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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" /***************************************************************************** * BID64_to_uint64_rnint ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_rnint (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_rnint (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 fstar; UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' if (x_sign) { // if n < 0 and q + exp = 20 then x is much less than -1/2 // => set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else { // if n > 0 and q + exp = 20 // if n >= 2^64 - 1/2 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1) // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=16 if (q == 1) { // C * 10^20 >= 0x9fffffffffffffffb __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] >= 0xfffffffffffffffbull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0x9fffffffffffffffb // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] >= 0xfffffffffffffffbull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } } // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2 // Note: some of the cases tested for above fall through to this point if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1) // res = 0 // else if x > 0 // res = +1 // else // if x < 0 // invalid exc ind = q - 1; // 0 <= ind <= 15 if (C1 <= midpoint64[ind]) { res = 0x0000000000000000ull; // return 0 } else if (!x_sign) { // n > 0 res = 0x0000000000000001ull; // return +1 } else { // if n < 0 res = 0x8000000000000000ull; *pfpsf |= INVALID_EXCEPTION; BID_RETURN (res); } } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded // to nearest to a 64-bit unsigned signed integer if (x_sign) { // x <= -1 // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // 1 <= x < 2^64-1/2 so x can be rounded // to nearest to a 64-bit unsigned integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; 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 64 bits C1 = C1 + midpoint64[ind - 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = (C1 + 1/2 * 10^x) * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; fstar.w[1] = P128.w[1] & maskhigh128[ind - 1]; fstar.w[0] = P128.w[0]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; // if the result was a midpoint it was rounded away from zero, so // it will need a correction // check for midpoints if ((fstar.w[1] == 0) && fstar.w[0] && (fstar.w[0] <= ten2mk128trunc[ind - 1].w[1])) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // the result is a midpoint; round to nearest if (Cstar & 0x01) { // Cstar is odd; MP in [EVEN, ODD] // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 Cstar--; // Cstar is now even } // else MP in [ODD, EVEN] } res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); } /***************************************************************************** * BID64_to_uint64_xrnint ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_xrnint (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_xrnint (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) UINT64 tmp64; BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 fstar; UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' if (x_sign) { // if n < 0 and q + exp = 20 then x is much less than -1/2 // => set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else { // if n > 0 and q + exp = 20 // if n >= 2^64 - 1/2 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1) // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=16 if (q == 1) { // C * 10^20 >= 0x9fffffffffffffffb __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] >= 0xfffffffffffffffbull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0x9fffffffffffffffb // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] >= 0xfffffffffffffffbull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } } // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2 // Note: some of the cases tested for above fall through to this point if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) // if 0.c(0)c(1)...c(q-1) <= 0.5 <=> c(0)c(1)...c(q-1) <= 5 * 10^(q-1) // res = 0 // else if x > 0 // res = +1 // else // if x < 0 // invalid exc ind = q - 1; // 0 <= ind <= 15 if (C1 <= midpoint64[ind]) { res = 0x0000000000000000ull; // return 0 } else if (!x_sign) { // n > 0 res = 0x0000000000000001ull; // return +1 } else { // if n < 0 res = 0x8000000000000000ull; *pfpsf |= INVALID_EXCEPTION; BID_RETURN (res); } // set inexact flag *pfpsf |= INEXACT_EXCEPTION; } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded // to nearest to a 64-bit unsigned signed integer if (x_sign) { // x <= -1 // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // 1 <= x < 2^64-1/2 so x can be rounded // to nearest to a 64-bit unsigned integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; 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 64 bits C1 = C1 + midpoint64[ind - 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = (C1 + 1/2 * 10^x) * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; fstar.w[1] = P128.w[1] & maskhigh128[ind - 1]; fstar.w[0] = P128.w[0]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; // 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) { // fstar.w[1] is 0 if (fstar.w[0] > 0x8000000000000000ull) { // f* > 1/2 and the result may be exact tmp64 = fstar.w[0] - 0x8000000000000000ull; // f* - 1/2 if ((tmp64 > ten2mk128trunc[ind - 1].w[1])) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // 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 3 <= ind - 1 <= 14 if (fstar.w[1] > onehalf128[ind - 1] || (fstar.w[1] == onehalf128[ind - 1] && fstar.w[0])) { // f2* > 1/2 and the result may be exact // Calculate f2* - 1/2 tmp64 = fstar.w[1] - onehalf128[ind - 1]; if (tmp64 || fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // 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[1] == 0) && fstar.w[0] && (fstar.w[0] <= ten2mk128trunc[ind - 1].w[1])) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // the result is a midpoint; round to nearest if (Cstar & 0x01) { // Cstar is odd; MP in [EVEN, ODD] // if floor(C*) is odd C = floor(C*) - 1; the result >= 1 Cstar--; // Cstar is now even } // else MP in [ODD, EVEN] } res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); } /***************************************************************************** * BID64_to_uint64_floor ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_floor (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_floor (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero if (x_sign) { // if n < 0 the conversion is invalid // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' // n > 0 and q + exp = 20 // if n >= 2^64 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65) // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=16 if (q == 1) { // C * 10^20 >= 0xa0000000000000000 __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] >= 0x0a) { // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0xa0000000000000000 // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] >= 0x0a) { // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } // n is not too large to be converted to int64 if -1 < n < 2^64 // Note: some of the cases tested for above fall through to this point if ((q + exp) <= 0) { // n = +0.[0...0]c(0)c(1)...c(q-1) // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // 1 <= x < 2^64 so x can be rounded // to nearest to a 64-bit unsigned signed integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 64 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); } /***************************************************************************** * BID64_to_uint64_xfloor ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_xfloor (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_xfloor (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 fstar; UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero if (x_sign) { // if n < 0 the conversion is invalid // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' // n > 0 and q + exp = 20 // if n >= 2^64 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65) // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=16 if (q == 1) { // C * 10^20 >= 0xa0000000000000000 __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] >= 0x0a) { // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0xa0000000000000000 // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] >= 0x0a) { // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } // n is not too large to be converted to int64 if -1 < n < 2^64 // Note: some of the cases tested for above fall through to this point if ((q + exp) <= 0) { // n = +0.[0...0]c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // 1 <= x < 2^64 so x can be rounded // to nearest to a 64-bit unsigned signed integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 64 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; fstar.w[1] = P128.w[1] & maskhigh128[ind - 1]; fstar.w[0] = P128.w[0]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; // 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) { // fstar.w[1] is 0 if (fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // if 3 <= ind - 1 <= 14 if (fstar.w[1] || fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); } /***************************************************************************** * BID64_to_uint64_ceil ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_ceil (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_ceil (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 fstar; UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' if (x_sign) { // if n < 0 and q + exp = 20 then x is much less than -1 // => set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else { // if n > 0 and q + exp = 20 // if n > 2^64 - 1 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) > 2^64 - 1 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 > 2^64 - 1 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65 - 2) // <=> C * 10^(21-q) > 0x9fffffffffffffff6, 1<=q<=16 if (q == 1) { // C * 10^20 > 0x9fffffffffffffff6 __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] > 0xfffffffffffffff6ull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0x9fffffffffffffff6 // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] > 0xfffffffffffffff6ull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } } // n is not too large to be converted to int64 if -1 < n < 2^64 // Note: some of the cases tested for above fall through to this point