818ab71a41
From-SVN: r232055
4459 lines
171 KiB
C
4459 lines
171 KiB
C
/* Copyright (C) 2007-2016 Free Software Foundation, Inc.
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This file is part of GCC.
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GCC is free software; you can redistribute it and/or modify it under
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the terms of the GNU General Public License as published by the Free
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Software Foundation; either version 3, or (at your option) any later
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version.
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GCC is distributed in the hope that it will be useful, but WITHOUT ANY
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WARRANTY; without even the implied warranty of MERCHANTABILITY or
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FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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for more details.
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Under Section 7 of GPL version 3, you are granted additional
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permissions described in the GCC Runtime Library Exception, version
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3.1, as published by the Free Software Foundation.
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You should have received a copy of the GNU General Public License and
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a copy of the GCC Runtime Library Exception along with this program;
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see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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<http://www.gnu.org/licenses/>. */
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/*****************************************************************************
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*
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* BID128 fma x * y + z
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*
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****************************************************************************/
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#include "bid_internal.h"
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static void
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rounding_correction (unsigned int rnd_mode,
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unsigned int is_inexact_lt_midpoint,
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unsigned int is_inexact_gt_midpoint,
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unsigned int is_midpoint_lt_even,
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unsigned int is_midpoint_gt_even,
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int unbexp,
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UINT128 * ptrres, _IDEC_flags * ptrfpsf) {
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// unbiased true exponent unbexp may be larger than emax
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UINT128 res = *ptrres; // expected to have the correct sign and coefficient
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// (the exponent field is ignored, as unbexp is used instead)
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UINT64 sign, exp;
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UINT64 C_hi, C_lo;
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// general correction from RN to RA, RM, RP, RZ
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// Note: if the result is negative, then is_inexact_lt_midpoint,
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// is_inexact_gt_midpoint, is_midpoint_lt_even, and is_midpoint_gt_even
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// have to be considered as if determined for the absolute value of the
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// result (so they seem to be reversed)
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if (is_inexact_lt_midpoint || is_inexact_gt_midpoint ||
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is_midpoint_lt_even || is_midpoint_gt_even) {
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*ptrfpsf |= INEXACT_EXCEPTION;
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}
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// apply correction to result calculated with unbounded exponent
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sign = res.w[1] & MASK_SIGN;
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exp = (UINT64) (unbexp + 6176) << 49; // valid only if expmin<=unbexp<=expmax
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C_hi = res.w[1] & MASK_COEFF;
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C_lo = res.w[0];
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if ((!sign && ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint) ||
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((rnd_mode == ROUNDING_TIES_AWAY || rnd_mode == ROUNDING_UP) &&
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is_midpoint_gt_even))) ||
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(sign && ((rnd_mode == ROUNDING_DOWN && is_inexact_lt_midpoint) ||
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((rnd_mode == ROUNDING_TIES_AWAY || rnd_mode == ROUNDING_DOWN) &&
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is_midpoint_gt_even)))) {
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// C = C + 1
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C_lo = C_lo + 1;
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if (C_lo == 0)
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C_hi = C_hi + 1;
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if (C_hi == 0x0001ed09bead87c0ull && C_lo == 0x378d8e6400000000ull) {
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// C = 10^34 => rounding overflow
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C_hi = 0x0000314dc6448d93ull;
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C_lo = 0x38c15b0a00000000ull; // 10^33
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// exp = exp + EXP_P1;
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unbexp = unbexp + 1;
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exp = (UINT64) (unbexp + 6176) << 49;
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}
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} else if ((is_midpoint_lt_even || is_inexact_gt_midpoint) &&
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((sign && (rnd_mode == ROUNDING_UP || rnd_mode == ROUNDING_TO_ZERO)) ||
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(!sign && (rnd_mode == ROUNDING_DOWN || rnd_mode == ROUNDING_TO_ZERO)))) {
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// C = C - 1
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C_lo = C_lo - 1;
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if (C_lo == 0xffffffffffffffffull)
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C_hi--;
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// check if we crossed into the lower decade
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if (C_hi == 0x0000314dc6448d93ull && C_lo == 0x38c15b09ffffffffull) {
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// C = 10^33 - 1
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if (exp > 0) {
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C_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
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C_lo = 0x378d8e63ffffffffull;
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// exp = exp - EXP_P1;
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unbexp = unbexp - 1;
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exp = (UINT64) (unbexp + 6176) << 49;
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} else { // if exp = 0 the result is tiny & inexact
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*ptrfpsf |= UNDERFLOW_EXCEPTION;
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}
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}
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} else {
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; // the result is already correct
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}
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if (unbexp > expmax) { // 6111
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*ptrfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
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exp = 0;
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if (!sign) { // result is positive
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if (rnd_mode == ROUNDING_UP || rnd_mode == ROUNDING_TIES_AWAY) { // +inf
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C_hi = 0x7800000000000000ull;
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C_lo = 0x0000000000000000ull;
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} else { // res = +MAXFP = (10^34-1) * 10^emax
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C_hi = 0x5fffed09bead87c0ull;
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C_lo = 0x378d8e63ffffffffull;
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}
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} else { // result is negative
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if (rnd_mode == ROUNDING_DOWN || rnd_mode == ROUNDING_TIES_AWAY) { // -inf
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C_hi = 0xf800000000000000ull;
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C_lo = 0x0000000000000000ull;
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} else { // res = -MAXFP = -(10^34-1) * 10^emax
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C_hi = 0xdfffed09bead87c0ull;
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C_lo = 0x378d8e63ffffffffull;
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}
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}
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}
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// assemble the result
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res.w[1] = sign | exp | C_hi;
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res.w[0] = C_lo;
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*ptrres = res;
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}
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static void
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add256 (UINT256 x, UINT256 y, UINT256 * pz) {
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// *z = x + yl assume the sum fits in 256 bits
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UINT256 z;
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z.w[0] = x.w[0] + y.w[0];
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if (z.w[0] < x.w[0]) {
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x.w[1]++;
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if (x.w[1] == 0x0000000000000000ull) {
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x.w[2]++;
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if (x.w[2] == 0x0000000000000000ull) {
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x.w[3]++;
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}
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}
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}
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z.w[1] = x.w[1] + y.w[1];
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if (z.w[1] < x.w[1]) {
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x.w[2]++;
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if (x.w[2] == 0x0000000000000000ull) {
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x.w[3]++;
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}
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}
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z.w[2] = x.w[2] + y.w[2];
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if (z.w[2] < x.w[2]) {
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x.w[3]++;
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}
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z.w[3] = x.w[3] + y.w[3]; // it was assumed that no carry is possible
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*pz = z;
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}
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static void
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sub256 (UINT256 x, UINT256 y, UINT256 * pz) {
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// *z = x - y; assume x >= y
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UINT256 z;
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z.w[0] = x.w[0] - y.w[0];
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if (z.w[0] > x.w[0]) {
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x.w[1]--;
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if (x.w[1] == 0xffffffffffffffffull) {
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x.w[2]--;
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if (x.w[2] == 0xffffffffffffffffull) {
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x.w[3]--;
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}
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}
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}
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z.w[1] = x.w[1] - y.w[1];
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if (z.w[1] > x.w[1]) {
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x.w[2]--;
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if (x.w[2] == 0xffffffffffffffffull) {
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x.w[3]--;
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}
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}
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z.w[2] = x.w[2] - y.w[2];
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if (z.w[2] > x.w[2]) {
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x.w[3]--;
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}
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z.w[3] = x.w[3] - y.w[3]; // no borrow possible, because x >= y
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*pz = z;
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}
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static int
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nr_digits256 (UINT256 R256) {
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int ind;
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// determine the number of decimal digits in R256
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if (R256.w[3] == 0x0 && R256.w[2] == 0x0 && R256.w[1] == 0x0) {
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// between 1 and 19 digits
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for (ind = 1; ind <= 19; ind++) {
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if (R256.w[0] < ten2k64[ind]) {
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break;
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}
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}
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// ind digits
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} else if (R256.w[3] == 0x0 && R256.w[2] == 0x0 &&
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(R256.w[1] < ten2k128[0].w[1] ||
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(R256.w[1] == ten2k128[0].w[1]
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&& R256.w[0] < ten2k128[0].w[0]))) {
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// 20 digits
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ind = 20;
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} else if (R256.w[3] == 0x0 && R256.w[2] == 0x0) {
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// between 21 and 38 digits
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for (ind = 1; ind <= 18; ind++) {
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if (R256.w[1] < ten2k128[ind].w[1] ||
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(R256.w[1] == ten2k128[ind].w[1] &&
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R256.w[0] < ten2k128[ind].w[0])) {
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break;
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}
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}
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// ind + 20 digits
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ind = ind + 20;
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} else if (R256.w[3] == 0x0 &&
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(R256.w[2] < ten2k256[0].w[2] ||
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(R256.w[2] == ten2k256[0].w[2] &&
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R256.w[1] < ten2k256[0].w[1]) ||
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(R256.w[2] == ten2k256[0].w[2] &&
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R256.w[1] == ten2k256[0].w[1] &&
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R256.w[0] < ten2k256[0].w[0]))) {
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// 39 digits
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ind = 39;
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} else {
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// between 40 and 68 digits
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for (ind = 1; ind <= 29; ind++) {
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if (R256.w[3] < ten2k256[ind].w[3] ||
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(R256.w[3] == ten2k256[ind].w[3] &&
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R256.w[2] < ten2k256[ind].w[2]) ||
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(R256.w[3] == ten2k256[ind].w[3] &&
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R256.w[2] == ten2k256[ind].w[2] &&
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R256.w[1] < ten2k256[ind].w[1]) ||
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(R256.w[3] == ten2k256[ind].w[3] &&
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R256.w[2] == ten2k256[ind].w[2] &&
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R256.w[1] == ten2k256[ind].w[1] &&
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R256.w[0] < ten2k256[ind].w[0])) {
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break;
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}
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}
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// ind + 39 digits
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ind = ind + 39;
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}
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return (ind);
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}
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// add/subtract C4 and C3 * 10^scale; this may follow a previous rounding, so
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// use the rounding information from ptr_is_* to avoid a double rounding error
