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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_int64_rnint ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_rnint, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) UINT64 tmp64; BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 fstar; UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n < -2^63 - 1/2 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) > 2^63+1/2 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 5*(2^64+1), 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 0x50000000000000005, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0000000000000005ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n >= 2^63 - 1/2 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63-1/2 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 5*(2^64-1), 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 0x4fffffffffffffffb, 1<=q<=34 C.w[1] = 0x0000000000000004ull; C.w[0] = 0xfffffffffffffffbull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1/2 <= n < 2^63-1/2 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1) // res = 0 // else // res = +/-1 ind = q - 1; if (ind <= 18) { // 0 <= ind <= 18 if ((C1.w[1] == 0) && (C1.w[0] <= midpoint64[ind])) { res = 0x0000000000000000ull; // return 0 } else if (x_sign) { // n < 0 res = 0xffffffffffffffffull; // return -1 } else { // n > 0 res = 0x0000000000000001ull; // return +1 } } else { // 19 <= ind <= 33 if ((C1.w[1] < midpoint128[ind - 19].w[1]) || ((C1.w[1] == midpoint128[ind - 19].w[1]) && (C1.w[0] <= midpoint128[ind - 19].w[0]))) { res = 0x0000000000000000ull; // return 0 } else if (x_sign) { // n < 0 res = 0xffffffffffffffffull; // return -1 } else { // n > 0 res = 0x0000000000000001ull; // return +1 } } } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63-1/2 <= x <= -1 or 1 <= x < 2^63-1/2 so x can be rounded // to nearest to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 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] } if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); } /***************************************************************************** * BID128_to_int64_xrnint ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_xrnint, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) UINT64 tmp64, tmp64A; BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 fstar; UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n < -2^63 - 1/2 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) > 2^63+1/2 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 5*(2^64+1), 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 0x50000000000000005, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0000000000000005ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n >= 2^63 - 1/2 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63-1/2 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 5*(2^64-1), 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 0x4fffffffffffffffb, 1<=q<=34 C.w[1] = 0x0000000000000004ull; C.w[0] = 0xfffffffffffffffbull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1/2 <= n < 2^63-1/2 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1) // res = 0 // else // res = +/-1 ind = q - 1; if (ind <= 18) { // 0 <= ind <= 18 if ((C1.w[1] == 0) && (C1.w[0] <= midpoint64[ind])) { res = 0x0000000000000000ull; // return 0 } else if (x_sign) { // n < 0 res = 0xffffffffffffffffull; // return -1 } else { // n > 0 res = 0x0000000000000001ull; // return +1 } } else { // 19 <= ind <= 33 if ((C1.w[1] < midpoint128[ind - 19].w[1]) || ((C1.w[1] == midpoint128[ind - 19].w[1]) && (C1.w[0] <= midpoint128[ind - 19].w[0]))) { res = 0x0000000000000000ull; // return 0 } else if (x_sign) { // n < 0 res = 0xffffffffffffffffull; // return -1 } else { // n > 0 res = 0x0000000000000001ull; // return +1 } } // set inexact flag *pfpsf |= INEXACT_EXCEPTION; } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63-1/2 <= x <= -1 or 1 <= x < 2^63-1/2 so x can be rounded // to nearest to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits tmp64 = C1.w[0]; if (ind <= 19) { C1.w[0] = C1.w[0] + midpoint64[ind - 1]; } else { C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; } if (C1.w[0] < tmp64) C1.w[1]++; // calculate C* and f* // C* is actually floor(C*) in this case // C* and f* need shifting and masking, as shown by // shiftright128[] and maskhigh128[] // 1 <= x <= 33 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = (C1 + 1/2 * 10^x) * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[1] = P256.w[3]; Cstar.w[0] = P256.w[2]; fstar.w[3] = 0; fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } else { // 22 <= ind - 1 <= 33 Cstar.w[1] = 0; Cstar.w[0] = P256.w[3]; fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; fstar.w[2] = P256.w[2]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 // if (0 < f* < 10^(-x)) then the result is a midpoint // if floor(C*) is