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/* Copyright (C) 2007-2021 Free Software Foundation, Inc.
This file is part of GCC.
GCC is free software; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
GCC is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
for more details.
Under Section 7 of GPL version 3, you are granted additional
permissions described in the GCC Runtime Library Exception, version
3.1, as published by the Free Software Foundation.
You should have received a copy of the GNU General Public License and
a copy of the GCC Runtime Library Exception along with this program;
see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
<http://www.gnu.org/licenses/>. */
#define BID_128RES
#include "bid_internal.h"
/*****************************************************************************
* BID128_round_integral_exact
****************************************************************************/
BID128_FUNCTION_ARG1 (bid128_round_integral_exact, x)
UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
};
UINT64 x_sign;
UINT64 x_exp;
int exp; // unbiased exponent
// Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64)
UINT64 tmp64;
BID_UI64DOUBLE tmp1;
unsigned int x_nr_bits;
int q, ind, shift;
UINT128 C1;
UINT256 fstar;
UINT256 P256;
// check for NaN or Infinity
if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) {
// x is special
if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
// if x = NaN, 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];
}
BID_RETURN (res)
} else { // x is not a NaN, so it must be infinity
if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf
// return +inf
res.w[1] = 0x7800000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else { // x is -inf
// return -inf
res.w[1] = 0xf800000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
BID_RETURN (res);
}
}
// unpack x
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];
// 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
;
}
}
// test for input equal to zero
if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
// x is 0
// return 0 preserving the sign bit and the preferred exponent
// of MAX(Q(x), 0)
if (x_exp <= (0x1820ull << 49)) {
res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull;
} else {
res.w[1] = x_sign | x_exp;
}
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
// x is not special and is not zero
switch (rnd_mode) {
case ROUNDING_TO_NEAREST:
case ROUNDING_TIES_AWAY:
// if (exp <= -(p+1)) return 0.0
if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
*pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
break;
case ROUNDING_DOWN:
// if (exp <= -p) return -1.0 or +0.0
if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffa000000000000ull == -34
if (x_sign) {
// if negative, return negative 1, because we know coefficient
// is non-zero (would have been caught above)
res.w[1] = 0xb040000000000000ull;
res.w[0] = 0x0000000000000001ull;
} else {
// if positive, return positive 0, because we know coefficient is
// non-zero (would have been caught above)
res.w[1] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
*pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
break;
case ROUNDING_UP:
// if (exp <= -p) return -0.0 or +1.0
if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34
if (x_sign) {
// if negative, return negative 0, because we know the coefficient
// is non-zero (would have been caught above)
res.w[1] = 0xb040000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else {
// if positive, return positive 1, because we know coefficient is
// non-zero (would have been caught above)
res.w[1] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000001ull;
}
*pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
break;
case ROUNDING_TO_ZERO:
// if (exp <= -p) return -0.0 or +0.0
if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
*pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
break;
}
// q = 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
tmp1.d = (double) (C1.w[0] >> 32); // exact conversion
x_nr_bits =
33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
} else { // x < 2^32
tmp1.d = (double) (C1.w[0]); // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // if x < 2^53
tmp1.d = (double) C1.w[0]; // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1])
tmp1.d = (double) C1.w[1]; // exact conversion
x_nr_bits =
65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
q = nr_digits[x_nr_bits - 1].digits;
if (q == 0) {
q = 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))
q++;
}
exp = (x_exp >> 49) - 6176;
if (exp >= 0) { // -exp <= 0
// the argument is an integer already
res.w[1] = x.w[1];
res.w[0] = x.w[0];
BID_RETURN (res);
}
// exp < 0
switch (rnd_mode) {
case ROUNDING_TO_NEAREST:
if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q
// need to shift right -exp digits from the coefficient; exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// chop off ind digits from the lower part of C1
// C1 = C1 + 1/2 * 10^x where the result C1 fits in 127 bits
tmp64 = C1.w[0];
if (ind <= 19) {
C1.w[0] = C1.w[0] + midpoint64[ind - 1];
} else {
C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
}
if (C1.w[0] < tmp64)
C1.w[1]++;
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = (C1 + 1/2 * 10^x) * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
// determine the value of res and fstar
// determine inexactness of the rounding of C*
// if (0 < f* - 1/2 < 10^(-x)) then
// the result is exact
// else // if (f* - 1/2 > T*) then
// the result is inexact
// Note: we are going to use ten2mk128[] instead of ten2mk128trunc[]
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
