Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA
02110-1301, USA. */
+#include "bid_internal.h"
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid64dq_add (UINT64 * pres, UINT64 * px, UINT128 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 x = *px;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT64
+bid64dq_add (UINT64 x, UINT128 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT64 res = 0xbaddbaddbaddbaddull;
+ UINT128 x1;
+
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid64qq_add (&res, &x1, py
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid64qq_add (x1, y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid64qd_add (UINT64 * pres, UINT128 * px, UINT64 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 y = *py;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT64
+bid64qd_add (UINT128 x, UINT64 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT64 res = 0xbaddbaddbaddbaddull;
+ UINT128 y1;
+
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid64qq_add (&res, px, &y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid64qq_add (x, y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid64qq_add (UINT64 * pres, UINT128 * px, UINT128 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT128 x = *px, y = *py;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT64
+bid64qq_add (UINT128 x, UINT128 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+
+ UINT128 one = { {0x0000000000000001ull, 0x3040000000000000ull}
+ };
+ UINT64 res = 0xbaddbaddbaddbaddull;
+
+ BID_SWAP128 (one);
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64qqq_fma (&res, &one, &x, &y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ res = bid64qqq_fma (one, x, y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid128dd_add (UINT128 * pres, UINT64 * px, UINT64 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 x = *px, y = *py;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT128
+bid128dd_add (UINT64 x, UINT64 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
+ };
+ UINT128 x1, y1;
+
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid128_add (&res, &x1, &y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid128_add (x1, y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid128dq_add (UINT128 * pres, UINT64 * px, UINT128 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 x = *px;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT128
+bid128dq_add (UINT64 x, UINT128 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
+ };
+ UINT128 x1;
+
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid128_add (&res, &x1, py
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid128_add (x1, y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid128qd_add (UINT128 * pres, UINT128 * px, UINT64 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 y = *py;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT128
+bid128qd_add (UINT128 x, UINT64 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
+ };
+ UINT128 y1;
+
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid128_add (&res, px, &y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid128_add (x, y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+// bid128_add stands for bid128qq_add
+
+
/*****************************************************************************
- * BID128 add
+ * BID64/BID128 sub
****************************************************************************/
-#include "bid_internal.h"
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid64dq_sub (UINT64 * pres, UINT64 * px, UINT128 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 x = *px;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT64
+bid64dq_sub (UINT64 x, UINT128 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT64 res = 0xbaddbaddbaddbaddull;
+ UINT128 x1;
+
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid64qq_sub (&res, &x1, py
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid64qq_sub (x1, y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid64qd_sub (UINT64 * pres, UINT128 * px, UINT64 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 y = *py;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT64
+bid64qd_sub (UINT128 x, UINT64 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT64 res = 0xbaddbaddbaddbaddull;
+ UINT128 y1;
+
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid64qq_sub (&res, px, &y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid64qq_sub (x, y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
#if DECIMAL_CALL_BY_REFERENCE
void
-__bid128_add (UINT128 * pres, UINT128 * px,
- UINT128 *
- py _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
- _EXC_INFO_PARAM) {
+bid64qq_sub (UINT64 * pres, UINT128 * px, UINT128 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
UINT128 x = *px, y = *py;
#if !DECIMAL_GLOBAL_ROUNDING
unsigned int rnd_mode = *prnd_mode;
#endif
#else
+UINT64
+bid64qq_sub (UINT128 x, UINT128 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+
+ UINT128 one = { {0x0000000000000001ull, 0x3040000000000000ull}
+ };
+ UINT64 res = 0xbaddbaddbaddbaddull;
+ UINT64 y_sign;
+
+ BID_SWAP128 (one);
+ if ((y.w[HIGH_128W] & MASK_NAN) != MASK_NAN) { // y is not NAN
+ // change its sign
+ y_sign = y.w[HIGH_128W] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
+ if (y_sign)
+ y.w[HIGH_128W] = y.w[HIGH_128W] & 0x7fffffffffffffffull;
+ else
+ y.w[HIGH_128W] = y.w[HIGH_128W] | 0x8000000000000000ull;
+ }
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64qqq_fma (&res, &one, &x, &y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ res = bid64qqq_fma (one, x, y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid128dd_sub (UINT128 * pres, UINT64 * px, UINT64 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 x = *px, y = *py;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
UINT128
-__bid128_add (UINT128 x,
- UINT128 y _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
- _EXC_INFO_PARAM) {
+bid128dd_sub (UINT64 x, UINT64 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
#endif
+ UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
+ };
+ UINT128 x1, y1;
- UINT128 res;
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid128_sub (&res, &x1, &y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid128_sub (x1, y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid128dq_sub (UINT128 * pres, UINT64 * px, UINT128 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 x = *px;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT128
+bid128dq_sub (UINT64 x, UINT128 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
+ };
+ UINT128 x1;
+
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid128_sub (&res, &x1, py
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ x1 = bid64_to_bid128 (x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid128_sub (x1, y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid128qd_sub (UINT128 * pres, UINT128 * px, UINT64 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT64 y = *py;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT128
+bid128qd_sub (UINT128 x, UINT64 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
+ };
+ UINT128 y1;
+
+#if DECIMAL_CALL_BY_REFERENCE
+ bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ bid128_sub (&res, px, &y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#else
+ y1 = bid64_to_bid128 (y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG);
+ res = bid128_sub (x, y1
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
+#endif
+ BID_RETURN (res);
+}
+
+#if DECIMAL_CALL_BY_REFERENCE
+void
+bid128_add (UINT128 * pres, UINT128 * px, UINT128 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+ UINT128 x = *px, y = *py;
+#if !DECIMAL_GLOBAL_ROUNDING
+ unsigned int rnd_mode = *prnd_mode;
+#endif
+#else
+UINT128
+bid128_add (UINT128 x, UINT128 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
+#endif
+ UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull}
+ };
UINT64 x_sign, y_sign, tmp_sign;
- UINT64 x_exp, y_exp, tmp_exp; // e1 = x_exp, e2 = y_exp
+ UINT64 x_exp, y_exp, tmp_exp; // e1 = x_exp, e2 = y_exp
UINT64 C1_hi, C2_hi, tmp_signif_hi;
UINT64 C1_lo, C2_lo, tmp_signif_lo;
- // Note: C1.w[1], C1.w[0] represent C1_hi, C1_lo (all are UINT64)
- // Note: C2.w[1], C2.w[0] represent C2_hi, C2_lo (all are UINT64)
+ // Note: C1.w[1], C1.w[0] represent C1_hi, C1_lo (all UINT64)
+ // Note: C2.w[1], C2.w[0] represent C2_hi, C2_lo (all UINT64)
UINT64 tmp64, tmp64A, tmp64B;
BID_UI64DOUBLE tmp1, tmp2;
int x_nr_bits, y_nr_bits;
UINT64 halfulp64;
UINT128 halfulp128;
UINT128 C1, C2;
- UINT128 __bid_ten2m1;
- UINT128 highf2star; // top 128 bits in f2*; low 128 bits in R256[1], R256[0]
+ UINT128 ten2m1;
+ UINT128 highf2star; // top 128 bits in f2*; low 128 bits in R256[1], R256[0]
UINT256 P256, Q256, R256;
int is_inexact = 0, is_midpoint_lt_even = 0, is_midpoint_gt_even = 0;
int is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0;
int second_pass = 0;
+ BID_SWAP128 (x);
+ BID_SWAP128 (y);
+ x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
+ y_sign = y.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
+
// check for NaN or Infinity
if (((x.w[1] & MASK_SPECIAL) == MASK_SPECIAL)
|| ((y.w[1] & MASK_SPECIAL) == MASK_SPECIAL)) {
// x is special or y is special
-
- if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
- if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return quiet (x)
- res.w[1] = x.w[1] & 0xfdffffffffffffffull;
- res.w[0] = x.w[0];
- } else { // x is QNaN
- if ((y.w[1] & MASK_SNAN) == MASK_SNAN) { // y is SNAN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- }
- // return x
- res.w[1] = x.w[1];
- res.w[0] = x.w[0];
+ if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN
+ // check first for non-canonical NaN payload
+ if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) ||
+ (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull)
+ && (x.w[0] > 0x38c15b09ffffffffull))) {
+ x.w[1] = x.w[1] & 0xffffc00000000000ull;
+ x.w[0] = 0x0ull;
}
+ if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return quiet (x)
+ res.w[1] = x.w[1] & 0xfc003fffffffffffull;
+ // clear out also G[6]-G[16]
+ res.w[0] = x.w[0];
+ } else { // x is QNaN
+ // return x
+ res.w[1] = x.w[1] & 0xfc003fffffffffffull;
+ // clear out G[6]-G[16]
+ res.w[0] = x.w[0];
+ // if y = SNaN signal invalid exception
+ if ((y.w[1] & MASK_SNAN) == MASK_SNAN) {
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ }
+ }
+ BID_SWAP128 (res);
BID_RETURN (res);
- } else if ((y.w[1] & MASK_NAN) == MASK_NAN) { // y is NAN
- if ((y.w[1] & MASK_SNAN) == MASK_SNAN) { // y is SNAN
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return quiet (y)
- res.w[1] = y.w[1] & 0xfdffffffffffffffull;
- res.w[0] = y.w[0];
- } else { // y is QNaN
- // return y
- res.w[1] = y.w[1];
- res.w[0] = y.w[0];
+ } else if ((y.w[1] & MASK_NAN) == MASK_NAN) { // y is NAN
+ // check first for non-canonical NaN payload
+ if (((y.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) ||
+ (((y.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull)
+ && (y.w[0] > 0x38c15b09ffffffffull))) {
+ y.w[1] = y.w[1] & 0xffffc00000000000ull;
+ y.w[0] = 0x0ull;
+ }
+ if ((y.w[1] & MASK_SNAN) == MASK_SNAN) { // y is SNAN
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return quiet (y)
+ res.w[1] = y.w[1] & 0xfc003fffffffffffull;
+ // clear out also G[6]-G[16]
+ res.w[0] = y.w[0];
+ } else { // y is QNaN
+ // return y
+ res.w[1] = y.w[1] & 0xfc003fffffffffffull;
+ // clear out G[6]-G[16]
+ res.w[0] = y.w[0];
}
+ BID_SWAP128 (res);
BID_RETURN (res);
- } else { // neither x not y is NaN; at least one is infinity
- if ((x.w[1] & MASK_ANY_INF) == MASK_INF) { // x is infinity
- if ((y.w[1] & MASK_ANY_INF) == MASK_INF) { // y is infinity
- // if same sign, return either of them
- if ((x.w[1] & MASK_SIGN) == (y.w[1] & MASK_SIGN)) {
- res.w[1] = x.w[1];
- res.w[0] = x.w[0];
- } else { // x and y are infinities of opposite signs
- // set invalid flag
- *pfpsf |= INVALID_EXCEPTION;
- // return QNaN Indefinite
- res.w[1] = 0x7c00000000000000ull;
- res.w[0] = 0x0000000000000000ull;
- }
- } else { // y is 0 or finite
- // return x
- res.w[1] = x.w[1];
- res.w[0] = x.w[0];
- }
- } else { // x is not NaN or infinity, so y must be infinity
- res.w[1] = y.w[1];
- res.w[0] = y.w[0];
+ } else { // neither x not y is NaN; at least one is infinity
+ if ((x.w[1] & MASK_ANY_INF) == MASK_INF) { // x is infinity
+ if ((y.w[1] & MASK_ANY_INF) == MASK_INF) { // y is infinity
+ // if same sign, return either of them
+ if ((x.w[1] & MASK_SIGN) == (y.w[1] & MASK_SIGN)) {
+ res.w[1] = x_sign | MASK_INF;
+ res.w[0] = 0x0ull;
+ } else { // x and y are infinities of opposite signs
+ // set invalid flag
+ *pfpsf |= INVALID_EXCEPTION;
+ // return QNaN Indefinite
+ res.w[1] = 0x7c00000000000000ull;
+ res.w[0] = 0x0000000000000000ull;
+ }
+ } else { // y is 0 or finite
+ // return x
+ res.w[1] = x_sign | MASK_INF;
+ res.w[0] = 0x0ull;
+ }
+ } else { // x is not NaN or infinity, so y must be infinity
+ res.w[1] = y_sign | MASK_INF;
+ res.w[0] = 0x0ull;
}
+ BID_SWAP128 (res);
BID_RETURN (res);
}
}
// unpack the arguments
+
// unpack x
- x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
- x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bit positions
C1_hi = x.w[1] & MASK_COEFF;
C1_lo = x.w[0];
- // unpack y
- y_sign = y.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
- y_exp = y.w[1] & MASK_EXP; // biased and shifted left 49 bit positions
- C2_hi = y.w[1] & MASK_COEFF;
- C2_lo = y.w[0];
-
// test for non-canonical values:
// - values whose encoding begins with x00, x01, or x10 and whose
// coefficient is larger than 10^34 -1, or
// and infinitis were eliminated already this test is reduced to
// checking for x10x)
- // test for non-canonical values of the argument x
- if ((((C1_hi > 0x0001ed09bead87c0ull)
- || ((C1_hi == 0x0001ed09bead87c0ull)
- && (C1_lo > 0x378d8e63ffffffffull)))
- && ((x.w[1] & 0x6000000000000000ull) != 0x6000000000000000ull))
- || ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- // check for the case where the exponent is shifted right by 2 bits!
- if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) {
- x_exp = (x.w[1] << 2) & MASK_EXP; // same position as for G[0]G[1] != 11
+ // x is not infinity; check for non-canonical values - treated as zero
+ if ((x.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) {
+ // G0_G1=11; non-canonical
+ x_exp = (x.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits
+ C1_hi = 0; // significand high
+ C1_lo = 0; // significand low
+ } else { // G0_G1 != 11
+ x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits
+ if (C1_hi > 0x0001ed09bead87c0ull ||
+ (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo > 0x378d8e63ffffffffull)) {
+ // x is non-canonical if coefficient is larger than 10^34 -1
+ C1_hi = 0;
+ C1_lo = 0;
+ } else { // canonical
+ ;
}
- x.w[1] = x.w[1] & 0x8000000000000000ull; // preserve the sign bit
- x.w[0] = 0;
- C1_hi = 0;
- C1_lo = 0;
}
- // test for non-canonical values of the argument y
- if ((((C2_hi > 0x0001ed09bead87c0ull)
- || ((C2_hi == 0x0001ed09bead87c0ull)
- && (C2_lo > 0x378d8e63ffffffffull)))
- && ((y.w[1] & 0x6000000000000000ull) != 0x6000000000000000ull))
- || ((y.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull)) {
- // check for the case where the exponent is shifted right by 2 bits!