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static void
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add_and_round (int q3,
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int q4,
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int e4,
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int delta,
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int p34,
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UINT64 z_sign,
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UINT64 p_sign,
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UINT128 C3,
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UINT256 C4,
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int rnd_mode,
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int *ptr_is_midpoint_lt_even,
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int *ptr_is_midpoint_gt_even,
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int *ptr_is_inexact_lt_midpoint,
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int *ptr_is_inexact_gt_midpoint,
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_IDEC_flags * ptrfpsf, UINT128 * ptrres) {
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int scale;
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int x0;
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int ind;
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UINT64 R64;
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UINT128 P128, R128;
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UINT192 P192, R192;
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UINT256 R256;
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int is_midpoint_lt_even = 0;
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int is_midpoint_gt_even = 0;
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int is_inexact_lt_midpoint = 0;
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int is_inexact_gt_midpoint = 0;
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int is_midpoint_lt_even0 = 0;
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int is_midpoint_gt_even0 = 0;
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int is_inexact_lt_midpoint0 = 0;
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int is_inexact_gt_midpoint0 = 0;
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int incr_exp = 0;
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int is_tiny = 0;
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int lt_half_ulp = 0;
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int eq_half_ulp = 0;
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// int gt_half_ulp = 0;
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UINT128 res = *ptrres;
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// scale C3 up by 10^(q4-delta-q3), 0 <= q4-delta-q3 <= 2*P34-2 = 66
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scale = q4 - delta - q3; // 0 <= scale <= 66 (or 0 <= scale <= 68 if this
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// comes from Cases (2), (3), (4), (5), (6), with 0 <= |delta| <= 1
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// calculate C3 * 10^scale in R256 (it has at most 67 decimal digits for
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// Cases (15),(16),(17) and at most 69 for Cases (2),(3),(4),(5),(6))
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if (scale == 0) {
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R256.w[3] = 0x0ull;
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R256.w[2] = 0x0ull;
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R256.w[1] = C3.w[1];
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R256.w[0] = C3.w[0];
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} else if (scale <= 19) { // 10^scale fits in 64 bits
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P128.w[1] = 0;
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P128.w[0] = ten2k64[scale];
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__mul_128x128_to_256 (R256, P128, C3);
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} else if (scale <= 38) { // 10^scale fits in 128 bits
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__mul_128x128_to_256 (R256, ten2k128[scale - 20], C3);
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} else if (scale <= 57) { // 39 <= scale <= 57
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// 10^scale fits in 192 bits but C3 * 10^scale fits in 223 or 230 bits
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// (10^67 has 223 bits; 10^69 has 230 bits);
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// must split the computation:
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// 10^scale * C3 = 10*38 * 10^(scale-38) * C3 where 10^38 takes 127
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// bits and so 10^(scale-38) * C3 fits in 128 bits with certainty
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// Note that 1 <= scale - 38 <= 19 => 10^(scale-38) fits in 64 bits
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__mul_64x128_to_128 (R128, ten2k64[scale - 38], C3);
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// now multiply R128 by 10^38
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__mul_128x128_to_256 (R256, R128, ten2k128[18]);
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} else { // 58 <= scale <= 66
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// 10^scale takes between 193 and 220 bits,
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// and C3 * 10^scale fits in 223 bits (10^67/10^69 has 223/230 bits)
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// must split the computation:
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// 10^scale * C3 = 10*38 * 10^(scale-38) * C3 where 10^38 takes 127
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// bits and so 10^(scale-38) * C3 fits in 128 bits with certainty
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// Note that 20 <= scale - 38 <= 30 => 10^(scale-38) fits in 128 bits
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// Calculate first 10^(scale-38) * C3, which fits in 128 bits; because
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// 10^(scale-38) takes more than 64 bits, C3 will take less than 64
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__mul_64x128_to_128 (R128, C3.w[0], ten2k128[scale - 58]);
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// now calculate 10*38 * 10^(scale-38) * C3
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__mul_128x128_to_256 (R256, R128, ten2k128[18]);
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}
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// C3 * 10^scale is now in R256
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// for Cases (15), (16), (17) C4 > C3 * 10^scale because C4 has at least
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// one extra digit; for Cases (2), (3), (4), (5), or (6) any order is
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// possible
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// add/subtract C4 and C3 * 10^scale; the exponent is e4
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if (p_sign == z_sign) { // R256 = C4 + R256
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// calculate R256 = C4 + C3 * 10^scale = C4 + R256 which is exact,
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// but may require rounding
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add256 (C4, R256, &R256);
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} else { // if (p_sign != z_sign) { // R256 = C4 - R256
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// calculate R256 = C4 - C3 * 10^scale = C4 - R256 or
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// R256 = C3 * 10^scale - C4 = R256 - C4 which is exact,
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// but may require rounding
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// compare first R256 = C3 * 10^scale and C4
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if (R256.w[3] > C4.w[3] || (R256.w[3] == C4.w[3] && R256.w[2] > C4.w[2]) ||
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(R256.w[3] == C4.w[3] && R256.w[2] == C4.w[2] && R256.w[1] > C4.w[1]) ||
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(R256.w[3] == C4.w[3] && R256.w[2] == C4.w[2] && R256.w[1] == C4.w[1] &&
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R256.w[0] >= C4.w[0])) { // C3 * 10^scale >= C4
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// calculate R256 = C3 * 10^scale - C4 = R256 - C4, which is exact,
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// but may require rounding
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sub256 (R256, C4, &R256);
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// flip p_sign too, because the result has the sign of z
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p_sign = z_sign;
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} else { // if C4 > C3 * 10^scale
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// calculate R256 = C4 - C3 * 10^scale = C4 - R256, which is exact,
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// but may require rounding
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sub256 (C4, R256, &R256);
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}
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// if the result is pure zero, the sign depends on the rounding mode
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// (x*y and z had opposite signs)
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if (R256.w[3] == 0x0ull && R256.w[2] == 0x0ull &&
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R256.w[1] == 0x0ull && R256.w[0] == 0x0ull) {
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if (rnd_mode != ROUNDING_DOWN)
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p_sign = 0x0000000000000000ull;
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else
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p_sign = 0x8000000000000000ull;
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// the exponent is max (e4, expmin)
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if (e4 < -6176)
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e4 = expmin;
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// assemble result
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res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49);
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res.w[0] = 0x0;
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*ptrres = res;
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return;
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}
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}
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// determine the number of decimal digits in R256
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ind = nr_digits256 (R256);
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// the exact result is (-1)^p_sign * R256 * 10^e4 where q (R256) = ind;
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// round to the destination precision, with unbounded exponent
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if (ind <= p34) {
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// result rounded to the destination precision with unbounded exponent
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// is exact
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if (ind + e4 < p34 + expmin) {
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is_tiny = 1; // applies to all rounding modes
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}
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res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | R256.w[1];
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res.w[0] = R256.w[0];
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// Note: res is correct only if expmin <= e4 <= expmax
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} else { // if (ind > p34)
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// if more than P digits, round to nearest to P digits
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// round R256 to p34 digits
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x0 = ind - p34; // 1 <= x0 <= 34 as 35 <= ind <= 68
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if (ind <= 38) {
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P128.w[1] = R256.w[1];
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P128.w[0] = R256.w[0];
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round128_19_38 (ind, x0, P128, &R128, &incr_exp,
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&is_midpoint_lt_even, &is_midpoint_gt_even,
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&is_inexact_lt_midpoint, &is_inexact_gt_midpoint);
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} else if (ind <= 57) {
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P192.w[2] = R256.w[2];
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P192.w[1] = R256.w[1];
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P192.w[0] = R256.w[0];
|
|
round192_39_57 (ind, x0, P192, &R192, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint);
|
|
R128.w[1] = R192.w[1];
|
|
R128.w[0] = R192.w[0];
|
|
} else { // if (ind <= 68)
|
|
round256_58_76 (ind, x0, R256, &R256, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint);
|
|
R128.w[1] = R256.w[1];
|
|
R128.w[0] = R256.w[0];
|
|
}
|
|
// the rounded result has p34 = 34 digits
|
|
e4 = e4 + x0 + incr_exp;
|
|
if (rnd_mode == ROUNDING_TO_NEAREST) {
|
|
if (e4 < expmin) {
|
|
is_tiny = 1; // for other rounding modes apply correction
|
|
}
|
|
} else {
|
|
// for RM, RP, RZ, RA apply correction in order to determine tininess
|
|
// but do not save the result; apply the correction to
|
|
// (-1)^p_sign * significand * 10^0
|
|
P128.w[1] = p_sign | 0x3040000000000000ull | R128.w[1];
|
|
P128.w[0] = R128.w[0];
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint, is_midpoint_lt_even,
|
|
is_midpoint_gt_even, 0, &P128, ptrfpsf);
|
|
scale = ((P128.w[1] & MASK_EXP) >> 49) - 6176; // -1, 0, or +1
|
|
// the number of digits in the significand is p34 = 34
|
|
if (e4 + scale < expmin) {
|
|
is_tiny = 1;
|
|
}
|
|
}
|
|
ind = p34; // the number of decimal digits in the signifcand of res
|
|
res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | R128.w[1]; // RN
|
|
res.w[0] = R128.w[0];
|
|
// Note: res is correct only if expmin <= e4 <= expmax
|
|
// set the inexact flag after rounding with bounded exponent, if any
|
|
}
|
|
// at this point we have the result rounded with unbounded exponent in
|
|
// res and we know its tininess:
|
|
// res = (-1)^p_sign * significand * 10^e4,
|
|
// where q (significand) = ind <= p34
|
|
// Note: res is correct only if expmin <= e4 <= expmax
|
|
|
|
// check for overflow if RN
|
|
if (rnd_mode == ROUNDING_TO_NEAREST && (ind + e4) > (p34 + expmax)) {
|
|
res.w[1] = p_sign | 0x7800000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
*ptrres = res;
|
|
*ptrfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
|
|
return; // BID_RETURN (res)
|
|
} // else not overflow or not RN, so continue
|
|
|
|
// if (e4 >= expmin) we have the result rounded with bounded exponent
|
|
if (e4 < expmin) {
|
|
x0 = expmin - e4; // x0 >= 1; the number of digits to chop off of res
|
|
// where the result rounded [at most] once is
|
|
// (-1)^p_sign * significand_res * 10^e4
|
|
|
|
// avoid double rounding error
|
|
is_inexact_lt_midpoint0 = is_inexact_lt_midpoint;
|
|
is_inexact_gt_midpoint0 = is_inexact_gt_midpoint;
|
|
is_midpoint_lt_even0 = is_midpoint_lt_even;
|
|
is_midpoint_gt_even0 = is_midpoint_gt_even;
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
|
|
if (x0 > ind) {
|
|
// nothing is left of res when moving the decimal point left x0 digits
|
|
is_inexact_lt_midpoint = 1;
|
|
res.w[1] = p_sign | 0x0000000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
e4 = expmin;
|
|
} else if (x0 == ind) { // 1 <= x0 = ind <= p34 = 34
|
|
// this is <, =, or > 1/2 ulp
|
|
// compare the ind-digit value in the significand of res with
|
|
// 1/2 ulp = 5*10^(ind-1), i.e. determine whether it is
|
|
// less than, equal to, or greater than 1/2 ulp (significand of res)
|
|
R128.w[1] = res.w[1] & MASK_COEFF;
|
|
R128.w[0] = res.w[0];
|
|
if (ind <= 19) {
|
|
if (R128.w[0] < midpoint64[ind - 1]) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (R128.w[0] == midpoint64[ind - 1]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
is_midpoint_gt_even = 1;
|
|
} else { // > 1/2 ulp
|
|
// gt_half_ulp = 1;
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
} else { // if (ind <= 38) {
|
|
if (R128.w[1] < midpoint128[ind - 20].w[1] ||
|
|
(R128.w[1] == midpoint128[ind - 20].w[1] &&
|
|
R128.w[0] < midpoint128[ind - 20].w[0])) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (R128.w[1] == midpoint128[ind - 20].w[1] &&
|
|
R128.w[0] == midpoint128[ind - 20].w[0]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
is_midpoint_gt_even = 1;
|
|
} else { // > 1/2 ulp
|
|
// gt_half_ulp = 1;
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
}
|
|
if (lt_half_ulp || eq_half_ulp) {
|
|
// res = +0.0 * 10^expmin
|
|
res.w[1] = 0x0000000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
} else { // if (gt_half_ulp)
|
|
// res = +1 * 10^expmin
|
|
res.w[1] = 0x0000000000000000ull;
|
|
res.w[0] = 0x0000000000000001ull;
|
|
}
|
|
res.w[1] = p_sign | res.w[1];
|
|
e4 = expmin;
|
|
} else { // if (1 <= x0 <= ind - 1 <= 33)
|
|
// round the ind-digit result to ind - x0 digits
|
|
|
|
if (ind <= 18) { // 2 <= ind <= 18
|
|
round64_2_18 (ind, x0, res.w[0], &R64, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint);
|
|
res.w[1] = 0x0;
|
|
res.w[0] = R64;
|
|
} else if (ind <= 38) {
|
|
P128.w[1] = res.w[1] & MASK_COEFF;
|
|
P128.w[0] = res.w[0];
|
|
round128_19_38 (ind, x0, P128, &res, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
}
|
|
e4 = e4 + x0; // expmin
|
|
// we want the exponent to be expmin, so if incr_exp = 1 then
|
|
// multiply the rounded result by 10 - it will still fit in 113 bits
|
|
if (incr_exp) {
|
|
// 64 x 128 -> 128
|
|
P128.w[1] = res.w[1] & MASK_COEFF;
|
|
P128.w[0] = res.w[0];
|
|
__mul_64x128_to_128 (res, ten2k64[1], P128);
|
|
}
|
|
res.w[1] =
|
|
p_sign | ((UINT64) (e4 + 6176) << 49) | (res.w[1] & MASK_COEFF);
|
|
// avoid a double rounding error
|
|
if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) &&
|
|
is_midpoint_lt_even) { // double rounding error upward
|
|
// res = res - 1
|
|
res.w[0]--;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
// Note: a double rounding error upward is not possible; for this
|
|
// the result after the first rounding would have to be 99...95
|
|
// (35 digits in all), possibly followed by a number of zeros; this
|
|
// is not possible in Cases (2)-(6) or (15)-(17) which may get here
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) &&
|
|
is_midpoint_gt_even) { // double rounding error downward
|
|
// res = res + 1
|
|
res.w[0]++;
|
|
if (res.w[0] == 0)
|
|
res.w[1]++;
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
// if this second rounding was exact the result may still be
