* BID128_to_uint64_rnint
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_rnint, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64,
+ bid128_to_uint64_rnint, x)
- UINT64 res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ UINT64 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, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 fstar;
- UINT256 P256;
+ UINT64 tmp64;
+ BID_UI64DOUBLE tmp1;
+ unsigned int x_nr_bits;
+ int q, ind, shift;
+ UINT128 C1, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 fstar;
+ UINT256 P256;
// 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
+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];
// 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 Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- }
+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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
}
+ BID_RETURN (res);
+}
+}
// check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // x is not special and is not zero
-
- // 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] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // 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 { // 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;
+ } 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- if (x_sign) { // if n < 0 and q + exp = 20
- // if n < -1/2 then n cannot be converted to uint64 with RN
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 1/2
- // <=> 0.c(0)c(1)...c(q-1) * 10^21 > 0x05, 1<=q<=34
- // <=> C * 10^(21-q) > 0x05, 1<=q<=34
- if (q == 21) {
- // C > 5
- if (C1.w[1] != 0 || C1.w[0] > 0x05ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to 64-bit unsigned int fall through
- // to '1 <= q + exp <= 20'
- } else {
- // if 1 <= q <= 20
- // C * 10^(21-q) > 5 is true because C >= 1 and 10^(21-q) >= 10
- // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C > 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64
+ if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // if n < -1/2 then n cannot be converted to uint64 with RN
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 1/2
+ // <=> 0.c(0)c(1)...c(q-1) * 10^21 > 0x05, 1<=q<=34
+ // <=> C * 10^(21-q) > 0x05, 1<=q<=34
+ if (q == 21) {
+ // C > 5
+ if (C1.w[1] != 0 || C1.w[0] > 0x05ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // if n > 0 and q + exp = 20
- // if n >= 2^64 - 1/2 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1)
- // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34
- if (q == 1) {
- // C * 10^20 >= 0x9fffffffffffffffb
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q <= 19) {
- // C * 10^(21-q) >= 0x9fffffffffffffffb
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 20) {
- // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff
- C.w[0] = C1.w[0] + C1.w[0];
- C.w[1] = C1.w[1] + C1.w[1];
- if (C.w[0] < C1.w[0])
- C.w[1]++;
- if (C.w[1] > 0x01 || (C.w[1] == 0x01
- && C.w[0] >= 0xffffffffffffffffull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 21) {
- // C >= 0x9fffffffffffffffb
- if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
- && C1.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits
- C.w[1] = 0x09;
- C.w[0] = 0xfffffffffffffffbull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- }
+ // else cases that can be rounded to 64-bit unsigned int fall through
+ // to '1 <= q + exp <= 20'
+ } else {
+ // if 1 <= q <= 20
+ // C * 10^(21-q) > 5 is true because C >= 1 and 10^(21-q) >= 10
+ // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C > 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- }
- // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1)
- // return 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1)
- // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1)
- // res = 0
- // else if x > 0
- // res = +1
- // else // if x < 0
- // invalid exc
- ind = q - 1;
- if (ind <= 18) { // 0 <= ind <= 18
- if ((C1.w[1] == 0) && (C1.w[0] <= __bid_midpoint64[ind])) {
- res = 0x0000000000000000ull; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ } else { // if n > 0 and q + exp = 20
+ // if n >= 2^64 - 1/2 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1)
+ // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34
+ if (q == 1) {
+ // C * 10^20 >= 0x9fffffffffffffffb
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q <= 19) {
+ // C * 10^(21-q) >= 0x9fffffffffffffffb
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 20) {
+ // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff
+ C.w[0] = C1.w[0] + C1.w[0];
+ C.w[1] = C1.w[1] + C1.w[1];
+ if (C.w[0] < C1.w[0])
+ C.w[1]++;
+ if (C.w[1] > 0x01 || (C.w[1] == 0x01
+ && C.w[0] >= 0xffffffffffffffffull)) {
+ // set invalid flag
*pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- } else { // 19 <= ind <= 33
- if ((C1.w[1] < __bid_midpoint128[ind - 19].w[1])
- || ((C1.w[1] == __bid_midpoint128[ind - 19].w[1])
- && (C1.w[0] <= __bid_midpoint128[ind - 19].w[0]))) {
- res = 0x0000000000000000ull; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 21) {
+ // C >= 0x9fffffffffffffffb
+ if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
+ && C1.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits
+ C.w[1] = 0x09;
+ C.w[0] = 0xfffffffffffffffbull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1]
+ || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
+ // set invalid flag
*pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
}
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded
- // to nearest to a 64-bit unsigned signed integer
- if (x_sign) { // x <= -1
- // set invalid flag
+ }
+ }
+ // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1)
+ // return 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1)
+ // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1)
+ // res = 0
+ // else if x > 0
+ // res = +1
+ // else // if x < 0
+ // invalid exc
+ ind = q - 1;
+ if (ind <= 18) { // 0 <= ind <= 18
+ if ((C1.w[1] == 0) && (C1.w[0] <= midpoint64[ind])) {
+ res = 0x0000000000000000ull; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
+ res = 0x8000000000000000ull;
*pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
+ }
+ } else { // 19 <= ind <= 33
+ if ((C1.w[1] < midpoint128[ind - 19].w[1])
+ || ((C1.w[1] == midpoint128[ind - 19].w[1])
+ && (C1.w[0] <= midpoint128[ind - 19].w[0]))) {
+ res = 0x0000000000000000ull; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
res = 0x8000000000000000ull;
- BID_RETURN (res);
+ *pfpsf |= INVALID_EXCEPTION;
}
- // 1 <= x < 2^64-1/2 so x can be rounded
- // to nearest to a 64-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 = C1 + 1/2 * 10^ind 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- 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];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
- 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];
- }
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
- }
- // if the result was a midpoint it was rounded away from zero, so
- // it will need a correction
- // check for midpoints
- if ((fstar.w[3] == 0) && (fstar.w[2] == 0)
- && (fstar.w[1] || fstar.w[0])
- && (fstar.w[1] < __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] <= __bid_ten2mk128trunc[ind - 1].w[0]))) {
- // the result is a midpoint; round to nearest
- if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD]
- // if floor(C*) is odd C = floor(C*) - 1; the result >= 1
- Cstar.w[0]--; // Cstar.w[0] is now even
- } // else MP in [ODD, EVEN]
- }
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
+ }
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded
+ // to nearest to a 64-bit unsigned signed integer
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^64-1/2 so x can be rounded
+ // to nearest to a 64-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 = C1 + 1/2 * 10^ind 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 {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
+ 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ 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];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
+ 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];
+ }
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ }
+ // if the result was a midpoint it was rounded away from zero, so
+ // it will need a correction
+ // check for midpoints
+ if ((fstar.w[3] == 0) && (fstar.w[2] == 0)
+ && (fstar.w[1] || fstar.w[0])
+ && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) {
+ // the result is a midpoint; round to nearest
+ if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD]
+ // if floor(C*) is odd C = floor(C*) - 1; the result >= 1
+ Cstar.w[0]--; // Cstar.w[0] is now even
+ } // else MP in [ODD, EVEN]
}
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint64_xrnint
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_xrnint, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64,
+ bid128_to_uint64_xrnint, x)
- UINT64 res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ UINT64 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, tmp64A;
- BID_UI64DOUBLE tmp1;
- unsigned int x_nr_bits;
- int q, ind, shift;
- UINT128 C1, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 fstar;
- UINT256 P256;
+ UINT64 tmp64, tmp64A;
+ BID_UI64DOUBLE tmp1;
+ unsigned int x_nr_bits;
+ int q, ind, shift;
+ UINT128 C1, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 fstar;
+ UINT256 P256;
// 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
+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];
// 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 Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- }
+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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
}
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+}
+}
// check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // x is not special and is not zero
-
- // 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] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // 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 { // 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);
- }
- 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++;
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- exp = (x_exp >> 49) - 6176;
+ } 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- if (x_sign) { // if n < 0 and q + exp = 20
- // if n < -1/2 then n cannot be converted to uint64 with RN
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 1/2
- // <=> 0.c(0)c(1)...c(q-1) * 10^21 > 0x05, 1<=q<=34
- // <=> C * 10^(21-q) > 0x05, 1<=q<=34
- if (q == 21) {
- // C > 5
- if (C1.w[1] != 0 || C1.w[0] > 0x05ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to 64-bit unsigned int fall through
- // to '1 <= q + exp <= 20'
- } else {
- // if 1 <= q <= 20
- // C * 10^(21-q) > 5 is true because C >= 1 and 10^(21-q) >= 10
- // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C > 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64
+ if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // if n < -1/2 then n cannot be converted to uint64 with RN
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 1/2
+ // <=> 0.c(0)c(1)...c(q-1) * 10^21 > 0x05, 1<=q<=34
+ // <=> C * 10^(21-q) > 0x05, 1<=q<=34
+ if (q == 21) {
+ // C > 5
+ if (C1.w[1] != 0 || C1.w[0] > 0x05ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // if n > 0 and q + exp = 20
- // if n >= 2^64 - 1/2 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1)
- // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34
- if (q == 1) {
- // C * 10^20 >= 0x9fffffffffffffffb
