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734 |
jeremybenn |
/* Copyright (C) 2007, 2009 Free Software Foundation, Inc.
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This file is part of GCC.
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GCC is free software; you can redistribute it and/or modify it under
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the terms of the GNU General Public License as published by the Free
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Software Foundation; either version 3, or (at your option) any later
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version.
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GCC is distributed in the hope that it will be useful, but WITHOUT ANY
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WARRANTY; without even the implied warranty of MERCHANTABILITY or
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FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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for more details.
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Under Section 7 of GPL version 3, you are granted additional
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permissions described in the GCC Runtime Library Exception, version
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3.1, as published by the Free Software Foundation.
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You should have received a copy of the GNU General Public License and
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a copy of the GCC Runtime Library Exception along with this program;
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see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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<http://www.gnu.org/licenses/>. */
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#include "bid_internal.h"
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#if DECIMAL_CALL_BY_REFERENCE
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void
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bid64dq_add (UINT64 * pres, UINT64 * px, UINT128 * py
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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UINT64 x = *px;
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#if !DECIMAL_GLOBAL_ROUNDING
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unsigned int rnd_mode = *prnd_mode;
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#endif
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#else
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UINT64
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bid64dq_add (UINT64 x, UINT128 y
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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#endif
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UINT64 res = 0xbaddbaddbaddbaddull;
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UINT128 x1;
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#if DECIMAL_CALL_BY_REFERENCE
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bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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bid64qq_add (&res, &x1, py
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#else
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x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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res = bid64qq_add (x1, y
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#endif
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BID_RETURN (res);
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}
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#if DECIMAL_CALL_BY_REFERENCE
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void
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bid64qd_add (UINT64 * pres, UINT128 * px, UINT64 * py
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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UINT64 y = *py;
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#if !DECIMAL_GLOBAL_ROUNDING
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unsigned int rnd_mode = *prnd_mode;
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#endif
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#else
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UINT64
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bid64qd_add (UINT128 x, UINT64 y
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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#endif
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UINT64 res = 0xbaddbaddbaddbaddull;
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UINT128 y1;
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#if DECIMAL_CALL_BY_REFERENCE
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bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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bid64qq_add (&res, px, &y1
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#else
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y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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res = bid64qq_add (x, y1
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#endif
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BID_RETURN (res);
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}
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#if DECIMAL_CALL_BY_REFERENCE
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void
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bid64qq_add (UINT64 * pres, UINT128 * px, UINT128 * py
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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UINT128 x = *px, y = *py;
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#if !DECIMAL_GLOBAL_ROUNDING
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unsigned int rnd_mode = *prnd_mode;
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#endif
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#else
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UINT64
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bid64qq_add (UINT128 x, UINT128 y
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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#endif
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UINT128 one = { {0x0000000000000001ull, 0x3040000000000000ull}
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};
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UINT64 res = 0xbaddbaddbaddbaddull;
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BID_SWAP128 (one);
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#if DECIMAL_CALL_BY_REFERENCE
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bid64qqq_fma (&res, &one, &x, &y
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#else
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res = bid64qqq_fma (one, x, y
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#endif
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BID_RETURN (res);
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}
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#if DECIMAL_CALL_BY_REFERENCE
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void
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bid128dd_add (UINT128 * pres, UINT64 * px, UINT64 * py
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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UINT64 x = *px, y = *py;
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#if !DECIMAL_GLOBAL_ROUNDING
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unsigned int rnd_mode = *prnd_mode;
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#endif
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#else
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UINT128
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bid128dd_add (UINT64 x, UINT64 y
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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#endif
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UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
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};
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UINT128 x1, y1;
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#if DECIMAL_CALL_BY_REFERENCE
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bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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bid128_add (&res, &x1, &y1
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#else
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x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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res = bid128_add (x1, y1
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#endif
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BID_RETURN (res);
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}
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#if DECIMAL_CALL_BY_REFERENCE
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void
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bid128dq_add (UINT128 * pres, UINT64 * px, UINT128 * py
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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UINT64 x = *px;
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#if !DECIMAL_GLOBAL_ROUNDING
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unsigned int rnd_mode = *prnd_mode;
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#endif
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#else
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UINT128
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bid128dq_add (UINT64 x, UINT128 y
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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#endif
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UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
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};
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UINT128 x1;
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#if DECIMAL_CALL_BY_REFERENCE
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bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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bid128_add (&res, &x1, py
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#else
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x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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res = bid128_add (x1, y
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#endif
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BID_RETURN (res);
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}
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#if DECIMAL_CALL_BY_REFERENCE
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void
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bid128qd_add (UINT128 * pres, UINT128 * px, UINT64 * py
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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UINT64 y = *py;
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#if !DECIMAL_GLOBAL_ROUNDING
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unsigned int rnd_mode = *prnd_mode;
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#endif
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#else
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UINT128
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bid128qd_add (UINT128 x, UINT64 y
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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_EXC_INFO_PARAM) {
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#endif
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UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
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};
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UINT128 y1;
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#if DECIMAL_CALL_BY_REFERENCE
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bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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bid128_add (&res, px, &y1
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#else
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y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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res = bid128_add (x, y1
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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_EXC_INFO_ARG);
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#endif
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BID_RETURN (res);
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}
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// bid128_add stands for bid128qq_add
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/*****************************************************************************
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* BID64/BID128 sub
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****************************************************************************/
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#if DECIMAL_CALL_BY_REFERENCE
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void
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bid64dq_sub (UINT64 * pres, UINT64 * px, UINT128 * py
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| 241 |
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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| 242 |
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_EXC_INFO_PARAM) {
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UINT64 x = *px;
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| 244 |
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#if !DECIMAL_GLOBAL_ROUNDING
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unsigned int rnd_mode = *prnd_mode;
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| 246 |
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#endif
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| 247 |
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#else
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| 248 |
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UINT64
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| 249 |
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bid64dq_sub (UINT64 x, UINT128 y
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| 250 |
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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| 251 |
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_EXC_INFO_PARAM) {
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#endif
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UINT64 res = 0xbaddbaddbaddbaddull;
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UINT128 x1;
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| 256 |
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#if DECIMAL_CALL_BY_REFERENCE
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bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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bid64qq_sub (&res, &x1, py
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| 259 |
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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| 260 |
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_EXC_INFO_ARG);
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#else
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x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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res = bid64qq_sub (x1, y
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| 264 |
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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| 265 |
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_EXC_INFO_ARG);
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#endif
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BID_RETURN (res);
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| 268 |
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}
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| 270 |
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| 271 |
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#if DECIMAL_CALL_BY_REFERENCE
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| 272 |
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void
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| 273 |
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bid64qd_sub (UINT64 * pres, UINT128 * px, UINT64 * py
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| 274 |
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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| 275 |
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_EXC_INFO_PARAM) {
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| 276 |
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UINT64 y = *py;
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| 277 |
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#if !DECIMAL_GLOBAL_ROUNDING
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| 278 |
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unsigned int rnd_mode = *prnd_mode;
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| 279 |
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#endif
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| 280 |
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#else
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| 281 |
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UINT64
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| 282 |
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bid64qd_sub (UINT128 x, UINT64 y
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| 283 |
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
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| 284 |
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_EXC_INFO_PARAM) {
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| 285 |
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#endif
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| 286 |
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UINT64 res = 0xbaddbaddbaddbaddull;
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| 287 |
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UINT128 y1;
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| 288 |
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| 289 |
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#if DECIMAL_CALL_BY_REFERENCE
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| 290 |
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bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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| 291 |
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bid64qq_sub (&res, px, &y1
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| 292 |
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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| 293 |
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_EXC_INFO_ARG);
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| 294 |
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#else
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| 295 |
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y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
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res = bid64qq_sub (x, y1
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| 297 |
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_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
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| 298 |
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_EXC_INFO_ARG);
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| 299 |
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#endif
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| 300 |
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BID_RETURN (res);
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| 301 |
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}
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| 302 |
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| 303 |
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| 304 |
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#if DECIMAL_CALL_BY_REFERENCE
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| 305 |
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void
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| 306 |
|
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bid64qq_sub (UINT64 * pres, UINT128 * px, UINT128 * py
|
| 307 |
|
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_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 308 |
|
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_EXC_INFO_PARAM) {
|
| 309 |
|
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UINT128 x = *px, y = *py;
|
| 310 |
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#if !DECIMAL_GLOBAL_ROUNDING
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| 311 |
|
|
unsigned int rnd_mode = *prnd_mode;
|
| 312 |
|
|
#endif
|
| 313 |
|
|
#else
|
| 314 |
|
|
UINT64
|
| 315 |
|
|
bid64qq_sub (UINT128 x, UINT128 y
|
| 316 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 317 |
|
|
_EXC_INFO_PARAM) {
|
| 318 |
|
|
#endif
|
| 319 |
|
|
|
| 320 |
|
|
UINT128 one = { {0x0000000000000001ull, 0x3040000000000000ull}
|
| 321 |
|
|
};
|
| 322 |
|
|
UINT64 res = 0xbaddbaddbaddbaddull;
|
| 323 |
|
|
UINT64 y_sign;
|
| 324 |
|
|
|
| 325 |
|
|
BID_SWAP128 (one);
|
| 326 |
|
|
if ((y.w[HIGH_128W] & MASK_NAN) != MASK_NAN) { // y is not NAN
|
| 327 |
|
|
// change its sign
|
| 328 |
|
|
y_sign = y.w[HIGH_128W] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
| 329 |
|
|
if (y_sign)
|
| 330 |
|
|
y.w[HIGH_128W] = y.w[HIGH_128W] & 0x7fffffffffffffffull;
|
| 331 |
|
|
else
|
| 332 |
|
|
y.w[HIGH_128W] = y.w[HIGH_128W] | 0x8000000000000000ull;
|
| 333 |
|
|
}
|
| 334 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 335 |
|
|
bid64qqq_fma (&res, &one, &x, &y
|
| 336 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 337 |
|
|
_EXC_INFO_ARG);
|
| 338 |
|
|
#else
|
| 339 |
|
|
res = bid64qqq_fma (one, x, y
|
| 340 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 341 |
|
|
_EXC_INFO_ARG);
|
| 342 |
|
|
#endif
|
| 343 |
|
|
BID_RETURN (res);
|
| 344 |
|
|
}
|
| 345 |
|
|
|
| 346 |
|
|
|
| 347 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 348 |
|
|
void
|
| 349 |
|
|
bid128dd_sub (UINT128 * pres, UINT64 * px, UINT64 * py
|
| 350 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 351 |
|
|
_EXC_INFO_PARAM) {
|
| 352 |
|
|
UINT64 x = *px, y = *py;
|
| 353 |
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
| 354 |
|
|
unsigned int rnd_mode = *prnd_mode;
|
| 355 |
|
|
#endif
|
| 356 |
|
|
#else
|
| 357 |
|
|
UINT128
|
| 358 |
|
|
bid128dd_sub (UINT64 x, UINT64 y
|
| 359 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 360 |
|
|
_EXC_INFO_PARAM) {
|
| 361 |
|
|
#endif
|
| 362 |
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
|
| 363 |
|
|
};
|
| 364 |
|
|
UINT128 x1, y1;
|
| 365 |
|
|
|
| 366 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 367 |
|
|
bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
| 368 |
|
|
bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
| 369 |
|
|
bid128_sub (&res, &x1, &y1
|
| 370 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 371 |
|
|
_EXC_INFO_ARG);
|
| 372 |
|
|
#else
|
| 373 |
|
|
x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
| 374 |
|
|
y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
| 375 |
|
|
res = bid128_sub (x1, y1
|
| 376 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 377 |
