudes (y, x, z, p);
|
|
Z[0] = -Y[0];
|
|
}
|
|
else
|
|
Z[0] = 0;
|
|
}
|
|
}
|
|
|
|
#ifndef NO__MUL
|
|
/* Multiply *X and *Y and store result in *Z. X and Y may overlap but not X
|
|
and Z or Y and Z. For P in [1, 2, 3], the exact result is truncated to P
|
|
digits. In case P > 3 the error is bounded by 1.001 ULP. */
|
|
void
|
|
SECTION
|
|
__mul (const mp_no *x, const mp_no *y, mp_no *z, int p)
|
|
{
|
|
long i, j, k, ip, ip2;
|
|
long p2 = p;
|
|
mantissa_store_t zk;
|
|
const mp_no *a;
|
|
mantissa_store_t *diag;
|
|
|
|
/* Is z=0? */
|
|
if (__glibc_unlikely (X[0] * Y[0] == 0))
|
|
{
|
|
Z[0] = 0;
|
|
return;
|
|
}
|
|
|
|
/* We need not iterate through all X's and Y's since it's pointless to
|
|
multiply zeroes. Here, both are zero... */
|
|
for (ip2 = p2; ip2 > 0; ip2--)
|
|
if (X[ip2] != 0 || Y[ip2] != 0)
|
|
break;
|
|
|
|
a = X[ip2] != 0 ? y : x;
|
|
|
|
/* ... and here, at least one of them is still zero. */
|
|
for (ip = ip2; ip > 0; ip--)
|
|
if (a->d[ip] != 0)
|
|
break;
|
|
|
|
/* The product looks like this for p = 3 (as an example):
|
|
|
|
|
|
a1 a2 a3
|
|
x b1 b2 b3
|
|
-----------------------------
|
|
a1*b3 a2*b3 a3*b3
|
|
a1*b2 a2*b2 a3*b2
|
|
a1*b1 a2*b1 a3*b1
|
|
|
|
So our K needs to ideally be P*2, but we're limiting ourselves to P + 3
|
|
for P >= 3. We compute the above digits in two parts; the last P-1
|
|
digits and then the first P digits. The last P-1 digits are a sum of
|
|
products of the input digits from P to P-k where K is 0 for the least
|
|
significant digit and increases as we go towards the left. The product
|
|
term is of the form X[k]*X[P-k] as can be seen in the above example.
|
|
|
|
The first P digits are also a sum of products with the same product term,
|
|
except that the sum is from 1 to k. This is also evident from the above
|
|
example.
|
|
|
|
Another thing that becomes evident is that only the most significant
|
|
ip+ip2 digits of the result are non-zero, where ip and ip2 are the
|
|
'internal precision' of the input numbers, i.e. digits after ip and ip2
|
|
are all 0. */
|
|
|
|
k = (__glibc_unlikely (p2 < 3)) ? p2 + p2 : p2 + 3;
|
|
|
|
while (k > ip + ip2 + 1)
|
|
Z[k--] = 0;
|
|
|
|
zk = 0;
|
|
|
|
/* Precompute sums of diagonal elements so that we can directly use them
|
|
later. See the next comment to know we why need them. */
|
|
diag = alloca (k * sizeof (mantissa_store_t));
|
|
mantissa_store_t d = 0;
|
|
for (i = 1; i <= ip; i++)
|
|
{
|
|
d += X[i] * (mantissa_store_t) Y[i];
|
|
diag[i] = d;
|
|
}
|
|
while (i < k)
|
|
diag[i++] = d;
|
|
|
|
while (k > p2)
|
|
{
|
|
long lim = k / 2;
|
|
|
|
if (k % 2 == 0)
|
|
/* We want to add this only once, but since we subtract it in the sum
|
|
of products above, we add twice. */
|
|
zk += 2 * X[lim] * (mantissa_store_t) Y[lim];
|
|
|
|
for (i = k - p2, j = p2; i < j; i++, j--)
|
|
zk += (X[i] + X[j]) * (mantissa_store_t) (Y[i] + Y[j]);
|
|
|
|
zk -= diag[k - 1];
|
|
|
|
DIV_RADIX (zk, Z[k]);
|
|
k--;
|
|
}
|
|
|
|
/* The real deal. Mantissa digit Z[k] is the sum of all X[i] * Y[j] where i
|
|
goes from 1 -> k - 1 and j goes the same range in reverse. To reduce the
|
|
number of multiplications, we halve the range and if k is an even number,
|
|
add the diagonal element X[k/2]Y[k/2]. Through the half range, we compute
|
|
X[i] * Y[j] as (X[i] + X[j]) * (Y[i] + Y[j]) - X[i] * Y[i] - X[j] * Y[j].