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q <= 19) {
- // C * 10^(21-q) >= 0x9fffffffffffffffb
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 20) {
- // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff
- C.w[0] = C1.w[0] + C1.w[0];
- C.w[1] = C1.w[1] + C1.w[1];
- if (C.w[0] < C1.w[0])
- C.w[1]++;
- if (C.w[1] > 0x01 || (C.w[1] == 0x01
- && C.w[0] >= 0xffffffffffffffffull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 21) {
- // C >= 0x9fffffffffffffffb
- if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
- && C1.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits
- C.w[1] = 0x09;
- C.w[0] = 0xfffffffffffffffbull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- }
+ // else cases that can be rounded to 64-bit unsigned int fall through
+ // to '1 <= q + exp <= 20'
+ } else {
+ // if 1 <= q <= 20
+ // C * 10^(21-q) > 5 is true because C >= 1 and 10^(21-q) >= 10
+ // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C > 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- }
- // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1)
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // return 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1)
- // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1)
- // res = 0
- // else if x > 0
- // res = +1
- // else // if x < 0
- // invalid exc
- ind = q - 1;
- if (ind <= 18) { // 0 <= ind <= 18
- if ((C1.w[1] == 0) && (C1.w[0] <= __bid_midpoint64[ind])) {
- res = 0x0000000000000000ull; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ } else { // if n > 0 and q + exp = 20
+ // if n >= 2^64 - 1/2 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1)
+ // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34
+ if (q == 1) {
+ // C * 10^20 >= 0x9fffffffffffffffb
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q <= 19) {
+ // C * 10^(21-q) >= 0x9fffffffffffffffb
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 20) {
+ // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff
+ C.w[0] = C1.w[0] + C1.w[0];
+ C.w[1] = C1.w[1] + C1.w[1];
+ if (C.w[0] < C1.w[0])
+ C.w[1]++;
+ if (C.w[1] > 0x01 || (C.w[1] == 0x01
+ && C.w[0] >= 0xffffffffffffffffull)) {
+ // set invalid flag
*pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // 19 <= ind <= 33
- if ((C1.w[1] < __bid_midpoint128[ind - 19].w[1])
- || ((C1.w[1] == __bid_midpoint128[ind - 19].w[1])
- && (C1.w[0] <= __bid_midpoint128[ind - 19].w[0]))) {
- res = 0x0000000000000000ull; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 21) {
+ // C >= 0x9fffffffffffffffb
+ if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
+ && C1.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits
+ C.w[1] = 0x09;
+ C.w[0] = 0xfffffffffffffffbull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1]
+ || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
+ // set invalid flag
*pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
BID_RETURN (res);
}
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
}
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded
- // to nearest to a 64-bit unsigned signed integer
- if (x_sign) { // x <= -1
- // set invalid flag
+ }
+ }
+ // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1)
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // return 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1)
+ // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1)
+ // res = 0
+ // else if x > 0
+ // res = +1
+ // else // if x < 0
+ // invalid exc
+ ind = q - 1;
+ if (ind <= 18) { // 0 <= ind <= 18
+ if ((C1.w[1] == 0) && (C1.w[0] <= midpoint64[ind])) {
+ res = 0x0000000000000000ull; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
+ res = 0x8000000000000000ull;
*pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
+ BID_RETURN (res);
+ }
+ } else { // 19 <= ind <= 33
+ if ((C1.w[1] < midpoint128[ind - 19].w[1])
+ || ((C1.w[1] == midpoint128[ind - 19].w[1])
+ && (C1.w[0] <= midpoint128[ind - 19].w[0]))) {
+ res = 0x0000000000000000ull; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
res = 0x8000000000000000ull;
+ *pfpsf |= INVALID_EXCEPTION;
BID_RETURN (res);
}
- // 1 <= x < 2^64-1/2 so x can be rounded
- // to nearest to a 64-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 = C1 + 1/2 * 10^ind 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- 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];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
- 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];
- }
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
- }
- // 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
- if (ind - 1 <= 2) {
- 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_ten2mk128trunc[ind - 1].w[1]
- || (tmp64 == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] >= __bid_ten2mk128trunc[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 inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded
+ // to nearest to a 64-bit unsigned signed integer
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^64-1/2 so x can be rounded
+ // to nearest to a 64-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 = C1 + 1/2 * 10^ind 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ 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];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
+ 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];
+ }
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ }
+ // 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
+ if (ind - 1 <= 2) {
+ 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 > ten2mk128trunc[ind - 1].w[1]
+ || (tmp64 == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
- }
- } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
- if (fstar.w[3] > 0x0 || (fstar.w[3] == 0x0
- && fstar.w[2] > __bid_one_half128[ind - 1]) || (fstar.w[3] == 0x0
- && 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];
- tmp64A = fstar.w[3];
- if (tmp64 > fstar.w[2])
- tmp64A--;
- if (tmp64A || tmp64
- || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ } // 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) { // if 3 <= ind <= 21
+ if (fstar.w[3] > 0x0 ||
+ (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) ||
+ (fstar.w[3] == 0x0 && 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];
+ tmp64A = fstar.w[3];
+ if (tmp64 > fstar.w[2])
+ tmp64A--;
+ if (tmp64A || tmp64
+ || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
- }
- } else { // if 22 <= ind <= 33
- 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_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ }
+ } else { // if 22 <= ind <= 33
+ 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] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[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 rounded away from zero, so
- // it will need a correction
- // check for midpoints
- if ((fstar.w[3] == 0) && (fstar.w[2] == 0)
- && (fstar.w[1] || fstar.w[0])
- && (fstar.w[1] < __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] <= __bid_ten2mk128trunc[ind - 1].w[0]))) {
- // the result is a midpoint; round to nearest
- if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD]
- // if floor(C*) is odd C = floor(C*) - 1; the result >= 1
- Cstar.w[0]--; // Cstar.w[0] is now even
- } // else MP in [ODD, EVEN]
- }
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
- } else {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
+ // if the result was a midpoint it was rounded away from zero, so
+ // it will need a correction
+ // check for midpoints
+ if ((fstar.w[3] == 0) && (fstar.w[2] == 0)
+ && (fstar.w[1] || fstar.w[0])
+ && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) {
+ // the result is a midpoint; round to nearest
+ if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD]
+ // if floor(C*) is odd C = floor(C*) - 1; the result >= 1
+ Cstar.w[0]--; // Cstar.w[0] is now even
+ } // else MP in [ODD, EVEN]
}
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint64_floor
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_floor, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64,
+ bid128_to_uint64_floor, x)
- UINT64 res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ UINT64 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, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 P256;
+ BID_UI64DOUBLE tmp1;
+ unsigned int x_nr_bits;
+ int q, ind, shift;
+ UINT128 C1, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 P256;
// 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
+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];
// 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
+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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+}
+}
+ // check for non-canonical values (after the check for special values)
+if ((C1.w[1] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // if n < 0 then n cannot be converted to uint64 with RM
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 0
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ // if n > 0 and q + exp = 20
+ // if n >= 2^64 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65
+ // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34
+ if (q == 1) {
+ // C * 10^20 >= 0xa0000000000000000
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] >= 0x0a) {
+ // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
- } else { // x is QNaN
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q <= 19) {
+ // C * 10^(21-q) >= 0xa0000000000000000
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] >= 0x0a) {
+ // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 20) {
+ // C >= 0x10000000000000000
+ if (C1.w[1] >= 0x01) {
+ // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
- } else { // x is -inf
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 21) {
+ // C >= 0xa0000000000000000
+ if (C1.w[1] >= 0x0a) {
+ // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- BID_RETURN (res);
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits
+ C.w[1] = 0x0a;
+ C.w[0] = 0x0000000000000000ull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
}
}
- // check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
+ // n is not too large to be converted to int64 if 0 <= n < 2^64
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) <= 0) { // n = +0.[0...0]c(0)c(1)...c(q-1)
+ // return 0
res = 0x0000000000000000ull;
BID_RETURN (res);
- } else { // x is not special and is not zero
-
- // if n < 0 then n cannot be converted to uint64 with RM
- if (x_sign) { // if n < 0 and q + exp = 20
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 0
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- // if n > 0 and q + exp = 20
- // if n >= 2^64 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65
- // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34
- if (q == 1) {
- // C * 10^20 >= 0xa0000000000000000
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] >= 0x0a) {
- // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q <= 19) {
- // C * 10^(21-q) >= 0xa0000000000000000
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] >= 0x0a) {
- // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 20) {
- // C >= 0x10000000000000000
- if (C1.w[1] >= 0x01) {
- // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 21) {
- // C >= 0xa0000000000000000
- if (C1.w[1] >= 0x0a) {
- // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits
- C.w[1] = 0x0a;
- C.w[0] = 0x0000000000000000ull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // 1 <= x < 2^64 so x can be rounded
