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/* 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); }
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