* BID128_to_uint32_rnint
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_rnint, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_rnint, x)
- unsigned int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ unsigned int 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
}
+ 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 = 0x00000000;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x00000000;
- 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ 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++;
- }
- exp = (x_exp >> 49) - 6176;
- if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in an unsigned 32-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 <= 10'
- if (x_sign) { // if n < 0 and q + exp = 10
- // if n < -1/2 then n cannot be converted to uint32 with RN
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 1/2
- // <=> 0.c(0)c(1)...c(q-1) * 10^11 > 0x05, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 > 0x05ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 > 0x05 <=>
- // C > 0x05 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 1/2 up)
- tmp64 = 0x05ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- } else { // if n > 0 and q + exp = 10
- // if n >= 2^32 - 1/2 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x9fffffffbull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=>
- // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32-1/2 up)
- tmp64 = 0x9fffffffbull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- }
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- // n is not too large to be converted to int32: -1/2 <= n < 2^32 - 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 = 0x00000000;
- 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 = 0x00000000; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in an unsigned 32-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 <= 10'
+ if (x_sign) { // if n < 0 and q + exp = 10
+ // if n < -1/2 then n cannot be converted to uint32 with RN
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 1/2
+ // <=> 0.c(0)c(1)...c(q-1) * 10^11 > 0x05, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 > 0x05ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 > 0x05 <=>
+ // C > 0x05 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 1/2 up)
+ tmp64 = 0x05ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ 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 = 0x00000000; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
+ } else { // if n > 0 and q + exp = 10
+ // if n >= 2^32 - 1/2 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x9fffffffbull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=>
+ // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32-1/2 up)
+ tmp64 = 0x9fffffffbull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
}
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
}
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- if (x_sign) { // x <= -1
- // set invalid flag
+ }
+ }
+ // n is not too large to be converted to int32: -1/2 <= n < 2^32 - 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 = 0x00000000;
+ 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 = 0x00000000; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
+ res = 0x80000000;
*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 = 0x00000000; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
res = 0x80000000;
- BID_RETURN (res);
+ *pfpsf |= INVALID_EXCEPTION;
+ }
+ }
+ } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^32-1/2 so x can be rounded
+ // to nearest to a 32-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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];
}
- // 1 <= x < 2^32-1/2 so x can be rounded
- // to nearest to a 32-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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 <= 10
- // res = C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
+ 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 <= 10
+ // res = C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint32_xrnint
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_xrnint, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_xrnint, x)
- unsigned int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ unsigned int 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;
- unsigned int tmp_inexact = 0;
+ 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;
+ unsigned int tmp_inexact = 0;
// 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
}
+ 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 = 0x00000000;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x00000000;
- 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ 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++;
- }
- exp = (x_exp >> 49) - 6176;
- if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in an unsigned 32-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 <= 10'
- if (x_sign) { // if n < 0 and q + exp = 10
- // if n < -1/2 then n cannot be converted to uint32 with RN
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 1/2
- // <=> 0.c(0)c(1)...c(q-1) * 10^11 > 0x05, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 > 0x05ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 > 0x05 <=>
- // C > 0x05 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 1/2 up)
- tmp64 = 0x05ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- } else { // if n > 0 and q + exp = 10
- // if n >= 2^32 - 1/2 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x9fffffffbull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=>
- // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32-1/2 up)
- tmp64 = 0x9fffffffbull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- }
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- // n is not too large to be converted to int32: -1/2 <= n < 2^32 - 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 = 0x00000000;
- 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 = 0x00000000; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in an unsigned 32-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 <= 10'
+ if (x_sign) { // if n < 0 and q + exp = 10
+ // if n < -1/2 then n cannot be converted to uint32 with RN
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 1/2
+ // <=> 0.c(0)c(1)...c(q-1) * 10^11 > 0x05, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 > 0x05ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 > 0x05 <=>
+ // C > 0x05 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 1/2 up)
+ tmp64 = 0x05ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
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 = 0x00000000; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
+ } else { // if n > 0 and q + exp = 10
+ // if n >= 2^32 - 1/2 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x9fffffffbull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=>
+ // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32-1/2 up)
+ tmp64 = 0x9fffffffbull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