- if ((y.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) {
- y_exp = (y.w[1] << 2) & MASK_EXP; // same position as for G[0]G[1] != 11
+
+ // unpack y
+ C2_hi = y.w[1] & MASK_COEFF;
+ C2_lo = y.w[0];
+ // y is not infinity; check for non-canonical values - treated as zero
+ if ((y.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) {
+ // G0_G1=11; non-canonical
+ y_exp = (y.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits
+ C2_hi = 0; // significand high
+ C2_lo = 0; // significand low
+ } else { // G0_G1 != 11
+ y_exp = y.w[1] & MASK_EXP; // biased and shifted left 49 bits
+ if (C2_hi > 0x0001ed09bead87c0ull ||
+ (C2_hi == 0x0001ed09bead87c0ull
+ && C2_lo > 0x378d8e63ffffffffull)) {
+ // y is non-canonical if coefficient is larger than 10^34 -1
+ C2_hi = 0;
+ C2_lo = 0;
+ } else { // canonical
+ ;
}
- y.w[1] = y.w[1] & 0x8000000000000000ull; // preserve the sign bit
- y.w[0] = 0;
- C2_hi = 0;
- C2_lo = 0;
}
if ((C1_hi == 0x0ull) && (C1_lo == 0x0ull)) {
// if y is 0 return 0 with the smaller exponent
if ((C2_hi == 0x0ull) && (C2_lo == 0x0ull)) {
if (x_exp < y_exp)
- res.w[1] = x_exp;
+ res.w[1] = x_exp;
else
- res.w[1] = y_exp;
+ res.w[1] = y_exp;
if (x_sign && y_sign)
- res.w[1] = res.w[1] | x_sign; // both negative
+ res.w[1] = res.w[1] | x_sign; // both negative
else if (rnd_mode == ROUNDING_DOWN && x_sign != y_sign)
- res.w[1] = res.w[1] | 0x8000000000000000ull; // -0
+ res.w[1] = res.w[1] | 0x8000000000000000ull; // -0
// else; // res = +0
res.w[0] = 0;
} else {
// for 0 + y return y, with the preferred exponent
if (y_exp <= x_exp) {
- res.w[1] = y.w[1];
- res.w[0] = y.w[0];
- } else { // if y_exp > x_exp
- // return (C2 * 10^scale) * 10^(y_exp - scale)
- // where scale = min (P34-q2, y_exp-x_exp)
- // determine q2 = nr. of decimal digits in y
- // determine first the nr. of bits in y (y_nr_bits)
-
- if (C2_hi == 0) { // y_bits is the nr. of bits in C2_lo
- if (C2_lo >= 0x0020000000000000ull) { // y >= 2^53
- // split the 64-bit value in two 32-bit halves to avoid
- // rounding errors
- if (C2_lo >= 0x0000000100000000ull) { // y >= 2^32
- tmp2.d = (double) (C2_lo >> 32); // exact conversion
- ///tmp2_i64 = *(UINT64 *) & tmp2_d;
- y_nr_bits =
- 32 +
- ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
- } else { // y < 2^32
- tmp2.d = (double) (C2_lo); // exact conversion
- ///tmp2_i64 = *(UINT64 *) & tmp2_d;
- y_nr_bits =
- ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
- }
- } else { // if y < 2^53
- tmp2.d = (double) C2_lo; // exact conversion
- ///tmp2_i64 = *(UINT64 *) & tmp2_d;
- y_nr_bits =
- ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
- }
- } else { // C2_hi != 0 => nr. bits = 64 + nr_bits (C2_hi)
- tmp2.d = (double) C2_hi; // exact conversion
- ///tmp2_i64 = *(UINT64 *) & tmp2_d;
- y_nr_bits =
- 64 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
- }
- q2 = __bid_nr_digits[y_nr_bits].digits;
- if (q2 == 0) {
- q2 = __bid_nr_digits[y_nr_bits].digits1;
- if (C2_hi > __bid_nr_digits[y_nr_bits].threshold_hi
- || (C2_hi == __bid_nr_digits[y_nr_bits].threshold_hi
- && C2_lo >= __bid_nr_digits[y_nr_bits].threshold_lo))
- q2++;
- }
- // return (C2 * 10^scale) * 10^(y_exp - scale)
- // where scale = min (P34-q2, y_exp-x_exp)
- scale = P34 - q2;
- ind = (y_exp - x_exp) >> 49;
- if (ind < scale)
- scale = ind;
- if (scale == 0) {
- res.w[1] = y.w[1];
- res.w[0] = y.w[0];
- } else if (q2 <= 19) { // y fits in 64 bits
- if (scale <= 19) { // 10^scale fits in 64 bits
- // 64 x 64 C2_lo * __bid_ten2k64[scale]
- __mul_64x64_to_128MACH (res, C2_lo, __bid_ten2k64[scale]);
- } else { // 10^scale fits in 128 bits
- // 64 x 128 C2_lo * __bid_ten2k128[scale - 20]
- __mul_128x64_to_128 (res, C2_lo, __bid_ten2k128[scale - 20]);
- }
- } else { // y fits in 128 bits, but 10^scale must fit in 64 bits
- // 64 x 128 __bid_ten2k64[scale] * C2
- C2.w[1] = C2_hi;
- C2.w[0] = C2_lo;
- __mul_128x64_to_128 (res, __bid_ten2k64[scale], C2);
- }
- // subtract scale from the exponent
- y_exp = y_exp - ((UINT64) scale << 49);
- res.w[1] = res.w[1] | y_sign | y_exp;
+ res.w[1] = y.w[1];
+ res.w[0] = y.w[0];
+ } else { // if y_exp > x_exp
+ // return (C2 * 10^scale) * 10^(y_exp - scale)
+ // where scale = min (P34-q2, y_exp-x_exp)
+ // determine q2 = nr. of decimal digits in y
+ // determine first the nr. of bits in y (y_nr_bits)
+
+ if (C2_hi == 0) { // y_bits is the nr. of bits in C2_lo
+ if (C2_lo >= 0x0020000000000000ull) { // y >= 2^53
+ // split the 64-bit value in two 32-bit halves to avoid
+ // rounding errors
+ if (C2_lo >= 0x0000000100000000ull) { // y >= 2^32
+ tmp2.d = (double) (C2_lo >> 32); // exact conversion
+ y_nr_bits =
+ 32 +
+ ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ } else { // y < 2^32
+ tmp2.d = (double) (C2_lo); // exact conversion
+ y_nr_bits =
+ ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ }
+ } else { // if y < 2^53
+ tmp2.d = (double) C2_lo; // exact conversion
+ y_nr_bits =
+ ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ }
+ } else { // C2_hi != 0 => nr. bits = 64 + nr_bits (C2_hi)
+ tmp2.d = (double) C2_hi; // exact conversion
+ y_nr_bits =
+ 64 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ }
+ q2 = nr_digits[y_nr_bits].digits;
+ if (q2 == 0) {
+ q2 = nr_digits[y_nr_bits].digits1;
+ if (C2_hi > nr_digits[y_nr_bits].threshold_hi ||
+ (C2_hi == nr_digits[y_nr_bits].threshold_hi &&
+ C2_lo >= nr_digits[y_nr_bits].threshold_lo))
+ q2++;
+ }
+ // return (C2 * 10^scale) * 10^(y_exp - scale)
+ // where scale = min (P34-q2, y_exp-x_exp)
+ scale = P34 - q2;
+ ind = (y_exp - x_exp) >> 49;
+ if (ind < scale)
+ scale = ind;
+ if (scale == 0) {
+ res.w[1] = y.w[1];
+ res.w[0] = y.w[0];
+ } else if (q2 <= 19) { // y fits in 64 bits
+ if (scale <= 19) { // 10^scale fits in 64 bits
+ // 64 x 64 C2_lo * ten2k64[scale]
+ __mul_64x64_to_128MACH (res, C2_lo, ten2k64[scale]);
+ } else { // 10^scale fits in 128 bits
+ // 64 x 128 C2_lo * ten2k128[scale - 20]
+ __mul_128x64_to_128 (res, C2_lo, ten2k128[scale - 20]);
+ }
+ } else { // y fits in 128 bits, but 10^scale must fit in 64 bits
+ // 64 x 128 ten2k64[scale] * C2
+ C2.w[1] = C2_hi;
+ C2.w[0] = C2_lo;
+ __mul_128x64_to_128 (res, ten2k64[scale], C2);
+ }
+ // subtract scale from the exponent
+ y_exp = y_exp - ((UINT64) scale << 49);
+ res.w[1] = res.w[1] | y_sign | y_exp;
}
}
+ BID_SWAP128 (res);
BID_RETURN (res);
} else if ((C2_hi == 0x0ull) && (C2_lo == 0x0ull)) {
// y is 0 and x is not special, and not zero
if (x_exp <= y_exp) {
res.w[1] = x.w[1];
res.w[0] = x.w[0];
- } else { // if x_exp > y_exp
+ } else { // if x_exp > y_exp
// return (C1 * 10^scale) * 10^(x_exp - scale)
// where scale = min (P34-q1, x_exp-y_exp)
// determine q1 = nr. of decimal digits in x
// determine first the nr. of bits in x
- if (C1_hi == 0) { // x_bits is the nr. of bits in C1_lo
- if (C1_lo >= 0x0020000000000000ull) { // x >= 2^53
- // split the 64-bit value in two 32-bit halves to avoid
- // rounding errors
- if (C1_lo >= 0x0000000100000000ull) { // x >= 2^32
- tmp1.d = (double) (C1_lo >> 32); // exact conversion
- x_nr_bits =
- 32 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) -
- 0x3ff);
- } else { // x < 2^32
- tmp1.d = (double) (C1_lo); // exact conversion
- x_nr_bits =
- ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
- }
- } else { // if x < 2^53
- tmp1.d = (double) C1_lo; // exact conversion
- x_nr_bits =
- ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
- }
- } else { // C1_hi != 0 => nr. bits = 64 + nr_bits (C1_hi)
- tmp1.d = (double) C1_hi; // exact conversion
- x_nr_bits =
- 64 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ if (C1_hi == 0) { // x_bits is the nr. of bits in C1_lo
+ if (C1_lo >= 0x0020000000000000ull) { // x >= 2^53
+ // split the 64-bit value in two 32-bit halves to avoid
+ // rounding errors
+ if (C1_lo >= 0x0000000100000000ull) { // x >= 2^32
+ tmp1.d = (double) (C1_lo >> 32); // exact conversion
+ x_nr_bits =
+ 32 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) -
+ 0x3ff);
+ } else { // x < 2^32
+ tmp1.d = (double) (C1_lo); // exact conversion
+ x_nr_bits =
+ ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ }
+ } else { // if x < 2^53
+ tmp1.d = (double) C1_lo; // exact conversion
+ x_nr_bits =
+ ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ }
+ } else { // C1_hi != 0 => nr. bits = 64 + nr_bits (C1_hi)
+ tmp1.d = (double) C1_hi; // exact conversion
+ x_nr_bits =
+ 64 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- q1 = __bid_nr_digits[x_nr_bits].digits;
+ q1 = nr_digits[x_nr_bits].digits;
if (q1 == 0) {
- q1 = __bid_nr_digits[x_nr_bits].digits1;
- if (C1_hi > __bid_nr_digits[x_nr_bits].threshold_hi
- || (C1_hi == __bid_nr_digits[x_nr_bits].threshold_hi
- && C1_lo >= __bid_nr_digits[x_nr_bits].threshold_lo))
- q1++;
+ q1 = nr_digits[x_nr_bits].digits1;
+ if (C1_hi > nr_digits[x_nr_bits].threshold_hi ||
+ (C1_hi == nr_digits[x_nr_bits].threshold_hi &&
+ C1_lo >= nr_digits[x_nr_bits].threshold_lo))
+ q1++;
}
// return (C1 * 10^scale) * 10^(x_exp - scale)
// where scale = min (P34-q1, x_exp-y_exp)
scale = P34 - q1;
ind = (x_exp - y_exp) >> 49;
if (ind < scale)
- scale = ind;
+ scale = ind;
if (scale == 0) {
- res.w[1] = x.w[1];
- res.w[0] = x.w[0];
- } else if (q1 <= 19) { // x fits in 64 bits
- if (scale <= 19) { // 10^scale fits in 64 bits
- // 64 x 64 C1_lo * __bid_ten2k64[scale]
- __mul_64x64_to_128MACH (res, C1_lo, __bid_ten2k64[scale]);
- } else { // 10^scale fits in 128 bits
- // 64 x 128 C1_lo * __bid_ten2k128[scale - 20]
- __mul_128x64_to_128 (res, C1_lo, __bid_ten2k128[scale - 20]);
- }
- } else { // x fits in 128 bits, but 10^scale must fit in 64 bits
- // 64 x 128 __bid_ten2k64[scale] * C1
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- __mul_128x64_to_128 (res, __bid_ten2k64[scale], C1);
+ res.w[1] = x.w[1];
+ res.w[0] = x.w[0];
+ } else if (q1 <= 19) { // x fits in 64 bits
+ if (scale <= 19) { // 10^scale fits in 64 bits
+ // 64 x 64 C1_lo * ten2k64[scale]
+ __mul_64x64_to_128MACH (res, C1_lo, ten2k64[scale]);
+ } else { // 10^scale fits in 128 bits
+ // 64 x 128 C1_lo * ten2k128[scale - 20]
+ __mul_128x64_to_128 (res, C1_lo, ten2k128[scale - 20]);
+ }
+ } else { // x fits in 128 bits, but 10^scale must fit in 64 bits
+ // 64 x 128 ten2k64[scale] * C1
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ __mul_128x64_to_128 (res, ten2k64[scale], C1);
}
// subtract scale from the exponent
x_exp = x_exp - ((UINT64) scale << 49);
res.w[1] = res.w[1] | x_sign | x_exp;
}
+ BID_SWAP128 (res);
BID_RETURN (res);
- } else { // x and y are not canonical, not special, and are not zero
+ } else { // x and y are not canonical, not special, and are not zero
// note that the result may still be zero, and then it has to have the
// preferred exponent
- if (x_exp < y_exp) { // if exp_x < exp_y then swap x and y
+ if (x_exp < y_exp) { // if exp_x < exp_y then swap x and y
tmp_sign = x_sign;
tmp_exp = x_exp;
tmp_signif_hi = C1_hi;
}
// q1 = nr. of decimal digits in x
// determine first the nr. of bits in x
- if (C1_hi == 0) { // x_bits is the nr. of bits in C1_lo
- if (C1_lo >= 0x0020000000000000ull) { // x >= 2^53
- //split the 64-bit value in two 32-bit halves to avoid rounding errors
- if (C1_lo >= 0x0000000100000000ull) { // x >= 2^32
- tmp1.d = (double) (C1_lo >> 32); // exact conversion
- x_nr_bits =
- 32 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
- } else { // x < 2^32
- tmp1.d = (double) (C1_lo); // exact conversion
- x_nr_bits =
- ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
- }
- } else { // if x < 2^53
- tmp1.d = (double) C1_lo; // exact conversion
- x_nr_bits =
- ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ if (C1_hi == 0) { // x_bits is the nr. of bits in C1_lo
+ if (C1_lo >= 0x0020000000000000ull) { // x >= 2^53
+ //split the 64-bit value in two 32-bit halves to avoid rounding errors
+ if (C1_lo >= 0x0000000100000000ull) { // x >= 2^32
+ tmp1.d = (double) (C1_lo >> 32); // exact conversion
+ x_nr_bits =
+ 32 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ } else { // x < 2^32
+ tmp1.d = (double) (C1_lo); // exact conversion
+ x_nr_bits =
+ ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ }
+ } else { // if x < 2^53
+ tmp1.d = (double) C1_lo; // exact conversion
+ x_nr_bits =
+ ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- } else { // C1_hi != 0 => nr. bits = 64 + nr_bits (C1_hi)
- tmp1.d = (double) C1_hi; // exact conversion
+ } else { // C1_hi != 0 => nr. bits = 64 + nr_bits (C1_hi)
+ tmp1.d = (double) C1_hi; // exact conversion
x_nr_bits =
- 64 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ 64 + ((((unsigned int) (tmp1.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- q1 = __bid_nr_digits[x_nr_bits].digits;
+ q1 = nr_digits[x_nr_bits].digits;
if (q1 == 0) {
- q1 = __bid_nr_digits[x_nr_bits].digits1;
- if (C1_hi > __bid_nr_digits[x_nr_bits].threshold_hi