// redundant shift = shiftright128[ind - 1]; // shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
// redundant fstar.w[3] = 0;
// redundant fstar.w[2] = 0;
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* < 10^(-x) <=> midpoint
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
// if 0 < fstar < 10^(-x), subtract 1 if odd (for rounding to even)
if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
((fstar.w[1] < (ten2mk128[ind - 1].w[1]))
|| ((fstar.w[1] == ten2mk128[ind - 1].w[1])
&& (fstar.w[0] < ten2mk128[ind - 1].w[0])))) {
// subract 1 to make even
if (res.w[0]-- == 0) {
res.w[1]--;
}
}
if (fstar.w[1] > 0x8000000000000000ull ||
(fstar.w[1] == 0x8000000000000000ull
&& fstar.w[0] > 0x0ull)) {
// f* > 1/2 and the result may be exact
tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2
if (tmp64 > ten2mk128[ind - 1].w[1] ||
(tmp64 == ten2mk128[ind - 1].w[1] &&
fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
} // else the result is exact
} else { // the result is inexact; f2* <= 1/2
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
}
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 3 <= shift <= 63
res.w[1] = (P256.w[3] >> shift);
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
// redundant fstar.w[3] = 0;
fstar.w[2] = P256.w[2] & maskhigh128[ind - 1];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* < 10^(-x) <=> midpoint
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
fstar.w[2] == 0 && (fstar.w[1] < ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] < ten2mk128[ind - 1].w[0]))) {
// subract 1 to make even
if (res.w[0]-- == 0) {
res.w[1]--;
}
}
if (fstar.w[2] > onehalf128[ind - 1] ||
(fstar.w[2] == onehalf128[ind - 1]
&& (fstar.w[1] || fstar.w[0]))) {
// f2* > 1/2 and the result may be exact
// Calculate f2* - 1/2
tmp64 = fstar.w[2] - onehalf128[ind - 1];
if (tmp64 || fstar.w[1] > ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
} // else the result is exact
} else { // the result is inexact; f2* <= 1/2
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
}
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = P256.w[3] >> shift;
fstar.w[3] = P256.w[3] & maskhigh128[ind - 1];
fstar.w[2] = P256.w[2];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* < 10^(-x) <=> midpoint
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
fstar.w[3] == 0 && fstar.w[2] == 0 &&
(fstar.w[1] < ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] < ten2mk128[ind - 1].w[0]))) {
// subract 1 to make even
if (res.w[0]-- == 0) {
res.w[1]--;
}
}
if (fstar.w[3] > onehalf128[ind - 1] ||
(fstar.w[3] == onehalf128[ind - 1] &&
(fstar.w[2] || fstar.w[1] || fstar.w[0]))) {
// f2* > 1/2 and the result may be exact
// Calculate f2* - 1/2
tmp64 = fstar.w[3] - onehalf128[ind - 1];
if (tmp64 || fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1]
|| (fstar.w[1] == ten2mk128[ind - 1].w[1]
&& fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
} // else the result is exact
} else { // the result is inexact; f2* <= 1/2
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
}
}
res.w[1] = x_sign | 0x3040000000000000ull | res.w[1];
BID_RETURN (res);
} else { // if ((q + exp) < 0) <=> q < -exp
// the result is +0 or -0
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
*pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
break;
case ROUNDING_TIES_AWAY:
if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q
// need to shift right -exp digits from the coefficient; exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// chop off ind digits from the lower part of C1
// C1 = C1 + 1/2 * 10^x where the result C1 fits in 127 bits
tmp64 = C1.w[0];
if (ind <= 19) {
C1.w[0] = C1.w[0] + midpoint64[ind - 1];
} else {
C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
}
if (C1.w[0] < tmp64)
C1.w[1]++;
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = (C1 + 1/2 * 10^x) * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
// the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g.
// if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
// if (0 < f* < 10^(-x)) then the result is a midpoint
// if floor(C*) is even then C* = floor(C*) - logical right
// shift; C* has p decimal digits, correct by Prop. 1)
// else if floor(C*) is odd C* = floor(C*)-1 (logical right
// shift; C* has p decimal digits, correct by Pr. 1)
// else
// C* = floor(C*) (logical right shift; C has p decimal digits,
// correct by Property 1)
// n = C* * 10^(e+x)
// determine also the inexactness of the rounding of C*
// if (0 < f* - 1/2 < 10^(-x)) then
// the result is exact
// else // if (f* - 1/2 > T*) then
// the result is inexact
// Note: we are going to use ten2mk128[] instead of ten2mk128trunc[]
// shift right C* by Ex-128 = shiftright128[ind]
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
// redundant shift = shiftright128[ind - 1]; // shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
// redundant fstar.w[3] = 0;
// redundant fstar.w[2] = 0;
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
if (fstar.w[1] > 0x8000000000000000ull ||
(fstar.w[1] == 0x8000000000000000ull
&& fstar.w[0] > 0x0ull)) {
// f* > 1/2 and the result may be exact
tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2
if ((tmp64 > ten2mk128[ind - 1].w[1] ||
(tmp64 == ten2mk128[ind - 1].w[1] &&
fstar.w[0] >= ten2mk128[ind - 1].w[0]))) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
} // else the result is exact
} else { // the result is inexact; f2* <= 1/2
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
}
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 3 <= shift <= 63
res.w[1] = (P256.w[3] >> shift);
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
// redundant fstar.w[3] = 0;