BID_RETURN (res);
}
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
}
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- if (x_sign) { // x <= -1
- // set invalid flag
+ }
+ }
+ // n is not too large to be converted to int32: -1/2 <= n < 2^32 - 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 = 0x00000000;
+ 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 = 0x00000000; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
+ res = 0x80000000;
*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 = 0x00000000; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
res = 0x80000000;
+ *pfpsf |= INVALID_EXCEPTION;
BID_RETURN (res);
}
- // 1 <= x < 2^32-1/2 so x can be rounded
- // to nearest to a 32-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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;
- tmp_inexact = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ }
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^32-1/2 so x can be rounded
+ // to nearest to a 32-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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;
tmp_inexact = 1;
- }
- } 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;
- tmp_inexact = 1;
- } // 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;
+ tmp_inexact = 1;
+ }
+ } 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;
tmp_inexact = 1;
- }
- } 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;
- tmp_inexact = 1;
- } // 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;
+ tmp_inexact = 1;
+ }
+ } 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;
tmp_inexact = 1;
- }
- }
-
- // 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
- if (tmp_inexact)
- *pfpsf |= INEXACT_EXCEPTION;
- } else if (exp == 0) {
- // 1 <= q <= 10
- // res = C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ // *pfpsf |= INEXACT_EXCEPTION;
+ tmp_inexact = 1;
+ }
+ }
+
+ // 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
+ if (tmp_inexact)
+ *pfpsf |= INEXACT_EXCEPTION;
+ } else if (exp == 0) {
+ // 1 <= q <= 10
+ // res = C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint32_floor
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_floor, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_floor, x)
- unsigned int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ unsigned int 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;
- int is_inexact_gt_midpoint = 0;
- int is_midpoint_lt_even = 0;
+ 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;
+ int is_inexact_gt_midpoint = 0;
+ int is_midpoint_lt_even = 0;
// 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
+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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+ // x < 0 is invalid
+ if (x_sign) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // x > 0 from this point on
+ // 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
+ } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
+ // n > 0 and q + exp = 10
+ // if n >= 2^32 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0xa00000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x80000000;
- } else { // x is -inf
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=>
+ // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32 up)
+ tmp64 = 0xa00000000ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
}
- BID_RETURN (res);
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
}
}
- // 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 = 0x00000000;
- 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 int32: 0 <= n < 2^31
+ // 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) or n = +0.c(0)c(1)...c(q-1)
+ // return 0
res = 0x00000000;
BID_RETURN (res);
- } else { // x is not special and is not zero
- // x < 0 is invalid
- if (x_sign) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // x > 0 from this point on
- // 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 { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ // 1 <= x < 2^32 so x can be rounded down to a 32-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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];
}
- } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
- // n > 0 and q + exp = 10
- // if n >= 2^32 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0xa00000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=>
- // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32 up)
- tmp64 = 0xa00000000ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
+ 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])) {
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ is_inexact_gt_midpoint = 1;
+ }
+ } 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])) {
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ is_inexact_gt_midpoint = 1;
+ }
+ } 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])) {
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ is_inexact_gt_midpoint = 1;
}
- 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 = 0x80000000;
- BID_RETURN (res);
+ }
+
+ // 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
+ is_inexact_gt_midpoint = 0;
+ } else { // else MP in [ODD, EVEN]
+ is_midpoint_lt_even = 1;
+ is_inexact_gt_midpoint = 0;
}
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
}
- }
- // n is not too large to be converted to int32: 0 <= n < 2^31
- // 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) or n = +0.c(0)c(1)...c(q-1)
- // return 0
- res = 0x00000000;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- // 1 <= x < 2^32 so x can be rounded down to a 32-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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]))) {
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- is_inexact_gt_midpoint = 1;
- }
- } 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])) {
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- is_inexact_gt_midpoint = 1;
- }
- } 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])) {
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- is_inexact_gt_midpoint = 1;
- }
- }
-
- // 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
- is_inexact_gt_midpoint = 0;
- } else { // else MP in [ODD, EVEN]
- is_midpoint_lt_even = 1;
- is_inexact_gt_midpoint = 0;
- }
- }
- // general correction for RM
- if (is_midpoint_lt_even || is_inexact_gt_midpoint) {
- Cstar.w[0] = Cstar.w[0] - 1;
- } else {
- ; // the result is already correct
- }
- res = Cstar.w[0];
- } else if (exp == 0) {
- // 1 <= q <= 10
- // res = +C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = +C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
+ // general correction for RM
+ if (is_midpoint_lt_even || is_inexact_gt_midpoint) {
+ Cstar.w[0] = Cstar.w[0] - 1;
+ } else {
+ ; // the result is already correct
}
+ res = Cstar.w[0];
+ } else if (exp == 0) {
+ // 1 <= q <= 10
+ // res = +C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = +C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint32_xfloor