|
|
_EXC_INFO_ARG);
|
| 378 |
|
|
#endif
|
| 379 |
|
|
BID_RETURN (res);
|
| 380 |
|
|
}
|
| 381 |
|
|
|
| 382 |
|
|
|
| 383 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 384 |
|
|
void
|
| 385 |
|
|
bid128dq_sub (UINT128 * pres, UINT64 * px, UINT128 * py
|
| 386 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 387 |
|
|
_EXC_INFO_PARAM) {
|
| 388 |
|
|
UINT64 x = *px;
|
| 389 |
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
| 390 |
|
|
unsigned int rnd_mode = *prnd_mode;
|
| 391 |
|
|
#endif
|
| 392 |
|
|
#else
|
| 393 |
|
|
UINT128
|
| 394 |
|
|
bid128dq_sub (UINT64 x, UINT128 y
|
| 395 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 396 |
|
|
_EXC_INFO_PARAM) {
|
| 397 |
|
|
#endif
|
| 398 |
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
|
| 399 |
|
|
};
|
| 400 |
|
|
UINT128 x1;
|
| 401 |
|
|
|
| 402 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 403 |
|
|
bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
| 404 |
|
|
bid128_sub (&res, &x1, py
|
| 405 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 406 |
|
|
_EXC_INFO_ARG);
|
| 407 |
|
|
#else
|
| 408 |
|
|
x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
| 409 |
|
|
res = bid128_sub (x1, y
|
| 410 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 411 |
|
|
_EXC_INFO_ARG);
|
| 412 |
|
|
#endif
|
| 413 |
|
|
BID_RETURN (res);
|
| 414 |
|
|
}
|
| 415 |
|
|
|
| 416 |
|
|
|
| 417 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 418 |
|
|
void
|
| 419 |
|
|
bid128qd_sub (UINT128 * pres, UINT128 * px, UINT64 * py
|
| 420 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 421 |
|
|
_EXC_INFO_PARAM) {
|
| 422 |
|
|
UINT64 y = *py;
|
| 423 |
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
| 424 |
|
|
unsigned int rnd_mode = *prnd_mode;
|
| 425 |
|
|
#endif
|
| 426 |
|
|
#else
|
| 427 |
|
|
UINT128
|
| 428 |
|
|
bid128qd_sub (UINT128 x, UINT64 y
|
| 429 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 430 |
|
|
_EXC_INFO_PARAM) {
|
| 431 |
|
|
#endif
|
| 432 |
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
|
| 433 |
|
|
};
|
| 434 |
|
|
UINT128 y1;
|
| 435 |
|
|
|
| 436 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 437 |
|
|
bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
| 438 |
|
|
bid128_sub (&res, px, &y1
|
| 439 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 440 |
|
|
_EXC_INFO_ARG);
|
| 441 |
|
|
#else
|
| 442 |
|
|
y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
| 443 |
|
|
res = bid128_sub (x, y1
|
| 444 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 445 |
|
|
_EXC_INFO_ARG);
|
| 446 |
|
|
#endif
|
| 447 |
|
|
BID_RETURN (res);
|
| 448 |
|
|
}
|
| 449 |
|
|
|
| 450 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 451 |
|
|
void
|
| 452 |
|
|
bid128_add (UINT128 * pres, UINT128 * px, UINT128 * py
|
| 453 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 454 |
|
|
_EXC_INFO_PARAM) {
|
| 455 |
|
|
UINT128 x = *px, y = *py;
|
| 456 |
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
| 457 |
|
|
unsigned int rnd_mode = *prnd_mode;
|
| 458 |
|
|
#endif
|
| 459 |
|
|
#else
|
| 460 |
|
|
UINT128
|
| 461 |
|
|
bid128_add (UINT128 x, UINT128 y
|
| 462 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 463 |
|
|
_EXC_INFO_PARAM) {
|
| 464 |
|
|
#endif
|
| 465 |
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
|
| 466 |
|
|
};
|
| 467 |
|
|
UINT64 x_sign, y_sign, tmp_sign;
|
| 468 |
|
|
UINT64 x_exp, y_exp, tmp_exp; // e1 = x_exp, e2 = y_exp
|
| 469 |
|
|
UINT64 C1_hi, C2_hi, tmp_signif_hi;
|
| 470 |
|
|
UINT64 C1_lo, C2_lo, tmp_signif_lo;
|
| 471 |
|
|
// Note: C1.w[1], C1.w[0] represent C1_hi, C1_lo (all UINT64)
|
| 472 |
|
|
// Note: C2.w[1], C2.w[0] represent C2_hi, C2_lo (all UINT64)
|
| 473 |
|
|
UINT64 tmp64, tmp64A, tmp64B;
|
| 474 |
|
|
BID_UI64DOUBLE tmp1, tmp2;
|
| 475 |
|
|
int x_nr_bits, y_nr_bits;
|
| 476 |
|
|
int q1, q2, delta, scale, x1, ind, shift, tmp_inexact = 0;
|
| 477 |
|
|
UINT64 halfulp64;
|
| 478 |
|
|
UINT128 halfulp128;
|
| 479 |
|
|
UINT128 C1, C2;
|
| 480 |
|
|
UINT128 ten2m1;
|
| 481 |
|
|
UINT128 highf2star; // top 128 bits in f2*; low 128 bits in R256[1], R256[0]
|
| 482 |
|
|
UINT256 P256, Q256, R256;
|
| 483 |
|
|
int is_inexact = 0, is_midpoint_lt_even = 0, is_midpoint_gt_even = 0;
|
| 484 |
|
|
int is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
| 485 |
|
|
int second_pass = 0;
|
| 486 |
|
|
|
| 487 |
|
|
BID_SWAP128 (x);
|
| 488 |
|
|
BID_SWAP128 (y);
|
| 489 |
|
|
x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
| 490 |
|
|
y_sign = y.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
| 491 |
|
|
|
| 492 |
|
|
// check for NaN or Infinity
|
| 493 |
|
|
if (((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL)
|
| 494 |
|
|
|| ((y.w[1] & MASK_SPECIAL) == MASK_SPECIAL)) {
|
| 495 |
|
|
// x is special or y is special
|
| 496 |
|
|
if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
|
| 497 |
|
|
// check first for non-canonical NaN payload
|
| 498 |
|
|
if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) ||
|
| 499 |
|
|
(((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull)
|
| 500 |
|
|
&& (x.w[0] > 0x38c15b09ffffffffull))) {
|
| 501 |
|
|
x.w[1] = x.w[1] & 0xffffc00000000000ull;
|
| 502 |
|
|
x.w[0] = 0x0ull;
|
| 503 |
|
|
}
|
| 504 |
|
|
if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
|
| 505 |
|
|
// set invalid flag
|
| 506 |
|
|
*pfpsf |= INVALID_EXCEPTION;
|
| 507 |
|
|
// return quiet (x)
|
| 508 |
|
|
res.w[1] = x.w[1] & 0xfc003fffffffffffull;
|
| 509 |
|
|
// clear out also G[6]-G[16]
|
| 510 |
|
|
res.w[0] = x.w[0];
|
| 511 |
|
|
} else { // x is QNaN
|
| 512 |
|
|
// return x
|
| 513 |
|
|
res.w[1] = x.w[1] & 0xfc003fffffffffffull;
|
| 514 |
|
|
// clear out G[6]-G[16]
|
| 515 |
|
|
res.w[0] = x.w[0];
|
| 516 |
|
|
// if y = SNaN signal invalid exception
|
| 517 |
|
|
if ((y.w[1] & MASK_SNAN) == MASK_SNAN) {
|
| 518 |
|
|
// set invalid flag
|
| 519 |
|
|
*pfpsf |= INVALID_EXCEPTION;
|
| 520 |
|
|
}
|
| 521 |
|
|
}
|
| 522 |
|
|
BID_SWAP128 (res);
|
| 523 |
|
|
BID_RETURN (res);
|
| 524 |
|
|
} else if ((y.w[1] & MASK_NAN) == MASK_NAN) { // y is NAN
|
| 525 |
|
|
// check first for non-canonical NaN payload
|
| 526 |
|
|
if (((y.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) ||
|
| 527 |
|
|
(((y.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull)
|
| 528 |
|
|
&& (y.w[0] > 0x38c15b09ffffffffull))) {
|
| 529 |
|
|
y.w[1] = y.w[1] & 0xffffc00000000000ull;
|
| 530 |
|
|
y.w[0] = 0x0ull;
|
| 531 |
|
|
}
|
| 532 |
|
|
if ((y.w[1] & MASK_SNAN) == MASK_SNAN) { // y is SNAN
|
| 533 |
|
|
// set invalid flag
|
| 534 |
|
|
*pfpsf |= INVALID_EXCEPTION;
|
| 535 |
|
|
// return quiet (y)
|
| 536 |
|
|
res.w[1] = y.w[1] & 0xfc003fffffffffffull;
|
| 537 |
|
|
// clear out also G[6]-G[16]
|
| 538 |
|
|
res.w[0] = y.w[0];
|
| 539 |
|
|
} else { // y is QNaN
|
| 540 |
|
|
// return y
|
| 541 |
|
|
res.w[1] = y.w[1] & 0xfc003fffffffffffull;
|
| 542 |
|
|
// clear out G[6]-G[16]
|
| 543 |
|
|
res.w[0] = y.w[0];
|
| 544 |
|
|
}
|
| 545 |
|
|
BID_SWAP128 (res);
|
| 546 |
|
|
BID_RETURN (res);
|
| 547 |
|
|
} else { // neither x not y is NaN; at least one is infinity
|
| 548 |
|
|
if ((x.w[1] & MASK_ANY_INF) == MASK_INF) { // x is infinity
|
| 549 |
|
|
if ((y.w[1] & MASK_ANY_INF) == MASK_INF) { // y is infinity
|
| 550 |
|
|
// if same sign, return either of them
|
| 551 |
|
|
if ((x.w[1] & MASK_SIGN) == (y.w[1] & MASK_SIGN)) {
|
| 552 |
|
|
res.w[1] = x_sign | MASK_INF;
|
| 553 |
|
|
res.w[0] = 0x0ull;
|
| 554 |
|
|
} else { // x and y are infinities of opposite signs
|
| 555 |
|
|
// set invalid flag
|
| 556 |
|
|
*pfpsf |= INVALID_EXCEPTION;
|
| 557 |
|
|
// return QNaN Indefinite
|
| 558 |
|
|
res.w[1] = 0x7c00000000000000ull;
|
| 559 |
|
|
res.w[0] = 0x0000000000000000ull;
|
| 560 |
|
|
}
|
| 561 |
|
|
} else { // y is 0 or finite
|
| 562 |
|
|
// return x
|
| 563 |
|
|
res.w[1] = x_sign | MASK_INF;
|
| 564 |
|
|
res.w[0] = 0x0ull;
|
| 565 |
|
|
}
|
| 566 |
|
|
} else { // x is not NaN or infinity, so y must be infinity
|
| 567 |
|
|
res.w[1] = y_sign | MASK_INF;
|
| 568 |
|
|
res.w[0] = 0x0ull;
|
| 569 |
|
|
}
|
| 570 |
|
|
BID_SWAP128 (res);
|
| 571 |
|
|
BID_RETURN (res);
|
| 572 |
|
|
}
|
| 573 |
|
|
}
|
| 574 |
|
|
// unpack the arguments
|
| 575 |
|
|
|
| 576 |
|
|
// unpack x
|
| 577 |
|
|
C1_hi = x.w[1] & MASK_COEFF;
|
| 578 |
|
|
C1_lo = x.w[0];
|
| 579 |
|
|
// test for non-canonical values:
|
| 580 |
|
|
// - values whose encoding begins with x00, x01, or x10 and whose
|
| 581 |
|
|
// coefficient is larger than 10^34 -1, or
|
| 582 |
|
|
// - values whose encoding begins with x1100, x1101, x1110 (if NaNs
|
| 583 |
|
|
// and infinitis were eliminated already this test is reduced to
|
| 584 |
|
|
// checking for x10x)
|
| 585 |
|
|
|
| 586 |
|
|
// x is not infinity; check for non-canonical values - treated as zero
|
| 587 |
|
|
if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) {
|
| 588 |
|
|
// G0_G1=11; non-canonical
|
| 589 |
|
|
x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits
|
| 590 |
|
|
C1_hi = 0; // significand high
|
| 591 |
|
|
C1_lo = 0; // significand low
|
| 592 |
|
|
} else { // G0_G1 != 11
|
| 593 |
|
|
x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits
|
| 594 |
|
|
if (C1_hi > 0x0001ed09bead87c0ull ||
|
| 595 |
|
|
(C1_hi == 0x0001ed09bead87c0ull
|
| 596 |
|
|
&& C1_lo > 0x378d8e63ffffffffull)) {
|
| 597 |
|
|
// x is non-canonical if coefficient is larger than 10^34 -1
|
| 598 |
|
|
C1_hi = 0;
|
| 599 |
|
|
C1_lo = 0;
|
| 600 |
|
|
} else { // canonical
|
| 601 |
|
|
;
|
| 602 |
|
|
}
|
| 603 |
|
|
}
|
| 604 |
|
|
|
| 605 |
|
|
// unpack y
|
| 606 |
|
|
C2_hi = y.w[1] & MASK_COEFF;
|
| 607 |
|
|
C2_lo = y.w[0];
|
| 608 |
|
|
// y is not infinity; check for non-canonical values - treated as zero
|
| 609 |
|
|
if ((y.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) {
|
| 610 |
|
|
// G0_G1=11; non-canonical
|
| 611 |
|
|
y_exp = (y.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits
|
| 612 |
|
|
C2_hi = 0; // significand high
|
| 613 |
|
|
C2_lo = 0; // significand low
|
| 614 |
|
|
} else { // G0_G1 != 11
|
| 615 |
|
|
y_exp = y.w[1] & MASK_EXP; // biased and shifted left 49 bits
|
| 616 |
|
|
if (C2_hi > 0x0001ed09bead87c0ull ||
|
| 617 |
|
|
(C2_hi == 0x0001ed09bead87c0ull
|
| 618 |
|
|
&& C2_lo > 0x378d8e63ffffffffull)) {
|
| 619 |
|
|
// y is non-canonical if coefficient is larger than 10^34 -1
|
| 620 |
|
|
C2_hi = 0;
|
| 621 |
|
|
C2_lo = 0;
|
| 622 |
|
|
} else { // canonical
|
| 623 |
|
|
;
|
| 624 |
|
|
}
|
| 625 |
|
|
}
|
| 626 |
|
|
|
| 627 |
|
|
if ((C1_hi == 0x0ull) && (C1_lo == 0x0ull)) {
|
| 628 |
|
|
// x is 0 and y is not special
|
| 629 |
|
|
// if y is 0 return 0 with the smaller exponent
|
| 630 |
|
|
if ((C2_hi == 0x0ull) && (C2_lo == 0x0ull)) {
|
| 631 |
|
|
if (x_exp < y_exp)
|
| 632 |
|
|
res.w[1] = x_exp;
|
| 633 |
|
|
else
|
| 634 |
|
|
res.w[1] = y_exp;
|
| 635 |
|
|
if (x_sign && y_sign)
|
| 636 |
|
|
res.w[1] = res.w[1] | x_sign; // both negative
|
| 637 |
|
|
else if (rnd_mode == ROUNDING_DOWN && x_sign != y_sign)
|
| 638 |
|
|
res.w[1] = res.w[1] | 0x8000000000000000ull; // -0
|
| 639 |
|
|
// else; // res = +0
|
| 640 |
|
|
res.w[0] = 0;
|
| 641 |
|
|
} else {
|
| 642 |
|
|
// for 0 + y return y, with the preferred exponent
|
| 643 |
|
|
if (y_exp <= x_exp) {
|
| 644 |
|
|
res.w[1] = y.w[1];
|
| 645 |
|
|
res.w[0] = y.w[0];
|
| 646 |
|
|
} else { // if y_exp > x_exp
|
| 647 |
|
|
// return (C2 * 10^scale) * 10^(y_exp - scale)
|
| 648 |
|
|
// where scale = min (P34-q2, y_exp-x_exp)
|
| 649 |
|
|
// determine q2 = nr. of decimal digits in y
|
| 650 |
|
|
// determine first the nr. of bits in y (y_nr_bits)
|
| 651 |
|
|
|
| 652 |
|
|
if (C2_hi == 0) { // y_bits is the nr. of bits in C2_lo
|
| 653 |
|
|
if (C2_lo >= 0x0020000000000000ull) { // y >= 2^53
|
| 654 |
|
|
// split the 64-bit value in two 32-bit halves to avoid
|
| 655 |
|
|
// rounding errors
|
| 656 |
|
|
if (C2_lo >= 0x0000000100000000ull) { // y >= 2^32
|
| 657 |
|
|
tmp2.d = (double) (C2_lo >> 32); // exact conversion
|
| 658 |
|
|
y_nr_bits =
|
| 659 |
|
|
32 +
|
| 660 |
|
|
((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 661 |
|
|
} else { // y < 2^32
|
| 662 |
|
|
tmp2.d = (double) (C2_lo); // exact conversion
|
| 663 |
|
|
y_nr_bits =
|
| 664 |
|
|
((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 665 |
|
|
}
|
| 666 |
|
|
} else { // if y < 2^53
|
| 667 |
|
|
tmp2.d = (double) C2_lo; // exact conversion
|
| 668 |
|
|
y_nr_bits =
|
| 669 |
|
|
((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 670 |
|
|
}
|
| 671 |
|
|
} else { // C2_hi != 0 => nr. bits = 64 + nr_bits (C2_hi)
|
| 672 |
|
|
tmp2.d = (double) C2_hi; // exact conversion
|
| 673 |
|
|
y_nr_bits =
|
| 674 |
|
|
64 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 675 |
|
|
}
|
| 676 |
|
|
q2 = nr_digits[y_nr_bits].digits;
|
| 677 |
|
|
if (q2 == 0) {
|
| 678 |
|
|
q2 = nr_digits[y_nr_bits].digits1;
|
| 679 |
|
|
if (C2_hi > nr_digits[y_nr_bits].threshold_hi ||
|
| 680 |
|
|
(C2_hi == nr_digits[y_nr_bits].threshold_hi &&
|
| 681 |
|
|
C2_lo >= nr_digits[y_nr_bits].threshold_lo))
|
| 682 |
|
|
q2++;
|
| 683 |
|
|
}
|
| 684 |
|
|
// return (C2 * 10^scale) * 10^(y_exp - scale)
|
| 685 |
|
|
// where scale = min (P34-q2, y_exp-x_exp)
|
| 686 |
|
|
scale = P34 - q2;
|
| 687 |
|
|
ind = (y_exp - x_exp) >> 49;
|
| 688 |
|
|
if (ind < scale)
|
| 689 |
|
|
scale = ind;
|
| 690 |
|
|
if (scale == 0) {
|
| 691 |
|
|
res.w[1] = y.w[1];
|
| 692 |
|
|
res.w[0] = y.w[0];
|
| 693 |
|
|
} else if (q2 <= 19) { // y fits in 64 bits
|
| 694 |
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
| 695 |
|
|
// 64 x 64 C2_lo * ten2k64[scale]
|
| 696 |
|
|
__mul_64x64_to_128MACH (res, C2_lo, ten2k64[scale]);
|
| 697 |
|
|
} else { // 10^scale fits in 128 bits
|
| 698 |
|
|
// 64 x 128 C2_lo * ten2k128[scale - 20]
|
| 699 |
|
|
__mul_128x64_to_128 (res, C2_lo, ten2k128[scale - 20]);
|
| 700 |
|
|
}
|
| 701 |
|
|
} else { // y fits in 128 bits, but 10^scale must fit in 64 bits
|
| 702 |
|
|
// 64 x 128 ten2k64[scale] * C2
|
| 703 |
|
|
C2.w[1] = C2_hi;
|
| 704 |
|
|
C2.w[0] = C2_lo;
|
| 705 |
|
|
__mul_128x64_to_128 (res, ten2k64[scale], C2);
|
| 706 |
|
|
}
|
| 707 |
|
|
// subtract scale from the exponent
|
| 708 |
|
|
y_exp = y_exp - ((UINT64) scale << 49);
|
| 709 |
|
|
res.w[1] = res.w[1] | y_sign | y_exp;
|
| 710 |
|
|
}
|
| 711 |
|
|
}
|
| 712 |
|
|
BID_SWAP128 (res);
|
| 713 |
|
|
BID_RETURN (res);
|
| 714 |
|
|
} else if ((C2_hi == 0x0ull) && (C2_lo == 0x0ull)) {
|
| 715 |
|
|
// y is 0 and x is not special, and not zero
|
| 716 |
|
|
// for x + 0 return x, with the preferred exponent
|
| 717 |
|
|
if (x_exp <= y_exp) {
|
| 718 |
|
|
res.w[1] = x.w[1];
|
| 719 |
|
|
res.w[0] = x.w[0];
|
| 720 |
|
|
} else { // if x_exp > y_exp
|
| 721 |
|
|
// return (C1 * 10^scale) * 10^(x_exp - scale)
|
| 722 |
|
|
// where scale = min (P34-q1, x_exp-y_exp)
|
| 723 |
|
|
// determine q1 = nr. of decimal digits in x
|
| 724 |
|
|
// determine first the nr. of bits in x
|
| 725 |
|
|
if (C1_hi == 0) { // x_bits is the nr. of bits in C1_lo
|
| 726 |
|
|
if (C1_lo >= 0x0020000000000000ull) { // x >= 2^53
|
| 727 |
|
|
// split the 64-bit value in two 32-bit halves to avoid
|
| 728 |
|
|
// rounding errors
|
| 729 |
|
|
if (C1_lo >= 0x0000000100000000ull) { // x >= 2^32
|
| 730 |
|
|
tmp1.d = (double) (C1_lo >> 32); // exact conversion
|
| 731 |
|
|
x_nr_bits =
|
| 732 |
|
|
32 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) -
|
| 733 |
|
|
0x3ff);
|
| 734 |
|
|
} else { // x < 2^32
|
| 735 |
|
|
tmp1.d = (double) (C1_lo); // exact conversion
|
| 736 |
|
|
x_nr_bits =
|
| 737 |
|
|
((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 738 |
|
|
}
|
| 739 |
|
|
} else { // if x < 2^53
|
| 740 |
|
|
tmp1.d = (double) C1_lo; // exact conversion
|
| 741 |
|
|
x_nr_bits =
|
| 742 |
|
|
((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 743 |
|
|
}
|
| 744 |
|
|
} else { // C1_hi != 0 => nr. bits = 64 + nr_bits (C1_hi)
|
| 745 |
|
|
tmp1.d = (double) C1_hi; // exact conversion
|
| 746 |
|
|
x_nr_bits =
|
| 747 |
|
|
64 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 748 |
|
|
}
|
| 749 |
|
|
q1 = nr_digits[x_nr_bits].digits;
|
| 750 |
|
|
if (q1 == 0) {
|
| 751 |
|
|
q1 = nr_digits[x_nr_bits].digits1;
|
| 752 |
|
|
if (C1_hi > nr_digits[x_nr_bits].threshold_hi ||
|
| 753 |
|
|
(C1_hi == nr_digits[x_nr_bits].threshold_hi &&
|
| 754 |
|
|
C1_lo >= nr_digits[x_nr_bits].threshold_lo))
|
| 755 |
|
|
q1++;
|
| 756 |
|
|
}
|
| 757 |
|
|
// return (C1 * 10^scale) * 10^(x_exp - scale)
|
| 758 |
|
|
// where scale = min (P34-q1, x_exp-y_exp)
|
| 759 |
|
|
scale = P34 - q1;
|
| 760 |
|
|
ind = (x_exp - y_exp) >> 49;
|
| 761 |
|
|
if (ind < scale)
|
| 762 |
|
|
scale = ind;
|
| 763 |
|
|
if (scale == 0) {
|
| 764 |
|
|
res.w[1] = x.w[1];
|
| 765 |
|
|
res.w[0] = x.w[0];
|
| 766 |
|
|
} else if (q1 <= 19) { // x fits in 64 bits
|
| 767 |
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
| 768 |
|
|
// 64 x 64 C1_lo * ten2k64[scale]
|
| 769 |
|
|
__mul_64x64_to_128MACH (res, C1_lo, ten2k64[scale]);
|
| 770 |
|
|
} else { // 10^scale fits in 128 bits
|
| 771 |
|
|
// 64 x 128 C1_lo * ten2k128[scale - 20]
|
| 772 |
|
|
__mul_128x64_to_128 (res, C1_lo, ten2k128[scale - 20]);
|
| 773 |
|
|
}
|
| 774 |
|
|
} else { // x fits in 128 bits, but 10^scale must fit in 64 bits
|
| 775 |
|
|
// 64 x 128 ten2k64[scale] * C1
|
| 776 |
|
|
C1.w[1] = C1_hi;
|
| 777 |
|
|
C1.w[0] = C1_lo;
|
| 778 |
|
|
__mul_128x64_to_128 (res, ten2k64[scale], C1);
|
| 779 |
|
|
}
|
| 780 |
|
|
// subtract scale from the exponent
|
| 781 |
|
|
x_exp = x_exp - ((UINT64) scale << 49);
|
| 782 |
|
|
res.w[1] = res.w[1] | x_sign | x_exp;
|
| 783 |
|
|
}
|
| 784 |
|
|
BID_SWAP128 (res);
|
| 785 |
|
|
BID_RETURN (res);
|
| 786 |
|
|
} else { // x and y are not canonical, not special, and are not zero
|
| 787 |
|
|
// note that the result may still be zero, and then it has to have the
|
| 788 |
|
|
// preferred exponent
|
| 789 |
|
|
if (x_exp < y_exp) { // if exp_x < exp_y then swap x and y
|
| 790 |
|
|
tmp_sign = x_sign;
|
| 791 |
|
|
tmp_exp = x_exp;
|
| 792 |
|
|
tmp_signif_hi = C1_hi;
|
| 793 |
|
|
tmp_signif_lo = C1_lo;
|
| 794 |
|
|
x_sign = y_sign;
|
| 795 |
|
|
x_exp = y_exp;
|
| 796 |
|
|
C1_hi = C2_hi;
|
| 797 |
|
|
C1_lo = C2_lo;
|
| 798 |
|
|
y_sign = tmp_sign;
|
| 799 |
|
|
y_exp = tmp_exp;
|
| 800 |
|
|
C2_hi = tmp_signif_hi;
|
| 801 |
|
|
C2_lo = tmp_signif_lo;
|
| 802 |
|
|
}
|
| 803 |
|
|
// q1 = nr. of decimal digits in x
|
| 804 |
|
|
// determine first the nr. of bits in x
|
| 805 |
|
|
if (C1_hi == 0) { // x_bits is the nr. of bits in C1_lo
|
| 806 |
|
|
if (C1_lo >= 0x0020000000000000ull) { // x >= 2^53
|
| 807 |
|
|
//split the 64-bit value in two 32-bit halves to avoid rounding errors
|
| 808 |
|
|
if (C1_lo >= 0x0000000100000000ull) { // x >= 2^32
|
| 809 |
|
|
tmp1.d = (double) (C1_lo >> 32); // exact conversion
|
| 810 |
|
|
x_nr_bits =
|
| 811 |
|
|
32 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 812 |
|
|
} else { // x < 2^32
|
| 813 |
|
|
tmp1.d = (double) (C1_lo); // exact conversion
|
| 814 |
|
|
x_nr_bits =
|
| 815 |
|
|
((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 816 |
|
|
}
|
| 817 |
|
|
} else { // if x < 2^53
|
| 818 |
|
|
tmp1.d = (double) C1_lo; // exact conversion
|
| 819 |
|
|
x_nr_bits =
|
| 820 |
|
|
((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 821 |
|
|
}
|
| 822 |
|
|
} else { // C1_hi != 0 => nr. bits = 64 + nr_bits (C1_hi)
|
| 823 |
|
|
tmp1.d = (double) C1_hi; // exact conversion
|
| 824 |
|
|
x_nr_bits =
|
| 825 |
|
|
64 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 826 |
|
|
}
|
| 827 |
|
|
|
| 828 |
|
|
q1 = nr_digits[x_nr_bits].digits;
|
| 829 |
|
|
if (q1 == 0) {
|
| 830 |
|
|
q1 = nr_digits[x_nr_bits].digits1;
|
| 831 |
|
|
if (C1_hi > nr_digits[x_nr_bits].threshold_hi ||
|
| 832 |
|
|
(C1_hi == nr_digits[x_nr_bits].threshold_hi &&
|
| 833 |
|
|
C1_lo >= nr_digits[x_nr_bits].threshold_lo))
|
| 834 |
|
|
q1++;
|
| 835 |
|
|
}
|
| 836 |
|
|
// q2 = nr. of decimal digits in y
|
| 837 |
|
|
// determine first the nr. of bits in y (y_nr_bits)
|
| 838 |
|
|
if (C2_hi == 0) { // y_bits is the nr. of bits in C2_lo
|
| 839 |
|
|
if (C2_lo >= 0x0020000000000000ull) { // y >= 2^53
|
| 840 |
|
|
//split the 64-bit value in two 32-bit halves to avoid rounding errors
|
| 841 |
|
|
if (C2_lo >= 0x0000000100000000ull) { // y >= 2^32
|
| 842 |
|
|
tmp2.d = (double) (C2_lo >> 32); // exact conversion
|
| 843 |
|
|
y_nr_bits =
|
| 844 |
|
|
32 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 845 |
|
|
} else { // y < 2^32
|
| 846 |
|
|
tmp2.d = (double) (C2_lo); // exact conversion
|
| 847 |
|
|
y_nr_bits =
|
| 848 |
|
|
((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 849 |
|
|
}
|
| 850 |
|
|
} else { // if y < 2^53
|
| 851 |
|
|
tmp2.d = (double) C2_lo; // exact conversion
|
| 852 |
|
|
y_nr_bits =
|
| 853 |
|
|
((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 854 |
|
|
}
|
| 855 |
|
|
} else { // C2_hi != 0 => nr. bits = 64 + nr_bits (C2_hi)
|
| 856 |
|
|
tmp2.d = (double) C2_hi; // exact conversion
|
| 857 |
|
|
y_nr_bits =
|
| 858 |
|
|
64 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
| 859 |
|
|
}
|
| 860 |
|
|
|
| 861 |
|
|
q2 = nr_digits[y_nr_bits].digits;
|
| 862 |
|
|
if (q2 == 0) {
|
| 863 |
|
|
q2 = nr_digits[y_nr_bits].digits1;
|
| 864 |
|
|
if (C2_hi > nr_digits[y_nr_bits].threshold_hi ||
|
| 865 |
|
|
(C2_hi == nr_digits[y_nr_bits].threshold_hi &&
|
| 866 |
|
|
C2_lo >= nr_digits[y_nr_bits].threshold_lo))
|
| 867 |
|
|
q2++;
|
| 868 |
|
|
}
|
| 869 |
|
|
|
| 870 |
|
|
delta = q1 + (int) (x_exp >> 49) - q2 - (int) (y_exp >> 49);
|
| 871 |
|
|
|
| 872 |
|
|
if (delta >= P34) {
|
| 873 |
|
|
// round the result directly because 0 < C2 < ulp (C1 * 10^(x_exp-e2))
|
| 874 |
|
|
// n = C1 * 10^e1 or n = C1 +/- 10^(q1-P34)) * 10^e1
|
| 875 |
|
|
// the result is inexact; the preferred exponent is the least possible
|
| 876 |
|
|
|
| 877 |
|
|
if (delta >= P34 + 1) {
|
| 878 |
|
|
// for RN the result is the operand with the larger magnitude,
|
| 879 |
|
|
// possibly scaled up by 10^(P34-q1)
|
| 880 |
|
|
// an overflow cannot occur in this case (rounding to nearest)
|
| 881 |
|
|
if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
| 882 |
|
|
// Note: because delta >= P34+1 it is certain that
|
| 883 |
|
|
// x_exp - ((UINT64)scale << 49) will stay above e_min
|
| 884 |
|
|
scale = P34 - q1;
|
| 885 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 886 |
|
|
// 1 <= q1 <= 19 => 15 <= scale <= 33
|
| 887 |
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
| 888 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
| 889 |
|
|
} else { // if 20 <= scale <= 33
|
| 890 |
|
|
// C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
| 891 |
|
|
// (C1 * 10^(scale-19)) fits in 64 bits
|
| 892 |
|
|
C1_lo = C1_lo * ten2k64[scale - 19];
|
| 893 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
| 894 |
|
|
}
|
| 895 |
|
|
} else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
| 896 |
|
|
// => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
| 897 |
|
|
C1.w[1] = C1_hi;
|
| 898 |
|
|
C1.w[0] = C1_lo;
|
| 899 |
|
|
// C1 = ten2k64[P34 - q1] * C1
|
| 900 |
|
|
__mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
| 901 |
|
|
}
|
| 902 |
|
|
x_exp = x_exp - ((UINT64) scale << 49);
|
| 903 |
|
|
C1_hi = C1.w[1];
|
| 904 |
|
|
C1_lo = C1.w[0];
|
| 905 |
|
|
}
|
| 906 |
|
|
// some special cases arise: if delta = P34 + 1 and C1 = 10^(P34-1)
|
| 907 |
|
|
// (after scaling) and x_sign != y_sign and C2 > 5*10^(q2-1) =>
|
| 908 |
|
|
// subtract 1 ulp
|
| 909 |
|
|
// Note: do this only for rounding to nearest; for other rounding
|
| 910 |
|
|
// modes the correction will be applied next
|
| 911 |
|
|
if ((rnd_mode == ROUNDING_TO_NEAREST
|
| 912 |
|
|
|| rnd_mode == ROUNDING_TIES_AWAY) && delta == (P34 + 1)
|
| 913 |
|
|
&& C1_hi == 0x0000314dc6448d93ull
|
| 914 |
|
|
&& C1_lo == 0x38c15b0a00000000ull && x_sign != y_sign
|
| 915 |
|
|
&& ((q2 <= 19 && C2_lo > midpoint64[q2 - 1]) || (q2 >= 20
|
| 916 |
|
|
&& (C2_hi >
|
| 917 |
|
|
midpoint128
|
| 918 |
|
|
[q2 -
|
| 919 |
|
|
20].