+ // down to a 64-bit unsigned signed integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 fits in 127 bits
+ // 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
}
- }
- // n is not too large to be converted to int64 if 0 <= n < 2^64
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) <= 0) { // n = +0.[0...0]c(0)c(1)...c(q-1)
- // return 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // 1 <= x < 2^64 so x can be rounded
- // down to a 64-bit unsigned signed integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 fits in 127 bits
- // 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
- }
- // the top Ex bits of 10^(-x) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x=1, T*=__bid_ten2mk128trunc[0]=0x19999999999999999999999999999999
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
- }
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
- } else {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
+ // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g.
+ // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
}
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint64_xfloor
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_xfloor, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64,
+ bid128_to_uint64_xfloor, x)
- UINT64 res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ UINT64 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, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 fstar;
- UINT256 P256;
+ BID_UI64DOUBLE tmp1;
+ unsigned int x_nr_bits;
+ int q, ind, shift;
+ UINT128 C1, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 fstar;
+ UINT256 P256;
// 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
+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];
// 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
+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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+}
+}
+ // check for non-canonical values (after the check for special values)
+if ((C1.w[1] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // if n < 0 then n cannot be converted to uint64 with RM
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 0
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ // if n > 0 and q + exp = 20
+ // if n >= 2^64 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65
+ // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34
+ if (q == 1) {
+ // C * 10^20 >= 0xa0000000000000000
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] >= 0x0a) {
+ // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
- } else { // x is QNaN
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q <= 19) {
+ // C * 10^(21-q) >= 0xa0000000000000000
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] >= 0x0a) {
+ // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 20) {
+ // C >= 0x10000000000000000
+ if (C1.w[1] >= 0x01) {
+ // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
- } else { // x is -inf
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 21) {
+ // C >= 0xa0000000000000000
+ if (C1.w[1] >= 0x0a) {
+ // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- BID_RETURN (res);
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits
+ C.w[1] = 0x0a;
+ C.w[0] = 0x0000000000000000ull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
}
}
- // check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
+ // n is not too large to be converted to int64 if 0 <= n < 2^64
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) <= 0) { // n = +0.[0...0]c(0)c(1)...c(q-1)
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // return 0
res = 0x0000000000000000ull;
BID_RETURN (res);
- } else { // x is not special and is not zero
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // 1 <= x < 2^64 so x can be rounded
+ // down to a 64-bit unsigned signed integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 fits in 127 bits
+ // 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ 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];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
+ 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];
+ }
+ // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g.
+ // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ }
+ // determine inexactness of the rounding of C*
+ // if (0 < f* < 10^(-x)) then
+ // the result is exact
+ // else // if (f* > T*) then
+ // the result is inexact
+ if (ind - 1 <= 2) {
+ if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] ||
+ (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] &&
+ fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } // else the result is exact
+ } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
+ if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } // else the result is exact
+ } else { // if 22 <= ind <= 33
+ if (fstar.w[3] || fstar.w[2]
+ || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } // else the result is exact
+ }
- // if n < 0 then n cannot be converted to uint64 with RM
- if (x_sign) { // if n < 0 and q + exp = 20
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 0
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
}
- // 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
+ }
+}
+
+BID_RETURN (res);
+}
+
+/*****************************************************************************
+ * BID128_to_uint64_ceil
+ ****************************************************************************/
+
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, bid128_to_uint64_ceil,
+ x)
+
+ UINT64 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, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 fstar;
+ UINT256 P256;
+
+ // 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];
+
+ // 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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+}
+}
+ // check for non-canonical values (after the check for special values)
+if ((C1.w[1] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // 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 { // 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);
- }
- 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++;
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- exp = (x_exp >> 49) - 6176;
- if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- // if n > 0 and q + exp = 20
- // if n >= 2^64 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65
- // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34
+ } 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // if n <= -1 then n cannot be converted to uint64 with RZ
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1
+ // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34
+ // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34
+ if (q == 21) {
+ // C >= a
+ if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to 64-bit unsigned int fall through
+ // to '1 <= q + exp <= 20'
+ } else {
+ // if 1 <= q <= 20
+ // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10
+ // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ } else { // if n > 0 and q + exp = 20
+ // if n > 2^64 - 1 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) > 2^64 - 1
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 > 2^64 - 1
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 > 10 * (2^64 - 1)
+ // <=> C * 10^(21-q) > 0x9fffffffffffffff6, 1<=q<=34
if (q == 1) {
- // C * 10^20 >= 0xa0000000000000000
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] >= 0x0a) {
- // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
+ // C * 10^20 > 0x9fffffffffffffff6
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] > 0xfffffffffffffff6ull)) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
// else cases that can be rounded to a 64-bit int fall through
// to '1 <= q + exp <= 20'
} else if (q <= 19) {
- // C * 10^(21-q) >= 0xa0000000000000000
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] >= 0x0a) {
- // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
+ // C * 10^(21-q) > 0x9fffffffffffffff6
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] > 0xfffffffffffffff6ull)) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
// else cases that can be rounded to a 64-bit int fall through
// to '1 <= q + exp <= 20'
} else if (q == 20) {
- // C >= 0x10000000000000000
- if (C1.w[1] >= 0x01) {
- // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) {
+ // C > 0xffffffffffffffff
+ if (C1.w[1]) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
// else cases that can be rounded to a 64-bit int fall through
// to '1 <= q + exp <= 20'
} else if (q == 21) {
- // C >= 0xa0000000000000000
- if (C1.w[1] >= 0x0a) {
- // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) {
+ // C > 0x9fffffffffffffff6
+ if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
+ && C1.w[0] > 0xfffffffffffffff6ull)) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
}
// else cases that can be rounded to a 64-bit int fall through
// to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits
- C.w[1] = 0x0a;
- C.w[0] = 0x0000000000000000ull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C > 10^(q-21) * 0x9fffffffffffffff6 max 44 bits x 68 bits
+ C.w[1] = 0x09;
+ C.w[0] = 0xfffffffffffffff6ull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
// to '1 <= q + exp <= 20'
}
}
- // n is not too large to be converted to int64 if 0 <= n < 2^64
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) <= 0) { // n = +0.[0...0]c(0)c(1)...c(q-1)
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // return 0
+ }
+ // n is not too large to be converted to int64 if -1 < n <= 2^64 - 1
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1)
+ // return 0 or 1
+ if (x_sign)
res = 0x0000000000000000ull;
+ else
+ res = 0x0000000000000001ull;
+ BID_RETURN (res);
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded
+ // to zero to a 64-bit unsigned signed integer
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
BID_RETURN (res);
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // 1 <= x < 2^64 so x can be rounded
- // down to a 64-bit unsigned signed integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 fits in 127 bits
- // 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- 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];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
- 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];
- }
- // the top Ex bits of 10^(-x) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x=1, T*=__bid_ten2mk128trunc[0]=0x19999999999999999999999999999999
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
- }
- // determine inexactness of the rounding of C*
- // if (0 < f* < 10^(-x)) then
- // the result is exact
- // else // if (f* > T*) then
- // the result is inexact
- if (ind - 1 <= 2) {
- if (fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
- if (fstar.w[2] || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else { // if 22 <= ind <= 33
- if (fstar.w[3] || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- }
-
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
- } else {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
+ }
+ // 1 <= x <= 2^64 - 1 so x can be rounded
+ // to zero to a 64-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 fits in 127 bits
+ // 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ 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];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
+ 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];
}
+ // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g.
+ // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ }
+ // if the result is positive and inexact, need to add 1 to it
+
+ // determine inexactness of the rounding of C*
+ // if (0 < f* < 10^(-x)) then
+ // the result is exact
+ // else // if (f* > T*) then
+ // the result is inexact
+ if (ind - 1 <= 2) {
+ if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ if (!x_sign) { // positive and inexact
+ Cstar.w[0]++;
+ if (Cstar.w[0] == 0x0)
+ Cstar.w[1]++;
+ }
+ } // else the result is exact
+ } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
+ if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ if (!x_sign) { // positive and inexact
+ Cstar.w[0]++;
+ if (Cstar.w[0] == 0x0)
+ Cstar.w[1]++;
+ }
+ } // else the result is exact
+ } else { // if 22 <= ind <= 33
+ if (fstar.w[3] || fstar.w[2]
+ || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ if (!x_sign) { // positive and inexact
+ Cstar.w[0]++;
+ if (Cstar.w[0] == 0x0)
+ Cstar.w[1]++;
+ }
+ } // else the result is exact
+ }
+
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
- * BID128_to_uint64_ceil
+ * BID128_to_uint64_xceil
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_ceil, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64,
+ bid128_to_uint64_xceil, x)
- UINT64 res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ UINT64 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, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 fstar;
- UINT256 P256;
+ BID_UI64DOUBLE tmp1;
+ unsigned int x_nr_bits;
+ int q, ind, shift;
+ UINT128 C1, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 fstar;
+ UINT256 P256;
// 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
+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];
// 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 Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- }
+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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
}
+ BID_RETURN (res);
+}
+}
// check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // x is not special and is not zero
-
- // 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] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // 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 { // 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);
- }
- 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++;
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- exp = (x_exp >> 49) - 6176;
- if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- if (x_sign) { // if n < 0 and q + exp = 20
- // if n <= -1 then n cannot be converted to uint64 with RZ
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1
- // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34
- // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34
- if (q == 21) {
- // C >= a
- if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to 64-bit unsigned int fall through
- // to '1 <= q + exp <= 20'
- } else {
- // if 1 <= q <= 20
- // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10
- // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64
+ } 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // if n <= -1 then n cannot be converted to uint64 with RZ
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1
+ // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34
+ // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34
+ if (q == 21) {
+ // C >= a
+ if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // if n > 0 and q + exp = 20
- // if n > 2^64 - 1 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) > 2^64 - 1
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 > 2^64 - 1
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 > 10 * (2^64 - 1)
- // <=> C * 10^(21-q) > 0x9fffffffffffffff6, 1<=q<=34
- if (q == 1) {
- // C * 10^20 > 0x9fffffffffffffff6
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] > 0xfffffffffffffff6ull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q <= 19) {
- // C * 10^(21-q) > 0x9fffffffffffffff6
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] > 0xfffffffffffffff6ull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 20) {
- // C > 0xffffffffffffffff
- if (C1.w[1] > 0x0 || (C1.w[1] == 0x0
- && C1.w[0] > 0xffffffffffffffffull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 21) {
- // C > 0x9fffffffffffffff6
- if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
- && C1.w[0] > 0xfffffffffffffff6ull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C > 10^(q-21) * 0x9fffffffffffffff6 max 44 bits x 68 bits
- C.w[1] = 0x09;
- C.w[0] = 0xfffffffffffffff6ull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- }
- }
- }
- // n is not too large to be converted to int64 if -1 < n <= 2^64 - 1
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1)
- // return 0 or 1
- if (x_sign)
- res = 0x0000000000000000ull;
- else
- res = 0x0000000000000001ull;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded
- // to zero to a 64-bit unsigned signed integer
- if (x_sign) { // x <= -1
+ // else cases that can be rounded to 64-bit unsigned int fall through
+ // to '1 <= q + exp <= 20'
+ } else {
+ // if 1 <= q <= 20
+ // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10
+ // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- // 1 <= x <= 2^64 - 1 so x can be rounded
- // to zero to a 64-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 fits in 127 bits
- // 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- 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];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
- 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];
+ } else { // if n > 0 and q + exp = 20
+ // if n > 2^64 - 1 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) > 2^64 - 1
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 > 2^64 - 1
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 > 10 * (2^64 - 1)
+ // <=> C * 10^(21-q) > 0x9fffffffffffffff6, 1<=q<=34
+ if (q == 1) {
+ // C * 10^20 > 0x9fffffffffffffff6
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] > 0xfffffffffffffff6ull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- // the top Ex bits of 10^(-x) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x=1, T*=__bid_ten2mk128trunc[0]=0x19999999999999999999999999999999
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q <= 19) {
+ // C * 10^(21-q) > 0x9fffffffffffffff6
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] > 0xfffffffffffffff6ull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- // if the result is positive and inexact, need to add 1 to it
-
- // determine inexactness of the rounding of C*
- // if (0 < f* < 10^(-x)) then
- // the result is exact
- // else // if (f* > T*) then
- // the result is inexact
- if (ind - 1 <= 2) {
- if (fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- if (!x_sign) { // positive and inexact
- Cstar.w[0]++;
- if (Cstar.w[0] == 0x0)
- Cstar.w[1]++;
- }
- } // else the result is exact
- } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
- if (fstar.w[2] || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- if (!x_sign) { // positive and inexact
- Cstar.w[0]++;
- if (Cstar.w[0] == 0x0)
- Cstar.w[1]++;
- }
- } // else the result is exact
- } else { // if 22 <= ind <= 33
- if (fstar.w[3] || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- if (!x_sign) { // positive and inexact
- Cstar.w[0]++;
- if (Cstar.w[0] == 0x0)
- Cstar.w[1]++;
- }
- } // else the result is exact
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 20) {
+ // C > 0xffffffffffffffff
+ if (C1.w[1]) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
-
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
- } else {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
- }
- }
- }
- BID_RETURN (res);
-}
-
-/*****************************************************************************
- * BID128_to_uint64_xceil
- ****************************************************************************/
-
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_xceil, x)
-
- UINT64 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, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 fstar;
- UINT256 P256;
-
- // 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];
-
- // 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 Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 21) {
+ // C > 0x9fffffffffffffff6
+ if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
+ && C1.w[0] > 0xfffffffffffffff6ull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C > 10^(q-21) * 0x9fffffffffffffff6 max 44 bits x 68 bits
+ C.w[1] = 0x09;
+ C.w[0] = 0xfffffffffffffff6ull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
}
- BID_RETURN (res);
}
}
- // check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x0000000000000000ull;
+ // n is not too large to be converted to int64 if -1 < n <= 2^64 - 1
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1)
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // return 0 or 1
+ if (x_sign)
+ res = 0x0000000000000000ull;
+ else
+ res = 0x0000000000000001ull;
BID_RETURN (res);
- } else { // x is not special and is not zero
-
- // 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded
+ // to zero to a 64-bit unsigned signed integer
+ if (x_sign) { // x <= -1
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- if (x_sign) { // if n < 0 and q + exp = 20
- // if n <= -1 then n cannot be converted to uint64 with RZ
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1
- // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34
- // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34
- if (q == 21) {
- // C >= a
- if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to 64-bit unsigned int fall through
- // to '1 <= q + exp <= 20'
- } else {
- // if 1 <= q <= 20
- // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10
- // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- } else { // if n > 0 and q + exp = 20
- // if n > 2^64 - 1 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) > 2^64 - 1
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 > 2^64 - 1
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 > 10 * (2^64 - 1)
- // <=> C * 10^(21-q) > 0x9fffffffffffffff6, 1<=q<=34