if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // return 0 or 1 if (x_sign) res = 0x0000000000000000ull; else res = 0x0000000000000001ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // x <= -1 or 1 <= x <= 2^64 - 1 so if positive x can be rounded // to nearest to a 64-bit unsigned signed integer if (x_sign) { // x <= -1 // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // 1 <= x <= 2^64 - 1 so x can be rounded // to nearest to a 64-bit unsigned integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 64 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; fstar.w[1] = P128.w[1] & maskhigh128[ind - 1]; fstar.w[0] = P128.w[0]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; // 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) { // fstar.w[1] is 0 if (fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] Cstar++; } // else the result is exact } else { // if 3 <= ind - 1 <= 14 if (fstar.w[1] || fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] Cstar++; } // else the result is exact } res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); } /***************************************************************************** * BID64_to_uint64_xceil ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_xceil (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_xceil (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 fstar; UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' if (x_sign) { // if n < 0 and q + exp = 20 then x is much less than -1 // => set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else { // if n > 0 and q + exp = 20 // if n > 2^64 - 1 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) > 2^64 - 1 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 > 2^64 - 1 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65 - 2) // <=> C * 10^(21-q) > 0x9fffffffffffffff6, 1<=q<=16 if (q == 1) { // C * 10^20 > 0x9fffffffffffffff6 __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] > 0xfffffffffffffff6ull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0x9fffffffffffffff6 // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] > 0xfffffffffffffff6ull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } } // n is not too large to be converted to int64 if -1 < n < 2^64 // Note: some of the cases tested for above fall through to this point if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return 0 or 1 if (x_sign) res = 0x0000000000000000ull; else res = 0x0000000000000001ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // x <= -1 or 1 <= x <= 2^64 - 1 so if positive x can be rounded // to nearest to a 64-bit unsigned signed integer if (x_sign) { // x <= -1 // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // 1 <= x <= 2^64 - 1 so x can be rounded // to nearest to a 64-bit unsigned integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 64 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; fstar.w[1] = P128.w[1] & maskhigh128[ind - 1]; fstar.w[0] = P128.w[0]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; // 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) { // fstar.w[1] is 0 if (fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] Cstar++; // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // if 3 <= ind - 1 <= 14 if (fstar.w[1] || fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] Cstar++; // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); } /***************************************************************************** * BID64_to_uint64_int ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_int (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_int (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' if (x_sign) { // if n < 0 and q + exp = 20 then x is much less than -1 // => set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else { // if n > 0 and q + exp = 20 // if n >= 2^64 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65) // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=16 if (q == 1) { // C * 10^20 >= 0xa0000000000000000 __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] >= 0x0a) { // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0xa0000000000000000 // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] >= 0x0a) { // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } } // n is not too large to be converted to int64 if -1 < n < 2^64 // Note: some of the cases tested for above fall through to this point if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded // to nearest to a 64-bit unsigned signed integer if (x_sign) { // x <= -1 // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // 1 <= x < 2^64 so x can be rounded // to nearest to a 64-bit unsigned integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 64 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); } /***************************************************************************** * BID64_to_uint64_xint ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_xint (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_xint (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 fstar; UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' if (x_sign) { // if n < 0 and q + exp = 20 then x is much less than -1 // => set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else { // if n > 0 and q + exp = 20 // if n >= 2^64 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65) // <=> C * 10^(21-q) >= 0xa0000000000000000, 1<=q<=16 if (q == 1) { // C * 10^20 >= 0xa0000000000000000 __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] >= 0x0a) { // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0xa0000000000000000 // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] >= 0x0a) { // actually C.w[1] == 0x0a && C.w[0] >= 0x0000000000000000ull) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } } // n is not too large to be converted to int64 if -1 < n < 2^64 // Note: some of the cases tested for above fall through to this point if ((q + exp) <= 0) { // n = +/-0.