- || (C1_hi == __bid_nr_digits[x_nr_bits].threshold_hi
- && C1_lo >= __bid_nr_digits[x_nr_bits].threshold_lo))
- q1++;
+ q1 = nr_digits[x_nr_bits].digits1;
+ if (C1_hi > nr_digits[x_nr_bits].threshold_hi ||
+ (C1_hi == nr_digits[x_nr_bits].threshold_hi &&
+ C1_lo >= nr_digits[x_nr_bits].threshold_lo))
+ q1++;
}
// q2 = nr. of decimal digits in y
// determine first the nr. of bits in y (y_nr_bits)
- if (C2_hi == 0) { // y_bits is the nr. of bits in C2_lo
- if (C2_lo >= 0x0020000000000000ull) { // y >= 2^53
- //split the 64-bit value in two 32-bit halves to avoid rounding errors
- if (C2_lo >= 0x0000000100000000ull) { // y >= 2^32
- tmp2.d = (double) (C2_lo >> 32); // exact conversion
- ///tmp2_i64 = *(UINT64 *) & tmp2_d;
- y_nr_bits =
- 32 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
- } else { // y < 2^32
- tmp2.d = (double) (C2_lo); // exact conversion
- ///tmp2_i64 = *(UINT64 *) & tmp2_d;
- y_nr_bits =
- ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
- }
- } else { // if y < 2^53
- tmp2.d = (double) C2_lo; // exact conversion
- ///tmp2_i64 = *(UINT64 *) & tmp2_d;
- y_nr_bits =
- ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ if (C2_hi == 0) { // y_bits is the nr. of bits in C2_lo
+ if (C2_lo >= 0x0020000000000000ull) { // y >= 2^53
+ //split the 64-bit value in two 32-bit halves to avoid rounding errors
+ if (C2_lo >= 0x0000000100000000ull) { // y >= 2^32
+ tmp2.d = (double) (C2_lo >> 32); // exact conversion
+ y_nr_bits =
+ 32 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ } else { // y < 2^32
+ tmp2.d = (double) (C2_lo); // exact conversion
+ y_nr_bits =
+ ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ }
+ } else { // if y < 2^53
+ tmp2.d = (double) C2_lo; // exact conversion
+ y_nr_bits =
+ ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- } else { // C2_hi != 0 => nr. bits = 64 + nr_bits (C2_hi)
- tmp2.d = (double) C2_hi; // exact conversion
- ///tmp2_i64 = *(UINT64 *) & tmp2_d;
+ } else { // C2_hi != 0 => nr. bits = 64 + nr_bits (C2_hi)
+ tmp2.d = (double) C2_hi; // exact conversion
y_nr_bits =
- 64 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
+ 64 + ((((unsigned int) (tmp2.ui64 >> 52)) & 0x7ff) - 0x3ff);
}
- q2 = __bid_nr_digits[y_nr_bits].digits;
+ q2 = nr_digits[y_nr_bits].digits;
if (q2 == 0) {
- q2 = __bid_nr_digits[y_nr_bits].digits1;
- if (C2_hi > __bid_nr_digits[y_nr_bits].threshold_hi
- || (C2_hi == __bid_nr_digits[y_nr_bits].threshold_hi
- && C2_lo >= __bid_nr_digits[y_nr_bits].threshold_lo))
- q2++;
+ q2 = nr_digits[y_nr_bits].digits1;
+ if (C2_hi > nr_digits[y_nr_bits].threshold_hi ||
+ (C2_hi == nr_digits[y_nr_bits].threshold_hi &&
+ C2_lo >= nr_digits[y_nr_bits].threshold_lo))
+ q2++;
}
delta = q1 + (int) (x_exp >> 49) - q2 - (int) (y_exp >> 49);
// the result is inexact; the preferred exponent is the least possible
if (delta >= P34 + 1) {
- // for RN the result is the operand with the larger magnitude,
- // possibly scaled up by 10^(P34-q1)
- // an overflow cannot occur in this case (rounding to nearest)
- if (q1 < P34) { // scale C1 up by 10^(P34-q1)
- // Note: because delta >= P34+1 it is certain that
- // x_exp - ((UINT64)scale << 49) will stay above e_min
- scale = P34 - q1;
- if (q1 <= 19) { // C1 fits in 64 bits
- // 1 <= q1 <= 19 => 15 <= scale <= 33
- if (scale <= 19) { // 10^scale fits in 64 bits
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[scale], C1_lo);
- } else { // if 20 <= scale <= 33
- // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
- // (C1 * 10^(scale-19)) fits in 64 bits
- C1_lo = C1_lo * __bid_ten2k64[scale - 19];
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[19], C1_lo);
- }
- } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
- // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- // C1 = __bid_ten2k64[P34 - q1] * C1
- __mul_128x64_to_128 (C1, __bid_ten2k64[P34 - q1], C1);
- }
- x_exp = x_exp - ((UINT64) scale << 49);
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- }
- // some special cases arise: if delta = P34 + 1 and C1 = 10^(P34-1)
- // (after scaling) and x_sign != y_sign and C2 > 5*10^(q2-1) =>
- // subtract 1 ulp
- // Note: do this only for rounding to nearest; for other rounding
- // modes the correction will be applied next
- if ((rnd_mode == ROUNDING_TO_NEAREST
- || rnd_mode == ROUNDING_TIES_AWAY) && delta == (P34 + 1)
- && C1_hi == 0x0000314dc6448d93ull
- && C1_lo == 0x38c15b0a00000000ull && x_sign != y_sign
- && ((q2 <= 19 && C2_lo > __bid_midpoint64[q2 - 1]) || (q2 >= 20
- && (C2_hi > __bid_midpoint128[q2 - 20].w[1]
- || (C2_hi == __bid_midpoint128[q2 - 20].w[1]
- && C2_lo > __bid_midpoint128[q2 - 20].w[0]))))) {
- // C1 = 10^34 - 1 and decrement x_exp by 1 (no underflow possible)
- C1_hi = 0x0001ed09bead87c0ull;
- C1_lo = 0x378d8e63ffffffffull;
- x_exp = x_exp - EXP_P1;
- }
- if (rnd_mode != ROUNDING_TO_NEAREST) {
- if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign)
- || (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
- // add 1 ulp and then check for overflow
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- x_exp = x_exp + EXP_P1;
- if (x_exp == EXP_MAX_P1) { // overflow
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- x_exp = 0; // x_sign is preserved
- // set overflow flag (the inexact flag was set too)
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- } else if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
- || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
- || (rnd_mode == ROUNDING_TO_ZERO && x_sign != y_sign)) {
- // subtract 1 ulp from C1
- // Note: because delta >= P34 + 1 the result cannot be zero
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi = C1_hi - 1;
- // if the coefficient is 10^33 - 1 then make it 10^34 - 1 and
- // decrease the exponent by 1 (because delta >= P34 + 1 the
- // exponent will not become less than e_min)
- // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
- // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
- if (C1_hi == 0x0000314dc6448d93ull
- && C1_lo == 0x38c15b09ffffffffull) {
- // make C1 = 10^34 - 1
- C1_hi = 0x0001ed09bead87c0ull;
- C1_lo = 0x378d8e63ffffffffull;
- x_exp = x_exp - EXP_P1;
- }
- } else {
- ; // the result is already correct
- }
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // assemble the result
- res.w[1] = x_sign | x_exp | C1_hi;
- res.w[0] = C1_lo;
- } else { // delta = P34
- // in most cases, the smaller operand may be < or = or > 1/2 ulp of the
- // larger operand
- // however, the case C1 = 10^(q1-1) and x_sign != y_sign is special due
- // to accuracy loss after subtraction, and will be treated separately
- if (x_sign == y_sign || (q1 <= 20
- && (C1_hi != 0 || C1_lo != __bid_ten2k64[q1 - 1])) || (q1 >= 21
- && (C1_hi != __bid_ten2k128[q1 - 21].w[1]
- || C1_lo != __bid_ten2k128[q1 - 21].w[0]))) {
- // if x_sign == y_sign or C1 != 10^(q1-1)
- // compare C2 with 1/2 ulp = 5 * 10^(q2-1), the latter read from table
- // Note: cases q1<=19 and q1>=20 can be coalesced at some latency cost
- if (q2 <= 19) { // C2 and 5*10^(q2-1) both fit in 64 bits
- halfulp64 = __bid_midpoint64[q2 - 1]; // 5 * 10^(q2-1)
- if (C2_lo < halfulp64) { // n2 < 1/2 ulp (n1)
- // for RN the result is the operand with the larger magnitude,
- // possibly scaled up by 10^(P34-q1)
- // an overflow cannot occur in this case (rounding to nearest)
- if (q1 < P34) { // scale C1 up by 10^(P34-q1)
- // Note: because delta = P34 it is certain that
- // x_exp - ((UINT64)scale << 49) will stay above e_min
- scale = P34 - q1;
- if (q1 <= 19) { // C1 fits in 64 bits
- // 1 <= q1 <= 19 => 15 <= scale <= 33
- if (scale <= 19) { // 10^scale fits in 64 bits
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[scale], C1_lo);
- } else { // if 20 <= scale <= 33
- // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
- // (C1 * 10^(scale-19)) fits in 64 bits
- C1_lo = C1_lo * __bid_ten2k64[scale - 19];
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[19], C1_lo);
- }
- } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
- // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- // C1 = __bid_ten2k64[P34 - q1] * C1
- __mul_128x64_to_128 (C1, __bid_ten2k64[P34 - q1], C1);
- }
- x_exp = x_exp - ((UINT64) scale << 49);
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- }
- if (rnd_mode != ROUNDING_TO_NEAREST) {
- if ((rnd_mode == ROUNDING_DOWN && x_sign
- && y_sign) || (rnd_mode == ROUNDING_UP
- && !x_sign && !y_sign)) {
- // add 1 ulp and then check for overflow
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- x_exp = x_exp + EXP_P1;
- if (x_exp == EXP_MAX_P1) { // overflow
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- x_exp = 0; // x_sign is preserved
- // set overflow flag (the inexact flag was set too)
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- } else if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign) ||
- (rnd_mode == ROUNDING_UP && x_sign && !y_sign) ||
- (rnd_mode == ROUNDING_TO_ZERO && x_sign != y_sign)) {
- // subtract 1 ulp from C1
- // Note: because delta >= P34 + 1 the result cannot be zero
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi = C1_hi - 1;
- // if the coefficient is 10^33-1 then make it 10^34-1 and
- // decrease the exponent by 1 (because delta >= P34 + 1 the
- // exponent will not become less than e_min)
- // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
- // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
- if (C1_hi == 0x0000314dc6448d93ull
- && C1_lo == 0x38c15b09ffffffffull) {
- // make C1 = 10^34 - 1
- C1_hi = 0x0001ed09bead87c0ull;
- C1_lo = 0x378d8e63ffffffffull;
- x_exp = x_exp - EXP_P1;
- }
- } else {
- ; // the result is already correct
- }
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // assemble the result
- res.w[1] = x_sign | x_exp | C1_hi;
- res.w[0] = C1_lo;
- } else if ((C2_lo == halfulp64)
- && (q1 < P34 || ((C1_lo & 0x1) == 0))) {
- // n2 = 1/2 ulp (n1) and C1 is even
- // the result is the operand with the larger magnitude,
- // possibly scaled up by 10^(P34-q1)
- // an overflow cannot occur in this case (rounding to nearest)
- if (q1 < P34) { // scale C1 up by 10^(P34-q1)
- // Note: because delta = P34 it is certain that
- // x_exp - ((UINT64)scale << 49) will stay above e_min
- scale = P34 - q1;
- if (q1 <= 19) { // C1 fits in 64 bits
- // 1 <= q1 <= 19 => 15 <= scale <= 33
- if (scale <= 19) { // 10^scale fits in 64 bits
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[scale], C1_lo);
- } else { // if 20 <= scale <= 33
- // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
- // (C1 * 10^(scale-19)) fits in 64 bits
- C1_lo = C1_lo * __bid_ten2k64[scale - 19];
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[19], C1_lo);
- }
- } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
- // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- // C1 = __bid_ten2k64[P34 - q1] * C1
- __mul_128x64_to_128 (C1, __bid_ten2k64[P34 - q1], C1);
- }
- x_exp = x_exp - ((UINT64) scale << 49);
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- }
- if ((rnd_mode == ROUNDING_TO_NEAREST
- && x_sign == y_sign && (C1_lo & 0x01))
- || (rnd_mode == ROUNDING_TIES_AWAY
- && x_sign == y_sign) || (rnd_mode == ROUNDING_UP
- && !x_sign && !y_sign)
- || (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)) {
- // add 1 ulp and then check for overflow
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- x_exp = x_exp + EXP_P1;
- if (x_exp == EXP_MAX_P1) { // overflow
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- x_exp = 0; // x_sign is preserved
- // set overflow flag (the inexact flag was set too)
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- } else if ((rnd_mode == ROUNDING_TO_NEAREST
- && x_sign != y_sign && (C1_lo & 0x01))
- || (rnd_mode == ROUNDING_DOWN && !x_sign
- && y_sign) || (rnd_mode == ROUNDING_UP
- && x_sign && !y_sign)
- || (rnd_mode == ROUNDING_TO_ZERO
- && x_sign != y_sign)) {
- // subtract 1 ulp from C1
- // Note: because delta >= P34 + 1 the result cannot be zero
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi = C1_hi - 1;
- // if the coefficient is 10^33 - 1 then make it 10^34 - 1
- // and decrease the exponent by 1 (because delta >= P34 + 1
- // the exponent will not become less than e_min)
- // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
- // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
- if (C1_hi == 0x0000314dc6448d93ull
- && C1_lo == 0x38c15b09ffffffffull) {
- // make C1 = 10^34 - 1
- C1_hi = 0x0001ed09bead87c0ull;
- C1_lo = 0x378d8e63ffffffffull;
- x_exp = x_exp - EXP_P1;
- }
- } else {
- ; // the result is already correct
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // assemble the result
- res.w[1] = x_sign | x_exp | C1_hi;
- res.w[0] = C1_lo;
- } else { // if C2_lo > halfulp64 ||
- // (C2_lo == halfulp64 && q1 == P34 && ((C1_lo & 0x1) == 1)), i.e.
