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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/>. */ #define BID_128RES #include "bid_internal.h" /***************************************************************************** * BID128_round_integral_exact ****************************************************************************/ BID128_FUNCTION_ARG1 (bid128_round_integral_exact, x) UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} }; 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; UINT256 fstar; UINT256 P256; // 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 = NaN, then res = Q (x) // check first for non-canonical NaN payload if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && (x.w[0] > 0x38c15b09ffffffffull))) { x.w[1] = x.w[1] & 0xffffc00000000000ull; x.w[0] = 0x0ull; } if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return quiet (x) res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] res.w[0] = x.w[0]; } else { // x is QNaN // return x res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] res.w[0] = x.w[0]; } BID_RETURN (res) } else { // x is not a NaN, so it must be infinity if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf // return +inf res.w[1] = 0x7800000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // x is -inf // return -inf res.w[1] = 0xf800000000000000ull; res.w[0] = 0x0000000000000000ull; } BID_RETURN (res); } } // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for non-canonical values (treated as zero) if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 // non-canonical x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits C1.w[1] = 0; // significand high C1.w[0] = 0; // significand low } else { // G0_G1 != 11 x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] > 0x378d8e63ffffffffull)) { // x is non-canonical if coefficient is larger than 10^34 -1 C1.w[1] = 0; C1.w[0] = 0; } else { // canonical ; } } // test for input equal to zero if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 // return 0 preserving the sign bit and the preferred exponent // of MAX(Q(x), 0) if (x_exp <= (0x1820ull << 49)) { res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; } else { res.w[1] = x_sign | x_exp; } res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero switch (rnd_mode) { case ROUNDING_TO_NEAREST: case ROUNDING_TIES_AWAY: // if (exp <= -(p+1)) return 0.0 if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; *pfpsf |= INEXACT_EXCEPTION; BID_RETURN (res); } break; case ROUNDING_DOWN: // if (exp <= -p) return -1.0 or +0.0 if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffa000000000000ull == -34 if (x_sign) { // if negative, return negative 1, because we know coefficient // is non-zero (would have been caught above) res.w[1] = 0xb040000000000000ull; res.w[0] = 0x0000000000000001ull; } else { // if positive, return positive 0, because we know coefficient is // non-zero (would have been caught above) res.w[1] = 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; } *pfpsf |= INEXACT_EXCEPTION; BID_RETURN (res); } break; case ROUNDING_UP: // if (exp <= -p) return -0.0 or +1.0 if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 if (x_sign) { // if negative, return negative 0, because we know the coefficient // is non-zero (would have been caught above) res.w[1] = 0xb040000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // if positive, return positive 1, because we know coefficient is // non-zero (would have been caught above) res.w[1] = 0x3040000000000000ull; res.w[0] = 0x0000000000000001ull; } *pfpsf |= INEXACT_EXCEPTION; BID_RETURN (res); } break; case ROUNDING_TO_ZERO: // if (exp <= -p) return -0.0 or +0.0 if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; *pfpsf |= INEXACT_EXCEPTION; BID_RETURN (res); } break; } // 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 (exp >= 0) { // -exp <= 0 // the argument is an integer already res.w[1] = x.w[1]; res.w[0] = x.w[0]; BID_RETURN (res); } // exp < 0 switch (rnd_mode) { case ROUNDING_TO_NEAREST: if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q // need to shift right -exp digits from the coefficient; exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 = C1 + 1/2 * 10^x 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 <= 34 // 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]); // determine the value of res and fstar // 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 // Note: we are going to use ten2mk128[] instead of ten2mk128trunc[] if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 // redundant shift = shiftright128[ind - 1]; // shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; // redundant fstar.w[3] = 0; // redundant fstar.w[2] = 0; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; // fraction f* < 10^(-x) <=> midpoint // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] // if 0 < fstar < 10^(-x), subtract 1 if odd (for rounding to even) if ((res.w[0] & 0x0000000000000001ull) && // is result odd? ((fstar.w[1] < (ten2mk128[ind - 1].w[1])) || ((fstar.w[1] == ten2mk128[ind - 1].w[1]) && (fstar.w[0] < ten2mk128[ind - 1].w[0])))) { // subract 1 to make even if (res.w[0]-- == 0) { res.w[1]--; } } 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 > ten2mk128[ind - 1].w[1] || (tmp64 == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 3 <= shift <= 63 res.w[1] = (P256.w[3] >> shift); res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); // redundant 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]; // fraction f* < 10^(-x) <=> midpoint // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if ((res.w[0] & 0x0000000000000001ull) && // is result odd? fstar.w[2] == 0 && (fstar.w[1] < ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] < ten2mk128[ind - 1].w[0]))) { // subract 1 to make even if (res.w[0]-- == 0) { res.w[1]--; } } if (fstar.w[2] > onehalf128[ind - 1] || (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]; if (tmp64 || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = P256.w[3] >> shift; 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]; // fraction f* < 10^(-x) <=> midpoint // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if ((res.w[0] & 0x0000000000000001ull) && // is result odd? fstar.w[3] == 0 && fstar.w[2] == 0 && (fstar.w[1] < ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] < ten2mk128[ind - 1].w[0]))) { // subract 1 to make even if (res.w[0]-- == 0) { res.w[1]--; } } 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] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[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; } } res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; BID_RETURN (res); } else { // if ((q + exp) < 0) <=> q < -exp // the result is +0 or -0 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; *pfpsf |= INEXACT_EXCEPTION; BID_RETURN (res); } break; case ROUNDING_TIES_AWAY: if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q // need to shift right -exp digits from the coefficient; exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 = C1 + 1/2 * 10^x 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 <= 34 // 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]); // 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) // determine also the 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 // Note: we are going to use ten2mk128[] instead of ten2mk128trunc[] // shift right C* by Ex-128 = shiftright128[ind] if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 // redundant shift = shiftright128[ind - 1]; // shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; // redundant fstar.w[3] = 0; // redundant fstar.w[2] = 0; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; 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 > ten2mk128[ind - 1].w[1] || (tmp64 == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0]))) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 3 <= shift <= 63 res.w[1] = (P256.w[3] >> shift); res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); // redundant 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]; if (fstar.w[2] > onehalf128[ind - 1] || (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]; if (tmp64 || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = P256.w[3] >> shift; 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]; 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] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } // if the result was a midpoint, it was already rounded away from zero res.w[1] |= x_sign | 0x3040000000000000ull; BID_RETURN (res); } else { // if ((q + exp) < 0) <=> q < -exp // the result is +0 or -0 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; *pfpsf |= INEXACT_EXCEPTION; BID_RETURN (res); } break; case ROUNDING_DOWN: if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q // need to shift right -exp digits from the coefficient; exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // (number of digits to be chopped off) // chop off ind digits from the lower part of C1 // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE // 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]++; // if carry-out from C1.w[0], increment 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 <= 34 // 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 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; // redundant fstar.w[3] = 0; // redundant fstar.w[2] = 0; // redundant fstar.w[1] = P256.w[1]; // redundant fstar.w[0] = P256.w[0]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if ((P256.w[1] > ten2mk128[ind - 1].w[1]) || (P256.w[1] == ten2mk128[ind - 1].w[1] && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { *pfpsf |= INEXACT_EXCEPTION; // if positive, the truncated value is already the correct result if (x_sign) { // if negative if (++res.w[0] == 0) { res.w[1]++; } } } } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 0 <= shift <= 102 res.w[1] = (P256.w[3] >> shift); res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); // redundant 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { *pfpsf |= INEXACT_EXCEPTION; // if positive, the truncated value is already the correct result if (x_sign) { // if negative if (++res.w[0] == 0) { res.w[1]++; } } } } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = P256.w[3] >> shift; 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { *pfpsf |= INEXACT_EXCEPTION; // if positive, the truncated value is already the correct result if (x_sign) { // if negative if (++res.w[0] == 