[0...0]c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // x <= -1 or 1 <= x < 2^64 so if positive x can be rounded // to nearest to a 64-bit unsigned signed integer if (x_sign) { // x <= -1 // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // 1 <= x < 2^64 so x can be rounded // to nearest to a 64-bit unsigned integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; ind is a synonym for 'x' // chop off ind digits from the lower part of C1 // C1 fits in 64 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = C1 * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; fstar.w[1] = P128.w[1] & maskhigh128[ind - 1]; fstar.w[0] = P128.w[0]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; // 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) { // fstar.w[1] is 0 if (fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // if 3 <= ind - 1 <= 14 if (fstar.w[1] || fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); } /***************************************************************************** * BID64_to_uint64_rninta ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_rninta (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_rninta (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' if (x_sign) { // if n < 0 and q + exp = 20 then x is much less than -1/2 // => set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else { // if n > 0 and q + exp = 20 // if n >= 2^64 - 1/2 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1) // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=16 if (q == 1) { // C * 10^20 >= 0x9fffffffffffffffb __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] >= 0xfffffffffffffffbull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0x9fffffffffffffffb // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] >= 0xfffffffffffffffbull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } } // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2 // Note: some of the cases tested for above fall through to this point if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1) // res = 0 // else if x > 0 // res = +1 // else // if x < 0 // invalid exc ind = q - 1; // 0 <= ind <= 15 if (C1 < midpoint64[ind]) { res = 0x0000000000000000ull; // return 0 } else if (!x_sign) { // n > 0 res = 0x0000000000000001ull; // return +1 } else { // if n < 0 res = 0x8000000000000000ull; *pfpsf |= INVALID_EXCEPTION; BID_RETURN (res); } } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded // to nearest to a 64-bit unsigned signed integer if (x_sign) { // x <= -1 // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // 1 <= x < 2^64-1/2 so x can be rounded // to nearest to a 64-bit unsigned integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; 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 64 bits C1 = C1 + midpoint64[ind - 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = (C1 + 1/2 * 10^x) * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; // if the result was a midpoint it was rounded away from zero res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); } /***************************************************************************** * BID64_to_uint64_xrninta ****************************************************************************/ #if DECIMAL_CALL_BY_REFERENCE void bid64_to_uint64_xrninta (UINT64 * pres, UINT64 * px _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px; #else UINT64 bid64_to_uint64_xrninta (UINT64 x _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res; UINT64 x_sign; UINT64 x_exp; int exp; // unbiased exponent // Note: C1 represents x_significand (UINT64) UINT64 tmp64; BID_UI64DOUBLE tmp1; unsigned int x_nr_bits; int q, ind, shift; UINT64 C1; UINT128 C; UINT64 Cstar; // C* represents up to 16 decimal digits ~ 54 bits UINT128 fstar; UINT128 P128; // check for NaN or Infinity if ((x & MASK_NAN) == MASK_NAN || (x & MASK_INF) == MASK_INF) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // unpack x x_sign = x & MASK_SIGN; // 0 for positive, MASK_SIGN for negative // if steering bits are 11 (condition will be 0), then exponent is G[0:w+1] => if ((x & MASK_STEERING_BITS) == MASK_STEERING_BITS) { x_exp = (x & MASK_BINARY_EXPONENT2) >> 51; // biased C1 = (x & MASK_BINARY_SIG2) | MASK_BINARY_OR2; if (C1 > 9999999999999999ull) { // non-canonical x_exp = 0; C1 = 0; } } else { x_exp = (x & MASK_BINARY_EXPONENT1) >> 53; // biased C1 = x & MASK_BINARY_SIG1; } // check for zeros (possibly from non-canonical values) if (C1 == 0x0ull) { // x is 0 res = 0x0000000000000000ull; BID_RETURN (res); } // x is not special and is not