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_xfloor, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_xfloor, x)
- unsigned int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ unsigned int 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;
- int is_inexact_gt_midpoint = 0;
- int is_midpoint_lt_even = 0;
+ 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;
+ int is_inexact_gt_midpoint = 0;
+ int is_midpoint_lt_even = 0;
// 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
+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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ BID_RETURN (res);
+} else { // x is not special and is not zero
+ // x < 0 is invalid
+ if (x_sign) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // x > 0 from this point on
+ // 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
+ } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
+ // n > 0 and q + exp = 10
+ // if n >= 2^32 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0xa00000000ull) {
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x80000000;
- } else { // x is -inf
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=>
+ // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32 up)
+ tmp64 = 0xa00000000ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
}
- BID_RETURN (res);
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
}
}
- // 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 = 0x00000000;
- 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 int32: 0 <= n < 2^31
+ // 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) or n = +0.c(0)c(1)...c(q-1)
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // return 0
res = 0x00000000;
BID_RETURN (res);
- } else { // x is not special and is not zero
- // x < 0 is invalid
- if (x_sign) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // x > 0 from this point on
- // 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 { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ // 1 <= x < 2^32 so x can be rounded down to a 32-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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];
}
- } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
- // n > 0 and q + exp = 10
- // if n >= 2^32 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0xa00000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=>
- // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32 up)
- tmp64 = 0xa00000000ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
+ 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];
}
- }
- // n is not too large to be converted to int32: 0 <= n < 2^31
- // 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) or n = +0.c(0)c(1)...c(q-1)
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // return 0
- res = 0x00000000;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- // 1 <= x < 2^32 so x can be rounded down to a 32-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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
+ // 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;
- is_inexact_gt_midpoint = 1;
- }
- } 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;
+ is_inexact_gt_midpoint = 1;
+ }
+ } 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;
- is_inexact_gt_midpoint = 1;
- }
- } 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;
+ is_inexact_gt_midpoint = 1;
+ }
+ } 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;
- is_inexact_gt_midpoint = 1;
- }
- }
-
- // 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
- is_inexact_gt_midpoint = 0;
- } else { // else MP in [ODD, EVEN]
- is_midpoint_lt_even = 1;
- is_inexact_gt_midpoint = 0;
- }
- }
- // general correction for RM
- if (is_midpoint_lt_even || is_inexact_gt_midpoint) {
- Cstar.w[0] = Cstar.w[0] - 1;
- } else {
- ; // the result is already correct
- }
- res = Cstar.w[0];
- } else if (exp == 0) {
- // 1 <= q <= 10
- // res = +C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = +C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ is_inexact_gt_midpoint = 1;
+ }
+ }
+
+ // 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
+ is_inexact_gt_midpoint = 0;
+ } else { // else MP in [ODD, EVEN]
+ is_midpoint_lt_even = 1;
+ is_inexact_gt_midpoint = 0;
+ }
}
+ // general correction for RM
+ if (is_midpoint_lt_even || is_inexact_gt_midpoint) {
+ Cstar.w[0] = Cstar.w[0] - 1;
+ } else {
+ ; // the result is already correct
+ }
+ res = Cstar.w[0];
+ } else if (exp == 0) {
+ // 1 <= q <= 10
+ // res = +C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = +C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint32_ceil
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_ceil, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_ceil, x)
- unsigned int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ unsigned int 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;
- int is_inexact_lt_midpoint = 0;
- int is_midpoint_gt_even = 0;
+ 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;
+ int is_inexact_lt_midpoint = 0;
+ int is_midpoint_gt_even = 0;
// 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
}
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x00000000;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x00000000;
- 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
+ if (x_sign) { // if n < 0 and q + exp = 10
+ // if n <= -1 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1
+ // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x50000000a, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x0aull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=>
+ // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 1 up)
+ tmp64 = 0x0aull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
+ } else { // if n > 0 and q + exp = 10
+ // if n > 2^32 - 1 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 2^32 - 1
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 > 0x9fffffff6ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6 <=>
+ // C > 0x9fffffff6 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32 up)
+ tmp64 = 0x9fffffff6ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
+ }
+ }
+ // n is not too large to be converted to int32: -2^32-1 < n <= 2^32-1
+ // 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) or n = +/-0.c(0)c(1)...c(q-1)
+ // return 0
+ if (x_sign)
+ res = 0x00000000;
+ else
+ res = 0x00000001;
+ BID_RETURN (res);
+ } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ // -2^32-1 < x <= -1 or 1 <= x <= 2^32-1 so x can be rounded
+ // toward positive infinity to a 32-bit signed integer
+ if (x_sign) { // x <= -1 is invalid
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x80000000;
BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
- if (x_sign) { // if n < 0 and q + exp = 10
- // if n <= -1 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1
- // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x50000000a, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x0aull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=>
- // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 1 up)
- tmp64 = 0x0aull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- } else { // if n > 0 and q + exp = 10
- // if n > 2^32 - 1 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 2^32 - 1
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 > 0x9fffffff6ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6 <=>
- // C > 0x9fffffff6 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32 up)
- tmp64 = 0x9fffffff6ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
+ }
+ // x > 0 from this point on
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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])) {
+ is_inexact_lt_midpoint = 1;
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ ;
+ }
+ } 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])) {
+ is_inexact_lt_midpoint = 1;
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ }
+ } 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])) {
+ is_inexact_lt_midpoint = 1;
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ ;
}
}
- }
- // n is not too large to be converted to int32: -2^32-1 < n <= 2^32-1
- // 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) or n = +/-0.c(0)c(1)...c(q-1)
- // return 0
- if (x_sign)
- res = 0x00000000;
- else
- res = 0x00000001;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- // -2^32-1 < x <= -1 or 1 <= x <= 2^32-1 so x can be rounded
- // toward positive infinity to a 32-bit signed integer
- if (x_sign) { // x <= -1 is invalid
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
+
+ // 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
+ is_midpoint_gt_even = 1;
+ is_inexact_lt_midpoint = 0;
+ } else { // else MP in [ODD, EVEN]
+ is_inexact_lt_midpoint = 0;
+ }
}
- // x > 0 from this point on
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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]))) {
- is_inexact_lt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- ;
- }
- } 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])) {
- is_inexact_lt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- }
- } 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])) {
- is_inexact_lt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- ;
- }
- }
-
- // 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
- is_midpoint_gt_even = 1;
- is_inexact_lt_midpoint = 0;
- } else { // else MP in [ODD, EVEN]
- is_inexact_lt_midpoint = 0;
- }
- }
- // general correction for RM
- if (is_midpoint_gt_even || is_inexact_lt_midpoint) {
- Cstar.w[0] = Cstar.w[0] + 1;
- } else {
- ; // the result is already correct
- }
- res = Cstar.w[0];
- } else if (exp == 0) {
- // 1 <= q <= 10
- // res = +C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = +C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
+ // general correction for RM
+ if (is_midpoint_gt_even || is_inexact_lt_midpoint) {
+ Cstar.w[0] = Cstar.w[0] + 1;
+ } else {
+ ; // the result is already correct
}
+ res = Cstar.w[0];
+ } else if (exp == 0) {
+ // 1 <= q <= 10
+ // res = +C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = +C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint32_xceil
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_xceil, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_xceil, x)
- unsigned int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ unsigned int 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;
- int is_inexact_lt_midpoint = 0;
- int is_midpoint_gt_even = 0;
+ 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;
+ int is_inexact_lt_midpoint = 0;
+ int is_midpoint_gt_even = 0;
// 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
}
+ 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 = 0x00000000;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x00000000;
- 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ 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++;
+ } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
+ if (x_sign) { // if n < 0 and q + exp = 10
+ // if n <= -1 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1
+ // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x50000000a, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x0aull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=>
+ // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 1 up)
+ tmp64 = 0x0aull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
+ } else { // if n > 0 and q + exp = 10
+ // if n > 2^32 - 1 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 2^32 - 1
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 > 0x9fffffff6ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6 <=>
+ // C > 0x9fffffff6 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32 up)
+ tmp64 = 0x9fffffff6ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
}
- exp = (x_exp >> 49) - 6176;
- if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ }
+ // n is not too large to be converted to int32: -2^32-1 < n <= 2^32-1
+ // 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) or n = +/-0.c(0)c(1)...c(q-1)
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // return 0
+ if (x_sign)
+ res = 0x00000000;
+ else
+ res = 0x00000001;
+ BID_RETURN (res);
+ } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ // -2^32-1 < x <= -1 or 1 <= x <= 2^32-1 so x can be rounded
+ // toward positive infinity to a 32-bit signed integer
+ if (x_sign) { // x <= -1 is invalid
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x80000000;
BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
- if (x_sign) { // if n < 0 and q + exp = 10
- // if n <= -1 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1
- // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x50000000a, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x0aull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=>
- // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 1 up)
- tmp64 = 0x0aull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- } else { // if n > 0 and q + exp = 10
- // if n > 2^32 - 1 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 2^32 - 1
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 > 0x9fffffff6ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6 <=>
- // C > 0x9fffffff6 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32 up)
- tmp64 = 0x9fffffff6ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- }
}
- // n is not too large to be converted to int32: -2^32-1 < n <= 2^32-1
- // 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) or n = +/-0.c(0)c(1)...c(q-1)
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // return 0
- if (x_sign)
- res = 0x00000000;
- else
- res = 0x00000001;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- // -2^32-1 < x <= -1 or 1 <= x <= 2^32-1 so x can be rounded
- // toward positive infinity to a 32-bit signed integer
- if (x_sign) { // x <= -1 is invalid
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
+ // x > 0 from this point on
