* BID128_round_integral_exact
****************************************************************************/
-BID128_FUNCTION_ARG1(__bid128_round_integral_exact, x)
+BID128_FUNCTION_ARG1 (bid128_round_integral_exact, x)
- UINT128 res = {{ 0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull }};
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ 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;
+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.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Quiet (SNaN)
- res.w[1] = x.w[1] & 0xfdffffffffffffffull;
- res.w[0] = x.w[0];
- } else { // x is QNaN
- // return the input QNaN
- res.w[1] = x.w[1];
- 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);
+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
- x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions
- C1.w[1] = x.w[1] & MASK_COEFF;
- C1.w[0] = x.w[0];
-
- // test for non-canonical values:
- // - values whose encoding begins with x00, x01, or x10 and whose
- // coefficient is larger than 10^34 -1, or
- // - values whose encoding begins with x1100, x1101, x1110 (if NaNs
- // and infinitis were eliminated already this test is reduced to
- // checking for x10x)
+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];
- // test for non-canonical values of the argument x
- if ((((C1.w[1] > 0x0001ed09bead87c0ull) ||
- ((C1.w[1] == 0x0001ed09bead87c0ull) &&
- (C1.w[0] > 0x378d8e63ffffffffull))) &&
- ((x.w[1] & 0x6000000000000000ull) != 0x6000000000000000ull)) ||
- ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- x.w[1] = x.w[1] & 0x8000000000000000ull; // preserve the sign bit
- x.w[0] = 0;
- x_exp = 0x1820ull << 49;
+ // 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.w[1] & 0xfffe000000000000ull;
- }
- res.w[0] = 0x0000000000000000ull;
- BID_RETURN (res);
+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 = __bid_nr_digits[x_nr_bits - 1].digits;
- if (q == 0) {
- q = __bid_nr_digits[x_nr_bits - 1].digits1;
- if (C1.w[1] > __bid_nr_digits[x_nr_bits - 1].threshold_hi
- || (C1.w[1] == __bid_nr_digits[x_nr_bits - 1].threshold_hi
- && C1.w[0] >= __bid_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];
+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);
}
- // 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] + __bid_midpoint64[ind - 1];
- } else {
- C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- C1.w[1] = C1.w[1] + __bid_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
- // __bid_shiftright128[] and __bid_maskhigh128[]
- // 1 <= x <= 34
- // kx = 10^(-x) = __bid_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, __bid_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 __bid_ten2mk128[] instead of __bid_ten2mk128trunc[]
-
- if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
- // redundant shift = __bid_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 __bid_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] < (__bid_ten2mk128[ind - 1].w[1]))
- || ((fstar.w[1] == __bid_ten2mk128[ind - 1].w[1])
- && (fstar.w[0] < __bid_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 > __bid_ten2mk128[ind - 1].w[1]
- || (tmp64 == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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 = __bid_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] & __bid_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 __bid_ten2mk128[]
- if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
- fstar.w[2] == 0 && (fstar.w[1] < __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] < __bid_ten2mk128[ind - 1].w[0]))) {
- // subract 1 to make even
- if (res.w[0]-- == 0) {
- res.w[1]--;
- }
- }
- if (fstar.w[2] > __bid_one_half128[ind - 1]
- || (fstar.w[2] == __bid_one_half128[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] - __bid_one_half128[ind - 1];
- if (tmp64 || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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 = __bid_shiftright128[ind - 1] - 64; // 2 <= shift <= 38
- res.w[1] = 0;
- res.w[0] = P256.w[3] >> shift;
- fstar.w[3] = P256.w[3] & __bid_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 __bid_ten2mk128[]
- if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
- fstar.w[3] == 0 && fstar.w[2] == 0
- && (fstar.w[1] < __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] < __bid_ten2mk128[ind - 1].w[0]))) {
- // subract 1 to make even
- if (res.w[0]-- == 0) {
- res.w[1]--;
- }
- }
- if (fstar.w[3] > __bid_one_half128[ind - 1]
- || (fstar.w[3] == __bid_one_half128[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] - __bid_one_half128[ind - 1];
- if (tmp64 || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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] + __bid_midpoint64[ind - 1];
- } else {