- // 1/2 ulp(n1) < n2 < 1 ulp(n1) or n2 = 1/2 ulp(n1) and C1 odd
- // res = x+1 ulp if n1*n2 > 0 and res = x-1 ulp if n1*n2 < 0
- if (q1 < P34) { // then 1 ulp = 10^(e1+q1-P34) < 10^e1
- // Note: if (q1 == P34) then 1 ulp = 10^(e1+q1-P34) = 10^e1
- // because q1 < P34 we must first replace C1 by
- // C1 * 10^(P34-q1), and must decrease the exponent by
- // (P34-q1) (it will still be at least e_min)
- scale = P34 - q1;
- if (q1 <= 19) { // C1 fits in 64 bits
- // 1 <= q1 <= 19 => 15 <= scale <= 33
- if (scale <= 19) { // 10^scale fits in 64 bits
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[scale], C1_lo);
- } else { // if 20 <= scale <= 33
- // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
- // (C1 * 10^(scale-19)) fits in 64 bits
- C1_lo = C1_lo * __bid_ten2k64[scale - 19];
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[19], C1_lo);
- }
- } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
- // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- // C1 = __bid_ten2k64[P34 - q1] * C1
- __mul_128x64_to_128 (C1, __bid_ten2k64[P34 - q1], C1);
- }
- x_exp = x_exp - ((UINT64) scale << 49);
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- // check for rounding overflow
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- x_exp = x_exp + EXP_P1;
- }
- }
- if ((rnd_mode == ROUNDING_TO_NEAREST
- && x_sign != y_sign)
- || (rnd_mode == ROUNDING_TIES_AWAY
- && x_sign != y_sign && C2_lo != halfulp64)
- || (rnd_mode == ROUNDING_DOWN && !x_sign
- && y_sign) || (rnd_mode == ROUNDING_UP && x_sign
- && !y_sign) || (rnd_mode == ROUNDING_TO_ZERO
- && x_sign != y_sign)) {
- // the result is x - 1
- // for RN n1 * n2 < 0; underflow not possible
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi--;
- // check if we crossed into the lower decade
- if (C1_hi == 0x0000314dc6448d93ull &&
- C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
- C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
- C1_lo = 0x378d8e63ffffffffull;
- x_exp = x_exp - EXP_P1; // no underflow, because n1 >> n2
- }
- } else if ((rnd_mode == ROUNDING_TO_NEAREST
- && x_sign == y_sign)
- || (rnd_mode == ROUNDING_TIES_AWAY
- && x_sign == y_sign)
- || (rnd_mode == ROUNDING_DOWN && x_sign
- && y_sign) || (rnd_mode == ROUNDING_UP
- && !x_sign && !y_sign)) {
- // the result is x + 1
- // for RN x_sign = y_sign, i.e. n1*n2 > 0
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- x_exp = x_exp + EXP_P1;
- if (x_exp == EXP_MAX_P1) { // overflow
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- x_exp = 0; // x_sign is preserved
- // set the overflow flag
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- } else {
- ; // the result is x
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // assemble the result
- res.w[1] = x_sign | x_exp | C1_hi;
- res.w[0] = C1_lo;
- }
- } else { // if q2 >= 20 then 5*10^(q2-1) and C2 (the latter in
- // most cases) fit only in more than 64 bits
- halfulp128 = __bid_midpoint128[q2 - 20]; // 5 * 10^(q2-1)
- if ((C2_hi < halfulp128.w[1])
- || (C2_hi == halfulp128.w[1]
- && C2_lo < halfulp128.w[0])) {
- // n2 < 1/2 ulp (n1)
- // the result is the operand with the larger magnitude,
- // possibly scaled up by 10^(P34-q1)
- // an overflow cannot occur in this case (rounding to nearest)
- if (q1 < P34) { // scale C1 up by 10^(P34-q1)
- // Note: because delta = P34 it is certain that
- // x_exp - ((UINT64)scale << 49) will stay above e_min
- scale = P34 - q1;
- if (q1 <= 19) { // C1 fits in 64 bits
- // 1 <= q1 <= 19 => 15 <= scale <= 33
- if (scale <= 19) { // 10^scale fits in 64 bits
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[scale], C1_lo);
- } else { // if 20 <= scale <= 33
- // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
- // (C1 * 10^(scale-19)) fits in 64 bits
- C1_lo = C1_lo * __bid_ten2k64[scale - 19];
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[19], C1_lo);
- }
- } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
- // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- // C1 = __bid_ten2k64[P34 - q1] * C1
- __mul_128x64_to_128 (C1, __bid_ten2k64[P34 - q1], C1);
- }
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- x_exp = x_exp - ((UINT64) scale << 49);
- }
- if (rnd_mode != ROUNDING_TO_NEAREST) {
- if ((rnd_mode == ROUNDING_DOWN && x_sign
- && y_sign) || (rnd_mode == ROUNDING_UP
- && !x_sign && !y_sign)) {
- // add 1 ulp and then check for overflow
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- x_exp = x_exp + EXP_P1;
- if (x_exp == EXP_MAX_P1) { // overflow
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- x_exp = 0; // x_sign is preserved
- // set overflow flag (the inexact flag was set too)
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- } else if ((rnd_mode == ROUNDING_DOWN && !x_sign
- && y_sign) || (rnd_mode == ROUNDING_UP
- && x_sign && !y_sign)
- || (rnd_mode == ROUNDING_TO_ZERO
- && x_sign != y_sign)) {
- // subtract 1 ulp from C1
- // Note: because delta >= P34 + 1 the result cannot be zero
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi = C1_hi - 1;
- // if the coefficient is 10^33-1 then make it 10^34-1 and
- // decrease the exponent by 1 (because delta >= P34 + 1 the
- // exponent will not become less than e_min)
- // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
- // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
- if (C1_hi == 0x0000314dc6448d93ull
- && C1_lo == 0x38c15b09ffffffffull) {
- // make C1 = 10^34 - 1
- C1_hi = 0x0001ed09bead87c0ull;
- C1_lo = 0x378d8e63ffffffffull;
- x_exp = x_exp - EXP_P1;
- }
- } else {
- ; // the result is already correct
- }
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // assemble the result
- res.w[1] = x_sign | x_exp | C1_hi;
- res.w[0] = C1_lo;
- } else if ((C2_hi == halfulp128.w[1]
- && C2_lo == halfulp128.w[0])
- && (q1 < P34 || ((C1_lo & 0x1) == 0))) {
- // midpoint & lsb in C1 is 0
- // n2 = 1/2 ulp (n1) and C1 is even
- // the result is the operand with the larger magnitude,
- // possibly scaled up by 10^(P34-q1)
- // an overflow cannot occur in this case (rounding to nearest)
- if (q1 < P34) { // scale C1 up by 10^(P34-q1)
- // Note: because delta = P34 it is certain that
- // x_exp - ((UINT64)scale << 49) will stay above e_min
- scale = P34 - q1;
- if (q1 <= 19) { // C1 fits in 64 bits
- // 1 <= q1 <= 19 => 15 <= scale <= 33
- if (scale <= 19) { // 10^scale fits in 64 bits
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[scale], C1_lo);
- } else { // if 20 <= scale <= 33
- // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
- // (C1 * 10^(scale-19)) fits in 64 bits
- C1_lo = C1_lo * __bid_ten2k64[scale - 19];
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[19], C1_lo);
- }
- } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
- // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- // C1 = __bid_ten2k64[P34 - q1] * C1
- __mul_128x64_to_128 (C1, __bid_ten2k64[P34 - q1], C1);
- }
- x_exp = x_exp - ((UINT64) scale << 49);
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- }
- if (rnd_mode != ROUNDING_TO_NEAREST) {
- if ((rnd_mode == ROUNDING_TIES_AWAY
- && x_sign == y_sign)
- || (rnd_mode == ROUNDING_UP && !y_sign)) {
- // add 1 ulp and then check for overflow
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- x_exp = x_exp + EXP_P1;
- if (x_exp == EXP_MAX_P1) { // overflow
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- x_exp = 0; // x_sign is preserved
- // set overflow flag (the inexact flag was set too)
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- } else if ((rnd_mode == ROUNDING_DOWN && y_sign)
- || (rnd_mode == ROUNDING_TO_ZERO
- && x_sign != y_sign)) {
- // subtract 1 ulp from C1
- // Note: because delta >= P34 + 1 the result cannot be zero
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi = C1_hi - 1;
- // if the coefficient is 10^33 - 1 then make it 10^34 - 1
- // and decrease the exponent by 1 (because delta >= P34 + 1
- // the exponent will not become less than e_min)
- // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
- // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
- if (C1_hi == 0x0000314dc6448d93ull
- && C1_lo == 0x38c15b09ffffffffull) {
- // make C1 = 10^34 - 1
- C1_hi = 0x0001ed09bead87c0ull;
- C1_lo = 0x378d8e63ffffffffull;
- x_exp = x_exp - EXP_P1;
- }
- } else {
- ; // the result is already correct
- }
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // assemble the result
- res.w[1] = x_sign | x_exp | C1_hi;
- res.w[0] = C1_lo;
- } else { // if C2 > halfulp128 ||
- // (C2 == halfulp128 && q1 == P34 && ((C1 & 0x1) == 1)), i.e.
- // 1/2 ulp(n1) < n2 < 1 ulp(n1) or n2 = 1/2 ulp(n1) and C1 odd
- // res = x+1 ulp if n1*n2 > 0 and res = x-1 ulp if n1*n2 < 0
- if (q1 < P34) { // then 1 ulp = 10^(e1+q1-P34) < 10^e1
- // Note: if (q1 == P34) then 1 ulp = 10^(e1+q1-P34) = 10^e1
- // because q1 < P34 we must first replace C1 by C1*10^(P34-q1),
- // and must decrease the exponent by (P34-q1) (it will still be
- // at least e_min)
- scale = P34 - q1;
- if (q1 <= 19) { // C1 fits in 64 bits
- // 1 <= q1 <= 19 => 15 <= scale <= 33
- if (scale <= 19) { // 10^scale fits in 64 bits
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[scale], C1_lo);
- } else { // if 20 <= scale <= 33
- // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
- // (C1 * 10^(scale-19)) fits in 64 bits
- C1_lo = C1_lo * __bid_ten2k64[scale - 19];
- __mul_64x64_to_128MACH (C1, __bid_ten2k64[19], C1_lo);
- }
- } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
- // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- // C1 = __bid_ten2k64[P34 - q1] * C1
- __mul_128x64_to_128 (C1, __bid_ten2k64[P34 - q1], C1);
- }
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- x_exp = x_exp - ((UINT64) scale << 49);
- }
- if ((rnd_mode == ROUNDING_TO_NEAREST
- && x_sign != y_sign) || (rnd_mode == ROUNDING_TIES_AWAY
- && x_sign != y_sign &&
- (C2_hi != halfulp128.w[1] || C2_lo != halfulp128.w[0])) ||
- (rnd_mode == ROUNDING_DOWN && !x_sign
- && y_sign) || (rnd_mode == ROUNDING_UP && x_sign
- && !y_sign) || (rnd_mode == ROUNDING_TO_ZERO
- && x_sign != y_sign)) {
- // the result is x - 1
- // for RN n1 * n2 < 0; underflow not possible
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi--;
- // check if we crossed into the lower decade
- if (C1_hi == 0x0000314dc6448d93ull &&
- C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
- C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
- C1_lo = 0x378d8e63ffffffffull;
- x_exp = x_exp - EXP_P1; // no underflow, because n1 >> n2
- }
- } else if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign == y_sign)
- || (rnd_mode == ROUNDING_TIES_AWAY && x_sign == y_sign) ||
- (rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
- (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
- // the result is x + 1
- // for RN x_sign = y_sign, i.e. n1*n2 > 0
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- x_exp = x_exp + EXP_P1;
- if (x_exp == EXP_MAX_P1) { // overflow
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- x_exp = 0; // x_sign is preserved
- // set the overflow flag
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- } else {
- ; // the result is x
- }
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // assemble the result
- res.w[1] = x_sign | x_exp | C1_hi;
- res.w[0] = C1_lo;
- }
- } // end q1 >= 20
- // end case where C1 != 10^(q1-1)
- } else { // C1 = 10^(q1-1) and x_sign != y_sign
- // instead of C' = (C1 * 10^(e1-e2) + C2)rnd,P34
- // calculate C' = C1 * 10^(e1-e2-x1) + (C2 * 10^(-x1))rnd,P34
- // where x1 = q2 - 1, 0 <= x1 <= P34 - 1
- // Because C1 = 10^(q1-1) and x_sign != y_sign, C' will have P34
- // digits and n = C' * 10^(e2+x1)
- // If the result has P34+1 digits, redo the steps above with x1+1
- // If the result has P34-1 digits or less, redo the steps above with
- // x1-1 but only if initially x1 >= 1
- // NOTE: these two steps can be improved, e.g we could guess if
- // P34+1 or P34-1 digits will be obtained by adding/subtracting
- // just the top 64 bits of the two operands
- // The result cannot be zero, and it cannot overflow
- x1 = q2 - 1; // 0 <= x1 <= P34-1
- // Calculate C1 * 10^(e1-e2-x1) where 1 <= e1-e2-x1 <= P34
- // scale = (int)(e1 >> 49) - (int)(e2 >> 49) - x1; 0 <= scale <= P34-1
- scale = P34 - q1 + 1; // scale=e1-e2-x1 = P34+1-q1; 1<=scale<=P34
- // either C1 or 10^(e1-e2-x1) may not fit is 64 bits,
- // but their product fits with certainty in 128 bits
- if (scale >= 20) { //10^(e1-e2-x1) doesn't fit in 64 bits, but C1 does
- __mul_128x64_to_128 (C1, C1_lo, __bid_ten2k128[scale - 20]);
- } else { // if (scale >= 1
- // if 1 <= scale <= 19 then 10^(e1-e2-x1) fits in 64 bits
- if (q1 <= 19) { // C1 fits in 64 bits
- __mul_64x64_to_128MACH (C1, C1_lo, __bid_ten2k64[scale]);
- } else { // q1 >= 20
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- __mul_128x64_to_128 (C1, __bid_ten2k64[scale], C1);
- }
- }
- tmp64 = C1.w[0]; // C1.w[1], C1.w[0] contains C1 * 10^(e1-e2-x1)
-
- // now round C2 to q2-x1 = 1 decimal digit
- // C2' = C2 + 1/2 * 10^x1 = C2 + 5 * 10^(x1-1)
- ind = x1 - 1; // -1 <= ind <= P34 - 2
- if (ind >= 0) { // if (x1 >= 1)
- C2.w[0] = C2_lo;
- C2.w[1] = C2_hi;
- if (ind <= 18) {
- C2.w[0] = C2.w[0] + __bid_midpoint64[ind];
- if (C2.w[0] < C2_lo)
- C2.w[1]++;
- } else { // 19 <= ind <= 32
- C2.w[0] = C2.w[0] + __bid_midpoint128[ind - 19].w[0];
- C2.w[1] = C2.w[1] + __bid_midpoint128[ind - 19].w[1];
- if (C2.w[0] < C2_lo)
- C2.w[1]++;
- }
- // the approximation of 10^(-x1) was rounded up to 118 bits
- __mul_128x128_to_256 (R256, C2, __bid_ten2mk128[ind]); // R256 = C2*, f2*
- // calculate C2* and f2*
- // C2* is actually floor(C2*) in this case
- // C2* and f2* need shifting and masking, as shown by
- // __bid_shiftright128[] and __bid_maskhigh128[]
- // the top Ex bits of 10^(-x1) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x1=1, T*=__bid_ten2mk128trunc[0]=0x19999999999999999999999999999999
- // if (0 < f2* < 10^(-x1)) then
- // if floor(C1+C2*) is even then C2* = floor(C2*) - logical right
- // shift; C2* has p decimal digits, correct by Prop. 1)
- // else if floor(C1+C2*) is odd C2* = floor(C2*)-1 (logical right
- // shift; C2* has p decimal digits, correct by Pr. 1)
- // else
- // C2* = floor(C2*) (logical right shift; C has p decimal digits,
- // correct by Property 1)
- // n = C2* * 10^(e2+x1)
-
- if (ind <= 2) {
- highf2star.w[1] = 0x0;
- highf2star.w[0] = 0x0; // low f2* ok
- } else if (ind <= 21) {
- highf2star.w[1] = 0x0;
- highf2star.w[0] = R256.w[2] & __bid_maskhigh128[ind]; // low f2* ok
- } else {
- highf2star.w[1] = R256.w[3] & __bid_maskhigh128[ind];
- highf2star.w[0] = R256.w[2]; // low f2* is ok
- }
- // shift right C2* by Ex-128 = __bid_shiftright128[ind]
- if (ind >= 3) {
- shift = __bid_shiftright128[ind];
- if (shift < 64) { // 3 <= shift <= 63
- R256.w[2] =
- (R256.w[2] >> shift) | (R256.w[3] << (64 - shift));
- R256.w[3] = (R256.w[3] >> shift);
- } else { // 66 <= shift <= 102
- R256.w[2] = (R256.w[3] >> (shift - 64));
- R256.w[3] = 0x0ULL;
- }
- }
- // redundant
- is_inexact_lt_midpoint = 0;
- is_inexact_gt_midpoint = 0;
- is_midpoint_lt_even = 0;
- is_midpoint_gt_even = 0;
- // determine inexactness of the rounding of C2*
- // (cannot be followed by a second rounding)
- // if (0 < f2* - 1/2 < 10^(-x1)) then
- // the result is exact
- // else (if f2* - 1/2 > T* then)
- // the result of is inexact
- if (ind <= 2) {
- if (R256.w[1] > 0x8000000000000000ull ||
- (R256.w[1] == 0x8000000000000000ull && R256.w[0] > 0x0ull)) {
- // f2* > 1/2 and the result may be exact
- tmp64A = R256.w[1] - 0x8000000000000000ull; // f* - 1/2
- if ((tmp64A > __bid_ten2mk128trunc[ind].w[1]
- || (tmp64A == __bid_ten2mk128trunc[ind].w[1]
- && R256.w[0] >= __bid_ten2mk128trunc[ind].w[0]))) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // this rounding is applied to C2 only!
- // x_sign != y_sign
- is_inexact_gt_midpoint = 1;
- } // else the result is exact
- // rounding down, unless a midpoint in [ODD, EVEN]
- } else { // the result is inexact; f2* <= 1/2
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // this rounding is applied to C2 only!
- // x_sign != y_sign
- is_inexact_lt_midpoint = 1;
- }
- } else if (ind <= 21) { // if 3 <= ind <= 21
- if (highf2star.w[1] > 0x0 || (highf2star.w[1] == 0x0
- && highf2star.w[0] > __bid_one_half128[ind])
- || (highf2star.w[1] == 0x0
- && highf2star.w[0] == __bid_one_half128[ind]
- && (R256.w[1] || R256.w[0]))) {
- // f2* > 1/2 and the result may be exact
- // Calculate f2* - 1/2
- tmp64A = highf2star.w[0] - __bid_one_half128[ind];
- tmp64B = highf2star.w[1];
- if (tmp64A > highf2star.w[0])
- tmp64B--;
- if (tmp64B || tmp64A
- || R256.w[1] > __bid_ten2mk128trunc[ind].w[1]
- || (R256.w[1] == __bid_ten2mk128trunc[ind].w[1]
- && R256.w[0] > __bid_ten2mk128trunc[ind].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // this rounding is applied to C2 only!
- // x_sign != y_sign
- is_inexact_gt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // this rounding is applied to C2 only!
- // x_sign != y_sign
- is_inexact_lt_midpoint = 1;
- }
- } else { // if 22 <= ind <= 33
- if (highf2star.w[1] > __bid_one_half128[ind]
- || (highf2star.w[1] == __bid_one_half128[ind]
- && (highf2star.w[0] || R256.w[1]
- || R256.w[0]))) {
- // f2* > 1/2 and the result may be exact
- // Calculate f2* - 1/2
- // tmp64A = highf2star.w[0];
- tmp64B = highf2star.w[1] - __bid_one_half128[ind];
- if (tmp64B || highf2star.w[0]
- || R256.w[1] > __bid_ten2mk128trunc[ind].w[1]
- || (R256.w[1] == __bid_ten2mk128trunc[ind].w[1]
- && R256.w[0] > __bid_ten2mk128trunc[ind].w[0])) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // this rounding is applied to C2 only!
- // x_sign != y_sign
- is_inexact_gt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // this rounding is applied to C2 only!
