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Diffstat (limited to 'gcc-4.9/libgcc/config/libbid/bid128_to_uint64.c')
| -rw-r--r-- | gcc-4.9/libgcc/config/libbid/bid128_to_uint64.c | 3401 |
1 files changed, 3401 insertions, 0 deletions
diff --git a/gcc-4.9/libgcc/config/libbid/bid128_to_uint64.c b/gcc-4.9/libgcc/config/libbid/bid128_to_uint64.c new file mode 100644 index 000000000..f723c31b2 --- /dev/null +++ b/gcc-4.9/libgcc/config/libbid/bid128_to_uint64.c @@ -0,0 +1,3401 @@ +/* Copyright (C) 2007-2014 Free Software Foundation, Inc. + +This file is part of GCC. + +GCC is free software; you can redistribute it and/or modify it under +the terms of the GNU General Public License as published by the Free +Software Foundation; either version 3, or (at your option) any later +version. + +GCC is distributed in the hope that it will be useful, but WITHOUT ANY +WARRANTY; without even the implied warranty of MERCHANTABILITY or +FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License +for more details. + +Under Section 7 of GPL version 3, you are granted additional +permissions described in the GCC Runtime Library Exception, version +3.1, as published by the Free Software Foundation. + +You should have received a copy of the GNU General Public License and +a copy of the GCC Runtime Library Exception along with this program; +see the files COPYING3 and COPYING.RUNTIME respectively. If not, see +<http://www.gnu.org/licenses/>. */ + +#include "bid_internal.h" + +/***************************************************************************** + * BID128_to_uint64_rnint + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, + bid128_to_uint64_rnint, x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + 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]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + if (x_sign) { // if n < 0 and q + exp = 20 + // if n < -1/2 then n cannot be converted to uint64 with RN + // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 1/2 + // <=> 0.c(0)c(1)...c(q-1) * 10^21 > 0x05, 1<=q<=34 + // <=> C * 10^(21-q) > 0x05, 1<=q<=34 + if (q == 21) { + // C > 5 + if (C1.w[1] != 0 || C1.w[0] > 0x05ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to 64-bit unsigned int fall through + // to '1 <= q + exp <= 20' + } else { + // if 1 <= q <= 20 + // C * 10^(21-q) > 5 is true because C >= 1 and 10^(21-q) >= 10 + // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C > 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // if n > 0 and q + exp = 20 + // if n >= 2^64 - 1/2 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1) + // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34 + if (q == 1) { + // C * 10^20 >= 0x9fffffffffffffffb + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) >= 0x9fffffffffffffffb + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff + C.w[0] = C1.w[0] + C1.w[0]; + C.w[1] = C1.w[1] + C1.w[1]; + if (C.w[0] < C1.w[0]) + C.w[1]++; + if (C.w[1] > 0x01 || (C.w[1] == 0x01 + && C.w[0] >= 0xffffffffffffffffull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C >= 0x9fffffffffffffffb + if (C1.w[1] > 0x09 || (C1.w[1] == 0x09 + && C1.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits + C.w[1] = 0x09; + C.w[0] = 0xfffffffffffffffbull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + } + // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) + // return 0 + res = 0x0000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) + // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1) + // res = 0 + // else if x > 0 + // res = +1 + // else // if x < 0 + // invalid exc + ind = q - 1; + if (ind <= 18) { // 0 <= ind <= 18 + if ((C1.w[1] == 0) && (C1.w[0] <= midpoint64[ind])) { + res = 0x0000000000000000ull; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x8000000000000000ull; + *pfpsf |= INVALID_EXCEPTION; + } + } else { // 19 <= ind <= 33 + if ((C1.w[1] < midpoint128[ind - 19].w[1]) + || ((C1.w[1] == midpoint128[ind - 19].w[1]) + && (C1.w[0] <= midpoint128[ind - 19].w[0]))) { + res = 0x0000000000000000ull; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x8000000000000000ull; + *pfpsf |= INVALID_EXCEPTION; + } + } + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded + // to nearest to a 64-bit unsigned signed integer + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // 1 <= x < 2^64-1/2 so x can be rounded + // to nearest to a 64-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + } // else MP in [ODD, EVEN] + } + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint64_xrnint + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, + bid128_to_uint64_xrnint, x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + + // unpack x +x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative +x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions +C1.w[1] = x.w[1] & MASK_COEFF; +C1.w[0] = x.w[0]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + if (x_sign) { // if n < 0 and q + exp = 20 + // if n < -1/2 then n cannot be converted to uint64 with RN + // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 1/2 + // <=> 0.c(0)c(1)...c(q-1) * 10^21 > 0x05, 1<=q<=34 + // <=> C * 10^(21-q) > 0x05, 1<=q<=34 + if (q == 21) { + // C > 5 + if (C1.w[1] != 0 || C1.w[0] > 0x05ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to 64-bit unsigned int fall through + // to '1 <= q + exp <= 20' + } else { + // if 1 <= q <= 20 + // C * 10^(21-q) > 5 is true because C >= 1 and 10^(21-q) >= 10 + // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C > 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // if n > 0 and q + exp = 20 + // if n >= 2^64 - 1/2 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1) + // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34 + if (q == 1) { + // C * 10^20 >= 0x9fffffffffffffffb + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) >= 0x9fffffffffffffffb + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff + C.w[0] = C1.w[0] + C1.w[0]; + C.w[1] = C1.w[1] + C1.w[1]; + if (C.w[0] < C1.w[0]) + C.w[1]++; + if (C.w[1] > 0x01 || (C.w[1] == 0x01 + && C.w[0] >= 0xffffffffffffffffull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C >= 0x9fffffffffffffffb + if (C1.w[1] > 0x09 || (C1.w[1] == 0x09 + && C1.