|
| 920 |
|
|
w[1]
|
| 921 |
|
|
||
|
| 922 |
|
|
(C2_hi
|
| 923 |
|
|
==
|
| 924 |
|
|
midpoint128
|
| 925 |
|
|
[q2 -
|
| 926 |
|
|
20].
|
| 927 |
|
|
w[1]
|
| 928 |
|
|
&&
|
| 929 |
|
|
C2_lo
|
| 930 |
|
|
>
|
| 931 |
|
|
midpoint128
|
| 932 |
|
|
[q2 -
|
| 933 |
|
|
20].
|
| 934 |
|
|
w
|
| 935 |
|
|
[0])))))
|
| 936 |
|
|
{
|
| 937 |
|
|
// C1 = 10^34 - 1 and decrement x_exp by 1 (no underflow possible)
|
| 938 |
|
|
C1_hi = 0x0001ed09bead87c0ull;
|
| 939 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 940 |
|
|
x_exp = x_exp - EXP_P1;
|
| 941 |
|
|
}
|
| 942 |
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
| 943 |
|
|
if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
|
| 944 |
|
|
(rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
|
| 945 |
|
|
// add 1 ulp and then check for overflow
|
| 946 |
|
|
C1_lo = C1_lo + 1;
|
| 947 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 948 |
|
|
C1_hi = C1_hi + 1;
|
| 949 |
|
|
}
|
| 950 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 951 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 952 |
|
|
// C1 = 10^34 => rounding overflow
|
| 953 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 954 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 955 |
|
|
x_exp = x_exp + EXP_P1;
|
| 956 |
|
|
if (x_exp == EXP_MAX_P1) { // overflow
|
| 957 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 958 |
|
|
C1_lo = 0x0ull;
|
| 959 |
|
|
x_exp = 0; // x_sign is preserved
|
| 960 |
|
|
// set overflow flag (the inexact flag was set too)
|
| 961 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 962 |
|
|
}
|
| 963 |
|
|
}
|
| 964 |
|
|
} else if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign) ||
|
| 965 |
|
|
(rnd_mode == ROUNDING_UP && x_sign && !y_sign) ||
|
| 966 |
|
|
(rnd_mode == ROUNDING_TO_ZERO
|
| 967 |
|
|
&& x_sign != y_sign)) {
|
| 968 |
|
|
// subtract 1 ulp from C1
|
| 969 |
|
|
// Note: because delta >= P34 + 1 the result cannot be zero
|
| 970 |
|
|
C1_lo = C1_lo - 1;
|
| 971 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 972 |
|
|
C1_hi = C1_hi - 1;
|
| 973 |
|
|
// if the coefficient is 10^33 - 1 then make it 10^34 - 1 and
|
| 974 |
|
|
// decrease the exponent by 1 (because delta >= P34 + 1 the
|
| 975 |
|
|
// exponent will not become less than e_min)
|
| 976 |
|
|
// 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
| 977 |
|
|
// 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
| 978 |
|
|
if (C1_hi == 0x0000314dc6448d93ull
|
| 979 |
|
|
&& C1_lo == 0x38c15b09ffffffffull) {
|
| 980 |
|
|
// make C1 = 10^34 - 1
|
| 981 |
|
|
C1_hi = 0x0001ed09bead87c0ull;
|
| 982 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 983 |
|
|
x_exp = x_exp - EXP_P1;
|
| 984 |
|
|
}
|
| 985 |
|
|
} else {
|
| 986 |
|
|
; // the result is already correct
|
| 987 |
|
|
}
|
| 988 |
|
|
}
|
| 989 |
|
|
// set the inexact flag
|
| 990 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 991 |
|
|
// assemble the result
|
| 992 |
|
|
res.w[1] = x_sign | x_exp | C1_hi;
|
| 993 |
|
|
res.w[0] = C1_lo;
|
| 994 |
|
|
} else { // delta = P34
|
| 995 |
|
|
// in most cases, the smaller operand may be < or = or > 1/2 ulp of the
|
| 996 |
|
|
// larger operand
|
| 997 |
|
|
// however, the case C1 = 10^(q1-1) and x_sign != y_sign is special due
|
| 998 |
|
|
// to accuracy loss after subtraction, and will be treated separately
|
| 999 |
|
|
if (x_sign == y_sign || (q1 <= 20
|
| 1000 |
|
|
&& (C1_hi != 0
|
| 1001 |
|
|
|| C1_lo != ten2k64[q1 - 1]))
|
| 1002 |
|
|
|| (q1 >= 21 && (C1_hi != ten2k128[q1 - 21].w[1]
|
| 1003 |
|
|
|| C1_lo != ten2k128[q1 - 21].w[0]))) {
|
| 1004 |
|
|
// if x_sign == y_sign or C1 != 10^(q1-1)
|
| 1005 |
|
|
// compare C2 with 1/2 ulp = 5 * 10^(q2-1), the latter read from table
|
| 1006 |
|
|
// Note: cases q1<=19 and q1>=20 can be coalesced at some latency cost
|
| 1007 |
|
|
if (q2 <= 19) { // C2 and 5*10^(q2-1) both fit in 64 bits
|
| 1008 |
|
|
halfulp64 = midpoint64[q2 - 1]; // 5 * 10^(q2-1)
|
| 1009 |
|
|
if (C2_lo < halfulp64) { // n2 < 1/2 ulp (n1)
|
| 1010 |
|
|
// for RN the result is the operand with the larger magnitude,
|
| 1011 |
|
|
// possibly scaled up by 10^(P34-q1)
|
| 1012 |
|
|
// an overflow cannot occur in this case (rounding to nearest)
|
| 1013 |
|
|
if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
| 1014 |
|
|
// Note: because delta = P34 it is certain that
|
| 1015 |
|
|
// x_exp - ((UINT64)scale << 49) will stay above e_min
|
| 1016 |
|
|
scale = P34 - q1;
|
| 1017 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 1018 |
|
|
// 1 <= q1 <= 19 => 15 <= scale <= 33
|
| 1019 |
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
| 1020 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
| 1021 |
|
|
} else { // if 20 <= scale <= 33
|
| 1022 |
|
|
// C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
| 1023 |
|
|
// (C1 * 10^(scale-19)) fits in 64 bits
|
| 1024 |
|
|
C1_lo = C1_lo * ten2k64[scale - 19];
|
| 1025 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
| 1026 |
|
|
}
|
| 1027 |
|
|
} else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
| 1028 |
|
|
// => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
| 1029 |
|
|
C1.w[1] = C1_hi;
|
| 1030 |
|
|
C1.w[0] = C1_lo;
|
| 1031 |
|
|
// C1 = ten2k64[P34 - q1] * C1
|
| 1032 |
|
|
__mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
| 1033 |
|
|
}
|
| 1034 |
|
|
x_exp = x_exp - ((UINT64) scale << 49);
|
| 1035 |
|
|
C1_hi = C1.w[1];
|
| 1036 |
|
|
C1_lo = C1.w[0];
|
| 1037 |
|
|
}
|
| 1038 |
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
| 1039 |
|
|
if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
|
| 1040 |
|
|
(rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
|
| 1041 |
|
|
// add 1 ulp and then check for overflow
|
| 1042 |
|
|
C1_lo = C1_lo + 1;
|
| 1043 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 1044 |
|
|
C1_hi = C1_hi + 1;
|
| 1045 |
|
|
}
|
| 1046 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 1047 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 1048 |
|
|
// C1 = 10^34 => rounding overflow
|
| 1049 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 1050 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 1051 |
|
|
x_exp = x_exp + EXP_P1;
|
| 1052 |
|
|
if (x_exp == EXP_MAX_P1) { // overflow
|
| 1053 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 1054 |
|
|
C1_lo = 0x0ull;
|
| 1055 |
|
|
x_exp = 0; // x_sign is preserved
|
| 1056 |
|
|
// set overflow flag (the inexact flag was set too)
|
| 1057 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 1058 |
|
|
}
|
| 1059 |
|
|
}
|
| 1060 |
|
|
} else
|
| 1061 |
|
|
if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
|
| 1062 |
|
|
|| (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
| 1063 |
|
|
|| (rnd_mode == ROUNDING_TO_ZERO
|
| 1064 |
|
|
&& x_sign != y_sign)) {
|
| 1065 |
|
|
// subtract 1 ulp from C1
|
| 1066 |
|
|
// Note: because delta >= P34 + 1 the result cannot be zero
|
| 1067 |
|
|
C1_lo = C1_lo - 1;
|
| 1068 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 1069 |
|
|
C1_hi = C1_hi - 1;
|
| 1070 |
|
|
// if the coefficient is 10^33-1 then make it 10^34-1 and
|
| 1071 |
|
|
// decrease the exponent by 1 (because delta >= P34 + 1 the
|
| 1072 |
|
|
// exponent will not become less than e_min)
|
| 1073 |
|
|
// 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
| 1074 |
|
|
// 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
| 1075 |
|
|
if (C1_hi == 0x0000314dc6448d93ull
|
| 1076 |
|
|
&& C1_lo == 0x38c15b09ffffffffull) {
|
| 1077 |
|
|
// make C1 = 10^34 - 1
|
| 1078 |
|
|
C1_hi = 0x0001ed09bead87c0ull;
|
| 1079 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 1080 |
|
|
x_exp = x_exp - EXP_P1;
|
| 1081 |
|
|
}
|
| 1082 |
|
|
} else {
|
| 1083 |
|
|
; // the result is already correct
|
| 1084 |
|
|
}
|
| 1085 |
|
|
}
|
| 1086 |
|
|
// set the inexact flag
|
| 1087 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1088 |
|
|
// assemble the result
|
| 1089 |
|
|
res.w[1] = x_sign | x_exp | C1_hi;
|
| 1090 |
|
|
res.w[0] = C1_lo;
|
| 1091 |
|
|
} else if ((C2_lo == halfulp64)
|
| 1092 |
|
|
&& (q1 < P34 || ((C1_lo & 0x1) == 0))) {
|
| 1093 |
|
|
// n2 = 1/2 ulp (n1) and C1 is even
|
| 1094 |
|
|
// the result is the operand with the larger magnitude,
|
| 1095 |
|
|
// possibly scaled up by 10^(P34-q1)
|
| 1096 |
|
|
// an overflow cannot occur in this case (rounding to nearest)
|
| 1097 |
|
|
if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
| 1098 |
|
|
// Note: because delta = P34 it is certain that
|
| 1099 |
|
|
// x_exp - ((UINT64)scale << 49) will stay above e_min
|
| 1100 |
|
|
scale = P34 - q1;
|
| 1101 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 1102 |
|
|
// 1 <= q1 <= 19 => 15 <= scale <= 33
|
| 1103 |
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
| 1104 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
| 1105 |
|
|
} else { // if 20 <= scale <= 33
|
| 1106 |
|
|
// C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
| 1107 |
|
|
// (C1 * 10^(scale-19)) fits in 64 bits
|
| 1108 |
|
|
C1_lo = C1_lo * ten2k64[scale - 19];
|
| 1109 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
| 1110 |
|
|
}
|
| 1111 |
|
|
} else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
| 1112 |
|
|
// => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
| 1113 |
|
|
C1.w[1] = C1_hi;
|
| 1114 |
|
|
C1.w[0] = C1_lo;
|
| 1115 |
|
|
// C1 = ten2k64[P34 - q1] * C1
|
| 1116 |
|
|
__mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
| 1117 |
|
|
}
|
| 1118 |
|
|
x_exp = x_exp - ((UINT64) scale << 49);
|
| 1119 |
|
|
C1_hi = C1.w[1];
|
| 1120 |
|
|
C1_lo = C1.w[0];
|
| 1121 |
|
|
}
|
| 1122 |
|
|
if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign == y_sign
|
| 1123 |
|
|
&& (C1_lo & 0x01)) || (rnd_mode == ROUNDING_TIES_AWAY
|
| 1124 |
|
|
&& x_sign == y_sign)
|
| 1125 |
|
|
|| (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)
|
| 1126 |
|
|
|| (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)) {
|
| 1127 |
|
|
// add 1 ulp and then check for overflow
|
| 1128 |
|
|
C1_lo = C1_lo + 1;
|
| 1129 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 1130 |
|
|
C1_hi = C1_hi + 1;
|
| 1131 |
|
|
}
|
| 1132 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 1133 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 1134 |
|
|
// C1 = 10^34 => rounding overflow
|
| 1135 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 1136 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 1137 |
|
|
x_exp = x_exp + EXP_P1;
|
| 1138 |
|
|
if (x_exp == EXP_MAX_P1) { // overflow
|
| 1139 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 1140 |
|
|
C1_lo = 0x0ull;
|
| 1141 |
|
|
x_exp = 0; // x_sign is preserved
|
| 1142 |
|
|
// set overflow flag (the inexact flag was set too)
|
| 1143 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 1144 |
|
|
}
|
| 1145 |
|
|
}
|
| 1146 |
|
|
} else
|
| 1147 |
|
|
if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign != y_sign
|
| 1148 |
|
|
&& (C1_lo & 0x01)) || (rnd_mode == ROUNDING_DOWN
|
| 1149 |
|
|
&& !x_sign && y_sign)
|
| 1150 |
|
|
|| (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
| 1151 |
|
|
|| (rnd_mode == ROUNDING_TO_ZERO
|
| 1152 |
|
|
&& x_sign != y_sign)) {
|
| 1153 |
|
|
// subtract 1 ulp from C1
|
| 1154 |
|
|
// Note: because delta >= P34 + 1 the result cannot be zero
|
| 1155 |
|
|
C1_lo = C1_lo - 1;
|
| 1156 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 1157 |
|
|
C1_hi = C1_hi - 1;
|
| 1158 |
|
|
// if the coefficient is 10^33 - 1 then make it 10^34 - 1
|
| 1159 |
|
|
// and decrease the exponent by 1 (because delta >= P34 + 1
|
| 1160 |
|
|
// the exponent will not become less than e_min)
|
| 1161 |
|
|
// 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
| 1162 |
|
|
// 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
| 1163 |
|
|
if (C1_hi == 0x0000314dc6448d93ull
|
| 1164 |
|
|
&& C1_lo == 0x38c15b09ffffffffull) {
|
| 1165 |
|
|
// make C1 = 10^34 - 1
|
| 1166 |
|
|
C1_hi = 0x0001ed09bead87c0ull;
|
| 1167 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 1168 |
|
|
x_exp = x_exp - EXP_P1;
|
| 1169 |
|
|
}
|
| 1170 |
|
|
} else {
|
| 1171 |
|
|
; // the result is already correct
|
| 1172 |
|
|
}
|
| 1173 |
|
|
// set the inexact flag
|
| 1174 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1175 |
|
|
// assemble the result
|
| 1176 |
|
|
res.w[1] = x_sign | x_exp | C1_hi;
|
| 1177 |
|
|
res.w[0] = C1_lo;
|
| 1178 |
|
|
} else { // if C2_lo > halfulp64 ||
|
| 1179 |
|
|
// (C2_lo == halfulp64 && q1 == P34 && ((C1_lo & 0x1) == 1)), i.e.