fstar.w[2] = P256.w[2] & maskhigh128[ind - 1];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
if (fstar.w[2] > onehalf128[ind - 1] ||
(fstar.w[2] == onehalf128[ind - 1]
&& (fstar.w[1] || fstar.w[0]))) {
// f2* > 1/2 and the result may be exact
// Calculate f2* - 1/2
tmp64 = fstar.w[2] - onehalf128[ind - 1];
if (tmp64 || fstar.w[1] > ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
} // else the result is exact
} else { // the result is inexact; f2* <= 1/2
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
}
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = P256.w[3] >> shift;
fstar.w[3] = P256.w[3] & maskhigh128[ind - 1];
fstar.w[2] = P256.w[2];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
if (fstar.w[3] > onehalf128[ind - 1] ||
(fstar.w[3] == onehalf128[ind - 1] &&
(fstar.w[2] || fstar.w[1] || fstar.w[0]))) {
// f2* > 1/2 and the result may be exact
// Calculate f2* - 1/2
tmp64 = fstar.w[3] - onehalf128[ind - 1];
if (tmp64 || fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1]
|| (fstar.w[1] == ten2mk128[ind - 1].w[1]
&& fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
} // else the result is exact
} else { // the result is inexact; f2* <= 1/2
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
}
}
// if the result was a midpoint, it was already rounded away from zero
res.w[1] |= x_sign | 0x3040000000000000ull;
BID_RETURN (res);
} else { // if ((q + exp) < 0) <=> q < -exp
// the result is +0 or -0
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
*pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
break;
case ROUNDING_DOWN:
if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q
// need to shift right -exp digits from the coefficient; exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// (number of digits to be chopped off)
// chop off ind digits from the lower part of C1
// FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate
// FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP
// FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE
// FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE
// tmp64 = C1.w[0];
// if (ind <= 19) {
// C1.w[0] = C1.w[0] + midpoint64[ind - 1];
// } else {
// C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
// C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
// }
// if (C1.w[0] < tmp64) C1.w[1]++;
// if carry-out from C1.w[0], increment C1.w[1]
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = (C1 + 1/2 * 10^x) * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
// redundant fstar.w[3] = 0;
// redundant fstar.w[2] = 0;
// redundant fstar.w[1] = P256.w[1];
// redundant fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if ((P256.w[1] > ten2mk128[ind - 1].w[1])
|| (P256.w[1] == ten2mk128[ind - 1].w[1]
&& (P256.w[0] >= ten2mk128[ind - 1].w[0]))) {
*pfpsf |= INEXACT_EXCEPTION;
// if positive, the truncated value is already the correct result
if (x_sign) { // if negative
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 0 <= shift <= 102
res.w[1] = (P256.w[3] >> shift);
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
// redundant fstar.w[3] = 0;
fstar.w[2] = P256.w[2] & maskhigh128[ind - 1];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
*pfpsf |= INEXACT_EXCEPTION;
// if positive, the truncated value is already the correct result
if (x_sign) { // if negative
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = P256.w[3] >> shift;
fstar.w[3] = P256.w[3] & maskhigh128[ind - 1];
fstar.w[2] = P256.w[2];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[3] || fstar.w[2]
|| fstar.w[1] > ten2mk128[ind - 1].w[1]
|| (fstar.w[1] == ten2mk128[ind - 1].w[1]
&& fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
*pfpsf |= INEXACT_EXCEPTION;
// if positive, the truncated value is already the correct result
if (x_sign) { // if negative
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
}
res.w[1] = x_sign | 0x3040000000000000ull | res.w[1];
BID_RETURN (res);
} else { // if exp < 0 and q + exp <= 0
if (x_sign) { // negative rounds down to -1.0
res.w[1] = 0xb040000000000000ull;
res.w[0] = 0x0000000000000001ull;
} else { // positive rpunds down to +0.0
res.w[1] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
*pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
break;
case ROUNDING_UP:
if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q
// need to shift right -exp digits from the coefficient; exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// (number of digits to be chopped off)
// chop off ind digits from the lower part of C1
// FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate
// FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP
// FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE
// FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE
// tmp64 = C1.w[0];
// if (ind <= 19) {
// C1.w[0] = C1.w[0] + midpoint64[ind - 1];
// } else {
// C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
// C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
// }
// if (C1.w[0] < tmp64) C1.w[1]++;
// if carry-out from C1.w[0], increment C1.w[1]
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = C1 * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
// redundant fstar.w[3] = 0;
// redundant fstar.w[2] = 0;
// redundant fstar.w[1] = P256.w[1];
// redundant fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if ((P256.w[1] > ten2mk128[ind - 1].w[1])
|| (P256.w[1] == ten2mk128[ind - 1].w[1]
&& (P256.w[0] >= ten2mk128[ind - 1].w[0]))) {
*pfpsf |= INEXACT_EXCEPTION;
// if negative, the truncated value is already the correct result
if (!x_sign) { // if positive
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 3 <= shift <= 63
res.w[1] = (P256.w[3] >> shift);