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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];
}
- // x > 0 from this point on
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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;
- is_inexact_lt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ // 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;
- is_inexact_lt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ is_inexact_lt_midpoint = 1;
+ } // 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;
- is_inexact_lt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ is_inexact_lt_midpoint = 1;
+ } // 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;
- }
- }
-
- // 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
- is_midpoint_gt_even = 1;
- is_inexact_lt_midpoint = 0;
- } else { // else MP in [ODD, EVEN]
- is_inexact_lt_midpoint = 0;
- }
- }
- // general correction for RM
- if (is_midpoint_gt_even || is_inexact_lt_midpoint) {
- Cstar.w[0] = Cstar.w[0] + 1;
- } else {
- ; // the result is already correct
- }
- res = Cstar.w[0];
- } else if (exp == 0) {
- // 1 <= q <= 10
- // res = +C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = +C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
+ is_inexact_lt_midpoint = 1;
+ } // 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] < 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
+ is_midpoint_gt_even = 1;
+ is_inexact_lt_midpoint = 0;
+ } else { // else MP in [ODD, EVEN]
+ is_inexact_lt_midpoint = 0;
+ }
}
+ // general correction for RM
+ if (is_midpoint_gt_even || is_inexact_lt_midpoint) {
+ Cstar.w[0] = Cstar.w[0] + 1;
+ } else {
+ ; // the result is already correct
+ }
+ res = Cstar.w[0];
+ } else if (exp == 0) {
+ // 1 <= q <= 10
+ // res = +C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = +C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint32_int
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_int, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_int, x)
- int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ int 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;
- int is_inexact_gt_midpoint = 0;
- int is_midpoint_lt_even = 0;
+ 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;
+ int is_inexact_gt_midpoint = 0;
+ int is_midpoint_lt_even = 0;
// 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
}
+ 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 = 0x00000000;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x00000000;
- 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ 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++;
+ } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
+ if (x_sign) { // if n < 0 and q + exp = 10
+ // if n <= -1 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1
+ // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x0aull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit uint fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=>
+ // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 1 up)
+ tmp64 = 0x0aull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
+ } else { // if n > 0 and q + exp = 10
+ // if n >= 2^32 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0xa00000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit uint fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=>
+ // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32 up)
+ tmp64 = 0xa00000000ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
}
- exp = (x_exp >> 49) - 6176;
- if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ }
+ // n is not too large to be converted to uint32: -2^32 < n < 2^32
+ // 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) or n = +/-0.c(0)c(1)...c(q-1)
+ // return 0
+ res = 0x00000000;
+ BID_RETURN (res);
+ } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ // x = d(0)...d(k).d(k+1)..., k >= 0, d(0) != 0
+ if (x_sign) { // x <= -1
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x80000000;
BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
- if (x_sign) { // if n < 0 and q + exp = 10
- // if n <= -1 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1
- // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x0aull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit uint fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=>
- // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 1 up)
- tmp64 = 0x0aull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- } else { // if n > 0 and q + exp = 10
- // if n >= 2^32 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0xa00000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit uint fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=>
- // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32 up)
- tmp64 = 0xa00000000ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
+ }
+ // x > 0 from this point on
+ // 1 <= x < 2^32 so x can be rounded to zero to a 32-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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])) {
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ is_inexact_gt_midpoint = 1;
+ }
+ } 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])) {
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ is_inexact_gt_midpoint = 1;
+ }
+ } 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])) {
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ is_inexact_gt_midpoint = 1;
}
}
- }
- // n is not too large to be converted to uint32: -2^32 < n < 2^32
- // 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) or n = +/-0.c(0)c(1)...c(q-1)
- // return 0
- res = 0x00000000;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- // x = d(0)...d(k).d(k+1)..., k >= 0, d(0) != 0
- if (x_sign) { // x <= -1
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
+
+ // 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
+ is_inexact_gt_midpoint = 0;
+ } else { // else MP in [ODD, EVEN]
+ is_midpoint_lt_even = 1;
+ is_inexact_gt_midpoint = 0;
+ }
}
- // x > 0 from this point on
- // 1 <= x < 2^32 so x can be rounded to zero to a 32-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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])) {
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- is_inexact_gt_midpoint = 1;
- }
- } 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])) {
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- is_inexact_gt_midpoint = 1;
- }
- } 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])) {
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- is_inexact_gt_midpoint = 1;
- }
- }
-
- // 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
- is_inexact_gt_midpoint = 0;
- } else { // else MP in [ODD, EVEN]
- is_midpoint_lt_even = 1;
- is_inexact_gt_midpoint = 0;
- }
- }
- // general correction for RZ
- if (is_midpoint_lt_even || is_inexact_gt_midpoint) {
- Cstar.w[0] = Cstar.w[0] - 1;
- } else {
- ; // exact, the result is already correct
- }
- res = Cstar.w[0];
- } else if (exp == 0) {
- // 1 <= q <= 10