- if (q == 1) {
- // C * 10^20 > 0x9fffffffffffffff6
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] > 0xfffffffffffffff6ull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q <= 19) {
- // C * 10^(21-q) > 0x9fffffffffffffff6
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] > 0xfffffffffffffff6ull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 20) {
- // C > 0xffffffffffffffff
- if (C1.w[1] > 0x0 || (C1.w[1] == 0x0
- && C1.w[0] > 0xffffffffffffffffull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
+ }
+ // 1 <= x <= 2^64 - 1 so x can be rounded
+ // to zero to a 64-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 fits in 127 bits
+ // 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ 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];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
+ 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];
+ }
+ // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g.
+ // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ }
+ // if the result is positive and inexact, need to add 1 to it
+
+ // determine inexactness of the rounding of C*
+ // if (0 < f* < 10^(-x)) then
+ // the result is exact
+ // else // if (f* > T*) then
+ // the result is inexact
+ if (ind - 1 <= 2) {
+ if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ if (!x_sign) { // positive and inexact
+ Cstar.w[0]++;
+ if (Cstar.w[0] == 0x0)
+ Cstar.w[1]++;
}
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 21) {
- // C > 0x9fffffffffffffff6
- if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
- && C1.w[0] > 0xfffffffffffffff6ull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } // else the result is exact
+ } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
+ if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ if (!x_sign) { // positive and inexact
+ Cstar.w[0]++;
+ if (Cstar.w[0] == 0x0)
+ Cstar.w[1]++;
}
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C > 10^(q-21) * 0x9fffffffffffffff6 max 44 bits x 68 bits
- C.w[1] = 0x09;
- C.w[0] = 0xfffffffffffffff6ull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } // else the result is exact
+ } else { // if 22 <= ind <= 33
+ if (fstar.w[3] || fstar.w[2]
+ || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ if (!x_sign) { // positive and inexact
+ Cstar.w[0]++;
+ if (Cstar.w[0] == 0x0)
+ Cstar.w[1]++;
}
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- }
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } // else the result is exact
}
- }
- // n is not too large to be converted to int64 if -1 < n <= 2^64 - 1
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1)
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // return 0 or 1
- if (x_sign)
- res = 0x0000000000000000ull;
- else
- res = 0x0000000000000001ull;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded
- // to zero to a 64-bit unsigned signed integer
- if (x_sign) { // x <= -1
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // 1 <= x <= 2^64 - 1 so x can be rounded
- // to zero to a 64-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 fits in 127 bits
- // 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- 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];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
- 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];
- }
- // the top Ex bits of 10^(-x) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x=1, T*=__bid_ten2mk128trunc[0]=0x19999999999999999999999999999999
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
- }
- // if the result is positive and inexact, need to add 1 to it
-
- // determine inexactness of the rounding of C*
- // if (0 < f* < 10^(-x)) then
- // the result is exact
- // else // if (f* > T*) then
- // the result is inexact
- if (ind - 1 <= 2) {
- if (fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- if (!x_sign) { // positive and inexact
- Cstar.w[0]++;
- if (Cstar.w[0] == 0x0)
- Cstar.w[1]++;
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
- if (fstar.w[2] || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- if (!x_sign) { // positive and inexact
- Cstar.w[0]++;
- if (Cstar.w[0] == 0x0)
- Cstar.w[1]++;
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else { // if 22 <= ind <= 33
- if (fstar.w[3] || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- if (!x_sign) { // positive and inexact
- Cstar.w[0]++;
- if (Cstar.w[0] == 0x0)
- Cstar.w[1]++;
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- }
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
- } else {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
- }
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint64_int
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_int, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, bid128_to_uint64_int,
+ x)
- UINT64 res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ UINT64 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, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 P256;
+ BID_UI64DOUBLE tmp1;
+ unsigned int x_nr_bits;
+ int q, ind, shift;
+ UINT128 C1, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 P256;
// 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
+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];
// 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 Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- }
+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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
}
+ BID_RETURN (res);
+}
+}
// check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // x is not special and is not zero
-
- // 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] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // 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 { // 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;
+ } 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- if (x_sign) { // if n < 0 and q + exp = 20
- // if n <= -1 then n cannot be converted to uint64 with RZ
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1
- // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34
- // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34
- if (q == 21) {
- // C >= a
- if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to 64-bit unsigned int fall through
- // to '1 <= q + exp <= 20'
- } else {
- // if 1 <= q <= 20
- // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10
- // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64
+ if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // if n <= -1 then n cannot be converted to uint64 with RZ
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1
+ // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34
+ // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34
+ if (q == 21) {
+ // C >= a
+ if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // if n > 0 and q + exp = 20
- // if n >= 2^64 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65
- // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34
- if (q == 1) {
- // C * 10^20 >= 0xa0000000000000000
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] >= 0x0a) {
- // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q <= 19) {
- // C * 10^(21-q) >= 0xa0000000000000000
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] >= 0x0a) {
- // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 20) {
- // C >= 0x10000000000000000
- if (C1.w[1] >= 0x01) {
- // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 21) {
- // C >= 0xa0000000000000000
- if (C1.w[1] >= 0x0a) {
- // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits
- C.w[1] = 0x0a;
- C.w[0] = 0x0000000000000000ull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1]
- && C1.w[0] >= C.w[0])) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- }
- }
- }
- // n is not too large to be converted to int64 if -1 < n < 2^64
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1)
- // return 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded
- // to zero to a 64-bit unsigned signed integer
- if (x_sign) { // x <= -1
+ // else cases that can be rounded to 64-bit unsigned int fall through
+ // to '1 <= q + exp <= 20'
+ } else {
+ // if 1 <= q <= 20
+ // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10
+ // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- // 1 <= x < 2^64 so x can be rounded
- // to zero to a 64-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 fits in 127 bits
- // 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
+ } else { // if n > 0 and q + exp = 20
+ // if n >= 2^64 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65
+ // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34
+ if (q == 1) {
+ // C * 10^20 >= 0xa0000000000000000
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] >= 0x0a) {
+ // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- // the top Ex bits of 10^(-x) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x=1, T*=__bid_ten2mk128trunc[0]=0x19999999999999999999999999999999
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q <= 19) {
+ // C * 10^(21-q) >= 0xa0000000000000000
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] >= 0x0a) {
+ // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
- } else {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 20) {
+ // C >= 0x10000000000000000
+ if (C1.w[1] >= 0x01) {
+ // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 21) {
+ // C >= 0xa0000000000000000
+ if (C1.w[1] >= 0x0a) {
+ // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits
+ C.w[1] = 0x0a;
+ C.w[0] = 0x0000000000000000ull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1]
+ || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
}
}
}
- BID_RETURN (res);
+ // n is not too large to be converted to int64 if -1 < n < 2^64
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1)
+ // return 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded
+ // to zero to a 64-bit unsigned signed integer
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^64 so x can be rounded
+ // to zero to a 64-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 fits in 127 bits
+ // 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
+ }
+ // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g.
+ // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ }
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
+ }
+ }
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint64_xint
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_xint, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, bid128_to_uint64_xint,
+ x)
- UINT64 res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ UINT64 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, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 fstar;
- UINT256 P256;
+ BID_UI64DOUBLE tmp1;
+ unsigned int x_nr_bits;
+ int q, ind, shift;
+ UINT128 C1, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 fstar;
+ UINT256 P256;
// 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
+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];
// 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 Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- }
+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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
}
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+}
+}
// check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // x is not special and is not zero
-
- // 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] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // 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 { // 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- if (x_sign) { // if n < 0 and q + exp = 20
- // if n <= -1 then n cannot be converted to uint64 with RZ
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1
- // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34
- // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34
- if (q == 21) {
- // C >= a
- if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to 64-bit unsigned int fall through
- // to '1 <= q + exp <= 20'
- } else {
- // if 1 <= q <= 20
- // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10
- // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64
+ } 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // if n <= -1 then n cannot be converted to uint64 with RZ
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1
+ // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34
+ // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34
+ if (q == 21) {
+ // C >= a
+ if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // if n > 0 and q + exp = 20
- // if n >= 2^64 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65
- // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34
- if (q == 1) {
- // C * 10^20 >= 0xa0000000000000000
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] >= 0x0a) {
- // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q <= 19) {
- // C * 10^(21-q) >= 0xa0000000000000000
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] >= 0x0a) {
- // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 20) {
- // C >= 0x10000000000000000
- if (C1.w[1] >= 0x01) {
- // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 21) {
- // C >= 0xa0000000000000000
- if (C1.w[1] >= 0x0a) {
- // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits
- C.w[1] = 0x0a;
- C.w[0] = 0x0000000000000000ull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- }
- }
- }
- // n is not too large to be converted to int64 if -1 < n < 2^64
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1)
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // return 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded
- // to zero to a 64-bit unsigned signed integer
- if (x_sign) { // x <= -1
+ // else cases that can be rounded to 64-bit unsigned int fall through
+ // to '1 <= q + exp <= 20'
+ } else {
+ // if 1 <= q <= 20
+ // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10
+ // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- // 1 <= x < 2^64 so x can be rounded
- // to zero to a 64-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 fits in 127 bits
- // 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- 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];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
- 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];
+ } else { // if n > 0 and q + exp = 20
+ // if n >= 2^64 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65
+ // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34
+ if (q == 1) {
+ // C * 10^20 >= 0xa0000000000000000
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] >= 0x0a) {
+ // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q <= 19) {
+ // C * 10^(21-q) >= 0xa0000000000000000
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] >= 0x0a) {
+ // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- // the top Ex bits of 10^(-x) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x=1, T*=__bid_ten2mk128trunc[0]=0x19999999999999999999999999999999
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 20) {
+ // C >= 0x10000000000000000
+ if (C1.w[1] >= 0x01) {
+ // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- // determine inexactness of the rounding of C*
- // if (0 < f* < 10^(-x)) then
- // the result is exact
- // else // if (f* > T*) then
- // the result is inexact
- if (ind - 1 <= 2) {
- if (fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
- if (fstar.w[2] || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else { // if 22 <= ind <= 33
- if (fstar.w[3] || fstar.w[2]
- || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 21) {
+ // C >= 0xa0000000000000000
+ if (C1.w[1] >= 0x0a) {
+ // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
-
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
- } else {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits
+ C.w[1] = 0x0a;
+ C.w[0] = 0x0000000000000000ull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1]
+ || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ }
+ }
+ }
+ // n is not too large to be converted to int64 if -1 < n < 2^64
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1)
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // return 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded
+ // to zero to a 64-bit unsigned signed integer
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^64 so x can be rounded
+ // to zero to a 64-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 fits in 127 bits
+ // 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ 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];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
+ 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];
+ }
+ // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g.
+ // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
}
+ // determine inexactness of the rounding of C*
+ // if (0 < f* < 10^(-x)) then
+ // the result is exact
+ // else // if (f* > T*) then
+ // the result is inexact
+ if (ind - 1 <= 2) {
+ if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } // else the result is exact
+ } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
+ if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } // else the result is exact
+ } else { // if 22 <= ind <= 33
+ if (fstar.w[3] || fstar.w[2]
+ || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } // else the result is exact
+ }
+
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint64_rninta
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_rninta, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64,
+ bid128_to_uint64_rninta, x)
- UINT64 res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ UINT64 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, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 P256;
+ UINT64 tmp64;
+ BID_UI64DOUBLE tmp1;
+ unsigned int x_nr_bits;
+ int q, ind, shift;
+ UINT128 C1, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 P256;
// 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
+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];
// 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 Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
+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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+}
+}
+ // check for non-canonical values (after the check for special values)
+if ((C1.w[1] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // 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);
}
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is -inf
+ } 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // if n <= -1/2 then n cannot be converted to uint64 with RN
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1/2
+ // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x05, 1<=q<=34
+ // <=> C * 10^(21-q) >= 0x05, 1<=q<=34
+ if (q == 21) {
+ // C >= 5
+ if (C1.w[1] != 0 || C1.w[0] >= 0x05ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to 64-bit unsigned int fall through
+ // to '1 <= q + exp <= 20'
+ } else {
+ // if 1 <= q <= 20
+ // C * 10^(21-q) >= 5 is true because C >= 1 and 10^(21-q) >= 10
+ // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- BID_RETURN (res);
- }
- }
- // check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // x is not special and is not zero
-
- // 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 n > 0 and q + exp = 20
+ // if n >= 2^64 - 1/2 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1)
+ // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34
+ if (q == 1) {
+ // C * 10^20 >= 0x9fffffffffffffffb
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- } 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- if (x_sign) { // if n < 0 and q + exp = 20
- // if n <= -1/2 then n cannot be converted to uint64 with RN
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1/2
- // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x05, 1<=q<=34
- // <=> C * 10^(21-q) >= 0x05, 1<=q<=34
- if (q == 21) {
- // C >= 5
- if (C1.w[1] != 0 || C1.w[0] >= 0x05ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to 64-bit unsigned int fall through
- // to '1 <= q + exp <= 20'
- } else {
- // if 1 <= q <= 20
- // C * 10^(21-q) >= 5 is true because C >= 1 and 10^(21-q) >= 10
- // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q <= 19) {
+ // C * 10^(21-q) >= 0x9fffffffffffffffb
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] >= 0xfffffffffffffffbull)) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // if n > 0 and q + exp = 20
- // if n >= 2^64 - 1/2 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1)
- // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34
- if (q == 1) {
- // C * 10^20 >= 0x9fffffffffffffffb
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q <= 19) {
- // C * 10^(21-q) >= 0x9fffffffffffffffb
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 20) {
- // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff
- C.w[0] = C1.w[0] + C1.w[0];
- C.w[1] = C1.w[1] + C1.w[1];
- if (C.w[0] < C1.w[0])
- C.w[1]++;
- if (C.w[1] > 0x01 || (C.w[1] == 0x01
- && C.w[0] >= 0xffffffffffffffffull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 21) {
- // C >= 0x9fffffffffffffffb
- if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
- && C1.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits
- C.w[1] = 0x09;
- C.w[0] = 0xfffffffffffffffbull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 20) {
+ // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff
+ C.w[0] = C1.w[0] + C1.w[0];
+ C.w[1] = C1.w[1] + C1.w[1];
+ if (C.w[0] < C1.w[0])
+ C.w[1]++;
+ if (C.w[1] > 0x01 || (C.w[1] == 0x01
+ && C.w[0] >= 0xffffffffffffffffull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- }
- }
- // n is not too large to be converted to int64 if -1/2 < n < 2^64 - 1/2
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1)
- // return 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1)
- // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1)
- // res = 0
- // else if x > 0
- // res = +1
- // else // if x < 0
- // invalid exc
- ind = q - 1;
- if (ind <= 18) { // 0 <= ind <= 18
- if ((C1.w[1] == 0) && (C1.w[0] < __bid_midpoint64[ind])) {
- res = 0x0000000000000000ull; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 21) {
+ // C >= 0x9fffffffffffffffb
+ if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
+ && C1.w[0] >= 0xfffffffffffffffbull)) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // 19 <= ind <= 33
- if ((C1.w[1] < __bid_midpoint128[ind - 19].w[1])
- || ((C1.w[1] == __bid_midpoint128[ind - 19].w[1])
- && (C1.w[0] < __bid_midpoint128[ind - 19].w[0]))) {
- res = 0x0000000000000000ull; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits
+ C.w[1] = 0x09;
+ C.w[0] = 0xfffffffffffffffbull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1]
+ || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
}
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded
- // to nearest to a 64-bit unsigned signed integer
- if (x_sign) { // x <= -1
+ }
+ }
+ // n is not too large to be converted to int64 if -1/2 < n < 2^64 - 1/2
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1)
+ // return 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1)
+ // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1)
+ // res = 0
+ // else if x > 0
+ // res = +1
+ // else // if x < 0
+ // invalid exc
+ ind = q - 1;
+ if (ind <= 18) { // 0 <= ind <= 18
+ if ((C1.w[1] == 0) && (C1.w[0] < midpoint64[ind])) {
+ res = 0x0000000000000000ull; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- // 1 <= x < 2^64-1/2 so x can be rounded
- // to nearest to a 64-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 = C1 + 1/2 * 10^ind 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
- }
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
- }
-
- // if the result was a midpoint it was rounded away from zero
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
+ } else { // 19 <= ind <= 33
+ if ((C1.w[1] < midpoint128[ind - 19].w[1])
+ || ((C1.w[1] == midpoint128[ind - 19].w[1])
+ && (C1.w[0] < midpoint128[ind - 19].w[0]))) {
+ res = 0x0000000000000000ull; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ }
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded
+ // to nearest to a 64-bit unsigned signed integer
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^64-1/2 so x can be rounded
+ // to nearest to a 64-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 = C1 + 1/2 * 10^ind 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 {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
+ 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
}
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ }
+
+ // if the result was a midpoint it was rounded away from zero
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint64_xrninta
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(UINT64, __bid128_to_uint64_xrninta, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64,
+ bid128_to_uint64_xrninta, x)
- UINT64 res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ UINT64 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, tmp64A;
- BID_UI64DOUBLE tmp1;
- unsigned int x_nr_bits;
- int q, ind, shift;
- UINT128 C1, C;
- UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
- UINT256 fstar;
- UINT256 P256;
+ UINT64 tmp64, tmp64A;
+ BID_UI64DOUBLE tmp1;
+ unsigned int x_nr_bits;
+ int q, ind, shift;
+ UINT128 C1, C;
+ UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits
+ UINT256 fstar;
+ UINT256 P256;
// 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
+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];
// 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 Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- } else { // x is not a NaN, so it must be infinity
- if (!x_sign) { // x is +inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- }
- BID_RETURN (res);
- }
+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 Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ }
+ BID_RETURN (res);
+} else { // x is not a NaN, so it must be infinity
+ if (!x_sign) { // x is +inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
}
+ BID_RETURN (res);
+}
+}
// check for non-canonical values (after the check for special values)
- if ((C1.w[1] > 0x0001ed09bead87c0ull)
- || (C1.w[1] == 0x0001ed09bead87c0ull
- && (C1.w[0] > 0x378d8e63ffffffffull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else { // x is not special and is not zero
-
- // 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] > 0x0001ed09bead87c0ull)
+ || (C1.w[1] == 0x0001ed09bead87c0ull
+ && (C1.w[0] > 0x378d8e63ffffffffull))
+ || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+
+ // 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 { // 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);
- }
- 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++;
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- exp = (x_exp >> 49) - 6176;
+ } 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 ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
- // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
- // so x rounded to an integer may or may not fit in an unsigned 64-bit int
- // the cases that do not fit are identified here; the ones that fit
- // fall through and will be handled with other cases further,
- // under '1 <= q + exp <= 20'
- if (x_sign) { // if n < 0 and q + exp = 20
- // if n <= -1/2 then n cannot be converted to uint64 with RN
- // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1/2
- // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x05, 1<=q<=34
- // <=> C * 10^(21-q) >= 0x05, 1<=q<=34
- if (q == 21) {
- // C >= 5
- if (C1.w[1] != 0 || C1.w[0] >= 0x05ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to 64-bit unsigned int fall through
- // to '1 <= q + exp <= 20'
- } else {
- // if 1 <= q <= 20
- // C * 10^(21-q) >= 5 is true because C >= 1 and 10^(21-q) >= 10
- // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64
+ if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1)
+ // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43...
