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362 lines
9.6 KiB
C
362 lines
9.6 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) 2001-2017 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:uexp.c */
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/* */
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/* FUNCTION:uexp */
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/* exp1 */
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/* */
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/* FILES NEEDED:dla.h endian.h mpa.h mydefs.h uexp.h */
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/* mpa.c mpexp.x slowexp.c */
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/* */
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/* An ultimate exp routine. Given an IEEE double machine number x */
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/* it computes the correctly rounded (to nearest) value of e^x */
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/* Assumption: Machine arithmetic operations are performed in */
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/* round to nearest mode of IEEE 754 standard. */
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/* */
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/***************************************************************************/
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#include <math.h>
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#include "endian.h"
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#include "uexp.h"
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#include "mydefs.h"
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#include "MathLib.h"
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#include "uexp.tbl"
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#include <math_private.h>
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#include <fenv.h>
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#include <float.h>
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#ifndef SECTION
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# define SECTION
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#endif
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double __slowexp (double);
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/* An ultimate exp routine. Given an IEEE double machine number x it computes
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the correctly rounded (to nearest) value of e^x. */
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double
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SECTION
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__ieee754_exp (double x)
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{
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double bexp, t, eps, del, base, y, al, bet, res, rem, cor;
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mynumber junk1, junk2, binexp = {{0, 0}};
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int4 i, j, m, n, ex;
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double retval;
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{
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SET_RESTORE_ROUND (FE_TONEAREST);
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junk1.x = x;
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m = junk1.i[HIGH_HALF];
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n = m & hugeint;
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if (n > smallint && n < bigint)
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{
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y = x * log2e.x + three51.x;
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bexp = y - three51.x; /* multiply the result by 2**bexp */
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junk1.x = y;
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eps = bexp * ln_two2.x; /* x = bexp*ln(2) + t - eps */
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t = x - bexp * ln_two1.x;
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y = t + three33.x;
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base = y - three33.x; /* t rounded to a multiple of 2**-18 */
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junk2.x = y;
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del = (t - base) - eps; /* x = bexp*ln(2) + base + del */
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eps = del + del * del * (p3.x * del + p2.x);
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binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 1023) << 20;
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i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
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j = (junk2.i[LOW_HALF] & 511) << 1;
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al = coar.x[i] * fine.x[j];
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bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
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+ coar.x[i + 1] * fine.x[j + 1]);
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rem = (bet + bet * eps) + al * eps;
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res = al + rem;
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cor = (al - res) + rem;
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if (res == (res + cor * err_0))
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{
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retval = res * binexp.x;
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goto ret;
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}
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else
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{
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retval = __slowexp (x);
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goto ret;
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} /*if error is over bound */
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}
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if (n <= smallint)
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{
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retval = 1.0;
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goto ret;
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}
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if (n >= badint)
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{
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if (n > infint)
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{
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retval = x + x;
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goto ret;
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} /* x is NaN */
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if (n < infint)
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{
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if (x > 0)
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goto ret_huge;
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else
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goto ret_tiny;
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}
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/* x is finite, cause either overflow or underflow */
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if (junk1.i[LOW_HALF] != 0)
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{
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retval = x + x;
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goto ret;
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} /* x is NaN */
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retval = (x > 0) ? inf.x : zero; /* |x| = inf; return either inf or 0 */
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goto ret;
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}
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y = x * log2e.x + three51.x;
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bexp = y - three51.x;
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junk1.x = y;
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eps = bexp * ln_two2.x;
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t = x - bexp * ln_two1.x;
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y = t + three33.x;
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base = y - three33.x;
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junk2.x = y;
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del = (t - base) - eps;
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eps = del + del * del * (p3.x * del + p2.x);
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i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
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j = (junk2.i[LOW_HALF] & 511) << 1;
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al = coar.x[i] * fine.x[j];
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bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
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+ coar.x[i + 1] * fine.x[j + 1]);
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rem = (bet + bet * eps) + al * eps;
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res = al + rem;
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cor = (al - res) + rem;
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if (m >> 31)
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{
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ex = junk1.i[LOW_HALF];
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if (res < 1.0)
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{
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res += res;
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cor += cor;
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ex -= 1;
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}
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if (ex >= -1022)
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{
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binexp.i[HIGH_HALF] = (1023 + ex) << 20;
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if (res == (res + cor * err_0))
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{
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retval = res * binexp.x;
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goto ret;
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}
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else
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{
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retval = __slowexp (x);
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goto check_uflow_ret;
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} /*if error is over bound */
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}
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ex = -(1022 + ex);
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binexp.i[HIGH_HALF] = (1023 - ex) << 20;
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res *= binexp.x;
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cor *= binexp.x;
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eps = 1.0000000001 + err_0 * binexp.x;
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t = 1.0 + res;
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y = ((1.0 - t) + res) + cor;
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res = t + y;
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cor = (t - res) + y;
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if (res == (res + eps * cor))
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{
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binexp.i[HIGH_HALF] = 0x00100000;
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retval = (res - 1.0) * binexp.x;
