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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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