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Diffstat (limited to 'gcc-4.4.3/libgcc/config/libbid/bid128_to_uint32.c')
| -rw-r--r-- | gcc-4.4.3/libgcc/config/libbid/bid128_to_uint32.c | 3588 |
1 files changed, 3588 insertions, 0 deletions
diff --git a/gcc-4.4.3/libgcc/config/libbid/bid128_to_uint32.c b/gcc-4.4.3/libgcc/config/libbid/bid128_to_uint32.c new file mode 100644 index 000000000..3d8f44c87 --- /dev/null +++ b/gcc-4.4.3/libgcc/config/libbid/bid128_to_uint32.c @@ -0,0 +1,3588 @@ +/* Copyright (C) 2007, 2009 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_uint32_rnint + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_rnint, x) + + unsigned int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in an unsigned 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + if (x_sign) { // if n < 0 and q + exp = 10 + // if n < -1/2 then n cannot be converted to uint32 with RN + // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 1/2 + // <=> 0.c(0)c(1)...c(q-1) * 10^11 > 0x05, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 > 0x05ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 > 0x05 <=> + // C > 0x05 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 1/2 up) + tmp64 = 0x05ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } else { // if n > 0 and q + exp = 10 + // if n >= 2^32 - 1/2 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x9fffffffbull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=> + // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32-1/2 up) + tmp64 = 0x9fffffffbull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + } + // n is not too large to be converted to int32: -1/2 <= n < 2^32 - 1/2 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) + // return 0 + res = 0x00000000; + BID_RETURN (res); + } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) + // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1) + // res = 0 + // else if x > 0 + // res = +1 + // else // if x < 0 + // invalid exc + ind = q - 1; + if (ind <= 18) { // 0 <= ind <= 18 + if ((C1.w[1] == 0) && (C1.w[0] <= midpoint64[ind])) { + res = 0x00000000; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x80000000; + *pfpsf |= INVALID_EXCEPTION; + } + } else { // 19 <= ind <= 33 + if ((C1.w[1] < midpoint128[ind - 19].w[1]) + || ((C1.w[1] == midpoint128[ind - 19].w[1]) + && (C1.w[0] <= midpoint128[ind - 19].w[0]))) { + res = 0x00000000; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x80000000; + *pfpsf |= INVALID_EXCEPTION; + } + } + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // 1 <= x < 2^32-1/2 so x can be rounded + // to nearest to a 32-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + } // else MP in [ODD, EVEN] + } + res = Cstar.w[0]; // the result is positive + } else if (exp == 0) { + // 1 <= q <= 10 + // res = C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint32_xrnint + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_xrnint, x) + + unsigned int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + unsigned int tmp_inexact = 0; + + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in an unsigned 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + if (x_sign) { // if n < 0 and q + exp = 10 + // if n < -1/2 then n cannot be converted to uint32 with RN + // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 1/2 + // <=> 0.c(0)c(1)...c(q-1) * 10^11 > 0x05, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 > 0x05ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 > 0x05 <=> + // C > 0x05 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 1/2 up) + tmp64 = 0x05ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } else { // if n > 0 and q + exp = 10 + // if n >= 2^32 - 1/2 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x9fffffffbull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=> + // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32-1/2 up) + tmp64 = 0x9fffffffbull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + } + // n is not too large to be converted to int32: -1/2 <= n < 2^32 - 1/2 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 + res = 0x00000000; + BID_RETURN (res); + } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) + // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1) + // res = 0 + // else if x > 0 + // res = +1 + // else // if x < 0 + // invalid exc + ind = q - 1; + if (ind <= 18) { // 0 <= ind <= 18 + if ((C1.w[1] == 0) && (C1.w[0] <= midpoint64[ind])) { + res = 0x00000000; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x80000000; + *pfpsf |= INVALID_EXCEPTION; + BID_RETURN (res); + } + } else { // 19 <= ind <= 33 + if ((C1.w[1] < midpoint128[ind - 19].w[1]) + || ((C1.w[1] == midpoint128[ind - 19].w[1]) + && (C1.w[0] <= midpoint128[ind - 19].w[0]))) { + res = 0x00000000; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x80000000; + *pfpsf |= INVALID_EXCEPTION; + BID_RETURN (res); + } + } + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // 1 <= x < 2^32-1/2 so x can be rounded + // to nearest to a 32-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // determine inexactness of the rounding of C* + // if (0 < f* - 1/2 < 10^(-x)) then + // the result is exact + // else // if (f* - 1/2 > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > 0x8000000000000000ull || + (fstar.w[1] == 0x8000000000000000ull + && fstar.w[0] > 