even then C* = floor(C*) - logical right // shift; C* has p decimal digits, correct by Prop. 1) // else if floor(C*) is odd C* = floor(C*)-1 (logical right // shift; C* has p decimal digits, correct by Pr. 1) // else // C* = floor(C*) (logical right shift; C has p decimal digits, // correct by Property 1) // n = C* * 10^(e+x) // shift right C* by Ex-128 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 102 if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[0] = (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); } else { // 22 <= ind - 1 <= 33 Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 } // determine inexactness of the rounding of C* // if (0 < f* - 1/2 < 10^(-x)) then // the result is exact // else // if (f* - 1/2 > T*) then // the result is inexact if (ind - 1 <= 2) { if (fstar.w[1] > 0x8000000000000000ull || (fstar.w[1] == 0x8000000000000000ull && fstar.w[0] > 0x0ull)) { // f* > 1/2 and the result may be exact tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 if (tmp64 > ten2mk128trunc[ind - 1].w[1] || (tmp64 == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 if (fstar.w[3] > 0x0 || (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && (fstar.w[1] || fstar.w[0]))) { // f2* > 1/2 and the result may be exact // Calculate f2* - 1/2 tmp64 = fstar.w[2] - onehalf128[ind - 1]; tmp64A = fstar.w[3]; if (tmp64 > fstar.w[2]) tmp64A--; if (tmp64A || tmp64 || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } else { // if 22 <= ind <= 33 if (fstar.w[3] > onehalf128[ind - 1] || (fstar.w[3] == onehalf128[ind - 1] && (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f2* > 1/2 and the result may be exact // Calculate f2* - 1/2 tmp64 = fstar.w[3] - onehalf128[ind - 1]; if (tmp64 || fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } // if the result was a midpoint it was rounded away from zero, so // it will need a correction // check for midpoints if ((fstar.w[3] == 0) && (fstar.w[2] == 0) && (fstar.w[1] || fstar.w[0]) && (fstar.w[1] < ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] <= ten2mk128trunc[ind - 1].w[0]))) { // the result is a midpoint; round to nearest if (Cstar.w[0] & 0x01) { // Cstar.w[0] is odd; MP in [EVEN, ODD] // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 Cstar.w[0]--; // Cstar.w[0] is now even } // else MP in [ODD, EVEN] } if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); } /***************************************************************************** * BID128_to_int64_floor ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_floor, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 fstar; UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n < -2^63 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) > 2^63 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 10*2^63, 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 0x50000000000000000, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x0000000000000000ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n >= 2^63 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 5*2^64, 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 0x50000000000000000, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x0000000000000000ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1 < n < 2^63 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // return -1 or 0 if (x_sign) res = 0xffffffffffffffffull; else res = 0x0000000000000000ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63 <= x <= -1 or 1 <= x < 2^63 so x can be rounded // toward zero to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 127 bits // calculate C* and f* // C* is actually floor(C*) in this case // C* and f* need shifting and masking, as shown by // shiftright128[] and maskhigh128[] // 1 <= x <= 33 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[1] = P256.w[3]; Cstar.w[0] = P256.w[2]; fstar.w[3] = 0; fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } else { // 22 <= ind - 1 <= 33 Cstar.w[1] = 0; Cstar.w[0] = P256.w[3]; fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; fstar.w[2] = P256.w[2]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 // C* = floor(C*) (logical right shift; C has p decimal digits, // correct by Property 1) // n = C* * 10^(e+x) // shift right C* by Ex-128 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 102 if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[0] = (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); } else { // 22 <= ind - 1 <= 33 Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 } // if the result is negative and inexact, need to add 1 to it // determine inexactness of the rounding of C* // if (0 < f* < 10^(-x)) then // the result is exact // else // if (f* > T*) then // the result is inexact if (ind - 1 <= 2) { if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } } // else the result is exact } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } } // else the result is exact } else { // if 22 <= ind <= 33 if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } } // else the result is exact } if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); } /***************************************************************************** * BID128_to_int64_xfloor ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_xfloor, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 fstar; UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n < -2^63 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) > 2^63 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 10*2^63, 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 0x50000000000000000, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x0000000000000000ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n >= 2^63 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 5*2^64, 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 0x50000000000000000, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x0000000000000000ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1 < n < 2^63 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return -1 or 0 if (x_sign) res = 0xffffffffffffffffull; else res = 0x0000000000000000ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63 <= x <= -1 or 1 <= x < 2^63 so x can be rounded // toward zero to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 127 bits // calculate C* and f* // C* is actually floor(C*) in this case // C* and f* need shifting and masking, as shown by // shiftright128[] and maskhigh128[] // 1 <= x <= 33 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[1] = P256.w[3]; Cstar.w[0] = P256.w[2]; fstar.w[3] = 0; fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } else { // 22 <= ind - 1 <= 33 Cstar.w[1] = 0; Cstar.w[0] = P256.w[3]; fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; fstar.w[2] = P256.w[2]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 // C* = floor(C*) (logical right shift; C has p decimal digits, // correct by Property 1) // n = C* * 10^(e+x) // shift right C* by Ex-128 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 102 if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[0] = (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); } else { // 22 <= ind - 1 <= 33 Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 } // if the result is negative and inexact, need to add 1 to it // determine inexactness of the rounding of C* // if (0 < f* < 10^(-x)) then // the result is exact // else // if (f* > T*) then // the result is inexact if (ind - 1 <= 2) { if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // if 22 <= ind <= 33 if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); } /***************************************************************************** * BID128_to_int64_ceil ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_ceil, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 fstar; UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n <= -2^63 - 1 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63+1 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 5*(2^64+2), 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 0x5000000000000000a, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x000000000000000aull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n > 2^63 - 1 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) > 2^63 - 1 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 > 10*(2^63-1), 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 > 0x4fffffffffffffff6, 1<=q<=34 C.w[1] = 0x0000000000000004ull; C.w[0] = 0xfffffffffffffff6ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1 < n <= 2^63 - 1 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // return 0 or 1 if (x_sign) res = 0x0000000000000000ull; else res = 0x0000000000000001ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63-1 < x <= -1 or 1 <= x <= 2^63 - 1 so x can be rounded // up to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 127 bits // calculate C* and f* // C* is actually floor(C*) in this case // C* and f* need shifting and masking, as shown by // shiftright128[] and maskhigh128[] // 1 <= x <= 33 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[1] = P256.w[3]; Cstar.w[0] = P256.w[2]; fstar.w[3] = 0; fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } else { // 22 <= ind - 1 <= 33 Cstar.w[1] = 0; Cstar.w[0] = P256.w[3]; fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; fstar.w[2] = P256.w[2]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 // C* = floor(C*) (logical right shift; C has p decimal digits, // correct by Property 1) // n = C* * 10^(e+x) // shift right C* by Ex-128 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 102 if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[0] = (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); } else { // 22 <= ind - 1 <= 33 Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 } // if the result is positive and inexact, need to add 1 to it // determine inexactness of the rounding of C* // if (0 < f* < 10^(-x)) then // the result is exact // else // if (f* > T*) then // the result is inexact if (ind - 1 <= 2) { if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (!x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } } // else the result is exact } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (!x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } } // else the result is exact } else { // if 22 <= ind <= 33 if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (!x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } } // else the result is exact } if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); } /***************************************************************************** * BID128_to_int64_xceil ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_xceil, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 fstar; UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n <= -2^63 - 1 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63+1 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 5*(2^64+2), 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 > 0x5000000000000000a, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x000000000000000aull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n > 2^63 - 1 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) > 2^63 - 1 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 > 10*(2^63-1), 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 > 0x4fffffffffffffff6, 1<=q<=34 C.w[1] = 0x0000000000000004ull; C.w[0] = 0xfffffffffffffff6ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] > C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1 < n <= 2^63 - 1 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return 0 or 1 if (x_sign) res = 0x0000000000000000ull; else res = 0x0000000000000001ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63-1 < x <= -1 or 1 <= x <= 2^63 - 1 so x can be rounded // up to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 127 bits // calculate C* and f* // C* is actually floor(C*) in this case // C* and f* need shifting and masking, as shown by // shiftright128[] and maskhigh128[] // 1 <= x <= 33 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[1] = P256.w[3]; Cstar.w[0] = P256.w[2]; fstar.w[3] = 0; fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } else { // 22 <= ind - 1 <= 33 Cstar.w[1] = 0; Cstar.w[0] = P256.w[3]; fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; fstar.w[2] = P256.w[2]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 // C* = floor(C*) (logical right shift; C has p decimal digits, // correct by Property 1) // n = C* * 10^(e+x) // shift right C* by Ex-128 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 102 if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[0] = (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); } else { // 22 <= ind - 1 <= 33 Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 } // if the result is positive and inexact, need to add 1 to it // determine inexactness of the rounding of C* // if (0 < f* < 10^(-x)) then // the result is exact // else // if (f* > T*) then // the result is inexact if (ind - 1 <= 2) { if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (!x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (!x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // if 22 <= ind <= 33 if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { if (!x_sign) { // positive and inexact Cstar.w[0]++; if (Cstar.w[0] == 0x0) Cstar.w[1]++; } // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); } /***************************************************************************** * BID128_to_int64_int ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_int, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n <= -2^63 - 1 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63+1 // <=> 0.c(0)c(1)...c(q-1) * 10^20 >= 5*(2^64+2), 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 >= 0x5000000000000000a, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x000000000000000aull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n >= 2^63 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 5*2^64, 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 0x50000000000000000, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x0000000000000000ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1 < n < 2^63 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63-1 < x <= -1 or 1 <= x < 2^63 so x can be rounded // toward zero to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 127 bits // calculate C* and f* // C* is actually floor(C*) in this case // C* and f* need shifting and masking, as shown by // shiftright128[] and maskhigh128[] // 1 <= x <= 33 