0) { res.w[1]++; } } } } res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; BID_RETURN (res); } else { // if exp < 0 and q + exp <= 0 if (x_sign) { // negative rounds down to -1.0 res.w[1] = 0xb040000000000000ull; res.w[0] = 0x0000000000000001ull; } else { // positive rpunds down to +0.0 res.w[1] = 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; } *pfpsf |= INEXACT_EXCEPTION; BID_RETURN (res); } break; case ROUNDING_UP: if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q // need to shift right -exp digits from the coefficient; exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // (number of digits to be chopped off) // chop off ind digits from the lower part of C1 // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE // 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]++; // if carry-out from C1.w[0], increment 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 <= 34 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; // redundant fstar.w[3] = 0; // redundant fstar.w[2] = 0; // redundant fstar.w[1] = P256.w[1]; // redundant fstar.w[0] = P256.w[0]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if ((P256.w[1] > ten2mk128[ind - 1].w[1]) || (P256.w[1] == ten2mk128[ind - 1].w[1] && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { *pfpsf |= INEXACT_EXCEPTION; // if negative, the truncated value is already the correct result if (!x_sign) { // if positive if (++res.w[0] == 0) { res.w[1]++; } } } } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 3 <= shift <= 63 res.w[1] = (P256.w[3] >> shift); res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); // redundant 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { *pfpsf |= INEXACT_EXCEPTION; // if negative, the truncated value is already the correct result if (!x_sign) { // if positive if (++res.w[0] == 0) { res.w[1]++; } } } } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = P256.w[3] >> shift; 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { *pfpsf |= INEXACT_EXCEPTION; // if negative, the truncated value is already the correct result if (!x_sign) { // if positive if (++res.w[0] == 0) { res.w[1]++; } } } } res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; BID_RETURN (res); } else { // if exp < 0 and q + exp <= 0 if (x_sign) { // negative rounds up to -0.0 res.w[1] = 0xb040000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // positive rpunds up to +1.0 res.w[1] = 0x3040000000000000ull; res.w[0] = 0x0000000000000001ull; } *pfpsf |= INEXACT_EXCEPTION; BID_RETURN (res); } break; case ROUNDING_TO_ZERO: if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q // need to shift right -exp digits from the coefficient; exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // (number of digits to be chopped off) // chop off ind digits from the lower part of C1 // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE //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]++; // if carry-out from C1.w[0], increment 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 <= 34 // 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 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; // redundant fstar.w[3] = 0; // redundant fstar.w[2] = 0; // redundant fstar.w[1] = P256.w[1]; // redundant fstar.w[0] = P256.w[0]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if ((P256.w[1] > ten2mk128[ind - 1].w[1]) || (P256.w[1] == ten2mk128[ind - 1].w[1] && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { *pfpsf |= INEXACT_EXCEPTION; } } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 3 <= shift <= 63 res.w[1] = (P256.w[3] >> shift); res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); // redundant 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { *pfpsf |= INEXACT_EXCEPTION; } } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = P256.w[3] >> shift; 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { *pfpsf |= INEXACT_EXCEPTION; } } res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; BID_RETURN (res); } else { // if exp < 0 and q + exp <= 0 the result is +0 or -0 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; *pfpsf |= INEXACT_EXCEPTION; BID_RETURN (res); } break; } BID_RETURN (res); } /***************************************************************************** * BID128_round_integral_nearest_even ****************************************************************************/ BID128_FUNCTION_ARG1_NORND (bid128_round_integral_nearest_even, x) UINT128 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; // UINT128 res is C* at first - represents up to 34 decimal digits ~ 113 bits UINT256 fstar; UINT256 P256; // 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 = NaN, then res = Q (x) // check first for non-canonical NaN payload if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && (x.w[0] > 0x38c15b09ffffffffull))) { x.w[1] = x.w[1] & 0xffffc00000000000ull; x.w[0] = 0x0ull; } if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return quiet (x) res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] res.w[0] = x.w[0]; } else { // x is QNaN // return x res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] res.w[0] = x.w[0]; } BID_RETURN (res) } else { // x is not a NaN, so it must be infinity if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf // return +inf res.w[1] = 