- // x_sign != y_sign
- is_inexact_lt_midpoint = 1;
- }
- }
- // check for midpoints after determining inexactness
- if ((R256.w[1] || R256.w[0]) && (highf2star.w[1] == 0)
- && (highf2star.w[0] == 0)
- && (R256.w[1] < __bid_ten2mk128trunc[ind].w[1]
- || (R256.w[1] == __bid_ten2mk128trunc[ind].w[1]
- && R256.w[0] <= __bid_ten2mk128trunc[ind].w[0]))) {
- // the result is a midpoint
- if ((tmp64 + R256.w[2]) & 0x01) { // MP in [EVEN, ODD]
- // if floor(C2*) is odd C = floor(C2*) - 1; the result may be 0
- R256.w[2]--;
- if (R256.w[2] == 0xffffffffffffffffull)
- R256.w[3]--;
- // this rounding is applied to C2 only!
- // x_sign != y_sign
- is_midpoint_lt_even = 1;
- is_inexact_lt_midpoint = 0;
- is_inexact_gt_midpoint = 0;
- } else {
- // else MP in [ODD, EVEN]
- // this rounding is applied to C2 only!
- // x_sign != y_sign
- is_midpoint_gt_even = 1;
- is_inexact_lt_midpoint = 0;
- is_inexact_gt_midpoint = 0;
- }
- }
- } else { // if (ind == -1) only when x1 = 0
- R256.w[2] = C2_lo;
- R256.w[3] = C2_hi;
- is_midpoint_lt_even = 0;
- is_midpoint_gt_even = 0;
- is_inexact_lt_midpoint = 0;
- is_inexact_gt_midpoint = 0;
- }
- // and now subtract C1 * 10^(e1-e2-x1) - (C2 * 10^(-x1))rnd,P34
- // because x_sign != y_sign this last operation is exact
- C1.w[0] = C1.w[0] - R256.w[2];
- C1.w[1] = C1.w[1] - R256.w[3];
- if (C1.w[0] > tmp64)
- C1.w[1]--; // borrow
- if (C1.w[1] >= 0x8000000000000000ull) { // negative coefficient!
- C1.w[0] = ~C1.w[0];
- C1.w[0]++;
- C1.w[1] = ~C1.w[1];
- if (C1.w[0] == 0x0)
- C1.w[1]++;
- tmp_sign = y_sign; // the result will have the sign of y
- } else {
- tmp_sign = x_sign;
- }
- // the difference has exactly P34 digits
- x_sign = tmp_sign;
- if (x1 >= 1)
- y_exp = y_exp + ((UINT64) x1 << 49);
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- // general correction from RN to RA, RM, RP, RZ; result uses y_exp
- if (rnd_mode != ROUNDING_TO_NEAREST) {
- if ((!x_sign &&
- ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint) ||
- ((rnd_mode == ROUNDING_TIES_AWAY || rnd_mode == ROUNDING_UP) &&
- is_midpoint_gt_even))) ||
- (x_sign &&
- ((rnd_mode == ROUNDING_DOWN && is_inexact_lt_midpoint) ||
- ((rnd_mode == ROUNDING_TIES_AWAY || rnd_mode == ROUNDING_DOWN)
- && is_midpoint_gt_even)))) {
- // C1 = C1 + 1
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- y_exp = y_exp + EXP_P1;
- }
- } else if ((is_midpoint_lt_even || is_inexact_gt_midpoint) &&
- ((x_sign &&
- (rnd_mode == ROUNDING_UP || rnd_mode == ROUNDING_TO_ZERO)) ||
- (!x_sign &&
- (rnd_mode == ROUNDING_DOWN || rnd_mode == ROUNDING_TO_ZERO)))) {
- // C1 = C1 - 1
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi--;
- // check if we crossed into the lower decade
- if (C1_hi == 0x0000314dc6448d93ull &&
- C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
- C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
- C1_lo = 0x378d8e63ffffffffull;
- y_exp = y_exp - EXP_P1;
- // no underflow, because delta + q2 >= P34 + 1
- }
- } else {
- ; // exact, the result is already correct
- }
- }
- // assemble the result
- res.w[1] = x_sign | y_exp | C1_hi;
- res.w[0] = C1_lo;
- }
- } // end delta = P34
- } else { // if (|delta| <= P34 - 1)
- if (delta >= 0) { // if (0 <= delta <= P34 - 1)
- if (delta <= P34 - 1 - q2) {
- // calculate C' directly; the result is exact
- // in this case 1<=q1<=P34-1, 1<=q2<=P34-1 and 0 <= e1-e2 <= P34-2
- // The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
- // exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
- // but their product fits with certainty in 128 bits (actually in 113)
- scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
-
- if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
- __mul_128x64_to_128 (C1, C1_lo, __bid_ten2k128[scale - 20]);
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- } else if (scale >= 1) {
- // if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
- if (q1 <= 19) { // C1 fits in 64 bits
- __mul_64x64_to_128MACH (C1, C1_lo, __bid_ten2k64[scale]);
- } else { // q1 >= 20
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- __mul_128x64_to_128 (C1, __bid_ten2k64[scale], C1);
- }
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- } else { // if (scale == 0) C1 is unchanged
- C1.w[0] = C1_lo; // C1.w[1] = C1_hi;
- }
- // now add C2
- if (x_sign == y_sign) {
- // the result cannot overflow
- C1_lo = C1_lo + C2_lo;
- C1_hi = C1_hi + C2_hi;
- if (C1_lo < C1.w[0])
- C1_hi++;
- } else { // if x_sign != y_sign
- C1_lo = C1_lo - C2_lo;
- C1_hi = C1_hi - C2_hi;
- if (C1_lo > C1.w[0])
- C1_hi--;
- // the result can be zero, but it cannot overflow
- if (C1_lo == 0 && C1_hi == 0) {
- // assemble the result
- if (x_exp < y_exp)
- res.w[1] = x_exp;
- else
- res.w[1] = y_exp;
- res.w[0] = 0;
- if (rnd_mode == ROUNDING_DOWN) {
- res.w[1] |= 0x8000000000000000ull;
- }
- BID_RETURN (res);
- }
- if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
- C1_lo = ~C1_lo;
- C1_lo++;
- C1_hi = ~C1_hi;
- if (C1_lo == 0x0)
- C1_hi++;
- x_sign = y_sign; // the result will have the sign of y
- }
- }
- // assemble the result
- res.w[1] = x_sign | y_exp | C1_hi;
- res.w[0] = C1_lo;
- } else if (delta == P34 - q2) {
- // calculate C' directly; the result may be inexact if it requires
- // P34+1 decimal digits; in this case the 'cutoff' point for addition
- // is at the position of the lsb of C2, so 0 <= e1-e2 <= P34-1
- // The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
- // exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
- // but their product fits with certainty in 128 bits (actually in 113)
- scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
- if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
- __mul_128x64_to_128 (C1, C1_lo, __bid_ten2k128[scale - 20]);
- } else if (scale >= 1) {
- // if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
- if (q1 <= 19) { // C1 fits in 64 bits
- __mul_64x64_to_128MACH (C1, C1_lo, __bid_ten2k64[scale]);
- } else { // q1 >= 20
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- __mul_128x64_to_128 (C1, __bid_ten2k64[scale], C1);
- }
- } else { // if (scale == 0) C1 is unchanged
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo; // only the low part is necessary
- }
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- // now add C2
- if (x_sign == y_sign) {
- // the result can overflow!
- C1_lo = C1_lo + C2_lo;
- C1_hi = C1_hi + C2_hi;
- if (C1_lo < C1.w[0])
- C1_hi++;
- // test for overflow, possible only when C1 >= 10^34
- if (C1_hi > 0x0001ed09bead87c0ull ||
- (C1_hi == 0x0001ed09bead87c0ull &&
- C1_lo >= 0x378d8e6400000000ull)) { // C1 >= 10^34
- // in this case q = P34 + 1 and x = q - P34 = 1, so multiply
- // C'' = C'+ 5 = C1 + 5 by k1 ~ 10^(-1) calculated for P34 + 1
- // decimal digits
- // Calculate C'' = C' + 1/2 * 10^x
- if (C1_lo >= 0xfffffffffffffffbull) { // low half add has carry
- C1_lo = C1_lo + 5;
- C1_hi = C1_hi + 1;
- } else {
- C1_lo = C1_lo + 5;
- }
- // the approximation of 10^(-1) was rounded up to 118 bits
- // 10^(-1) =~ 33333333333333333333333333333400 * 2^-129
- // 10^(-1) =~ 19999999999999999999999999999a00 * 2^-128
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo; // C''
- __bid_ten2m1.w[1] = 0x1999999999999999ull;
- __bid_ten2m1.w[0] = 0x9999999999999a00ull;
- __mul_128x128_to_256 (P256, C1, __bid_ten2m1); // P256 = C*, f*
- // C* is actually floor(C*) in this case
- // the top Ex = 128 bits of 10^(-1) are
- // T* = 0x00199999999999999999999999999999
- // if (0 < f* < 10^(-x)) then
- // 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^(e2+x)
- if ((P256.w[1] || P256.w[0])
- && (P256.w[1] < 0x1999999999999999ull
- || (P256.w[1] == 0x1999999999999999ull
- && P256.w[0] <= 0x9999999999999999ull))) {
- // the result is a midpoint
- if (P256.w[2] & 0x01) {
- is_midpoint_gt_even = 1;
- // if floor(C*) is odd C = floor(C*) - 1; the result is not 0
- P256.w[2]--;
- if (P256.w[2] == 0xffffffffffffffffull)
- P256.w[3]--;
- } else {
- is_midpoint_lt_even = 1;
- }
- }
- // n = Cstar * 10^(e2+1)
- y_exp = y_exp + EXP_P1;
- // C* != 10^P because C* has P34 digits
- // check for overflow
- if (y_exp == EXP_MAX_P1
- && (rnd_mode == ROUNDING_TO_NEAREST
- || rnd_mode == ROUNDING_TIES_AWAY)) {
- // overflow for RN
- res.w[1] = x_sign | 0x7800000000000000ull; // +/-inf
- res.w[0] = 0x0ull;
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // set the overflow flag
- *pfpsf |= OVERFLOW_EXCEPTION;
- BID_RETURN (res);
- }
- // if (0 < f* - 1/2 < 10^(-x)) then
- // the result of the addition is exact
- // else
- // the result of the addition is inexact
- if (P256.w[1] > 0x8000000000000000ull ||
- (P256.w[1] == 0x8000000000000000ull &&
- P256.w[0] > 0x0ull)) { // the result may be exact
- tmp64 = P256.w[1] - 0x8000000000000000ull; // f* - 1/2
- if ((tmp64 > 0x1999999999999999ull
- || (tmp64 == 0x1999999999999999ull
- && P256.w[0] >= 0x9999999999999999ull))) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- is_inexact = 1;
- } // else the result is exact
- } else { // the result is inexact
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- is_inexact = 1;
- }
- C1_hi = P256.w[3];
- C1_lo = P256.w[2];
- if (!is_midpoint_gt_even && !is_midpoint_lt_even) {
- is_inexact_lt_midpoint = is_inexact
- && (P256.w[1] & 0x8000000000000000ull);
- is_inexact_gt_midpoint = is_inexact
- && !(P256.w[1] & 0x8000000000000000ull);
- }
- // general correction from RN to RA, RM, RP, RZ;
- // result uses y_exp
- if (rnd_mode != ROUNDING_TO_NEAREST) {
- if ((!x_sign &&
- ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint) ||
- ((rnd_mode == ROUNDING_TIES_AWAY || rnd_mode == ROUNDING_UP)
- && is_midpoint_gt_even))) ||
- (x_sign &&
- ((rnd_mode == ROUNDING_DOWN && is_inexact_lt_midpoint) ||
- ((rnd_mode == ROUNDING_TIES_AWAY ||
- rnd_mode == ROUNDING_DOWN) && is_midpoint_gt_even)))) {
- // C1 = C1 + 1
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- y_exp = y_exp + EXP_P1;
- }
- } else if ((is_midpoint_lt_even || is_inexact_gt_midpoint) &&
- ((x_sign && (rnd_mode == ROUNDING_UP ||
- rnd_mode == ROUNDING_TO_ZERO)) || (!x_sign &&
- (rnd_mode == ROUNDING_DOWN ||
- rnd_mode == ROUNDING_TO_ZERO)))) {
- // C1 = C1 - 1
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi--;
- // check if we crossed into the lower decade
- if (C1_hi == 0x0000314dc6448d93ull &&
- C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
- C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
- C1_lo = 0x378d8e63ffffffffull;
- y_exp = y_exp - EXP_P1;
- // no underflow, because delta + q2 >= P34 + 1
- }
- } else {
- ; // exact, the result is already correct
- }
- // in all cases check for overflow (RN and RA solved already)
- if (y_exp == EXP_MAX_P1) { // overflow
- if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
- (rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- } else { // RM and res > 0, RP and res < 0, or RZ
- C1_hi = 0x5fffed09bead87c0ull;
- C1_lo = 0x378d8e63ffffffffull;
- }
- y_exp = 0; // x_sign is preserved
- // set the inexact flag (in case the exact addition was exact)
- *pfpsf |= INEXACT_EXCEPTION;
- // set the overflow flag
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- } // else if (C1 < 10^34) then C1 is the coeff.; the result is exact
- } else { // if x_sign != y_sign the result is exact
- C1_lo = C1_lo - C2_lo;
- C1_hi = C1_hi - C2_hi;
- if (C1_lo > C1.w[0])
- C1_hi--;
- // the result can be zero, but it cannot overflow
- if (C1_lo == 0 && C1_hi == 0) {
- // assemble the result
- if (x_exp < y_exp)
- res.w[1] = x_exp;
- else
- res.w[1] = y_exp;
- res.w[0] = 0;
- if (rnd_mode == ROUNDING_DOWN) {
- res.w[1] |= 0x8000000000000000ull;
- }
- BID_RETURN (res);
- }
- if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
- C1_lo = ~C1_lo;
- C1_lo++;
- C1_hi = ~C1_hi;
- if (C1_lo == 0x0)
- C1_hi++;
- x_sign = y_sign; // the result will have the sign of y
- }
- }
- // assemble the result
- res.w[1] = x_sign | y_exp | C1_hi;
- res.w[0] = C1_lo;
- } else { // if (delta >= P34 + 1 - q2)
- // instead of C' = (C1 * 10^(e1-e2) + C2)rnd,P34
- // calculate C' = C1 * 10^(e1-e2-x1) + (C2 * 10^(-x1))rnd,P34
- // where x1 = q1 + e1 - e2 - P34, 1 <= x1 <= P34 - 1
- // In most cases C' will have P34 digits, and n = C' * 10^(e2+x1)
- // If the result has P34+1 digits, redo the steps above with x1+1
- // If the result has P34-1 digits or less, redo the steps above with
- // x1-1 but only if initially x1 >= 1
- // NOTE: these two steps can be improved, e.g we could guess if
- // P34+1 or P34-1 digits will be obtained by adding/subtracting just
- // the top 64 bits of the two operands
- // The result cannot be zero, but it can overflow
- x1 = delta + q2 - P34; // 1 <= x1 <= P34-1
- roundC2:
- // Calculate C1 * 10^(e1-e2-x1) where 0 <= e1-e2-x1 <= P34 - 1
- // scale = (int)(e1 >> 49) - (int)(e2 >> 49) - x1; 0 <= scale <= P34-1
- scale = delta - q1 + q2 - x1; // scale = e1 - e2 - x1 = P34 - q1
- // either C1 or 10^(e1-e2-x1) may not fit is 64 bits,
- // but their product fits with certainty in 128 bits (actually in 113)
- if (scale >= 20) { //10^(e1-e2-x1) doesn't fit in 64 bits, but C1 does
- __mul_128x64_to_128 (C1, C1_lo, __bid_ten2k128[scale - 20]);
- } else if (scale >= 1) {
- // if 1 <= scale <= 19 then 10^(e1-e2-x1) fits in 64 bits
- if (q1 <= 19) { // C1 fits in 64 bits
- __mul_64x64_to_128MACH (C1, C1_lo, __bid_ten2k64[scale]);
- } else { // q1 >= 20
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- __mul_128x64_to_128 (C1, __bid_ten2k64[scale], C1);
- }
- } else { // if (scale == 0) C1 is unchanged
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- }
- tmp64 = C1.w[0]; // C1.w[1], C1.w[0] contains C1 * 10^(e1-e2-x1)
-
- // now round C2 to q2-x1 decimal digits, where 1<=x1<=q2-1<=P34-1
- // (but if we got here a second time after x1 = x1 - 1, then
- // x1 >= 0; note that for x1 = 0 C2 is unchanged)
- // C2' = C2 + 1/2 * 10^x1 = C2 + 5 * 10^(x1-1)
- ind = x1 - 1; // 0 <= ind <= q2-2<=P34-2=32; but note that if x1 = 0
- // during a second pass, then ind = -1
- if (ind >= 0) { // if (x1 >= 1)
- C2.w[0] = C2_lo;
- C2.w[1] = C2_hi;
- if (ind <= 18) {
- C2.w[0] = C2.w[0] + __bid_midpoint64[ind];
- if (C2.w[0] < C2_lo)
- C2.w[1]++;
- } else { // 19 <= ind <= 32
- C2.w[0] = C2.w[0] + __bid_midpoint128[ind - 19].w[0];
- C2.w[1] = C2.w[1] + __bid_midpoint128[ind - 19].w[1];
- if (C2.w[0] < C2_lo)
- C2.w[1]++;
- }
- // the approximation of 10^(-x1) was rounded up to 118 bits
- __mul_128x128_to_256 (R256, C2, __bid_ten2mk128[ind]); // R256 = C2*, f2*
- // calculate C2* and f2*
- // C2* is actually floor(C2*) in this case
- // C2* and f2* need shifting and masking, as shown by
- // __bid_shiftright128[] and __bid_maskhigh128[]
- // the top Ex bits of 10^(-x1) are T* = __bid_ten2mk128trunc[ind], e.g.
