gcc/libgcc/config/libbid/bid128_round_integral.c

/* Copyright (C) 2007-2022 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);
}
}