0x7800000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // x is -inf // return -inf res.w[1] = 0xf800000000000000ull; res.w[0] = 0x0000000000000000ull; } BID_RETURN (res); } } // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for non-canonical values (treated as zero) if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 // non-canonical x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits C1.w[1] = 0; // significand high C1.w[0] = 0; // significand low } else { // G0_G1 != 11 x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] > 0x378d8e63ffffffffull)) { // x is non-canonical if coefficient is larger than 10^34 -1 C1.w[1] = 0; C1.w[0] = 0; } else { // canonical ; } } // test for input equal to zero if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 // return 0 preserving the sign bit and the preferred exponent // of MAX(Q(x), 0) if (x_exp <= (0x1820ull << 49)) { res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; } else { res.w[1] = x_sign | x_exp; } res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // if (exp <= -(p+1)) return 0 if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if (exp >= 0) { // -exp <= 0 // the argument is an integer already res.w[1] = x.w[1]; res.w[0] = x.w[0]; BID_RETURN (res); } else if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q // need to shift right -exp digits from the coefficient; the exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 = C1 + 1/2 * 10^x 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 <= 34 // 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]); // determine the value of res and fstar if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 // redundant shift = shiftright128[ind - 1]; // shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; // redundant fstar.w[3] = 0; // redundant fstar.w[2] = 0; // redundant fstar.w[1] = P256.w[1]; // redundant fstar.w[0] = P256.w[0]; // fraction f* < 10^(-x) <=> midpoint // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] // if 0 < fstar < 10^(-x), subtract 1 if odd (for rounding to even) if ((res.w[0] & 0x0000000000000001ull) && // is result odd? ((P256.w[1] < (ten2mk128[ind - 1].w[1])) || ((P256.w[1] == ten2mk128[ind - 1].w[1]) && (P256.w[0] < ten2mk128[ind - 1].w[0])))) { // subract 1 to make even if (res.w[0]-- == 0) { res.w[1]--; } } } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 3 <= shift <= 63 res.w[1] = (P256.w[3] >> shift); res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); // redundant 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]; // fraction f* < 10^(-x) <=> midpoint // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if ((res.w[0] & 0x0000000000000001ull) && // is result odd? fstar.w[2] == 0 && (fstar.w[1] < ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] < ten2mk128[ind - 1].w[0]))) { // subract 1 to make even if (res.w[0]-- == 0) { res.w[1]--; } } } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = P256.w[3] >> shift; 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]; // fraction f* < 10^(-x) <=> midpoint // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if ((res.w[0] & 0x0000000000000001ull) && // is result odd? fstar.w[3] == 0 && fstar.w[2] == 0 && (fstar.w[1] < ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] < ten2mk128[ind - 1].w[0]))) { // subract 1 to make even if (res.w[0]-- == 0) { res.w[1]--; } } } res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; BID_RETURN (res); } else { // if ((q + exp) < 0) <=> q < -exp // the result is +0 or -0 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } } /***************************************************************************** * BID128_round_integral_negative ****************************************************************************/ BID128_FUNCTION_ARG1_NORND (bid128_round_integral_negative, x) UINT128 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo // (all are UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1; // UINT128 res is C* at first - represents up to 34 decimal digits ~ // 113 bits UINT256 fstar; UINT256 P256; // 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 = NaN, then res = Q (x) // check first for non-canonical NaN payload if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && (x.w[0] > 0x38c15b09ffffffffull))) { x.w[1] = x.w[1] & 0xffffc00000000000ull; x.w[0] = 0x0ull; } if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return quiet (x) res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] res.w[0] = x.w[0]; } else { // x is QNaN // return x res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] res.w[0] = x.w[0]; } BID_RETURN (res) } else { // x is not a NaN, so it must be infinity if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf // return +inf res.w[1] = 0x7800000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // x is -inf // return -inf res.w[1] = 0xf800000000000000ull; res.w[0] = 0x0000000000000000ull; } BID_RETURN (res); } } // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for non-canonical values (treated as zero) if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 // non-canonical x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits C1.w[1] = 0; // significand high C1.w[0] = 0; // significand low } else { // G0_G1 != 11 