+ // so x rounded to an integer may or may not fit in an unsigned 64-bit int
+ // the cases that do not fit are identified here; the ones that fit
+ // fall through and will be handled with other cases further,
+ // under '1 <= q + exp <= 20'
+ if (x_sign) { // if n < 0 and q + exp = 20
+ // if n <= -1/2 then n cannot be converted to uint64 with RN
+ // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1/2
+ // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x05, 1<=q<=34
+ // <=> C * 10^(21-q) >= 0x05, 1<=q<=34
+ if (q == 21) {
+ // C >= 5
+ if (C1.w[1] != 0 || C1.w[0] >= 0x05ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // if n > 0 and q + exp = 20
- // if n >= 2^64 - 1/2 then n is too large
- // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2
- // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1)
- // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34
- if (q == 1) {
- // C * 10^20 >= 0x9fffffffffffffffb
- __mul_128x64_to_128 (C, C1.w[0], __bid_ten2k128[0]); // 10^20 * C
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q <= 19) {
- // C * 10^(21-q) >= 0x9fffffffffffffffb
- __mul_64x64_to_128MACH (C, C1.w[0], __bid_ten2k64[21 - q]);
- if (C.w[1] > 0x09 || (C.w[1] == 0x09
- && C.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 20) {
- // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff
- C.w[0] = C1.w[0] + C1.w[0];
- C.w[1] = C1.w[1] + C1.w[1];
- if (C.w[0] < C1.w[0])
- C.w[1]++;
- if (C.w[1] > 0x01 || (C.w[1] == 0x01
- && C.w[0] >= 0xffffffffffffffffull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else if (q == 21) {
- // C >= 0x9fffffffffffffffb
- if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
- && C1.w[0] >= 0xfffffffffffffffbull)) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
- // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits
- C.w[1] = 0x09;
- C.w[0] = 0xfffffffffffffffbull;
- __mul_128x64_to_128 (C, __bid_ten2k64[q - 21], C);
- if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x8000000000000000ull;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 64-bit int fall through
- // to '1 <= q + exp <= 20'
- }
+ // else cases that can be rounded to 64-bit unsigned int fall through
+ // to '1 <= q + exp <= 20'
+ } else {
+ // if 1 <= q <= 20
+ // C * 10^(21-q) >= 5 is true because C >= 1 and 10^(21-q) >= 10
+ // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
}
- }
- // n is not too large to be converted to int64 if -1/2 < n < 2^64 - 1/2
- // Note: some of the cases tested for above fall through to this point
- if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1)
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // return 0
- res = 0x0000000000000000ull;
- BID_RETURN (res);
- } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1)
- // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1)
- // res = 0
- // else if x > 0
- // res = +1
- // else // if x < 0
- // invalid exc
- ind = q - 1;
- if (ind <= 18) { // 0 <= ind <= 18
- if ((C1.w[1] == 0) && (C1.w[0] < __bid_midpoint64[ind])) {
- res = 0x0000000000000000ull; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ } else { // if n > 0 and q + exp = 20
+ // if n >= 2^64 - 1/2 then n is too large
+ // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2
+ // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1)
+ // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34
+ if (q == 1) {
+ // C * 10^20 >= 0x9fffffffffffffffb
+ __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q <= 19) {
+ // C * 10^(21-q) >= 0x9fffffffffffffffb
+ __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]);
+ if (C.w[1] > 0x09 || (C.w[1] == 0x09
+ && C.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 20) {
+ // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff
+ C.w[0] = C1.w[0] + C1.w[0];
+ C.w[1] = C1.w[1] + C1.w[1];
+ if (C.w[0] < C1.w[0])
+ C.w[1]++;
+ if (C.w[1] > 0x01 || (C.w[1] == 0x01
+ && C.w[0] >= 0xffffffffffffffffull)) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- } else { // 19 <= ind <= 33
- if ((C1.w[1] < __bid_midpoint128[ind - 19].w[1])
- || ((C1.w[1] == __bid_midpoint128[ind - 19].w[1])
- && (C1.w[0] < __bid_midpoint128[ind - 19].w[0]))) {
- res = 0x0000000000000000ull; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else if (q == 21) {
+ // C >= 0x9fffffffffffffffb
+ if (C1.w[1] > 0x09 || (C1.w[1] == 0x09
+ && C1.w[0] >= 0xfffffffffffffffbull)) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
+ } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13
+ // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits
+ C.w[1] = 0x09;
+ C.w[0] = 0xfffffffffffffffbull;
+ __mul_128x64_to_128 (C, ten2k64[q - 21], C);
+ if (C1.w[1] > C.w[1]
+ || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) {
+ // set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
+ // else cases that can be rounded to a 64-bit int fall through
+ // to '1 <= q + exp <= 20'
}
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
- // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded
- // to nearest to a 64-bit unsigned signed integer
- if (x_sign) { // x <= -1
+ }
+ }
+ // n is not too large to be converted to int64 if -1/2 < n < 2^64 - 1/2
+ // Note: some of the cases tested for above fall through to this point
+ if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1)
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // return 0
+ res = 0x0000000000000000ull;
+ BID_RETURN (res);
+ } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1)
+ // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1)
+ // res = 0
+ // else if x > 0
+ // res = +1
+ // else // if x < 0
+ // invalid exc
+ ind = q - 1;
+ if (ind <= 18) { // 0 <= ind <= 18
+ if ((C1.w[1] == 0) && (C1.w[0] < midpoint64[ind])) {
+ res = 0x0000000000000000ull; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
+ res = 0x8000000000000000ull;
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x8000000000000000ull;
BID_RETURN (res);
}
- // 1 <= x < 2^64-1/2 so x can be rounded
- // to nearest to a 64-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
- ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
- // chop off ind digits from the lower part of C1
- // C1 = C1 + 1/2 * 10^ind 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 <= 33
- // 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 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[1] = P256.w[3];
- Cstar.w[0] = P256.w[2];
- 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];
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[1] = 0;
- Cstar.w[0] = P256.w[3];
- 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];
- }
- // 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]
- shift = __bid_shiftright128[ind - 1]; // 0 <= shift <= 102
- if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
- Cstar.w[0] =
- (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
- // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
- } else { // 22 <= ind - 1 <= 33
- Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
- }
- // 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
- if (ind - 1 <= 2) {
- 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_ten2mk128trunc[ind - 1].w[1]
- || (tmp64 == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] >= __bid_ten2mk128trunc[ind - 1].w[0]))) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ } else { // 19 <= ind <= 33
+ if ((C1.w[1] < midpoint128[ind - 19].w[1])
+ || ((C1.w[1] == midpoint128[ind - 19].w[1])
+ && (C1.w[0] < midpoint128[ind - 19].w[0]))) {
+ res = 0x0000000000000000ull; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
+ res = 0x8000000000000000ull;
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ }
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19)
+ // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded
+ // to nearest to a 64-bit unsigned signed integer
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x8000000000000000ull;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^64-1/2 so x can be rounded
+ // to nearest to a 64-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20
+ ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x'
+ // chop off ind digits from the lower part of C1
+ // C1 = C1 + 1/2 * 10^ind 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 <= 33
+ // 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 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[1] = P256.w[3];
+ Cstar.w[0] = P256.w[2];
+ 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];
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[1] = 0;
+ Cstar.w[0] = P256.w[3];
+ 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];
+ }
+ // 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]
+ shift = shiftright128[ind - 1]; // 0 <= shift <= 102
+ if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21
+ Cstar.w[0] =
+ (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift));
+ // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift);
+ } else { // 22 <= ind - 1 <= 33
+ Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38
+ }
+ // 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
+ if (ind - 1 <= 2) {
+ 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 > ten2mk128trunc[ind - 1].w[1]
+ || (tmp64 == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
- }
- } else if (ind - 1 <= 21) { // if 3 <= ind <= 21
- if (fstar.w[3] > 0x0 || (fstar.w[3] == 0x0
- && fstar.w[2] > __bid_one_half128[ind - 1]) || (fstar.w[3] == 0x0
- && 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];
- tmp64A = fstar.w[3];
- if (tmp64 > fstar.w[2])
- tmp64A--;
- if (tmp64A || tmp64
- || fstar.w[1] > __bid_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ } // 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) { // if 3 <= ind <= 21
+ if (fstar.w[3] > 0x0 ||
+ (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) ||
+ (fstar.w[3] == 0x0 && 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];
+ tmp64A = fstar.w[3];
+ if (tmp64 > fstar.w[2])
+ tmp64A--;
+ if (tmp64A || tmp64
+ || fstar.w[1] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) {
// set the inexact flag
*pfpsf |= INEXACT_EXCEPTION;
- }
- } else { // if 22 <= ind <= 33
- 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_ten2mk128trunc[ind - 1].w[1]
- || (fstar.w[1] == __bid_ten2mk128trunc[ind - 1].w[1]
- && fstar.w[0] > __bid_ten2mk128trunc[ind - 1].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ }
+ } else { // if 22 <= ind <= 33
+ 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] > ten2mk128trunc[ind - 1].w[1]
+ || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1]
+ && fstar.w[0] > ten2mk128trunc[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 rounded away from zero
- res = Cstar.w[0]; // the result is positive
- } else if (exp == 0) {
- // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
- // res = C (exact)
- res = C1.w[0];
- } else {
- // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
- // res = C * 10^exp (exact) - must fit in 64 bits
- res = C1.w[0] * __bid_ten2k64[exp];
}
+
+ // if the result was a midpoint it was rounded away from zero
+ res = Cstar.w[0]; // the result is positive
+ } else if (exp == 0) {
+ // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0
+ // res = C (exact)
+ res = C1.w[0];
+ } else {
+ // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20
+ // res = C * 10^exp (exact) - must fit in 64 bits
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
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