- C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- C1.w[1] = C1.w[1] + __bid_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
- // __bid_shiftright128[] and __bid_maskhigh128[]
- // 1 <= x <= 34
- // kx = 10^(-x) = __bid_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, __bid_ten2mk128[ind - 1]);
- // the top Ex bits of 10^(-x) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x=1, T*=__bid_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 __bid_ten2mk128[] instead of __bid_ten2mk128trunc[]
- // shift right C* by Ex-128 = __bid_shiftright128[ind]
- if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0
- // redundant shift = __bid_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 > __bid_ten2mk128[ind - 1].w[1]
- || (tmp64 == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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 = __bid_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] & __bid_maskhigh128[ind - 1];
- fstar.w[1] = P256.w[1];
- fstar.w[0] = P256.w[0];
- if (fstar.w[2] > __bid_one_half128[ind - 1]
- || (fstar.w[2] == __bid_one_half128[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] - __bid_one_half128[ind - 1];
- if (tmp64 || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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 = __bid_shiftright128[ind - 1] - 64; // 2 <= shift <= 38
- res.w[1] = 0;
- res.w[0] = P256.w[3] >> shift;
- fstar.w[3] = P256.w[3] & __bid_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] > __bid_one_half128[ind - 1]
- || (fstar.w[3] == __bid_one_half128[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] - __bid_one_half128[ind - 1];
- if (tmp64 || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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] + __bid_midpoint64[ind - 1];
- // } else {
- // C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- // C1.w[1] = C1.w[1] + __bid_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
- // __bid_shiftright128[] and __bid_maskhigh128[]
- // 1 <= x <= 34
- // kx = 10^(-x) = __bid_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, __bid_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 __bid_ten2mk128[]
- if ((P256.w[1] > __bid_ten2mk128[ind - 1].w[1])
- || (P256.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && (P256.w[0] >= __bid_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 = __bid_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] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[2] || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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 = __bid_shiftright128[ind - 1] - 64; // 2 <= shift <= 38
- res.w[1] = 0;
- res.w[0] = P256.w[3] >> shift;
- fstar.w[3] = P256.w[3] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[3] || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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] + __bid_midpoint64[ind - 1];
- // } else {
- // C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- // C1.w[1] = C1.w[1] + __bid_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
- // __bid_shiftright128[] and __bid_maskhigh128[]
- // 1 <= x <= 34
- // kx = 10^(-x) = __bid_ten2mk128[ind - 1]
- // C* = C1 * 10^(-x)
- // the approximation of 10^(-x) was rounded up to 118 bits
- __mul_128x128_to_256 (P256, C1, __bid_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 __bid_ten2mk128[]
- if ((P256.w[1] > __bid_ten2mk128[ind - 1].w[1])
- || (P256.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && (P256.w[0] >= __bid_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 = __bid_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] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[2] || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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 = __bid_shiftright128[ind - 1] - 64; // 2 <= shift <= 38
- res.w[1] = 0;
- res.w[0] = P256.w[3] >> shift;
- fstar.w[3] = P256.w[3] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[3] || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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] + __bid_midpoint64[ind - 1];
- // } else {
- // C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- // C1.w[1] = C1.w[1] + __bid_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
- // __bid_shiftright128[] and __bid_maskhigh128[]
- // 1 <= x <= 34
- // kx = 10^(-x) = __bid_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, __bid_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 __bid_ten2mk128[]
- if ((P256.w[1] > __bid_ten2mk128[ind - 1].w[1])
- || (P256.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && (P256.w[0] >= __bid_ten2mk128[ind - 1].w[0]))) {
- *pfpsf |= INEXACT_EXCEPTION;
- }
- } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63
- shift = __bid_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] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[2] || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_ten2mk128[ind - 1].w[0])) {