|
|
|
|
This reduction tells us that we're summing two things, the first term
|
|
through the half range and the negative of the sum of the product of all
|
|
terms of X and Y in the full range. i.e.
|
|
|
|
SUM(X[i] * Y[i]) for k terms. This is precalculated above for each k in
|
|
a single loop so that it completes in O(n) time and can hence be directly
|
|
used in the loop below. */
|
|
while (k > 1)
|
|
{
|
|
long lim = k / 2;
|
|
|
|
if (k % 2 == 0)
|
|
/* We want to add this only once, but since we subtract it in the sum
|
|
of products above, we add twice. */
|
|
zk += 2 * X[lim] * (mantissa_store_t) Y[lim];
|
|
|
|
for (i = 1, j = k - 1; i < j; i++, j--)
|
|
zk += (X[i] + X[j]) * (mantissa_store_t) (Y[i] + Y[j]);
|
|
|
|
zk -= diag[k - 1];
|
|
|
|
DIV_RADIX (zk, Z[k]);
|
|
k--;
|
|
}
|
|
Z[k] = zk;
|
|
|
|
/* Get the exponent sum into an intermediate variable. This is a subtle
|
|
optimization, where given enough registers, all operations on the exponent
|
|
happen in registers and the result is written out only once into EZ. */
|
|
int e = EX + EY;
|
|
|
|
/* Is there a carry beyond the most significant digit? */
|
|
if (__glibc_unlikely (Z[1] == 0))
|
|
{
|
|
for (i = 1; i <= p2; i++)
|
|
Z[i] = Z[i + 1];
|
|
e--;
|
|
}
|
|
|
|
EZ = e;
|
|
Z[0] = X[0] * Y[0];
|
|
}
|
|
#endif
|
|
|
|
#ifndef NO__SQR
|
|
/* Square *X and store result in *Y. X and Y may not overlap. For P in
|
|
[1, 2, 3], the exact result is truncated to P digits. In case P > 3 the
|
|
error is bounded by 1.001 ULP. This is a faster special case of
|
|
multiplication. */
|
|
void
|
|
SECTION
|
|
__sqr (const mp_no *x, mp_no *y, int p)
|
|
{
|
|
long i, j, k, ip;
|
|
mantissa_store_t yk;
|
|
|
|
/* Is z=0? */
|
|
if (__glibc_unlikely (X[0] == 0))
|
|
{
|
|
Y[0] = 0;
|
|
return;
|
|
}
|
|
|
|
/* We need not iterate through all X's since it's pointless to
|
|
multiply zeroes. */
|
|
for (ip = p; ip > 0; ip--)
|
|
if (X[ip] != 0)
|
|
break;
|
|
|
|
k = (__glibc_unlikely (p < 3)) ? p + p : p + 3;
|
|
|
|
while (k > 2 * ip + 1)
|
|
Y[k--] = 0;
|
|
|
|
yk = 0;
|
|
|
|
while (k > p)
|
|
{
|
|
mantissa_store_t yk2 = 0;
|
|
long lim = k / 2;
|
|
|
|
if (k % 2 == 0)
|
|
yk += X[lim] * (mantissa_store_t) X[lim];
|
|
|
|
/* In __mul, this loop (and the one within the next while loop) run
|
|
between a range to calculate the mantissa as follows:
|
|
|
|
Z[k] = X[k] * Y[n] + X[k+1] * Y[n-1] ... + X[n-1] * Y[k+1]
|
|
+ X[n] * Y[k]
|
|
|
|
For X == Y, we can get away with summing halfway and doubling the
|
|
result. For cases where the range size is even, the mid-point needs
|