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
// redundant fstar.w[3] = 0;
fstar.w[2] = P256.w[2] & maskhigh128[ind - 1];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
*pfpsf |= INEXACT_EXCEPTION;
// if negative, the truncated value is already the correct result
if (!x_sign) { // if positive
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = P256.w[3] >> shift;
fstar.w[3] = P256.w[3] & maskhigh128[ind - 1];
fstar.w[2] = P256.w[2];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[3] || fstar.w[2]
|| fstar.w[1] > ten2mk128[ind - 1].w[1]
|| (fstar.w[1] == ten2mk128[ind - 1].w[1]
&& fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
*pfpsf |= INEXACT_EXCEPTION;
// if negative, the truncated value is already the correct result
if (!x_sign) { // if positive
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
}
res.w[1] = x_sign | 0x3040000000000000ull | res.w[1];
BID_RETURN (res);
} else { // if exp < 0 and q + exp <= 0
if (x_sign) { // negative rounds up to -0.0
res.w[1] = 0xb040000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else { // positive rpunds up to +1.0
res.w[1] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000001ull;
}
*pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
break;
case ROUNDING_TO_ZERO:
if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q
// need to shift right -exp digits from the coefficient; exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// (number of digits to be chopped off)
// chop off ind digits from the lower part of C1
// FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate
// FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP
// FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE
// FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE
//tmp64 = C1.w[0];
// if (ind <= 19) {
// C1.w[0] = C1.w[0] + midpoint64[ind - 1];
// } else {
// C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
// C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
// }
// if (C1.w[0] < tmp64) C1.w[1]++;
// if carry-out from C1.w[0], increment C1.w[1]
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = (C1 + 1/2 * 10^x) * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
// redundant fstar.w[3] = 0;
// redundant fstar.w[2] = 0;
// redundant fstar.w[1] = P256.w[1];
// redundant fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if ((P256.w[1] > ten2mk128[ind - 1].w[1])
|| (P256.w[1] == ten2mk128[ind - 1].w[1]
&& (P256.w[0] >= ten2mk128[ind - 1].w[0]))) {
*pfpsf |= INEXACT_EXCEPTION;
}
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 3 <= shift <= 63
res.w[1] = (P256.w[3] >> shift);
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
// redundant fstar.w[3] = 0;
fstar.w[2] = P256.w[2] & maskhigh128[ind - 1];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
*pfpsf |= INEXACT_EXCEPTION;
}
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = P256.w[3] >> shift;
fstar.w[3] = P256.w[3] & maskhigh128[ind - 1];
fstar.w[2] = P256.w[2];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[3] || fstar.w[2]
|| fstar.w[1] > ten2mk128[ind - 1].w[1]
|| (fstar.w[1] == ten2mk128[ind - 1].w[1]
&& fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
*pfpsf |= INEXACT_EXCEPTION;
}
}
res.w[1] = x_sign | 0x3040000000000000ull | res.w[1];
BID_RETURN (res);
} else { // if exp < 0 and q + exp <= 0 the result is +0 or -0
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
*pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
break;
}
BID_RETURN (res);
}
/*****************************************************************************
* BID128_round_integral_nearest_even
****************************************************************************/
BID128_FUNCTION_ARG1_NORND (bid128_round_integral_nearest_even, x)
UINT128 res;
UINT64 x_sign;
UINT64 x_exp;
int exp; // unbiased exponent
// Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64)
UINT64 tmp64;
BID_UI64DOUBLE tmp1;
unsigned int x_nr_bits;
int q, ind, shift;
UINT128 C1;
// UINT128 res is C* at first - represents up to 34 decimal digits ~ 113 bits
UINT256 fstar;
UINT256 P256;
// check for NaN or Infinity
if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) {
// x is special
if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
// if x = NaN, 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];
}
BID_RETURN (res)
} else { // x is not a NaN, so it must be infinity
if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf
// return +inf
res.w[1] = 0x7800000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else { // x is -inf
// return -inf
res.w[1] = 0xf800000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
BID_RETURN (res);
}
}
// unpack x
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];
// 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
;
}
}
// test for input equal to zero
if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
// x is 0
// return 0 preserving the sign bit and the preferred exponent
// of MAX(Q(x), 0)
if (x_exp <= (0x1820ull << 49)) {
res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull;
} else {
res.w[1] = x_sign | x_exp;
}
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
// x is not special and is not zero
// if (exp <= -(p+1)) return 0
if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
// q = 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
tmp1.d = (double) (C1.w[0] >> 32); // exact conversion
x_nr_bits =
33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
} else { // x < 2^32
tmp1.d = (double) (C1.w[0]); // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // if x < 2^53
tmp1.d = (double) C1.w[0]; // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1])
tmp1.d = (double) C1.w[1]; // exact conversion