|
|
// inexact because of the first rounding
|
|
if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) {
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else if (is_midpoint_gt_even &&
|
|
(is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) {
|
|
// pulled up to a midpoint
|
|
is_inexact_lt_midpoint = 1;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else if (is_midpoint_lt_even &&
|
|
(is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) {
|
|
// pulled down to a midpoint
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else {
|
|
;
|
|
}
|
|
}
|
|
}
|
|
// res contains the correct result
|
|
// apply correction if not rounding to nearest
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint, is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e4, &res, ptrfpsf);
|
|
}
|
|
if (is_midpoint_lt_even || is_midpoint_gt_even ||
|
|
is_inexact_lt_midpoint || is_inexact_gt_midpoint) {
|
|
// set the inexact flag
|
|
*ptrfpsf |= INEXACT_EXCEPTION;
|
|
if (is_tiny)
|
|
*ptrfpsf |= UNDERFLOW_EXCEPTION;
|
|
}
|
|
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
*ptrres = res;
|
|
return;
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
static void
|
|
bid128_ext_fma (int *ptr_is_midpoint_lt_even,
|
|
int *ptr_is_midpoint_gt_even,
|
|
int *ptr_is_inexact_lt_midpoint,
|
|
int *ptr_is_inexact_gt_midpoint, UINT128 * pres,
|
|
UINT128 * px, UINT128 * py,
|
|
UINT128 *
|
|
pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT128 x = *px, y = *py, z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
static UINT128
|
|
bid128_ext_fma (int *ptr_is_midpoint_lt_even,
|
|
int *ptr_is_midpoint_gt_even,
|
|
int *ptr_is_inexact_lt_midpoint,
|
|
int *ptr_is_inexact_gt_midpoint, UINT128 x, UINT128 y,
|
|
UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM
|
|
_EXC_MASKS_PARAM _EXC_INFO_PARAM) {
|
|
#endif
|
|
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT64 x_sign, y_sign, z_sign, p_sign, tmp_sign;
|
|
UINT64 x_exp = 0, y_exp = 0, z_exp = 0, p_exp;
|
|
int true_p_exp;
|
|
UINT128 C1, C2, C3;
|
|
UINT256 C4;
|
|
int q1 = 0, q2 = 0, q3 = 0, q4;
|
|
int e1, e2, e3, e4;
|
|
int scale, ind, delta, x0;
|
|
int p34 = P34; // used to modify the limit on the number of digits
|
|
BID_UI64DOUBLE tmp;
|
|
int x_nr_bits, y_nr_bits, z_nr_bits;
|
|
unsigned int save_fpsf;
|
|
int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0;
|
|
int is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
int is_midpoint_lt_even0 = 0, is_midpoint_gt_even0 = 0;
|
|
int is_inexact_lt_midpoint0 = 0, is_inexact_gt_midpoint0 = 0;
|
|
int incr_exp = 0;
|
|
int lsb;
|
|
int lt_half_ulp = 0;
|
|
int eq_half_ulp = 0;
|
|
int gt_half_ulp = 0;
|
|
int is_tiny = 0;
|
|
UINT64 R64, tmp64;
|
|
UINT128 P128, R128;
|
|
UINT192 P192, R192;
|
|
UINT256 R256;
|
|
|
|
// the following are based on the table of special cases for fma; the NaN
|
|
// behavior is similar to that of the IA-64 Architecture fma
|
|
|
|
// identify cases where at least one operand is NaN
|
|
|
|
BID_SWAP128 (x);
|
|
BID_SWAP128 (y);
|
|
BID_SWAP128 (z);
|
|
if ((y.w[1] & MASK_NAN) == MASK_NAN) { // y is NAN
|
|
// if x = {0, f, inf, NaN}, y = NaN, z = {0, f, inf, NaN} then res = Q (y)
|
|
// check first for non-canonical NaN payload
|
|
if (((y.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) ||
|
|
(((y.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) &&
|
|
(y.w[0] > 0x38c15b09ffffffffull))) {
|
|
y.w[1] = y.w[1] & 0xffffc00000000000ull;
|
|
y.w[0] = 0x0ull;
|
|
}
|
|
if ((y.w[1] & MASK_SNAN) == MASK_SNAN) { // y is SNAN
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
// return quiet (y)
|
|
res.w[1] = y.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16]
|
|
res.w[0] = y.w[0];
|
|
} else { // y is QNaN
|
|
// return y
|
|
res.w[1] = y.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16]
|
|
res.w[0] = y.w[0];
|
|
// if z = SNaN or x = SNaN signal invalid exception
|
|
if ((z.w[1] & MASK_SNAN) == MASK_SNAN ||
|
|
(x.w[1] & MASK_SNAN) == MASK_SNAN) {
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
}
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
} else if ((z.w[1] & MASK_NAN) == MASK_NAN) { // z is NAN
|
|
// if x = {0, f, inf, NaN}, y = {0, f, inf}, z = NaN then res = Q (z)
|
|
// check first for non-canonical NaN payload
|
|
if (((z.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) ||
|
|
(((z.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) &&
|
|
(z.w[0] > 0x38c15b09ffffffffull))) {
|
|
z.w[1] = z.w[1] & 0xffffc00000000000ull;
|
|
z.w[0] = 0x0ull;
|
|
}
|
|
if ((z.w[1] & MASK_SNAN) == MASK_SNAN) { // z is SNAN
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
// return quiet (z)
|
|
res.w[1] = z.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16]
|
|
res.w[0] = z.w[0];
|
|
} else { // z is QNaN
|
|
// return z
|
|
res.w[1] = z.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16]
|
|
res.w[0] = z.w[0];
|
|
// if x = SNaN signal invalid exception
|
|
if ((x.w[1] & MASK_SNAN) == MASK_SNAN) {
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
}
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
} else if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
|
|
// if x = NaN, y = {0, f, inf}, z = {0, f, inf} then res = Q (x)
|
|
// check first for non-canonical NaN payload
|
|
if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) ||
|
|
(((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) &&
|
|
(x.w[0] > 0x38c15b09ffffffffull))) {
|
|
x.w[1] = x.w[1] & 0xffffc00000000000ull;
|
|
x.w[0] = 0x0ull;
|
|
}
|
|
if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
// return quiet (x)
|
|
res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16]
|
|
res.w[0] = x.w[0];
|
|
} else { // x is QNaN
|
|
// return x
|
|
res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16]
|
|
res.w[0] = x.w[0];
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
// x, y, z are 0, f, or inf but not NaN => unpack the arguments and check
|
|
// for non-canonical values
|
|
|
|
x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
|
C1.w[1] = x.w[1] & MASK_COEFF;
|
|
C1.w[0] = x.w[0];
|
|
if ((x.w[1] & MASK_ANY_INF) != MASK_INF) { // x != inf
|
|
// if x is not infinity check for non-canonical values - treated as zero
|
|
if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11
|
|
// non-canonical
|
|
x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits
|
|
C1.w[1] = 0; // significand high
|
|
C1.w[0] = 0; // significand low
|
|
} else { // G0_G1 != 11
|
|
x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits
|
|
if (C1.w[1] > 0x0001ed09bead87c0ull ||
|
|
(C1.w[1] == 0x0001ed09bead87c0ull &&
|
|
C1.w[0] > 0x378d8e63ffffffffull)) {
|
|
// x is non-canonical if coefficient is larger than 10^34 -1
|
|
C1.w[1] = 0;
|
|
C1.w[0] = 0;
|
|
} else { // canonical
|
|
;
|
|
}
|
|
}
|
|
}
|
|
y_sign = y.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
|
C2.w[1] = y.w[1] & MASK_COEFF;
|
|
C2.w[0] = y.w[0];
|
|
if ((y.w[1] & MASK_ANY_INF) != MASK_INF) { // y != inf
|
|
// if y is not infinity check for non-canonical values - treated as zero
|
|
if ((y.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11
|
|
// non-canonical
|
|
y_exp = (y.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits
|
|
C2.w[1] = 0; // significand high
|
|
C2.w[0] = 0; // significand low
|
|
} else { // G0_G1 != 11
|
|
y_exp = y.w[1] & MASK_EXP; // biased and shifted left 49 bits
|
|
if (C2.w[1] > 0x0001ed09bead87c0ull ||
|
|
(C2.w[1] == 0x0001ed09bead87c0ull &&
|
|
C2.w[0] > 0x378d8e63ffffffffull)) {
|
|
// y is non-canonical if coefficient is larger than 10^34 -1
|
|
C2.w[1] = 0;
|
|
C2.w[0] = 0;
|
|
} else { // canonical
|
|
;
|
|
}
|
|
}
|
|
}
|
|
z_sign = z.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
|
C3.w[1] = z.w[1] & MASK_COEFF;
|
|
C3.w[0] = z.w[0];
|
|
if ((z.w[1] & MASK_ANY_INF) != MASK_INF) { // z != inf
|
|
// if z is not infinity check for non-canonical values - treated as zero
|
|
if ((z.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11
|
|
// non-canonical
|
|
z_exp = (z.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits
|
|
C3.w[1] = 0; // significand high
|
|
C3.w[0] = 0; // significand low
|
|
} else { // G0_G1 != 11
|
|
z_exp = z.w[1] & MASK_EXP; // biased and shifted left 49 bits
|
|
if (C3.w[1] > 0x0001ed09bead87c0ull ||
|
|
(C3.w[1] == 0x0001ed09bead87c0ull &&
|
|
C3.w[0] > 0x378d8e63ffffffffull)) {
|
|
// z is non-canonical if coefficient is larger than 10^34 -1
|
|
C3.w[1] = 0;
|
|
C3.w[0] = 0;
|
|
} else { // canonical
|
|
;
|
|
}
|
|
}
|
|
}
|
|
|
|
p_sign = x_sign ^ y_sign; // sign of the product
|
|
|
|
// identify cases where at least one operand is infinity
|
|
|
|
if ((x.w[1] & MASK_ANY_INF) == MASK_INF) { // x = inf
|
|
if ((y.w[1] & MASK_ANY_INF) == MASK_INF) { // y = inf
|
|
if ((z.w[1] & MASK_ANY_INF) == MASK_INF) { // z = inf
|
|
if (p_sign == z_sign) {
|
|
res.w[1] = z_sign | MASK_INF;
|
|
res.w[0] = 0x0;
|
|
} else {
|
|
// return QNaN Indefinite
|
|
res.w[1] = 0x7c00000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
}
|
|
} else { // z = 0 or z = f
|
|
res.w[1] = p_sign | MASK_INF;
|
|
res.w[0] = 0x0;
|
|
}
|
|
} else if (C2.w[1] != 0 || C2.w[0] != 0) { // y = f
|
|
if ((z.w[1] & MASK_ANY_INF) == MASK_INF) { // z = inf
|
|
if (p_sign == z_sign) {
|
|
res.w[1] = z_sign | MASK_INF;
|
|
res.w[0] = 0x0;
|
|
} else {
|
|
// return QNaN Indefinite
|
|
res.w[1] = 0x7c00000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
}
|
|
} else { // z = 0 or z = f
|
|
res.w[1] = p_sign | MASK_INF;
|
|
res.w[0] = 0x0;
|
|
}
|
|
} else { // y = 0
|
|
// return QNaN Indefinite
|
|
res.w[1] = 0x7c00000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
} else if ((y.w[1] & MASK_ANY_INF) == MASK_INF) { // y = inf
|
|
if ((z.w[1] & MASK_ANY_INF) == MASK_INF) { // z = inf
|
|
// x = f, necessarily
|
|
if ((p_sign != z_sign)
|
|
|| (C1.w[1] == 0x0ull && C1.w[0] == 0x0ull)) {
|
|
// return QNaN Indefinite
|
|
res.w[1] = 0x7c00000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
} else {
|
|
res.w[1] = z_sign | MASK_INF;
|
|
res.w[0] = 0x0;
|
|
}
|
|
} else if (C1.w[1] == 0x0 && C1.w[0] == 0x0) { // x = 0
|
|
// z = 0, f, inf
|
|
// return QNaN Indefinite
|
|
res.w[1] = 0x7c00000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
// set invalid flag
|
|
*pfpsf |= INVALID_EXCEPTION;
|
|
} else {
|
|
// x = f and z = 0, f, necessarily
|
|
res.w[1] = p_sign | MASK_INF;
|
|
res.w[0] = 0x0;
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
} else if ((z.w[1] & MASK_ANY_INF) == MASK_INF) { // z = inf
|
|
// x = 0, f and y = 0, f, necessarily
|
|
res.w[1] = z_sign | MASK_INF;
|
|
res.w[0] = 0x0;
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
|
|
true_p_exp = (x_exp >> 49) - 6176 + (y_exp >> 49) - 6176;
|
|
if (true_p_exp < -6176)
|
|
p_exp = 0; // cannot be less than EXP_MIN
|
|
else
|
|
p_exp = (UINT64) (true_p_exp + 6176) << 49;
|
|
|
|
if (((C1.w[1] == 0x0 && C1.w[0] == 0x0) || (C2.w[1] == 0x0 && C2.w[0] == 0x0)) && C3.w[1] == 0x0 && C3.w[0] == 0x0) { // (x = 0 or y = 0) and z = 0
|
|
// the result is 0
|
|
if (p_exp < z_exp)
|
|
res.w[1] = p_exp; // preferred exponent
|
|
else
|
|
res.w[1] = z_exp; // preferred exponent
|
|
if (p_sign == z_sign) {
|
|
res.w[1] |= z_sign;
|
|
res.w[0] = 0x0;
|
|
} else { // x * y and z have opposite signs
|
|
if (rnd_mode == ROUNDING_DOWN) {
|
|
// res = -0.0
|
|
res.w[1] |= MASK_SIGN;
|
|
res.w[0] = 0x0;
|
|
} else {
|
|
// res = +0.0
|
|
// res.w[1] |= 0x0;
|
|
res.w[0] = 0x0;
|
|
}
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
// from this point on, we may need to know the number of decimal digits
|
|
// in the significands of x, y, z when x, y, z != 0
|
|
|
|
if (C1.w[1] != 0 || C1.w[0] != 0) { // x = f (non-zero finite)
|
|
// q1 = nr. of decimal digits in x
|
|
// determine first the nr. of bits in x
|
|
if (C1.w[1] == 0) {
|
|
if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53
|
|
// split the 64-bit value in two 32-bit halves to avoid rounding errors
|
|
if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32
|
|
tmp.d = (double) (C1.w[0] >> 32); // exact conversion
|
|
x_nr_bits =
|
|
33 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
} else { // x < 2^32
|
|
tmp.d = (double) (C1.w[0]); // exact conversion
|
|
x_nr_bits =
|
|
1 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
} else { // if x < 2^53
|
|
tmp.d = (double) C1.w[0]; // exact conversion
|
|
x_nr_bits =
|
|
1 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
} else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1])
|
|
tmp.d = (double) C1.w[1]; // exact conversion
|
|
x_nr_bits =
|
|
65 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
q1 = nr_digits[x_nr_bits - 1].digits;
|
|
if (q1 == 0) {
|
|
q1 = nr_digits[x_nr_bits - 1].digits1;
|
|
if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi ||
|
|
(C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi &&
|
|
C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo))
|
|
q1++;
|
|
}
|
|
}
|
|
|
|
if (C2.w[1] != 0 || C2.w[0] != 0) { // y = f (non-zero finite)
|
|
if (C2.w[1] == 0) {
|
|
if (C2.w[0] >= 0x0020000000000000ull) { // y >= 2^53
|
|
// split the 64-bit value in two 32-bit halves to avoid rounding errors
|
|
if (C2.w[0] >= 0x0000000100000000ull) { // y >= 2^32
|
|
tmp.d = (double) (C2.w[0] >> 32); // exact conversion
|
|
y_nr_bits =
|
|
32 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
} else { // y < 2^32
|
|
tmp.d = (double) C2.w[0]; // exact conversion
|
|
y_nr_bits =
|
|
((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
} else { // if y < 2^53
|
|
tmp.d = (double) C2.w[0]; // exact conversion
|
|
y_nr_bits =
|
|
((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
} else { // C2.w[1] != 0 => nr. bits = 64 + nr_bits (C2.w[1])
|
|
tmp.d = (double) C2.w[1]; // exact conversion
|
|
y_nr_bits =
|
|
64 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
|
|
q2 = nr_digits[y_nr_bits].digits;
|
|
if (q2 == 0) {
|
|
q2 = nr_digits[y_nr_bits].digits1;
|
|
if (C2.w[1] > nr_digits[y_nr_bits].threshold_hi ||
|
|
(C2.w[1] == nr_digits[y_nr_bits].threshold_hi &&
|
|
C2.w[0] >= nr_digits[y_nr_bits].threshold_lo))
|
|
q2++;
|
|
}
|
|
}
|
|
|
|
if (C3.w[1] != 0 || C3.w[0] != 0) { // z = f (non-zero finite)
|
|
if (C3.w[1] == 0) {
|
|
if (C3.w[0] >= 0x0020000000000000ull) { // z >= 2^53
|
|
// split the 64-bit value in two 32-bit halves to avoid rounding errors
|
|
if (C3.w[0] >= 0x0000000100000000ull) { // z >= 2^32
|
|
tmp.d = (double) (C3.w[0] >> 32); // exact conversion
|
|
z_nr_bits =
|
|
32 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
} else { // z < 2^32
|
|
tmp.d = (double) C3.w[0]; // exact conversion
|
|
z_nr_bits =
|
|
((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
} else { // if z < 2^53
|
|
tmp.d = (double) C3.w[0]; // exact conversion
|
|
z_nr_bits =
|
|
((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
} else { // C3.w[1] != 0 => nr. bits = 64 + nr_bits (C3.w[1])
|
|
tmp.d = (double) C3.w[1]; // exact conversion
|
|
z_nr_bits =
|
|
64 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
|
|
q3 = nr_digits[z_nr_bits].digits;
|
|
if (q3 == 0) {
|
|
q3 = nr_digits[z_nr_bits].digits1;
|
|
if (C3.w[1] > nr_digits[z_nr_bits].threshold_hi ||
|
|
(C3.w[1] == nr_digits[z_nr_bits].threshold_hi &&
|
|
C3.w[0] >= nr_digits[z_nr_bits].threshold_lo))
|
|
q3++;
|
|
}
|
|
}
|
|
|
|
if ((C1.w[1] == 0x0 && C1.w[0] == 0x0) ||
|
|
(C2.w[1] == 0x0 && C2.w[0] == 0x0)) {
|
|
// x = 0 or y = 0
|
|
// z = f, necessarily; for 0 + z return z, with the preferred exponent
|
|
// the result is z, but need to get the preferred exponent
|
|
if (z_exp <= p_exp) { // the preferred exponent is z_exp
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | C3.w[1];
|
|
res.w[0] = C3.w[0];
|
|
} else { // if (p_exp < z_exp) the preferred exponent is p_exp
|
|
// return (C3 * 10^scale) * 10^(z_exp - scale)
|
|
// where scale = min (p34-q3, (z_exp-p_exp) >> 49)
|
|
scale = p34 - q3;
|
|
ind = (z_exp - p_exp) >> 49;
|
|
if (ind < scale)
|
|
scale = ind;
|
|
if (scale == 0) {
|
|
res.w[1] = z.w[1]; // & MASK_COEFF, which is redundant
|
|
res.w[0] = z.w[0];
|
|
} else if (q3 <= 19) { // z fits in 64 bits
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
|
// 64 x 64 C3.w[0] * ten2k64[scale]
|
|
__mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]);
|
|
} else { // 10^scale fits in 128 bits
|
|
// 64 x 128 C3.w[0] * ten2k128[scale - 20]
|
|
__mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]);
|
|
}
|
|
} else { // z fits in 128 bits, but 10^scale must fit in 64 bits
|
|
// 64 x 128 ten2k64[scale] * C3
|
|
__mul_128x64_to_128 (res, ten2k64[scale], C3);
|
|
}
|
|
// subtract scale from the exponent
|
|
z_exp = z_exp - ((UINT64) scale << 49);
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
} else {
|
|
; // continue with x = f, y = f, z = 0 or x = f, y = f, z = f
|
|
}
|
|
|
|
e1 = (x_exp >> 49) - 6176; // unbiased exponent of x
|
|
e2 = (y_exp >> 49) - 6176; // unbiased exponent of y
|
|
e3 = (z_exp >> 49) - 6176; // unbiased exponent of z
|
|
e4 = e1 + e2; // unbiased exponent of the exact x * y
|
|
|
|
// calculate C1 * C2 and its number of decimal digits, q4
|
|
|
|
// the exact product has either q1 + q2 - 1 or q1 + q2 decimal digits
|
|
// where 2 <= q1 + q2 <= 68
|
|
// calculate C4 = C1 * C2 and determine q
|
|
C4.w[3] = C4.w[2] = C4.w[1] = C4.w[0] = 0;
|
|
if (q1 + q2 <= 19) { // if 2 <= q1 + q2 <= 19, C4 = C1 * C2 fits in 64 bits
|
|
C4.w[0] = C1.w[0] * C2.w[0];
|
|
// if C4 < 10^(q1+q2-1) then q4 = q1 + q2 - 1 else q4 = q1 + q2
|
|
if (C4.w[0] < ten2k64[q1 + q2 - 1])
|
|
q4 = q1 + q2 - 1; // q4 in [1, 18]
|
|
else
|
|
q4 = q1 + q2; // q4 in [2, 19]
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 64;
|
|
} else if (q1 + q2 == 20) { // C4 = C1 * C2 fits in 64 or 128 bits
|
|
// q1 <= 19 and q2 <= 19 so both C1 and C2 fit in 64 bits
|
|
__mul_64x64_to_128MACH (C4, C1.w[0], C2.w[0]);
|
|
// if C4 < 10^(q1+q2-1) = 10^19 then q4 = q1+q2-1 = 19 else q4 = q1+q2 = 20
|
|
if (C4.w[1] == 0 && C4.w[0] < ten2k64[19]) { // 19 = q1+q2-1
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 64;
|
|
q4 = 19; // 19 = q1 + q2 - 1
|
|
} else {
|
|
// if (C4.w[1] == 0)
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 64;
|
|
// else
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 128;
|
|
q4 = 20; // 20 = q1 + q2
|
|
}
|
|
} else if (q1 + q2 <= 38) { // 21 <= q1 + q2 <= 38
|
|
// C4 = C1 * C2 fits in 64 or 128 bits
|
|
// (64 bits possibly, but only when q1 + q2 = 21 and C4 has 20 digits)
|
|
// at least one of C1, C2 has at most 19 decimal digits & fits in 64 bits
|
|
if (q1 <= 19) {
|
|
__mul_128x64_to_128 (C4, C1.w[0], C2);
|
|
} else { // q2 <= 19
|
|
__mul_128x64_to_128 (C4, C2.w[0], C1);
|
|
}
|
|
// if C4 < 10^(q1+q2-1) then q4 = q1 + q2 - 1 else q4 = q1 + q2
|
|
if (C4.w[1] < ten2k128[q1 + q2 - 21].w[1] ||
|
|
(C4.w[1] == ten2k128[q1 + q2 - 21].w[1] &&
|
|
C4.w[0] < ten2k128[q1 + q2 - 21].w[0])) {
|
|
// if (C4.w[1] == 0) // q4 = 20, necessarily
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 64;
|
|
// else
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 128;
|
|
q4 = q1 + q2 - 1; // q4 in [20, 37]
|
|
} else {
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 128;
|
|
q4 = q1 + q2; // q4 in [21, 38]
|
|
}
|
|
} else if (q1 + q2 == 39) { // C4 = C1 * C2 fits in 128 or 192 bits
|
|
// both C1 and C2 fit in 128 bits (actually in 113 bits)
|
|