|
| 1180 |
|
|
// 1/2 ulp(n1) < n2 < 1 ulp(n1) or n2 = 1/2 ulp(n1) and C1 odd
|
| 1181 |
|
|
// res = x+1 ulp if n1*n2 > 0 and res = x-1 ulp if n1*n2 < 0
|
| 1182 |
|
|
if (q1 < P34) { // then 1 ulp = 10^(e1+q1-P34) < 10^e1
|
| 1183 |
|
|
// Note: if (q1 == P34) then 1 ulp = 10^(e1+q1-P34) = 10^e1
|
| 1184 |
|
|
// because q1 < P34 we must first replace C1 by
|
| 1185 |
|
|
// C1 * 10^(P34-q1), and must decrease the exponent by
|
| 1186 |
|
|
// (P34-q1) (it will still be at least e_min)
|
| 1187 |
|
|
scale = P34 - q1;
|
| 1188 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 1189 |
|
|
// 1 <= q1 <= 19 => 15 <= scale <= 33
|
| 1190 |
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
| 1191 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
| 1192 |
|
|
} else { // if 20 <= scale <= 33
|
| 1193 |
|
|
// C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
| 1194 |
|
|
// (C1 * 10^(scale-19)) fits in 64 bits
|
| 1195 |
|
|
C1_lo = C1_lo * ten2k64[scale - 19];
|
| 1196 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
| 1197 |
|
|
}
|
| 1198 |
|
|
} else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
| 1199 |
|
|
// => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
| 1200 |
|
|
C1.w[1] = C1_hi;
|
| 1201 |
|
|
C1.w[0] = C1_lo;
|
| 1202 |
|
|
// C1 = ten2k64[P34 - q1] * C1
|
| 1203 |
|
|
__mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
| 1204 |
|
|
}
|
| 1205 |
|
|
x_exp = x_exp - ((UINT64) scale << 49);
|
| 1206 |
|
|
C1_hi = C1.w[1];
|
| 1207 |
|
|
C1_lo = C1.w[0];
|
| 1208 |
|
|
// check for rounding overflow
|
| 1209 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 1210 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 1211 |
|
|
// C1 = 10^34 => rounding overflow
|
| 1212 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 1213 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 1214 |
|
|
x_exp = x_exp + EXP_P1;
|
| 1215 |
|
|
}
|
| 1216 |
|
|
}
|
| 1217 |
|
|
if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign != y_sign)
|
| 1218 |
|
|
|| (rnd_mode == ROUNDING_TIES_AWAY && x_sign != y_sign
|
| 1219 |
|
|
&& C2_lo != halfulp64)
|
| 1220 |
|
|
|| (rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
|
| 1221 |
|
|
|| (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
| 1222 |
|
|
|| (rnd_mode == ROUNDING_TO_ZERO
|
| 1223 |
|
|
&& x_sign != y_sign)) {
|
| 1224 |
|
|
// the result is x - 1
|
| 1225 |
|
|
// for RN n1 * n2 < 0; underflow not possible
|
| 1226 |
|
|
C1_lo = C1_lo - 1;
|
| 1227 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 1228 |
|
|
C1_hi--;
|
| 1229 |
|
|
// check if we crossed into the lower decade
|
| 1230 |
|
|
if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
| 1231 |
|
|
C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
| 1232 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 1233 |
|
|
x_exp = x_exp - EXP_P1; // no underflow, because n1 >> n2
|
| 1234 |
|
|
}
|
| 1235 |
|
|
} else
|
| 1236 |
|
|
if ((rnd_mode == ROUNDING_TO_NEAREST
|
| 1237 |
|
|
&& x_sign == y_sign)
|
| 1238 |
|
|
|| (rnd_mode == ROUNDING_TIES_AWAY
|
| 1239 |
|
|
&& x_sign == y_sign)
|
| 1240 |
|
|
|| (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)
|
| 1241 |
|
|
|| (rnd_mode == ROUNDING_UP && !x_sign
|
| 1242 |
|
|
&& !y_sign)) {
|
| 1243 |
|
|
// the result is x + 1
|
| 1244 |
|
|
// for RN x_sign = y_sign, i.e. n1*n2 > 0
|
| 1245 |
|
|
C1_lo = C1_lo + 1;
|
| 1246 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 1247 |
|
|
C1_hi = C1_hi + 1;
|
| 1248 |
|
|
}
|
| 1249 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 1250 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 1251 |
|
|
// C1 = 10^34 => rounding overflow
|
| 1252 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 1253 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 1254 |
|
|
x_exp = x_exp + EXP_P1;
|
| 1255 |
|
|
if (x_exp == EXP_MAX_P1) { // overflow
|
| 1256 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 1257 |
|
|
C1_lo = 0x0ull;
|
| 1258 |
|
|
x_exp = 0; // x_sign is preserved
|
| 1259 |
|
|
// set the overflow flag
|
| 1260 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 1261 |
|
|
}
|
| 1262 |
|
|
}
|
| 1263 |
|
|
} else {
|
| 1264 |
|
|
; // the result is x
|
| 1265 |
|
|
}
|
| 1266 |
|
|
// set the inexact flag
|
| 1267 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1268 |
|
|
// assemble the result
|
| 1269 |
|
|
res.w[1] = x_sign | x_exp | C1_hi;
|
| 1270 |
|
|
res.w[0] = C1_lo;
|
| 1271 |
|
|
}
|
| 1272 |
|
|
} else { // if q2 >= 20 then 5*10^(q2-1) and C2 (the latter in
|
| 1273 |
|
|
// most cases) fit only in more than 64 bits
|
| 1274 |
|
|
halfulp128 = midpoint128[q2 - 20]; // 5 * 10^(q2-1)
|
| 1275 |
|
|
if ((C2_hi < halfulp128.w[1])
|
| 1276 |
|
|
|| (C2_hi == halfulp128.w[1]
|
| 1277 |
|
|
&& C2_lo < halfulp128.w[0])) {
|
| 1278 |
|
|
// n2 < 1/2 ulp (n1)
|
| 1279 |
|
|
// the result is the operand with the larger magnitude,
|
| 1280 |
|
|
// possibly scaled up by 10^(P34-q1)
|
| 1281 |
|
|
// an overflow cannot occur in this case (rounding to nearest)
|
| 1282 |
|
|
if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
| 1283 |
|
|
// Note: because delta = P34 it is certain that
|
| 1284 |
|
|
// x_exp - ((UINT64)scale << 49) will stay above e_min
|
| 1285 |
|
|
scale = P34 - q1;
|
| 1286 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 1287 |
|
|
// 1 <= q1 <= 19 => 15 <= scale <= 33
|
| 1288 |
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
| 1289 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
| 1290 |
|
|
} else { // if 20 <= scale <= 33
|
| 1291 |
|
|
// C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
| 1292 |
|
|
// (C1 * 10^(scale-19)) fits in 64 bits
|
| 1293 |
|
|
C1_lo = C1_lo * ten2k64[scale - 19];
|
| 1294 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
| 1295 |
|
|
}
|
| 1296 |
|
|
} else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
| 1297 |
|
|
// => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
| 1298 |
|
|
C1.w[1] = C1_hi;
|
| 1299 |
|
|
C1.w[0] = C1_lo;
|
| 1300 |
|
|
// C1 = ten2k64[P34 - q1] * C1
|
| 1301 |
|
|
__mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
| 1302 |
|
|
}
|
| 1303 |
|
|
C1_hi = C1.w[1];
|
| 1304 |
|
|
C1_lo = C1.w[0];
|
| 1305 |
|
|
x_exp = x_exp - ((UINT64) scale << 49);
|
| 1306 |
|
|
}
|
| 1307 |
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
| 1308 |
|
|
if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
|
| 1309 |
|
|
(rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
|
| 1310 |
|
|
// add 1 ulp and then check for overflow
|
| 1311 |
|
|
C1_lo = C1_lo + 1;
|
| 1312 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 1313 |
|
|
C1_hi = C1_hi + 1;
|
| 1314 |
|
|
}
|
| 1315 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 1316 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 1317 |
|
|
// C1 = 10^34 => rounding overflow
|
| 1318 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 1319 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 1320 |
|
|
x_exp = x_exp + EXP_P1;
|
| 1321 |
|
|
if (x_exp == EXP_MAX_P1) { // overflow
|
| 1322 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 1323 |
|
|
C1_lo = 0x0ull;
|
| 1324 |
|
|
x_exp = 0; // x_sign is preserved
|
| 1325 |
|
|
// set overflow flag (the inexact flag was set too)
|
| 1326 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 1327 |
|
|
}
|
| 1328 |
|
|
}
|
| 1329 |
|
|
} else
|
| 1330 |
|
|
if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
|
| 1331 |
|
|
|| (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
| 1332 |
|
|
|| (rnd_mode == ROUNDING_TO_ZERO
|
| 1333 |
|
|
&& x_sign != y_sign)) {
|
| 1334 |
|
|
// subtract 1 ulp from C1
|
| 1335 |
|
|
// Note: because delta >= P34 + 1 the result cannot be zero
|
| 1336 |
|
|
C1_lo = C1_lo - 1;
|
| 1337 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 1338 |
|
|
C1_hi = C1_hi - 1;
|
| 1339 |
|
|
// if the coefficient is 10^33-1 then make it 10^34-1 and
|
| 1340 |
|
|
// decrease the exponent by 1 (because delta >= P34 + 1 the
|
| 1341 |
|
|
// exponent will not become less than e_min)
|
| 1342 |
|
|
// 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
| 1343 |
|
|
// 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
| 1344 |
|
|
if (C1_hi == 0x0000314dc6448d93ull
|
| 1345 |
|
|
&& C1_lo == 0x38c15b09ffffffffull) {
|
| 1346 |
|
|
// make C1 = 10^34 - 1
|
| 1347 |
|
|
C1_hi = 0x0001ed09bead87c0ull;
|
| 1348 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 1349 |
|
|
x_exp = x_exp - EXP_P1;
|
| 1350 |
|
|
}
|
| 1351 |
|
|
} else {
|
| 1352 |
|
|
; // the result is already correct
|
| 1353 |
|
|
}
|
| 1354 |
|
|
}
|
| 1355 |
|
|
// set the inexact flag
|
| 1356 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1357 |
|
|
// assemble the result
|
| 1358 |
|
|
res.w[1] = x_sign | x_exp | C1_hi;
|
| 1359 |
|
|
res.w[0] = C1_lo;
|
| 1360 |
|
|
} else if ((C2_hi == halfulp128.w[1]
|
| 1361 |
|
|
&& C2_lo == halfulp128.w[0])
|
| 1362 |
|
|
&& (q1 < P34 || ((C1_lo & 0x1) == 0))) {
|
| 1363 |
|
|
// midpoint & lsb in C1 is 0
|
| 1364 |
|
|
// n2 = 1/2 ulp (n1) and C1 is even
|
| 1365 |
|
|
// the result is the operand with the larger magnitude,
|
| 1366 |
|
|
// possibly scaled up by 10^(P34-q1)
|
| 1367 |
|
|
// an overflow cannot occur in this case (rounding to nearest)
|
| 1368 |
|
|
if (q1 < P34) { // scale C1 up by 10^(P34-q1)
|
| 1369 |
|
|
// Note: because delta = P34 it is certain that
|
| 1370 |
|
|
// x_exp - ((UINT64)scale << 49) will stay above e_min
|
| 1371 |
|
|
scale = P34 - q1;
|
| 1372 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 1373 |
|
|
// 1 <= q1 <= 19 => 15 <= scale <= 33
|
| 1374 |
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
| 1375 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
| 1376 |
|
|
} else { // if 20 <= scale <= 33
|
| 1377 |
|
|
// C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
| 1378 |
|
|
// (C1 * 10^(scale-19)) fits in 64 bits
|
| 1379 |
|
|
C1_lo = C1_lo * ten2k64[scale - 19];
|
| 1380 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
| 1381 |
|
|
}
|
| 1382 |
|
|
} else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
| 1383 |
|
|
// => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
| 1384 |
|
|
C1.w[1] = C1_hi;
|
| 1385 |
|
|
C1.w[0] = C1_lo;
|
| 1386 |
|
|
// C1 = ten2k64[P34 - q1] * C1
|
| 1387 |
|
|
__mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
| 1388 |
|
|
}
|
| 1389 |
|
|
x_exp = x_exp - ((UINT64) scale << 49);
|
| 1390 |
|
|
C1_hi = C1.w[1];
|
| 1391 |
|
|
C1_lo = C1.w[0];
|
| 1392 |
|
|
}
|
| 1393 |
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
| 1394 |
|
|
if ((rnd_mode == ROUNDING_TIES_AWAY && x_sign == y_sign)
|
| 1395 |
|
|
|| (rnd_mode == ROUNDING_UP && !y_sign)) {
|
| 1396 |
|
|
// add 1 ulp and then check for overflow
|
| 1397 |
|
|
C1_lo = C1_lo + 1;
|
| 1398 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 1399 |
|
|
C1_hi = C1_hi + 1;
|
| 1400 |
|
|
}
|
| 1401 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 1402 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 1403 |
|
|
// C1 = 10^34 => rounding overflow
|
| 1404 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 1405 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 1406 |
|
|
x_exp = x_exp + EXP_P1;
|
| 1407 |
|
|
if (x_exp == EXP_MAX_P1) { // overflow
|
| 1408 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 1409 |
|
|
C1_lo = 0x0ull;
|
| 1410 |
|
|
x_exp = 0; // x_sign is preserved
|
| 1411 |
|
|
// set overflow flag (the inexact flag was set too)
|
| 1412 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 1413 |
|
|
}
|
| 1414 |
|
|
}
|
| 1415 |
|
|
} else if ((rnd_mode == ROUNDING_DOWN && y_sign)
|
| 1416 |
|
|
|| (rnd_mode == ROUNDING_TO_ZERO
|
| 1417 |
|
|
&& x_sign != y_sign)) {
|
| 1418 |
|
|
// subtract 1 ulp from C1
|
| 1419 |
|
|
// Note: because delta >= P34 + 1 the result cannot be zero
|
| 1420 |
|
|
C1_lo = C1_lo - 1;
|
| 1421 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 1422 |
|
|
C1_hi = C1_hi - 1;
|
| 1423 |
|
|
// if the coefficient is 10^33 - 1 then make it 10^34 - 1
|
| 1424 |
|
|
// and decrease the exponent by 1 (because delta >= P34 + 1
|
| 1425 |
|
|
// the exponent will not become less than e_min)
|
| 1426 |
|
|
// 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
|
| 1427 |
|
|
// 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
|
| 1428 |
|
|
if (C1_hi == 0x0000314dc6448d93ull
|
| 1429 |
|
|
&& C1_lo == 0x38c15b09ffffffffull) {
|
| 1430 |
|
|
// make C1 = 10^34 - 1
|
| 1431 |
|
|
C1_hi = 0x0001ed09bead87c0ull;
|
| 1432 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 1433 |
|
|
x_exp = x_exp - EXP_P1;
|
| 1434 |
|
|
}
|
| 1435 |
|
|
} else {
|
| 1436 |
|
|
; // the result is already correct
|
| 1437 |
|
|
}
|
| 1438 |
|
|
}
|
| 1439 |
|
|
// set the inexact flag
|
| 1440 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1441 |
|
|
// assemble the result
|
| 1442 |
|
|
res.w[1] = x_sign | x_exp | C1_hi;
|
| 1443 |
|
|
res.w[0] = C1_lo;
|
| 1444 |
|
|
} else { // if C2 > halfulp128 ||
|
| 1445 |
|
|
// (C2 == halfulp128 && q1 == P34 && ((C1 & 0x1) == 1)), i.e.
|
| 1446 |
|
|
// 1/2 ulp(n1) < n2 < 1 ulp(n1) or n2 = 1/2 ulp(n1) and C1 odd
|
| 1447 |
|
|
// res = x+1 ulp if n1*n2 > 0 and res = x-1 ulp if n1*n2 < 0
|
| 1448 |
|
|
if (q1 < P34) { // then 1 ulp = 10^(e1+q1-P34) < 10^e1
|
| 1449 |
|
|
// Note: if (q1 == P34) then 1 ulp = 10^(e1+q1-P34) = 10^e1
|
| 1450 |
|
|
// because q1 < P34 we must first replace C1 by C1*10^(P34-q1),
|
| 1451 |
|
|
// and must decrease the exponent by (P34-q1) (it will still be
|
| 1452 |
|
|
// at least e_min)
|
| 1453 |
|
|
scale = P34 - q1;
|
| 1454 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 1455 |
|
|
// 1 <= q1 <= 19 => 15 <= scale <= 33
|
| 1456 |
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
| 1457 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
|
| 1458 |
|
|
} else { // if 20 <= scale <= 33
|
| 1459 |
|
|
// C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
|
| 1460 |
|
|
// (C1 * 10^(scale-19)) fits in 64 bits
|
| 1461 |
|
|
C1_lo = C1_lo * ten2k64[scale - 19];
|
| 1462 |
|
|
__mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
|
| 1463 |
|
|
}
|
| 1464 |
|
|
} else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
|
| 1465 |
|
|
// => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
|
| 1466 |
|
|
C1.w[1] = C1_hi;
|
| 1467 |
|
|
C1.w[0] = C1_lo;
|
| 1468 |
|
|
// C1 = ten2k64[P34 - q1] * C1
|
| 1469 |
|
|
__mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
|
| 1470 |
|
|
}
|
| 1471 |
|
|
C1_hi = C1.w[1];
|
| 1472 |
|
|
C1_lo = C1.w[0];
|
| 1473 |
|
|
x_exp = x_exp - ((UINT64) scale << 49);
|
| 1474 |
|
|
}
|
| 1475 |
|
|
if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign != y_sign)
|
| 1476 |
|
|
|| (rnd_mode == ROUNDING_TIES_AWAY && x_sign != y_sign
|
| 1477 |
|
|
&& (C2_hi != halfulp128.w[1]
|
| 1478 |
|
|
|| C2_lo != halfulp128.w[0]))
|
| 1479 |
|
|
|| (rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
|
| 1480 |
|
|
|| (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
|
| 1481 |
|
|
|| (rnd_mode == ROUNDING_TO_ZERO
|
| 1482 |
|
|
&& x_sign != y_sign)) {
|
| 1483 |
|
|
// the result is x - 1
|
| 1484 |
|
|
// for RN n1 * n2 < 0; underflow not possible
|
| 1485 |
|
|
C1_lo = C1_lo - 1;
|
| 1486 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 1487 |
|
|
C1_hi--;
|
| 1488 |
|
|
// check if we crossed into the lower decade
|
| 1489 |
|
|
if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
| 1490 |
|
|
C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
| 1491 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 1492 |
|
|
x_exp = x_exp - EXP_P1; // no underflow, because n1 >> n2
|
| 1493 |
|
|
}
|
| 1494 |
|
|
} else
|
| 1495 |
|
|
if ((rnd_mode == ROUNDING_TO_NEAREST
|
| 1496 |
|
|
&& x_sign == y_sign)
|
| 1497 |
|
|
|| (rnd_mode == ROUNDING_TIES_AWAY
|
| 1498 |
|
|
&& x_sign == y_sign)
|
| 1499 |
|
|
|| (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)
|
| 1500 |
|
|
|| (rnd_mode == ROUNDING_UP && !x_sign
|
| 1501 |
|
|
&& !y_sign)) {
|
| 1502 |
|
|
// the result is x + 1
|
| 1503 |
|
|
// for RN x_sign = y_sign, i.e. n1*n2 > 0
|
| 1504 |
|
|
C1_lo = C1_lo + 1;
|
| 1505 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 1506 |
|
|
C1_hi = C1_hi + 1;
|
| 1507 |
|
|
}
|
| 1508 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 1509 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 1510 |
|
|
// C1 = 10^34 => rounding overflow
|
| 1511 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 1512 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 1513 |
|
|
x_exp = x_exp + EXP_P1;
|
| 1514 |
|
|
if (x_exp == EXP_MAX_P1) { // overflow
|
| 1515 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 1516 |
|
|
C1_lo = 0x0ull;
|
| 1517 |
|
|
x_exp = 0; // x_sign is preserved
|
| 1518 |
|
|
// set the overflow flag
|
| 1519 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 1520 |
|
|
}
|
| 1521 |
|
|
}
|
| 1522 |
|
|
} else {
|
| 1523 |
|
|
; // the result is x
|
| 1524 |
|
|
}
|
| 1525 |
|
|
// set the inexact flag
|
| 1526 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1527 |
|
|
// assemble the result
|
| 1528 |
|
|
res.w[1] = x_sign | x_exp | C1_hi;
|
| 1529 |
|
|
res.w[0] = C1_lo;
|
| 1530 |
|
|
}
|
| 1531 |
|
|
} // end q1 >= 20
|
| 1532 |
|
|
// end case where C1 != 10^(q1-1)
|
| 1533 |
|
|
} else { // C1 = 10^(q1-1) and x_sign != y_sign
|
| 1534 |
|
|
// instead of C' = (C1 * 10^(e1-e2) + C2)rnd,P34
|
| 1535 |
|
|
// calculate C' = C1 * 10^(e1-e2-x1) + (C2 * 10^(-x1))rnd,P34
|
| 1536 |
|
|
// where x1 = q2 - 1, 0 <= x1 <= P34 - 1
|
| 1537 |
|
|
// Because C1 = 10^(q1-1) and x_sign != y_sign, C' will have P34
|
| 1538 |
|
|
// digits and n = C' * 10^(e2+x1)
|
| 1539 |
|
|
// If the result has P34+1 digits, redo the steps above with x1+1
|
| 1540 |
|
|
// If the result has P34-1 digits or less, redo the steps above with
|
| 1541 |
|
|
// x1-1 but only if initially x1 >= 1
|
| 1542 |
|
|
// NOTE: these two steps can be improved, e.g we could guess if
|
| 1543 |
|
|
// P34+1 or P34-1 digits will be obtained by adding/subtracting
|
| 1544 |
|
|
// just the top 64 bits of the two operands
|
| 1545 |
|
|
// The result cannot be zero, and it cannot overflow
|
| 1546 |
|
|
x1 = q2 - 1; // 0 <= x1 <= P34-1
|
| 1547 |
|
|
// Calculate C1 * 10^(e1-e2-x1) where 1 <= e1-e2-x1 <= P34
|
| 1548 |
|
|
// scale = (int)(e1 >> 49) - (int)(e2 >> 49) - x1; 0 <= scale <= P34-1
|
| 1549 |
|
|
scale = P34 - q1 + 1; // scale=e1-e2-x1 = P34+1-q1; 1<=scale<=P34
|
| 1550 |
|
|
// either C1 or 10^(e1-e2-x1) may not fit is 64 bits,
|
| 1551 |
|
|
// but their product fits with certainty in 128 bits
|
| 1552 |
|
|
if (scale >= 20) { //10^(e1-e2-x1) doesn't fit in 64 bits, but C1 does
|
| 1553 |
|
|
__mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
| 1554 |
|
|
} else { // if (scale >= 1
|
| 1555 |
|
|
// if 1 <= scale <= 19 then 10^(e1-e2-x1) fits in 64 bits
|
| 1556 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 1557 |
|
|
__mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
| 1558 |
|
|
} else { // q1 >= 20
|
| 1559 |
|
|
C1.w[1] = C1_hi;
|
| 1560 |
|
|
C1.w[0] = C1_lo;
|
| 1561 |
|
|
__mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
| 1562 |
|
|
}
|
| 1563 |
|
|
}
|
| 1564 |
|
|
tmp64 = C1.w[0]; // C1.w[1], C1.w[0] contains C1 * 10^(e1-e2-x1)
|
| 1565 |
|
|
|
| 1566 |
|
|
// now round C2 to q2-x1 = 1 decimal digit
|
| 1567 |
|
|
// C2' = C2 + 1/2 * 10^x1 = C2 + 5 * 10^(x1-1)
|
| 1568 |
|
|
ind = x1 - 1; // -1 <= ind <= P34 - 2
|
| 1569 |
|
|
if (ind >= 0) { // if (x1 >= 1)
|
| 1570 |
|
|
C2.w[0] = C2_lo;
|
| 1571 |
|
|
C2.w[1] = C2_hi;
|
| 1572 |
|
|
if (ind <= 18) {
|
| 1573 |
|
|
C2.w[0] = C2.w[0] + midpoint64[ind];
|
| 1574 |
|
|
if (C2.w[0] < C2_lo)
|
| 1575 |
|
|
C2.w[1]++;
|
| 1576 |
|
|
} else { // 19 <= ind <= 32
|
| 1577 |
|
|
C2.w[0] = C2.w[0] + midpoint128[ind - 19].w[0];
|
| 1578 |
|
|
C2.w[1] = C2.w[1] + midpoint128[ind - 19].w[1];
|
| 1579 |
|
|
if (C2.w[0] < C2_lo)
|
| 1580 |
|
|
C2.w[1]++;
|
| 1581 |
|
|
}
|
| 1582 |
|
|
// the approximation of 10^(-x1) was rounded up to 118 bits
|
| 1583 |
|
|
__mul_128x128_to_256 (R256, C2, ten2mk128[ind]); // R256 = C2*, f2*
|
| 1584 |
|
|
// calculate C2* and f2*
|
| 1585 |
|
|
// C2* is actually floor(C2*) in this case
|
| 1586 |
|
|
// C2* and f2* need shifting and masking, as shown by
|
| 1587 |
|
|
// shiftright128[] and maskhigh128[]
|
| 1588 |
|
|
// the top Ex bits of 10^(-x1) are T* = ten2mk128trunc[ind], e.g.
|
| 1589 |
|
|
// if x1=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
|
| 1590 |
|
|
// if (0 < f2* < 10^(-x1)) then
|
| 1591 |
|
|
// if floor(C1+C2*) is even then C2* = floor(C2*) - logical right
|
| 1592 |
|
|
// shift; C2* has p decimal digits, correct by Prop. 1)
|
| 1593 |
|
|
// else if floor(C1+C2*) is odd C2* = floor(C2*)-1 (logical right
|
| 1594 |
|
|
// shift; C2* has p decimal digits, correct by Pr. 1)
|
| 1595 |
|
|
// else
|
| 1596 |
|
|
// C2* = floor(C2*) (logical right shift; C has p decimal digits,
|
| 1597 |
|
|
// correct by Property 1)
|
| 1598 |
|
|
// n = C2* * 10^(e2+x1)
|
| 1599 |
|
|
|
| 1600 |
|
|
if (ind <= 2) {
|
| 1601 |
|
|
highf2star.w[1] = 0x0;
|
| 1602 |
|
|
highf2star.w[0] = 0x0; // low f2* ok
|
| 1603 |
|
|
} else if (ind <= 21) {
|
| 1604 |
|
|
highf2star.w[1] = 0x0;
|
| 1605 |
|
|
highf2star.w[0] = R256.w[2] & maskhigh128[ind]; // low f2* ok
|
| 1606 |
|
|
} else {
|
| 1607 |
|
|
highf2star.w[1] = R256.w[3] & maskhigh128[ind];
|
| 1608 |
|
|
highf2star.w[0] = R256.w[2]; // low f2* is ok
|
| 1609 |
|
|
}
|
| 1610 |
|
|
// shift right C2* by Ex-128 = shiftright128[ind]
|
| 1611 |
|
|
if (ind >= 3) {
|
| 1612 |
|
|
shift = shiftright128[ind];
|
| 1613 |
|
|
if (shift < 64) { // 3 <= shift <= 63
|
| 1614 |
|
|
R256.w[2] =
|
| 1615 |
|
|
(R256.w[2] >> shift) | (R256.w[3] << (64 - shift));
|
| 1616 |
|
|
R256.w[3] = (R256.w[3] >> shift);
|
| 1617 |
|
|
} else { // 66 <= shift <= 102
|
| 1618 |
|
|
R256.w[2] = (R256.w[3] >> (shift - 64));
|
| 1619 |
|
|
R256.w[3] = 0x0ULL;
|
| 1620 |
|
|
}
|
| 1621 |
|
|
}
|
| 1622 |
|
|
// redundant
|
| 1623 |
|
|
is_inexact_lt_midpoint = 0;
|
| 1624 |
|
|
is_inexact_gt_midpoint = 0;
|
| 1625 |
|
|
is_midpoint_lt_even = 0;
|
| 1626 |
|
|
is_midpoint_gt_even = 0;
|
| 1627 |
|
|
// determine inexactness of the rounding of C2*
|
| 1628 |
|
|
// (cannot be followed by a second rounding)
|
| 1629 |
|
|
// if (0 < f2* - 1/2 < 10^(-x1)) then
|
| 1630 |
|
|
// the result is exact
|
| 1631 |
|
|
// else (if f2* - 1/2 > T* then)
|
| 1632 |
|
|
// the result of is inexact
|
| 1633 |
|
|
if (ind <= 2) {
|
| 1634 |
|
|
if (R256.w[1] > 0x8000000000000000ull ||
|
| 1635 |
|
|
(R256.w[1] == 0x8000000000000000ull
|
| 1636 |
|
|
&& R256.w[0] > 0x0ull)) {
|
| 1637 |
|
|
// f2* > 1/2 and the result may be exact
|
| 1638 |
|
|
tmp64A = R256.w[1] - 0x8000000000000000ull; // f* - 1/2
|
| 1639 |
|
|
if ((tmp64A > ten2mk128trunc[ind].w[1]
|
| 1640 |
|
|
|| (tmp64A == ten2mk128trunc[ind].w[1]
|
| 1641 |
|
|
&& R256.w[0] >= ten2mk128trunc[ind].w[0]))) {
|
| 1642 |
|
|
// set the inexact flag
|
| 1643 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1644 |
|
|
// this rounding is applied to C2 only!