- // if x1=1, T*=__bid_ten2mk128trunc[0]=0x19999999999999999999999999999999
- // if (0 < f2* < 10^(-x1)) then
- // if floor(C1+C2*) is even then C2* = floor(C2*) - logical right
- // shift; C2* has p decimal digits, correct by Prop. 1)
- // else if floor(C1+C2*) is odd C2* = floor(C2*)-1 (logical right
- // shift; C2* has p decimal digits, correct by Pr. 1)
- // else
- // C2* = floor(C2*) (logical right shift; C has p decimal digits,
- // correct by Property 1)
- // n = C2* * 10^(e2+x1)
-
- if (ind <= 2) {
- highf2star.w[1] = 0x0;
- highf2star.w[0] = 0x0; // low f2* ok
- } else if (ind <= 21) {
- highf2star.w[1] = 0x0;
- highf2star.w[0] = R256.w[2] & __bid_maskhigh128[ind]; // low f2* ok
- } else {
- highf2star.w[1] = R256.w[3] & __bid_maskhigh128[ind];
- highf2star.w[0] = R256.w[2]; // low f2* is ok
- }
- // shift right C2* by Ex-128 = __bid_shiftright128[ind]
- if (ind >= 3) {
- shift = __bid_shiftright128[ind];
- if (shift < 64) { // 3 <= shift <= 63
- R256.w[2] =
- (R256.w[2] >> shift) | (R256.w[3] << (64 - shift));
- R256.w[3] = (R256.w[3] >> shift);
- } else { // 66 <= shift <= 102
- R256.w[2] = (R256.w[3] >> (shift - 64));
- R256.w[3] = 0x0ULL;
- }
- }
- if (second_pass) {
- is_inexact_lt_midpoint = 0;
- is_inexact_gt_midpoint = 0;
- is_midpoint_lt_even = 0;
- is_midpoint_gt_even = 0;
- }
- // determine inexactness of the rounding of C2* (this may be
- // followed by a second rounding only if we get P34+1
- // decimal digits)
- // if (0 < f2* - 1/2 < 10^(-x1)) then
- // the result is exact
- // else (if f2* - 1/2 > T* then)
- // the result of is inexact
- if (ind <= 2) {
- if (R256.w[1] > 0x8000000000000000ull ||
- (R256.w[1] == 0x8000000000000000ull && R256.w[0] > 0x0ull)) {
- // f2* > 1/2 and the result may be exact
- tmp64A = R256.w[1] - 0x8000000000000000ull; // f* - 1/2
- if ((tmp64A > __bid_ten2mk128trunc[ind].w[1]
- || (tmp64A == __bid_ten2mk128trunc[ind].w[1]
- && R256.w[0] >= __bid_ten2mk128trunc[ind].w[0]))) {
- // set the inexact flag
- // *pfpsf |= INEXACT_EXCEPTION;
- tmp_inexact = 1; // may be set again during a second pass
- // this rounding is applied to C2 only!
- if (x_sign == y_sign)
- is_inexact_lt_midpoint = 1;
- else // if (x_sign != y_sign)
- is_inexact_gt_midpoint = 1;
- } // else the result is exact
- // rounding down, unless a midpoint in [ODD, EVEN]
- } else { // the result is inexact; f2* <= 1/2
- // set the inexact flag
- // *pfpsf |= INEXACT_EXCEPTION;
- tmp_inexact = 1; // just in case we will round a second time
- // rounding up, unless a midpoint in [EVEN, ODD]
- // this rounding is applied to C2 only!
- if (x_sign == y_sign)
- is_inexact_gt_midpoint = 1;
- else // if (x_sign != y_sign)
- is_inexact_lt_midpoint = 1;
- }
- } else if (ind <= 21) { // if 3 <= ind <= 21
- if (highf2star.w[1] > 0x0 || (highf2star.w[1] == 0x0
- && highf2star.w[0] > __bid_one_half128[ind])
- || ((highf2star.w[1] == 0x0
- && highf2star.w[0] == __bid_one_half128[ind])
- && (R256.w[1] || R256.w[0]))) {
- // f2* > 1/2 and the result may be exact
- // Calculate f2* - 1/2
- tmp64A = highf2star.w[0] - __bid_one_half128[ind];
- tmp64B = highf2star.w[1];
- if (tmp64A > highf2star.w[0])
- tmp64B--;
- if (tmp64B || tmp64A
- || R256.w[1] > __bid_ten2mk128trunc[ind].w[1]
- || (R256.w[1] == __bid_ten2mk128trunc[ind].w[1]
- && R256.w[0] > __bid_ten2mk128trunc[ind].w[0])) {
- // set the inexact flag
- // *pfpsf |= INEXACT_EXCEPTION;
- tmp_inexact = 1; // may be set again during a second pass
- // this rounding is applied to C2 only!
- if (x_sign == y_sign)
- is_inexact_lt_midpoint = 1;
- else // if (x_sign != y_sign)
- is_inexact_gt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- // set the inexact flag
- // *pfpsf |= INEXACT_EXCEPTION;
- tmp_inexact = 1; // may be set again during a second pass
- // rounding up, unless a midpoint in [EVEN, ODD]
- // this rounding is applied to C2 only!
- if (x_sign == y_sign)
- is_inexact_gt_midpoint = 1;
- else // if (x_sign != y_sign)
- is_inexact_lt_midpoint = 1;
- }
- } else { // if 22 <= ind <= 33
- if (highf2star.w[1] > __bid_one_half128[ind]
- || (highf2star.w[1] == __bid_one_half128[ind]
- && (highf2star.w[0] || R256.w[1]
- || R256.w[0]))) {
- // f2* > 1/2 and the result may be exact
- // Calculate f2* - 1/2
- // tmp64A = highf2star.w[0];
- tmp64B = highf2star.w[1] - __bid_one_half128[ind];
- if (tmp64B || highf2star.w[0]
- || R256.w[1] > __bid_ten2mk128trunc[ind].w[1]
- || (R256.w[1] == __bid_ten2mk128trunc[ind].w[1]
- && R256.w[0] > __bid_ten2mk128trunc[ind].w[0])) {
- // set the inexact flag
- // *pfpsf |= INEXACT_EXCEPTION;
- tmp_inexact = 1; // may be set again during a second pass
- // this rounding is applied to C2 only!
- if (x_sign == y_sign)
- is_inexact_lt_midpoint = 1;
- else // if (x_sign != y_sign)
- is_inexact_gt_midpoint = 1;
- } // else the result is exact
- } else { // the result is inexact; f2* <= 1/2
- // set the inexact flag
- // *pfpsf |= INEXACT_EXCEPTION;
- tmp_inexact = 1; // may be set again during a second pass
- // rounding up, unless a midpoint in [EVEN, ODD]
- // this rounding is applied to C2 only!
- if (x_sign == y_sign)
- is_inexact_gt_midpoint = 1;
- else // if (x_sign != y_sign)
- is_inexact_lt_midpoint = 1;
- }
- }
- // check for midpoints
- if ((R256.w[1] || R256.w[0]) && (highf2star.w[1] == 0)
- && (highf2star.w[0] == 0)
- && (R256.w[1] < __bid_ten2mk128trunc[ind].w[1]
- || (R256.w[1] == __bid_ten2mk128trunc[ind].w[1]
- && R256.w[0] <= __bid_ten2mk128trunc[ind].w[0]))) {
- // the result is a midpoint
- if ((tmp64 + R256.w[2]) & 0x01) { // MP in [EVEN, ODD]
- // if floor(C2*) is odd C = floor(C2*) - 1; the result may be 0
- R256.w[2]--;
- if (R256.w[2] == 0xffffffffffffffffull)
- R256.w[3]--;
- // this rounding is applied to C2 only!
- if (x_sign == y_sign)
- is_midpoint_gt_even = 1;
- else // if (x_sign != y_sign)
- is_midpoint_lt_even = 1;
- is_inexact_lt_midpoint = 0;
- is_inexact_gt_midpoint = 0;
- } else {
- // else MP in [ODD, EVEN]
- // this rounding is applied to C2 only!
- if (x_sign == y_sign)
- is_midpoint_lt_even = 1;
- else // if (x_sign != y_sign)
- is_midpoint_gt_even = 1;
- is_inexact_lt_midpoint = 0;
- is_inexact_gt_midpoint = 0;
- }
- }
- // end if (ind >= 0)
- } else { // if (ind == -1); only during a 2nd pass, and when x1 = 0
- R256.w[2] = C2_lo;
- R256.w[3] = C2_hi;
- tmp_inexact = 0;
- // to correct a possible setting to 1 from 1st pass
- if (second_pass) {
- is_midpoint_lt_even = 0;
- is_midpoint_gt_even = 0;
- is_inexact_lt_midpoint = 0;
- is_inexact_gt_midpoint = 0;
- }
- }
- // and now add/subtract C1 * 10^(e1-e2-x1) +/- (C2 * 10^(-x1))rnd,P34
- if (x_sign == y_sign) { // addition; could overflow
- // no second pass is possible this way (only for x_sign != y_sign)
- C1.w[0] = C1.w[0] + R256.w[2];
- C1.w[1] = C1.w[1] + R256.w[3];
- if (C1.w[0] < tmp64)
- C1.w[1]++; // carry
- // if the sum has P34+1 digits, i.e. C1>=10^34 redo the calculation
- // with x1=x1+1
- if (C1.w[1] > 0x0001ed09bead87c0ull ||
- (C1.w[1] == 0x0001ed09bead87c0ull &&
- C1.w[0] >= 0x378d8e6400000000ull)) { // C1 >= 10^34
- // chop off one more digit from the sum, but make sure there is
- // no double-rounding error (see table - double rounding logic)
- // now round C1 from P34+1 to P34 decimal digits
- // C1' = C1 + 1/2 * 10 = C1 + 5
- if (C1.w[0] >= 0xfffffffffffffffbull) { // low half add has carry
- C1.w[0] = C1.w[0] + 5;
- C1.w[1] = C1.w[1] + 1;
- } else {
- C1.w[0] = C1.w[0] + 5;
- }
- // the approximation of 10^(-1) was rounded up to 118 bits
- __mul_128x128_to_256 (Q256, C1, __bid_ten2mk128[0]); // Q256 = C1*, f1*
- // C1* is actually floor(C1*) in this case
- // the top 128 bits of 10^(-1) are
- // T* = __bid_ten2mk128trunc[0]=0x19999999999999999999999999999999
- // if (0 < f1* < 10^(-1)) then
- // if floor(C1*) is even then C1* = floor(C1*) - logical right
- // shift; C1* has p decimal digits, correct by Prop. 1)
- // else if floor(C1*) is odd C1* = floor(C1*) - 1 (logical right
- // shift; C1* has p decimal digits, correct by Pr. 1)
- // else
- // C1* = floor(C1*) (logical right shift; C has p decimal digits
- // correct by Property 1)
- // n = C1* * 10^(e2+x1+1)
- if ((Q256.w[1] || Q256.w[0])
- && (Q256.w[1] < __bid_ten2mk128trunc[0].w[1]
- || (Q256.w[1] == __bid_ten2mk128trunc[0].w[1]
- && Q256.w[0] <= __bid_ten2mk128trunc[0].w[0]))) {
- // the result is a midpoint
- if (is_inexact_lt_midpoint) { // for the 1st rounding
- is_inexact_gt_midpoint = 1;
- is_inexact_lt_midpoint = 0;
- is_midpoint_gt_even = 0;
- is_midpoint_lt_even = 0;
- } else if (is_inexact_gt_midpoint) { // for the 1st rounding
- Q256.w[2]--;
- if (Q256.w[2] == 0xffffffffffffffffull)
- Q256.w[3]--;
- is_inexact_gt_midpoint = 0;
- is_inexact_lt_midpoint = 1;
- is_midpoint_gt_even = 0;
- is_midpoint_lt_even = 0;
- } else if (is_midpoint_gt_even) { // for the 1st rounding
- // Note: cannot have is_midpoint_lt_even
- is_inexact_gt_midpoint = 0;
- is_inexact_lt_midpoint = 1;
- is_midpoint_gt_even = 0;
- is_midpoint_lt_even = 0;
- } else { // the first rounding must have been exact
- if (Q256.w[2] & 0x01) { // MP in [EVEN, ODD]
- // the truncated result is correct
- Q256.w[2]--;
- if (Q256.w[2] == 0xffffffffffffffffull)
- Q256.w[3]--;
- is_inexact_gt_midpoint = 0;
- is_inexact_lt_midpoint = 0;
- is_midpoint_gt_even = 1;
- is_midpoint_lt_even = 0;
- } else { // MP in [ODD, EVEN]
- is_inexact_gt_midpoint = 0;
- is_inexact_lt_midpoint = 0;
- is_midpoint_gt_even = 0;
- is_midpoint_lt_even = 1;
- }
- }
- tmp_inexact = 1; // in all cases
- } else { // the result is not a midpoint
- // determine inexactness of the rounding of C1 (the sum C1+C2*)
- // if (0 < f1* - 1/2 < 10^(-1)) then
- // the result is exact
- // else (if f1* - 1/2 > T* then)
- // the result of is inexact
- // ind = 0
- if (Q256.w[1] > 0x8000000000000000ull
- || (Q256.w[1] == 0x8000000000000000ull
- && Q256.w[0] > 0x0ull)) {
- // f1* > 1/2 and the result may be exact
- Q256.w[1] = Q256.w[1] - 0x8000000000000000ull; // f1* - 1/2
- if ((Q256.w[1] > __bid_ten2mk128trunc[0].w[1]
- || (Q256.w[1] == __bid_ten2mk128trunc[0].w[1]
- && Q256.w[0] > __bid_ten2mk128trunc[0].w[0]))) {
- is_inexact_gt_midpoint = 0;
- is_inexact_lt_midpoint = 1;
- is_midpoint_gt_even = 0;
- is_midpoint_lt_even = 0;
- // set the inexact flag
- tmp_inexact = 1;
- // *pfpsf |= INEXACT_EXCEPTION;
- } else { // else the result is exact for the 2nd rounding
- if (tmp_inexact) { // if the previous rounding was inexact
- if (is_midpoint_lt_even) {
- is_inexact_gt_midpoint = 1;
- is_midpoint_lt_even = 0;
- } else if (is_midpoint_gt_even) {
- is_inexact_lt_midpoint = 1;
- is_midpoint_gt_even = 0;
- } else {
- ; // no change
- }
- }
- }
- // rounding down, unless a midpoint in [ODD, EVEN]
- } else { // the result is inexact; f1* <= 1/2
- is_inexact_gt_midpoint = 1;
- is_inexact_lt_midpoint = 0;
- is_midpoint_gt_even = 0;
- is_midpoint_lt_even = 0;
- // set the inexact flag
- tmp_inexact = 1;
- // *pfpsf |= INEXACT_EXCEPTION;
- }
- } // end 'the result is not a midpoint'
- // n = C1 * 10^(e2+x1)
- C1.w[1] = Q256.w[3];
- C1.w[0] = Q256.w[2];
- y_exp = y_exp + ((UINT64) (x1 + 1) << 49);
- } else { // C1 < 10^34
- // C1.w[1] and C1.w[0] already set
- // n = C1 * 10^(e2+x1)
- y_exp = y_exp + ((UINT64) x1 << 49);
- }
- // check for overflow
- if (y_exp == EXP_MAX_P1
- && (rnd_mode == ROUNDING_TO_NEAREST
- || rnd_mode == ROUNDING_TIES_AWAY)) {
- res.w[1] = 0x7800000000000000ull | x_sign; // +/-inf
- res.w[0] = 0x0ull;
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // set the overflow flag
- *pfpsf |= OVERFLOW_EXCEPTION;
- BID_RETURN (res);
- } // else no overflow
- } else { // if x_sign != y_sign the result of this subtract. is exact
- C1.w[0] = C1.w[0] - R256.w[2];
- C1.w[1] = C1.w[1] - R256.w[3];
- if (C1.w[0] > tmp64)
- C1.w[1]--; // borrow
- if (C1.w[1] >= 0x8000000000000000ull) { // negative coefficient!