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits + C.w[1] = 0x09; + C.w[0] = 0xfffffffffffffffbull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + } + // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 + res = 0x0000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) + // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1) + // res = 0 + // else if x > 0 + // res = +1 + // else // if x < 0 + // invalid exc + ind = q - 1; + if (ind <= 18) { // 0 <= ind <= 18 + if ((C1.w[1] == 0) && (C1.w[0] <= midpoint64[ind])) { + res = 0x0000000000000000ull; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x8000000000000000ull; + *pfpsf |= INVALID_EXCEPTION; + BID_RETURN (res); + } + } else { // 19 <= ind <= 33 + if ((C1.w[1] < midpoint128[ind - 19].w[1]) + || ((C1.w[1] == midpoint128[ind - 19].w[1]) + && (C1.w[0] <= midpoint128[ind - 19].w[0]))) { + res = 0x0000000000000000ull; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x8000000000000000ull; + *pfpsf |= INVALID_EXCEPTION; + BID_RETURN (res); + } + } + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded + // to nearest to a 64-bit unsigned signed integer + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // 1 <= x < 2^64-1/2 so x can be rounded + // to nearest to a 64-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // determine inexactness of the rounding of C* + // if (0 < f* - 1/2 < 10^(-x)) then + // the result is exact + // else // if (f* - 1/2 > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > 0x8000000000000000ull || + (fstar.w[1] == 0x8000000000000000ull + && fstar.w[0] > 0x0ull)) { + // f* > 1/2 and the result may be exact + tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 + if (tmp64 > ten2mk128trunc[ind - 1].w[1] + || (tmp64 == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else 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 the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } + } + + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + } // else MP in [ODD, EVEN] + } + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint64_floor + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, + bid128_to_uint64_floor, x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 P256; + + // unpack x +x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative +x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions +C1.w[1] = x.w[1] & MASK_COEFF; +C1.w[0] = x.w[0]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // if n < 0 then n cannot be converted to uint64 with RM + if (x_sign) { // if n < 0 and q + exp = 20 + // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 0 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + // if n > 0 and q + exp = 20 + // if n >= 2^64 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65 + // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34 + if (q == 1) { + // C * 10^20 >= 0xa0000000000000000 + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] >= 0x0a) { + // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) >= 0xa0000000000000000 + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] >= 0x0a) { + // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C >= 0x10000000000000000 + if (C1.w[1] >= 0x01) { + // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C >= 0xa0000000000000000 + if (C1.w[1] >= 0x0a) { + // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits + C.w[1] = 0x0a; + C.w[0] = 0x0000000000000000ull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + // n is not too large to be converted to int64 if 0 <= n < 2^64 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { // n = +0.[0...0]c(0)c(1)...c(q-1) + // return 0 + res = 0x0000000000000000ull; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // 1 <= x < 2^64 so x can be rounded + // down to a 64-bit unsigned signed integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 fits in 127 bits + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = C1 * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint64_xfloor + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, + bid128_to_uint64_xfloor, x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + + // unpack x +x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative +x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions +C1.w[1] = x.w[1] & MASK_COEFF; +C1.w[0] = x.w[0]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // if n < 0 then n cannot be converted to uint64 with RM + if (x_sign) { // if n < 0 and q + exp = 20 + // too large if c(0)c(1)...c(19).c(20)...c(q-1) > 0 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + // if n > 0 and q + exp = 20 + // if n >= 2^64 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65 + // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34 + if (q == 1) { + // C * 10^20 >= 0xa0000000000000000 + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] >= 0x0a) { + // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) >= 0xa0000000000000000 + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] >= 0x0a) { + // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C >= 0x10000000000000000 + if (C1.w[1] >= 0x01) { + // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C >= 0xa0000000000000000 + if (C1.w[1] >= 0x0a) { + // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits + C.w[1] = 0x0a; + C.w[0] = 0x0000000000000000ull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + // n is not too large to be converted to int64 if 0 <= n < 2^64 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { // n = +0.