x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] > 0x378d8e63ffffffffull)) { // x is non-canonical if coefficient is larger than 10^34 -1 C1.w[1] = 0; C1.w[0] = 0; } else { // canonical ; } } // test for input equal to zero if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 // return 0 preserving the sign bit and the preferred exponent // of MAX(Q(x), 0) if (x_exp <= (0x1820ull << 49)) { res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; } else { res.w[1] = x_sign | x_exp; } res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // if (exp <= -p) return -1.0 or +0.0 if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 if (x_sign) { // if negative, return negative 1, because we know the coefficient // is non-zero (would have been caught above) res.w[1] = 0xb040000000000000ull; res.w[0] = 0x0000000000000001ull; } else { // if positive, return positive 0, because we know coefficient is // non-zero (would have been caught above) res.w[1] = 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; } BID_RETURN (res); } // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if (exp >= 0) { // -exp <= 0 // the argument is an integer already res.w[1] = x.w[1]; res.w[0] = x.w[0]; BID_RETURN (res); } else if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q // need to shift right -exp digits from the coefficient; the exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // (number of digits to be chopped off) // chop off ind digits from the lower part of C1 // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE //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]++; // if carry-out from C1.w[0], increment 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 <= 34 // 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 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; // if positive, the truncated value is already the correct result if (x_sign) { // if negative // redundant fstar.w[3] = 0; // redundant fstar.w[2] = 0; // redundant fstar.w[1] = P256.w[1]; // redundant fstar.w[0] = P256.w[0]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if ((P256.w[1] > ten2mk128[ind - 1].w[1]) || (P256.w[1] == ten2mk128[ind - 1].w[1] && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { if (++res.w[0] == 0) { res.w[1]++; } } } } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 0 <= shift <= 102 res.w[1] = (P256.w[3] >> shift); res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); // if positive, the truncated value is already the correct result if (x_sign) { // if negative // redundant 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { if (++res.w[0] == 0) { res.w[1]++; } } } } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = P256.w[3] >> shift; // if positive, the truncated value is already the correct result if (x_sign) { // if negative 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { if (++res.w[0] == 0) { res.w[1]++; } } } } res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; BID_RETURN (res); } else { // if exp < 0 and q + exp <= 0 if (x_sign) { // negative rounds down to -1.0 res.w[1] = 0xb040000000000000ull; res.w[0] = 0x0000000000000001ull; } else { // positive rpunds down to +0.0 res.w[1] = 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; } BID_RETURN (res); } } /***************************************************************************** * BID128_round_integral_positive ****************************************************************************/ BID128_FUNCTION_ARG1_NORND (bid128_round_integral_positive, x) UINT128 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo // (all are UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1; // UINT128 res is C* at first - represents up to 34 decimal digits ~ // 113 bits UINT256 fstar; UINT256 P256; // 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 = NaN, then res = Q (x) // check first for non-canonical NaN payload if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && (x.w[0] > 0x38c15b09ffffffffull))) { x.w[1] = x.w[1] & 0xffffc00000000000ull; x.w[0] = 0x0ull; } if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return quiet (x) res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] res.w[0] = x.w[0]; } else { // x is QNaN // return x res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] res.w[0] = x.w[0]; } BID_RETURN (res) } else { // x is not a NaN, so it must be infinity if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf // return +inf res.w[1] = 0x7800000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // x is -inf // return -inf res.w[1] = 0xf800000000000000ull; res.w[0] = 0x0000000000000000ull; } BID_RETURN (res); } } // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for non-canonical values (treated as zero) if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 // non-canonical x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits C1.w[1] = 0; // significand high C1.w[0] = 0; // significand low } else { // G0_G1 != 11 x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] > 0x378d8e63ffffffffull)) { // x is non-canonical if coefficient is larger than 10^34 -1 C1.w[1] = 0; C1.w[0] = 0; } else { // canonical ; } } // test for input equal to zero if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 // return 0 preserving the sign bit and the preferred exponent // of MAX(Q(x), 0) if (x_exp <= (0x1820ull << 49)) { res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; } else { res.w[1] = x_sign | x_exp; } res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // if (exp <= -p) return -0.0 or +1.0 if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 if (x_sign) { // if negative, return negative 0, because we know the coefficient // is non-zero (would have been caught above) res.w[1] = 0xb040000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // if positive, return positive 1, because we know coefficient is // non-zero (would have been caught above) res.w[1] = 0x3040000000000000ull; res.w[0] = 0x0000000000000001ull; } BID_RETURN (res); } // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split 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 (exp >= 0) { // -exp <= 0 // the argument is an integer already res.w[1] = x.w[1]; res.w[0] = x.w[0]; BID_RETURN (res); } else if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q // need to shift right -exp digits from the coefficient; exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // (number of digits to be chopped off) // chop off ind digits from the lower part of C1 // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE // 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]++; // if carry-out from C1.w[0], increment 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 <= 34 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; // if negative, the truncated value is already the correct result if (!x_sign) { // if positive // redundant fstar.w[3] = 0; // redundant fstar.w[2] = 0; // redundant fstar.w[1] = P256.w[1]; // redundant fstar.w[0] = P256.w[0]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if ((P256.w[1] > ten2mk128[ind - 1].w[1]) || (P256.w[1] == ten2mk128[ind - 1].w[1] && (P256.w[0] >= ten2mk128[ind - 1].w[0]))) { if (++res.w[0] == 0) { res.w[1]++; } } } } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 3 <= shift <= 63 res.w[1] = (P256.w[3] >> shift); res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); // if negative, the truncated value is already the correct result if (!x_sign) { // if positive // redundant 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { if (++res.w[0] == 0) { res.w[1]++; } } } } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = P256.w[3] >> shift; // if negative, the truncated value is already the correct result if (!x_sign) { // if positive 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]; // fraction f* > 10^(-x) <=> inexact // f* is in the right position to be compared with // 10^(-x) from ten2mk128[] if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128[ind - 1].w[1] || (fstar.w[1] == ten2mk128[ind - 1].w[1] && fstar.w[0] >= ten2mk128[ind - 1].w[0])) { if (++res.w[0] == 0) { res.w[1]++; } } } } res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; BID_RETURN (res); } else { // if exp < 0 and q + exp <= 0 if (x_sign) { // negative rounds up to -0.0 res.w[1] = 0xb040000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // positive rpunds up to +1.0 res.w[1] = 0x3040000000000000ull; res.w[0] = 0x0000000000000001ull; } BID_RETURN (res); } } /***************************************************************************** * BID128_round_integral_zero ****************************************************************************/ BID128_FUNCTION_ARG1_NORND (bid128_round_integral_zero, x) UINT128 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo // (all are UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1; // UINT128 res is C* at first - represents up to 34 decimal digits ~ // 113 bits UINT256 P256; // 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 = NaN, then res = Q (x) // check first for non-canonical NaN payload if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && (x.w[0] > 0x38c15b09ffffffffull))) { x.w[1] = x.w[1] & 0xffffc00000000000ull; x.w[0] = 0x0ull; } if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return quiet (x) res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] res.w[0] = x.w[0]; } else { // x is QNaN // return x res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] res.w[0] = x.w[0]; } BID_RETURN (res) } else { // x is not a NaN, so it must be infinity if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf // return +inf res.w[1] = 0x7800000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // x is -inf // return -inf res.w[1] = 0xf800000000000000ull; res.w[0] = 0x0000000000000000ull; } BID_RETURN (res); } } // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for non-canonical values (treated as zero) if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 // non-canonical x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits C1.w[1] = 0; // significand high C1.w[0] = 0; // significand low } else { // G0_G1 != 11 x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] > 0x378d8e63ffffffffull)) { // x is non-canonical if coefficient is larger than 10^34 -1 C1.w[1] = 0; C1.w[0] = 0; } else { // canonical ; } } // test for input equal to zero if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 // return 0 preserving the sign bit and the preferred exponent // of MAX(Q(x), 0) if (x_exp <= (0x1820ull << 49)) { res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; } else { res.w[1] = x_sign | x_exp; } res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // if (exp <= -p) return -0.0 or +0.0 if (x_exp <= 0x2ffc000000000000ull) { // 0x2ffc000000000000ull == -34 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if (exp >= 0) { // -exp <= 0 // the argument is an integer already res.w[1] = x.w[1]; res.w[0] = x.w[0]; BID_RETURN (res); } else if ((q + exp) > 0) { // exp < 0 and 1 <= -exp < q // need to shift right -exp digits from the coefficient; the exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // (number of digits to be chopped off) // chop off ind digits from the lower part of C1 // FOR ROUND_TO_NEAREST, WE ADD 1/2 ULP(y) then truncate // FOR ROUND_TO_ZERO, WE DON'T NEED TO ADD 1/2 ULP // FOR ROUND_TO_POSITIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF POSITIVE // FOR ROUND_TO_NEGATIVE_INFINITY, WE TRUNCATE, THEN ADD 1 IF NEGATIVE //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]++; // if carry-out from C1.w[0], increment 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 <= 34 // 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 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 3 <= shift <= 63 res.w[1] = (P256.w[3] >> shift); res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1] - 64; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = P256.w[3] >> shift; } res.w[1] = x_sign | 0x3040000000000000ull | res.w[1]; BID_RETURN (res); } else { // if exp < 0 and q + exp <= 0 the result is +0 or -0 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } } /***************************************************************************** * BID128_round_integral_nearest_away ****************************************************************************/ BID128_FUNCTION_ARG1_NORND (bid128_round_integral_nearest_away, x) UINT128 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; // UINT128 res is C* at first - represents up to 34 decimal digits ~ // 113 bits // UINT256 fstar; UINT256 P256; // 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 = NaN, then res = Q (x) // check first for non-canonical NaN payload if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && (x.w[0] > 0x38c15b09ffffffffull))) { x.w[1] = x.w[1] & 0xffffc00000000000ull; x.w[0] = 0x0ull; } if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return quiet (x) res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] res.w[0] = x.w[0]; } else { // x is QNaN // return x res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] res.w[0] = x.w[0]; } BID_RETURN (res) } else { // x is not a NaN, so it must be infinity if ((x.w[1] & MASK_SIGN) == 0x0ull) { // x is +inf // return +inf res.w[1] = 0x7800000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // x is -inf // return -inf res.w[1] = 0xf800000000000000ull; res.w[0] = 0x0000000000000000ull; } BID_RETURN (res); } } // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for non-canonical values (treated as zero) if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 // non-canonical x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits C1.w[1] = 0; // significand high C1.w[0] = 0; // significand low } else { // G0_G1 != 11 x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] > 0x378d8e63ffffffffull)) { // x is non-canonical if coefficient is larger than 10^34 -1 C1.w[1] = 0; C1.w[0] = 0; } else { // canonical ; } } // test for input equal to zero if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 // return 0 preserving the sign bit and the preferred exponent // of MAX(Q(x), 0) if (x_exp <= (0x1820ull << 49)) { res.w[1] = (x.w[1] & 0x8000000000000000ull) | 0x3040000000000000ull; } else { res.w[1] = x_sign | x_exp; } res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // if (exp <= -(p+1)) return 0.0 if (x_exp <= 0x2ffa000000000000ull) { // 0x2ffa000000000000ull == -35 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if (exp >= 0) { // -exp <= 0 // the argument is an integer already res.w[1] = x.w[1]; res.w[0] = x.w[0]; BID_RETURN (res); } else if ((q + exp) >= 0) { // exp < 0 and 1 <= -exp <= q // need to shift right -exp digits from the coefficient; the exp will be 0 ind = -exp; // 1 <= ind <= 34; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 = C1 + 1/2 * 10^x 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 <= 34 // 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]); // 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] if (ind - 1 <= 2) { // 0 <= ind - 1 <= 2 => shift = 0 res.w[1] = P256.w[3]; res.w[0] = P256.w[2]; } else if (ind - 1 <= 21) { // 3 <= ind - 1 <= 21 => 3 <= shift <= 63 shift = shiftright128[ind - 1]; // 3 <= shift <= 63 res.w[0] = (P256.w[3] << (64 - shift)) | (P256.w[2] >> shift); res.w[1] = (P256.w[3] >> shift); } else { // 22 <= ind - 1 <= 33 shift = shiftright128[ind - 1]; // 2 <= shift <= 38 res.w[1] = 0; res.w[0] = (P256.w[3] >> (shift - 64)); // 2 <= shift - 64 <= 38 } // if the result was a midpoint, it was already rounded away from zero res.w[1] |= x_sign | 0x3040000000000000ull; BID_RETURN (res); } else { // if ((q + exp) < 0) <=> q < -exp // the result is +0 or -0 res.w[1] = x_sign | 0x3040000000000000ull; res.w[0] = 0x0000000000000000ull; BID_RETURN (res); } }
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