x_nr_bits =
65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
q = nr_digits[x_nr_bits - 1].digits;
if (q == 0) {
q = 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))
q++;
}
exp = (x_exp >> 49) - 6176;
if (exp >= 0) { // -exp <= 0
// the argument is an integer already
res.w[1] = x.w[1];
res.w[0] = x.w[0];
BID_RETURN (res);
} else if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q
// need to shift right -exp digits from the coefficient; the exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// chop off ind digits from the lower part of C1
// C1 = C1 + 1/2 * 10^x where the result C1 fits in 127 bits
tmp64 = C1.w[0];
if (ind <= 19) {
C1.w[0] = C1.w[0] + midpoint64[ind - 1];
} else {
C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
}
if (C1.w[0] < tmp64)
C1.w[1]++;
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = (C1 + 1/2 * 10^x) * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
// determine the value of res and fstar
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
// redundant shift = shiftright128[ind - 1]; // shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
// redundant fstar.w[3] = 0;
// redundant fstar.w[2] = 0;
// redundant fstar.w[1] = P256.w[1];
// redundant fstar.w[0] = P256.w[0];
// fraction f* < 10^(-x) <=> midpoint
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
// if 0 < fstar < 10^(-x), subtract 1 if odd (for rounding to even)
if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
((P256.w[1] < (ten2mk128[ind - 1].w[1]))
|| ((P256.w[1] == ten2mk128[ind - 1].w[1])
&& (P256.w[0] < ten2mk128[ind - 1].w[0])))) {
// subract 1 to make even
if (res.w[0]-- == 0) {
res.w[1]--;
}
}
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 3 <= shift <= 63
res.w[1] = (P256.w[3] >> shift);
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
// redundant fstar.w[3] = 0;
fstar.w[2] = P256.w[2] & maskhigh128[ind - 1];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* < 10^(-x) <=> midpoint
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
fstar.w[2] == 0 && (fstar.w[1] < ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] < ten2mk128[ind - 1].w[0]))) {
// subract 1 to make even
if (res.w[0]-- == 0) {
res.w[1]--;
}
}
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = P256.w[3] >> shift;
fstar.w[3] = P256.w[3] & maskhigh128[ind - 1];
fstar.w[2] = P256.w[2];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* < 10^(-x) <=> midpoint
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
fstar.w[3] == 0 && fstar.w[2] == 0
&& (fstar.w[1] < ten2mk128[ind - 1].w[1]
|| (fstar.w[1] == ten2mk128[ind - 1].w[1]
&& fstar.w[0] < ten2mk128[ind - 1].w[0]))) {
// subract 1 to make even
if (res.w[0]-- == 0) {
res.w[1]--;
}
}
}
res.w[1] = x_sign | 0x3040000000000000ull | res.w[1];
BID_RETURN (res);
} else { // if ((q + exp) < 0) <=> q < -exp
// the result is +0 or -0
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
}
/*****************************************************************************
* BID128_round_integral_negative
****************************************************************************/
BID128_FUNCTION_ARG1_NORND (bid128_round_integral_negative, x)
UINT128 res;
UINT64 x_sign;
UINT64 x_exp;
int exp; // unbiased exponent
// Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo
// (all are UINT64)
BID_UI64DOUBLE tmp1;
unsigned int x_nr_bits;
int q, ind, shift;
UINT128 C1;
// UINT128 res is C* at first - represents up to 34 decimal digits ~
// 113 bits
UINT256 fstar;
UINT256 P256;
// check for NaN or Infinity
if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) {
// x is special
if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
// if x = NaN, 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];
}
BID_RETURN (res)
} else { // x is not a NaN, so it must be infinity
if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf
// return +inf
res.w[1] = 0x7800000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else { // x is -inf
// return -inf
res.w[1] = 0xf800000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
BID_RETURN (res);
}
}
// unpack x
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];
// 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
;
}
}
// test for input equal to zero
if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
// x is 0
// return 0 preserving the sign bit and the preferred exponent
// of MAX(Q(x), 0)
if (x_exp <= (0x1820ull << 49)) {
res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull;
} else {
res.w[1] = x_sign | x_exp;
}
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
// x is not special and is not zero
// if (exp <= -p) return -1.0 or +0.0
if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34
if (x_sign) {
// if negative, return negative 1, because we know the coefficient
// is non-zero (would have been caught above)
res.w[1] = 0xb040000000000000ull;
res.w[0] = 0x0000000000000001ull;
} else {
// if positive, return positive 0, because we know coefficient is
// non-zero (would have been caught above)
res.w[1] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
BID_RETURN (res);
}
// q = 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
tmp1.d = (double) (C1.w[0] >> 32); // exact conversion
x_nr_bits =
33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
} else { // x < 2^32
tmp1.d = (double) (C1.w[0]); // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // if x < 2^53
tmp1.d = (double) C1.w[0]; // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1])
tmp1.d = (double) C1.w[1]; // exact conversion
x_nr_bits =
65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
q = nr_digits[x_nr_bits - 1].digits;
if (q == 0) {
q = 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))