// may replace this by 128x128_to192
|
|
__mul_128x128_to_256 (C4, C1, C2); // C4.w[3] is 0
|
|
// if C4 < 10^(q1+q2-1) = 10^38 then q4 = q1+q2-1 = 38 else q4 = q1+q2 = 39
|
|
if (C4.w[2] == 0 && (C4.w[1] < ten2k128[18].w[1] ||
|
|
(C4.w[1] == ten2k128[18].w[1]
|
|
&& C4.w[0] < ten2k128[18].w[0]))) {
|
|
// 18 = 38 - 20 = q1+q2-1 - 20
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 128;
|
|
q4 = 38; // 38 = q1 + q2 - 1
|
|
} else {
|
|
// if (C4.w[2] == 0)
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 128;
|
|
// else
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 192;
|
|
q4 = 39; // 39 = q1 + q2
|
|
}
|
|
} else if (q1 + q2 <= 57) { // 40 <= q1 + q2 <= 57
|
|
// C4 = C1 * C2 fits in 128 or 192 bits
|
|
// (128 bits possibly, but only when q1 + q2 = 40 and C4 has 39 digits)
|
|
// both C1 and C2 fit in 128 bits (actually in 113 bits); at most one
|
|
// may fit in 64 bits
|
|
if (C1.w[1] == 0) { // C1 fits in 64 bits
|
|
// __mul_64x128_full (REShi64, RESlo128, A64, B128)
|
|
__mul_64x128_full (C4.w[2], C4, C1.w[0], C2);
|
|
} else if (C2.w[1] == 0) { // C2 fits in 64 bits
|
|
// __mul_64x128_full (REShi64, RESlo128, A64, B128)
|
|
__mul_64x128_full (C4.w[2], C4, C2.w[0], C1);
|
|
} else { // both C1 and C2 require 128 bits
|
|
// may use __mul_128x128_to_192 (C4.w[2], C4.w[0], C2.w[0], C1);
|
|
__mul_128x128_to_256 (C4, C1, C2); // C4.w[3] = 0
|
|
}
|
|
// if C4 < 10^(q1+q2-1) then q4 = q1 + q2 - 1 else q4 = q1 + q2
|
|
if (C4.w[2] < ten2k256[q1 + q2 - 40].w[2] ||
|
|
(C4.w[2] == ten2k256[q1 + q2 - 40].w[2] &&
|
|
(C4.w[1] < ten2k256[q1 + q2 - 40].w[1] ||
|
|
(C4.w[1] == ten2k256[q1 + q2 - 40].w[1] &&
|
|
C4.w[0] < ten2k256[q1 + q2 - 40].w[0])))) {
|
|
// if (C4.w[2] == 0) // q4 = 39, necessarily
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 128;
|
|
// else
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 192;
|
|
q4 = q1 + q2 - 1; // q4 in [39, 56]
|
|
} else {
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 192;
|
|
q4 = q1 + q2; // q4 in [40, 57]
|
|
}
|
|
} else if (q1 + q2 == 58) { // C4 = C1 * C2 fits in 192 or 256 bits
|
|
// both C1 and C2 fit in 128 bits (actually in 113 bits); at most one
|
|
// may fit in 64 bits
|
|
if (C1.w[1] == 0) { // C1 * C2 will fit in 192 bits
|
|
__mul_64x128_full (C4.w[2], C4, C1.w[0], C2); // may use 64x128_to_192
|
|
} else if (C2.w[1] == 0) { // C1 * C2 will fit in 192 bits
|
|
__mul_64x128_full (C4.w[2], C4, C2.w[0], C1); // may use 64x128_to_192
|
|
} else { // C1 * C2 will fit in 192 bits or in 256 bits
|
|
__mul_128x128_to_256 (C4, C1, C2);
|
|
}
|
|
// if C4 < 10^(q1+q2-1) = 10^57 then q4 = q1+q2-1 = 57 else q4 = q1+q2 = 58
|
|
if (C4.w[3] == 0 && (C4.w[2] < ten2k256[18].w[2] ||
|
|
(C4.w[2] == ten2k256[18].w[2]
|
|
&& (C4.w[1] < ten2k256[18].w[1]
|
|
|| (C4.w[1] == ten2k256[18].w[1]
|
|
&& C4.w[0] < ten2k256[18].w[0]))))) {
|
|
// 18 = 57 - 39 = q1+q2-1 - 39
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 192;
|
|
q4 = 57; // 57 = q1 + q2 - 1
|
|
} else {
|
|
// if (C4.w[3] == 0)
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 192;
|
|
// else
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 256;
|
|
q4 = 58; // 58 = q1 + q2
|
|
}
|
|
} else { // if 59 <= q1 + q2 <= 68
|
|
// C4 = C1 * C2 fits in 192 or 256 bits
|
|
// (192 bits possibly, but only when q1 + q2 = 59 and C4 has 58 digits)
|
|
// both C1 and C2 fit in 128 bits (actually in 113 bits); none fits in
|
|
// 64 bits
|
|
// may use __mul_128x128_to_192 (C4.w[2], C4.w[0], C2.w[0], C1);
|
|
__mul_128x128_to_256 (C4, C1, C2); // C4.w[3] = 0
|
|
// if C4 < 10^(q1+q2-1) then q4 = q1 + q2 - 1 else q4 = q1 + q2
|
|
if (C4.w[3] < ten2k256[q1 + q2 - 40].w[3] ||
|
|
(C4.w[3] == ten2k256[q1 + q2 - 40].w[3] &&
|
|
(C4.w[2] < ten2k256[q1 + q2 - 40].w[2] ||
|
|
(C4.w[2] == ten2k256[q1 + q2 - 40].w[2] &&
|
|
(C4.w[1] < ten2k256[q1 + q2 - 40].w[1] ||
|
|
(C4.w[1] == ten2k256[q1 + q2 - 40].w[1] &&
|
|
C4.w[0] < ten2k256[q1 + q2 - 40].w[0])))))) {
|
|
// if (C4.w[3] == 0) // q4 = 58, necessarily
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 192;
|
|
// else
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 256;
|
|
q4 = q1 + q2 - 1; // q4 in [58, 67]
|
|
} else {
|
|
// length of C1 * C2 rounded up to a multiple of 64 bits is len = 256;
|
|
q4 = q1 + q2; // q4 in [59, 68]
|
|
}
|
|
}
|
|
|
|
if (C3.w[1] == 0x0 && C3.w[0] == 0x0) { // x = f, y = f, z = 0
|
|
save_fpsf = *pfpsf; // sticky bits - caller value must be preserved
|
|
*pfpsf = 0;
|
|
|
|
if (q4 > p34) {
|
|
|
|
// truncate C4 to p34 digits into res
|
|
// x = q4-p34, 1 <= x <= 34 because 35 <= q4 <= 68
|
|
x0 = q4 - p34;
|
|
if (q4 <= 38) {
|
|
P128.w[1] = C4.w[1];
|
|
P128.w[0] = C4.w[0];
|
|
round128_19_38 (q4, x0, P128, &res, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
} else if (q4 <= 57) { // 35 <= q4 <= 57
|
|
P192.w[2] = C4.w[2];
|
|
P192.w[1] = C4.w[1];
|
|
P192.w[0] = C4.w[0];
|
|
round192_39_57 (q4, x0, P192, &R192, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
res.w[0] = R192.w[0];
|
|
res.w[1] = R192.w[1];
|
|
} else { // if (q4 <= 68)
|
|
round256_58_76 (q4, x0, C4, &R256, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
res.w[0] = R256.w[0];
|
|
res.w[1] = R256.w[1];
|
|
}
|
|
e4 = e4 + x0;
|
|
if (incr_exp) {
|
|
e4 = e4 + 1;
|
|
}
|
|
q4 = p34;
|
|
// res is now the coefficient of the result rounded to the destination
|
|
// precision, with unbounded exponent; the exponent is e4; q4=digits(res)
|
|
} else { // if (q4 <= p34)
|
|
// C4 * 10^e4 is the result rounded to the destination precision, with
|
|
// unbounded exponent (which is exact)
|
|
|
|
if ((q4 + e4 <= p34 + expmax) && (e4 > expmax)) {
|
|
// e4 is too large, but can be brought within range by scaling up C4
|
|
scale = e4 - expmax; // 1 <= scale < P-q4 <= P-1 => 1 <= scale <= P-2
|
|
// res = (C4 * 10^scale) * 10^expmax
|
|
if (q4 <= 19) { // C4 fits in 64 bits
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
|
// 64 x 64 C4.w[0] * ten2k64[scale]
|
|
__mul_64x64_to_128MACH (res, C4.w[0], ten2k64[scale]);
|
|
} else { // 10^scale fits in 128 bits
|
|
// 64 x 128 C4.w[0] * ten2k128[scale - 20]
|
|
__mul_128x64_to_128 (res, C4.w[0], ten2k128[scale - 20]);
|
|
}
|
|
} else { // C4 fits in 128 bits, but 10^scale must fit in 64 bits
|
|
// 64 x 128 ten2k64[scale] * CC43
|
|
__mul_128x64_to_128 (res, ten2k64[scale], C4);
|
|
}
|
|
e4 = e4 - scale; // expmax
|
|
q4 = q4 + scale;
|
|
} else {
|
|
res.w[1] = C4.w[1];
|
|
res.w[0] = C4.w[0];
|
|
}
|
|
// res is the coefficient of the result rounded to the destination
|
|
// precision, with unbounded exponent (it has q4 digits); the exponent
|
|
// is e4 (exact result)
|
|
}
|
|
|
|
// check for overflow
|
|
if (q4 + e4 > p34 + expmax) {
|
|
if (rnd_mode == ROUNDING_TO_NEAREST) {
|
|
res.w[1] = p_sign | 0x7800000000000000ull; // +/-inf
|
|
res.w[0] = 0x0000000000000000ull;
|
|
*pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
|
|
} else {
|
|
res.w[1] = p_sign | res.w[1];
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e4, &res, pfpsf);
|
|
}
|
|
*pfpsf |= save_fpsf;
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
// check for underflow
|
|
if (q4 + e4 < expmin + P34) {
|
|
is_tiny = 1; // the result is tiny
|
|
if (e4 < expmin) {
|
|
// if e4 < expmin, we must truncate more of res
|
|
x0 = expmin - e4; // x0 >= 1
|
|
is_inexact_lt_midpoint0 = is_inexact_lt_midpoint;
|
|
is_inexact_gt_midpoint0 = is_inexact_gt_midpoint;
|
|
is_midpoint_lt_even0 = is_midpoint_lt_even;
|
|
is_midpoint_gt_even0 = is_midpoint_gt_even;
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
// the number of decimal digits in res is q4
|
|
if (x0 < q4) { // 1 <= x0 <= q4-1 => round res to q4 - x0 digits
|
|
if (q4 <= 18) { // 2 <= q4 <= 18, 1 <= x0 <= 17
|
|
round64_2_18 (q4, x0, res.w[0], &R64, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
if (incr_exp) {
|
|
// R64 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 1, 1 <= q4 - x0 <= 17
|
|
R64 = ten2k64[q4 - x0];
|
|
}
|
|
// res.w[1] = 0; (from above)
|
|
res.w[0] = R64;
|
|
} else { // if (q4 <= 34)
|
|
// 19 <= q4 <= 38
|
|
P128.w[1] = res.w[1];
|
|
P128.w[0] = res.w[0];
|
|
round128_19_38 (q4, x0, P128, &res, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
if (incr_exp) {
|
|
// increase coefficient by a factor of 10; this will be <= 10^33
|
|
// R128 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 1, 1 <= q4 - x0 <= 37
|
|
if (q4 - x0 <= 19) { // 1 <= q4 - x0 <= 19
|
|
// res.w[1] = 0;
|
|
res.w[0] = ten2k64[q4 - x0];
|
|
} else { // 20 <= q4 - x0 <= 37
|
|
res.w[0] = ten2k128[q4 - x0 - 20].w[0];
|
|
res.w[1] = ten2k128[q4 - x0 - 20].w[1];
|
|
}
|
|
}
|
|
}
|
|
e4 = e4 + x0; // expmin
|
|
} else if (x0 == q4) {
|
|
// the second rounding is for 0.d(0)d(1)...d(q4-1) * 10^emin
|
|
// determine relationship with 1/2 ulp
|
|
if (q4 <= 19) {
|
|
if (res.w[0] < midpoint64[q4 - 1]) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (res.w[0] == midpoint64[q4 - 1]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
is_midpoint_gt_even = 1;
|
|
} else { // > 1/2 ulp
|
|
// gt_half_ulp = 1;
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
} else { // if (q4 <= 34)
|
|
if (res.w[1] < midpoint128[q4 - 20].w[1] ||
|
|
(res.w[1] == midpoint128[q4 - 20].w[1] &&
|
|
res.w[0] < midpoint128[q4 - 20].w[0])) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (res.w[1] == midpoint128[q4 - 20].w[1] &&
|
|
res.w[0] == midpoint128[q4 - 20].w[0]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
is_midpoint_gt_even = 1;
|
|
} else { // > 1/2 ulp
|
|
// gt_half_ulp = 1;
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
}
|
|
if (lt_half_ulp || eq_half_ulp) {
|
|
// res = +0.0 * 10^expmin
|
|
res.w[1] = 0x0000000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
} else { // if (gt_half_ulp)
|
|
// res = +1 * 10^expmin
|
|
res.w[1] = 0x0000000000000000ull;
|
|
res.w[0] = 0x0000000000000001ull;
|
|
}
|
|
e4 = expmin;
|
|
} else { // if (x0 > q4)
|
|
// the second rounding is for 0.0...d(0)d(1)...d(q4-1) * 10^emin
|
|
res.w[1] = 0;
|
|
res.w[0] = 0;
|
|
e4 = expmin;
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
// avoid a double rounding error
|
|
if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) &&
|
|
is_midpoint_lt_even) { // double rounding error upward
|
|
// res = res - 1
|
|
res.w[0]--;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
// Note: a double rounding error upward is not possible; for this
|
|
// the result after the first rounding would have to be 99...95
|
|
// (35 digits in all), possibly followed by a number of zeros; this
|
|
// not possible for f * f + 0
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) &&
|
|
is_midpoint_gt_even) { // double rounding error downward
|
|
// res = res + 1
|
|
res.w[0]++;
|
|
if (res.w[0] == 0)
|
|
res.w[1]++;
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
// if this second rounding was exact the result may still be
|
|
// inexact because of the first rounding
|
|
if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) {
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else if (is_midpoint_gt_even &&
|
|
(is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) {
|
|
// pulled up to a midpoint
|
|
is_inexact_lt_midpoint = 1;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else if (is_midpoint_lt_even &&
|
|
(is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) {
|
|
// pulled down to a midpoint
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else {
|
|
;
|
|
}
|
|
} else { // if e4 >= emin then q4 < P and the result is tiny and exact
|
|
if (e3 < e4) {
|
|
// if (e3 < e4) the preferred exponent is e3
|
|
// return (C4 * 10^scale) * 10^(e4 - scale)
|
|
// where scale = min (p34-q4, (e4 - e3))
|
|
scale = p34 - q4;
|
|
ind = e4 - e3;
|
|
if (ind < scale)
|
|
scale = ind;
|
|
if (scale == 0) {
|
|
; // res and e4 are unchanged
|
|
} else if (q4 <= 19) { // C4 fits in 64 bits
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
|
// 64 x 64 res.w[0] * ten2k64[scale]
|
|
__mul_64x64_to_128MACH (res, res.w[0], ten2k64[scale]);
|
|
} else { // 10^scale fits in 128 bits
|
|
// 64 x 128 res.w[0] * ten2k128[scale - 20]
|
|
__mul_128x64_to_128 (res, res.w[0], ten2k128[scale - 20]);
|
|
}
|
|
} else { // res fits in 128 bits, but 10^scale must fit in 64 bits
|
|
// 64 x 128 ten2k64[scale] * C3
|
|
__mul_128x64_to_128 (res, ten2k64[scale], res);
|
|
}
|
|
// subtract scale from the exponent
|
|
e4 = e4 - scale;
|
|
}
|
|
}
|
|
|
|
// check for inexact result
|
|
if (is_inexact_lt_midpoint || is_inexact_gt_midpoint ||
|
|
is_midpoint_lt_even || is_midpoint_gt_even) {
|
|
// set the inexact flag and the underflow flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
*pfpsf |= UNDERFLOW_EXCEPTION;
|
|
}
|
|
res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | res.w[1];
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e4, &res, pfpsf);
|
|
}
|
|
*pfpsf |= save_fpsf;
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
// no overflow, and no underflow for rounding to nearest
|
|
res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | res.w[1];
|
|
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e4, &res, pfpsf);
|
|
// if e4 = expmin && significand < 10^33 => result is tiny (for RD, RZ)
|
|
if (e4 == expmin) {
|
|
if ((res.w[1] & MASK_COEFF) < 0x0000314dc6448d93ull ||
|
|
((res.w[1] & MASK_COEFF) == 0x0000314dc6448d93ull &&
|
|
res.w[0] < 0x38c15b0a00000000ull)) {
|
|
is_tiny = 1;
|
|
}
|
|
}
|
|
}
|
|
|
|
if (is_inexact_lt_midpoint || is_inexact_gt_midpoint ||
|
|
is_midpoint_lt_even || is_midpoint_gt_even) {
|
|
// set the inexact flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
if (is_tiny)
|
|
*pfpsf |= UNDERFLOW_EXCEPTION;
|
|
}
|
|
|
|
if ((*pfpsf & INEXACT_EXCEPTION) == 0) { // x * y is exact
|
|
// need to ensure that the result has the preferred exponent
|
|
p_exp = res.w[1] & MASK_EXP;
|
|
if (z_exp < p_exp) { // the preferred exponent is z_exp
|
|
// signficand of res in C3
|
|
C3.w[1] = res.w[1] & MASK_COEFF;
|
|
C3.w[0] = res.w[0];
|
|
// the number of decimal digits of x * y is q4 <= 34
|
|
// Note: the coefficient fits in 128 bits
|
|
|
|
// return (C3 * 10^scale) * 10^(p_exp - scale)
|
|
// where scale = min (p34-q4, (p_exp-z_exp) >> 49)
|
|
scale = p34 - q4;
|
|
ind = (p_exp - z_exp) >> 49;
|
|
if (ind < scale)
|
|
scale = ind;
|
|
// subtract scale from the exponent
|
|
p_exp = p_exp - ((UINT64) scale << 49);
|
|
if (scale == 0) {
|
|
; // leave res unchanged
|
|
} else if (q4 <= 19) { // x * y fits in 64 bits
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
|
// 64 x 64 C3.w[0] * ten2k64[scale]
|
|
__mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]);
|
|
} else { // 10^scale fits in 128 bits
|
|
// 64 x 128 C3.w[0] * ten2k128[scale - 20]
|
|
__mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]);
|
|
}
|
|
res.w[1] = p_sign | (p_exp & MASK_EXP) | res.w[1];
|
|
} else { // x * y fits in 128 bits, but 10^scale must fit in 64 bits
|
|
// 64 x 128 ten2k64[scale] * C3
|
|
__mul_128x64_to_128 (res, ten2k64[scale], C3);
|
|
res.w[1] = p_sign | (p_exp & MASK_EXP) | res.w[1];
|
|
}
|
|
} // else leave the result as it is, because p_exp <= z_exp
|
|
}
|
|
*pfpsf |= save_fpsf;
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
} // else we have f * f + f
|
|
|
|
// continue with x = f, y = f, z = f
|
|
|
|
delta = q3 + e3 - q4 - e4;
|
|
delta_ge_zero:
|
|
if (delta >= 0) {
|
|
|
|
if (p34 <= delta - 1 || // Case (1')
|
|
(p34 == delta && e3 + 6176 < p34 - q3)) { // Case (1''A)
|
|
// check for overflow, which can occur only in Case (1')
|
|
if ((q3 + e3) > (p34 + expmax) && p34 <= delta - 1) {
|
|
// e3 > expmax implies p34 <= delta-1 and e3 > expmax is a necessary
|
|
// condition for (q3 + e3) > (p34 + expmax)
|
|
if (rnd_mode == ROUNDING_TO_NEAREST) {
|
|
res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf
|
|
res.w[0] = 0x0000000000000000ull;
|
|
*pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
|
|
} else {
|
|
if (p_sign == z_sign) {
|
|
is_inexact_lt_midpoint = 1;
|
|
} else {
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
// q3 <= p34; if (q3 < p34) scale C3 up by 10^(p34-q3)
|
|
scale = p34 - q3;
|
|
if (scale == 0) {
|
|
res.w[1] = z_sign | C3.w[1];
|
|
res.w[0] = C3.w[0];
|
|
} else {
|
|
if (q3 <= 19) { // C3 fits in 64 bits
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
|
// 64 x 64 C3.w[0] * ten2k64[scale]
|
|
__mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]);
|
|
} else { // 10^scale fits in 128 bits
|
|
// 64 x 128 C3.w[0] * ten2k128[scale - 20]
|
|
__mul_128x64_to_128 (res, C3.w[0],
|
|
ten2k128[scale - 20]);
|
|
}
|
|
} else { // C3 fits in 128 bits, but 10^scale must fit in 64 bits
|
|
// 64 x 128 ten2k64[scale] * C3
|
|
__mul_128x64_to_128 (res, ten2k64[scale], C3);
|
|
}
|
|
// the coefficient in res has q3 + scale = p34 digits
|
|
}
|
|
e3 = e3 - scale;
|
|
res.w[1] = z_sign | res.w[1];
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e3, &res, pfpsf);
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
// res = z
|
|
if (q3 < p34) { // the preferred exponent is z_exp - (p34 - q3)
|
|
// return (C3 * 10^scale) * 10^(z_exp - scale)
|
|
// where scale = min (p34-q3, z_exp-EMIN)
|
|
scale = p34 - q3;
|
|
ind = e3 + 6176;
|
|
if (ind < scale)
|
|
scale = ind;
|
|
if (scale == 0) {
|
|
res.w[1] = C3.w[1];
|
|
res.w[0] = C3.w[0];
|
|
} else if (q3 <= 19) { // z fits in 64 bits
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
|
// 64 x 64 C3.w[0] * ten2k64[scale]
|
|
__mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]);
|
|
} else { // 10^scale fits in 128 bits
|
|
// 64 x 128 C3.w[0] * ten2k128[scale - 20]
|
|
__mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]);
|
|
}
|
|
} else { // z fits in 128 bits, but 10^scale must fit in 64 bits
|
|
// 64 x 128 ten2k64[scale] * C3
|
|
__mul_128x64_to_128 (res, ten2k64[scale], C3);
|
|
}
|
|
// the coefficient in res has q3 + scale digits
|
|
// subtract scale from the exponent
|
|
z_exp = z_exp - ((UINT64) scale << 49);
|
|
e3 = e3 - scale;
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
if (scale + q3 < p34)
|
|
*pfpsf |= UNDERFLOW_EXCEPTION;
|
|
} else {
|
|
scale = 0;
|
|
res.w[1] = z_sign | ((UINT64) (e3 + 6176) << 49) | C3.w[1];
|
|
res.w[0] = C3.w[0];
|
|
}
|
|
|
|
// use the following to avoid double rounding errors when operating on
|
|
// mixed formats in rounding to nearest, and for correcting the result
|
|
// if not rounding to nearest
|
|
if ((p_sign != z_sign) && (delta == (q3 + scale + 1))) {
|
|
// there is a gap of exactly one digit between the scaled C3 and C4
|
|
// C3 * 10^ scale = 10^(q3+scale-1) <=> C3 = 10^(q3-1) is special case
|
|
if ((q3 <= 19 && C3.w[0] != ten2k64[q3 - 1]) ||
|
|
(q3 == 20 && (C3.w[1] != 0 || C3.w[0] != ten2k64[19])) ||
|
|
(q3 >= 21 && (C3.w[1] != ten2k128[q3 - 21].w[1] ||
|
|
C3.w[0] != ten2k128[q3 - 21].w[0]))) {
|
|
// C3 * 10^ scale != 10^(q3-1)
|
|
// if ((res.w[1] & MASK_COEFF) != 0x0000314dc6448d93ull ||
|
|
// res.w[0] != 0x38c15b0a00000000ull) { // C3 * 10^scale != 10^33
|
|
is_inexact_gt_midpoint = 1; // if (z_sign), set as if for abs. value
|
|
} else { // if C3 * 10^scale = 10^(q3+scale-1)
|
|
// ok from above e3 = (z_exp >> 49) - 6176;
|
|
// the result is always inexact
|
|
if (q4 == 1) {
|
|
R64 = C4.w[0];
|
|
} else {
|
|
// if q4 > 1 then truncate C4 from q4 digits to 1 digit;
|
|
// x = q4-1, 1 <= x <= 67 and check if this operation is exact
|
|