[0...0]c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 + res = 0x0000000000000000ull; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // 1 <= x < 2^64 so x can be rounded + // down to a 64-bit unsigned signed integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 fits in 127 bits + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = C1 * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // determine inexactness of the rounding of C* + // if (0 < f* < 10^(-x)) then + // the result is exact + // else // if (f* > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || + (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && + fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else { // if 22 <= ind <= 33 + if (fstar.w[3] || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } + + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint64_ceil + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, bid128_to_uint64_ceil, + x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + + // unpack x +x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative +x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions +C1.w[1] = x.w[1] & MASK_COEFF; +C1.w[0] = x.w[0]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + if (x_sign) { // if n < 0 and q + exp = 20 + // if n <= -1 then n cannot be converted to uint64 with RZ + // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1 + // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34 + // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34 + if (q == 21) { + // C >= a + if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to 64-bit unsigned int fall through + // to '1 <= q + exp <= 20' + } else { + // if 1 <= q <= 20 + // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10 + // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // if n > 0 and q + exp = 20 + // if n > 2^64 - 1 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) > 2^64 - 1 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 > 2^64 - 1 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 > 10 * (2^64 - 1) + // <=> C * 10^(21-q) > 0x9fffffffffffffff6, 1<=q<=34 + if (q == 1) { + // C * 10^20 > 0x9fffffffffffffff6 + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] > 0xfffffffffffffff6ull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) > 0x9fffffffffffffff6 + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] > 0xfffffffffffffff6ull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C > 0xffffffffffffffff + if (C1.w[1]) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C > 0x9fffffffffffffff6 + if (C1.w[1] > 0x09 || (C1.w[1] == 0x09 + && C1.w[0] > 0xfffffffffffffff6ull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C > 10^(q-21) * 0x9fffffffffffffff6 max 44 bits x 68 bits + C.w[1] = 0x09; + C.w[0] = 0xfffffffffffffff6ull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + } + // n is not too large to be converted to int64 if -1 < n <= 2^64 - 1 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) + // return 0 or 1 + if (x_sign) + res = 0x0000000000000000ull; + else + res = 0x0000000000000001ull; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded + // to zero to a 64-bit unsigned signed integer + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // 1 <= x <= 2^64 - 1 so x can be rounded + // to zero to a 64-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 fits in 127 bits + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = C1 * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // if the result is positive and inexact, need to add 1 to it + + // determine inexactness of the rounding of C* + // if (0 < f* < 10^(-x)) then + // the result is exact + // else // if (f* > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + if (!x_sign) { // positive and inexact + Cstar.w[0]++; + if (Cstar.w[0] == 0x0) + Cstar.w[1]++; + } + } // else the result is exact + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + if (!x_sign) { // positive and inexact + Cstar.w[0]++; + if (Cstar.w[0] == 0x0) + Cstar.w[1]++; + } + } // else the result is exact + } else { // if 22 <= ind <= 33 + if (fstar.w[3] || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + if (!x_sign) { // positive and inexact + Cstar.w[0]++; + if (Cstar.w[0] == 0x0) + Cstar.w[1]++; + } + } // else the result is exact + } + + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint64_xceil + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, + bid128_to_uint64_xceil, x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + + // unpack x +x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative +x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions +C1.w[1] = x.w[1] & MASK_COEFF; +C1.w[0] = x.w[0]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + if (x_sign) { // if n < 0 and q + exp = 20 + // if n <= -1 then n cannot be converted to uint64 with RZ + // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1 + // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34 + // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34 + if (q == 21) { + // C >= a + if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to 64-bit unsigned int fall through + // to '1 <= q + exp <= 20' + } else { + // if 1 <= q <= 20 + // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10 + // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // if n > 0 and q + exp = 20 + // if n > 2^64 - 1 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) > 2^64 - 1 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 > 2^64 - 1 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 > 10 * (2^64 - 1) + // <=> C * 10^(21-q) > 0x9fffffffffffffff6, 1<=q<=34 + if (q == 1) { + // C * 10^20 > 0x9fffffffffffffff6 + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] > 0xfffffffffffffff6ull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) > 0x9fffffffffffffff6 + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] > 0xfffffffffffffff6ull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C > 0xffffffffffffffff + if (C1.w[1]) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C > 0x9fffffffffffffff6 + if (C1.w[1] > 0x09 || (C1.w[1] == 0x09 + && C1.w[0] > 0xfffffffffffffff6ull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C > 10^(q-21) * 0x9fffffffffffffff6 max 44 bits x 68 bits + C.w[1] = 0x09; + C.w[0] = 0xfffffffffffffff6ull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + } + // n is not too large to be converted to int64 if -1 < n <= 2^64 - 1 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 or 1 + if (x_sign) + res = 0x0000000000000000ull; + else + res = 0x0000000000000001ull; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded + // to zero to a 64-bit unsigned signed integer + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // 1 <= x <= 2^64 - 1 so x can be rounded + // to zero to a 64-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 fits in 127 bits + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = C1 * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // if the result is positive and inexact, need to add 1 to it + + // determine inexactness of the rounding of C* + // if (0 < f* < 10^(-x)) then + // the result is exact + // else // if (f* > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + if (!x_sign) { // positive and inexact + Cstar.w[0]++; + if (Cstar.w[0] == 0x0) + Cstar.w[1]++; + } + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + if (!x_sign) { // positive and inexact + Cstar.w[0]++; + if (Cstar.w[0] == 0x0) + Cstar.w[1]++; + } + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else { // if 22 <= ind <= 33 + if (fstar.w[3] || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + if (!x_sign) { // positive and inexact + Cstar.w[0]++; + if (Cstar.w[0] == 0x0) + Cstar.w[1]++; + } + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } + + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint64_int + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, bid128_to_uint64_int, + x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 P256; + + // unpack x +x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative +x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions +C1.w[1] = x.w[1] & MASK_COEFF; +C1.w[0] = x.w[0]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + if (x_sign) { // if n < 0 and q + exp = 20 + // if n <= -1 then n cannot be converted to uint64 with RZ + // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1 + // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34 + // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34 + if (q == 21) { + // C >= a + if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to 64-bit unsigned int fall through + // to '1 <= q + exp <= 20' + } else { + // if 1 <= q <= 20 + // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10 + // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // if n > 0 and q + exp = 20 + // if n >= 2^64 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65 + // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34 + if (q == 1) { + // C * 10^20 >= 0xa0000000000000000 + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] >= 0x0a) { + // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) >= 0xa0000000000000000 + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] >= 0x0a) { + // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C >= 0x10000000000000000 + if (C1.w[1] >= 0x01) { + // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C >= 0xa0000000000000000 + if (C1.w[1] >= 0x0a) { + // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits + C.w[1] = 0x0a; + C.w[0] = 0x0000000000000000ull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + } + // n is not too large to be converted to int64 if -1 < n < 2^64 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) + // return 0 + res = 0x0000000000000000ull; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded + // to zero to a 64-bit unsigned signed integer + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // 1 <= x < 2^64 so x can be rounded + // to zero to a 64-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 fits in 127 bits + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = C1 * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint64_xint + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, bid128_to_uint64_xint, + x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + + // unpack x +x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative +x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions +C1.w[1] = x.w[1] & MASK_COEFF; +C1.w[0] = x.w[0]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + if (x_sign) { // if n < 0 and q + exp = 20 + // if n <= -1 then n cannot be converted to uint64 with RZ + // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1 + // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x0a, 1<=q<=34 + // <=> C * 10^(21-q) >= 0x0a, 1<=q<=34 + if (q == 21) { + // C >= a + if (C1.w[1] != 0 || C1.w[0] >= 0x0aull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to 64-bit unsigned int fall through + // to '1 <= q + exp <= 20' + } else { + // if 1 <= q <= 20 + // C * 10^(21-q) >= a is true because C >= 1 and 10^(21-q) >= 10 + // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= a * 10^(q-21) is true because C > 2^64 and a*10^(q-21) < 2^64 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // if n > 0 and q + exp = 20 + // if n >= 2^64 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*2^65 + // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=34 + if (q == 1) { + // C * 10^20 >= 0xa0000000000000000 + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] >= 0x0a) { + // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) >= 0xa0000000000000000 + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] >= 0x0a) { + // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C >= 0x10000000000000000 + if (C1.w[1] >= 0x01) { + // actually C1.w[1] == 0x01 && C1.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C >= 0xa0000000000000000 + if (C1.w[1] >= 0x0a) { + // actually C1.w[1] == 0x0a && C1.w[0] >= 0x0000000000000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 10^(q-21) * 0xa0000000000000000 max 44 bits x 68 bits + C.w[1] = 0x0a; + C.w[0] = 0x0000000000000000ull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + } + // n is not too large to be converted to int64 if -1 < n < 2^64 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 + res = 0x0000000000000000ull; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded + // to zero to a 64-bit unsigned signed integer + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // 1 <= x < 2^64 so x can be rounded + // to zero to a 64-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 fits in 127 bits + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = C1 * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // determine inexactness of the rounding of C* + // if (0 < f* < 10^(-x)) then + // the result is exact + // else // if (f* > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else { // if 22 <= ind <= 33 + if (fstar.w[3] || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } + + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint64_rninta + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, + bid128_to_uint64_rninta, x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + 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]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + if (x_sign) { // if n < 0 and q + exp = 20 + // if n <= -1/2 then n cannot be converted to uint64 with RN + // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1/2 + // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x05, 1<=q<=34 + // <=> C * 10^(21-q) >= 0x05, 1<=q<=34 + if (q == 21) { + // C >= 5 + if (C1.w[1] != 0 || C1.w[0] >= 0x05ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to 64-bit unsigned int fall through + // to '1 <= q + exp <= 20' + } else { + // if 1 <= q <= 20 + // C * 10^(21-q) >= 5 is true because C >= 1 and 10^(21-q) >= 10 + // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // if n > 0 and q + exp = 20 + // if n >= 2^64 - 1/2 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1) + // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34 + if (q == 1) { + // C * 10^20 >= 0x9fffffffffffffffb + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) >= 0x9fffffffffffffffb + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff + C.w[0] = C1.w[0] + C1.w[0]; + C.w[1] = C1.w[1] + C1.w[1]; + if (C.w[0] < C1.w[0]) + C.w[1]++; + if (C.w[1] > 0x01 || (C.w[1] == 0x01 + && C.w[0] >= 0xffffffffffffffffull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C >= 0x9fffffffffffffffb + if (C1.w[1] > 0x09 || (C1.w[1] == 0x09 + && C1.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits + C.w[1] = 0x09; + C.w[0] = 0xfffffffffffffffbull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + } + // n is not too large to be converted to int64 if -1/2 < n < 2^64 - 1/2 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) + // return 0 + res = 0x0000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) + // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1) + // res = 0 + // else if x > 0 + // res = +1 + // else // if x < 0 + // invalid exc + ind = q - 1; + if (ind <= 18) { // 0 <= ind <= 18 + if ((C1.w[1] == 0) && (C1.w[0] < midpoint64[ind])) { + res = 0x0000000000000000ull; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // 19 <= ind <= 33 + if ((C1.w[1] < midpoint128[ind - 19].w[1]) + || ((C1.w[1] == midpoint128[ind - 19].w[1]) + && (C1.w[0] < midpoint128[ind - 19].w[0]))) { + res = 0x0000000000000000ull; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded + // to nearest to a 64-bit unsigned signed integer + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // 1 <= x < 2^64-1/2 so x can be rounded + // to nearest to a 64-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + + // if the result was a midpoint it was rounded away from zero + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint64_xrninta + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (UINT64, + bid128_to_uint64_xrninta, x) + + UINT64 