- *pfpsf |= INEXACT_EXCEPTION;
- }
- } else { // 22 <= ind - 1 <= 33
- shift = __bid_shiftright128[ind - 1] - 64; // 2 <= shift <= 38
- res.w[1] = 0;
- res.w[0] = P256.w[3] >> shift;
- fstar.w[3] = P256.w[3] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[3] || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_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;
+ 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;
- }
- 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.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Quiet (SNaN)
- res.w[1] = x.w[1] & 0xfdffffffffffffffull;
- res.w[0] = x.w[0];
- } else { // x is QNaN
- // return the input QNaN
- res.w[1] = x.w[1];
- 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);
}
+ *pfpsf |= INEXACT_EXCEPTION;
+ BID_RETURN (res);
}
- // unpack x
- x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
- x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions
- C1.w[1] = x.w[1] & MASK_COEFF;
- C1.w[0] = x.w[0];
-
- // test for non-canonical values:
- // - values whose encoding begins with x00, x01, or x10 and whose
- // coefficient is larger than 10^34 -1, or
- // - values whose encoding begins with x1100, x1101, x1110 (if NaNs
- // and infinitis were eliminated already this test is reduced to
- // checking for x10x)
-
- // test for non-canonical values of the argument x
- if ((((C1.w[1] > 0x0001ed09bead87c0ull)
- || ((C1.w[1] == 0x0001ed09bead87c0ull)
- && (C1.w[0] > 0x378d8e63ffffffffull)))
- && ((x.w[1] & 0x6000000000000000ull) != 0x6000000000000000ull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- x.w[1] = x.w[1] & 0x8000000000000000ull; // preserve the sign bit
- x.w[0] = 0;
- x_exp = 0x1820ull << 49;
- C1.w[1] = 0;
- C1.w[0] = 0;
- }
- // 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;
+ 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 {
- res.w[1] = x.w[1] & 0xfffe000000000000ull;
+ // 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;
}
- res.w[0] = 0x0000000000000000ull;
+ *pfpsf |= INEXACT_EXCEPTION;
BID_RETURN (res);
}
- // x is not special and is not zero
-
- // if (exp <= -(p+1)) return 0
- if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35
+ 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
+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 { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1])
- tmp1.d = (double) C1.w[1]; // exact conversion
+ } else { // if x < 2^53
+ tmp1.d = (double) C1.w[0]; // exact conversion
x_nr_bits =
- 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- q = __bid_nr_digits[x_nr_bits - 1].digits;
- if (q == 0) {
- q = __bid_nr_digits[x_nr_bits - 1].digits1;
- if (C1.w[1] > __bid_nr_digits[x_nr_bits - 1].threshold_hi
- || (C1.w[1] == __bid_nr_digits[x_nr_bits - 1].threshold_hi
- && C1.w[0] >= __bid_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
+} 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] + __bid_midpoint64[ind - 1];
+ C1.w[0] = C1.w[0] + midpoint64[ind - 1];
} else {
- C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- C1.w[1] = C1.w[1] + __bid_midpoint128[ind - 20].w[1];
+ 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
- // __bid_shiftright128[] and __bid_maskhigh128[]
+ // shiftright128[] and maskhigh128[]
// 1 <= x <= 34
- // kx = 10^(-x) = __bid_ten2mk128[ind - 1]
+ // 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, __bid_ten2mk128[ind - 1]);
+ __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 = __bid_shiftright128[ind - 1]; // 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];
+ 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 __bid_ten2mk128[]
+ // 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] < (__bid_ten2mk128[ind - 1].w[1]))
- || ((P256.w[1] == __bid_ten2mk128[ind - 1].w[1])
- && (P256.w[0] < __bid_ten2mk128[ind - 1].w[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[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 = __bid_shiftright128[ind - 1]; // 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] & __bid_maskhigh128[ind - 1];
+ 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 __bid_ten2mk128[]
+ // 10^(-x) from ten2mk128[]
if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
- fstar.w[2] == 0 && (fstar.w[1] < __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] < __bid_ten2mk128[ind - 1].w[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[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 = __bid_shiftright128[ind - 1] - 64; // 2 <= shift <= 38
+ 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] & __bid_maskhigh128[ind - 1];
+ 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 __bid_ten2mk128[]
+ // 10^(-x) from ten2mk128[]
if ((res.w[0] & 0x0000000000000001ull) && // is result odd?