if (q4 <= 18) { // 2 <= q4 <= 18
|
|
round64_2_18 (q4, q4 - 1, C4.w[0], &R64, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
} else if (q4 <= 38) {
|
|
P128.w[1] = C4.w[1];
|
|
P128.w[0] = C4.w[0];
|
|
round128_19_38 (q4, q4 - 1, P128, &R128, &incr_exp,
|
|
&is_midpoint_lt_even,
|
|
&is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
R64 = R128.w[0]; // one decimal digit
|
|
} else if (q4 <= 57) {
|
|
P192.w[2] = C4.w[2];
|
|
P192.w[1] = C4.w[1];
|
|
P192.w[0] = C4.w[0];
|
|
round192_39_57 (q4, q4 - 1, P192, &R192, &incr_exp,
|
|
&is_midpoint_lt_even,
|
|
&is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
R64 = R192.w[0]; // one decimal digit
|
|
} else { // if (q4 <= 68)
|
|
round256_58_76 (q4, q4 - 1, C4, &R256, &incr_exp,
|
|
&is_midpoint_lt_even,
|
|
&is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
R64 = R256.w[0]; // one decimal digit
|
|
}
|
|
if (incr_exp) {
|
|
R64 = 10;
|
|
}
|
|
}
|
|
if (q4 == 1 && C4.w[0] == 5) {
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 1;
|
|
is_midpoint_gt_even = 0;
|
|
} else if ((e3 == expmin) ||
|
|
R64 < 5 || (R64 == 5 && is_inexact_gt_midpoint)) {
|
|
// result does not change
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else {
|
|
is_inexact_lt_midpoint = 1;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
// result decremented is 10^(q3+scale) - 1
|
|
if ((q3 + scale) <= 19) {
|
|
res.w[1] = 0;
|
|
res.w[0] = ten2k64[q3 + scale];
|
|
} else { // if ((q3 + scale + 1) <= 35)
|
|
res.w[1] = ten2k128[q3 + scale - 20].w[1];
|
|
res.w[0] = ten2k128[q3 + scale - 20].w[0];
|
|
}
|
|
res.w[0] = res.w[0] - 1; // borrow never occurs
|
|
z_exp = z_exp - EXP_P1;
|
|
e3 = e3 - 1;
|
|
res.w[1] = z_sign | ((UINT64) (e3 + 6176) << 49) | res.w[1];
|
|
}
|
|
if (e3 == expmin) {
|
|
if (R64 < 5 || (R64 == 5 && !is_inexact_lt_midpoint)) {
|
|
; // result not tiny (in round-to-nearest mode)
|
|
} else {
|
|
*pfpsf |= UNDERFLOW_EXCEPTION;
|
|
}
|
|
}
|
|
} // end 10^(q3+scale-1)
|
|
// set the inexact flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
} else {
|
|
if (p_sign == z_sign) {
|
|
// if (z_sign), set as if for absolute value
|
|
is_inexact_lt_midpoint = 1;
|
|
} else { // if (p_sign != z_sign)
|
|
// if (z_sign), set as if for absolute value
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
}
|
|
// the result is always inexact => set the inexact flag
|
|
// Determine tininess:
|
|
// if (exp > expmin)
|
|
// the result is not tiny
|
|
// else // if exp = emin
|
|
// if (q3 + scale < p34)
|
|
// the result is tiny
|
|
// else // if (q3 + scale = p34)
|
|
// if (C3 * 10^scale > 10^33)
|
|
// the result is not tiny
|
|
// else // if C3 * 10^scale = 10^33
|
|
// if (xy * z > 0)
|
|
// the result is not tiny
|
|
// else // if (xy * z < 0)
|
|
// if (z > 0)
|
|
// if rnd_mode != RP
|
|
// the result is tiny
|
|
// else // if RP
|
|
// the result is not tiny
|
|
// else // if (z < 0)
|
|
// if rnd_mode != RM
|
|
// the result is tiny
|
|
// else // if RM
|
|
// the result is not tiny
|
|
// endif
|
|
// endif
|
|
// endif
|
|
// endif
|
|
// endif
|
|
// endif
|
|
if ((e3 == expmin && (q3 + scale) < p34) ||
|
|
(e3 == expmin && (q3 + scale) == p34 &&
|
|
(res.w[1] & MASK_COEFF) == 0x0000314dc6448d93ull && // 10^33_high
|
|
res.w[0] == 0x38c15b0a00000000ull && // 10^33_low
|
|
z_sign != p_sign && ((!z_sign && rnd_mode != ROUNDING_UP) ||
|
|
(z_sign && rnd_mode != ROUNDING_DOWN)))) {
|
|
*pfpsf |= UNDERFLOW_EXCEPTION;
|
|
}
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e3, &res, pfpsf);
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
|
|
} else if (p34 == delta) { // Case (1''B)
|
|
|
|
// because Case (1''A) was treated above, e3 + 6176 >= p34 - q3
|
|
// and C3 can be scaled up to p34 digits if needed
|
|
|
|
// scale C3 to p34 digits if needed
|
|
scale = p34 - q3; // 0 <= scale <= p34 - 1
|
|
if (scale == 0) {
|
|
res.w[1] = C3.w[1];
|
|
res.w[0] = C3.w[0];
|
|
} else if (q3 <= 19) { // z fits in 64 bits
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
|
// 64 x 64 C3.w[0] * ten2k64[scale]
|
|
__mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]);
|
|
} else { // 10^scale fits in 128 bits
|
|
// 64 x 128 C3.w[0] * ten2k128[scale - 20]
|
|
__mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]);
|
|
}
|
|
} else { // z fits in 128 bits, but 10^scale must fit in 64 bits
|
|
// 64 x 128 ten2k64[scale] * C3
|
|
__mul_128x64_to_128 (res, ten2k64[scale], C3);
|
|
}
|
|
// subtract scale from the exponent
|
|
z_exp = z_exp - ((UINT64) scale << 49);
|
|
e3 = e3 - scale;
|
|
// now z_sign, z_exp, and res correspond to a z scaled to p34 = 34 digits
|
|
|
|
// determine whether x * y is less than, equal to, or greater than
|
|
// 1/2 ulp (z)
|
|
if (q4 <= 19) {
|
|
if (C4.w[0] < midpoint64[q4 - 1]) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
} else if (C4.w[0] == midpoint64[q4 - 1]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
} else { // > 1/2 ulp
|
|
gt_half_ulp = 1;
|
|
}
|
|
} else if (q4 <= 38) {
|
|
if (C4.w[2] == 0 && (C4.w[1] < midpoint128[q4 - 20].w[1] ||
|
|
(C4.w[1] == midpoint128[q4 - 20].w[1] &&
|
|
C4.w[0] < midpoint128[q4 - 20].w[0]))) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
} else if (C4.w[2] == 0 && C4.w[1] == midpoint128[q4 - 20].w[1] &&
|
|
C4.w[0] == midpoint128[q4 - 20].w[0]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
} else { // > 1/2 ulp
|
|
gt_half_ulp = 1;
|
|
}
|
|
} else if (q4 <= 58) {
|
|
if (C4.w[3] == 0 && (C4.w[2] < midpoint192[q4 - 39].w[2] ||
|
|
(C4.w[2] == midpoint192[q4 - 39].w[2] &&
|
|
C4.w[1] < midpoint192[q4 - 39].w[1]) ||
|
|
(C4.w[2] == midpoint192[q4 - 39].w[2] &&
|
|
C4.w[1] == midpoint192[q4 - 39].w[1] &&
|
|
C4.w[0] < midpoint192[q4 - 39].w[0]))) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
} else if (C4.w[3] == 0 && C4.w[2] == midpoint192[q4 - 39].w[2] &&
|
|
C4.w[1] == midpoint192[q4 - 39].w[1] &&
|
|
C4.w[0] == midpoint192[q4 - 39].w[0]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
} else { // > 1/2 ulp
|
|
gt_half_ulp = 1;
|
|
}
|
|
} else {
|
|
if (C4.w[3] < midpoint256[q4 - 59].w[3] ||
|
|
(C4.w[3] == midpoint256[q4 - 59].w[3] &&
|
|
C4.w[2] < midpoint256[q4 - 59].w[2]) ||
|
|
(C4.w[3] == midpoint256[q4 - 59].w[3] &&
|
|
C4.w[2] == midpoint256[q4 - 59].w[2] &&
|
|
C4.w[1] < midpoint256[q4 - 59].w[1]) ||
|
|
(C4.w[3] == midpoint256[q4 - 59].w[3] &&
|
|
C4.w[2] == midpoint256[q4 - 59].w[2] &&
|
|
C4.w[1] == midpoint256[q4 - 59].w[1] &&
|
|
C4.w[0] < midpoint256[q4 - 59].w[0])) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
} else if (C4.w[3] == midpoint256[q4 - 59].w[3] &&
|
|
C4.w[2] == midpoint256[q4 - 59].w[2] &&
|
|
C4.w[1] == midpoint256[q4 - 59].w[1] &&
|
|
C4.w[0] == midpoint256[q4 - 59].w[0]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
} else { // > 1/2 ulp
|
|
gt_half_ulp = 1;
|
|
}
|
|
}
|
|
|
|
if (p_sign == z_sign) {
|
|
if (lt_half_ulp) {
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
// use the following to avoid double rounding errors when operating on
|
|
// mixed formats in rounding to nearest
|
|
is_inexact_lt_midpoint = 1; // if (z_sign), as if for absolute value
|
|
} else if ((eq_half_ulp && (res.w[0] & 0x01)) || gt_half_ulp) {
|
|
// add 1 ulp to the significand
|
|
res.w[0]++;
|
|
if (res.w[0] == 0x0ull)
|
|
res.w[1]++;
|
|
// check for rounding overflow, when coeff == 10^34
|
|
if ((res.w[1] & MASK_COEFF) == 0x0001ed09bead87c0ull &&
|
|
res.w[0] == 0x378d8e6400000000ull) { // coefficient = 10^34
|
|
e3 = e3 + 1;
|
|
// coeff = 10^33
|
|
z_exp = ((UINT64) (e3 + 6176) << 49) & MASK_EXP;
|
|
res.w[1] = 0x0000314dc6448d93ull;
|
|
res.w[0] = 0x38c15b0a00000000ull;
|
|
}
|
|
// end add 1 ulp
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
if (eq_half_ulp) {
|
|
is_midpoint_lt_even = 1; // if (z_sign), as if for absolute value
|
|
} else {
|
|
is_inexact_gt_midpoint = 1; // if (z_sign), as if for absolute value
|
|
}
|
|
} else { // if (eq_half_ulp && !(res.w[0] & 0x01))
|
|
// leave unchanged
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
is_midpoint_gt_even = 1; // if (z_sign), as if for absolute value
|
|
}
|
|
// the result is always inexact, and never tiny
|
|
// set the inexact flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
// check for overflow
|
|
if (e3 > expmax && rnd_mode == ROUNDING_TO_NEAREST) {
|
|
res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf
|
|
res.w[0] = 0x0000000000000000ull;
|
|
*pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e3, &res, pfpsf);
|
|
z_exp = res.w[1] & MASK_EXP;
|
|
}
|
|
} else { // if (p_sign != z_sign)
|
|
// consider two cases, because C3 * 10^scale = 10^33 is a special case
|
|
if (res.w[1] != 0x0000314dc6448d93ull ||
|
|
res.w[0] != 0x38c15b0a00000000ull) { // C3 * 10^scale != 10^33
|
|
if (lt_half_ulp) {
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
// use the following to avoid double rounding errors when operating
|
|
// on mixed formats in rounding to nearest
|
|
is_inexact_gt_midpoint = 1; // if (z_sign), as if for absolute value
|
|
} else if ((eq_half_ulp && (res.w[0] & 0x01)) || gt_half_ulp) {
|
|
// subtract 1 ulp from the significand
|
|
res.w[0]--;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
if (eq_half_ulp) {
|
|
is_midpoint_gt_even = 1; // if (z_sign), as if for absolute value
|
|
} else {
|
|
is_inexact_lt_midpoint = 1; //if(z_sign), as if for absolute value
|
|
}
|
|
} else { // if (eq_half_ulp && !(res.w[0] & 0x01))
|
|
// leave unchanged
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
is_midpoint_lt_even = 1; // if (z_sign), as if for absolute value
|
|
}
|
|
// the result is always inexact, and never tiny
|
|
// check for overflow for RN
|
|
if (e3 > expmax) {
|
|
if (rnd_mode == ROUNDING_TO_NEAREST) {
|
|
res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf
|
|
res.w[0] = 0x0000000000000000ull;
|
|
*pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
|
|
} else {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even,
|
|
is_midpoint_gt_even, e3, &res,
|
|
pfpsf);
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
// set the inexact flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even,
|
|
is_midpoint_gt_even, e3, &res, pfpsf);
|
|
}
|
|
z_exp = res.w[1] & MASK_EXP;
|
|
} else { // if C3 * 10^scale = 10^33
|
|
e3 = (z_exp >> 49) - 6176;
|
|
if (e3 > expmin) {
|
|
// the result is exact if exp > expmin and C4 = d*10^(q4-1),
|
|
// where d = 1, 2, 3, ..., 9; it could be tiny too, but exact
|
|
if (q4 == 1) {
|
|
// if q4 = 1 the result is exact
|
|
// result coefficient = 10^34 - C4
|
|
res.w[1] = 0x0001ed09bead87c0ull;
|
|
res.w[0] = 0x378d8e6400000000ull - C4.w[0];
|
|
z_exp = z_exp - EXP_P1;
|
|
e3 = e3 - 1;
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
} else {
|
|
// if q4 > 1 then truncate C4 from q4 digits to 1 digit;
|
|
// x = q4-1, 1 <= x <= 67 and check if this operation is exact
|
|
if (q4 <= 18) { // 2 <= q4 <= 18
|
|
round64_2_18 (q4, q4 - 1, C4.w[0], &R64, &incr_exp,
|
|
&is_midpoint_lt_even,
|
|
&is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
} else if (q4 <= 38) {
|
|
P128.w[1] = C4.w[1];
|
|
P128.w[0] = C4.w[0];
|
|
round128_19_38 (q4, q4 - 1, P128, &R128, &incr_exp,
|
|
&is_midpoint_lt_even,
|
|
&is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
R64 = R128.w[0]; // one decimal digit
|
|
} else if (q4 <= 57) {
|
|
P192.w[2] = C4.w[2];
|
|
P192.w[1] = C4.w[1];
|
|
P192.w[0] = C4.w[0];
|
|
round192_39_57 (q4, q4 - 1, P192, &R192, &incr_exp,
|
|
&is_midpoint_lt_even,
|
|
&is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
R64 = R192.w[0]; // one decimal digit
|
|
} else { // if (q4 <= 68)
|
|
round256_58_76 (q4, q4 - 1, C4, &R256, &incr_exp,
|
|
&is_midpoint_lt_even,
|
|
&is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
R64 = R256.w[0]; // one decimal digit
|
|
}
|
|
if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
// the result is exact: 10^34 - R64
|
|
// incr_exp = 0 with certainty
|
|
z_exp = z_exp - EXP_P1;
|
|
e3 = e3 - 1;
|
|
res.w[1] =
|
|
z_sign | (z_exp & MASK_EXP) | 0x0001ed09bead87c0ull;
|
|
res.w[0] = 0x378d8e6400000000ull - R64;
|
|
} else {
|
|
// We want R64 to be the top digit of C4, but we actually
|
|
// obtained (C4 * 10^(-q4+1))RN; a correction may be needed,
|
|
// because the top digit is (C4 * 10^(-q4+1))RZ
|
|
// however, if incr_exp = 1 then R64 = 10 with certainty
|
|
if (incr_exp) {
|
|
R64 = 10;
|
|
}
|
|
// the result is inexact as C4 has more than 1 significant digit
|
|
// and C3 * 10^scale = 10^33
|
|
// example of case that is treated here:
|
|
// 100...0 * 10^e3 - 0.41 * 10^e3 =
|
|
// 0999...9.59 * 10^e3 -> rounds to 99...96*10^(e3-1)
|
|
// note that (e3 > expmin}
|
|
// in order to round, subtract R64 from 10^34 and then compare
|
|
// C4 - R64 * 10^(q4-1) with 1/2 ulp
|
|
// calculate 10^34 - R64
|
|
res.w[1] = 0x0001ed09bead87c0ull;
|
|
res.w[0] = 0x378d8e6400000000ull - R64;
|
|
z_exp = z_exp - EXP_P1; // will be OR-ed with sign & significand
|
|
// calculate C4 - R64 * 10^(q4-1); this is a rare case and
|
|
// R64 is small, 1 <= R64 <= 9
|
|
e3 = e3 - 1;
|
|
if (is_inexact_lt_midpoint) {
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (is_inexact_gt_midpoint) {
|
|
is_inexact_gt_midpoint = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (is_midpoint_lt_even) {
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 1;
|
|
} else if (is_midpoint_gt_even) {
|
|
is_midpoint_gt_even = 0;
|
|
is_midpoint_lt_even = 1;
|
|
} else {
|
|
;
|
|
}
|
|
// the result is always inexact, and never tiny
|
|
// check for overflow for RN
|
|
if (e3 > expmax) {
|
|
if (rnd_mode == ROUNDING_TO_NEAREST) {
|
|
res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf
|
|
res.w[0] = 0x0000000000000000ull;
|
|
*pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
|
|
} else {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even,
|
|
is_midpoint_gt_even, e3, &res,
|
|
pfpsf);
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
// set the inexact flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
res.w[1] =
|
|
z_sign | ((UINT64) (e3 + 6176) << 49) | res.w[1];
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even,
|
|
is_midpoint_gt_even, e3, &res,
|
|
pfpsf);
|
|
}
|
|
z_exp = res.w[1] & MASK_EXP;
|
|
} // end result is inexact
|
|
} // end q4 > 1
|
|
} else { // if (e3 = emin)
|
|
// if e3 = expmin the result is also tiny (the condition for
|
|
// tininess is C4 > 050...0 [q4 digits] which is met because
|
|
// the msd of C4 is not zero)
|
|
// the result is tiny and inexact in all rounding modes;
|
|
// it is either 100...0 or 0999...9 (use lt_half_ulp, eq_half_ulp,
|
|
// gt_half_ulp to calculate)
|
|
// if (lt_half_ulp || eq_half_ulp) res = 10^33 stays unchanged
|
|
|
|
// p_sign != z_sign so swap gt_half_ulp and lt_half_ulp
|
|
if (gt_half_ulp) { // res = 10^33 - 1
|
|
res.w[1] = 0x0000314dc6448d93ull;
|
|
res.w[0] = 0x38c15b09ffffffffull;
|
|
} else {
|
|
res.w[1] = 0x0000314dc6448d93ull;
|
|
res.w[0] = 0x38c15b0a00000000ull;
|
|
}
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
*pfpsf |= UNDERFLOW_EXCEPTION; // inexact is set later
|
|
|
|
if (eq_half_ulp) {
|
|
is_midpoint_lt_even = 1; // if (z_sign), as if for absolute value
|
|
} else if (lt_half_ulp) {
|
|
is_inexact_gt_midpoint = 1; //if(z_sign), as if for absolute value
|
|
} else { // if (gt_half_ulp)
|
|
is_inexact_lt_midpoint = 1; //if(z_sign), as if for absolute value
|
|
}
|
|
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even,
|
|
is_midpoint_gt_even, e3, &res,
|
|
pfpsf);
|
|
z_exp = res.w[1] & MASK_EXP;
|
|
}
|
|
} // end e3 = emin
|
|
// set the inexact flag (if the result was not exact)
|
|
if (is_inexact_lt_midpoint || is_inexact_gt_midpoint ||
|
|
is_midpoint_lt_even || is_midpoint_gt_even)
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
} // end 10^33
|
|
} // end if (p_sign != z_sign)
|
|
res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1];
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
|
|
} else if (((q3 <= delta && delta < p34 && p34 < delta + q4) || // Case (2)
|
|
(q3 <= delta && delta + q4 <= p34) || // Case (3)
|
|
(delta < q3 && p34 < delta + q4) || // Case (4)
|
|
(delta < q3 && q3 <= delta + q4 && delta + q4 <= p34) || // Case (5)
|
|
(delta + q4 < q3)) && // Case (6)
|
|
!(delta <= 1 && p_sign != z_sign)) { // Case (2), (3), (4), (5) or (6)
|
|
|
|
// the result has the sign of z
|
|
|
|
if ((q3 <= delta && delta < p34 && p34 < delta + q4) || // Case (2)
|
|
(delta < q3 && p34 < delta + q4)) { // Case (4)
|
|
// round first the sum x * y + z with unbounded exponent
|
|
// scale C3 up by scale = p34 - q3, 1 <= scale <= p34-1,
|
|
// 1 <= scale <= 33
|
|
// calculate res = C3 * 10^scale
|
|
scale = p34 - q3;
|
|
x0 = delta + q4 - p34;
|
|
} else if (delta + q4 < q3) { // Case (6)
|
|
// make Case (6) look like Case (3) or Case (5) with scale = 0
|
|
// by scaling up C4 by 10^(q3 - delta - q4)
|
|
scale = q3 - delta - q4; // 1 <= scale <= 33
|
|
if (q4 <= 19) { // 1 <= scale <= 19; C4 fits in 64 bits
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
|
// 64 x 64 C4.w[0] * ten2k64[scale]
|
|
__mul_64x64_to_128MACH (P128, C4.w[0], ten2k64[scale]);
|
|
} else { // 10^scale fits in 128 bits
|
|
// 64 x 128 C4.w[0] * ten2k128[scale - 20]
|
|
__mul_128x64_to_128 (P128, C4.w[0], ten2k128[scale - 20]);
|
|
}
|
|
} else { // C4 fits in 128 bits, but 10^scale must fit in 64 bits
|
|
// 64 x 128 ten2k64[scale] * C4
|
|
__mul_128x64_to_128 (P128, ten2k64[scale], C4);
|
|
}
|
|
C4.w[0] = P128.w[0];
|
|
C4.w[1] = P128.w[1];
|
|
// e4 does not need adjustment, as it is not used from this point on
|
|
scale = 0;
|
|
x0 = 0;
|
|
// now Case (6) looks like Case (3) or Case (5) with scale = 0
|
|
} else { // if Case (3) or Case (5)
|
|
// Note: Case (3) is similar to Case (2), but scale differs and the
|
|
// result is exact, unless it is tiny (so x0 = 0 when calculating the
|
|
// result with unbounded exponent)
|
|
|
|
// calculate first the sum x * y + z with unbounded exponent (exact)
|
|
// scale C3 up by scale = delta + q4 - q3, 1 <= scale <= p34-1,
|
|
// 1 <= scale <= 33
|
|
// calculate res = C3 * 10^scale
|
|
scale = delta + q4 - q3;
|
|
x0 = 0;
|
|
// Note: the comments which follow refer [mainly] to Case (2)]
|
|
}
|
|
|
|
case2_repeat:
|
|
if (scale == 0) { // this could happen e.g. if we return to case2_repeat
|
|
// or in Case (4)
|
|
res.w[1] = C3.w[1];
|
|
res.w[0] = C3.w[0];
|
|
} else if (q3 <= 19) { // 1 <= scale <= 19; z fits in 64 bits
|
|
if (scale <= 19) { // 10^scale fits in 64 bits
|
|
// 64 x 64 C3.w[0] * ten2k64[scale]
|
|
__mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]);
|
|
} else { // 10^scale fits in 128 bits
|
|
// 64 x 128 C3.w[0] * ten2k128[scale - 20]
|
|
__mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]);
|
|
}
|
|
} else { // z fits in 128 bits, but 10^scale must fit in 64 bits
|
|
// 64 x 128 ten2k64[scale] * C3
|
|
__mul_128x64_to_128 (res, ten2k64[scale], C3);
|
|
}
|
|
// e3 is already calculated
|
|
e3 = e3 - scale;
|
|
// now res = C3 * 10^scale and e3 = e3 - scale
|
|
// Note: C3 * 10^scale could be 10^34 if we returned to case2_repeat
|
|
// because the result was too small
|
|
|
|
// round C4 to nearest to q4 - x0 digits, where x0 = delta + q4 - p34,
|
|
// 1 <= x0 <= min (q4 - 1, 2 * p34 - 1) <=> 1 <= x0 <= min (q4 - 1, 67)
|
|
// Also: 1 <= q4 - x0 <= p34 -1 => 1 <= q4 - x0 <= 33 (so the result of
|
|
// the rounding fits in 128 bits!)