|
| 1645 |
|
|
// x_sign != y_sign
|
| 1646 |
|
|
is_inexact_gt_midpoint = 1;
|
| 1647 |
|
|
} // else the result is exact
|
| 1648 |
|
|
// rounding down, unless a midpoint in [ODD, EVEN]
|
| 1649 |
|
|
} else { // the result is inexact; f2* <= 1/2
|
| 1650 |
|
|
// set the inexact flag
|
| 1651 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1652 |
|
|
// this rounding is applied to C2 only!
|
| 1653 |
|
|
// x_sign != y_sign
|
| 1654 |
|
|
is_inexact_lt_midpoint = 1;
|
| 1655 |
|
|
}
|
| 1656 |
|
|
} else if (ind <= 21) { // if 3 <= ind <= 21
|
| 1657 |
|
|
if (highf2star.w[1] > 0x0 || (highf2star.w[1] == 0x0
|
| 1658 |
|
|
&& highf2star.w[0] >
|
| 1659 |
|
|
onehalf128[ind])
|
| 1660 |
|
|
|| (highf2star.w[1] == 0x0
|
| 1661 |
|
|
&& highf2star.w[0] == onehalf128[ind]
|
| 1662 |
|
|
&& (R256.w[1] || R256.w[0]))) {
|
| 1663 |
|
|
// f2* > 1/2 and the result may be exact
|
| 1664 |
|
|
// Calculate f2* - 1/2
|
| 1665 |
|
|
tmp64A = highf2star.w[0] - onehalf128[ind];
|
| 1666 |
|
|
tmp64B = highf2star.w[1];
|
| 1667 |
|
|
if (tmp64A > highf2star.w[0])
|
| 1668 |
|
|
tmp64B--;
|
| 1669 |
|
|
if (tmp64B || tmp64A
|
| 1670 |
|
|
|| R256.w[1] > ten2mk128trunc[ind].w[1]
|
| 1671 |
|
|
|| (R256.w[1] == ten2mk128trunc[ind].w[1]
|
| 1672 |
|
|
&& R256.w[0] > ten2mk128trunc[ind].w[0])) {
|
| 1673 |
|
|
// set the inexact flag
|
| 1674 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1675 |
|
|
// this rounding is applied to C2 only!
|
| 1676 |
|
|
// x_sign != y_sign
|
| 1677 |
|
|
is_inexact_gt_midpoint = 1;
|
| 1678 |
|
|
} // else the result is exact
|
| 1679 |
|
|
} else { // the result is inexact; f2* <= 1/2
|
| 1680 |
|
|
// set the inexact flag
|
| 1681 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1682 |
|
|
// this rounding is applied to C2 only!
|
| 1683 |
|
|
// x_sign != y_sign
|
| 1684 |
|
|
is_inexact_lt_midpoint = 1;
|
| 1685 |
|
|
}
|
| 1686 |
|
|
} else { // if 22 <= ind <= 33
|
| 1687 |
|
|
if (highf2star.w[1] > onehalf128[ind]
|
| 1688 |
|
|
|| (highf2star.w[1] == onehalf128[ind]
|
| 1689 |
|
|
&& (highf2star.w[0] || R256.w[1]
|
| 1690 |
|
|
|| R256.w[0]))) {
|
| 1691 |
|
|
// f2* > 1/2 and the result may be exact
|
| 1692 |
|
|
// Calculate f2* - 1/2
|
| 1693 |
|
|
// tmp64A = highf2star.w[0];
|
| 1694 |
|
|
tmp64B = highf2star.w[1] - onehalf128[ind];
|
| 1695 |
|
|
if (tmp64B || highf2star.w[0]
|
| 1696 |
|
|
|| R256.w[1] > ten2mk128trunc[ind].w[1]
|
| 1697 |
|
|
|| (R256.w[1] == ten2mk128trunc[ind].w[1]
|
| 1698 |
|
|
&& R256.w[0] > ten2mk128trunc[ind].w[0])) {
|
| 1699 |
|
|
// set the inexact flag
|
| 1700 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1701 |
|
|
// this rounding is applied to C2 only!
|
| 1702 |
|
|
// x_sign != y_sign
|
| 1703 |
|
|
is_inexact_gt_midpoint = 1;
|
| 1704 |
|
|
} // else the result is exact
|
| 1705 |
|
|
} else { // the result is inexact; f2* <= 1/2
|
| 1706 |
|
|
// set the inexact flag
|
| 1707 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1708 |
|
|
// this rounding is applied to C2 only!
|
| 1709 |
|
|
// x_sign != y_sign
|
| 1710 |
|
|
is_inexact_lt_midpoint = 1;
|
| 1711 |
|
|
}
|
| 1712 |
|
|
}
|
| 1713 |
|
|
// check for midpoints after determining inexactness
|
| 1714 |
|
|
if ((R256.w[1] || R256.w[0]) && (highf2star.w[1] == 0)
|
| 1715 |
|
|
&& (highf2star.w[0] == 0)
|
| 1716 |
|
|
&& (R256.w[1] < ten2mk128trunc[ind].w[1]
|
| 1717 |
|
|
|| (R256.w[1] == ten2mk128trunc[ind].w[1]
|
| 1718 |
|
|
&& R256.w[0] <= ten2mk128trunc[ind].w[0]))) {
|
| 1719 |
|
|
// the result is a midpoint
|
| 1720 |
|
|
if ((tmp64 + R256.w[2]) & 0x01) { // MP in [EVEN, ODD]
|
| 1721 |
|
|
// if floor(C2*) is odd C = floor(C2*) - 1; the result may be 0
|
| 1722 |
|
|
R256.w[2]--;
|
| 1723 |
|
|
if (R256.w[2] == 0xffffffffffffffffull)
|
| 1724 |
|
|
R256.w[3]--;
|
| 1725 |
|
|
// this rounding is applied to C2 only!
|
| 1726 |
|
|
// x_sign != y_sign
|
| 1727 |
|
|
is_midpoint_lt_even = 1;
|
| 1728 |
|
|
is_inexact_lt_midpoint = 0;
|
| 1729 |
|
|
is_inexact_gt_midpoint = 0;
|
| 1730 |
|
|
} else {
|
| 1731 |
|
|
// else MP in [ODD, EVEN]
|
| 1732 |
|
|
// this rounding is applied to C2 only!
|
| 1733 |
|
|
// x_sign != y_sign
|
| 1734 |
|
|
is_midpoint_gt_even = 1;
|
| 1735 |
|
|
is_inexact_lt_midpoint = 0;
|
| 1736 |
|
|
is_inexact_gt_midpoint = 0;
|
| 1737 |
|
|
}
|
| 1738 |
|
|
}
|
| 1739 |
|
|
} else { // if (ind == -1) only when x1 = 0
|
| 1740 |
|
|
R256.w[2] = C2_lo;
|
| 1741 |
|
|
R256.w[3] = C2_hi;
|
| 1742 |
|
|
is_midpoint_lt_even = 0;
|
| 1743 |
|
|
is_midpoint_gt_even = 0;
|
| 1744 |
|
|
is_inexact_lt_midpoint = 0;
|
| 1745 |
|
|
is_inexact_gt_midpoint = 0;
|
| 1746 |
|
|
}
|
| 1747 |
|
|
// and now subtract C1 * 10^(e1-e2-x1) - (C2 * 10^(-x1))rnd,P34
|
| 1748 |
|
|
// because x_sign != y_sign this last operation is exact
|
| 1749 |
|
|
C1.w[0] = C1.w[0] - R256.w[2];
|
| 1750 |
|
|
C1.w[1] = C1.w[1] - R256.w[3];
|
| 1751 |
|
|
if (C1.w[0] > tmp64)
|
| 1752 |
|
|
C1.w[1]--; // borrow
|
| 1753 |
|
|
if (C1.w[1] >= 0x8000000000000000ull) { // negative coefficient!
|
| 1754 |
|
|
C1.w[0] = ~C1.w[0];
|
| 1755 |
|
|
C1.w[0]++;
|
| 1756 |
|
|
C1.w[1] = ~C1.w[1];
|
| 1757 |
|
|
if (C1.w[0] == 0x0)
|
| 1758 |
|
|
C1.w[1]++;
|
| 1759 |
|
|
tmp_sign = y_sign; // the result will have the sign of y
|
| 1760 |
|
|
} else {
|
| 1761 |
|
|
tmp_sign = x_sign;
|
| 1762 |
|
|
}
|
| 1763 |
|
|
// the difference has exactly P34 digits
|
| 1764 |
|
|
x_sign = tmp_sign;
|
| 1765 |
|
|
if (x1 >= 1)
|
| 1766 |
|
|
y_exp = y_exp + ((UINT64) x1 << 49);
|
| 1767 |
|
|
C1_hi = C1.w[1];
|
| 1768 |
|
|
C1_lo = C1.w[0];
|
| 1769 |
|
|
// general correction from RN to RA, RM, RP, RZ; result uses y_exp
|
| 1770 |
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
| 1771 |
|
|
if ((!x_sign
|
| 1772 |
|
|
&& ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint)
|
| 1773 |
|
|
||
|
| 1774 |
|
|
((rnd_mode == ROUNDING_TIES_AWAY
|
| 1775 |
|
|
|| rnd_mode == ROUNDING_UP)
|
| 1776 |
|
|
&& is_midpoint_gt_even))) || (x_sign
|
| 1777 |
|
|
&&
|
| 1778 |
|
|
((rnd_mode ==
|
| 1779 |
|
|
ROUNDING_DOWN
|
| 1780 |
|
|
&&
|
| 1781 |
|
|
is_inexact_lt_midpoint)
|
| 1782 |
|
|
||
|
| 1783 |
|
|
((rnd_mode ==
|
| 1784 |
|
|
ROUNDING_TIES_AWAY
|
| 1785 |
|
|
|| rnd_mode ==
|
| 1786 |
|
|
ROUNDING_DOWN)
|
| 1787 |
|
|
&&
|
| 1788 |
|
|
is_midpoint_gt_even))))
|
| 1789 |
|
|
{
|
| 1790 |
|
|
// C1 = C1 + 1
|
| 1791 |
|
|
C1_lo = C1_lo + 1;
|
| 1792 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 1793 |
|
|
C1_hi = C1_hi + 1;
|
| 1794 |
|
|
}
|
| 1795 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 1796 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 1797 |
|
|
// C1 = 10^34 => rounding overflow
|
| 1798 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 1799 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 1800 |
|
|
y_exp = y_exp + EXP_P1;
|
| 1801 |
|
|
}
|
| 1802 |
|
|
} else if ((is_midpoint_lt_even || is_inexact_gt_midpoint)
|
| 1803 |
|
|
&&
|
| 1804 |
|
|
((x_sign
|
| 1805 |
|
|
&& (rnd_mode == ROUNDING_UP
|
| 1806 |
|
|
|| rnd_mode == ROUNDING_TO_ZERO))
|
| 1807 |
|
|
|| (!x_sign
|
| 1808 |
|
|
&& (rnd_mode == ROUNDING_DOWN
|
| 1809 |
|
|
|| rnd_mode == ROUNDING_TO_ZERO)))) {
|
| 1810 |
|
|
// C1 = C1 - 1
|
| 1811 |
|
|
C1_lo = C1_lo - 1;
|
| 1812 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 1813 |
|
|
C1_hi--;
|
| 1814 |
|
|
// check if we crossed into the lower decade
|
| 1815 |
|
|
if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
| 1816 |
|
|
C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
| 1817 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 1818 |
|
|
y_exp = y_exp - EXP_P1;
|
| 1819 |
|
|
// no underflow, because delta + q2 >= P34 + 1
|
| 1820 |
|
|
}
|
| 1821 |
|
|
} else {
|
| 1822 |
|
|
; // exact, the result is already correct
|
| 1823 |
|
|
}
|
| 1824 |
|
|
}
|
| 1825 |
|
|
// assemble the result
|
| 1826 |
|
|
res.w[1] = x_sign | y_exp | C1_hi;
|
| 1827 |
|
|
res.w[0] = C1_lo;
|
| 1828 |
|
|
}
|
| 1829 |
|
|
} // end delta = P34
|
| 1830 |
|
|
} else { // if (|delta| <= P34 - 1)
|
| 1831 |
|
|
if (delta >= 0) { // if (0 <= delta <= P34 - 1)
|
| 1832 |
|
|
if (delta <= P34 - 1 - q2) {
|
| 1833 |
|
|
// calculate C' directly; the result is exact
|
| 1834 |
|
|
// in this case 1<=q1<=P34-1, 1<=q2<=P34-1 and 0 <= e1-e2 <= P34-2
|
| 1835 |
|
|
// The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
|
| 1836 |
|
|
// exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
|
| 1837 |
|
|
// but their product fits with certainty in 128 bits (actually in 113)
|
| 1838 |
|
|
scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
|
| 1839 |
|
|
|
| 1840 |
|
|
if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
|
| 1841 |
|
|
__mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
| 1842 |
|
|
C1_hi = C1.w[1];
|
| 1843 |
|
|
C1_lo = C1.w[0];
|
| 1844 |
|
|
} else if (scale >= 1) {
|
| 1845 |
|
|
// if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
|
| 1846 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 1847 |
|
|
__mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
| 1848 |
|
|
} else { // q1 >= 20
|
| 1849 |
|
|
C1.w[1] = C1_hi;
|
| 1850 |
|
|
C1.w[0] = C1_lo;
|
| 1851 |
|
|
__mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
| 1852 |
|
|
}
|
| 1853 |
|
|
C1_hi = C1.w[1];
|
| 1854 |
|
|
C1_lo = C1.w[0];
|
| 1855 |
|
|
} else { // if (scale == 0) C1 is unchanged
|
| 1856 |
|
|
C1.w[0] = C1_lo; // C1.w[1] = C1_hi;
|
| 1857 |
|
|
}
|
| 1858 |
|
|
// now add C2
|
| 1859 |
|
|
if (x_sign == y_sign) {
|
| 1860 |
|
|
// the result cannot overflow
|
| 1861 |
|
|
C1_lo = C1_lo + C2_lo;
|
| 1862 |
|
|
C1_hi = C1_hi + C2_hi;
|
| 1863 |
|
|
if (C1_lo < C1.w[0])
|
| 1864 |
|
|
C1_hi++;
|
| 1865 |
|
|
} else { // if x_sign != y_sign
|
| 1866 |
|
|
C1_lo = C1_lo - C2_lo;
|
| 1867 |
|
|
C1_hi = C1_hi - C2_hi;
|
| 1868 |
|
|
if (C1_lo > C1.w[0])
|
| 1869 |
|
|
C1_hi--;
|
| 1870 |
|
|
// the result can be zero, but it cannot overflow
|
| 1871 |
|
|
if (C1_lo == 0 && C1_hi == 0) {
|
| 1872 |
|
|
// assemble the result
|
| 1873 |
|
|
if (x_exp < y_exp)
|
| 1874 |
|
|
res.w[1] = x_exp;
|
| 1875 |
|
|
else
|
| 1876 |
|
|
res.w[1] = y_exp;
|
| 1877 |
|
|
res.w[0] = 0;
|
| 1878 |
|
|
if (rnd_mode == ROUNDING_DOWN) {
|
| 1879 |
|
|
res.w[1] |= 0x8000000000000000ull;
|
| 1880 |
|
|
}
|
| 1881 |
|
|
BID_SWAP128 (res);
|
| 1882 |
|
|
BID_RETURN (res);
|
| 1883 |
|
|
}
|
| 1884 |
|
|
if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
|
| 1885 |
|
|
C1_lo = ~C1_lo;
|
| 1886 |
|
|
C1_lo++;
|
| 1887 |
|
|
C1_hi = ~C1_hi;
|
| 1888 |
|
|
if (C1_lo == 0x0)
|
| 1889 |
|
|
C1_hi++;
|
| 1890 |
|
|
x_sign = y_sign; // the result will have the sign of y
|
| 1891 |
|
|
}
|
| 1892 |
|
|
}
|
| 1893 |
|
|
// assemble the result
|
| 1894 |
|
|
res.w[1] = x_sign | y_exp | C1_hi;
|
| 1895 |
|
|
res.w[0] = C1_lo;
|
| 1896 |
|
|
} else if (delta == P34 - q2) {
|
| 1897 |
|
|
// calculate C' directly; the result may be inexact if it requires
|
| 1898 |
|
|
// P34+1 decimal digits; in this case the 'cutoff' point for addition
|
| 1899 |
|
|
// is at the position of the lsb of C2, so 0 <= e1-e2 <= P34-1
|
| 1900 |
|
|
// The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
|
| 1901 |
|
|
// exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
|
| 1902 |
|
|
// but their product fits with certainty in 128 bits (actually in 113)
|
| 1903 |
|
|
scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
|
| 1904 |
|
|
if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
|
| 1905 |
|
|
__mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
| 1906 |
|
|
} else if (scale >= 1) {
|
| 1907 |
|
|
// if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
|
| 1908 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 1909 |
|
|
__mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
| 1910 |
|
|
} else { // q1 >= 20
|
| 1911 |
|
|
C1.w[1] = C1_hi;
|
| 1912 |
|
|
C1.w[0] = C1_lo;
|
| 1913 |
|
|
__mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
| 1914 |
|
|
}
|
| 1915 |
|
|
} else { // if (scale == 0) C1 is unchanged
|
| 1916 |
|
|
C1.w[1] = C1_hi;
|
| 1917 |
|
|
C1.w[0] = C1_lo; // only the low part is necessary
|
| 1918 |
|
|
}
|
| 1919 |
|
|
C1_hi = C1.w[1];
|
| 1920 |
|
|
C1_lo = C1.w[0];
|
| 1921 |
|
|
// now add C2
|
| 1922 |
|
|
if (x_sign == y_sign) {
|
| 1923 |
|
|
// the result can overflow!