res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + + // unpack x +x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative +x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions +C1.w[1] = x.w[1] & MASK_COEFF; +C1.w[0] = x.w[0]; + + // check for NaN or Infinity +if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { + // x is special +if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN + if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x0000000000000000ull; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x0000000000000000ull; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + + if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) + // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... + // so x rounded to an integer may or may not fit in an unsigned 64-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 20' + if (x_sign) { // if n < 0 and q + exp = 20 + // if n <= -1/2 then n cannot be converted to uint64 with RN + // too large if c(0)c(1)...c(19).c(20)...c(q-1) >= 1/2 + // <=> 0.c(0)c(1)...c(q-1) * 10^21 >= 0x05, 1<=q<=34 + // <=> C * 10^(21-q) >= 0x05, 1<=q<=34 + if (q == 21) { + // C >= 5 + if (C1.w[1] != 0 || C1.w[0] >= 0x05ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to 64-bit unsigned int fall through + // to '1 <= q + exp <= 20' + } else { + // if 1 <= q <= 20 + // C * 10^(21-q) >= 5 is true because C >= 1 and 10^(21-q) >= 10 + // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 5 * 10^(q-21) is true because C > 2^64 and 5*10^(q-21) < 2^64 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // if n > 0 and q + exp = 20 + // if n >= 2^64 - 1/2 then n is too large + // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2 + // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1) + // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=34 + if (q == 1) { + // C * 10^20 >= 0x9fffffffffffffffb + __mul_128x64_to_128 (C, C1.w[0], ten2k128[0]); // 10^20 * C + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q <= 19) { + // C * 10^(21-q) >= 0x9fffffffffffffffb + __mul_64x64_to_128MACH (C, C1.w[0], ten2k64[21 - q]); + if (C.w[1] > 0x09 || (C.w[1] == 0x09 + && C.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 20) { + // C * 10 >= 0x9fffffffffffffffb <=> C * 2 > 1ffffffffffffffff + C.w[0] = C1.w[0] + C1.w[0]; + C.w[1] = C1.w[1] + C1.w[1]; + if (C.w[0] < C1.w[0]) + C.w[1]++; + if (C.w[1] > 0x01 || (C.w[1] == 0x01 + && C.w[0] >= 0xffffffffffffffffull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else if (q == 21) { + // C >= 0x9fffffffffffffffb + if (C1.w[1] > 0x09 || (C1.w[1] == 0x09 + && C1.w[0] >= 0xfffffffffffffffbull)) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } else { // if 22 <= q <= 34 => 1 <= q - 21 <= 13 + // C >= 10^(q-21) * 0x9fffffffffffffffb max 44 bits x 68 bits + C.w[1] = 0x09; + C.w[0] = 0xfffffffffffffffbull; + __mul_128x64_to_128 (C, ten2k64[q - 21], C); + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // else cases that can be rounded to a 64-bit int fall through + // to '1 <= q + exp <= 20' + } + } + } + // n is not too large to be converted to int64 if -1/2 < n < 2^64 - 1/2 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 + res = 0x0000000000000000ull; + BID_RETURN (res); + } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) + // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1) + // res = 0 + // else if x > 0 + // res = +1 + // else // if x < 0 + // invalid exc + ind = q - 1; + if (ind <= 18) { // 0 <= ind <= 18 + if ((C1.w[1] == 0) && (C1.w[0] < midpoint64[ind])) { + res = 0x0000000000000000ull; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x8000000000000000ull; + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } else { // 19 <= ind <= 33 + if ((C1.w[1] < midpoint128[ind - 19].w[1]) + || ((C1.w[1] == midpoint128[ind - 19].w[1]) + && (C1.w[0] < midpoint128[ind - 19].w[0]))) { + res = 0x0000000000000000ull; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x8000000000000000ull; + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + } + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } else { // if (1 <= q + exp <= 20, 1 <= q <= 34, -33 <= exp <= 19) + // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded + // to nearest to a 64-bit unsigned signed integer + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x8000000000000000ull; + BID_RETURN (res); + } + // 1 <= x < 2^64-1/2 so x can be rounded + // to nearest to a 64-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 20 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // determine inexactness of the rounding of C* + // if (0 < f* - 1/2 < 10^(-x)) then + // the result is exact + // else // if (f* - 1/2 > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > 0x8000000000000000ull || + (fstar.w[1] == 0x8000000000000000ull + && fstar.w[0] > 0x0ull)) { + // f* > 1/2 and the result may be exact + tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 + if (tmp64 > ten2mk128trunc[ind - 1].w[1] + || (tmp64 == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else 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 the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } + } + + // if the result was a midpoint it was rounded away from zero + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 20, but x < 2^64 - 1/2 so in this case C1.w[1] has to be 0 + // res = C (exact) + res = C1.w[0]; + } else { + // if (exp > 0) => 1 <= exp <= 19, 1 <= q < 19, 2 <= q + exp <= 20 + // res = C * 10^exp (exact) - must fit in 64 bits + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} |