q++;
}
exp = (x_exp >> 49) - 6176;
if (exp >= 0) { // -exp <= 0
// the argument is an integer already
res.w[1] = x.w[1];
res.w[0] = x.w[0];
BID_RETURN (res);
} else if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q
// need to shift right -exp digits from the coefficient; the exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// (number of digits to be chopped off)
// chop off ind digits from the lower part of C1
// FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate
// FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP
// FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE
// FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE
//tmp64 = C1.w[0];
// if (ind <= 19) {
// C1.w[0] = C1.w[0] + midpoint64[ind - 1];
// } else {
// C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
// C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
// }
// if (C1.w[0] < tmp64) C1.w[1]++;
// if carry-out from C1.w[0], increment C1.w[1]
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = (C1 + 1/2 * 10^x) * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
// if positive, the truncated value is already the correct result
if (x_sign) { // if negative
// redundant fstar.w[3] = 0;
// redundant fstar.w[2] = 0;
// redundant fstar.w[1] = P256.w[1];
// redundant fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if ((P256.w[1] > ten2mk128[ind - 1].w[1])
|| (P256.w[1] == ten2mk128[ind - 1].w[1]
&& (P256.w[0] >= ten2mk128[ind - 1].w[0]))) {
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 0 <= shift <= 102
res.w[1] = (P256.w[3] >> shift);
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
// if positive, the truncated value is already the correct result
if (x_sign) { // if negative
// redundant fstar.w[3] = 0;
fstar.w[2] = P256.w[2] & maskhigh128[ind - 1];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = P256.w[3] >> shift;
// if positive, the truncated value is already the correct result
if (x_sign) { // if negative
fstar.w[3] = P256.w[3] & maskhigh128[ind - 1];
fstar.w[2] = P256.w[2];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[3] || fstar.w[2]
|| fstar.w[1] > ten2mk128[ind - 1].w[1]
|| (fstar.w[1] == ten2mk128[ind - 1].w[1]
&& fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
}
res.w[1] = x_sign | 0x3040000000000000ull | res.w[1];
BID_RETURN (res);
} else { // if exp < 0 and q + exp <= 0
if (x_sign) { // negative rounds down to -1.0
res.w[1] = 0xb040000000000000ull;
res.w[0] = 0x0000000000000001ull;
} else { // positive rpunds down to +0.0
res.w[1] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
BID_RETURN (res);
}
}
/*****************************************************************************
* BID128_round_integral_positive
****************************************************************************/
BID128_FUNCTION_ARG1_NORND (bid128_round_integral_positive, x)
UINT128 res;
UINT64 x_sign;
UINT64 x_exp;
int exp; // unbiased exponent
// Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo
// (all are UINT64)
BID_UI64DOUBLE tmp1;
unsigned int x_nr_bits;
int q, ind, shift;
UINT128 C1;
// UINT128 res is C* at first - represents up to 34 decimal digits ~
// 113 bits
UINT256 fstar;
UINT256 P256;
// check for NaN or Infinity
if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) {
// x is special
if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
// if x = NaN, 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];
}
BID_RETURN (res)
} else { // x is not a NaN, so it must be infinity
if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf
// return +inf
res.w[1] = 0x7800000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else { // x is -inf
// return -inf
res.w[1] = 0xf800000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
BID_RETURN (res);
}
}
// unpack x
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];
// 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
;
}
}
// test for input equal to zero
if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
// x is 0
// return 0 preserving the sign bit and the preferred exponent
// of MAX(Q(x), 0)
if (x_exp <= (0x1820ull << 49)) {
res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull;
} else {
res.w[1] = x_sign | x_exp;
}
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
// x is not special and is not zero
// if (exp <= -p) return -0.0 or +1.0
if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34
if (x_sign) {
// if negative, return negative 0, because we know the coefficient
// is non-zero (would have been caught above)
res.w[1] = 0xb040000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else {
// if positive, return positive 1, because we know coefficient is
// non-zero (would have been caught above)
res.w[1] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000001ull;
}
BID_RETURN (res);
}
// q = 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 64-bit value in two 32-bit halves to avoid rounding errors
if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32
tmp1.d = (double) (C1.w[0] >> 32); // exact conversion
x_nr_bits =
33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
} else { // x < 2^32
tmp1.d = (double) (C1.w[0]); // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // if x < 2^53
tmp1.d = (double) C1.w[0]; // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1])
tmp1.d = (double) C1.w[1]; // exact conversion
x_nr_bits =
65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
q = nr_digits[x_nr_bits - 1].digits;
if (q == 0) {
q = 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))
q++;
}
exp = (x_exp >> 49) - 6176;
if (exp >= 0) { // -exp <= 0
// the argument is an integer already
res.w[1] = x.w[1];
res.w[0] = x.w[0];