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goto check_uflow_ret;
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}
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else
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{
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retval = __slowexp (x);
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goto check_uflow_ret;
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} /* if error is over bound */
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check_uflow_ret:
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if (retval < DBL_MIN)
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{
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double force_underflow = tiny * tiny;
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math_force_eval (force_underflow);
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}
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if (retval == 0)
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goto ret_tiny;
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goto ret;
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}
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else
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{
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binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 767) << 20;
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if (res == (res + cor * err_0))
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retval = res * binexp.x * t256.x;
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else
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retval = __slowexp (x);
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if (isinf (retval))
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goto ret_huge;
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else
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goto ret;
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}
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}
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ret:
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return retval;
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ret_huge:
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return hhuge * hhuge;
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ret_tiny:
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return tiny * tiny;
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}
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#ifndef __ieee754_exp
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strong_alias (__ieee754_exp, __exp_finite)
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#endif
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/* Compute e^(x+xx). The routine also receives bound of error of previous
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calculation. If after computing exp the error exceeds the allowed bounds,
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the routine returns a non-positive number. Otherwise it returns the
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computed result, which is always positive. */
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double
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SECTION
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__exp1 (double x, double xx, double error)
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{
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double bexp, t, eps, del, base, y, al, bet, res, rem, cor;
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mynumber junk1, junk2, binexp = {{0, 0}};
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int4 i, j, m, n, ex;
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junk1.x = x;
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m = junk1.i[HIGH_HALF];
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n = m & hugeint; /* no sign */
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if (n > smallint && n < bigint)
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{
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y = x * log2e.x + three51.x;
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bexp = y - three51.x; /* multiply the result by 2**bexp */
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junk1.x = y;
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eps = bexp * ln_two2.x; /* x = bexp*ln(2) + t - eps */
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t = x - bexp * ln_two1.x;
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y = t + three33.x;
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base = y - three33.x; /* t rounded to a multiple of 2**-18 */
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junk2.x = y;
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del = (t - base) + (xx - eps); /* x = bexp*ln(2) + base + del */
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eps = del + del * del * (p3.x * del + p2.x);
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binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 1023) << 20;
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i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
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j = (junk2.i[LOW_HALF] & 511) << 1;
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al = coar.x[i] * fine.x[j];
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bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
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+ coar.x[i + 1] * fine.x[j + 1]);
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rem = (bet + bet * eps) + al * eps;
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res = al + rem;
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cor = (al - res) + rem;
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if (res == (res + cor * (1.0 + error + err_1)))
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return res * binexp.x;
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else
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return -10.0;
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}
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if (n <= smallint)
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return 1.0; /* if x->0 e^x=1 */
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if (n >= badint)
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{
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if (n > infint)
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return (zero / zero); /* x is NaN, return invalid */
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if (n < infint)
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return ((x > 0) ? (hhuge * hhuge) : (tiny * tiny));
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/* x is finite, cause either overflow or underflow */
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if (junk1.i[LOW_HALF] != 0)
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return (zero / zero); /* x is NaN */
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return ((x > 0) ? inf.x : zero); /* |x| = inf; return either inf or 0 */
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}
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y = x * log2e.x + three51.x;
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bexp = y - three51.x;
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junk1.x = y;
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eps = bexp * ln_two2.x;
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t = x - bexp * ln_two1.x;
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y = t + three33.x;
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base = y - three33.x;
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junk2.x = y;
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del = (t - base) + (xx - eps);
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eps = del + del * del * (p3.x * del + p2.x);
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i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
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j = (junk2.i[LOW_HALF] & 511) << 1;
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al = coar.x[i] * fine.x[j];
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bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
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+ coar.x[i + 1] * fine.x[j + 1]);
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rem = (bet + bet * eps) + al * eps;
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res = al + rem;
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cor = (al - res) + rem;
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if (m >> 31)
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{
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ex = junk1.i[LOW_HALF];
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if (res < 1.0)
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{
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res += res;
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cor += cor;
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ex -= 1;
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}
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if (ex >= -1022)
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{
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binexp.i[HIGH_HALF] = (1023 + ex) << 20;
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if (res == (res + cor * (1.0 + error + err_1)))
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return res * binexp.x;
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else
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return -10.0;
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}
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ex = -(1022 + ex);
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binexp.i[HIGH_HALF] = (1023 - ex) << 20;
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res *= binexp.x;
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cor *= binexp.x;
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eps = 1.00000000001 + (error + err_1) * binexp.x;
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t = 1.0 + res;
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y = ((1.0 - t) + res) + cor;
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res = t + y;
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cor = (t - res) + y;
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if (res == (res + eps * cor))
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{
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binexp.i[HIGH_HALF] = 0x00100000;
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return (res - 1.0) * binexp.x;
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}
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else
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return -10.0;
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}
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else
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{
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binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 767) << 20;
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if (res == (res + cor * (1.0 + error + err_1)))
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return res * binexp.x * t256.x;
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else
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return -10.0;
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}
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}
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