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712 lines
13 KiB
C
712 lines
13 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) 2001-2013 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 <http://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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#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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int i;
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for (i = 1; i <= p; 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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int i;
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if (X[0] == ZERO)
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{
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if (Y[0] == ZERO)
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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] == ZERO)
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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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EY = EX;
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for (int 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))). */
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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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int i;
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double a, c, 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 if (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 = ONE, z[1] = X[1]; z[1] < TWO23;)
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{
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a *= TWO;
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z[1] *= TWO;
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}
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for (i = 2; i < 5; i++)
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{
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z[i] = X[i] * a;
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u = (z[i] + CUTTER) - CUTTER;
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if (u > z[i])
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u -= RADIX;
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z[i] -= u;
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z[i - 1] += u * RADIXI;
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}
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u = (z[3] + TWO71) - TWO71;
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if (u > z[3])
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u -= 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] == ZERO)
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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] == ZERO)
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continue;
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else
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{
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z[3] += ONE;
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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] += ONE;
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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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int i, k;
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double c, 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 = ZERO;
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return;
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}
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if (p == 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] = ZERO;
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z[3] = ZERO;
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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] = ZERO;
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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] = ZERO;
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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 (p == 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] = ZERO;
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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] = ZERO;
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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] = ZERO;
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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 = (z[3] + TWO57) - TWO57;
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if (u > z[3])
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u -= TWO5;
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if (u == z[3])
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{
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for (i = k + 1; i <= p; i++)
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{
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if (X[i] == ZERO)
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continue;
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else
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{
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z[3] += ONE;
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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] == ZERO)
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{
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*y = ZERO;
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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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int i, n;
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double u;
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/* Sign. */
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if (x == ZERO)
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{
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Y[0] = ZERO;
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return;
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}
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else if (x > ZERO)
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Y[0] = ONE;
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else
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{
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Y[0] = MONE;
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x = -x;
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}
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/* Exponent. */
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for (EY = ONE; x >= RADIX; EY += ONE)
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x *= RADIXI;
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for (; x < ONE; EY -= ONE)
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x *= RADIX;
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/* Digits. */
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n = MIN (p, 4);
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for (i = 1; i <= n; i++)
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{
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u = (x + TWO52) - TWO52;
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if (u > x)
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u -= ONE;
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Y[i] = u;
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x -= u;
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x *= RADIX;
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}
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for (; i <= p; i++)
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Y[i] = ZERO;
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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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int i, j, k;
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EZ = EX;
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i = p;
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j = p + EY - EX;
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k = p + 1;
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if (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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else
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Z[k] = ZERO;
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for (; j > 0; i--, j--)
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{
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Z[k] += X[i] + Y[j];
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if (Z[k] >= RADIX)
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{
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Z[k] -= RADIX;
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Z[--k] = ONE;
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}
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else
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Z[--k] = ZERO;
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}
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for (; i > 0; i--)
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{
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Z[k] += X[i];
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if (Z[k] >= RADIX)
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{
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Z[k] -= RADIX;
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Z[--k] = ONE;
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}
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else
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Z[--k] = ZERO;
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}
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if (Z[1] == ZERO)
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{
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for (i = 1; i <= p; i++)
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Z[i] = Z[i + 1];
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}
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else
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EZ += ONE;
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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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int i, j, k;
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EZ = EX;
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if (EX == EY)
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{
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i = j = k = p;
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Z[k] = Z[k + 1] = ZERO;
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}
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else
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{
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j = EX - EY;
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if (j > p)
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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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else
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{
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i = p;
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j = p + 1 - j;
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k = p;
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if (Y[j] > ZERO)
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{
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Z[k + 1] = RADIX - Y[j--];
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Z[k] = MONE;
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}
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else
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{
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Z[k + 1] = ZERO;
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Z[k] = ZERO;
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j--;