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[1] = P256.w[3]; Cstar.w[0] = P256.w[2]; } else { // 22 <= ind - 1 <= 33 Cstar.w[1] = 0; Cstar.w[0] = P256.w[3]; } // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 // C* = floor(C*) (logical right shift; C has p decimal digits, // correct by Property 1) // n = C* * 10^(e+x) // shift right C* by Ex-128 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 102 if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[0] = (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); } else { // 22 <= ind - 1 <= 33 Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 } if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); } /***************************************************************************** * BID128_to_xint64_xint ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_xint, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 fstar; UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n <= -2^63 - 1 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63+1 // <=> 0.c(0)c(1)...c(q-1) * 10^20 >= 5*(2^64+2), 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 >= 0x5000000000000000a, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x000000000000000aull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n >= 2^63 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 5*2^64, 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 0x50000000000000000, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0x0000000000000000ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1 < n < 2^63 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63-1 < x <= -1 or 1 <= x < 2^63 so x can be rounded // toward zero to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 127 bits // calculate C* and f* // C* is actually floor(C*) in this case // C* and f* need shifting and masking, as shown by // shiftright128[] and maskhigh128[] // 1 <= x <= 33 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[1] = P256.w[3]; Cstar.w[0] = P256.w[2]; fstar.w[3] = 0; fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } else { // 22 <= ind - 1 <= 33 Cstar.w[1] = 0; Cstar.w[0] = P256.w[3]; fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; fstar.w[2] = P256.w[2]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 // C* = floor(C*) (logical right shift; C has p decimal digits, // correct by Property 1) // n = C* * 10^(e+x) // shift right C* by Ex-128 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 102 if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[0] = (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); } else { // 22 <= ind - 1 <= 33 Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 } // determine inexactness of the rounding of C* // if (0 < f* < 10^(-x)) then // the result is exact // else // if (f* > T*) then // the result is inexact if (ind - 1 <= 2) { if (fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 if (fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } else { // if 22 <= ind <= 33 if (fstar.w[3] || fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); } /***************************************************************************** * BID128_to_int64_rninta ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_rninta, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) UINT64 tmp64; BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n <= -2^63 - 1/2 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63+1/2 // <=> 0.c(0)c(1)...c(q-1) * 10^20 >= 5*(2^64+1), 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 >= 0x50000000000000005, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0000000000000005ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n >= 2^63 - 1/2 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63-1/2 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 5*(2^64-1), 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 0x4fffffffffffffffb, 1<=q<=34 C.w[1] = 0x0000000000000004ull; C.w[0] = 0xfffffffffffffffbull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1/2 <= n < 2^63-1/2 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1) // res = 0 // else // res = +/-1 ind = q - 1; if (ind <= 18) { // 0 <= ind <= 18 if ((C1.w[1] == 0) && (C1.w[0] < midpoint64[ind])) { res = 0x0000000000000000ull; // return 0 } else if (x_sign) { // n < 0 res = 0xffffffffffffffffull; // return -1 } else { // n > 0 res = 0x0000000000000001ull; // return +1 } } else { // 19 <= ind <= 33 if ((C1.w[1] < midpoint128[ind - 19].w[1]) || ((C1.w[1] == midpoint128[ind - 19].w[1]) && (C1.w[0] < midpoint128[ind - 19].w[0]))) { res = 0x0000000000000000ull; // return 0 } else if (x_sign) { // n < 0 res = 0xffffffffffffffffull; // return -1 } else { // n > 0 res = 0x0000000000000001ull; // return +1 } } } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63-1/2 <= x <= -1 or 1 <= x < 2^63-1/2 so x can be rounded // to nearest to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits tmp64 = C1.w[0]; if (ind <= 19) { C1.w[0] = C1.w[0] + midpoint64[ind - 1]; } else { C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; } if (C1.w[0] < tmp64) C1.w[1]++; // calculate C* and f* // C* is actually floor(C*) in this case // C* and f* need shifting and masking, as shown by // shiftright128[] and maskhigh128[] // 1 <= x <= 33 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = (C1 + 1/2 * 10^x) * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[1] = P256.w[3]; Cstar.w[0] = P256.w[2]; } else { // 22 <= ind - 1 <= 33 Cstar.w[1] = 0; Cstar.w[0] = P256.w[3]; } // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 // if (0 < f* < 10^(-x)) then the result is a midpoint // if floor(C*) is even then C* = floor(C*) - logical right // shift; C* has p decimal digits, correct by Prop. 1) // else if floor(C*) is odd C* = floor(C*)-1 (logical right // shift; C* has p decimal digits, correct by Pr. 1) // else // C* = floor(C*) (logical right shift; C has p decimal digits, // correct by Property 1) // n = C* * 10^(e+x) // shift right C* by Ex-128 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 102 if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[0] = (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); } else { // 22 <= ind - 1 <= 33 Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 } // if the result was a midpoint it was rounded away from zero if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); } /***************************************************************************** * BID128_to_int64_xrninta ****************************************************************************/ BID128_FUNCTION_ARG1_NORND_CUSTOMRESTYPE (SINT64, bid128_to_int64_xrninta, x) SINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1.w[1], C1.w[0] represent x_signif_hi, x_signif_lo (all are UINT64) UINT64 tmp64, tmp64A; BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT128 C1, C; UINT128 Cstar; // C* represents up to 34 decimal digits ~ 113 bits UINT256 fstar; UINT256 P256; // unpack x x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; // check for NaN or Infinity if ((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL) { // x is special if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is QNaN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } else { // x is not a NaN, so it must be infinity if (!x_sign) { // x is +inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } else { // x is -inf // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; } BID_RETURN (res); } } // check for non-canonical values (after the check for special values) if ((C1.w[1] > 0x0001ed09bead87c0ull) || (C1.w[1] == 0x0001ed09bead87c0ull && (C1.w[0] > 0x378d8e63ffffffffull)) || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) { res = 0x0000000000000000ull; BID_RETURN (res); } else if ((C1.w[1] == 0x0ull) && (C1.w[0] == 0x0ull)) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // x is not special and is not zero // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp1.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[x_nr_bits - 1].digits; if (q == 0) { q = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q++; } exp = (x_exp >> 49) - 6176; if ((q + exp) > 19) { // x >= 10^19 ~= 2^63.11... (cannot fit in SINT64) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 19) { // x = c(0)c(1)...c(18).c(19)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in a signed 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 19' if (x_sign) { // if n < 0 and q + exp = 19 // if n <= -2^63 - 1/2 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63+1/2 // <=> 0.c(0)c(1)...c(q-1) * 10^20 >= 5*(2^64+1), 1<=q<=34 // <=> 0.c(0)c(1)...c(q-1) * 10^20 >= 0x50000000000000005, 1<=q<=34 C.w[1] = 0x0000000000000005ull; C.w[0] = 0000000000000005ull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } else { // if n > 0 and q + exp = 19 // if n >= 2^63 - 1/2 then n is too large // too large if c(0)c(1)...c(18).c(19)...c(q-1) >= 2^63-1/2 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 5*(2^64-1), 1<=q<=34 // <=> if 0.c(0)c(1)...c(q-1) * 10^20 >= 0x4fffffffffffffffb, 1<=q<=34 C.w[1] = 0x0000000000000004ull; C.w[0] = 0xfffffffffffffffbull; if (q <= 19) { // 1 <= q <= 19 => 1 <= 20-q <= 19 => // 10^(20-q) is 64-bit, and so is C1 __mul_64x64_to_128MACH (C1, C1.w[0], ten2k64[20 - q]); } else if (q == 20) { ; // C1 * 10^0 = C1 } else { // if 21 <= q <= 34 __mul_128x64_to_128 (C, ten2k64[q - 20], C); // max 47-bit x 67-bit } if (C1.w[1] > C.w[1] || (C1.w[1] == C.w[1] && C1.w[0] >= C.w[0])) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 19' } } // n is not too large to be converted to int64: -2^63-1/2 <= n < 2^63-1/2 // Note: some of the cases tested for above fall through to this point // Restore C1 which may have been modified above C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1) // res = 0 // else // res = +/-1 ind = q - 1; if (ind <= 18) { // 0 <= ind <= 18 if ((C1.w[1] == 0) && (C1.w[0] < midpoint64[ind])) { res = 0x0000000000000000ull; // return 0 } else if (x_sign) { // n < 0 res = 0xffffffffffffffffull; // return -1 } else { // n > 0 res = 0x0000000000000001ull; // return +1 } } else { // 19 <= ind <= 33 if ((C1.w[1] < midpoint128[ind - 19].w[1]) || ((C1.w[1] == midpoint128[ind - 19].w[1]) && (C1.w[0] < midpoint128[ind - 19].w[0]))) { res = 0x0000000000000000ull; // return 0 } else if (x_sign) { // n < 0 res = 0xffffffffffffffffull; // return -1 } else { // n > 0 res = 0x0000000000000001ull; // return +1 } } // set inexact flag *pfpsf |= INEXACT_EXCEPTION; } else { // if (1 <= q + exp <= 19, 1 <= q <= 34, -33 <= exp <= 18) // -2^63-1/2 <= x <= -1 or 1 <= x < 2^63-1/2 so x can be rounded // to nearest to a 64-bit signed integer if (exp < 0) { // 2 <= q <= 34, -33 <= exp <= -1, 1 <= q + exp <= 19 ind = -exp; // 1 <= ind <= 33; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 = C1 + 1/2 * 10^ind where the result C1 fits in 127 bits tmp64 = C1.w[0]; if (ind <= 19) { C1.w[0] = C1.w[0] + midpoint64[ind - 1]; } else { C1.w[0] = C1.w[0] + midpoint128[ind - 20].w[0]; C1.w[1] = C1.w[1] + midpoint128[ind - 20].w[1]; } if (C1.w[0] < tmp64) C1.w[1]++; // calculate C* and f* // C* is actually floor(C*) in this case // C* and f* need shifting and masking, as shown by // shiftright128[] and maskhigh128[] // 1 <= x <= 33 // kx = 10^(-x) = ten2mk128[ind - 1] // C* = (C1 + 1/2 * 10^x) * 10^(-x) // the approximation of 10^(-x) was rounded up to 118 bits __mul_128x128_to_256 (P256, C1, ten2mk128[ind - 1]); if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[1] = P256.w[3]; Cstar.w[0] = P256.w[2]; fstar.w[3] = 0; fstar.w[2] = P256.w[2] & maskhigh128[ind - 1]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } else { // 22 <= ind - 1 <= 33 Cstar.w[1] = 0; Cstar.w[0] = P256.w[3]; fstar.w[3] = P256.w[3] & maskhigh128[ind - 1]; fstar.w[2] = P256.w[2]; fstar.w[1] = P256.w[1]; fstar.w[0] = P256.w[0]; } // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind], e.g. // if x=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999 // if (0 < f* < 10^(-x)) then the result is a midpoint // if floor(C*) is even then C* = floor(C*) - logical right // shift; C* has p decimal digits, correct by Prop. 1) // else if floor(C*) is odd C* = floor(C*)-1 (logical right // shift; C* has p decimal digits, correct by Pr. 1) // else // C* = floor(C*) (logical right shift; C has p decimal digits, // correct by Property 1) // n = C* * 10^(e+x) // shift right C* by Ex-128 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 102 if (ind - 1 <= 21) { // 0 <= ind - 1 <= 21 Cstar.w[0] = (Cstar.w[0] >> shift) | (Cstar.w[1] << (64 - shift)); // redundant, it will be 0! Cstar.w[1] = (Cstar.w[1] >> shift); } else { // 22 <= ind - 1 <= 33 Cstar.w[0] = (Cstar.w[0] >> (shift - 64)); // 2 <= shift - 64 <= 38 } // determine inexactness of the rounding of C* // if (0 < f* - 1/2 < 10^(-x)) then // the result is exact // else // if (f* - 1/2 > T*) then // the result is inexact if (ind - 1 <= 2) { if (fstar.w[1] > 0x8000000000000000ull || (fstar.w[1] == 0x8000000000000000ull && fstar.w[0] > 0x0ull)) { // f* > 1/2 and the result may be exact tmp64 = fstar.w[1] - 0x8000000000000000ull; // f* - 1/2 if (tmp64 > ten2mk128trunc[ind - 1].w[1] || (tmp64 == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] >= ten2mk128trunc[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } else if (ind - 1 <= 21) { // if 3 <= ind <= 21 if (fstar.w[3] > 0x0 || (fstar.w[3] == 0x0 && fstar.w[2] > onehalf128[ind - 1]) || (fstar.w[3] == 0x0 && fstar.w[2] == onehalf128[ind - 1] && (fstar.w[1] || fstar.w[0]))) { // f2* > 1/2 and the result may be exact // Calculate f2* - 1/2 tmp64 = fstar.w[2] - onehalf128[ind - 1]; tmp64A = fstar.w[3]; if (tmp64 > fstar.w[2]) tmp64A--; if (tmp64A || tmp64 || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } else { // if 22 <= ind <= 33 if (fstar.w[3] > onehalf128[ind - 1] || (fstar.w[3] == onehalf128[ind - 1] && (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f2* > 1/2 and the result may be exact // Calculate f2* - 1/2 tmp64 = fstar.w[3] - onehalf128[ind - 1]; if (tmp64 || fstar.w[2] || fstar.w[1] > ten2mk128trunc[ind - 1].w[1] || (fstar.w[1] == ten2mk128trunc[ind - 1].w[1] && fstar.w[0] > ten2mk128trunc[ind - 1].w[0])) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } // if the result was a midpoint it was rounded away from zero if (x_sign) res = -Cstar.w[0]; else res = Cstar.w[0]; } else if (exp == 0) { // 1 <= q <= 19 // res = +/-C (exact) if (x_sign) res = -C1.w[0]; else res = C1.w[0]; } else { // if (exp>0) => 1 <= exp <= 18, 1 <= q < 18, 2 <= q + exp <= 19 // res = +/-C * 10^exp (exact) where this fits in 64-bit integer if (x_sign) res = -C1.w[0] * ten2k64[exp]; else res = C1.w[0] * ten2k64[exp]; } } } BID_RETURN (res); }
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