- fstar.w[3] == 0 && fstar.w[2] == 0
- && (fstar.w[1] < __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] < __bid_ten2mk128[ind - 1].w[0]))) {
+ 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);
// the result is +0 or -0
res.w[1] = x_sign | 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
+ *pfpsf |= INEXACT_EXCEPTION;
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;
+ 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)
- // 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.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Quiet (SNaN)
- res.w[1] = x.w[1] & 0xfdffffffffffffffull;
- res.w[0] = x.w[0];
- } else { // x is QNaN
- // return the input QNaN
- res.w[1] = x.w[1];
- res.w[0] = x.w[0];
+ // 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;
}
- 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;
+ } 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;
}
- BID_RETURN (res);
}
- }
- // unpack x
- x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
- x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions
- C1.w[1] = x.w[1] & MASK_COEFF;
- C1.w[0] = x.w[0];
-
- // test for non-canonical values:
- // - values whose encoding begins with x00, x01, or x10 and whose
- // coefficient is larger than 10^34 -1, or
- // - values whose encoding begins with x1100, x1101, x1110 (if NaNs
- // and infinitis were eliminated already this test is reduced to
- // checking for x10x)
-
- // test for non-canonical values of the argument x
- if ((((C1.w[1] > 0x0001ed09bead87c0ull)
- || ((C1.w[1] == 0x0001ed09bead87c0ull)
- && (C1.w[0] > 0x378d8e63ffffffffull)))
- && ((x.w[1] & 0x6000000000000000ull) != 0x6000000000000000ull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- x.w[1] = x.w[1] & 0x8000000000000000ull; // preserve the sign bit
- x.w[0] = 0;
- x_exp = 0x1820ull << 49;
- C1.w[1] = 0;
- C1.w[0] = 0;
- }
- // 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.w[1] & 0xfffe000000000000ull;
- }
- res.w[0] = 0x0000000000000000ull;
+ // if the result was a midpoint, it was already rounded away from zero
+ res.w[1] |= x_sign | 0x3040000000000000ull;
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;
- }
+ } 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);
}
- // 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 = __bid_nr_digits[x_nr_bits - 1].digits;
- if (q == 0) {
- q = __bid_nr_digits[x_nr_bits - 1].digits1;
- if (C1.w[1] > __bid_nr_digits[x_nr_bits - 1].threshold_hi
- || (C1.w[1] == __bid_nr_digits[x_nr_bits - 1].threshold_hi
- && C1.w[0] >= __bid_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
+ 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_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];
+ // tmp64 = C1.w[0];
// if (ind <= 19) {
- // C1.w[0] = C1.w[0] + __bid_midpoint64[ind - 1];
+ // C1.w[0] = C1.w[0] + midpoint64[ind - 1];
// } else {
- // C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- // C1.w[1] = C1.w[1] + __bid_midpoint128[ind - 20].w[1];
+ // 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
- // __bid_shiftright128[] and __bid_maskhigh128[]
+ // shiftright128[] and maskhigh128[]
// 1 <= x <= 34
- // kx = 10^(-x) = __bid_ten2mk128[ind - 1]
+ // 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, __bid_ten2mk128[ind - 1]);
+ __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 __bid_ten2mk128[]
- if ((P256.w[1] > __bid_ten2mk128[ind - 1].w[1])
- || (P256.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && (P256.w[0] >= __bid_ten2mk128[ind - 1].w[0]))) {
+ // 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 = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
+ 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] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[2] || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_ten2mk128[ind - 1].w[0])) {
+ // 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 = __bid_shiftright128[ind - 1] - 64; // 2 <= shift <= 38
+ 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] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[3] || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_ten2mk128[ind - 1].w[0])) {
+ 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] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000000ull;
}
+ *pfpsf |= INEXACT_EXCEPTION;
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.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Quiet (SNaN)
- res.w[1] = x.w[1] & 0xfdffffffffffffffull;
- res.w[0] = x.w[0];
- } else { // x is QNaN
- // return the input QNaN
- res.w[1] = x.w[1];
- 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