|
to be added separately (above). */
|
|
for (i = k - p, j = p; i < j; i++, j--)
|
|
yk2 += X[i] * (mantissa_store_t) X[j];
|
|
|
|
yk += 2 * yk2;
|
|
|
|
DIV_RADIX (yk, Y[k]);
|
|
k--;
|
|
}
|
|
|
|
while (k > 1)
|
|
{
|
|
mantissa_store_t yk2 = 0;
|
|
long lim = k / 2;
|
|
|
|
if (k % 2 == 0)
|
|
yk += X[lim] * (mantissa_store_t) X[lim];
|
|
|
|
/* Likewise for this loop. */
|
|
for (i = 1, j = k - 1; i < j; i++, j--)
|
|
yk2 += X[i] * (mantissa_store_t) X[j];
|
|
|
|
yk += 2 * yk2;
|
|
|
|
DIV_RADIX (yk, Y[k]);
|
|
k--;
|
|
}
|
|
Y[k] = yk;
|
|
|
|
/* Squares are always positive. */
|
|
Y[0] = 1;
|
|
|
|
/* Get the exponent sum into an intermediate variable. This is a subtle
|
|
optimization, where given enough registers, all operations on the exponent
|
|
happen in registers and the result is written out only once into EZ. */
|
|
int e = EX * 2;
|
|
|
|
/* Is there a carry beyond the most significant digit? */
|
|
if (__glibc_unlikely (Y[1] == 0))
|
|
{
|
|
for (i = 1; i <= p; i++)
|
|
Y[i] = Y[i + 1];
|
|
e--;
|
|
}
|
|
|
|
EY = e;
|
|
}
|
|
#endif
|
|
|
|
/* Invert *X and store in *Y. Relative error bound:
|
|
- For P = 2: 1.001 * R ^ (1 - P)
|
|
- For P = 3: 1.063 * R ^ (1 - P)
|
|
- For P > 3: 2.001 * R ^ (1 - P)
|
|
|
|
*X = 0 is not permissible. */
|
|
static void
|
|
SECTION
|
|
__inv (const mp_no *x, mp_no *y, int p)
|
|
{
|
|
long i;
|
|
double t;
|
|
mp_no z, w;
|
|
static const int np1[] =
|
|
{ 0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3,
|
|
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
|
|
};
|
|
|
|
__cpy (x, &z, p);
|
|
z.e = 0;
|
|
__mp_dbl (&z, &t, p);
|
|
t = 1 / t;
|
|
|
|
/* t == 0 will never happen at this point, since 1/t can only be 0 if t is
|
|
infinity, but before the division t == mantissa of x (exponent is 0). We
|
|
can instruct the compiler to ignore this case. */
|
|
if (t == 0)
|
|
__builtin_unreachable ();
|
|
|
|
__dbl_mp (t, y, p);
|
|
EY -= EX;
|
|
|
|
for (i = 0; i < np1[p]; i++)
|
|
{
|
|
__cpy (y, &w, p);
|
|
__mul (x, &w, y, p);
|
|
__sub (&__mptwo, y, &z, p);
|
|
__mul (&w, &z, y, p);
|
|
}
|
|
}
|
|
|
|
/* Divide *X by *Y and store result in *Z. X and Y may overlap but not X and Z
|
|
or Y and Z. Relative error bound:
|
|
- For P = 2: 2.001 * R ^ (1 - P)
|
|
- For P = 3: 2.063 * R ^ (1 - P)
|
|
- For P > 3: 3.001 * R ^ (1 - P)
|
|
|
|
*X = 0 is not permissible. */
|
|
void
|
|
SECTION
|
|
__dvd (const mp_no *x, const mp_no *y, mp_no *z, int p)
|
|
{
|
|
mp_no w;
|
|
|
|
if (X[0] == 0)
|
|
Z[0] = 0;
|
|
else
|
|
{
|
|
__inv (y, &w, p);
|
|
__mul (x, &w, z, p);
|
|
}
|
|
}
|