|
| 1924 |
|
|
C1_lo = C1_lo + C2_lo;
|
| 1925 |
|
|
C1_hi = C1_hi + C2_hi;
|
| 1926 |
|
|
if (C1_lo < C1.w[0])
|
| 1927 |
|
|
C1_hi++;
|
| 1928 |
|
|
// test for overflow, possible only when C1 >= 10^34
|
| 1929 |
|
|
if (C1_hi > 0x0001ed09bead87c0ull || (C1_hi == 0x0001ed09bead87c0ull && C1_lo >= 0x378d8e6400000000ull)) { // C1 >= 10^34
|
| 1930 |
|
|
// in this case q = P34 + 1 and x = q - P34 = 1, so multiply
|
| 1931 |
|
|
// C'' = C'+ 5 = C1 + 5 by k1 ~ 10^(-1) calculated for P34 + 1
|
| 1932 |
|
|
// decimal digits
|
| 1933 |
|
|
// Calculate C'' = C' + 1/2 * 10^x
|
| 1934 |
|
|
if (C1_lo >= 0xfffffffffffffffbull) { // low half add has carry
|
| 1935 |
|
|
C1_lo = C1_lo + 5;
|
| 1936 |
|
|
C1_hi = C1_hi + 1;
|
| 1937 |
|
|
} else {
|
| 1938 |
|
|
C1_lo = C1_lo + 5;
|
| 1939 |
|
|
}
|
| 1940 |
|
|
// the approximation of 10^(-1) was rounded up to 118 bits
|
| 1941 |
|
|
// 10^(-1) =~ 33333333333333333333333333333400 * 2^-129
|
| 1942 |
|
|
// 10^(-1) =~ 19999999999999999999999999999a00 * 2^-128
|
| 1943 |
|
|
C1.w[1] = C1_hi;
|
| 1944 |
|
|
C1.w[0] = C1_lo; // C''
|
| 1945 |
|
|
ten2m1.w[1] = 0x1999999999999999ull;
|
| 1946 |
|
|
ten2m1.w[0] = 0x9999999999999a00ull;
|
| 1947 |
|
|
__mul_128x128_to_256 (P256, C1, ten2m1); // P256 = C*, f*
|
| 1948 |
|
|
// C* is actually floor(C*) in this case
|
| 1949 |
|
|
// the top Ex = 128 bits of 10^(-1) are
|
| 1950 |
|
|
// T* = 0x00199999999999999999999999999999
|
| 1951 |
|
|
// if (0 < f* < 10^(-x)) then
|
| 1952 |
|
|
// if floor(C*) is even then C = floor(C*) - logical right
|
| 1953 |
|
|
// shift; C has p decimal digits, correct by Prop. 1)
|
| 1954 |
|
|
// else if floor(C*) is odd C = floor(C*) - 1 (logical right
|
| 1955 |
|
|
// shift; C has p decimal digits, correct by Pr. 1)
|
| 1956 |
|
|
// else
|
| 1957 |
|
|
// C = floor(C*) (logical right shift; C has p decimal digits,
|
| 1958 |
|
|
// correct by Property 1)
|
| 1959 |
|
|
// n = C * 10^(e2+x)
|
| 1960 |
|
|
if ((P256.w[1] || P256.w[0])
|
| 1961 |
|
|
&& (P256.w[1] < 0x1999999999999999ull
|
| 1962 |
|
|
|| (P256.w[1] == 0x1999999999999999ull
|
| 1963 |
|
|
&& P256.w[0] <= 0x9999999999999999ull))) {
|
| 1964 |
|
|
// the result is a midpoint
|
| 1965 |
|
|
if (P256.w[2] & 0x01) {
|
| 1966 |
|
|
is_midpoint_gt_even = 1;
|
| 1967 |
|
|
// if floor(C*) is odd C = floor(C*) - 1; the result is not 0
|
| 1968 |
|
|
P256.w[2]--;
|
| 1969 |
|
|
if (P256.w[2] == 0xffffffffffffffffull)
|
| 1970 |
|
|
P256.w[3]--;
|
| 1971 |
|
|
} else {
|
| 1972 |
|
|
is_midpoint_lt_even = 1;
|
| 1973 |
|
|
}
|
| 1974 |
|
|
}
|
| 1975 |
|
|
// n = Cstar * 10^(e2+1)
|
| 1976 |
|
|
y_exp = y_exp + EXP_P1;
|
| 1977 |
|
|
// C* != 10^P because C* has P34 digits
|
| 1978 |
|
|
// check for overflow
|
| 1979 |
|
|
if (y_exp == EXP_MAX_P1
|
| 1980 |
|
|
&& (rnd_mode == ROUNDING_TO_NEAREST
|
| 1981 |
|
|
|| rnd_mode == ROUNDING_TIES_AWAY)) {
|
| 1982 |
|
|
// overflow for RN
|
| 1983 |
|
|
res.w[1] = x_sign | 0x7800000000000000ull; // +/-inf
|
| 1984 |
|
|
res.w[0] = 0x0ull;
|
| 1985 |
|
|
// set the inexact flag
|
| 1986 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 1987 |
|
|
// set the overflow flag
|
| 1988 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 1989 |
|
|
BID_SWAP128 (res);
|
| 1990 |
|
|
BID_RETURN (res);
|
| 1991 |
|
|
}
|
| 1992 |
|
|
// if (0 < f* - 1/2 < 10^(-x)) then
|
| 1993 |
|
|
// the result of the addition is exact
|
| 1994 |
|
|
// else
|
| 1995 |
|
|
// the result of the addition is inexact
|
| 1996 |
|
|
if (P256.w[1] > 0x8000000000000000ull || (P256.w[1] == 0x8000000000000000ull && P256.w[0] > 0x0ull)) { // the result may be exact
|
| 1997 |
|
|
tmp64 = P256.w[1] - 0x8000000000000000ull; // f* - 1/2
|
| 1998 |
|
|
if ((tmp64 > 0x1999999999999999ull
|
| 1999 |
|
|
|| (tmp64 == 0x1999999999999999ull
|
| 2000 |
|
|
&& P256.w[0] >= 0x9999999999999999ull))) {
|
| 2001 |
|
|
// set the inexact flag
|
| 2002 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2003 |
|
|
is_inexact = 1;
|
| 2004 |
|
|
} // else the result is exact
|
| 2005 |
|
|
} else { // the result is inexact
|
| 2006 |
|
|
// set the inexact flag
|
| 2007 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2008 |
|
|
is_inexact = 1;
|
| 2009 |
|
|
}
|
| 2010 |
|
|
C1_hi = P256.w[3];
|
| 2011 |
|
|
C1_lo = P256.w[2];
|
| 2012 |
|
|
if (!is_midpoint_gt_even && !is_midpoint_lt_even) {
|
| 2013 |
|
|
is_inexact_lt_midpoint = is_inexact
|
| 2014 |
|
|
&& (P256.w[1] & 0x8000000000000000ull);
|
| 2015 |
|
|
is_inexact_gt_midpoint = is_inexact
|
| 2016 |
|
|
&& !(P256.w[1] & 0x8000000000000000ull);
|
| 2017 |
|
|
}
|
| 2018 |
|
|
// general correction from RN to RA, RM, RP, RZ;
|
| 2019 |
|
|
// result uses y_exp
|
| 2020 |
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
| 2021 |
|
|
if ((!x_sign
|
| 2022 |
|
|
&&
|
| 2023 |
|
|
((rnd_mode == ROUNDING_UP
|
| 2024 |
|
|
&& is_inexact_lt_midpoint)
|
| 2025 |
|
|
||
|
| 2026 |
|
|
((rnd_mode == ROUNDING_TIES_AWAY
|
| 2027 |
|
|
|| rnd_mode == ROUNDING_UP)
|
| 2028 |
|
|
&& is_midpoint_gt_even))) || (x_sign
|
| 2029 |
|
|
&&
|
| 2030 |
|
|
((rnd_mode ==
|
| 2031 |
|
|
ROUNDING_DOWN
|
| 2032 |
|
|
&&
|
| 2033 |
|
|
is_inexact_lt_midpoint)
|
| 2034 |
|
|
||
|
| 2035 |
|
|
((rnd_mode ==
|
| 2036 |
|
|
ROUNDING_TIES_AWAY
|
| 2037 |
|
|
|| rnd_mode ==
|
| 2038 |
|
|
ROUNDING_DOWN)
|
| 2039 |
|
|
&&
|
| 2040 |
|
|
is_midpoint_gt_even))))
|
| 2041 |
|
|
{
|
| 2042 |
|
|
// C1 = C1 + 1
|
| 2043 |
|
|
C1_lo = C1_lo + 1;
|
| 2044 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 2045 |
|
|
C1_hi = C1_hi + 1;
|
| 2046 |
|
|
}
|
| 2047 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 2048 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 2049 |
|
|
// C1 = 10^34 => rounding overflow
|
| 2050 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 2051 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 2052 |
|
|
y_exp = y_exp + EXP_P1;
|
| 2053 |
|
|
}
|
| 2054 |
|
|
} else
|
| 2055 |
|
|
if ((is_midpoint_lt_even || is_inexact_gt_midpoint)
|
| 2056 |
|
|
&&
|
| 2057 |
|
|
((x_sign
|
| 2058 |
|
|
&& (rnd_mode == ROUNDING_UP
|
| 2059 |
|
|
|| rnd_mode == ROUNDING_TO_ZERO))
|
| 2060 |
|
|
|| (!x_sign
|
| 2061 |
|
|
&& (rnd_mode == ROUNDING_DOWN
|
| 2062 |
|
|
|| rnd_mode == ROUNDING_TO_ZERO)))) {
|
| 2063 |
|
|
// C1 = C1 - 1
|
| 2064 |
|
|
C1_lo = C1_lo - 1;
|
| 2065 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 2066 |
|
|
C1_hi--;
|
| 2067 |
|
|
// check if we crossed into the lower decade
|
| 2068 |
|
|
if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
| 2069 |
|
|
C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
| 2070 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 2071 |
|
|
y_exp = y_exp - EXP_P1;
|
| 2072 |
|
|
// no underflow, because delta + q2 >= P34 + 1
|
| 2073 |
|
|
}
|
| 2074 |
|
|
} else {
|
| 2075 |
|
|
; // exact, the result is already correct
|
| 2076 |
|
|
}
|
| 2077 |
|
|
// in all cases check for overflow (RN and RA solved already)
|
| 2078 |
|
|
if (y_exp == EXP_MAX_P1) { // overflow
|
| 2079 |
|
|
if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
|
| 2080 |
|
|
(rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
|
| 2081 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 2082 |
|
|
C1_lo = 0x0ull;
|
| 2083 |
|
|
} else { // RM and res > 0, RP and res < 0, or RZ
|
| 2084 |
|
|
C1_hi = 0x5fffed09bead87c0ull;
|
| 2085 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 2086 |
|
|
}
|
| 2087 |
|
|
y_exp = 0; // x_sign is preserved
|
| 2088 |
|
|
// set the inexact flag (in case the exact addition was exact)
|
| 2089 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2090 |
|
|
// set the overflow flag
|
| 2091 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 2092 |
|
|
}
|
| 2093 |
|
|
}
|
| 2094 |
|
|
} // else if (C1 < 10^34) then C1 is the coeff.; the result is exact
|
| 2095 |
|
|
} else { // if x_sign != y_sign the result is exact
|
| 2096 |
|
|
C1_lo = C1_lo - C2_lo;
|
| 2097 |
|
|
C1_hi = C1_hi - C2_hi;
|
| 2098 |
|
|
if (C1_lo > C1.w[0])
|
| 2099 |
|
|
C1_hi--;
|
| 2100 |
|
|
// the result can be zero, but it cannot overflow
|
| 2101 |
|
|
if (C1_lo == 0 && C1_hi == 0) {
|
| 2102 |
|
|
// assemble the result
|
| 2103 |
|
|
if (x_exp < y_exp)
|
| 2104 |
|
|
res.w[1] = x_exp;
|
| 2105 |
|
|
else
|
| 2106 |
|
|
res.w[1] = y_exp;
|
| 2107 |
|
|
res.w[0] = 0;
|
| 2108 |
|
|
if (rnd_mode == ROUNDING_DOWN) {
|
| 2109 |
|
|
res.w[1] |= 0x8000000000000000ull;
|
| 2110 |
|
|
}
|
| 2111 |
|
|
BID_SWAP128 (res);
|
| 2112 |
|
|
BID_RETURN (res);
|
| 2113 |
|
|
}
|
| 2114 |
|
|
if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
|
| 2115 |
|
|
C1_lo = ~C1_lo;
|
| 2116 |
|
|
C1_lo++;
|
| 2117 |
|
|
C1_hi = ~C1_hi;
|
| 2118 |
|
|
if (C1_lo == 0x0)
|
| 2119 |
|
|
C1_hi++;
|
| 2120 |
|
|
x_sign = y_sign; // the result will have the sign of y
|
| 2121 |
|
|
}
|
| 2122 |
|
|
}
|
| 2123 |
|
|
// assemble the result
|
| 2124 |
|
|
res.w[1] = x_sign | y_exp | C1_hi;
|
| 2125 |
|
|
res.w[0] = C1_lo;
|
| 2126 |
|
|
} else { // if (delta >= P34 + 1 - q2)
|
| 2127 |
|
|
// instead of C' = (C1 * 10^(e1-e2) + C2)rnd,P34
|
| 2128 |
|
|
// calculate C' = C1 * 10^(e1-e2-x1) + (C2 * 10^(-x1))rnd,P34
|
| 2129 |
|
|
// where x1 = q1 + e1 - e2 - P34, 1 <= x1 <= P34 - 1
|
| 2130 |
|
|
// In most cases C' will have P34 digits, and n = C' * 10^(e2+x1)
|
| 2131 |
|
|
// If the result has P34+1 digits, redo the steps above with x1+1
|
| 2132 |
|
|
// If the result has P34-1 digits or less, redo the steps above with
|
| 2133 |
|
|
// x1-1 but only if initially x1 >= 1
|
| 2134 |
|
|
// NOTE: these two steps can be improved, e.g we could guess if
|
| 2135 |
|
|
// P34+1 or P34-1 digits will be obtained by adding/subtracting just
|
| 2136 |
|
|
// the top 64 bits of the two operands
|
| 2137 |
|
|
// The result cannot be zero, but it can overflow
|
| 2138 |
|
|
x1 = delta + q2 - P34; // 1 <= x1 <= P34-1
|
| 2139 |
|
|
roundC2:
|
| 2140 |
|
|
// Calculate C1 * 10^(e1-e2-x1) where 0 <= e1-e2-x1 <= P34 - 1
|
| 2141 |
|
|
// scale = (int)(e1 >> 49) - (int)(e2 >> 49) - x1; 0 <= scale <= P34-1
|
| 2142 |
|
|
scale = delta - q1 + q2 - x1; // scale = e1 - e2 - x1 = P34 - q1
|
| 2143 |
|
|
// either C1 or 10^(e1-e2-x1) may not fit is 64 bits,
|
| 2144 |
|
|
// but their product fits with certainty in 128 bits (actually in 113)
|
| 2145 |
|
|
if (scale >= 20) { //10^(e1-e2-x1) doesn't fit in 64 bits, but C1 does
|
| 2146 |
|
|
__mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
| 2147 |
|
|
} else if (scale >= 1) {
|
| 2148 |
|
|
// if 1 <= scale <= 19 then 10^(e1-e2-x1) fits in 64 bits
|
| 2149 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 2150 |
|
|
__mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
| 2151 |
|
|
} else { // q1 >= 20
|
| 2152 |
|
|
C1.w[1] = C1_hi;
|
| 2153 |
|
|
C1.w[0] = C1_lo;
|
| 2154 |
|
|
__mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
| 2155 |
|
|
}
|
| 2156 |
|
|
} else { // if (scale == 0) C1 is unchanged
|
| 2157 |
|
|
C1.w[1] = C1_hi;
|
| 2158 |
|
|
C1.w[0] = C1_lo;
|
| 2159 |
|
|
}
|
| 2160 |
|
|
tmp64 = C1.w[0]; // C1.w[1], C1.w[0] contains C1 * 10^(e1-e2-x1)
|
| 2161 |
|
|
|
| 2162 |
|
|
// now round C2 to q2-x1 decimal digits, where 1<=x1<=q2-1<=P34-1
|
| 2163 |
|
|
// (but if we got here a second time after x1 = x1 - 1, then
|
| 2164 |
|
|
// x1 >= 0; note that for x1 = 0 C2 is unchanged)
|
| 2165 |
|
|
// C2' = C2 + 1/2 * 10^x1 = C2 + 5 * 10^(x1-1)
|
| 2166 |
|
|
ind = x1 - 1; // 0 <= ind <= q2-2<=P34-2=32; but note that if x1 = 0
|
| 2167 |
|
|
// during a second pass, then ind = -1
|
| 2168 |
|
|
if (ind >= 0) { // if (x1 >= 1)
|
| 2169 |
|
|
C2.w[0] = C2_lo;
|
| 2170 |
|
|
C2.w[1] = C2_hi;
|
| 2171 |
|
|
if (ind <= 18) {
|
| 2172 |
|
|
C2.w[0] = C2.w[0] + midpoint64[ind];
|
| 2173 |
|
|
if (C2.w[0] < C2_lo)
|
| 2174 |
|
|
C2.w[1]++;
|
| 2175 |
|
|
} else { // 19 <= ind <= 32
|
| 2176 |
|
|
C2.w[0] = C2.w[0] + midpoint128[ind - 19].w[0];
|
| 2177 |
|
|
C2.w[1] = C2.w[1] + midpoint128[ind - 19].w[1];
|
| 2178 |
|
|
if (C2.w[0] < C2_lo)
|
| 2179 |
|
|
C2.w[1]++;
|
| 2180 |
|
|
}
|
| 2181 |
|
|
// the approximation of 10^(-x1) was rounded up to 118 bits
|
| 2182 |
|
|
__mul_128x128_to_256 (R256, C2, ten2mk128[ind]); // R256 = C2*, f2*
|
| 2183 |
|
|
// calculate C2* and f2*
|
| 2184 |
|
|
// C2* is actually floor(C2*) in this case
|
| 2185 |
|
|
// C2* and f2* need shifting and masking, as shown by
|
| 2186 |
|
|
// shiftright128[] and maskhigh128[]
|
| 2187 |
|
|
// the top Ex bits of 10^(-x1) are T* = ten2mk128trunc[ind], e.g.
|
| 2188 |
|
|
// if x1=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
|
| 2189 |
|
|
// if (0 < f2* < 10^(-x1)) then
|
| 2190 |
|
|
// if floor(C1+C2*) is even then C2* = floor(C2*) - logical right
|
| 2191 |
|
|
// shift; C2* has p decimal digits, correct by Prop. 1)
|
| 2192 |
|
|
// else if floor(C1+C2*) is odd C2* = floor(C2*)-1 (logical right
|
| 2193 |
|
|
// shift; C2* has p decimal digits, correct by Pr. 1)
|
| 2194 |
|
|
// else
|
| 2195 |
|
|
// C2* = floor(C2*) (logical right shift; C has p decimal digits,
|
| 2196 |
|
|
// correct by Property 1)
|
| 2197 |
|
|
// n = C2* * 10^(e2+x1)
|
| 2198 |
|
|
|
| 2199 |
|
|
if (ind <= 2) {
|
| 2200 |
|
|
highf2star.w[1] = 0x0;
|
| 2201 |
|
|
highf2star.w[0] = 0x0; // low f2* ok
|
| 2202 |
|
|
} else if (ind <= 21) {
|
| 2203 |
|
|
highf2star.w[1] = 0x0;
|
| 2204 |
|
|
highf2star.w[0] = R256.w[2] & maskhigh128[ind]; // low f2* ok
|
| 2205 |
|
|
} else {
|
| 2206 |
|
|
highf2star.w[1] = R256.w[3] & maskhigh128[ind];
|
| 2207 |
|
|
highf2star.w[0] = R256.w[2]; // low f2* is ok
|
| 2208 |
|
|
}
|
| 2209 |
|
|
// shift right C2* by Ex-128 = shiftright128[ind]
|
| 2210 |
|
|
if (ind >= 3) {
|
| 2211 |
|
|
shift = shiftright128[ind];
|
| 2212 |
|
|
if (shift < 64) { // 3 <= shift <= 63
|
| 2213 |
|
|
R256.w[2] =
|
| 2214 |
|
|
(R256.w[2] >> shift) | (R256.w[3] << (64 - shift));
|
| 2215 |
|
|
R256.w[3] = (R256.w[3] >> shift);
|
| 2216 |
|
|
} else { // 66 <= shift <= 102
|
| 2217 |
|
|
R256.w[2] = (R256.w[3] >> (shift - 64));
|
| 2218 |
|
|
R256.w[3] = 0x0ULL;
|
| 2219 |
|
|
}
|
| 2220 |
|
|
}
|
| 2221 |
|
|
if (second_pass) {
|
| 2222 |
|
|
is_inexact_lt_midpoint = 0;
|
| 2223 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2224 |
|
|
is_midpoint_lt_even = 0;
|
| 2225 |
|
|
is_midpoint_gt_even = 0;
|
| 2226 |
|
|
}
|
| 2227 |
|
|
// determine inexactness of the rounding of C2* (this may be
|
| 2228 |
|
|
// followed by a second rounding only if we get P34+1
|
| 2229 |
|
|
// decimal digits)
|
| 2230 |
|
|
// if (0 < f2* - 1/2 < 10^(-x1)) then
|
| 2231 |
|
|
// the result is exact
|
| 2232 |
|
|
// else (if f2* - 1/2 > T* then)
|
| 2233 |
|
|
// the result of is inexact
|
| 2234 |
|
|
if (ind <= 2) {
|
| 2235 |
|
|
if (R256.w[1] > 0x8000000000000000ull ||
|
| 2236 |
|
|
(R256.w[1] == 0x8000000000000000ull
|
| 2237 |
|
|
&& R256.w[0] > 0x0ull)) {
|
| 2238 |
|
|
// f2* > 1/2 and the result may be exact
|
| 2239 |
|
|
tmp64A = R256.w[1] - 0x8000000000000000ull; // f* - 1/2
|
| 2240 |
|
|
if ((tmp64A > ten2mk128trunc[ind].w[1]
|
| 2241 |
|
|
|| (tmp64A == ten2mk128trunc[ind].w[1]
|
| 2242 |
|
|
&& R256.w[0] >= ten2mk128trunc[ind].w[0]))) {
|
| 2243 |
|
|
// set the inexact flag
|
| 2244 |
|
|
// *pfpsf |= INEXACT_EXCEPTION;
|
| 2245 |
|
|
tmp_inexact = 1; // may be set again during a second pass
|
| 2246 |
|
|
// this rounding is applied to C2 only!
|
| 2247 |
|
|
if (x_sign == y_sign)
|
| 2248 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2249 |
|
|
else // if (x_sign != y_sign)
|
| 2250 |
|
|
is_inexact_gt_midpoint = 1;
|
| 2251 |
|
|
} // else the result is exact
|
| 2252 |
|
|
// rounding down, unless a midpoint in [ODD, EVEN]
|
| 2253 |
|
|
} else { // the result is inexact; f2* <= 1/2
|
| 2254 |
|
|
// set the inexact flag
|
| 2255 |
|
|
// *pfpsf |= INEXACT_EXCEPTION;
|
| 2256 |
|
|
tmp_inexact = 1; // just in case we will round a second time
|
| 2257 |
|
|
// rounding up, unless a midpoint in [EVEN, ODD]
|
| 2258 |
|
|
// this rounding is applied to C2 only!
|
| 2259 |
|
|
if (x_sign == y_sign)
|
| 2260 |
|
|
is_inexact_gt_midpoint = 1;
|
| 2261 |
|
|
else // if (x_sign != y_sign)
|
| 2262 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2263 |
|
|
}
|
| 2264 |
|
|
} else if (ind <= 21) { // if 3 <= ind <= 21
|
| 2265 |
|
|
if (highf2star.w[1] > 0x0 || (highf2star.w[1] == 0x0
|
| 2266 |
|
|
&& highf2star.w[0] >
|
| 2267 |
|
|
onehalf128[ind])
|
| 2268 |
|
|
|| (highf2star.w[1] == 0x0
|
| 2269 |
|
|
&& highf2star.w[0] == onehalf128[ind]
|
| 2270 |
|
|
&& (R256.w[1] || R256.w[0]))) {
|
| 2271 |
|
|
// f2* > 1/2 and the result may be exact
|
| 2272 |
|
|
// Calculate f2* - 1/2
|
| 2273 |
|
|
tmp64A = highf2star.w[0] - onehalf128[ind];
|
| 2274 |
|
|
tmp64B = highf2star.w[1];
|
| 2275 |
|
|
if (tmp64A > highf2star.w[0])
|
| 2276 |
|
|
tmp64B--;
|
| 2277 |
|
|
if (tmp64B || tmp64A
|
| 2278 |
|
|
|| R256.w[1] > ten2mk128trunc[ind].w[1]
|
| 2279 |
|
|
|| (R256.w[1] == ten2mk128trunc[ind].w[1]
|
| 2280 |
|
|
&& R256.w[0] > ten2mk128trunc[ind].w[0])) {
|
| 2281 |
|
|
// set the inexact flag
|
| 2282 |
|
|
// *pfpsf |= INEXACT_EXCEPTION;
|
| 2283 |
|
|
tmp_inexact = 1; // may be set again during a second pass
|
| 2284 |
|
|
// this rounding is applied to C2 only!