- C1.w[0] = ~C1.w[0];
- C1.w[0]++;
- C1.w[1] = ~C1.w[1];
- if (C1.w[0] == 0x0)
- C1.w[1]++;
- tmp_sign = y_sign;
- // the result will have the sign of y if last rnd
- } else {
- tmp_sign = x_sign;
- }
- // if the difference has P34-1 digits or less, i.e. C1 < 10^33 then
- // redo the calculation with x1=x1-1;
- // redo the calculation also if C1 = 10^33 and
- // (is_inexact_gt_midpoint or is_midpoint_lt_even);
- // (the last part should have really been
- // (is_inexact_lt_midpoint or is_midpoint_gt_even) from
- // the rounding of C2, but the position flags have been reversed)
- // 10^33 = 0x0000314dc6448d93 0x38c15b0a00000000
- if ((C1.w[1] < 0x0000314dc6448d93ull ||
- (C1.w[1] == 0x0000314dc6448d93ull &&
- C1.w[0] < 0x38c15b0a00000000ull)) ||
- (C1.w[1] == 0x0000314dc6448d93ull &&
- C1.w[0] == 0x38c15b0a00000000ull &&
- (is_inexact_gt_midpoint || is_midpoint_lt_even))) { // C1=10^33
- x1 = x1 - 1; // x1 >= 0
- if (x1 >= 0) {
- // clear position flags and tmp_inexact
- is_midpoint_lt_even = 0;
- is_midpoint_gt_even = 0;
- is_inexact_lt_midpoint = 0;
- is_inexact_gt_midpoint = 0;
- tmp_inexact = 0;
- second_pass = 1;
- goto roundC2; // else result has less than P34 digits
- }
- }
- // if the coefficient of the result is 10^34 it means that this
- // must be the second pass, and we are done
- if (C1.w[1] == 0x0001ed09bead87c0ull &&
- C1.w[0] == 0x378d8e6400000000ull) { // if C1 = 10^34
- C1.w[1] = 0x0000314dc6448d93ull; // C1 = 10^33
- C1.w[0] = 0x38c15b0a00000000ull;
- y_exp = y_exp + ((UINT64) 1 << 49);
- }
- x_sign = tmp_sign;
- if (x1 >= 1)
- y_exp = y_exp + ((UINT64) x1 << 49);
- // x1 = -1 is possible at the end of a second pass when the
- // first pass started with x1 = 1
- }
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- // general correction from RN to RA, RM, RP, RZ; result uses y_exp
- if (rnd_mode != ROUNDING_TO_NEAREST) {
- if ((!x_sign
- && ((rnd_mode == ROUNDING_UP
- && is_inexact_lt_midpoint)
- || ((rnd_mode == ROUNDING_TIES_AWAY
- || rnd_mode == ROUNDING_UP)
- && is_midpoint_gt_even))) ||
- (x_sign
- && ((rnd_mode == ROUNDING_DOWN
- && is_inexact_lt_midpoint)
- || ((rnd_mode == ROUNDING_TIES_AWAY
- || rnd_mode == ROUNDING_DOWN)
- && is_midpoint_gt_even)))) {
- // C1 = C1 + 1
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- y_exp = y_exp + EXP_P1;
- }
- } else if ((is_midpoint_lt_even || is_inexact_gt_midpoint) &&
- ((x_sign &&
- (rnd_mode == ROUNDING_UP || rnd_mode == ROUNDING_TO_ZERO)) ||
- (!x_sign &&
- (rnd_mode == ROUNDING_DOWN || rnd_mode == ROUNDING_TO_ZERO)))) {
- // C1 = C1 - 1
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi--;
- // check if we crossed into the lower decade
- if (C1_hi == 0x0000314dc6448d93ull &&
- C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
- C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
- C1_lo = 0x378d8e63ffffffffull;
- y_exp = y_exp - EXP_P1;
- // no underflow, because delta + q2 >= P34 + 1
- }
- } else {
- ; // exact, the result is already correct
- }
- // in all cases check for overflow (RN and RA solved already)
- if (y_exp == EXP_MAX_P1) { // overflow
- if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
- (rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- } else { // RM and res > 0, RP and res < 0, or RZ
- C1_hi = 0x5fffed09bead87c0ull;
- C1_lo = 0x378d8e63ffffffffull;
- }
- y_exp = 0; // x_sign is preserved
- // set the inexact flag (in case the exact addition was exact)
- *pfpsf |= INEXACT_EXCEPTION;
- // set the overflow flag
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- // assemble the result
- res.w[1] = x_sign | y_exp | C1_hi;
- res.w[0] = C1_lo;
- if (tmp_inexact)
- *pfpsf |= INEXACT_EXCEPTION;
- }
- } else { // if (-P34 + 1 <= delta <= -1) <=> 1 <= -delta <= P34 - 1
- // NOTE: the following, up to "} else { // if x_sign != y_sign
- // the result is exact" is identical to "else if (delta == P34 - q2) {"
- // from above; also, the code is not symmetric: a+b and b+a may take
- // different paths (need to unify eventually!)
- // calculate C' = C2 + C1 * 10^(e1-e2) directly; the result may be
- // inexact if it requires P34 + 1 decimal digits; in either case the
- // 'cutoff' point for addition is at the position of the lsb of C2
- // The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
- // exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
- // but their product fits with certainty in 128 bits (actually in 113)
- // Note that 0 <= e1 - e2 <= P34 - 2
- // -P34 + 1 <= delta <= -1 <=> -P34 + 1 <= delta <= -1 <=>
- // -P34 + 1 <= q1 + e1 - q2 - e2 <= -1 <=>
- // q2 - q1 - P34 + 1 <= e1 - e2 <= q2 - q1 - 1 <=>
- // 1 - P34 - P34 + 1 <= e1-e2 <= P34 - 1 - 1 => 0 <= e1-e2 <= P34 - 2
- scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
- if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
- __mul_128x64_to_128 (C1, C1_lo, __bid_ten2k128[scale - 20]);
- } else if (scale >= 1) {
- // if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
- if (q1 <= 19) { // C1 fits in 64 bits
- __mul_64x64_to_128MACH (C1, C1_lo, __bid_ten2k64[scale]);
- } else { // q1 >= 20
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo;
- __mul_128x64_to_128 (C1, __bid_ten2k64[scale], C1);
- }
- } else { // if (scale == 0) C1 is unchanged
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo; // only the low part is necessary
- }
- C1_hi = C1.w[1];
- C1_lo = C1.w[0];
- // now add C2
- if (x_sign == y_sign) {
- // the result can overflow!
- C1_lo = C1_lo + C2_lo;
- C1_hi = C1_hi + C2_hi;
- if (C1_lo < C1.w[0])
- C1_hi++;
- // test for overflow, possible only when C1 >= 10^34
- if (C1_hi > 0x0001ed09bead87c0ull ||
- (C1_hi == 0x0001ed09bead87c0ull &&
- C1_lo >= 0x378d8e6400000000ull)) { // C1 >= 10^34
- // in this case q = P34 + 1 and x = q - P34 = 1, so multiply
- // C'' = C'+ 5 = C1 + 5 by k1 ~ 10^(-1) calculated for P34 + 1
- // decimal digits
- // Calculate C'' = C' + 1/2 * 10^x
- if (C1_lo >= 0xfffffffffffffffbull) { // low half add has carry
- C1_lo = C1_lo + 5;
- C1_hi = C1_hi + 1;
- } else {
- C1_lo = C1_lo + 5;
- }
- // the approximation of 10^(-1) was rounded up to 118 bits
- // 10^(-1) =~ 33333333333333333333333333333400 * 2^-129
- // 10^(-1) =~ 19999999999999999999999999999a00 * 2^-128
- C1.w[1] = C1_hi;
- C1.w[0] = C1_lo; // C''
- __bid_ten2m1.w[1] = 0x1999999999999999ull;
- __bid_ten2m1.w[0] = 0x9999999999999a00ull;
- __mul_128x128_to_256 (P256, C1, __bid_ten2m1); // P256 = C*, f*
- // C* is actually floor(C*) in this case
- // the top Ex = 128 bits of 10^(-1) are
- // T* = 0x00199999999999999999999999999999
- // if (0 < f* < 10^(-x)) then
- // 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^(e2+x)
- if ((P256.w[1] || P256.w[0])
- && (P256.w[1] < 0x1999999999999999ull
- || (P256.w[1] == 0x1999999999999999ull
- && P256.w[0] <= 0x9999999999999999ull))) {
- // the result is a midpoint
- if (P256.w[2] & 0x01) {
- is_midpoint_gt_even = 1;
- // if floor(C*) is odd C = floor(C*) - 1; the result is not 0
- P256.w[2]--;
- if (P256.w[2] == 0xffffffffffffffffull)
- P256.w[3]--;
- } else {
- is_midpoint_lt_even = 1;
- }
- }
- // n = Cstar * 10^(e2+1)
- y_exp = y_exp + EXP_P1;
- // C* != 10^P34 because C* has P34 digits
- // check for overflow
- if (y_exp == EXP_MAX_P1
- && (rnd_mode == ROUNDING_TO_NEAREST
- || rnd_mode == ROUNDING_TIES_AWAY)) {
- // overflow for RN
- res.w[1] = x_sign | 0x7800000000000000ull; // +/-inf
- res.w[0] = 0x0ull;
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- // set the overflow flag
- *pfpsf |= OVERFLOW_EXCEPTION;
- BID_RETURN (res);
- }
- // if (0 < f* - 1/2 < 10^(-x)) then
- // the result of the addition is exact
- // else
- // the result of the addition is inexact
- if (P256.w[1] > 0x8000000000000000ull ||
- (P256.w[1] == 0x8000000000000000ull &&
- P256.w[0] > 0x0ull)) { // the result may be exact
- tmp64 = P256.w[1] - 0x8000000000000000ull; // f* - 1/2
- if ((tmp64 > 0x1999999999999999ull
- || (tmp64 == 0x1999999999999999ull
- && P256.w[0] >= 0x9999999999999999ull))) {
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- is_inexact = 1;
- } // else the result is exact
- } else { // the result is inexact
- // set the inexact flag
- *pfpsf |= INEXACT_EXCEPTION;
- is_inexact = 1;
- }
- C1_hi = P256.w[3];
- C1_lo = P256.w[2];
- if (!is_midpoint_gt_even && !is_midpoint_lt_even) {
- is_inexact_lt_midpoint = is_inexact
- && (P256.w[1] & 0x8000000000000000ull);
- is_inexact_gt_midpoint = is_inexact
- && !(P256.w[1] & 0x8000000000000000ull);
- }
- // general correction from RN to RA, RM, RP, RZ; result uses y_exp
- if (rnd_mode != ROUNDING_TO_NEAREST) {
- if ((!x_sign &&
- ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint) ||
- ((rnd_mode == ROUNDING_TIES_AWAY || rnd_mode == ROUNDING_UP)
- && is_midpoint_gt_even))) ||
- (x_sign &&
- ((rnd_mode == ROUNDING_DOWN && is_inexact_lt_midpoint) ||
- ((rnd_mode == ROUNDING_TIES_AWAY || rnd_mode == ROUNDING_DOWN)
- && is_midpoint_gt_even)))) {
- // C1 = C1 + 1
- C1_lo = C1_lo + 1;
- if (C1_lo == 0) { // rounding overflow in the low 64 bits
- C1_hi = C1_hi + 1;
- }
- if (C1_hi == 0x0001ed09bead87c0ull
- && C1_lo == 0x378d8e6400000000ull) {
- // C1 = 10^34 => rounding overflow
- C1_hi = 0x0000314dc6448d93ull;
- C1_lo = 0x38c15b0a00000000ull; // 10^33
- y_exp = y_exp + EXP_P1;
- }
- } else
- if ((is_midpoint_lt_even || is_inexact_gt_midpoint) &&
- ((x_sign &&
- (rnd_mode == ROUNDING_UP ||
- rnd_mode == ROUNDING_TO_ZERO)) ||
- (!x_sign &&
- (rnd_mode == ROUNDING_DOWN ||
- rnd_mode == ROUNDING_TO_ZERO)))) {
- // C1 = C1 - 1
- C1_lo = C1_lo - 1;
- if (C1_lo == 0xffffffffffffffffull)
- C1_hi--;
- // check if we crossed into the lower decade
- if (C1_hi == 0x0000314dc6448d93ull &&
- C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
- C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
- C1_lo = 0x378d8e63ffffffffull;
- y_exp = y_exp - EXP_P1;
- // no underflow, because delta + q2 >= P34 + 1
- }
- } else {
- ; // exact, the result is already correct
- }
- // in all cases check for overflow (RN and RA solved already)
- if (y_exp == EXP_MAX_P1) { // overflow
- if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
- (rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
- C1_hi = 0x7800000000000000ull; // +inf
- C1_lo = 0x0ull;
- } else { // RM and res > 0, RP and res < 0, or RZ
- C1_hi = 0x5fffed09bead87c0ull;
- C1_lo = 0x378d8e63ffffffffull;
- }
- y_exp = 0; // x_sign is preserved
- // set the inexact flag (in case the exact addition was exact)
- *pfpsf |= INEXACT_EXCEPTION;
- // set the overflow flag
- *pfpsf |= OVERFLOW_EXCEPTION;
- }
- }
- } // else if (C1 < 10^34) then C1 is the coeff.; the result is exact
- // assemble the result
- res.w[1] = x_sign | y_exp | C1_hi;
- res.w[0] = C1_lo;
- } else { // if x_sign != y_sign the result is exact
- C1_lo = C2_lo - C1_lo;
- C1_hi = C2_hi - C1_hi;
- if (C1_lo > C2_lo)
- C1_hi--;
- if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
- C1_lo = ~C1_lo;
- C1_lo++;
- C1_hi = ~C1_hi;
- if (C1_lo == 0x0)
- C1_hi++;
- x_sign = y_sign; // the result will have the sign of y
- }
- // the result can be zero, but it cannot overflow
- if (C1_lo == 0 && C1_hi == 0) {
- // assemble the result
- if (x_exp < y_exp)
- res.w[1] = x_exp;
- else
- res.w[1] = y_exp;
- res.w[0] = 0;
- if (rnd_mode == ROUNDING_DOWN) {
- res.w[1] |= 0x8000000000000000ull;
- }
- BID_RETURN (res);
- }
- // assemble the result
- res.w[1] = y_sign | y_exp | C1_hi;
- res.w[0] = C1_lo;
- }
+ // for RN the result is the operand with the larger magnitude,
+ // possibly scaled up by 10^(P34-q1)
+ // an overflow cannot occur in this case (rounding to nearest)
+ if (q1 < P34) { // scale C1 up by 10^(P34-q1)
+ // Note: because delta >= P34+1 it is certain that
+ // x_exp - ((UINT64)scale << 49) will stay above e_min
+ scale = P34 - q1;
+ if (q1 <= 19) { // C1 fits in 64 bits
+ // 1 <= q1 <= 19 => 15 <= scale <= 33
+ if (scale <= 19) { // 10^scale fits in 64 bits
+ __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
+ } else { // if 20 <= scale <= 33
+ // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
+ // (C1 * 10^(scale-19)) fits in 64 bits
+ C1_lo = C1_lo * ten2k64[scale - 19];
+ __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
+ }
+ } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
+ // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ // C1 = ten2k64[P34 - q1] * C1
+ __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
+ }
+ x_exp = x_exp - ((UINT64) scale << 49);
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ }
+ // some special cases arise: if delta = P34 + 1 and C1 = 10^(P34-1)
+ // (after scaling) and x_sign != y_sign and C2 > 5*10^(q2-1) =>
+ // subtract 1 ulp
+ // Note: do this only for rounding to nearest; for other rounding
+ // modes the correction will be applied next
+ if ((rnd_mode == ROUNDING_TO_NEAREST
+ || rnd_mode == ROUNDING_TIES_AWAY) && delta == (P34 + 1)
+ && C1_hi == 0x0000314dc6448d93ull
+ && C1_lo == 0x38c15b0a00000000ull && x_sign != y_sign
+ && ((q2 <= 19 && C2_lo > midpoint64[q2 - 1]) || (q2 >= 20
+ && (C2_hi >
+ midpoint128
+ [q2 -
+ 20].