- x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions
- C1.w[1] = x.w[1] & MASK_COEFF;
- C1.w[0] = x.w[0];
-
- // test for non-canonical values:
- // - values whose encoding begins with x00, x01, or x10 and whose
- // coefficient is larger than 10^34 -1, or
- // - values whose encoding begins with x1100, x1101, x1110 (if NaNs
- // and infinitis were eliminated already this test is reduced to
- // checking for x10x)
-
- // test for non-canonical values of the argument x
- if ((((C1.w[1] > 0x0001ed09bead87c0ull)
- || ((C1.w[1] == 0x0001ed09bead87c0ull)
- && (C1.w[0] > 0x378d8e63ffffffffull)))
- && ((x.w[1] & 0x6000000000000000ull) != 0x6000000000000000ull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- x.w[1] = x.w[1] & 0x8000000000000000ull; // preserve the sign bit
- x.w[0] = 0;
- x_exp = 0x1820ull << 49;
- C1.w[1] = 0;
- C1.w[0] = 0;
- }
- // 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.w[1] & 0xfffe000000000000ull;
- }
- 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 = __bid_nr_digits[x_nr_bits - 1].digits;
- if (q == 0) {
- q = __bid_nr_digits[x_nr_bits - 1].digits1;
- if (C1.w[1] > __bid_nr_digits[x_nr_bits - 1].threshold_hi
- || (C1.w[1] == __bid_nr_digits[x_nr_bits - 1].threshold_hi
- && C1.w[0] >= __bid_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
+ 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)
// 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] + __bid_midpoint64[ind - 1];
+ // C1.w[0] = C1.w[0] + midpoint64[ind - 1];
// } else {
- // C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- // C1.w[1] = C1.w[1] + __bid_midpoint128[ind - 20].w[1];
+ // 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
- // __bid_shiftright128[] and __bid_maskhigh128[]
+ // shiftright128[] and maskhigh128[]
// 1 <= x <= 34
- // kx = 10^(-x) = __bid_ten2mk128[ind - 1]
+ // 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, __bid_ten2mk128[ind - 1]);
+ __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 __bid_ten2mk128[]
- if ((P256.w[1] > __bid_ten2mk128[ind - 1].w[1])
- || (P256.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && (P256.w[0] >= __bid_ten2mk128[ind - 1].w[0]))) {
+ // 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 = __bid_shiftright128[ind - 1]; // 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] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[2] || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_ten2mk128[ind - 1].w[0])) {
+ // 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 = __bid_shiftright128[ind - 1] - 64; // 2 <= shift <= 38
+ 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] & __bid_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 __bid_ten2mk128[]
- if (fstar.w[3] || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128[ind - 1].w[1]
- && fstar.w[0] >= __bid_ten2mk128[ind - 1].w[0])) {
+ 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] = 0x3040000000000000ull;
res.w[0] = 0x0000000000000001ull;
}
+ *pfpsf |= INEXACT_EXCEPTION;
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.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Quiet (SNaN)
- res.w[1] = x.w[1] & 0xfdffffffffffffffull;
- res.w[0] = x.w[0];
- } else { // x is QNaN
- // return the input QNaN
- res.w[1] = x.w[1];
- 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
- x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions
- C1.w[1] = x.w[1] & MASK_COEFF;
- C1.w[0] = x.w[0];
-
- // test for non-canonical values:
- // - values whose encoding begins with x00, x01, or x10 and whose
- // coefficient is larger than 10^34 -1, or
- // - values whose encoding begins with x1100, x1101, x1110 (if NaNs
- // and infinitis were eliminated already this test is reduced to
- // checking for x10x)
-
- // test for non-canonical values of the argument x
- if ((((C1.w[1] > 0x0001ed09bead87c0ull)
- || ((C1.w[1] == 0x0001ed09bead87c0ull)
- && (C1.w[0] > 0x378d8e63ffffffffull)))
- && ((x.w[1] & 0x6000000000000000ull) != 0x6000000000000000ull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- x.w[1] = x.w[1] & 0x8000000000000000ull; // preserve the sign bit
- x.w[0] = 0;
- x_exp = 0x1820ull << 49;
- C1.w[1] = 0;
- C1.w[0] = 0;
- }
- // 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.w[1] & 0xfffe000000000000ull;
- }
- 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 = __bid_nr_digits[x_nr_bits - 1].digits;
- if (q == 0) {
- q = __bid_nr_digits[x_nr_bits - 1].digits1;
- if (C1.w[1] > __bid_nr_digits[x_nr_bits - 1].threshold_hi
- || (C1.w[1] == __bid_nr_digits[x_nr_bits - 1].threshold_hi
- && C1.w[0] >= __bid_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