zero // q = nr. of decimal digits in x (1 <= q <= 54) // determine first the nr. of bits in x if (C1 >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1 >= 0x0000000100000000ull) { // x >= 2^32 tmp1.d = (double) (C1 >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp1.d = (double) C1; // exact conversion x_nr_bits = 1 + ((((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 >= nr_digits[x_nr_bits - 1].threshold_lo) q++; } exp = x_exp - 398; // unbiased exponent if ((q + exp) > 20) { // x >= 10^20 ~= 2^66.45... (cannot fit in 64 bits) // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 20) { // x = c(0)c(1)...c(19).c(20)...c(q-1) // in this case 2^63.11... ~= 10^19 <= x < 10^20 ~= 2^66.43... // so x rounded to an integer may or may not fit in an unsigned 64-bit int // the cases that do not fit are identified here; the ones that fit // fall through and will be handled with other cases further, // under '1 <= q + exp <= 20' if (x_sign) { // if n < 0 and q + exp = 20 then x is much less than -1/2 // => set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } else { // if n > 0 and q + exp = 20 // if n >= 2^64 - 1/2 then n is too large // <=> c(0)c(1)...c(19).c(20)...c(q-1) >= 2^64-1/2 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^20 >= 2^64-1/2 // <=> 0.c(0)c(1)...c(19)c(20)...c(q-1) * 10^21 >= 5*(2^65-1) // <=> C * 10^(21-q) >= 0x9fffffffffffffffb, 1<=q<=16 if (q == 1) { // C * 10^20 >= 0x9fffffffffffffffb __mul_128x64_to_128 (C, C1, ten2k128[0]); // 10^20 * C if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] >= 0xfffffffffffffffbull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } else { // if (2 <= q <= 16) => 5 <= 21 - q <= 19 // Note: C * 10^(21-q) has 20 or 21 digits; 0x9fffffffffffffffb // has 21 digits __mul_64x64_to_128MACH (C, C1, ten2k64[21 - q]); if (C.w[1] > 0x09 || (C.w[1] == 0x09 && C.w[0] >= 0xfffffffffffffffbull)) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // else cases that can be rounded to a 64-bit int fall through // to '1 <= q + exp <= 20' } } } // n is not too large to be converted to int64 if -1/2 <= n < 2^64 - 1/2 // Note: some of the cases tested for above fall through to this point if ((q + exp) < 0) { // n = +/-0.0...c(0)c(1)...c(q-1) // set inexact flag *pfpsf |= INEXACT_EXCEPTION; // return 0 res = 0x0000000000000000ull; BID_RETURN (res); } else if ((q + exp) == 0) { // n = +/-0.c(0)c(1)...c(q-1) // if 0.c(0)c(1)...c(q-1) < 0.5 <=> c(0)c(1)...c(q-1) < 5 * 10^(q-1) // res = 0 // else if x > 0 // res = +1 // else // if x < 0 // invalid exc ind = q - 1; // 0 <= ind <= 15 if (C1 < midpoint64[ind]) { res = 0x0000000000000000ull; // return 0 } else if (!x_sign) { // n > 0 res = 0x0000000000000001ull; // return +1 } else { // if n < 0 res = 0x8000000000000000ull; *pfpsf |= INVALID_EXCEPTION; BID_RETURN (res); } // set inexact flag *pfpsf |= INEXACT_EXCEPTION; } else { // if (1 <= q + exp <= 20, 1 <= q <= 16, -15 <= exp <= 19) // x <= -1 or 1 <= x < 2^64-1/2 so if positive x can be rounded // to nearest to a 64-bit unsigned signed integer if (x_sign) { // x <= -1 // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return Integer Indefinite res = 0x8000000000000000ull; BID_RETURN (res); } // 1 <= x < 2^64-1/2 so x can be rounded // to nearest to a 64-bit unsigned integer if (exp < 0) { // 2 <= q <= 16, -15 <= exp <= -1, 1 <= q + exp <= 20 ind = -exp; // 1 <= ind <= 15; 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 64 bits C1 = C1 + midpoint64[ind - 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 <= 15 // kx = 10^(-x) = ten2mk64[ind - 1] // C* = (C1 + 1/2 * 10^x) * 10^(-x) // the approximation of 10^(-x) was rounded up to 54 bits __mul_64x64_to_128MACH (P128, C1, ten2mk64[ind - 1]); Cstar = P128.w[1]; fstar.w[1] = P128.w[1] & maskhigh128[ind - 1]; fstar.w[0] = P128.w[0]; // the top Ex bits of 10^(-x) are T* = ten2mk128trunc[ind].w[0], e.g. // if x=1, T*=ten2mk128trunc[0].w[0]=0x1999999999999999 // 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-64 = shiftright128[ind] shift = shiftright128[ind - 1]; // 0 <= shift <= 39 Cstar = Cstar >> shift; // 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) { // fstar.w[1] is 0 if (fstar.w[0] > 0x8000000000000000ull) { // f* > 1/2 and the result may be exact tmp64 = fstar.w[0] - 0x8000000000000000ull; // f* - 1/2 if ((tmp64 > ten2mk128trunc[ind - 1].w[1])) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // 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 3 <= ind - 1 <= 14 if (fstar.w[1] > onehalf128[ind - 1] || (fstar.w[1] == onehalf128[ind - 1] && fstar.w[0])) { // f2* > 1/2 and the result may be exact // Calculate f2* - 1/2 tmp64 = fstar.w[1] - onehalf128[ind - 1]; if (tmp64 || fstar.w[0] > ten2mk128trunc[ind - 1].w[1]) { // ten2mk128trunc[ind -1].w[1] is identical to // ten2mk128[ind -1].w[1] // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } } // if the result was a midpoint it was rounded away from zero res = Cstar; // the result is positive } else if (exp == 0) { // 1 <= q <= 10 // res = +C (exact) res = C1; // the result is positive } else { // if (exp > 0) => 1 <= exp <= 9, 1 <= q < 9, 2 <= q + exp <= 10 // res = +C * 10^exp (exact) res = C1 * ten2k64[exp]; // the result is positive } } BID_RETURN (res); }
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