|
| 2285 |
|
|
if (x_sign == y_sign)
|
| 2286 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2287 |
|
|
else // if (x_sign != y_sign)
|
| 2288 |
|
|
is_inexact_gt_midpoint = 1;
|
| 2289 |
|
|
} // else the result is exact
|
| 2290 |
|
|
} else { // the result is inexact; f2* <= 1/2
|
| 2291 |
|
|
// set the inexact flag
|
| 2292 |
|
|
// *pfpsf |= INEXACT_EXCEPTION;
|
| 2293 |
|
|
tmp_inexact = 1; // may be set again during a second pass
|
| 2294 |
|
|
// rounding up, unless a midpoint in [EVEN, ODD]
|
| 2295 |
|
|
// this rounding is applied to C2 only!
|
| 2296 |
|
|
if (x_sign == y_sign)
|
| 2297 |
|
|
is_inexact_gt_midpoint = 1;
|
| 2298 |
|
|
else // if (x_sign != y_sign)
|
| 2299 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2300 |
|
|
}
|
| 2301 |
|
|
} else { // if 22 <= ind <= 33
|
| 2302 |
|
|
if (highf2star.w[1] > onehalf128[ind]
|
| 2303 |
|
|
|| (highf2star.w[1] == onehalf128[ind]
|
| 2304 |
|
|
&& (highf2star.w[0] || R256.w[1]
|
| 2305 |
|
|
|| R256.w[0]))) {
|
| 2306 |
|
|
// f2* > 1/2 and the result may be exact
|
| 2307 |
|
|
// Calculate f2* - 1/2
|
| 2308 |
|
|
// tmp64A = highf2star.w[0];
|
| 2309 |
|
|
tmp64B = highf2star.w[1] - onehalf128[ind];
|
| 2310 |
|
|
if (tmp64B || highf2star.w[0]
|
| 2311 |
|
|
|| R256.w[1] > ten2mk128trunc[ind].w[1]
|
| 2312 |
|
|
|| (R256.w[1] == ten2mk128trunc[ind].w[1]
|
| 2313 |
|
|
&& R256.w[0] > ten2mk128trunc[ind].w[0])) {
|
| 2314 |
|
|
// set the inexact flag
|
| 2315 |
|
|
// *pfpsf |= INEXACT_EXCEPTION;
|
| 2316 |
|
|
tmp_inexact = 1; // may be set again during a second pass
|
| 2317 |
|
|
// this rounding is applied to C2 only!
|
| 2318 |
|
|
if (x_sign == y_sign)
|
| 2319 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2320 |
|
|
else // if (x_sign != y_sign)
|
| 2321 |
|
|
is_inexact_gt_midpoint = 1;
|
| 2322 |
|
|
} // else the result is exact
|
| 2323 |
|
|
} else { // the result is inexact; f2* <= 1/2
|
| 2324 |
|
|
// set the inexact flag
|
| 2325 |
|
|
// *pfpsf |= INEXACT_EXCEPTION;
|
| 2326 |
|
|
tmp_inexact = 1; // may be set again during a second pass
|
| 2327 |
|
|
// rounding up, unless a midpoint in [EVEN, ODD]
|
| 2328 |
|
|
// this rounding is applied to C2 only!
|
| 2329 |
|
|
if (x_sign == y_sign)
|
| 2330 |
|
|
is_inexact_gt_midpoint = 1;
|
| 2331 |
|
|
else // if (x_sign != y_sign)
|
| 2332 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2333 |
|
|
}
|
| 2334 |
|
|
}
|
| 2335 |
|
|
// check for midpoints
|
| 2336 |
|
|
if ((R256.w[1] || R256.w[0]) && (highf2star.w[1] == 0)
|
| 2337 |
|
|
&& (highf2star.w[0] == 0)
|
| 2338 |
|
|
&& (R256.w[1] < ten2mk128trunc[ind].w[1]
|
| 2339 |
|
|
|| (R256.w[1] == ten2mk128trunc[ind].w[1]
|
| 2340 |
|
|
&& R256.w[0] <= ten2mk128trunc[ind].w[0]))) {
|
| 2341 |
|
|
// the result is a midpoint
|
| 2342 |
|
|
if ((tmp64 + R256.w[2]) & 0x01) { // MP in [EVEN, ODD]
|
| 2343 |
|
|
// if floor(C2*) is odd C = floor(C2*) - 1; the result may be 0
|
| 2344 |
|
|
R256.w[2]--;
|
| 2345 |
|
|
if (R256.w[2] == 0xffffffffffffffffull)
|
| 2346 |
|
|
R256.w[3]--;
|
| 2347 |
|
|
// this rounding is applied to C2 only!
|
| 2348 |
|
|
if (x_sign == y_sign)
|
| 2349 |
|
|
is_midpoint_gt_even = 1;
|
| 2350 |
|
|
else // if (x_sign != y_sign)
|
| 2351 |
|
|
is_midpoint_lt_even = 1;
|
| 2352 |
|
|
is_inexact_lt_midpoint = 0;
|
| 2353 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2354 |
|
|
} else {
|
| 2355 |
|
|
// else MP in [ODD, EVEN]
|
| 2356 |
|
|
// this rounding is applied to C2 only!
|
| 2357 |
|
|
if (x_sign == y_sign)
|
| 2358 |
|
|
is_midpoint_lt_even = 1;
|
| 2359 |
|
|
else // if (x_sign != y_sign)
|
| 2360 |
|
|
is_midpoint_gt_even = 1;
|
| 2361 |
|
|
is_inexact_lt_midpoint = 0;
|
| 2362 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2363 |
|
|
}
|
| 2364 |
|
|
}
|
| 2365 |
|
|
// end if (ind >= 0)
|
| 2366 |
|
|
} else { // if (ind == -1); only during a 2nd pass, and when x1 = 0
|
| 2367 |
|
|
R256.w[2] = C2_lo;
|
| 2368 |
|
|
R256.w[3] = C2_hi;
|
| 2369 |
|
|
tmp_inexact = 0;
|
| 2370 |
|
|
// to correct a possible setting to 1 from 1st pass
|
| 2371 |
|
|
if (second_pass) {
|
| 2372 |
|
|
is_midpoint_lt_even = 0;
|
| 2373 |
|
|
is_midpoint_gt_even = 0;
|
| 2374 |
|
|
is_inexact_lt_midpoint = 0;
|
| 2375 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2376 |
|
|
}
|
| 2377 |
|
|
}
|
| 2378 |
|
|
// and now add/subtract C1 * 10^(e1-e2-x1) +/- (C2 * 10^(-x1))rnd,P34
|
| 2379 |
|
|
if (x_sign == y_sign) { // addition; could overflow
|
| 2380 |
|
|
// no second pass is possible this way (only for x_sign != y_sign)
|
| 2381 |
|
|
C1.w[0] = C1.w[0] + R256.w[2];
|
| 2382 |
|
|
C1.w[1] = C1.w[1] + R256.w[3];
|
| 2383 |
|
|
if (C1.w[0] < tmp64)
|
| 2384 |
|
|
C1.w[1]++; // carry
|
| 2385 |
|
|
// if the sum has P34+1 digits, i.e. C1>=10^34 redo the calculation
|
| 2386 |
|
|
// with x1=x1+1
|
| 2387 |
|
|
if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] >= 0x378d8e6400000000ull)) { // C1 >= 10^34
|
| 2388 |
|
|
// chop off one more digit from the sum, but make sure there is
|
| 2389 |
|
|
// no double-rounding error (see table - double rounding logic)
|
| 2390 |
|
|
// now round C1 from P34+1 to P34 decimal digits
|
| 2391 |
|
|
// C1' = C1 + 1/2 * 10 = C1 + 5
|
| 2392 |
|
|
if (C1.w[0] >= 0xfffffffffffffffbull) { // low half add has carry
|
| 2393 |
|
|
C1.w[0] = C1.w[0] + 5;
|
| 2394 |
|
|
C1.w[1] = C1.w[1] + 1;
|
| 2395 |
|
|
} else {
|
| 2396 |
|
|
C1.w[0] = C1.w[0] + 5;
|
| 2397 |
|
|
}
|
| 2398 |
|
|
// the approximation of 10^(-1) was rounded up to 118 bits
|
| 2399 |
|
|
__mul_128x128_to_256 (Q256, C1, ten2mk128[0]); // Q256 = C1*, f1*
|
| 2400 |
|
|
// C1* is actually floor(C1*) in this case
|
| 2401 |
|
|
// the top 128 bits of 10^(-1) are
|
| 2402 |
|
|
// T* = ten2mk128trunc[0]=0x19999999999999999999999999999999
|
| 2403 |
|
|
// if (0 < f1* < 10^(-1)) then
|
| 2404 |
|
|
// if floor(C1*) is even then C1* = floor(C1*) - logical right
|
| 2405 |
|
|
// shift; C1* has p decimal digits, correct by Prop. 1)
|
| 2406 |
|
|
// else if floor(C1*) is odd C1* = floor(C1*) - 1 (logical right
|
| 2407 |
|
|
// shift; C1* has p decimal digits, correct by Pr. 1)
|
| 2408 |
|
|
// else
|
| 2409 |
|
|
// C1* = floor(C1*) (logical right shift; C has p decimal digits
|
| 2410 |
|
|
// correct by Property 1)
|
| 2411 |
|
|
// n = C1* * 10^(e2+x1+1)
|
| 2412 |
|
|
if ((Q256.w[1] || Q256.w[0])
|
| 2413 |
|
|
&& (Q256.w[1] < ten2mk128trunc[0].w[1]
|
| 2414 |
|
|
|| (Q256.w[1] == ten2mk128trunc[0].w[1]
|
| 2415 |
|
|
&& Q256.w[0] <= ten2mk128trunc[0].w[0]))) {
|
| 2416 |
|
|
// the result is a midpoint
|
| 2417 |
|
|
if (is_inexact_lt_midpoint) { // for the 1st rounding
|
| 2418 |
|
|
is_inexact_gt_midpoint = 1;
|
| 2419 |
|
|
is_inexact_lt_midpoint = 0;
|
| 2420 |
|
|
is_midpoint_gt_even = 0;
|
| 2421 |
|
|
is_midpoint_lt_even = 0;
|
| 2422 |
|
|
} else if (is_inexact_gt_midpoint) { // for the 1st rounding
|
| 2423 |
|
|
Q256.w[2]--;
|
| 2424 |
|
|
if (Q256.w[2] == 0xffffffffffffffffull)
|
| 2425 |
|
|
Q256.w[3]--;
|
| 2426 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2427 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2428 |
|
|
is_midpoint_gt_even = 0;
|
| 2429 |
|
|
is_midpoint_lt_even = 0;
|
| 2430 |
|
|
} else if (is_midpoint_gt_even) { // for the 1st rounding
|
| 2431 |
|
|
// Note: cannot have is_midpoint_lt_even
|
| 2432 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2433 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2434 |
|
|
is_midpoint_gt_even = 0;
|
| 2435 |
|
|
is_midpoint_lt_even = 0;
|
| 2436 |
|
|
} else { // the first rounding must have been exact
|
| 2437 |
|
|
if (Q256.w[2] & 0x01) { // MP in [EVEN, ODD]
|
| 2438 |
|
|
// the truncated result is correct
|
| 2439 |
|
|
Q256.w[2]--;
|
| 2440 |
|
|
if (Q256.w[2] == 0xffffffffffffffffull)
|
| 2441 |
|
|
Q256.w[3]--;
|
| 2442 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2443 |
|
|
is_inexact_lt_midpoint = 0;
|
| 2444 |
|
|
is_midpoint_gt_even = 1;
|
| 2445 |
|
|
is_midpoint_lt_even = 0;
|
| 2446 |
|
|
} else { // MP in [ODD, EVEN]
|
| 2447 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2448 |
|
|
is_inexact_lt_midpoint = 0;
|
| 2449 |
|
|
is_midpoint_gt_even = 0;
|
| 2450 |
|
|
is_midpoint_lt_even = 1;
|
| 2451 |
|
|
}
|
| 2452 |
|
|
}
|
| 2453 |
|
|
tmp_inexact = 1; // in all cases
|
| 2454 |
|
|
} else { // the result is not a midpoint
|
| 2455 |
|
|
// determine inexactness of the rounding of C1 (the sum C1+C2*)
|
| 2456 |
|
|
// if (0 < f1* - 1/2 < 10^(-1)) then
|
| 2457 |
|
|
// the result is exact
|
| 2458 |
|
|
// else (if f1* - 1/2 > T* then)
|
| 2459 |
|
|
// the result of is inexact
|
| 2460 |
|
|
// ind = 0
|
| 2461 |
|
|
if (Q256.w[1] > 0x8000000000000000ull
|
| 2462 |
|
|
|| (Q256.w[1] == 0x8000000000000000ull
|
| 2463 |
|
|
&& Q256.w[0] > 0x0ull)) {
|
| 2464 |
|
|
// f1* > 1/2 and the result may be exact
|
| 2465 |
|
|
Q256.w[1] = Q256.w[1] - 0x8000000000000000ull; // f1* - 1/2
|
| 2466 |
|
|
if ((Q256.w[1] > ten2mk128trunc[0].w[1]
|
| 2467 |
|
|
|| (Q256.w[1] == ten2mk128trunc[0].w[1]
|
| 2468 |
|
|
&& Q256.w[0] > ten2mk128trunc[0].w[0]))) {
|
| 2469 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2470 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2471 |
|
|
is_midpoint_gt_even = 0;
|
| 2472 |
|
|
is_midpoint_lt_even = 0;
|
| 2473 |
|
|
// set the inexact flag
|
| 2474 |
|
|
tmp_inexact = 1;
|
| 2475 |
|
|
// *pfpsf |= INEXACT_EXCEPTION;
|
| 2476 |
|
|
} else { // else the result is exact for the 2nd rounding
|
| 2477 |
|
|
if (tmp_inexact) { // if the previous rounding was inexact
|
| 2478 |
|
|
if (is_midpoint_lt_even) {
|
| 2479 |
|
|
is_inexact_gt_midpoint = 1;
|
| 2480 |
|
|
is_midpoint_lt_even = 0;
|
| 2481 |
|
|
} else if (is_midpoint_gt_even) {
|
| 2482 |
|
|
is_inexact_lt_midpoint = 1;
|
| 2483 |
|
|
is_midpoint_gt_even = 0;
|
| 2484 |
|
|
} else {
|
| 2485 |
|
|
; // no change
|
| 2486 |
|
|
}
|
| 2487 |
|
|
}
|
| 2488 |
|
|
}
|
| 2489 |
|
|
// rounding down, unless a midpoint in [ODD, EVEN]
|
| 2490 |
|
|
} else { // the result is inexact; f1* <= 1/2
|
| 2491 |
|
|
is_inexact_gt_midpoint = 1;
|
| 2492 |
|
|
is_inexact_lt_midpoint = 0;
|
| 2493 |
|
|
is_midpoint_gt_even = 0;
|
| 2494 |
|
|
is_midpoint_lt_even = 0;
|
| 2495 |
|
|
// set the inexact flag
|
| 2496 |
|
|
tmp_inexact = 1;
|
| 2497 |
|
|
// *pfpsf |= INEXACT_EXCEPTION;
|
| 2498 |
|
|
}
|
| 2499 |
|
|
} // end 'the result is not a midpoint'
|
| 2500 |
|
|
// n = C1 * 10^(e2+x1)
|
| 2501 |
|
|
C1.w[1] = Q256.w[3];
|
| 2502 |
|
|
C1.w[0] = Q256.w[2];
|
| 2503 |
|
|
y_exp = y_exp + ((UINT64) (x1 + 1) << 49);
|
| 2504 |
|
|
} else { // C1 < 10^34
|
| 2505 |
|
|
// C1.w[1] and C1.w[0] already set
|
| 2506 |
|
|
// n = C1 * 10^(e2+x1)
|
| 2507 |
|
|
y_exp = y_exp + ((UINT64) x1 << 49);
|
| 2508 |
|
|
}
|
| 2509 |
|
|
// check for overflow
|
| 2510 |
|
|
if (y_exp == EXP_MAX_P1
|
| 2511 |
|
|
&& (rnd_mode == ROUNDING_TO_NEAREST
|
| 2512 |
|
|
|| rnd_mode == ROUNDING_TIES_AWAY)) {
|
| 2513 |
|
|
res.w[1] = 0x7800000000000000ull | x_sign; // +/-inf
|
| 2514 |
|
|
res.w[0] = 0x0ull;
|
| 2515 |
|
|
// set the inexact flag
|
| 2516 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2517 |
|
|
// set the overflow flag
|
| 2518 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 2519 |
|
|
BID_SWAP128 (res);
|
| 2520 |
|
|
BID_RETURN (res);
|
| 2521 |
|
|
} // else no overflow
|
| 2522 |
|
|
} else { // if x_sign != y_sign the result of this subtract. is exact
|
| 2523 |
|
|
C1.w[0] = C1.w[0] - R256.w[2];
|
| 2524 |
|
|
C1.w[1] = C1.w[1] - R256.w[3];
|
| 2525 |
|
|
if (C1.w[0] > tmp64)
|
| 2526 |
|
|
C1.w[1]--; // borrow
|
| 2527 |
|
|
if (C1.w[1] >= 0x8000000000000000ull) { // negative coefficient!
|
| 2528 |
|
|
C1.w[0] = ~C1.w[0];
|
| 2529 |
|
|
C1.w[0]++;
|
| 2530 |
|
|
C1.w[1] = ~C1.w[1];
|
| 2531 |
|
|
if (C1.w[0] == 0x0)
|
| 2532 |
|
|
C1.w[1]++;
|
| 2533 |
|
|
tmp_sign = y_sign;
|
| 2534 |
|
|
// the result will have the sign of y if last rnd
|
| 2535 |
|
|
} else {
|
| 2536 |
|
|
tmp_sign = x_sign;
|
| 2537 |
|
|
}
|
| 2538 |
|
|
// if the difference has P34-1 digits or less, i.e. C1 < 10^33 then
|
| 2539 |
|
|
// redo the calculation with x1=x1-1;
|
| 2540 |
|
|
// redo the calculation also if C1 = 10^33 and
|
| 2541 |
|
|
// (is_inexact_gt_midpoint or is_midpoint_lt_even);
|
| 2542 |
|
|
// (the last part should have really been
|
| 2543 |
|
|
// (is_inexact_lt_midpoint or is_midpoint_gt_even) from
|
| 2544 |
|
|
// the rounding of C2, but the position flags have been reversed)
|
| 2545 |
|
|
// 10^33 = 0x0000314dc6448d93 0x38c15b0a00000000
|
| 2546 |
|
|
if ((C1.w[1] < 0x0000314dc6448d93ull || (C1.w[1] == 0x0000314dc6448d93ull && C1.w[0] < 0x38c15b0a00000000ull)) || (C1.w[1] == 0x0000314dc6448d93ull && C1.w[0] == 0x38c15b0a00000000ull && (is_inexact_gt_midpoint || is_midpoint_lt_even))) { // C1=10^33
|
| 2547 |
|
|
x1 = x1 - 1; // x1 >= 0
|
| 2548 |
|
|
if (x1 >= 0) {
|
| 2549 |
|
|
// clear position flags and tmp_inexact
|
| 2550 |
|
|
is_midpoint_lt_even = 0;
|
| 2551 |
|
|
is_midpoint_gt_even = 0;
|
| 2552 |
|
|
is_inexact_lt_midpoint = 0;
|
| 2553 |
|
|
is_inexact_gt_midpoint = 0;
|
| 2554 |
|
|
tmp_inexact = 0;
|
| 2555 |
|
|
second_pass = 1;
|
| 2556 |
|
|
goto roundC2; // else result has less than P34 digits
|
| 2557 |
|
|
}
|
| 2558 |
|
|
}
|
| 2559 |
|
|
// if the coefficient of the result is 10^34 it means that this
|
| 2560 |
|
|
// must be the second pass, and we are done
|
| 2561 |
|
|
if (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] == 0x378d8e6400000000ull) { // if C1 = 10^34
|
| 2562 |
|
|
C1.w[1] = 0x0000314dc6448d93ull; // C1 = 10^33
|
| 2563 |
|
|
C1.w[0] = 0x38c15b0a00000000ull;
|
| 2564 |
|
|
y_exp = y_exp + ((UINT64) 1 << 49);
|
| 2565 |
|
|
}
|
| 2566 |
|
|
x_sign = tmp_sign;
|
| 2567 |
|
|
if (x1 >= 1)
|
| 2568 |
|
|
y_exp = y_exp + ((UINT64) x1 << 49);
|
| 2569 |
|
|
// x1 = -1 is possible at the end of a second pass when the
|
| 2570 |
|
|
// first pass started with x1 = 1
|
| 2571 |
|
|
}
|
| 2572 |
|
|
C1_hi = C1.w[1];
|
| 2573 |
|
|
C1_lo = C1.w[0];
|
| 2574 |
|
|
// general correction from RN to RA, RM, RP, RZ; result uses y_exp
|
| 2575 |
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
| 2576 |
|
|
if ((!x_sign
|
| 2577 |
|
|
&& ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint)
|
| 2578 |
|
|
||
|
| 2579 |
|
|
((rnd_mode == ROUNDING_TIES_AWAY
|
| 2580 |
|
|
|| rnd_mode == ROUNDING_UP)
|
| 2581 |
|
|
&& is_midpoint_gt_even))) || (x_sign
|
| 2582 |
|
|
&&
|
| 2583 |
|
|
((rnd_mode ==
|
| 2584 |
|
|
ROUNDING_DOWN
|
| 2585 |
|
|
&&
|
| 2586 |
|
|
is_inexact_lt_midpoint)
|
| 2587 |
|
|
||
|
| 2588 |
|
|
((rnd_mode ==
|
| 2589 |
|
|
ROUNDING_TIES_AWAY
|
| 2590 |
|
|
|| rnd_mode ==
|
| 2591 |
|
|
ROUNDING_DOWN)
|
| 2592 |
|
|
&&
|
| 2593 |
|
|
is_midpoint_gt_even))))
|
| 2594 |
|
|
{
|
| 2595 |
|
|
// C1 = C1 + 1
|
| 2596 |
|
|
C1_lo = C1_lo + 1;
|
| 2597 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 2598 |
|
|
C1_hi = C1_hi + 1;
|
| 2599 |
|
|
}
|
| 2600 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 2601 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 2602 |
|
|
// C1 = 10^34 => rounding overflow
|
| 2603 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 2604 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 2605 |
|
|
y_exp = y_exp + EXP_P1;
|
| 2606 |
|
|
}
|
| 2607 |
|
|
} else if ((is_midpoint_lt_even || is_inexact_gt_midpoint)
|
| 2608 |
|
|
&&
|
| 2609 |
|
|
((x_sign
|
| 2610 |
|
|
&& (rnd_mode == ROUNDING_UP
|
| 2611 |
|
|
|| rnd_mode == ROUNDING_TO_ZERO))
|
| 2612 |
|
|
|| (!x_sign
|
| 2613 |
|
|
&& (rnd_mode == ROUNDING_DOWN
|
| 2614 |
|
|
|| rnd_mode == ROUNDING_TO_ZERO)))) {
|
| 2615 |
|
|
// C1 = C1 - 1
|
| 2616 |
|
|
C1_lo = C1_lo - 1;
|
| 2617 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 2618 |
|
|
C1_hi--;
|
| 2619 |
|
|
// check if we crossed into the lower decade
|
| 2620 |
|
|
if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
| 2621 |
|
|
C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
| 2622 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 2623 |
|
|
y_exp = y_exp - EXP_P1;
|
| 2624 |
|
|
// no underflow, because delta + q2 >= P34 + 1
|
| 2625 |
|
|
}
|
| 2626 |
|
|
} else {
|
| 2627 |
|
|
; // exact, the result is already correct
|
| 2628 |
|
|
}
|
| 2629 |
|
|
// in all cases check for overflow (RN and RA solved already)
|
| 2630 |
|
|
if (y_exp == EXP_MAX_P1) { // overflow
|
| 2631 |
|
|
if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
|
| 2632 |
|
|
(rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
|
| 2633 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 2634 |
|
|
C1_lo = 0x0ull;
|
| 2635 |
|
|
} else { // RM and res > 0, RP and res < 0, or RZ
|
| 2636 |
|
|
C1_hi = 0x5fffed09bead87c0ull;
|
| 2637 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 2638 |
|
|
}
|
| 2639 |
|
|
y_exp = 0; // x_sign is preserved
|
| 2640 |
|
|
// set the inexact flag (in case the exact addition was exact)
|
| 2641 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2642 |
|
|
// set the overflow flag
|
| 2643 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 2644 |
|
|
}
|
| 2645 |
|
|
}
|
| 2646 |
|
|
// assemble the result
|
| 2647 |
|
|
res.w[1] = x_sign | y_exp | C1_hi;
|
| 2648 |
|
|
res.w[0] = C1_lo;
|
| 2649 |
|
|
if (tmp_inexact)
|
| 2650 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2651 |
|
|
}
|
| 2652 |
|
|
} else { // if (-P34 + 1 <= delta <= -1) <=> 1 <= -delta <= P34 - 1
|
| 2653 |
|
|
// NOTE: the following, up to "} else { // if x_sign != y_sign
|
| 2654 |
|
|
// the result is exact" is identical to "else if (delta == P34 - q2) {"
|
| 2655 |
|
|
// from above; also, the code is not symmetric: a+b and b+a may take
|
| 2656 |
|
|
// different paths (need to unify eventually!)