+ 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_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE
//tmp64 = C1.w[0];
// if (ind <= 19) {
- // C1.w[0] = C1.w[0] + __bid_midpoint64[ind - 1];
+ // C1.w[0] = C1.w[0] + midpoint64[ind - 1];
// } else {
- // C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- // C1.w[1] = C1.w[1] + __bid_midpoint128[ind - 20].w[1];
+ // 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
- // __bid_shiftright128[] and __bid_maskhigh128[]
+ // shiftright128[] and maskhigh128[]
// 1 <= x <= 34
- // kx = 10^(-x) = __bid_ten2mk128[ind - 1]
+ // 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, __bid_ten2mk128[ind - 1]);
+ __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 = __bid_shiftright128[ind - 1]; // 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 = __bid_shiftright128[ind - 1] - 64; // 2 <= shift <= 38
+ 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_away
+ * BID128_round_integral_nearest_even
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND(__bid128_round_integral_nearest_away, x)
+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;
+ 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) {
+if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) {
// x is special
- if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
- if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Quiet (SNaN)
- res.w[1] = x.w[1] & 0xfdffffffffffffffull;
- res.w[0] = x.w[0];
- } else { // x is QNaN
- // return the input QNaN
- res.w[1] = x.w[1];
- 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);
- }
+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
- x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions
- C1.w[1] = x.w[1] & MASK_COEFF;
- C1.w[0] = x.w[0];
+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];
- // test for non-canonical values:
- // - values whose encoding begins with x00, x01, or x10 and whose
- // coefficient is larger than 10^34 -1, or
- // - values whose encoding begins with x1100, x1101, x1110 (if NaNs
- // and infinitis were eliminated already this test is reduced to
- // checking for x10x)
-
- // test for non-canonical values of the argument x
- if ((((C1.w[1] > 0x0001ed09bead87c0ull)
- || ((C1.w[1] == 0x0001ed09bead87c0ull)
- && (C1.w[0] > 0x378d8e63ffffffffull)))
- && ((x.w[1] & 0x6000000000000000ull) != 0x6000000000000000ull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- x.w[1] = x.w[1] & 0x8000000000000000ull; // preserve the sign bit
- x.w[0] = 0;
- x_exp = 0x1820ull << 49;
+ // 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.w[1] & 0xfffe000000000000ull;
- }
- res.w[0] = 0x0000000000000000ull;
- BID_RETURN (res);
+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);
- }
+ // 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
+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 { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1])
- tmp1.d = (double) C1.w[1]; // exact conversion
+ } else { // if x < 2^53
+ tmp1.d = (double) C1.w[0]; // exact conversion
x_nr_bits =
- 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- q = __bid_nr_digits[x_nr_bits - 1].digits;
- if (q == 0) {
- q = __bid_nr_digits[x_nr_bits - 1].digits1;
- if (C1.w[1] > __bid_nr_digits[x_nr_bits - 1].threshold_hi
- || (C1.w[1] == __bid_nr_digits[x_nr_bits - 1].threshold_hi
- && C1.w[0] >= __bid_nr_digits[x_nr_bits - 1].threshold_lo))
- q++;
+} 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];
}
- 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] + __bid_midpoint64[ind - 1];
- } else {
- C1.w[0] = C1.w[0] + __bid_midpoint128[ind - 20].w[0];
- C1.w[1] = C1.w[1] + __bid_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]--;
+ }
}
- 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
- // __bid_shiftright128[] and __bid_maskhigh128[]
- // 1 <= x <= 34
- // kx = 10^(-x) = __bid_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, __bid_ten2mk128[ind - 1]);
- // the top Ex bits of 10^(-x) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x=1, T*=__bid_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 = __bid_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 = __bid_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 = __bid_shiftright128[ind - 1]; // 2 <= shift <= 38
- res.w[1] = 0;
- res.w[0] = (P256.w[3] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ } 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 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;
+ } 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;
- BID_RETURN (res);
+ } 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);
+}
}
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