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/* Copyright (C) 2007-2015 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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. */ |
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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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_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_sub (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_sub (&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_sub (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_sub (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_sub (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_sub (&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_sub (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_sub (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_sub (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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UINT64 y_sign; |
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BID_SWAP128 (one); |
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if ((y.w[HIGH_128W] & MASK_NAN) != MASK_NAN) { // y is not NAN |
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// change its sign |
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y_sign = y.w[HIGH_128W] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative |
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if (y_sign) |
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y.w[HIGH_128W] = y.w[HIGH_128W] & 0x7fffffffffffffffull; |
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else |
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y.w[HIGH_128W] = y.w[HIGH_128W] | 0x8000000000000000ull; |
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} |
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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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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 |
} |