|
|
// x0 = delta + q4 - p34 (calculated before reaching case2_repeat)
|
|
// because q3 + q4 - x0 <= P => x0 >= q3 + q4 - p34
|
|
if (x0 == 0) { // this could happen only if we return to case2_repeat, or
|
|
// for Case (3) or Case (6)
|
|
R128.w[1] = C4.w[1];
|
|
R128.w[0] = C4.w[0];
|
|
} else if (q4 <= 18) {
|
|
// 2 <= q4 <= 18, max(1, q3+q4-p34) <= x0 <= q4 - 1, 1 <= x0 <= 17
|
|
round64_2_18 (q4, x0, C4.w[0], &R64, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint);
|
|
if (incr_exp) {
|
|
// R64 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 1, 1 <= q4 - x0 <= 17
|
|
R64 = ten2k64[q4 - x0];
|
|
}
|
|
R128.w[1] = 0;
|
|
R128.w[0] = R64;
|
|
} else if (q4 <= 38) {
|
|
// 19 <= q4 <= 38, max(1, q3+q4-p34) <= x0 <= q4 - 1, 1 <= x0 <= 37
|
|
P128.w[1] = C4.w[1];
|
|
P128.w[0] = C4.w[0];
|
|
round128_19_38 (q4, x0, P128, &R128, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
if (incr_exp) {
|
|
// R128 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 1, 1 <= q4 - x0 <= 37
|
|
if (q4 - x0 <= 19) { // 1 <= q4 - x0 <= 19
|
|
R128.w[0] = ten2k64[q4 - x0];
|
|
// R128.w[1] stays 0
|
|
} else { // 20 <= q4 - x0 <= 37
|
|
R128.w[0] = ten2k128[q4 - x0 - 20].w[0];
|
|
R128.w[1] = ten2k128[q4 - x0 - 20].w[1];
|
|
}
|
|
}
|
|
} else if (q4 <= 57) {
|
|
// 38 <= q4 <= 57, max(1, q3+q4-p34) <= x0 <= q4 - 1, 5 <= x0 <= 56
|
|
P192.w[2] = C4.w[2];
|
|
P192.w[1] = C4.w[1];
|
|
P192.w[0] = C4.w[0];
|
|
round192_39_57 (q4, x0, P192, &R192, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
// R192.w[2] is always 0
|
|
if (incr_exp) {
|
|
// R192 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 5, 1 <= q4 - x0 <= 52
|
|
if (q4 - x0 <= 19) { // 1 <= q4 - x0 <= 19
|
|
R192.w[0] = ten2k64[q4 - x0];
|
|
// R192.w[1] stays 0
|
|
// R192.w[2] stays 0
|
|
} else { // 20 <= q4 - x0 <= 33
|
|
R192.w[0] = ten2k128[q4 - x0 - 20].w[0];
|
|
R192.w[1] = ten2k128[q4 - x0 - 20].w[1];
|
|
// R192.w[2] stays 0
|
|
}
|
|
}
|
|
R128.w[1] = R192.w[1];
|
|
R128.w[0] = R192.w[0];
|
|
} else {
|
|
// 58 <= q4 <= 68, max(1, q3+q4-p34) <= x0 <= q4 - 1, 25 <= x0 <= 67
|
|
round256_58_76 (q4, x0, C4, &R256, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
// R256.w[3] and R256.w[2] are always 0
|
|
if (incr_exp) {
|
|
// R256 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 25, 1 <= q4 - x0 <= 43
|
|
if (q4 - x0 <= 19) { // 1 <= q4 - x0 <= 19
|
|
R256.w[0] = ten2k64[q4 - x0];
|
|
// R256.w[1] stays 0
|
|
// R256.w[2] stays 0
|
|
// R256.w[3] stays 0
|
|
} else { // 20 <= q4 - x0 <= 33
|
|
R256.w[0] = ten2k128[q4 - x0 - 20].w[0];
|
|
R256.w[1] = ten2k128[q4 - x0 - 20].w[1];
|
|
// R256.w[2] stays 0
|
|
// R256.w[3] stays 0
|
|
}
|
|
}
|
|
R128.w[1] = R256.w[1];
|
|
R128.w[0] = R256.w[0];
|
|
}
|
|
// now add C3 * 10^scale in res and the signed top (q4-x0) digits of C4,
|
|
// rounded to nearest, which were copied into R128
|
|
if (z_sign == p_sign) {
|
|
lsb = res.w[0] & 0x01; // lsb of C3 * 10^scale
|
|
// the sum can result in [up to] p34 or p34 + 1 digits
|
|
res.w[0] = res.w[0] + R128.w[0];
|
|
res.w[1] = res.w[1] + R128.w[1];
|
|
if (res.w[0] < R128.w[0])
|
|
res.w[1]++; // carry
|
|
// if res > 10^34 - 1 need to increase x0 and decrease scale by 1
|
|
if (res.w[1] > 0x0001ed09bead87c0ull ||
|
|
(res.w[1] == 0x0001ed09bead87c0ull &&
|
|
res.w[0] > 0x378d8e63ffffffffull)) {
|
|
// avoid double rounding error
|
|
is_inexact_lt_midpoint0 = is_inexact_lt_midpoint;
|
|
is_inexact_gt_midpoint0 = is_inexact_gt_midpoint;
|
|
is_midpoint_lt_even0 = is_midpoint_lt_even;
|
|
is_midpoint_gt_even0 = is_midpoint_gt_even;
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
P128.w[1] = res.w[1];
|
|
P128.w[0] = res.w[0];
|
|
round128_19_38 (35, 1, P128, &res, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
// incr_exp is 0 with certainty in this case
|
|
// avoid a double rounding error
|
|
if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) &&
|
|
is_midpoint_lt_even) { // double rounding error upward
|
|
// res = res - 1
|
|
res.w[0]--;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
// Note: a double rounding error upward is not possible; for this
|
|
// the result after the first rounding would have to be 99...95
|
|
// (35 digits in all), possibly followed by a number of zeros; this
|
|
// not possible in Cases (2)-(6) or (15)-(17) which may get here
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) &&
|
|
is_midpoint_gt_even) { // double rounding error downward
|
|
// res = res + 1
|
|
res.w[0]++;
|
|
if (res.w[0] == 0)
|
|
res.w[1]++;
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint
|
|
&& !is_inexact_gt_midpoint) {
|
|
// if this second rounding was exact the result may still be
|
|
// inexact because of the first rounding
|
|
if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) {
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else if (is_midpoint_gt_even &&
|
|
(is_inexact_gt_midpoint0
|
|
|| is_midpoint_lt_even0)) {
|
|
// pulled up to a midpoint
|
|
is_inexact_lt_midpoint = 1;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else if (is_midpoint_lt_even &&
|
|
(is_inexact_lt_midpoint0
|
|
|| is_midpoint_gt_even0)) {
|
|
// pulled down to a midpoint
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else {
|
|
;
|
|
}
|
|
// adjust exponent
|
|
e3 = e3 + 1;
|
|
if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
if (is_midpoint_lt_even0 || is_midpoint_gt_even0 ||
|
|
is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
}
|
|
} else {
|
|
// this is the result rounded with unbounded exponent, unless a
|
|
// correction is needed
|
|
res.w[1] = res.w[1] & MASK_COEFF;
|
|
if (lsb == 1) {
|
|
if (is_midpoint_gt_even) {
|
|
// res = res + 1
|
|
is_midpoint_gt_even = 0;
|
|
is_midpoint_lt_even = 1;
|
|
res.w[0]++;
|
|
if (res.w[0] == 0x0)
|
|
res.w[1]++;
|
|
// check for rounding overflow
|
|
if (res.w[1] == 0x0001ed09bead87c0ull &&
|
|
res.w[0] == 0x378d8e6400000000ull) {
|
|
// res = 10^34 => rounding overflow
|
|
res.w[1] = 0x0000314dc6448d93ull;
|
|
res.w[0] = 0x38c15b0a00000000ull; // 10^33
|
|
e3++;
|
|
}
|
|
} else if (is_midpoint_lt_even) {
|
|
// res = res - 1
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 1;
|
|
res.w[0]--;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
// if the result is pure zero, the sign depends on the rounding
|
|
// mode (x*y and z had opposite signs)
|
|
if (res.w[1] == 0x0ull && res.w[0] == 0x0ull) {
|
|
if (rnd_mode != ROUNDING_DOWN)
|
|
z_sign = 0x0000000000000000ull;
|
|
else
|
|
z_sign = 0x8000000000000000ull;
|
|
// the exponent is max (e3, expmin)
|
|
res.w[1] = 0x0;
|
|
res.w[0] = 0x0;
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
} else {
|
|
;
|
|
}
|
|
}
|
|
}
|
|
} else { // if (z_sign != p_sign)
|
|
lsb = res.w[0] & 0x01; // lsb of C3 * 10^scale; R128 contains rounded C4
|
|
// used to swap rounding indicators if p_sign != z_sign
|
|
// the sum can result in [up to] p34 or p34 - 1 digits
|
|
tmp64 = res.w[0];
|
|
res.w[0] = res.w[0] - R128.w[0];
|
|
res.w[1] = res.w[1] - R128.w[1];
|
|
if (res.w[0] > tmp64)
|
|
res.w[1]--; // borrow
|
|
// if res < 10^33 and exp > expmin need to decrease x0 and
|
|
// increase scale by 1
|
|
if (e3 > expmin && ((res.w[1] < 0x0000314dc6448d93ull ||
|
|
(res.w[1] == 0x0000314dc6448d93ull &&
|
|
res.w[0] < 0x38c15b0a00000000ull)) ||
|
|
(is_inexact_lt_midpoint
|
|
&& res.w[1] == 0x0000314dc6448d93ull
|
|
&& res.w[0] == 0x38c15b0a00000000ull))
|
|
&& x0 >= 1) {
|
|
x0 = x0 - 1;
|
|
// first restore e3, otherwise it will be too small
|
|
e3 = e3 + scale;
|
|
scale = scale + 1;
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
incr_exp = 0;
|
|
goto case2_repeat;
|
|
}
|
|
// else this is the result rounded with unbounded exponent;
|
|
// because the result has opposite sign to that of C4 which was
|
|
// rounded, need to change the rounding indicators
|
|
if (is_inexact_lt_midpoint) {
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (is_inexact_gt_midpoint) {
|
|
is_inexact_gt_midpoint = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (lsb == 0) {
|
|
if (is_midpoint_lt_even) {
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 1;
|
|
} else if (is_midpoint_gt_even) {
|
|
is_midpoint_gt_even = 0;
|
|
is_midpoint_lt_even = 1;
|
|
} else {
|
|
;
|
|
}
|
|
} else if (lsb == 1) {
|
|
if (is_midpoint_lt_even) {
|
|
// res = res + 1
|
|
res.w[0]++;
|
|
if (res.w[0] == 0x0)
|
|
res.w[1]++;
|
|
// check for rounding overflow
|
|
if (res.w[1] == 0x0001ed09bead87c0ull &&
|
|
res.w[0] == 0x378d8e6400000000ull) {
|
|
// res = 10^34 => rounding overflow
|
|
res.w[1] = 0x0000314dc6448d93ull;
|
|
res.w[0] = 0x38c15b0a00000000ull; // 10^33
|
|
e3++;
|
|
}
|
|
} else if (is_midpoint_gt_even) {
|
|
// res = res - 1
|
|
res.w[0]--;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
// if the result is pure zero, the sign depends on the rounding
|
|
// mode (x*y and z had opposite signs)
|
|
if (res.w[1] == 0x0ull && res.w[0] == 0x0ull) {
|
|
if (rnd_mode != ROUNDING_DOWN)
|
|
z_sign = 0x0000000000000000ull;
|
|
else
|
|
z_sign = 0x8000000000000000ull;
|
|
// the exponent is max (e3, expmin)
|
|
res.w[1] = 0x0;
|
|
res.w[0] = 0x0;
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
}
|
|
} else {
|
|
;
|
|
}
|
|
} else {
|
|
;
|
|
}
|
|
}
|
|
// check for underflow
|
|
if (e3 == expmin) { // and if significand < 10^33 => result is tiny
|
|
if ((res.w[1] & MASK_COEFF) < 0x0000314dc6448d93ull ||
|
|
((res.w[1] & MASK_COEFF) == 0x0000314dc6448d93ull &&
|
|
res.w[0] < 0x38c15b0a00000000ull)) {
|
|
is_tiny = 1;
|
|
}
|
|
} else if (e3 < expmin) {
|
|
// the result is tiny, so we must truncate more of res
|
|
is_tiny = 1;
|
|
x0 = expmin - e3;
|
|
is_inexact_lt_midpoint0 = is_inexact_lt_midpoint;
|
|
is_inexact_gt_midpoint0 = is_inexact_gt_midpoint;
|
|
is_midpoint_lt_even0 = is_midpoint_lt_even;
|
|
is_midpoint_gt_even0 = is_midpoint_gt_even;
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
// determine the number of decimal digits in res
|
|
if (res.w[1] == 0x0) {
|
|
// between 1 and 19 digits
|
|
for (ind = 1; ind <= 19; ind++) {
|
|
if (res.w[0] < ten2k64[ind]) {
|
|
break;
|
|
}
|
|
}
|
|
// ind digits
|
|
} else if (res.w[1] < ten2k128[0].w[1] ||
|
|
(res.w[1] == ten2k128[0].w[1]
|
|
&& res.w[0] < ten2k128[0].w[0])) {
|
|
// 20 digits
|
|
ind = 20;
|
|
} else { // between 21 and 38 digits
|
|
for (ind = 1; ind <= 18; ind++) {
|
|
if (res.w[1] < ten2k128[ind].w[1] ||
|
|
(res.w[1] == ten2k128[ind].w[1] &&
|
|
res.w[0] < ten2k128[ind].w[0])) {
|
|
break;
|
|
}
|
|
}
|
|
// ind + 20 digits
|
|
ind = ind + 20;
|
|
}
|
|
|
|
// at this point ind >= x0; because delta >= 2 on this path, the case
|
|
// ind = x0 can occur only in Case (2) or case (3), when C3 has one
|
|
// digit (q3 = 1) equal to 1 (C3 = 1), e3 is expmin (e3 = expmin),
|
|
// the signs of x * y and z are opposite, and through cancellation
|
|
// the most significant decimal digit in res has the weight
|
|
// 10^(emin-1); however, it is clear that in this case the most
|
|
// significant digit is 9, so the result before rounding is
|
|
// 0.9... * 10^emin
|
|
// Otherwise, ind > x0 because there are non-zero decimal digits in the
|
|
// result with weight of at least 10^emin, and correction for underflow
|
|
// can be carried out using the round*_*_2_* () routines
|
|
if (x0 == ind) { // the result before rounding is 0.9... * 10^emin
|
|
res.w[1] = 0x0;
|
|
res.w[0] = 0x1;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (ind <= 18) { // check that 2 <= ind
|
|
// 2 <= ind <= 18, 1 <= x0 <= 17
|
|
round64_2_18 (ind, x0, res.w[0], &R64, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
if (incr_exp) {
|
|
// R64 = 10^(ind-x0), 1 <= ind - x0 <= ind - 1, 1 <= ind - x0 <= 17
|
|
R64 = ten2k64[ind - x0];
|
|
}
|
|
res.w[1] = 0;
|
|
res.w[0] = R64;
|
|
} else if (ind <= 38) {
|
|
// 19 <= ind <= 38
|
|
P128.w[1] = res.w[1];
|
|
P128.w[0] = res.w[0];
|
|
round128_19_38 (ind, x0, P128, &res, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
if (incr_exp) {
|
|
// R128 = 10^(ind-x0), 1 <= ind - x0 <= ind - 1, 1 <= ind - x0 <= 37
|
|
if (ind - x0 <= 19) { // 1 <= ind - x0 <= 19
|
|
res.w[0] = ten2k64[ind - x0];
|
|
// res.w[1] stays 0
|
|
} else { // 20 <= ind - x0 <= 37
|
|
res.w[0] = ten2k128[ind - x0 - 20].w[0];
|
|
res.w[1] = ten2k128[ind - x0 - 20].w[1];
|
|
}
|
|
}
|
|
}
|
|
// avoid a double rounding error
|
|
if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) &&
|
|
is_midpoint_lt_even) { // double rounding error upward
|
|
// res = res - 1
|
|
res.w[0]--;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
// Note: a double rounding error upward is not possible; for this
|
|
// the result after the first rounding would have to be 99...95
|
|
// (35 digits in all), possibly followed by a number of zeros; this
|
|
// not possible in Cases (2)-(6) which may get here
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) &&
|
|
is_midpoint_gt_even) { // double rounding error downward
|
|
// res = res + 1
|
|
res.w[0]++;
|
|
if (res.w[0] == 0)
|
|
res.w[1]++;
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
// if this second rounding was exact the result may still be
|
|
// inexact because of the first rounding
|
|
if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) {
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else if (is_midpoint_gt_even &&
|
|
(is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) {
|
|
// pulled up to a midpoint
|
|
is_inexact_lt_midpoint = 1;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else if (is_midpoint_lt_even &&
|
|
(is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) {
|
|
// pulled down to a midpoint
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else {
|
|
;
|
|
}
|
|
// adjust exponent
|
|
e3 = e3 + x0;
|
|
if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
if (is_midpoint_lt_even0 || is_midpoint_gt_even0 ||
|
|
is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
}
|
|
} else {
|
|
; // not underflow
|
|
}
|
|
// check for inexact result
|
|
if (is_inexact_lt_midpoint || is_inexact_gt_midpoint ||
|
|
is_midpoint_lt_even || is_midpoint_gt_even) {
|
|
// set the inexact flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
if (is_tiny)
|
|
*pfpsf |= UNDERFLOW_EXCEPTION;
|
|
}
|
|
// now check for significand = 10^34 (may have resulted from going
|
|
// back to case2_repeat)
|
|
if (res.w[1] == 0x0001ed09bead87c0ull &&
|
|
res.w[0] == 0x378d8e6400000000ull) { // if res = 10^34
|
|
res.w[1] = 0x0000314dc6448d93ull; // res = 10^33
|
|
res.w[0] = 0x38c15b0a00000000ull;
|
|
e3 = e3 + 1;
|
|
}
|
|
res.w[1] = z_sign | ((UINT64) (e3 + 6176) << 49) | res.w[1];
|
|
// check for overflow
|
|
if (rnd_mode == ROUNDING_TO_NEAREST && e3 > expmax) {
|
|
res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf
|
|
res.w[0] = 0x0000000000000000ull;
|
|
*pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
|
|
}
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e3, &res, pfpsf);
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
|
|
} else {
|
|
|
|
// we get here only if delta <= 1 in Cases (2), (3), (4), (5), or (6) and
|
|
// the signs of x*y and z are opposite; in these cases massive
|
|
// cancellation can occur, so it is better to scale either C3 or C4 and
|
|
// to perform the subtraction before rounding; rounding is performed
|
|
// next, depending on the number of decimal digits in the result and on
|
|
// the exponent value
|
|
// Note: overlow is not possible in this case
|
|
// this is similar to Cases (15), (16), and (17)
|
|
|
|
if (delta + q4 < q3) { // from Case (6)
|
|
// Case (6) with 0<= delta <= 1 is similar to Cases (15), (16), and
|
|
// (17) if we swap (C3, C4), (q3, q4), (e3, e4), (z_sign, p_sign)
|
|
// and call add_and_round; delta stays positive
|
|
// C4.w[3] = 0 and C4.w[2] = 0, so swap just the low part of C4 with C3
|
|
P128.w[1] = C3.w[1];
|
|
P128.w[0] = C3.w[0];
|
|
C3.w[1] = C4.w[1];
|
|
C3.w[0] = C4.w[0];
|
|
C4.w[1] = P128.w[1];
|
|
C4.w[0] = P128.w[0];
|
|
ind = q3;
|
|
q3 = q4;
|
|
q4 = ind;
|
|
ind = e3;
|
|
e3 = e4;
|
|
e4 = ind;
|
|
tmp_sign = z_sign;
|
|
z_sign = p_sign;
|
|
p_sign = tmp_sign;
|
|
} else { // from Cases (2), (3), (4), (5)
|
|
// In Cases (2), (3), (4), (5) with 0 <= delta <= 1 C3 has to be
|
|
// scaled up by q4 + delta - q3; this is the same as in Cases (15),
|
|
// (16), and (17) if we just change the sign of delta
|
|
delta = -delta;
|
|
}
|
|
add_and_round (q3, q4, e4, delta, p34, z_sign, p_sign, C3, C4,
|
|
rnd_mode, &is_midpoint_lt_even,
|
|
&is_midpoint_gt_even, &is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, pfpsf, &res);
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
|
|
}
|
|
|
|
} else { // if delta < 0
|
|
|
|
delta = -delta;
|
|
|
|
if (p34 < q4 && q4 <= delta) { // Case (7)
|
|
|
|
// truncate C4 to p34 digits into res
|
|
// x = q4-p34, 1 <= x <= 34 because 35 <= q4 <= 68
|
|
x0 = q4 - p34;
|
|
if (q4 <= 38) {
|
|