- // res = +C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = +C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
+ // general correction for RZ
+ if (is_midpoint_lt_even || is_inexact_gt_midpoint) {
+ Cstar.w[0] = Cstar.w[0] - 1;
+ } else {
+ ; // exact, the result is already correct
}
+ res = Cstar.w[0];
+ } else if (exp == 0) {
+ // 1 <= q <= 10
+ // res = +C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = +C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint32_xint
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_xint, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_xint, x)
- int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ int 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;
- int is_inexact_gt_midpoint = 0;
- int is_midpoint_lt_even = 0;
+ 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;
+ int is_inexact_gt_midpoint = 0;
+ int is_midpoint_lt_even = 0;
// 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
+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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ 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 = 0x80000000;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
+ } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
+ if (x_sign) { // if n < 0 and q + exp = 10
+ // if n <= -1 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1
+ // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x0aull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit uint fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=>
+ // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 1 up)
+ tmp64 = 0x0aull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
+ } else { // if n > 0 and q + exp = 10
+ // if n >= 2^32 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0xa00000000ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit uint fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=>
+ // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32 up)
+ tmp64 = 0xa00000000ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
}
- 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 = 0x00000000;
- 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 uint32: -2^32 < n < 2^32
+ // 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) or n = +/-0.c(0)c(1)...c(q-1)
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // return 0
res = 0x00000000;
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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ // x = d(0)...d(k).d(k+1)..., k >= 0, d(0) != 0
+ if (x_sign) { // x <= -1
// set invalid flag
*pfpsf |= INVALID_EXCEPTION;
// return Integer Indefinite
res = 0x80000000;
BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
- if (x_sign) { // if n < 0 and q + exp = 10
- // if n <= -1 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1
- // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x0aull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit uint fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=>
- // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 1 up)
- tmp64 = 0x0aull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- } else { // if n > 0 and q + exp = 10
- // if n >= 2^32 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0xa00000000ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit uint fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=>
- // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32 up)
- tmp64 = 0xa00000000ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- }
}
- // n is not too large to be converted to uint32: -2^32 < n < 2^32
- // 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) or n = +/-0.c(0)c(1)...c(q-1)
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // return 0
- res = 0x00000000;
- BID_RETURN (res);
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- // x = d(0)...d(k).d(k+1)..., k >= 0, d(0) != 0
- if (x_sign) { // x <= -1
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
+ // x > 0 from this point on
+ // 1 <= x < 2^32 so x can be rounded to zero to a 32-bit unsigned integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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];
}
- // x > 0 from this point on
- // 1 <= x < 2^32 so x can be rounded to zero to a 32-bit unsigned integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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
+ // 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;
- is_inexact_gt_midpoint = 1;
- }
- } 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;
+ is_inexact_gt_midpoint = 1;
+ }
+ } 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;
- is_inexact_gt_midpoint = 1;
- }
- } 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;
+ is_inexact_gt_midpoint = 1;
+ }
+ } 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;
- is_inexact_gt_midpoint = 1;
- }
- }
-
- // 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
- is_inexact_gt_midpoint = 0;
- } else { // else MP in [ODD, EVEN]
- is_midpoint_lt_even = 1;
- is_inexact_gt_midpoint = 0;
- }
- }
- // general correction for RZ
- if (is_midpoint_lt_even || is_inexact_gt_midpoint) {
- Cstar.w[0] = Cstar.w[0] - 1;
- } else {
- ; // exact, the result is already correct
- }
- res = Cstar.w[0];
- } else if (exp == 0) {
- // 1 <= q <= 10
- // res = +C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = +C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ is_inexact_gt_midpoint = 1;
+ }
+ }
+
+ // 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
+ is_inexact_gt_midpoint = 0;
+ } else { // else MP in [ODD, EVEN]
+ is_midpoint_lt_even = 1;
+ is_inexact_gt_midpoint = 0;
+ }
}
+ // general correction for RZ
+ if (is_midpoint_lt_even || is_inexact_gt_midpoint) {
+ Cstar.w[0] = Cstar.w[0] - 1;
+ } else {
+ ; // exact, the result is already correct
+ }
+ res = Cstar.w[0];
+ } else if (exp == 0) {
+ // 1 <= q <= 10
+ // res = +C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = +C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint32_rninta
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_rninta, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_rninta, x)
- unsigned int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ unsigned int 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
}
+ 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 = 0x00000000;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x00000000;
- 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ 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++;
- }
- exp = (x_exp >> 49) - 6176;
- if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
- if (x_sign) { // if n < 0 and q + exp = 10
- // if n <= -1/2 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1/2
- // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x05ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05 <=>
- // C >= 0x05 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 1/2 up)
- tmp64 = 0x05ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- } else { // if n > 0 and q + exp = 10