0x0ull)) { + // f* > 1/2 and the result may be exact + tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 + if (tmp64 > ten2mk128trunc[ind - 1].w[1] + || (tmp64 == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[3] > 0x0 || + (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || + (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && + (fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[2] - onehalf128[ind - 1]; + tmp64A = fstar.w[3]; + if (tmp64 > fstar.w[2]) + tmp64A--; + if (tmp64A || tmp64 + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } + } + + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + } // else MP in [ODD, EVEN] + } + res = Cstar.w[0]; // the result is positive + if (tmp_inexact) + *pfpsf |= INEXACT_EXCEPTION; + } else if (exp == 0) { + // 1 <= q <= 10 + // res = C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint32_floor + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_floor, x) + + unsigned int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + int is_inexact_gt_midpoint = 0; + int is_midpoint_lt_even = 0; + + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + // x < 0 is invalid + if (x_sign) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // x > 0 from this point on + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in a signed 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + // n > 0 and q + exp = 10 + // if n >= 2^32 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0xa00000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=> + // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32 up) + tmp64 = 0xa00000000ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + // n is not too large to be converted to int32: 0 <= n < 2^31 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { + // n = +0.0...c(0)c(1)...c(q-1) or n = +0.c(0)c(1)...c(q-1) + // return 0 + res = 0x00000000; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + // 1 <= x < 2^32 so x can be rounded down to a 32-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // determine inexactness of the rounding of C* + // if (0 < f* - 1/2 < 10^(-x)) then + // the result is exact + // else // if (f* - 1/2 > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > 0x8000000000000000ull || + (fstar.w[1] == 0x8000000000000000ull + && fstar.w[0] > 0x0ull)) { + // f* > 1/2 and the result may be exact + tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 + if (tmp64 > ten2mk128trunc[ind - 1].w[1] + || (tmp64 == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + is_inexact_gt_midpoint = 1; + } + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[3] > 0x0 || + (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || + (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && + (fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[2] - onehalf128[ind - 1]; + tmp64A = fstar.w[3]; + if (tmp64 > fstar.w[2]) + tmp64A--; + if (tmp64A || tmp64 + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + is_inexact_gt_midpoint = 1; + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + is_inexact_gt_midpoint = 1; + } + } + + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + is_inexact_gt_midpoint = 0; + } else { // else MP in [ODD, EVEN] + is_midpoint_lt_even = 1; + is_inexact_gt_midpoint = 0; + } + } + // general correction for RM + if (is_midpoint_lt_even || is_inexact_gt_midpoint) { + Cstar.w[0] = Cstar.w[0] - 1; + } else { + ; // the result is already correct + } + res = Cstar.w[0]; + } else if (exp == 0) { + // 1 <= q <= 10 + // res = +C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = +C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint32_xfloor + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_xfloor, x) + + unsigned int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + int is_inexact_gt_midpoint = 0; + int is_midpoint_lt_even = 0; + + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + // x < 0 is invalid + if (x_sign) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // x > 0 from this point on + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in a signed 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + // n > 0 and q + exp = 10 + // if n >= 2^32 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0xa00000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=> + // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32 up) + tmp64 = 0xa00000000ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + // n is not too large to be converted to int32: 0 <= n < 2^31 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { + // n = +0.0...c(0)c(1)...c(q-1) or n = +0.c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 + res = 0x00000000; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + // 1 <= x < 2^32 so x can be rounded down to a 32-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + 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; + is_inexact_gt_midpoint = 1; + } + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[3] > 0x0 || + (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || + (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && + (fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[2] - onehalf128[ind - 1]; + tmp64A = fstar.w[3]; + if (tmp64 > fstar.w[2]) + tmp64A--; + if (tmp64A || tmp64 + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + is_inexact_gt_midpoint = 1; + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + is_inexact_gt_midpoint = 1; + } + } + + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + is_inexact_gt_midpoint = 0; + } else { // else MP in [ODD, EVEN] + is_midpoint_lt_even = 1; + is_inexact_gt_midpoint = 0; + } + } + // general correction for RM + if (is_midpoint_lt_even || is_inexact_gt_midpoint) { + Cstar.w[0] = Cstar.w[0] - 1; + } else { + ; // the result is already correct + } + res = Cstar.w[0]; + } else if (exp == 0) { + // 1 <= q <= 10 + // res = +C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = +C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint32_ceil + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_ceil, x) + + unsigned int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + int is_inexact_lt_midpoint = 0; + int is_midpoint_gt_even = 0; + + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in a signed 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + if (x_sign) { // if n < 0 and q + exp = 10 + // if n <= -1 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1 + // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x50000000a, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x0aull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=> + // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 1 up) + tmp64 = 0x0aull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } else { // if n > 0 and q + exp = 10 + // if n > 2^32 - 1 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 2^32 - 1 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 > 0x9fffffff6ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6 <=> + // C > 0x9fffffff6 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32 up) + tmp64 = 0x9fffffff6ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + } + // n is not too large to be converted to int32: -2^32-1 < n <= 2^32-1 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { + // n = +/-0.0...c(0)c(1)...c(q-1) or n = +/-0.c(0)c(1)...c(q-1) + // return 0 + if (x_sign) + res = 0x00000000; + else + res = 0x00000001; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + // -2^32-1 < x <= -1 or 1 <= x <= 2^32-1 so x can be rounded + // toward positive infinity to a 32-bit signed integer + if (x_sign) { // x <= -1 is invalid + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // x > 0 from this point on + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // determine inexactness of the rounding of C* + // if (0 < f* - 1/2 < 10^(-x)) then + // the result is exact + // else // if (f* - 1/2 > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > 0x8000000000000000ull || + (fstar.w[1] == 0x8000000000000000ull + && fstar.w[0] > 0x0ull)) { + // f* > 1/2 and the result may be exact + tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 + if (tmp64 > ten2mk128trunc[ind - 1].w[1] + || (tmp64 == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { + is_inexact_lt_midpoint = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + ; + } + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[3] > 0x0 || + (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || + (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && + (fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[2] - onehalf128[ind - 1]; + tmp64A = fstar.w[3]; + if (tmp64 > fstar.w[2]) + tmp64A--; + if (tmp64A || tmp64 + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + is_inexact_lt_midpoint = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + is_inexact_lt_midpoint = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + ; + } + } + + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + is_midpoint_gt_even = 1; + is_inexact_lt_midpoint = 0; + } else { // else MP in [ODD, EVEN] + is_inexact_lt_midpoint = 0; + } + } + // general correction for RM + if (is_midpoint_gt_even || is_inexact_lt_midpoint) { + Cstar.w[0] = Cstar.w[0] + 1; + } else { + ; // the result is already correct + } + res = Cstar.w[0]; + } else if (exp == 0) { + // 1 <= q <= 10 + // res = +C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = +C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint32_xceil + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_xceil, x) + + unsigned int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + int is_inexact_lt_midpoint = 0; + int is_midpoint_gt_even = 0; + + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in a signed 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + if (x_sign) { // if n < 0 and q + exp = 10 + // if n <= -1 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1 + // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x50000000a, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x0aull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=> + // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 1 up) + tmp64 = 0x0aull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } else { // if n > 0 and q + exp = 10 + // if n > 2^32 - 1 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) > 2^32 - 1 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 > 0x9fffffff6ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 > 0x9fffffff6 <=> + // C > 0x9fffffff6 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32 up) + tmp64 = 0x9fffffff6ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + } + // n is not too large to be converted to int32: -2^32-1 < n <= 2^32-1 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { + // n = +/-0.0...c(0)c(1)...c(q-1) or n = +/-0.c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 + if (x_sign) + res = 0x00000000; + else + res = 0x00000001; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + // -2^32-1 < x <= -1 or 1 <= x <= 2^32-1 so x can be rounded + // toward positive infinity to a 32-bit signed integer + if (x_sign) { // x <= -1 is invalid + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // x > 0 from this point on + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // determine inexactness of the rounding of C* + // if (0 < f* - 1/2 < 10^(-x)) then + // the result is exact + // else // if (f* - 1/2 > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > 0x8000000000000000ull || + (fstar.w[1] == 0x8000000000000000ull + && fstar.w[0] > 0x0ull)) { + // f* > 1/2 and the result may be exact + tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 + if (tmp64 > ten2mk128trunc[ind - 1].w[1] + || (tmp64 == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + is_inexact_lt_midpoint = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[3] > 0x0 || + (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || + (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && + (fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[2] - onehalf128[ind - 1]; + tmp64A = fstar.w[3]; + if (tmp64 > fstar.w[2]) + tmp64A--; + if (tmp64A || tmp64 + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + is_inexact_lt_midpoint = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + is_inexact_lt_midpoint = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } + } + + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + is_midpoint_gt_even = 1; + is_inexact_lt_midpoint = 0; + } else { // else MP in [ODD, EVEN] + is_inexact_lt_midpoint = 0; + } + } + // general correction for RM + if (is_midpoint_gt_even || is_inexact_lt_midpoint) { + Cstar.w[0] = Cstar.w[0] + 1; + } else { + ; // the result is already correct + } + res = Cstar.w[0]; + } else if (exp == 0) { + // 1 <= q <= 10 + // res = +C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = +C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint32_int + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_int, x) + + int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + int is_inexact_gt_midpoint = 0; + int is_midpoint_lt_even = 0; + + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in a signed 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + if (x_sign) { // if n < 0 and q + exp = 10 + // if n <= -1 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1 + // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x0aull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit uint fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=> + // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 1 up) + tmp64 = 0x0aull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } else { // if n > 0 and q + exp = 10 + // if n >= 2^32 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0xa00000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit uint fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=> + // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32 up) + tmp64 = 0xa00000000ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + } + // n is not too large to be converted to uint32: -2^32 < n < 2^32 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { + // n = +/-0.0...c(0)c(1)...c(q-1) or n = +/-0.c(0)c(1)...c(q-1) + // return 0 + res = 0x00000000; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + // x = d(0)...d(k).d(k+1)..., k >= 0, d(0) != 0 + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // x > 0 from this point on + // 1 <= x < 2^32 so x can be rounded to zero to a 32-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // determine inexactness of the rounding of C* + // if (0 < f* - 1/2 < 10^(-x)) then + // the result is exact + // else // if (f* - 1/2 > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > 0x8000000000000000ull || + (fstar.w[1] == 0x8000000000000000ull + && fstar.w[0] > 0x0ull)) { + // f* > 1/2 and the result may be exact + tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 + if (tmp64 > ten2mk128trunc[ind - 1].w[1] + || (tmp64 == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + is_inexact_gt_midpoint = 1; + } + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[3] > 0x0 || + (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || + (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && + (fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[2] - onehalf128[ind - 1]; + tmp64A = fstar.w[3]; + if (tmp64 > fstar.w[2]) + tmp64A--; + if (tmp64A || tmp64 + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + is_inexact_gt_midpoint = 1; + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + is_inexact_gt_midpoint = 1; + } + } + + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + is_inexact_gt_midpoint = 0; + } else { // else MP in [ODD, EVEN] + is_midpoint_lt_even = 1; + is_inexact_gt_midpoint = 0; + } + } + // general correction for RZ + if (is_midpoint_lt_even || is_inexact_gt_midpoint) { + Cstar.w[0] = Cstar.w[0] - 1; + } else { + ; // exact, the result is already correct + } + res = Cstar.w[0]; + } else if (exp == 0) { + // 1 <= q <= 10 + // res = +C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = +C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint32_xint + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_xint, x) + + int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + int is_inexact_gt_midpoint = 0; + int is_midpoint_lt_even = 0; + + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in a signed 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + if (x_sign) { // if n < 0 and q + exp = 10 + // if n <= -1 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1 + // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x0aull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit uint fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x0a <=> + // C >= 0x0a * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 1 up) + tmp64 = 0x0aull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } else { // if n > 0 and q + exp = 10 + // if n >= 2^32 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0xa00000000ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit uint fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0xa00000000 <=> + // C >= 0xa00000000 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32 up) + tmp64 = 0xa00000000ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + } + // n is not too large to be converted to uint32: -2^32 < n < 2^32 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) <= 0) { + // n = +/-0.0...c(0)c(1)...c(q-1) or n = +/-0.c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 + res = 0x00000000; + BID_RETURN (res); + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + // x = d(0)...d(k).d(k+1)..., k >= 0, d(0) != 0 + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // x > 0 from this point on + // 1 <= x < 2^32 so x can be rounded to zero to a 32-bit unsigned integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // 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; + is_inexact_gt_midpoint = 1; + } + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[3] > 0x0 || + (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || + (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && + (fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[2] - onehalf128[ind - 1]; + tmp64A = fstar.w[3]; + if (tmp64 > fstar.w[2]) + tmp64A--; + if (tmp64A || tmp64 + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + is_inexact_gt_midpoint = 1; + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + *pfpsf |= INEXACT_EXCEPTION; + is_inexact_gt_midpoint = 1; + } + } + + // if the result was a midpoint it was rounded away from zero, so + // it will need a correction + // check for midpoints + if ((fstar.w[3] == 0) && (fstar.w[2] == 0) + && (fstar.w[1] || fstar.w[0]) + && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { + // the result is a midpoint; round to nearest + if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] + // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 + Cstar.w[0]--; // Cstar.w[0] is now even + is_inexact_gt_midpoint = 0; + } else { // else MP in [ODD, EVEN] + is_midpoint_lt_even = 1; + is_inexact_gt_midpoint = 0; + } + } + // general correction for RZ + if (is_midpoint_lt_even || is_inexact_gt_midpoint) { + Cstar.w[0] = Cstar.w[0] - 1; + } else { + ; // exact, the result is already correct + } + res = Cstar.w[0]; + } else if (exp == 0) { + // 1 <= q <= 10 + // res = +C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = +C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint32_rninta + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_rninta, x) + + unsigned int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in a signed 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + if (x_sign) { // if n < 0 and q + exp = 10 + // if n <= -1/2 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1/2 + // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x05ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05 <=> + // C >= 0x05 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 1/2 up) + tmp64 = 0x05ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } else { // if n > 0 and q + exp = 10 + // if n >= 2^32 - 1/2 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x9fffffffbull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=> + // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32-1/2 up) + tmp64 = 0x9fffffffbull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + } + // n is not too large to be converted to int32: -1/2 < n < 2^32 - 1/2 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) + // return 0 + res = 0x00000000; + BID_RETURN (res); + } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) + // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1) + // res = 0 + // else if x > 0 + // res = +1 + // else // if x < 0 + // invalid exc + ind = q - 1; + if (ind <= 18) { // 0 <= ind <= 18 + if ((C1.w[1] == 0) && (C1.w[0] < midpoint64[ind])) { + res = 0x00000000; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x80000000; + *pfpsf |= INVALID_EXCEPTION; + } + } else { // 19 <= ind <= 33 + if ((C1.w[1] < midpoint128[ind - 19].w[1]) + || ((C1.w[1] == midpoint128[ind - 19].w[1]) + && (C1.w[0] < midpoint128[ind - 19].w[0]))) { + res = 0x00000000; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x80000000; + *pfpsf |= INVALID_EXCEPTION; + } + } + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // 1 <= x < 2^31-1/2 so x can be rounded + // to nearest-away to a 32-bit signed integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // if the result was a midpoint, it was already rounded away from zero + res = Cstar.w[0]; // always positive + // no need to check for midpoints - already rounded away from zero! + } else if (exp == 0) { + // 1 <= q <= 10 + // res = +C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = +C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} + +/***************************************************************************** + * BID128_to_uint32_xrninta + ****************************************************************************/ + +BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (unsigned int, + bid128_to_uint32_xrninta, x) + + unsigned int res; + UINT64 x_sign; + UINT64 x_exp; + int exp; // unbiased exponent + // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) + UINT64 tmp64, tmp64A; + BID_UI64DOUBLE tmp1; + unsigned int x_nr_bits; + int q, ind, shift; + UINT128 C1, C; + UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits + UINT256 fstar; + UINT256 P256; + unsigned int tmp_inexact = 0; + + // 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 = 0x80000000; + } else { // x is QNaN + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} else { // x is not a NaN, so it must be infinity + if (!x_sign) { // x is +inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } else { // x is -inf + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + } + BID_RETURN (res); +} +} + // check for non-canonical values (after the check for special values) +if ((C1.w[1] > 0x0001ed09bead87c0ull) + || (C1.w[1] == 0x0001ed09bead87c0ull + && (C1.w[0] > 0x378d8e63ffffffffull)) + || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { + res = 0x00000000; + BID_RETURN (res); +} else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { + // x is 0 + res = 0x00000000; + BID_RETURN (res); +} else { // x is not special and is not zero + + // q = nr. of decimal digits in x + // determine first the nr. of bits in x + if (C1.w[1] == 0) { + if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 + // split the 64-bit value in two 32-bit halves to avoid rounding errors + if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 + tmp1.d = (double) (C1.w[0] >> 32); // exact conversion + x_nr_bits = + 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } else { // x < 2^32 + tmp1.d = (double) (C1.w[0]); // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // if x < 2^53 + tmp1.d = (double) C1.w[0]; // exact conversion + x_nr_bits = + 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) + tmp1.d = (double) C1.w[1]; // exact conversion + x_nr_bits = + 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); + } + q = nr_digits[x_nr_bits - 1].digits; + if (q == 0) { + q = nr_digits[x_nr_bits - 1].digits1; + if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi + || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi + && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) + q++; + } + exp = (x_exp >> 49) - 6176; + if ((q + exp) > 10) { // x >= 10^10 ~= 2^33.2... (cannot fit in 32 bits) + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } else if ((q + exp) == 10) { // x = c(0)c(1)...c(9).c(10)...c(q-1) + // in this case 2^29.89... ~= 10^9 <= x < 10^10 ~= 2^33.2... + // so x rounded to an integer may or may not fit in a signed 32-bit int + // the cases that do not fit are identified here; the ones that fit + // fall through and will be handled with other cases further, + // under '1 <= q + exp <= 10' + if (x_sign) { // if n < 0 and q + exp = 10 + // if n <= -1/2 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 1/2 + // <=> 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x05ull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x05 <=> + // C >= 0x05 * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 1/2 up) + tmp64 = 0x05ull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } else { // if n > 0 and q + exp = 10 + // if n >= 2^32 - 1/2 then n is too large + // too large if c(0)c(1)...c(9).c(10)...c(q-1) >= 2^32-1/2 + // too large if 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb, 1<=q<=34 + if (q <= 11) { + tmp64 = C1.w[0] * ten2k64[11 - q]; // C scaled up to 11-digit int + // c(0)c(1)...c(9)c(10) or c(0)c(1)...c(q-1)0...0 (11 digits) + if (tmp64 >= 0x9fffffffbull) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } else { // if (q > 11), i.e. 12 <= q <= 34 and so -24 <= exp <= -2 + // 0.c(0)c(1)...c(q-1) * 10^11 >= 0x9fffffffb <=> + // C >= 0x9fffffffb * 10^(q-11) where 1 <= q - 11 <= 23 + // (scale 2^32-1/2 up) + tmp64 = 0x9fffffffbull; + if (q - 11 <= 19) { // 1 <= q - 11 <= 19; 10^(q-11) requires 64 bits + __mul_64x64_to_128MACH (C, tmp64, ten2k64[q - 11]); + } else { // 20 <= q - 11 <= 23, and 10^(q-11) requires 128 bits + __mul_128x64_to_128 (C, tmp64, ten2k128[q - 31]); + } + if (C1.w[1] > C.w[1] + || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // else cases that can be rounded to a 32-bit int fall through + // to '1 <= q + exp <= 10' + } + } + } + // n is not too large to be converted to int32: -1/2 < n < 2^32 - 1/2 + // Note: some of the cases tested for above fall through to this point + if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + // return 0 + res = 0x00000000; + BID_RETURN (res); + } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) + // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1) + // res = 0 + // else if x > 0 + // res = +1 + // else // if x < 0 + // invalid exc + ind = q - 1; + if (ind <= 18) { // 0 <= ind <= 18 + if ((C1.w[1] == 0) && (C1.w[0] < midpoint64[ind])) { + res = 0x00000000; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x80000000; + *pfpsf |= INVALID_EXCEPTION; + BID_RETURN (res); + } + } else { // 19 <= ind <= 33 + if ((C1.w[1] < midpoint128[ind - 19].w[1]) + || ((C1.w[1] == midpoint128[ind - 19].w[1]) + && (C1.w[0] < midpoint128[ind - 19].w[0]))) { + res = 0x00000000; // return 0 + } else if (!x_sign) { // n > 0 + res = 0x00000001; // return +1 + } else { + res = 0x80000000; + *pfpsf |= INVALID_EXCEPTION; + BID_RETURN (res); + } + } + // set inexact flag + *pfpsf |= INEXACT_EXCEPTION; + } else { // if (1 <= q + exp <= 10, 1 <= q <= 34, -33 <= exp <= 9) + if (x_sign) { // x <= -1 + // set invalid flag + *pfpsf |= INVALID_EXCEPTION; + // return Integer Indefinite + res = 0x80000000; + BID_RETURN (res); + } + // 1 <= x < 2^31-1/2 so x can be rounded + // to nearest-away to a 32-bit signed integer + if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 10 + ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' + // chop off ind digits from the lower part of C1 + // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits + tmp64 = C1.w[0]; + if (ind <= 19) { + C1.w[0] = C1.w[0] + midpoint64[ind - 1]; + } else { + C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; + C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; + } + if (C1.w[0] < tmp64) + C1.w[1]++; + // calculate C* and f* + // C* is actually floor(C*) in this case + // C* and f* need shifting and masking, as shown by + // shiftright128[] and maskhigh128[] + // 1 <= x <= 33 + // kx = 10^(-x) = ten2mk128[ind - 1] + // C* = (C1 + 1/2 * 10^x) * 10^(-x) + // the approximation of 10^(-x) was rounded up to 118 bits + __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[1] = P256.w[3]; + Cstar.w[0] = P256.w[2]; + fstar.w[3] = 0; + fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } else { // 22 <= ind - 1 <= 33 + Cstar.w[1] = 0; + Cstar.w[0] = P256.w[3]; + fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; + fstar.w[2] = P256.w[2]; + fstar.w[1] = P256.w[1]; + fstar.w[0] = P256.w[0]; + } + // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. + // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 + // if (0 < f* < 10^(-x)) then the result is a midpoint + // if floor(C*) is even then C* = floor(C*) - logical right + // shift; C* has p decimal digits, correct by Prop. 1) + // else if floor(C*) is odd C* = floor(C*)-1 (logical right + // shift; C* has p decimal digits, correct by Pr. 1) + // else + // C* = floor(C*) (logical right shift; C has p decimal digits, + // correct by Property 1) + // n = C* * 10^(e+x) + + // shift right C* by Ex-128 = shiftright128[ind] + shift = shiftright128[ind - 1]; // 0 <= shift <= 102 + if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 + Cstar.w[0] = + (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); + // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); + } else { // 22 <= ind - 1 <= 33 + Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 + } + // if the result was a midpoint, it was already rounded away from zero + // determine inexactness of the rounding of C* + // if (0 < f* - 1/2 < 10^(-x)) then + // the result is exact + // else // if (f* - 1/2 > T*) then + // the result is inexact + if (ind - 1 <= 2) { + if (fstar.w[1] > 0x8000000000000000ull || + (fstar.w[1] == 0x8000000000000000ull + && fstar.w[0] > 0x0ull)) { + // f* > 1/2 and the result may be exact + tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 + if (tmp64 > ten2mk128trunc[ind - 1].w[1] + || (tmp64 == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } + } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 + if (fstar.w[3] > 0x0 || + (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || + (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && + (fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[2] - onehalf128[ind - 1]; + tmp64A = fstar.w[3]; + if (tmp64 > fstar.w[2]) + tmp64A--; + if (tmp64A || tmp64 + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } + } else { // if 22 <= ind <= 33 + if (fstar.w[3] > onehalf128[ind - 1] || + (fstar.w[3] == onehalf128[ind - 1] && + (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { + // f2* > 1/2 and the result may be exact + // Calculate f2* - 1/2 + tmp64 = fstar.w[3] - onehalf128[ind - 1]; + if (tmp64 || fstar.w[2] + || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] + || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] + && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } // else the result is exact + } else { // the result is inexact; f2* <= 1/2 + // set the inexact flag + // *pfpsf |= INEXACT_EXCEPTION; + tmp_inexact = 1; + } + } + // no need to check for midpoints - already rounded away from zero! + res = Cstar.w[0]; // the result is positive + if (tmp_inexact) + *pfpsf |= INEXACT_EXCEPTION; + } else if (exp == 0) { + // 1 <= q <= 10 + // res = +C (exact) + res = C1.w[0]; + } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 + // res = +C * 10^exp (exact) + res = C1.w[0] * ten2k64[exp]; + } + } +} + +BID_RETURN (res); +} |