P128.w[1] = C4.w[1];
|
|
P128.w[0] = C4.w[0];
|
|
round128_19_38 (q4, x0, P128, &res, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
} else if (q4 <= 57) { // 35 <= q4 <= 57
|
|
P192.w[2] = C4.w[2];
|
|
P192.w[1] = C4.w[1];
|
|
P192.w[0] = C4.w[0];
|
|
round192_39_57 (q4, x0, P192, &R192, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
res.w[0] = R192.w[0];
|
|
res.w[1] = R192.w[1];
|
|
} else { // if (q4 <= 68)
|
|
round256_58_76 (q4, x0, C4, &R256, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
res.w[0] = R256.w[0];
|
|
res.w[1] = R256.w[1];
|
|
}
|
|
e4 = e4 + x0;
|
|
if (incr_exp) {
|
|
e4 = e4 + 1;
|
|
}
|
|
if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
// if C4 rounded to p34 digits is exact then the result is inexact,
|
|
// in a way that depends on the signs of x * y and z
|
|
if (p_sign == z_sign) {
|
|
is_inexact_lt_midpoint = 1;
|
|
} else { // if (p_sign != z_sign)
|
|
if (res.w[1] != 0x0000314dc6448d93ull ||
|
|
res.w[0] != 0x38c15b0a00000000ull) { // res != 10^33
|
|
is_inexact_gt_midpoint = 1;
|
|
} else { // res = 10^33 and exact is a special case
|
|
// if C3 < 1/2 ulp then res = 10^33 and is_inexact_gt_midpoint = 1
|
|
// if C3 = 1/2 ulp then res = 10^33 and is_midpoint_lt_even = 1
|
|
// if C3 > 1/2 ulp then res = 10^34-1 and is_inexact_lt_midpoint = 1
|
|
// Note: ulp is really ulp/10 (after borrow which propagates to msd)
|
|
if (delta > p34 + 1) { // C3 < 1/2
|
|
// res = 10^33, unchanged
|
|
is_inexact_gt_midpoint = 1;
|
|
} else { // if (delta == p34 + 1)
|
|
if (q3 <= 19) {
|
|
if (C3.w[0] < midpoint64[q3 - 1]) { // C3 < 1/2 ulp
|
|
// res = 10^33, unchanged
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (C3.w[0] == midpoint64[q3 - 1]) { // C3 = 1/2 ulp
|
|
// res = 10^33, unchanged
|
|
is_midpoint_lt_even = 1;
|
|
} else { // if (C3.w[0] > midpoint64[q3-1]), C3 > 1/2 ulp
|
|
res.w[1] = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
res.w[0] = 0x378d8e63ffffffffull;
|
|
e4 = e4 - 1;
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else { // if (20 <= q3 <=34)
|
|
if (C3.w[1] < midpoint128[q3 - 20].w[1] ||
|
|
(C3.w[1] == midpoint128[q3 - 20].w[1] &&
|
|
C3.w[0] < midpoint128[q3 - 20].w[0])) { // C3 < 1/2 ulp
|
|
// res = 10^33, unchanged
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (C3.w[1] == midpoint128[q3 - 20].w[1] &&
|
|
C3.w[0] == midpoint128[q3 - 20].w[0]) { // C3 = 1/2 ulp
|
|
// res = 10^33, unchanged
|
|
is_midpoint_lt_even = 1;
|
|
} else { // if (C3 > midpoint128[q3-20]), C3 > 1/2 ulp
|
|
res.w[1] = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
res.w[0] = 0x378d8e63ffffffffull;
|
|
e4 = e4 - 1;
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
} else if (is_midpoint_lt_even) {
|
|
if (z_sign != p_sign) {
|
|
// needs correction: res = res - 1
|
|
res.w[0] = res.w[0] - 1;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
// if it is (10^33-1)*10^e4 then the corect result is
|
|
// (10^34-1)*10(e4-1)
|
|
if (res.w[1] == 0x0000314dc6448d93ull &&
|
|
res.w[0] == 0x38c15b09ffffffffull) {
|
|
res.w[1] = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
res.w[0] = 0x378d8e63ffffffffull;
|
|
e4 = e4 - 1;
|
|
}
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else { // if (z_sign == p_sign)
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
} else if (is_midpoint_gt_even) {
|
|
if (z_sign == p_sign) {
|
|
// needs correction: res = res + 1 (cannot cross in the next binade)
|
|
res.w[0] = res.w[0] + 1;
|
|
if (res.w[0] == 0x0000000000000000ull)
|
|
res.w[1]++;
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else { // if (z_sign != p_sign)
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else {
|
|
; // the rounded result is already correct
|
|
}
|
|
// check for overflow
|
|
if (rnd_mode == ROUNDING_TO_NEAREST && e4 > expmax) {
|
|
res.w[1] = p_sign | 0x7800000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
*pfpsf |= (OVERFLOW_EXCEPTION | INEXACT_EXCEPTION);
|
|
} else { // no overflow or not RN
|
|
p_exp = ((UINT64) (e4 + 6176) << 49);
|
|
res.w[1] = p_sign | (p_exp & MASK_EXP) | res.w[1];
|
|
}
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e4, &res, pfpsf);
|
|
}
|
|
if (is_inexact_lt_midpoint || is_inexact_gt_midpoint ||
|
|
is_midpoint_lt_even || is_midpoint_gt_even) {
|
|
// set the inexact flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
|
|
} else if ((q4 <= p34 && p34 <= delta) || // Case (8)
|
|
(q4 <= delta && delta < p34 && p34 < delta + q3) || // Case (9)
|
|
(q4 <= delta && delta + q3 <= p34) || // Case (10)
|
|
(delta < q4 && q4 <= p34 && p34 < delta + q3) || // Case (13)
|
|
(delta < q4 && q4 <= delta + q3 && delta + q3 <= p34) || // Case (14)
|
|
(delta + q3 < q4 && q4 <= p34)) { // Case (18)
|
|
|
|
// Case (8) is similar to Case (1), with C3 and C4 swapped
|
|
// Case (9) is similar to Case (2), with C3 and C4 swapped
|
|
// Case (10) is similar to Case (3), with C3 and C4 swapped
|
|
// Case (13) is similar to Case (4), with C3 and C4 swapped
|
|
// Case (14) is similar to Case (5), with C3 and C4 swapped
|
|
// Case (18) is similar to Case (6), with C3 and C4 swapped
|
|
|
|
// swap (C3, C4), (q3, q4), (e3, 34), (z_sign, p_sign), (z_exp, p_exp)
|
|
// and go back to delta_ge_zero
|
|
// C4.w[3] = 0 and C4.w[2] = 0, so swap just the low part of C4 with C3
|
|
P128.w[1] = C3.w[1];
|
|
P128.w[0] = C3.w[0];
|
|
C3.w[1] = C4.w[1];
|
|
C3.w[0] = C4.w[0];
|
|
C4.w[1] = P128.w[1];
|
|
C4.w[0] = P128.w[0];
|
|
ind = q3;
|
|
q3 = q4;
|
|
q4 = ind;
|
|
ind = e3;
|
|
e3 = e4;
|
|
e4 = ind;
|
|
tmp_sign = z_sign;
|
|
z_sign = p_sign;
|
|
p_sign = tmp_sign;
|
|
tmp.ui64 = z_exp;
|
|
z_exp = p_exp;
|
|
p_exp = tmp.ui64;
|
|
goto delta_ge_zero;
|
|
|
|
} else if ((p34 <= delta && delta < q4 && q4 < delta + q3) || // Case (11)
|
|
(delta < p34 && p34 < q4 && q4 < delta + q3)) { // Case (12)
|
|
|
|
// round C3 to nearest to q3 - x0 digits, where x0 = e4 - e3,
|
|
// 1 <= x0 <= q3 - 1 <= p34 - 1
|
|
x0 = e4 - e3; // or x0 = delta + q3 - q4
|
|
if (q3 <= 18) { // 2 <= q3 <= 18
|
|
round64_2_18 (q3, x0, C3.w[0], &R64, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint);
|
|
// C3.w[1] = 0;
|
|
C3.w[0] = R64;
|
|
} else if (q3 <= 38) {
|
|
round128_19_38 (q3, x0, C3, &R128, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
C3.w[1] = R128.w[1];
|
|
C3.w[0] = R128.w[0];
|
|
}
|
|
// the rounded result has q3 - x0 digits
|
|
// we want the exponent to be e4, so if incr_exp = 1 then
|
|
// multiply the rounded result by 10 - it will still fit in 113 bits
|
|
if (incr_exp) {
|
|
// 64 x 128 -> 128
|
|
P128.w[1] = C3.w[1];
|
|
P128.w[0] = C3.w[0];
|
|
__mul_64x128_to_128 (C3, ten2k64[1], P128);
|
|
}
|
|
e3 = e3 + x0; // this is e4
|
|
// now add/subtract the 256-bit C4 and the new (and shorter) 128-bit C3;
|
|
// the result will have the sign of x * y; the exponent is e4
|
|
R256.w[3] = 0;
|
|
R256.w[2] = 0;
|
|
R256.w[1] = C3.w[1];
|
|
R256.w[0] = C3.w[0];
|
|
if (p_sign == z_sign) { // R256 = C4 + R256
|
|
add256 (C4, R256, &R256);
|
|
} else { // if (p_sign != z_sign) { // R256 = C4 - R256
|
|
sub256 (C4, R256, &R256); // the result cannot be pure zero
|
|
// because the result has opposite sign to that of R256 which was
|
|
// rounded, need to change the rounding indicators
|
|
lsb = C4.w[0] & 0x01;
|
|
if (is_inexact_lt_midpoint) {
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (is_inexact_gt_midpoint) {
|
|
is_inexact_gt_midpoint = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (lsb == 0) {
|
|
if (is_midpoint_lt_even) {
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 1;
|
|
} else if (is_midpoint_gt_even) {
|
|
is_midpoint_gt_even = 0;
|
|
is_midpoint_lt_even = 1;
|
|
} else {
|
|
;
|
|
}
|
|
} else if (lsb == 1) {
|
|
if (is_midpoint_lt_even) {
|
|
// res = res + 1
|
|
R256.w[0]++;
|
|
if (R256.w[0] == 0x0) {
|
|
R256.w[1]++;
|
|
if (R256.w[1] == 0x0) {
|
|
R256.w[2]++;
|
|
if (R256.w[2] == 0x0) {
|
|
R256.w[3]++;
|
|
}
|
|
}
|
|
}
|
|
// no check for rounding overflow - R256 was a difference
|
|
} else if (is_midpoint_gt_even) {
|
|
// res = res - 1
|
|
R256.w[0]--;
|
|
if (R256.w[0] == 0xffffffffffffffffull) {
|
|
R256.w[1]--;
|
|
if (R256.w[1] == 0xffffffffffffffffull) {
|
|
R256.w[2]--;
|
|
if (R256.w[2] == 0xffffffffffffffffull) {
|
|
R256.w[3]--;
|
|
}
|
|
}
|
|
}
|
|
} else {
|
|
;
|
|
}
|
|
} else {
|
|
;
|
|
}
|
|
}
|
|
// determine the number of decimal digits in R256
|
|
ind = nr_digits256 (R256); // ind >= p34
|
|
// if R256 is sum, then ind > p34; if R256 is a difference, then
|
|
// ind >= p34; this means that we can calculate the result rounded to
|
|
// the destination precision, with unbounded exponent, starting from R256
|
|
// and using the indicators from the rounding of C3 to avoid a double
|
|
// rounding error
|
|
|
|
if (ind < p34) {
|
|
;
|
|
} else if (ind == p34) {
|
|
// the result rounded to the destination precision with
|
|
// unbounded exponent
|
|
// is (-1)^p_sign * R256 * 10^e4
|
|
res.w[1] = R256.w[1];
|
|
res.w[0] = R256.w[0];
|
|
} else { // if (ind > p34)
|
|
// if more than P digits, round to nearest to P digits
|
|
// round R256 to p34 digits
|
|
x0 = ind - p34; // 1 <= x0 <= 34 as 35 <= ind <= 68
|
|
// save C3 rounding indicators to help avoid double rounding error
|
|
is_inexact_lt_midpoint0 = is_inexact_lt_midpoint;
|
|
is_inexact_gt_midpoint0 = is_inexact_gt_midpoint;
|
|
is_midpoint_lt_even0 = is_midpoint_lt_even;
|
|
is_midpoint_gt_even0 = is_midpoint_gt_even;
|
|
// initialize rounding indicators
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
// round to p34 digits; the result fits in 113 bits
|
|
if (ind <= 38) {
|
|
P128.w[1] = R256.w[1];
|
|
P128.w[0] = R256.w[0];
|
|
round128_19_38 (ind, x0, P128, &R128, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
} else if (ind <= 57) {
|
|
P192.w[2] = R256.w[2];
|
|
P192.w[1] = R256.w[1];
|
|
P192.w[0] = R256.w[0];
|
|
round192_39_57 (ind, x0, P192, &R192, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
R128.w[1] = R192.w[1];
|
|
R128.w[0] = R192.w[0];
|
|
} else { // if (ind <= 68)
|
|
round256_58_76 (ind, x0, R256, &R256, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
R128.w[1] = R256.w[1];
|
|
R128.w[0] = R256.w[0];
|
|
}
|
|
// the rounded result has p34 = 34 digits
|
|
e4 = e4 + x0 + incr_exp;
|
|
|
|
res.w[1] = R128.w[1];
|
|
res.w[0] = R128.w[0];
|
|
|
|
// avoid a double rounding error
|
|
if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) &&
|
|
is_midpoint_lt_even) { // double rounding error upward
|
|
// res = res - 1
|
|
res.w[0]--;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
// Note: a double rounding error upward is not possible; for this
|
|
// the result after the first rounding would have to be 99...95
|
|
// (35 digits in all), possibly followed by a number of zeros; this
|
|
// not possible in Cases (2)-(6) or (15)-(17) which may get here
|
|
// if this is 10^33 - 1 make it 10^34 - 1 and decrement exponent
|
|
if (res.w[1] == 0x0000314dc6448d93ull &&
|
|
res.w[0] == 0x38c15b09ffffffffull) { // 10^33 - 1
|
|
res.w[1] = 0x0001ed09bead87c0ull; // 10^34 - 1
|
|
res.w[0] = 0x378d8e63ffffffffull;
|
|
e4--;
|
|
}
|
|
} else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) &&
|
|
is_midpoint_gt_even) { // double rounding error downward
|
|
// res = res + 1
|
|
res.w[0]++;
|
|
if (res.w[0] == 0)
|
|
res.w[1]++;
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
// if this second rounding was exact the result may still be
|
|
// inexact because of the first rounding
|
|
if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) {
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else if (is_midpoint_gt_even &&
|
|
(is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) {
|
|
// pulled up to a midpoint
|
|
is_inexact_lt_midpoint = 1;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else if (is_midpoint_lt_even &&
|
|
(is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) {
|
|
// pulled down to a midpoint
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else {
|
|
;
|
|
}
|
|
}
|
|
|
|
// determine tininess
|
|
if (rnd_mode == ROUNDING_TO_NEAREST) {
|
|
if (e4 < expmin) {
|
|
is_tiny = 1; // for other rounding modes apply correction
|
|
}
|
|
} else {
|
|
// for RM, RP, RZ, RA apply correction in order to determine tininess
|
|
// but do not save the result; apply the correction to
|
|
// (-1)^p_sign * res * 10^0
|
|
P128.w[1] = p_sign | 0x3040000000000000ull | res.w[1];
|
|
P128.w[0] = res.w[0];
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
0, &P128, pfpsf);
|
|
scale = ((P128.w[1] & MASK_EXP) >> 49) - 6176; // -1, 0, or +1
|
|
// the number of digits in the significand is p34 = 34
|
|
if (e4 + scale < expmin) {
|
|
is_tiny = 1;
|
|
}
|
|
}
|
|
|
|
// the result rounded to the destination precision with unbounded exponent
|
|
// is (-1)^p_sign * res * 10^e4
|
|
res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | res.w[1]; // RN
|
|
// res.w[0] unchanged;
|
|
// Note: res is correct only if expmin <= e4 <= expmax
|
|
ind = p34; // the number of decimal digits in the signifcand of res
|
|
|
|
// at this point we have the result rounded with unbounded exponent in
|
|
// res and we know its tininess:
|
|
// res = (-1)^p_sign * significand * 10^e4,
|
|
// where q (significand) = ind = p34
|
|
// Note: res is correct only if expmin <= e4 <= expmax
|
|
|
|
// check for overflow if RN
|
|
if (rnd_mode == ROUNDING_TO_NEAREST
|
|
&& (ind + e4) > (p34 + expmax)) {
|
|
res.w[1] = p_sign | 0x7800000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
*pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
} // else not overflow or not RN, so continue
|
|
|
|
// from this point on this is similar to the last part of the computation
|
|
// for Cases (15), (16), (17)
|
|
|
|
// if (e4 >= expmin) we have the result rounded with bounded exponent
|
|
if (e4 < expmin) {
|
|
x0 = expmin - e4; // x0 >= 1; the number of digits to chop off of res
|
|
// where the result rounded [at most] once is
|
|
// (-1)^p_sign * significand_res * 10^e4
|
|
|
|
// avoid double rounding error
|
|
is_inexact_lt_midpoint0 = is_inexact_lt_midpoint;
|
|
is_inexact_gt_midpoint0 = is_inexact_gt_midpoint;
|
|
is_midpoint_lt_even0 = is_midpoint_lt_even;
|
|
is_midpoint_gt_even0 = is_midpoint_gt_even;
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
|
|
if (x0 > ind) {
|
|
// nothing is left of res when moving the decimal point left x0 digits
|
|
is_inexact_lt_midpoint = 1;
|
|
res.w[1] = p_sign | 0x0000000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
e4 = expmin;
|
|
} else if (x0 == ind) { // 1 <= x0 = ind <= p34 = 34
|
|
// this is <, =, or > 1/2 ulp
|
|
// compare the ind-digit value in the significand of res with
|
|
// 1/2 ulp = 5*10^(ind-1), i.e. determine whether it is
|
|
// less than, equal to, or greater than 1/2 ulp (significand of res)
|
|
R128.w[1] = res.w[1] & MASK_COEFF;
|
|
R128.w[0] = res.w[0];
|
|
if (ind <= 19) {
|
|
if (R128.w[0] < midpoint64[ind - 1]) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (R128.w[0] == midpoint64[ind - 1]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
is_midpoint_gt_even = 1;
|
|
} else { // > 1/2 ulp
|
|
gt_half_ulp = 1;
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
} else { // if (ind <= 38)
|
|
if (R128.w[1] < midpoint128[ind - 20].w[1] ||
|
|
(R128.w[1] == midpoint128[ind - 20].w[1] &&
|
|
R128.w[0] < midpoint128[ind - 20].w[0])) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (R128.w[1] == midpoint128[ind - 20].w[1] &&
|
|
R128.w[0] == midpoint128[ind - 20].w[0]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
is_midpoint_gt_even = 1;
|
|
} else { // > 1/2 ulp
|
|
gt_half_ulp = 1;
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
}
|
|
if (lt_half_ulp || eq_half_ulp) {
|
|
// res = +0.0 * 10^expmin
|
|
res.w[1] = 0x0000000000000000ull;
|
|
res.w[0] = 0x0000000000000000ull;
|
|
} else { // if (gt_half_ulp)
|
|
// res = +1 * 10^expmin
|
|
res.w[1] = 0x0000000000000000ull;
|
|
res.w[0] = 0x0000000000000001ull;
|
|
}
|
|
res.w[1] = p_sign | res.w[1];
|
|
e4 = expmin;
|
|
} else { // if (1 <= x0 <= ind - 1 <= 33)
|
|
// round the ind-digit result to ind - x0 digits
|
|
|
|
if (ind <= 18) { // 2 <= ind <= 18
|
|
round64_2_18 (ind, x0, res.w[0], &R64, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
res.w[1] = 0x0;
|
|
res.w[0] = R64;
|
|
} else if (ind <= 38) {
|
|
P128.w[1] = res.w[1] & MASK_COEFF;
|
|
P128.w[0] = res.w[0];
|
|
round128_19_38 (ind, x0, P128, &res, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint);
|
|
}
|
|
e4 = e4 + x0; // expmin
|
|
// we want the exponent to be expmin, so if incr_exp = 1 then
|
|
// multiply the rounded result by 10 - it will still fit in 113 bits
|
|
if (incr_exp) {
|
|
// 64 x 128 -> 128
|
|
P128.w[1] = res.w[1] & MASK_COEFF;
|
|
P128.w[0] = res.w[0];
|
|
__mul_64x128_to_128 (res, ten2k64[1], P128);
|
|
}
|
|
res.w[1] =
|
|
p_sign | ((UINT64) (e4 + 6176) << 49) | (res.