- // if n >= 2^32 - 1/2 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x9fffffffbull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=>
- // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32-1/2 up)
- tmp64 = 0x9fffffffbull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- }
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- // n is not too large to be converted to int32: -1/2 < n < 2^32 - 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 = 0x00000000;
- 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 = 0x00000000; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
+ if (x_sign) { // if n < 0 and q + exp = 10
+ // if n <= -1/2 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1/2
+ // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x05ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05 <=>
+ // C >= 0x05 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 1/2 up)
+ tmp64 = 0x05ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ 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 = 0x00000000; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
+ } else { // if n > 0 and q + exp = 10
+ // if n >= 2^32 - 1/2 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x9fffffffbull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=>
+ // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32-1/2 up)
+ tmp64 = 0x9fffffffbull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
+ BID_RETURN (res);
}
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
}
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- if (x_sign) { // x <= -1
- // set invalid flag
+ }
+ }
+ // n is not too large to be converted to int32: -1/2 < n < 2^32 - 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 = 0x00000000;
+ 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 = 0x00000000; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
+ res = 0x80000000;
*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 = 0x00000000; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
res = 0x80000000;
- BID_RETURN (res);
+ *pfpsf |= INVALID_EXCEPTION;
}
- // 1 <= x < 2^31-1/2 so x can be rounded
- // to nearest-away to a 32-bit signed integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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 already rounded away from zero
- res = Cstar.w[0]; // always positive
- // no need to check for midpoints - already rounded away from zero!
- } else if (exp == 0) {
- // 1 <= q <= 10
- // res = +C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = +C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
+ }
+ } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^31-1/2 so x can be rounded
+ // to nearest-away to a 32-bit signed integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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];
+ } 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 already rounded away from zero
+ res = Cstar.w[0]; // always positive
+ // no need to check for midpoints - already rounded away from zero!
+ } else if (exp == 0) {
+ // 1 <= q <= 10
+ // res = +C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = +C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
/*****************************************************************************
* BID128_to_uint32_xrninta
****************************************************************************/
-BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE(unsigned int, __bid128_to_uint32_xrninta, x)
+BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int,
+ bid128_to_uint32_xrninta, x)
- unsigned int res;
- UINT64 x_sign;
- UINT64 x_exp;
- int exp; // unbiased exponent
+ unsigned int 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;
- unsigned int tmp_inexact = 0;
+ 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;
+ unsigned int tmp_inexact = 0;
// 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 = 0x80000000;
- } else { // x is QNaN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
- } else { // x is -inf
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- }
- 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 = 0x80000000;
+ } else { // x is QNaN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ }
+ 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 = 0x80000000;
+ } else { // x is -inf
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
}
+ 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 = 0x00000000;
- BID_RETURN (res);
- } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
- // x is 0
- res = 0x00000000;
- 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 = 0x00000000;
+ BID_RETURN (res);
+} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) {
+ // x is 0
+ res = 0x00000000;
+ 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++;
- }
- exp = (x_exp >> 49) - 6176;
- if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
- // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
- // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
- if (x_sign) { // if n < 0 and q + exp = 10
- // if n <= -1/2 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1/2
- // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x05ull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05 <=>
- // C >= 0x05 * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 1/2 up)
- tmp64 = 0x05ull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- } else { // if n > 0 and q + exp = 10
- // if n >= 2^32 - 1/2 then n is too large
- // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2
- // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34
- if (q <= 11) {
- tmp64 = C1.w[0] * __bid_ten2k64[11 - q]; // C scaled up to 11-digit int
- // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
- if (tmp64 >= 0x9fffffffbull) {
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return Integer Indefinite
- res = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
- // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=>
- // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23
- // (scale 2^32-1/2 up)
- tmp64 = 0x9fffffffbull;
- if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
- __mul_64x64_to_128MACH (C, tmp64, __bid_ten2k64[q - 11]);
- } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
- __mul_128x64_to_128 (C, tmp64, __bid_ten2k128[q - 31]);
- }
- 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 = 0x80000000;
- BID_RETURN (res);
- }
- // else cases that can be rounded to a 32-bit int fall through
- // to '1 <= q + exp <= 10'
- }
- }
+ 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- // n is not too large to be converted to int32: -1/2 < n < 2^32 - 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 = 0x00000000;
- 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 = 0x00000000; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ } 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) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits)
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1)
+ // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2...
