--- /dev/null
+/* Copyright (C) 2007, 2009 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);
+}
+}
projects / msp430-gcc.git / blobdiff