BID_RETURN (res);
} else if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q
// need to shift right -exp digits from the coefficient; exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// (number of digits to be chopped off)
// chop off ind digits from the lower part of C1
// FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate
// FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP
// FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE
// FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE
// tmp64 = C1.w[0];
// if (ind <= 19) {
// C1.w[0] = C1.w[0] + midpoint64[ind - 1];
// } else {
// C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
// C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
// }
// if (C1.w[0] < tmp64) C1.w[1]++;
// if carry-out from C1.w[0], increment C1.w[1]
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = C1 * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
// if negative, the truncated value is already the correct result
if (!x_sign) { // if positive
// redundant fstar.w[3] = 0;
// redundant fstar.w[2] = 0;
// redundant fstar.w[1] = P256.w[1];
// redundant fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if ((P256.w[1] > ten2mk128[ind - 1].w[1])
|| (P256.w[1] == ten2mk128[ind - 1].w[1]
&& (P256.w[0] >= ten2mk128[ind - 1].w[0]))) {
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 3 <= shift <= 63
res.w[1] = (P256.w[3] >> shift);
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
// if negative, the truncated value is already the correct result
if (!x_sign) { // if positive
// redundant fstar.w[3] = 0;
fstar.w[2] = P256.w[2] & maskhigh128[ind - 1];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] ||
(fstar.w[1] == ten2mk128[ind - 1].w[1] &&
fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = P256.w[3] >> shift;
// if negative, the truncated value is already the correct result
if (!x_sign) { // if positive
fstar.w[3] = P256.w[3] & maskhigh128[ind - 1];
fstar.w[2] = P256.w[2];
fstar.w[1] = P256.w[1];
fstar.w[0] = P256.w[0];
// fraction f* > 10^(-x) <=> inexact
// f* is in the right position to be compared with
// 10^(-x) from ten2mk128[]
if (fstar.w[3] || fstar.w[2]
|| fstar.w[1] > ten2mk128[ind - 1].w[1]
|| (fstar.w[1] == ten2mk128[ind - 1].w[1]
&& fstar.w[0] >= ten2mk128[ind - 1].w[0])) {
if (++res.w[0] == 0) {
res.w[1]++;
}
}
}
}
res.w[1] = x_sign | 0x3040000000000000ull | res.w[1];
BID_RETURN (res);
} else { // if exp < 0 and q + exp <= 0
if (x_sign) { // negative rounds up to -0.0
res.w[1] = 0xb040000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else { // positive rpunds up to +1.0
res.w[1] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000001ull;
}
BID_RETURN (res);
}
}
/*****************************************************************************
* BID128_round_integral_zero
****************************************************************************/
BID128_FUNCTION_ARG1_NORND (bid128_round_integral_zero, x)
UINT128 res;
UINT64 x_sign;
UINT64 x_exp;
int exp; // unbiased exponent
// Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo
// (all are UINT64)
BID_UI64DOUBLE tmp1;
unsigned int x_nr_bits;
int q, ind, shift;
UINT128 C1;
// UINT128 res is C* at first - represents up to 34 decimal digits ~
// 113 bits
UINT256 P256;
// check for NaN or Infinity
if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) {
// x is special
if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
// if x = NaN, 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];
}
BID_RETURN (res)
} else { // x is not a NaN, so it must be infinity
if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf
// return +inf
res.w[1] = 0x7800000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else { // x is -inf
// return -inf
res.w[1] = 0xf800000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
BID_RETURN (res);
}
}
// unpack x
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];
// 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
;
}
}
// test for input equal to zero
if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
// x is 0
// return 0 preserving the sign bit and the preferred exponent
// of MAX(Q(x), 0)
if (x_exp <= (0x1820ull << 49)) {
res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull;
} else {
res.w[1] = x_sign | x_exp;
}
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
// x is not special and is not zero
// if (exp <= -p) return -0.0 or +0.0
if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
// q = 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
tmp1.d = (double) (C1.w[0] >> 32); // exact conversion
x_nr_bits =
33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
} else { // x < 2^32
tmp1.d = (double) (C1.w[0]); // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // if x < 2^53
tmp1.d = (double) C1.w[0]; // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1])
tmp1.d = (double) C1.w[1]; // exact conversion
x_nr_bits =
65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
q = nr_digits[x_nr_bits - 1].digits;
if (q == 0) {
q = 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))
q++;
}
exp = (x_exp >> 49) - 6176;
if (exp >= 0) { // -exp <= 0
// the argument is an integer already
res.w[1] = x.w[1];
res.w[0] = x.w[0];
BID_RETURN (res);
} else if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q
// need to shift right -exp digits from the coefficient; the exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// (number of digits to be chopped off)
// chop off ind digits from the lower part of C1
// FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate
// FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP
// FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE
// FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE
//tmp64 = C1.w[0];
// if (ind <= 19) {
// C1.w[0] = C1.w[0] + midpoint64[ind - 1];
// } else {
// C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
// C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
// }
// if (C1.w[0] < tmp64) C1.w[1]++;
// if carry-out from C1.w[0], increment C1.w[1]
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = (C1 + 1/2 * 10^x) * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 3 <= shift <= 63
res.w[1] = (P256.w[3] >> shift);
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = P256.w[3] >> shift;
}
res.w[1] = x_sign | 0x3040000000000000ull | res.w[1];
BID_RETURN (res);
} else { // if exp < 0 and q + exp <= 0 the result is +0 or -0
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
}
/*****************************************************************************
* BID128_round_integral_nearest_away
****************************************************************************/
BID128_FUNCTION_ARG1_NORND (bid128_round_integral_nearest_away, x)
UINT128 res;
UINT64 x_sign;
UINT64 x_exp;
int exp; // unbiased exponent
// Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo
// (all are UINT64)
UINT64 tmp64;
BID_UI64DOUBLE tmp1;
unsigned int x_nr_bits;
int q, ind, shift;
UINT128 C1;
// UINT128 res is C* at first - represents up to 34 decimal digits ~
// 113 bits
// UINT256 fstar;
UINT256 P256;
// check for NaN or Infinity
if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) {
// x is special
if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
// if x = NaN, 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];
}
BID_RETURN (res)
} else { // x is not a NaN, so it must be infinity
if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf
// return +inf
res.w[1] = 0x7800000000000000ull;
res.w[0] = 0x0000000000000000ull;
} else { // x is -inf
// return -inf
res.w[1] = 0xf800000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
BID_RETURN (res);
}
}
// unpack x
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];
// 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
;
}
}
// test for input equal to zero
if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
// x is 0
// return 0 preserving the sign bit and the preferred exponent
// of MAX(Q(x), 0)
if (x_exp <= (0x1820ull << 49)) {
res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull;
} else {
res.w[1] = x_sign | x_exp;
}
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
// x is not special and is not zero
// if (exp <= -(p+1)) return 0.0
if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
// q = 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
tmp1.d = (double) (C1.w[0] >> 32); // exact conversion
x_nr_bits =
33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
} else { // x < 2^32
tmp1.d = (double) (C1.w[0]); // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // if x < 2^53
tmp1.d = (double) C1.w[0]; // exact conversion
x_nr_bits =
1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
} else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1])
tmp1.d = (double) C1.w[1]; // exact conversion
x_nr_bits =
65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
q = nr_digits[x_nr_bits - 1].digits;
if (q == 0) {
q = 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))
q++;
}
exp = (x_exp >> 49) - 6176;
if (exp >= 0) { // -exp <= 0
// the argument is an integer already
res.w[1] = x.w[1];
res.w[0] = x.w[0];
BID_RETURN (res);
} else if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q
// need to shift right -exp digits from the coefficient; the exp will be 0
ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x'
// chop off ind digits from the lower part of C1
// C1 = C1 + 1/2 * 10^x where the result C1 fits in 127 bits
tmp64 = C1.w[0];
if (ind <= 19) {
C1.w[0] = C1.w[0] + midpoint64[ind - 1];
} else {
C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0];
C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1];
}
if (C1.w[0] < tmp64)
C1.w[1]++;
// calculate C* and f*
// C* is actually floor(C*) in this case
// C* and f* need shifting and masking, as shown by
// shiftright128[] and maskhigh128[]
// 1 <= x <= 34
// kx = 10^(-x) = ten2mk128[ind - 1]
// C* = (C1 + 1/2 * 10^x) * 10^(-x)
// the approximation of 10^(-x) was rounded up to 118 bits
__mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]);
// the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g.
// if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
// if (0 < f* < 10^(-x)) then the result is a midpoint
// if floor(C*) is even then C* = floor(C*) - logical right
// shift; C* has p decimal digits, correct by Prop. 1)
// else if floor(C*) is odd C* = floor(C*)-1 (logical right
// shift; C* has p decimal digits, correct by Pr. 1)
// else
// C* = floor(C*) (logical right shift; C has p decimal digits,
// correct by Property 1)
// n = C* * 10^(e+x)
// shift right C* by Ex-128 = shiftright128[ind]
if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
res.w[1] = P256.w[3];
res.w[0] = P256.w[2];
} else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
shift = shiftright128[ind - 1]; // 3 <= shift <= 63
res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift);
res.w[1] = (P256.w[3] >> shift);
} else { // 22 <= ind - 1 <= 33
shift = shiftright128[ind - 1]; // 2 <= shift <= 38
res.w[1] = 0;
res.w[0] = (P256.w[3] >> (shift - 64)); // 2 <= shift - 64 <= 38
}
// if the result was a midpoint, it was already rounded away from zero
res.w[1] |= x_sign | 0x3040000000000000ull;
BID_RETURN (res);
} else { // if ((q + exp) < 0) <=> q < -exp
// the result is +0 or -0
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
BID_RETURN (res);
}
}