|
|
w[1] & MASK_COEFF);
|
|
// avoid a double rounding error
|
|
if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) &&
|
|
is_midpoint_lt_even) { // double rounding error upward
|
|
// res = res - 1
|
|
res.w[0]--;
|
|
if (res.w[0] == 0xffffffffffffffffull)
|
|
res.w[1]--;
|
|
// Note: a double rounding error upward is not possible; for this
|
|
// the result after the first rounding would have to be 99...95
|
|
// (35 digits in all), possibly followed by a number of zeros; this
|
|
// not possible in this underflow case
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) &&
|
|
is_midpoint_gt_even) { // double rounding error downward
|
|
// res = res + 1
|
|
res.w[0]++;
|
|
if (res.w[0] == 0)
|
|
res.w[1]++;
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint
|
|
&& !is_inexact_gt_midpoint) {
|
|
// if this second rounding was exact the result may still be
|
|
// inexact because of the first rounding
|
|
if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) {
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else if (is_midpoint_gt_even &&
|
|
(is_inexact_gt_midpoint0
|
|
|| is_midpoint_lt_even0)) {
|
|
// pulled up to a midpoint
|
|
is_inexact_lt_midpoint = 1;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else if (is_midpoint_lt_even &&
|
|
(is_inexact_lt_midpoint0
|
|
|| is_midpoint_gt_even0)) {
|
|
// pulled down to a midpoint
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else {
|
|
;
|
|
}
|
|
}
|
|
}
|
|
// res contains the correct result
|
|
// apply correction if not rounding to nearest
|
|
if (rnd_mode != ROUNDING_TO_NEAREST) {
|
|
rounding_correction (rnd_mode,
|
|
is_inexact_lt_midpoint,
|
|
is_inexact_gt_midpoint,
|
|
is_midpoint_lt_even, is_midpoint_gt_even,
|
|
e4, &res, pfpsf);
|
|
}
|
|
if (is_midpoint_lt_even || is_midpoint_gt_even ||
|
|
is_inexact_lt_midpoint || is_inexact_gt_midpoint) {
|
|
// set the inexact flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
if (is_tiny)
|
|
*pfpsf |= UNDERFLOW_EXCEPTION;
|
|
}
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
|
|
} else if ((p34 <= delta && delta + q3 <= q4) || // Case (15)
|
|
(delta < p34 && p34 < delta + q3 && delta + q3 <= q4) || //Case (16)
|
|
(delta + q3 <= p34 && p34 < q4)) { // Case (17)
|
|
|
|
// calculate first the result rounded to the destination precision, with
|
|
// unbounded exponent
|
|
|
|
add_and_round (q3, q4, e4, delta, p34, z_sign, p_sign, C3, C4,
|
|
rnd_mode, &is_midpoint_lt_even,
|
|
&is_midpoint_gt_even, &is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, pfpsf, &res);
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
|
|
} else {
|
|
;
|
|
}
|
|
|
|
} // end if delta < 0
|
|
|
|
*ptr_is_midpoint_lt_even = is_midpoint_lt_even;
|
|
*ptr_is_midpoint_gt_even = is_midpoint_gt_even;
|
|
*ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint;
|
|
*ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint;
|
|
BID_SWAP128 (res);
|
|
BID_RETURN (res)
|
|
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid128_fma (UINT128 * pres, UINT128 * px, UINT128 * py, UINT128 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT128 x = *px, y = *py, z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT128
|
|
bid128_fma (UINT128 x, UINT128 y, UINT128 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
int is_midpoint_lt_even, is_midpoint_gt_even,
|
|
is_inexact_lt_midpoint, is_inexact_gt_midpoint;
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint,
|
|
&res, &x, &y, &z
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, x, y,
|
|
z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid128ddd_fma (UINT128 * pres, UINT64 * px, UINT64 * py, UINT64 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 x = *px, y = *py, z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT128
|
|
bid128ddd_fma (UINT64 x, UINT64 y, UINT64 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0,
|
|
is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT128 x1, y1, z1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint,
|
|
&res, &x1, &y1, &z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, x1, y1,
|
|
z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid128ddq_fma (UINT128 * pres, UINT64 * px, UINT64 * py, UINT128 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 x = *px, y = *py;
|
|
UINT128 z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT128
|
|
bid128ddq_fma (UINT64 x, UINT64 y, UINT128 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0,
|
|
is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT128 x1, y1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint,
|
|
&res, &x1, &y1, &z
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, x1, y1,
|
|
z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid128dqd_fma (UINT128 * pres, UINT64 * px, UINT128 * py, UINT64 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 x = *px, z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT128
|
|
bid128dqd_fma (UINT64 x, UINT128 y, UINT64 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0,
|
|
is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT128 x1, z1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint,
|
|
&res, &x1, py, &z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, x1, y,
|
|
z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid128dqq_fma (UINT128 * pres, UINT64 * px, UINT128 * py, UINT128 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 x = *px;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT128
|
|
bid128dqq_fma (UINT64 x, UINT128 y, UINT128 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0,
|
|
is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT128 x1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint,
|
|
&res, &x1, py, pz
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, x1, y,
|
|
z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid128qdd_fma (UINT128 * pres, UINT128 * px, UINT64 * py, UINT64 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 y = *py, z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT128
|
|
bid128qdd_fma (UINT128 x, UINT64 y, UINT64 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0,
|
|
is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT128 y1, z1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint,
|
|
&res, px, &y1, &z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, x, y1,
|
|
z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid128qdq_fma (UINT128 * pres, UINT128 * px, UINT64 * py, UINT128 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 y = *py;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT128
|
|
bid128qdq_fma (UINT128 x, UINT64 y, UINT128 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0,
|
|
is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT128 y1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint,
|
|
&res, px, &y1, pz
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, x, y1,
|
|
z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid128qqd_fma (UINT128 * pres, UINT128 * px, UINT128 * py, UINT64 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT128
|
|
bid128qqd_fma (UINT128 x, UINT128 y, UINT64 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0,
|
|
is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT128 z1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint,
|
|
&res, px, py, &z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint,
|
|
&is_inexact_gt_midpoint, x, y,
|
|
z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res);
|
|
}
|
|
|
|
// Note: bid128qqq_fma is represented by bid128_fma
|
|
|
|
// Note: bid64ddd_fma is represented by bid64_fma
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid64ddq_fma (UINT64 * pres, UINT64 * px, UINT64 * py, UINT128 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 x = *px, y = *py;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT64
|
|
bid64ddq_fma (UINT64 x, UINT64 y, UINT128 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
UINT64 res1 = 0xbaddbaddbaddbaddull;
|
|
UINT128 x1, y1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64qqq_fma (&res1, &x1, &y1, pz
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res1 = bid64qqq_fma (x1, y1, z
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res1);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid64dqd_fma (UINT64 * pres, UINT64 * px, UINT128 * py, UINT64 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 x = *px, z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT64
|
|
bid64dqd_fma (UINT64 x, UINT128 y, UINT64 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
UINT64 res1 = 0xbaddbaddbaddbaddull;
|
|
UINT128 x1, z1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64qqq_fma (&res1, &x1, py, &z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res1 = bid64qqq_fma (x1, y, z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res1);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid64dqq_fma (UINT64 * pres, UINT64 * px, UINT128 * py, UINT128 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 x = *px;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT64
|
|
bid64dqq_fma (UINT64 x, UINT128 y, UINT128 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
UINT64 res1 = 0xbaddbaddbaddbaddull;
|
|
UINT128 x1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64qqq_fma (&res1, &x1, py, pz
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res1 = bid64qqq_fma (x1, y, z
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res1);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid64qdd_fma (UINT64 * pres, UINT128 * px, UINT64 * py, UINT64 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 y = *py, z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT64
|
|
bid64qdd_fma (UINT128 x, UINT64 y, UINT64 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
UINT64 res1 = 0xbaddbaddbaddbaddull;
|
|
UINT128 y1, z1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64qqq_fma (&res1, px, &y1, &z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res1 = bid64qqq_fma (x, y1, z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res1);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid64qdq_fma (UINT64 * pres, UINT128 * px, UINT64 * py, UINT128 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 y = *py;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT64
|
|
bid64qdq_fma (UINT128 x, UINT64 y, UINT128 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
UINT64 res1 = 0xbaddbaddbaddbaddull;
|
|
UINT128 y1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64qqq_fma (&res1, px, &y1, pz
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res1 = bid64qqq_fma (x, y1, z
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res1);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid64qqd_fma (UINT64 * pres, UINT128 * px, UINT128 * py, UINT64 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT64 z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT64
|
|
bid64qqd_fma (UINT128 x, UINT128 y, UINT64 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
UINT64 res1 = 0xbaddbaddbaddbaddull;
|
|
UINT128 z1;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
bid64qqq_fma (&res1, px, py, &z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
|
|
res1 = bid64qqq_fma (x, y, z1
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
BID_RETURN (res1);
|
|
}
|
|
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
void
|
|
bid64qqq_fma (UINT64 * pres, UINT128 * px, UINT128 * py, UINT128 * pz
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
UINT128 x = *px, y = *py, z = *pz;
|
|
#if !DECIMAL_GLOBAL_ROUNDING
|
|
unsigned int rnd_mode = *prnd_mode;
|
|
#endif
|
|
#else
|
|
UINT64
|
|
bid64qqq_fma (UINT128 x, UINT128 y, UINT128 z
|
|
_RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
|
|
_EXC_INFO_PARAM) {
|
|
#endif
|
|
int is_midpoint_lt_even0 = 0, is_midpoint_gt_even0 = 0,
|
|
is_inexact_lt_midpoint0 = 0, is_inexact_gt_midpoint0 = 0;
|
|
int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0,
|
|
is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
|
|
int incr_exp;
|
|
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT128 res128 = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} };
|
|
UINT64 res1 = 0xbaddbaddbaddbaddull;
|
|
unsigned int save_fpsf; // needed because of the call to bid128_ext_fma
|
|
UINT64 sign;
|
|
UINT64 exp;
|
|
int unbexp;
|
|
UINT128 C;
|
|
BID_UI64DOUBLE tmp;
|
|
int nr_bits;
|
|
int q, x0;
|
|
int scale;
|
|
int lt_half_ulp = 0, eq_half_ulp = 0;
|
|
|
|
// Note: for rounding modes other than RN or RA, the result can be obtained
|
|
// by rounding first to BID128 and then to BID64
|
|
|
|
save_fpsf = *pfpsf; // sticky bits - caller value must be preserved
|
|
*pfpsf = 0;
|
|
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid128_ext_fma (&is_midpoint_lt_even0, &is_midpoint_gt_even0,
|
|
&is_inexact_lt_midpoint0, &is_inexact_gt_midpoint0,
|
|
&res, &x, &y, &z
|
|
_RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#else
|
|
res = bid128_ext_fma (&is_midpoint_lt_even0, &is_midpoint_gt_even0,
|
|
&is_inexact_lt_midpoint0,
|
|
&is_inexact_gt_midpoint0, x, y,
|
|
z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
|
|
_EXC_INFO_ARG);
|
|
#endif
|
|
|
|
if ((rnd_mode == ROUNDING_DOWN) || (rnd_mode == ROUNDING_UP) ||
|
|
(rnd_mode == ROUNDING_TO_ZERO) || // no double rounding error is possible
|
|
((res.w[HIGH_128W] & MASK_NAN) == MASK_NAN) || //res=QNaN (cannot be SNaN)
|
|
((res.w[HIGH_128W] & MASK_ANY_INF) == MASK_INF)) { // result is infinity
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid128_to_bid64 (&res1, &res _RND_MODE_ARG _EXC_FLAGS_ARG);
|
|
#else
|
|
res1 = bid128_to_bid64 (res _RND_MODE_ARG _EXC_FLAGS_ARG);
|
|
#endif
|
|
// determine the unbiased exponent of the result
|
|
unbexp = ((res1 >> 53) & 0x3ff) - 398;
|
|
|
|
// if subnormal, res1 must have exp = -398
|
|
// if tiny and inexact set underflow and inexact status flags
|
|
if (!((res1 & MASK_NAN) == MASK_NAN) && // res1 not NaN
|
|
(unbexp == -398)
|
|
&& ((res1 & MASK_BINARY_SIG1) < 1000000000000000ull)
|
|
&& (is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0
|
|
|| is_midpoint_lt_even0 || is_midpoint_gt_even0)) {
|
|
// set the inexact flag and the underflow flag
|
|
*pfpsf |= (INEXACT_EXCEPTION | UNDERFLOW_EXCEPTION);
|
|
} else if (is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0 ||
|
|
is_midpoint_lt_even0 || is_midpoint_gt_even0) {
|
|
// set the inexact flag and the underflow flag
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
}
|
|
|
|
*pfpsf |= save_fpsf;
|
|
BID_RETURN (res1);
|
|
} // else continue, and use rounding to nearest to round to 16 digits
|
|
|
|
// at this point the result is rounded to nearest (even or away) to 34 digits
|
|
// (or less if exact), and it is zero or finite non-zero canonical [sub]normal
|
|
sign = res.w[HIGH_128W] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
|
|
exp = res.w[HIGH_128W] & MASK_EXP; // biased and shifted left 49 bits
|
|
unbexp = (exp >> 49) - 6176;
|
|
C.w[1] = res.w[HIGH_128W] & MASK_COEFF;
|
|
C.w[0] = res.w[LOW_128W];
|
|
|
|
if ((C.w[1] == 0x0 && C.w[0] == 0x0) || // result is zero
|
|
(unbexp <= (-398 - 35)) || (unbexp >= (369 + 16))) {
|
|
// clear under/overflow
|
|
#if DECIMAL_CALL_BY_REFERENCE
|
|
bid128_to_bid64 (&res1, &res _RND_MODE_ARG _EXC_FLAGS_ARG);
|
|
#else
|
|
res1 = bid128_to_bid64 (res _RND_MODE_ARG _EXC_FLAGS_ARG);
|
|
#endif
|
|
*pfpsf |= save_fpsf;
|
|
BID_RETURN (res1);
|
|
} // else continue
|
|
|
|
// -398 - 34 <= unbexp <= 369 + 15
|
|
if (rnd_mode == ROUNDING_TIES_AWAY) {
|
|
// apply correction, if needed, to make the result rounded to nearest-even
|
|
if (is_midpoint_gt_even) {
|
|
// res = res - 1
|
|
res1--; // res1 is now even
|
|
} // else the result is already correctly rounded to nearest-even
|
|
}
|
|
// at this point the result is finite, non-zero canonical normal or subnormal,
|
|
// and in most cases overflow or underflow will not occur
|
|
|
|
// determine the number of digits q in the result
|
|
// q = nr. of decimal digits in x
|
|
// determine first the nr. of bits in x
|
|
if (C.w[1] == 0) {
|
|
if (C.w[0] >= 0x0020000000000000ull) { // x >= 2^53
|
|
// split the 64-bit value in two 32-bit halves to avoid rounding errors
|
|
if (C.w[0] >= 0x0000000100000000ull) { // x >= 2^32
|
|
tmp.d = (double) (C.w[0] >> 32); // exact conversion
|
|
nr_bits =
|
|
33 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
} else { // x < 2^32
|
|
tmp.d = (double) (C.w[0]); // exact conversion
|
|
nr_bits =
|
|
1 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
} else { // if x < 2^53
|
|
tmp.d = (double) C.w[0]; // exact conversion
|
|
nr_bits =
|
|
1 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
} else { // C.w[1] != 0 => nr. bits = 64 + nr_bits (C.w[1])
|
|
tmp.d = (double) C.w[1]; // exact conversion
|
|
nr_bits =
|
|
65 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff);
|
|
}
|
|
q = nr_digits[nr_bits - 1].digits;
|
|
if (q == 0) {
|
|
q = nr_digits[nr_bits - 1].digits1;
|
|
if (C.w[1] > nr_digits[nr_bits - 1].threshold_hi ||
|
|
(C.w[1] == nr_digits[nr_bits - 1].threshold_hi &&
|
|
C.w[0] >= nr_digits[nr_bits - 1].threshold_lo))
|
|
q++;
|
|
}
|
|
// if q > 16, round to nearest even to 16 digits (but for underflow it may
|
|
// have to be truncated even more)
|
|
if (q > 16) {
|
|
x0 = q - 16;
|
|
if (q <= 18) {
|
|
round64_2_18 (q, x0, C.w[0], &res1, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint);
|
|
} else { // 19 <= q <= 34
|
|
round128_19_38 (q, x0, C, &res128, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint);
|
|
res1 = res128.w[0]; // the result fits in 64 bits
|
|
}
|
|
unbexp = unbexp + x0;
|
|
if (incr_exp)
|
|
unbexp++;
|
|
q = 16; // need to set in case denormalization is necessary
|
|
} else {
|
|
// the result does not require a second rounding (and it must have
|
|
// been exact in the first rounding, since q <= 16)
|
|
res1 = C.w[0];
|
|
}
|
|
|
|
// avoid a double rounding error
|
|
if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) &&
|
|
is_midpoint_lt_even) { // double rounding error upward
|
|
// res = res - 1
|
|
res1--; // res1 becomes odd
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
if (res1 == 0x00038d7ea4c67fffull) { // 10^15 - 1
|
|
res1 = 0x002386f26fc0ffffull; // 10^16 - 1
|
|
unbexp--;
|
|
}
|
|
} else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) &&
|
|
is_midpoint_gt_even) { // double rounding error downward
|
|
// res = res + 1
|
|
res1++; // res1 becomes odd (so it cannot be 10^16)
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
// if this second rounding was exact the result may still be
|
|
// inexact because of the first rounding
|
|
if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) {
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else if (is_midpoint_gt_even &&
|
|
(is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) {
|
|
// pulled up to a midpoint
|
|
is_inexact_lt_midpoint = 1;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else if (is_midpoint_lt_even &&
|
|
(is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) {
|
|
// pulled down to a midpoint
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else {
|
|
;
|
|
}
|
|
// this is the result rounded correctly to nearest even, with unbounded exp.
|
|
|
|
// check for overflow
|
|
if (q + unbexp > P16 + expmax16) {
|
|
res1 = sign | 0x7800000000000000ull;
|
|
*pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION);
|
|
*pfpsf |= save_fpsf;
|
|
BID_RETURN (res1)
|
|
} else if (unbexp > expmax16) { // q + unbexp <= P16 + expmax16
|
|
// not overflow; the result must be exact, and we can multiply res1 by
|
|
// 10^(unbexp - expmax16) and the product will fit in 16 decimal digits
|
|
scale = unbexp - expmax16;
|
|
res1 = res1 * ten2k64[scale]; // res1 * 10^scale
|
|
unbexp = expmax16; // unbexp - scale
|
|
} else {
|
|
; // continue
|
|
}
|
|
|
|
// check for underflow
|
|
if (q + unbexp < P16 + expmin16) {
|
|
if (unbexp < expmin16) {
|
|
// we must truncate more of res
|
|
x0 = expmin16 - unbexp; // x0 >= 1
|
|
is_inexact_lt_midpoint0 = is_inexact_lt_midpoint;
|
|
is_inexact_gt_midpoint0 = is_inexact_gt_midpoint;
|
|
is_midpoint_lt_even0 = is_midpoint_lt_even;
|
|
is_midpoint_gt_even0 = is_midpoint_gt_even;
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
// the number of decimal digits in res1 is q
|
|
if (x0 < q) { // 1 <= x0 <= q-1 => round res to q - x0 digits
|
|
// 2 <= q <= 16, 1 <= x0 <= 15
|
|
round64_2_18 (q, x0, res1, &res1, &incr_exp,
|
|
&is_midpoint_lt_even, &is_midpoint_gt_even,
|
|
&is_inexact_lt_midpoint, &is_inexact_gt_midpoint);
|
|
if (incr_exp) {
|
|
// res1 = 10^(q-x0), 1 <= q - x0 <= q - 1, 1 <= q - x0 <= 15
|
|
res1 = ten2k64[q - x0];
|
|
}
|
|
unbexp = unbexp + x0; // expmin16
|
|
} else if (x0 == q) {
|
|
// the second rounding is for 0.d(0)d(1)...d(q-1) * 10^emin
|
|
// determine relationship with 1/2 ulp
|
|
// q <= 16
|
|
if (res1 < midpoint64[q - 1]) { // < 1/2 ulp
|
|
lt_half_ulp = 1;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if (res1 == midpoint64[q - 1]) { // = 1/2 ulp
|
|
eq_half_ulp = 1;
|
|
is_midpoint_gt_even = 1;
|
|
} else { // > 1/2 ulp
|
|
// gt_half_ulp = 1;
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
if (lt_half_ulp || eq_half_ulp) {
|
|
// res = +0.0 * 10^expmin16
|
|
res1 = 0x0000000000000000ull;
|
|
} else { // if (gt_half_ulp)
|
|
// res = +1 * 10^expmin16
|
|
res1 = 0x0000000000000001ull;
|
|
}
|
|
unbexp = expmin16;
|
|
} else { // if (x0 > q)
|
|
// the second rounding is for 0.0...d(0)d(1)...d(q-1) * 10^emin
|
|
res1 = 0x0000000000000000ull;
|
|
unbexp = expmin16;
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
// avoid a double rounding error
|
|
if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) &&
|
|
is_midpoint_lt_even) { // double rounding error upward
|
|
// res = res - 1
|
|
res1--; // res1 becomes odd
|
|
is_midpoint_lt_even = 0;
|
|
is_inexact_lt_midpoint = 1;
|
|
} else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) &&
|
|
is_midpoint_gt_even) { // double rounding error downward
|
|
// res = res + 1
|
|
res1++; // res1 becomes odd
|
|
is_midpoint_gt_even = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
} else if (!is_midpoint_lt_even && !is_midpoint_gt_even &&
|
|
!is_inexact_lt_midpoint && !is_inexact_gt_midpoint) {
|
|
// if this rounding was exact the result may still be
|
|
// inexact because of the previous roundings
|
|
if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) {
|
|
is_inexact_gt_midpoint = 1;
|
|
}
|
|
if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) {
|
|
is_inexact_lt_midpoint = 1;
|
|
}
|
|
} else if (is_midpoint_gt_even &&
|
|
(is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) {
|
|
// pulled up to a midpoint
|
|
is_inexact_lt_midpoint = 1;
|
|
is_inexact_gt_midpoint = 0;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else if (is_midpoint_lt_even &&
|
|
(is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) {
|
|
// pulled down to a midpoint
|
|
is_inexact_lt_midpoint = 0;
|
|
is_inexact_gt_midpoint = 1;
|
|
is_midpoint_lt_even = 0;
|
|
is_midpoint_gt_even = 0;
|
|
} else {
|
|
;
|
|
}
|
|
}
|
|
// else if unbexp >= emin then q < P (because q + unbexp < P16 + expmin16)
|
|
// and the result is tiny and exact
|
|
|
|
// check for inexact result
|
|
if (is_inexact_lt_midpoint || is_inexact_gt_midpoint ||
|
|
is_midpoint_lt_even || is_midpoint_gt_even ||
|
|
is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0 ||
|
|
is_midpoint_lt_even0 || is_midpoint_gt_even0) {
|
|
// set the inexact flag and the underflow flag
|
|
*pfpsf |= (INEXACT_EXCEPTION | UNDERFLOW_EXCEPTION);
|
|
}
|
|
} else if (is_inexact_lt_midpoint || is_inexact_gt_midpoint ||
|
|
is_midpoint_lt_even || is_midpoint_gt_even) {
|
|
*pfpsf |= INEXACT_EXCEPTION;
|
|
}
|
|
// this is the result rounded correctly to nearest, with bounded exponent
|
|
|
|
if (rnd_mode == ROUNDING_TIES_AWAY && is_midpoint_gt_even) { // correction
|
|
// res = res + 1
|
|
res1++; // res1 is now odd
|
|
} // else the result is already correct
|
|
|
|
// assemble the result
|
|
if (res1 < 0x0020000000000000ull) { // res < 2^53
|
|
res1 = sign | ((UINT64) (unbexp + 398) << 53) | res1;
|
|
} else { // res1 >= 2^53
|
|
res1 = sign | MASK_STEERING_BITS |
|
|
((UINT64) (unbexp + 398) << 51) | (res1 & MASK_BINARY_SIG2);
|
|
}
|
|
*pfpsf |= save_fpsf;
|
|
BID_RETURN (res1);
|
|
}
|