+ // so x rounded to an integer may or may not fit in a signed 32-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 <= 10'
+ if (x_sign) { // if n < 0 and q + exp = 10
+ // if n <= -1/2 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1/2
+ // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x05ull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05 <=>
+ // C >= 0x05 * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 1/2 up)
+ tmp64 = 0x05ull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
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 = 0x00000000; // return 0
- } else if (!x_sign) { // n > 0
- res = 0x00000001; // return +1
- } else {
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ }
+ } else { // if n > 0 and q + exp = 10
+ // if n >= 2^32 - 1/2 then n is too large
+ // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2
+ // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34
+ if (q <= 11) {
+ tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int
+ // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits)
+ if (tmp64 >= 0x9fffffffbull) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
+ } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2
+ // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=>
+ // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23
+ // (scale 2^32-1/2 up)
+ tmp64 = 0x9fffffffbull;
+ if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits
+ __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]);
+ } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits
+ __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]);
+ }
+ 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 = 0x80000000;
BID_RETURN (res);
}
+ // else cases that can be rounded to a 32-bit int fall through
+ // to '1 <= q + exp <= 10'
}
- // set inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
- if (x_sign) { // x <= -1
- // set invalid flag
+ }
+ }
+ // n is not too large to be converted to int32: -1/2 < n < 2^32 - 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 = 0x00000000;
+ 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 = 0x00000000; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
+ res = 0x80000000;
*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 = 0x00000000; // return 0
+ } else if (!x_sign) { // n > 0
+ res = 0x00000001; // return +1
+ } else {
res = 0x80000000;
+ *pfpsf |= INVALID_EXCEPTION;
BID_RETURN (res);
}
- // 1 <= x < 2^31-1/2 so x can be rounded
- // to nearest-away to a 32-bit signed integer
- if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
- 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 already rounded away from zero
- // 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;
- tmp_inexact = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
+ }
+ // set inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9)
+ if (x_sign) { // x <= -1
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return Integer Indefinite
+ res = 0x80000000;
+ BID_RETURN (res);
+ }
+ // 1 <= x < 2^31-1/2 so x can be rounded
+ // to nearest-away to a 32-bit signed integer
+ if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10
+ 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
+ }
+ // if the result was a midpoint, it was already rounded away from zero
+ // 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;
tmp_inexact = 1;
- }
- } 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;
- tmp_inexact = 1;
- } // 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;
+ tmp_inexact = 1;
+ }
+ } 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;
tmp_inexact = 1;
- }
- } 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;
- tmp_inexact = 1;
- } // 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;
+ tmp_inexact = 1;
+ }
+ } 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;
tmp_inexact = 1;
- }
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ // *pfpsf |= INEXACT_EXCEPTION;
+ tmp_inexact = 1;
}
- // no need to check for midpoints - already rounded away from zero!
- res = Cstar.w[0]; // the result is positive
- if (tmp_inexact)
- *pfpsf |= INEXACT_EXCEPTION;
- } else if (exp == 0) {
- // 1 <= q <= 10
- // res = +C (exact)
- res = C1.w[0];
- } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
- // res = +C * 10^exp (exact)
- res = C1.w[0] * __bid_ten2k64[exp];
}
+ // no need to check for midpoints - already rounded away from zero!
+ res = Cstar.w[0]; // the result is positive
+ if (tmp_inexact)
+ *pfpsf |= INEXACT_EXCEPTION;
+ } else if (exp == 0) {
+ // 1 <= q <= 10
+ // res = +C (exact)
+ res = C1.w[0];
+ } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10
+ // res = +C * 10^exp (exact)
+ res = C1.w[0] * ten2k64[exp];
}
}
- BID_RETURN (res);
+}
+
+BID_RETURN (res);
}
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