|
| 2657 |
|
|
// calculate C' = C2 + C1 * 10^(e1-e2) directly; the result may be
|
| 2658 |
|
|
// inexact if it requires P34 + 1 decimal digits; in either case the
|
| 2659 |
|
|
// 'cutoff' point for addition is at the position of the lsb of C2
|
| 2660 |
|
|
// The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
|
| 2661 |
|
|
// exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
|
| 2662 |
|
|
// but their product fits with certainty in 128 bits (actually in 113)
|
| 2663 |
|
|
// Note that 0 <= e1 - e2 <= P34 - 2
|
| 2664 |
|
|
// -P34 + 1 <= delta <= -1 <=> -P34 + 1 <= delta <= -1 <=>
|
| 2665 |
|
|
// -P34 + 1 <= q1 + e1 - q2 - e2 <= -1 <=>
|
| 2666 |
|
|
// q2 - q1 - P34 + 1 <= e1 - e2 <= q2 - q1 - 1 <=>
|
| 2667 |
|
|
// 1 - P34 - P34 + 1 <= e1-e2 <= P34 - 1 - 1 => 0 <= e1-e2 <= P34 - 2
|
| 2668 |
|
|
scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
|
| 2669 |
|
|
if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
|
| 2670 |
|
|
__mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
|
| 2671 |
|
|
} else if (scale >= 1) {
|
| 2672 |
|
|
// if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
|
| 2673 |
|
|
if (q1 <= 19) { // C1 fits in 64 bits
|
| 2674 |
|
|
__mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
|
| 2675 |
|
|
} else { // q1 >= 20
|
| 2676 |
|
|
C1.w[1] = C1_hi;
|
| 2677 |
|
|
C1.w[0] = C1_lo;
|
| 2678 |
|
|
__mul_128x64_to_128 (C1, ten2k64[scale], C1);
|
| 2679 |
|
|
}
|
| 2680 |
|
|
} else { // if (scale == 0) C1 is unchanged
|
| 2681 |
|
|
C1.w[1] = C1_hi;
|
| 2682 |
|
|
C1.w[0] = C1_lo; // only the low part is necessary
|
| 2683 |
|
|
}
|
| 2684 |
|
|
C1_hi = C1.w[1];
|
| 2685 |
|
|
C1_lo = C1.w[0];
|
| 2686 |
|
|
// now add C2
|
| 2687 |
|
|
if (x_sign == y_sign) {
|
| 2688 |
|
|
// the result can overflow!
|
| 2689 |
|
|
C1_lo = C1_lo + C2_lo;
|
| 2690 |
|
|
C1_hi = C1_hi + C2_hi;
|
| 2691 |
|
|
if (C1_lo < C1.w[0])
|
| 2692 |
|
|
C1_hi++;
|
| 2693 |
|
|
// test for overflow, possible only when C1 >= 10^34
|
| 2694 |
|
|
if (C1_hi > 0x0001ed09bead87c0ull || (C1_hi == 0x0001ed09bead87c0ull && C1_lo >= 0x378d8e6400000000ull)) { // C1 >= 10^34
|
| 2695 |
|
|
// in this case q = P34 + 1 and x = q - P34 = 1, so multiply
|
| 2696 |
|
|
// C'' = C'+ 5 = C1 + 5 by k1 ~ 10^(-1) calculated for P34 + 1
|
| 2697 |
|
|
// decimal digits
|
| 2698 |
|
|
// Calculate C'' = C' + 1/2 * 10^x
|
| 2699 |
|
|
if (C1_lo >= 0xfffffffffffffffbull) { // low half add has carry
|
| 2700 |
|
|
C1_lo = C1_lo + 5;
|
| 2701 |
|
|
C1_hi = C1_hi + 1;
|
| 2702 |
|
|
} else {
|
| 2703 |
|
|
C1_lo = C1_lo + 5;
|
| 2704 |
|
|
}
|
| 2705 |
|
|
// the approximation of 10^(-1) was rounded up to 118 bits
|
| 2706 |
|
|
// 10^(-1) =~ 33333333333333333333333333333400 * 2^-129
|
| 2707 |
|
|
// 10^(-1) =~ 19999999999999999999999999999a00 * 2^-128
|
| 2708 |
|
|
C1.w[1] = C1_hi;
|
| 2709 |
|
|
C1.w[0] = C1_lo; // C''
|
| 2710 |
|
|
ten2m1.w[1] = 0x1999999999999999ull;
|
| 2711 |
|
|
ten2m1.w[0] = 0x9999999999999a00ull;
|
| 2712 |
|
|
__mul_128x128_to_256 (P256, C1, ten2m1); // P256 = C*, f*
|
| 2713 |
|
|
// C* is actually floor(C*) in this case
|
| 2714 |
|
|
// the top Ex = 128 bits of 10^(-1) are
|
| 2715 |
|
|
// T* = 0x00199999999999999999999999999999
|
| 2716 |
|
|
// if (0 < f* < 10^(-x)) then
|
| 2717 |
|
|
// if floor(C*) is even then C = floor(C*) - logical right
|
| 2718 |
|
|
// shift; C has p decimal digits, correct by Prop. 1)
|
| 2719 |
|
|
// else if floor(C*) is odd C = floor(C*) - 1 (logical right
|
| 2720 |
|
|
// shift; C has p decimal digits, correct by Pr. 1)
|
| 2721 |
|
|
// else
|
| 2722 |
|
|
// C = floor(C*) (logical right shift; C has p decimal digits,
|
| 2723 |
|
|
// correct by Property 1)
|
| 2724 |
|
|
// n = C * 10^(e2+x)
|
| 2725 |
|
|
if ((P256.w[1] || P256.w[0])
|
| 2726 |
|
|
&& (P256.w[1] < 0x1999999999999999ull
|
| 2727 |
|
|
|| (P256.w[1] == 0x1999999999999999ull
|
| 2728 |
|
|
&& P256.w[0] <= 0x9999999999999999ull))) {
|
| 2729 |
|
|
// the result is a midpoint
|
| 2730 |
|
|
if (P256.w[2] & 0x01) {
|
| 2731 |
|
|
is_midpoint_gt_even = 1;
|
| 2732 |
|
|
// if floor(C*) is odd C = floor(C*) - 1; the result is not 0
|
| 2733 |
|
|
P256.w[2]--;
|
| 2734 |
|
|
if (P256.w[2] == 0xffffffffffffffffull)
|
| 2735 |
|
|
P256.w[3]--;
|
| 2736 |
|
|
} else {
|
| 2737 |
|
|
is_midpoint_lt_even = 1;
|
| 2738 |
|
|
}
|
| 2739 |
|
|
}
|
| 2740 |
|
|
// n = Cstar * 10^(e2+1)
|
| 2741 |
|
|
y_exp = y_exp + EXP_P1;
|
| 2742 |
|
|
// C* != 10^P34 because C* has P34 digits
|
| 2743 |
|
|
// check for overflow
|
| 2744 |
|
|
if (y_exp == EXP_MAX_P1
|
| 2745 |
|
|
&& (rnd_mode == ROUNDING_TO_NEAREST
|
| 2746 |
|
|
|| rnd_mode == ROUNDING_TIES_AWAY)) {
|
| 2747 |
|
|
// overflow for RN
|
| 2748 |
|
|
res.w[1] = x_sign | 0x7800000000000000ull; // +/-inf
|
| 2749 |
|
|
res.w[0] = 0x0ull;
|
| 2750 |
|
|
// set the inexact flag
|
| 2751 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2752 |
|
|
// set the overflow flag
|
| 2753 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 2754 |
|
|
BID_SWAP128 (res);
|
| 2755 |
|
|
BID_RETURN (res);
|
| 2756 |
|
|
}
|
| 2757 |
|
|
// if (0 < f* - 1/2 < 10^(-x)) then
|
| 2758 |
|
|
// the result of the addition is exact
|
| 2759 |
|
|
// else
|
| 2760 |
|
|
// the result of the addition is inexact
|
| 2761 |
|
|
if (P256.w[1] > 0x8000000000000000ull || (P256.w[1] == 0x8000000000000000ull && P256.w[0] > 0x0ull)) { // the result may be exact
|
| 2762 |
|
|
tmp64 = P256.w[1] - 0x8000000000000000ull; // f* - 1/2
|
| 2763 |
|
|
if ((tmp64 > 0x1999999999999999ull
|
| 2764 |
|
|
|| (tmp64 == 0x1999999999999999ull
|
| 2765 |
|
|
&& P256.w[0] >= 0x9999999999999999ull))) {
|
| 2766 |
|
|
// set the inexact flag
|
| 2767 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2768 |
|
|
is_inexact = 1;
|
| 2769 |
|
|
} // else the result is exact
|
| 2770 |
|
|
} else { // the result is inexact
|
| 2771 |
|
|
// set the inexact flag
|
| 2772 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2773 |
|
|
is_inexact = 1;
|
| 2774 |
|
|
}
|
| 2775 |
|
|
C1_hi = P256.w[3];
|
| 2776 |
|
|
C1_lo = P256.w[2];
|
| 2777 |
|
|
if (!is_midpoint_gt_even && !is_midpoint_lt_even) {
|
| 2778 |
|
|
is_inexact_lt_midpoint = is_inexact
|
| 2779 |
|
|
&& (P256.w[1] & 0x8000000000000000ull);
|
| 2780 |
|
|
is_inexact_gt_midpoint = is_inexact
|
| 2781 |
|
|
&& !(P256.w[1] & 0x8000000000000000ull);
|
| 2782 |
|
|
}
|
| 2783 |
|
|
// general correction from RN to RA, RM, RP, RZ; result uses y_exp
|
| 2784 |
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
| 2785 |
|
|
if ((!x_sign
|
| 2786 |
|
|
&& ((rnd_mode == ROUNDING_UP
|
| 2787 |
|
|
&& is_inexact_lt_midpoint)
|
| 2788 |
|
|
|| ((rnd_mode == ROUNDING_TIES_AWAY
|
| 2789 |
|
|
|| rnd_mode == ROUNDING_UP)
|
| 2790 |
|
|
&& is_midpoint_gt_even))) || (x_sign
|
| 2791 |
|
|
&&
|
| 2792 |
|
|
((rnd_mode ==
|
| 2793 |
|
|
ROUNDING_DOWN
|
| 2794 |
|
|
&&
|
| 2795 |
|
|
is_inexact_lt_midpoint)
|
| 2796 |
|
|
||
|
| 2797 |
|
|
((rnd_mode ==
|
| 2798 |
|
|
ROUNDING_TIES_AWAY
|
| 2799 |
|
|
|| rnd_mode
|
| 2800 |
|
|
==
|
| 2801 |
|
|
ROUNDING_DOWN)
|
| 2802 |
|
|
&&
|
| 2803 |
|
|
is_midpoint_gt_even))))
|
| 2804 |
|
|
{
|
| 2805 |
|
|
// C1 = C1 + 1
|
| 2806 |
|
|
C1_lo = C1_lo + 1;
|
| 2807 |
|
|
if (C1_lo == 0) { // rounding overflow in the low 64 bits
|
| 2808 |
|
|
C1_hi = C1_hi + 1;
|
| 2809 |
|
|
}
|
| 2810 |
|
|
if (C1_hi == 0x0001ed09bead87c0ull
|
| 2811 |
|
|
&& C1_lo == 0x378d8e6400000000ull) {
|
| 2812 |
|
|
// C1 = 10^34 => rounding overflow
|
| 2813 |
|
|
C1_hi = 0x0000314dc6448d93ull;
|
| 2814 |
|
|
C1_lo = 0x38c15b0a00000000ull; // 10^33
|
| 2815 |
|
|
y_exp = y_exp + EXP_P1;
|
| 2816 |
|
|
}
|
| 2817 |
|
|
} else
|
| 2818 |
|
|
if ((is_midpoint_lt_even || is_inexact_gt_midpoint) &&
|
| 2819 |
|
|
((x_sign && (rnd_mode == ROUNDING_UP ||
|
| 2820 |
|
|
rnd_mode == ROUNDING_TO_ZERO)) ||
|
| 2821 |
|
|
(!x_sign && (rnd_mode == ROUNDING_DOWN ||
|
| 2822 |
|
|
rnd_mode == ROUNDING_TO_ZERO)))) {
|
| 2823 |
|
|
// C1 = C1 - 1
|
| 2824 |
|
|
C1_lo = C1_lo - 1;
|
| 2825 |
|
|
if (C1_lo == 0xffffffffffffffffull)
|
| 2826 |
|
|
C1_hi--;
|
| 2827 |
|
|
// check if we crossed into the lower decade
|
| 2828 |
|
|
if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
|
| 2829 |
|
|
C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
|
| 2830 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 2831 |
|
|
y_exp = y_exp - EXP_P1;
|
| 2832 |
|
|
// no underflow, because delta + q2 >= P34 + 1
|
| 2833 |
|
|
}
|
| 2834 |
|
|
} else {
|
| 2835 |
|
|
; // exact, the result is already correct
|
| 2836 |
|
|
}
|
| 2837 |
|
|
// in all cases check for overflow (RN and RA solved already)
|
| 2838 |
|
|
if (y_exp == EXP_MAX_P1) { // overflow
|
| 2839 |
|
|
if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
|
| 2840 |
|
|
(rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
|
| 2841 |
|
|
C1_hi = 0x7800000000000000ull; // +inf
|
| 2842 |
|
|
C1_lo = 0x0ull;
|
| 2843 |
|
|
} else { // RM and res > 0, RP and res < 0, or RZ
|
| 2844 |
|
|
C1_hi = 0x5fffed09bead87c0ull;
|
| 2845 |
|
|
C1_lo = 0x378d8e63ffffffffull;
|
| 2846 |
|
|
}
|
| 2847 |
|
|
y_exp = 0; // x_sign is preserved
|
| 2848 |
|
|
// set the inexact flag (in case the exact addition was exact)
|
| 2849 |
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
| 2850 |
|
|
// set the overflow flag
|
| 2851 |
|
|
*pfpsf |= OVERFLOW_EXCEPTION;
|
| 2852 |
|
|
}
|
| 2853 |
|
|
}
|
| 2854 |
|
|
} // else if (C1 < 10^34) then C1 is the coeff.; the result is exact
|
| 2855 |
|
|
// assemble the result
|
| 2856 |
|
|
res.w[1] = x_sign | y_exp | C1_hi;
|
| 2857 |
|
|
res.w[0] = C1_lo;
|
| 2858 |
|
|
} else { // if x_sign != y_sign the result is exact
|
| 2859 |
|
|
C1_lo = C2_lo - C1_lo;
|
| 2860 |
|
|
C1_hi = C2_hi - C1_hi;
|
| 2861 |
|
|
if (C1_lo > C2_lo)
|
| 2862 |
|
|
C1_hi--;
|
| 2863 |
|
|
if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
|
| 2864 |
|
|
C1_lo = ~C1_lo;
|
| 2865 |
|
|
C1_lo++;
|
| 2866 |
|
|
C1_hi = ~C1_hi;
|
| 2867 |
|
|
if (C1_lo == 0x0)
|
| 2868 |
|
|
C1_hi++;
|
| 2869 |
|
|
x_sign = y_sign; // the result will have the sign of y
|
| 2870 |
|
|
}
|
| 2871 |
|
|
// the result can be zero, but it cannot overflow
|
| 2872 |
|
|
if (C1_lo == 0 && C1_hi == 0) {
|
| 2873 |
|
|
// assemble the result
|
| 2874 |
|
|
if (x_exp < y_exp)
|
| 2875 |
|
|
res.w[1] = x_exp;
|
| 2876 |
|
|
else
|
| 2877 |
|
|
res.w[1] = y_exp;
|
| 2878 |
|
|
res.w[0] = 0;
|
| 2879 |
|
|
if (rnd_mode == ROUNDING_DOWN) {
|
| 2880 |
|
|
res.w[1] |= 0x8000000000000000ull;
|
| 2881 |
|
|
}
|
| 2882 |
|
|
BID_SWAP128 (res);
|
| 2883 |
|
|
BID_RETURN (res);
|
| 2884 |
|
|
}
|
| 2885 |
|
|
// assemble the result
|
| 2886 |
|
|
res.w[1] = y_sign | y_exp | C1_hi;
|
| 2887 |
|
|
res.w[0] = C1_lo;
|
| 2888 |
|
|
}
|
| 2889 |
|
|
}
|
| 2890 |
|
|
}
|
| 2891 |
|
|
BID_SWAP128 (res);
|
| 2892 |
|
|
BID_RETURN (res)
|
| 2893 |
|
|
}
|
| 2894 |
|
|
}
|
| 2895 |
|
|
|
| 2896 |
|
|
|
| 2897 |
|
|
|
| 2898 |
|
|
// bid128_sub stands for bid128qq_sub
|
| 2899 |
|
|
|
| 2900 |
|
|
/*****************************************************************************
|
| 2901 |
|
|
* BID128 sub
|
| 2902 |
|
|
****************************************************************************/
|
| 2903 |
|
|
|
| 2904 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 2905 |
|
|
void
|
| 2906 |
|
|
bid128_sub (UINT128 * pres, UINT128 * px, UINT128 * py
|
| 2907 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 2908 |
|
|
_EXC_INFO_PARAM) {
|
| 2909 |
|
|
UINT128 x = *px, y = *py;
|
| 2910 |
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
| 2911 |
|
|
unsigned int rnd_mode = *prnd_mode;
|
| 2912 |
|
|
#endif
|
| 2913 |
|
|
#else
|
| 2914 |
|
|
UINT128
|
| 2915 |
|
|
bid128_sub (UINT128 x, UINT128 y
|
| 2916 |
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
| 2917 |
|
|
_EXC_INFO_PARAM) {
|
| 2918 |
|
|
#endif
|
| 2919 |
|
|
|
| 2920 |
|
|
UINT128 res;
|
| 2921 |
|
|
UINT64 y_sign;
|
| 2922 |
|
|
|
| 2923 |
|
|
if ((y.w[HIGH_128W] & MASK_NAN) != MASK_NAN) { // y is not NAN
|
| 2924 |
|
|
// change its sign
|
| 2925 |
|
|
y_sign = y.w[HIGH_128W] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
| 2926 |
|
|
if (y_sign)
|
| 2927 |
|
|
y.w[HIGH_128W] = y.w[HIGH_128W] & 0x7fffffffffffffffull;
|
| 2928 |
|
|
else
|
| 2929 |
|
|
y.w[HIGH_128W] = y.w[HIGH_128W] | 0x8000000000000000ull;
|
| 2930 |
|
|
}
|
| 2931 |
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
| 2932 |
|
|
bid128_add (&res, &x, &y
|
| 2933 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 2934 |
|
|
_EXC_INFO_ARG);
|
| 2935 |
|
|
#else
|
| 2936 |
|
|
res = bid128_add (x, y
|
| 2937 |
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
| 2938 |
|
|
_EXC_INFO_ARG);
|
| 2939 |
|
|
#endif
|
| 2940 |
|
|
BID_RETURN (res);
|
| 2941 |
|
|
}
|