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}
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}
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}
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for (; j > 0; i--, j--)
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{
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Z[k] += (X[i] - Y[j]);
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if (Z[k] < ZERO)
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{
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Z[k] += RADIX;
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Z[--k] = MONE;
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}
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else
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Z[--k] = ZERO;
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}
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for (; i > 0; i--)
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{
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Z[k] += X[i];
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if (Z[k] < ZERO)
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{
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Z[k] += RADIX;
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Z[--k] = MONE;
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}
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else
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Z[--k] = ZERO;
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}
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for (i = 1; Z[i] == ZERO; i++);
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EZ = EZ - i + 1;
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for (k = 1; i <= p + 1;)
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Z[k++] = Z[i++];
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for (; k <= p;)
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Z[k++] = ZERO;
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}
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/* Add *X and *Y and store the result in *Z. 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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void
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SECTION
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__add (const mp_no *x, const mp_no *y, mp_no *z, int p)
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{
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int n;
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if (X[0] == ZERO)
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{
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__cpy (y, z, p);
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return;
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}
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else if (Y[0] == ZERO)
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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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if (X[0] == Y[0])
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{
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if (__acr (x, y, p) > 0)
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{
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add_magnitudes (x, y, z, p);
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Z[0] = X[0];
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}
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else
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{
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add_magnitudes (y, x, z, p);
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Z[0] = Y[0];
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}
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}
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else
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{
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if ((n = __acr (x, y, p)) == 1)
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{
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sub_magnitudes (x, y, z, p);
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Z[0] = X[0];
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}
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else if (n == -1)
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{
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sub_magnitudes (y, x, z, p);
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Z[0] = Y[0];
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}
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else
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Z[0] = ZERO;
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}
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}
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/* Subtract *Y from *X and return the result in *Z. X and Y may overlap but
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not X and Z or Y and Z. One guard digit is used. The error is less than
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one ULP. */
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void
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SECTION
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__sub (const mp_no *x, const mp_no *y, mp_no *z, int p)
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{
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int n;
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if (X[0] == ZERO)
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{
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__cpy (y, z, p);
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Z[0] = -Z[0];
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return;
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}
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else if (Y[0] == ZERO)
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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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if (X[0] != Y[0])
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{
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if (__acr (x, y, p) > 0)
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{
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add_magnitudes (x, y, z, p);
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Z[0] = X[0];
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}
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else
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{
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add_magnitudes (y, x, z, p);
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Z[0] = -Y[0];
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}
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}
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else
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{
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if ((n = __acr (x, y, p)) == 1)
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{
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sub_magnitudes (x, y, z, p);
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Z[0] = X[0];
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}
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else if (n == -1)
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{
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sub_magnitudes (y, x, z, p);
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Z[0] = -Y[0];
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}
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else
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Z[0] = ZERO;
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}
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}
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/* Multiply *X and *Y and store result in *Z. X and Y may overlap but not X
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and Z or Y and Z. For P in [1, 2, 3], the exact result is truncated to P
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digits. In case P > 3 the error is bounded by 1.001 ULP. */
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void
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SECTION
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__mul (const mp_no *x, const mp_no *y, mp_no *z, int p)
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{
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int i, j, k, k2;
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double u, zk;
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/* Is z=0? */
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if (__glibc_unlikely (X[0] * Y[0] == ZERO))
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{
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Z[0] = ZERO;
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return;
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}
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|
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/* Multiply, add and carry. */
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k2 = (__glibc_unlikely (p < 3)) ? p + p : p + 3;
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zk = Z[k2] = ZERO;
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for (k = k2; k > p; k--)
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{
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for (i = k - p, j = p; i < p + 1; i++, j--)
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zk += X[i] * Y[j];
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u = (zk + CUTTER) - CUTTER;
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if (u > zk)
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u -= RADIX;
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Z[k] = zk - u;
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zk = u * RADIXI;
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}
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|
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while (k > 1)
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|
{
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for (i = 1, j = k - 1; i < k; i++, j--)
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zk += X[i] * Y[j];
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|
|
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u = (zk + CUTTER) - CUTTER;
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if (u > zk)
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u -= RADIX;
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Z[k] = zk - u;
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zk = u * RADIXI;
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|
k--;
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}
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|
Z[k] = zk;
|
|
|
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EZ = EX + EY;
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|
/* Is there a carry beyond the most significant digit? */
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|
if (__glibc_unlikely (Z[1] == ZERO))
|
|
{
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|
for (i = 1; i <= p; i++)
|
|
Z[i] = Z[i + 1];
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|
EZ--;
|
|
}
|
|
|
|
Z[0] = X[0] * Y[0];
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|
}
|
|
|
|
/* Invert *X and store in *Y. Relative error bound:
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|
- 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)
|
|
{
|
|
int 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);
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|
t = ONE / t;
|
|
__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] == ZERO)
|
|
Z[0] = ZERO;
|
|
else
|
|
{
|
|
__inv (y, &w, p);
|
|
__mul (x, &w, z, p);
|
|
}
|
|
}
|