+ w[1]
+ ||
+ (C2_hi
+ ==
+ midpoint128
+ [q2 -
+ 20].
+ w[1]
+ &&
+ C2_lo
+ >
+ midpoint128
+ [q2 -
+ 20].
+ w
+ [0])))))
+ {
+ // C1 = 10^34 - 1 and decrement x_exp by 1 (no underflow possible)
+ C1_hi = 0x0001ed09bead87c0ull;
+ C1_lo = 0x378d8e63ffffffffull;
+ x_exp = x_exp - EXP_P1;
+ }
+ if (rnd_mode != ROUNDING_TO_NEAREST) {
+ if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
+ (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
+ // add 1 ulp and then check for overflow
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ x_exp = x_exp + EXP_P1;
+ if (x_exp == EXP_MAX_P1) { // overflow
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ x_exp = 0; // x_sign is preserved
+ // set overflow flag (the inexact flag was set too)
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ } else if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign) ||
+ (rnd_mode == ROUNDING_UP && x_sign && !y_sign) ||
+ (rnd_mode == ROUNDING_TO_ZERO
+ && x_sign != y_sign)) {
+ // subtract 1 ulp from C1
+ // Note: because delta >= P34 + 1 the result cannot be zero
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi = C1_hi - 1;
+ // if the coefficient is 10^33 - 1 then make it 10^34 - 1 and
+ // decrease the exponent by 1 (because delta >= P34 + 1 the
+ // exponent will not become less than e_min)
+ // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
+ // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
+ if (C1_hi == 0x0000314dc6448d93ull
+ && C1_lo == 0x38c15b09ffffffffull) {
+ // make C1 = 10^34 - 1
+ C1_hi = 0x0001ed09bead87c0ull;
+ C1_lo = 0x378d8e63ffffffffull;
+ x_exp = x_exp - EXP_P1;
+ }
+ } else {
+ ; // the result is already correct
+ }
+ }
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // assemble the result
+ res.w[1] = x_sign | x_exp | C1_hi;
+ res.w[0] = C1_lo;
+ } else { // delta = P34
+ // in most cases, the smaller operand may be < or = or > 1/2 ulp of the
+ // larger operand
+ // however, the case C1 = 10^(q1-1) and x_sign != y_sign is special due
+ // to accuracy loss after subtraction, and will be treated separately
+ if (x_sign == y_sign || (q1 <= 20
+ && (C1_hi != 0
+ || C1_lo != ten2k64[q1 - 1]))
+ || (q1 >= 21 && (C1_hi != ten2k128[q1 - 21].w[1]
+ || C1_lo != ten2k128[q1 - 21].w[0]))) {
+ // if x_sign == y_sign or C1 != 10^(q1-1)
+ // compare C2 with 1/2 ulp = 5 * 10^(q2-1), the latter read from table
+ // Note: cases q1<=19 and q1>=20 can be coalesced at some latency cost
+ if (q2 <= 19) { // C2 and 5*10^(q2-1) both fit in 64 bits
+ halfulp64 = midpoint64[q2 - 1]; // 5 * 10^(q2-1)
+ if (C2_lo < halfulp64) { // n2 < 1/2 ulp (n1)
+ // for RN the result is the operand with the larger magnitude,
+ // possibly scaled up by 10^(P34-q1)
+ // an overflow cannot occur in this case (rounding to nearest)
+ if (q1 < P34) { // scale C1 up by 10^(P34-q1)
+ // Note: because delta = P34 it is certain that
+ // x_exp - ((UINT64)scale << 49) will stay above e_min
+ scale = P34 - q1;
+ if (q1 <= 19) { // C1 fits in 64 bits
+ // 1 <= q1 <= 19 => 15 <= scale <= 33
+ if (scale <= 19) { // 10^scale fits in 64 bits
+ __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
+ } else { // if 20 <= scale <= 33
+ // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
+ // (C1 * 10^(scale-19)) fits in 64 bits
+ C1_lo = C1_lo * ten2k64[scale - 19];
+ __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
+ }
+ } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
+ // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ // C1 = ten2k64[P34 - q1] * C1
+ __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
+ }
+ x_exp = x_exp - ((UINT64) scale << 49);
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ }
+ if (rnd_mode != ROUNDING_TO_NEAREST) {
+ if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
+ (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
+ // add 1 ulp and then check for overflow
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ x_exp = x_exp + EXP_P1;
+ if (x_exp == EXP_MAX_P1) { // overflow
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ x_exp = 0; // x_sign is preserved
+ // set overflow flag (the inexact flag was set too)
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ } else
+ if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
+ || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
+ || (rnd_mode == ROUNDING_TO_ZERO
+ && x_sign != y_sign)) {
+ // subtract 1 ulp from C1
+ // Note: because delta >= P34 + 1 the result cannot be zero
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi = C1_hi - 1;
+ // if the coefficient is 10^33-1 then make it 10^34-1 and
+ // decrease the exponent by 1 (because delta >= P34 + 1 the
+ // exponent will not become less than e_min)
+ // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
+ // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
+ if (C1_hi == 0x0000314dc6448d93ull
+ && C1_lo == 0x38c15b09ffffffffull) {
+ // make C1 = 10^34 - 1
+ C1_hi = 0x0001ed09bead87c0ull;
+ C1_lo = 0x378d8e63ffffffffull;
+ x_exp = x_exp - EXP_P1;
+ }
+ } else {
+ ; // the result is already correct
+ }
+ }
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // assemble the result
+ res.w[1] = x_sign | x_exp | C1_hi;
+ res.w[0] = C1_lo;
+ } else if ((C2_lo == halfulp64)
+ && (q1 < P34 || ((C1_lo & 0x1) == 0))) {
+ // n2 = 1/2 ulp (n1) and C1 is even
+ // the result is the operand with the larger magnitude,
+ // possibly scaled up by 10^(P34-q1)
+ // an overflow cannot occur in this case (rounding to nearest)
+ if (q1 < P34) { // scale C1 up by 10^(P34-q1)
+ // Note: because delta = P34 it is certain that
+ // x_exp - ((UINT64)scale << 49) will stay above e_min
+ scale = P34 - q1;
+ if (q1 <= 19) { // C1 fits in 64 bits
+ // 1 <= q1 <= 19 => 15 <= scale <= 33
+ if (scale <= 19) { // 10^scale fits in 64 bits
+ __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
+ } else { // if 20 <= scale <= 33
+ // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
+ // (C1 * 10^(scale-19)) fits in 64 bits
+ C1_lo = C1_lo * ten2k64[scale - 19];
+ __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
+ }
+ } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
+ // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ // C1 = ten2k64[P34 - q1] * C1
+ __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
+ }
+ x_exp = x_exp - ((UINT64) scale << 49);
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ }
+ if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign == y_sign
+ && (C1_lo & 0x01)) || (rnd_mode == ROUNDING_TIES_AWAY
+ && x_sign == y_sign)
+ || (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)
+ || (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)) {
+ // add 1 ulp and then check for overflow
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ x_exp = x_exp + EXP_P1;
+ if (x_exp == EXP_MAX_P1) { // overflow
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ x_exp = 0; // x_sign is preserved
+ // set overflow flag (the inexact flag was set too)
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ } else
+ if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign != y_sign
+ && (C1_lo & 0x01)) || (rnd_mode == ROUNDING_DOWN
+ && !x_sign && y_sign)
+ || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
+ || (rnd_mode == ROUNDING_TO_ZERO
+ && x_sign != y_sign)) {
+ // subtract 1 ulp from C1
+ // Note: because delta >= P34 + 1 the result cannot be zero
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi = C1_hi - 1;
+ // if the coefficient is 10^33 - 1 then make it 10^34 - 1
+ // and decrease the exponent by 1 (because delta >= P34 + 1
+ // the exponent will not become less than e_min)
+ // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
+ // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
+ if (C1_hi == 0x0000314dc6448d93ull
+ && C1_lo == 0x38c15b09ffffffffull) {
+ // make C1 = 10^34 - 1
+ C1_hi = 0x0001ed09bead87c0ull;
+ C1_lo = 0x378d8e63ffffffffull;
+ x_exp = x_exp - EXP_P1;
+ }
+ } else {
+ ; // the result is already correct
+ }
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // assemble the result
+ res.w[1] = x_sign | x_exp | C1_hi;
+ res.w[0] = C1_lo;
+ } else { // if C2_lo > halfulp64 ||
+ // (C2_lo == halfulp64 && q1 == P34 && ((C1_lo & 0x1) == 1)), i.e.
+ // 1/2 ulp(n1) < n2 < 1 ulp(n1) or n2 = 1/2 ulp(n1) and C1 odd
+ // res = x+1 ulp if n1*n2 > 0 and res = x-1 ulp if n1*n2 < 0
+ if (q1 < P34) { // then 1 ulp = 10^(e1+q1-P34) < 10^e1
+ // Note: if (q1 == P34) then 1 ulp = 10^(e1+q1-P34) = 10^e1
+ // because q1 < P34 we must first replace C1 by
+ // C1 * 10^(P34-q1), and must decrease the exponent by
+ // (P34-q1) (it will still be at least e_min)
+ scale = P34 - q1;
+ if (q1 <= 19) { // C1 fits in 64 bits
+ // 1 <= q1 <= 19 => 15 <= scale <= 33
+ if (scale <= 19) { // 10^scale fits in 64 bits
+ __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
+ } else { // if 20 <= scale <= 33
+ // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
+ // (C1 * 10^(scale-19)) fits in 64 bits
+ C1_lo = C1_lo * ten2k64[scale - 19];
+ __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
+ }
+ } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
+ // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ // C1 = ten2k64[P34 - q1] * C1
+ __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
+ }
+ x_exp = x_exp - ((UINT64) scale << 49);
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ // check for rounding overflow
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ x_exp = x_exp + EXP_P1;
+ }
+ }
+ if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign != y_sign)
+ || (rnd_mode == ROUNDING_TIES_AWAY && x_sign != y_sign
+ && C2_lo != halfulp64)
+ || (rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
+ || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
+ || (rnd_mode == ROUNDING_TO_ZERO
+ && x_sign != y_sign)) {
+ // the result is x - 1
+ // for RN n1 * n2 < 0; underflow not possible
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi--;
+ // check if we crossed into the lower decade
+ if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
+ C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
+ C1_lo = 0x378d8e63ffffffffull;
+ x_exp = x_exp - EXP_P1; // no underflow, because n1 >> n2
+ }
+ } else
+ if ((rnd_mode == ROUNDING_TO_NEAREST
+ && x_sign == y_sign)
+ || (rnd_mode == ROUNDING_TIES_AWAY
+ && x_sign == y_sign)
+ || (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)
+ || (rnd_mode == ROUNDING_UP && !x_sign
+ && !y_sign)) {
+ // the result is x + 1
+ // for RN x_sign = y_sign, i.e. n1*n2 > 0
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ x_exp = x_exp + EXP_P1;
+ if (x_exp == EXP_MAX_P1) { // overflow
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ x_exp = 0; // x_sign is preserved
+ // set the overflow flag
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ } else {
+ ; // the result is x
+ }
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // assemble the result
+ res.w[1] = x_sign | x_exp | C1_hi;
+ res.w[0] = C1_lo;
+ }
+ } else { // if q2 >= 20 then 5*10^(q2-1) and C2 (the latter in
+ // most cases) fit only in more than 64 bits
+ halfulp128 = midpoint128[q2 - 20]; // 5 * 10^(q2-1)
+ if ((C2_hi < halfulp128.w[1])
+ || (C2_hi == halfulp128.w[1]
+ && C2_lo < halfulp128.w[0])) {
+ // n2 < 1/2 ulp (n1)
+ // the result is the operand with the larger magnitude,
+ // possibly scaled up by 10^(P34-q1)
+ // an overflow cannot occur in this case (rounding to nearest)
+ if (q1 < P34) { // scale C1 up by 10^(P34-q1)
+ // Note: because delta = P34 it is certain that
+ // x_exp - ((UINT64)scale << 49) will stay above e_min
+ scale = P34 - q1;
+ if (q1 <= 19) { // C1 fits in 64 bits
+ // 1 <= q1 <= 19 => 15 <= scale <= 33
+ if (scale <= 19) { // 10^scale fits in 64 bits
+ __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
+ } else { // if 20 <= scale <= 33
+ // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
+ // (C1 * 10^(scale-19)) fits in 64 bits
+ C1_lo = C1_lo * ten2k64[scale - 19];
+ __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
+ }
+ } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
+ // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ // C1 = ten2k64[P34 - q1] * C1
+ __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
+ }
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ x_exp = x_exp - ((UINT64) scale << 49);
+ }
+ if (rnd_mode != ROUNDING_TO_NEAREST) {
+ if ((rnd_mode == ROUNDING_DOWN && x_sign && y_sign) ||
+ (rnd_mode == ROUNDING_UP && !x_sign && !y_sign)) {
+ // add 1 ulp and then check for overflow
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ x_exp = x_exp + EXP_P1;
+ if (x_exp == EXP_MAX_P1) { // overflow
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ x_exp = 0; // x_sign is preserved
+ // set overflow flag (the inexact flag was set too)
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ } else
+ if ((rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
+ || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
+ || (rnd_mode == ROUNDING_TO_ZERO
+ && x_sign != y_sign)) {
+ // subtract 1 ulp from C1
+ // Note: because delta >= P34 + 1 the result cannot be zero
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi = C1_hi - 1;
+ // if the coefficient is 10^33-1 then make it 10^34-1 and
+ // decrease the exponent by 1 (because delta >= P34 + 1 the
+ // exponent will not become less than e_min)
+ // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
+ // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
+ if (C1_hi == 0x0000314dc6448d93ull
+ && C1_lo == 0x38c15b09ffffffffull) {
+ // make C1 = 10^34 - 1
+ C1_hi = 0x0001ed09bead87c0ull;
+ C1_lo = 0x378d8e63ffffffffull;
+ x_exp = x_exp - EXP_P1;
+ }
+ } else {
+ ; // the result is already correct
+ }
+ }
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // assemble the result
+ res.w[1] = x_sign | x_exp | C1_hi;
+ res.w[0] = C1_lo;
+ } else if ((C2_hi == halfulp128.w[1]
+ && C2_lo == halfulp128.w[0])
+ && (q1 < P34 || ((C1_lo & 0x1) == 0))) {
+ // midpoint & lsb in C1 is 0
+ // n2 = 1/2 ulp (n1) and C1 is even
+ // the result is the operand with the larger magnitude,
+ // possibly scaled up by 10^(P34-q1)
+ // an overflow cannot occur in this case (rounding to nearest)
+ if (q1 < P34) { // scale C1 up by 10^(P34-q1)
+ // Note: because delta = P34 it is certain that
+ // x_exp - ((UINT64)scale << 49) will stay above e_min
+ scale = P34 - q1;
+ if (q1 <= 19) { // C1 fits in 64 bits
+ // 1 <= q1 <= 19 => 15 <= scale <= 33
+ if (scale <= 19) { // 10^scale fits in 64 bits
+ __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
+ } else { // if 20 <= scale <= 33
+ // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
+ // (C1 * 10^(scale-19)) fits in 64 bits
+ C1_lo = C1_lo * ten2k64[scale - 19];
+ __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
+ }
+ } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
+ // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ // C1 = ten2k64[P34 - q1] * C1
+ __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
+ }
+ x_exp = x_exp - ((UINT64) scale << 49);
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ }
+ if (rnd_mode != ROUNDING_TO_NEAREST) {
+ if ((rnd_mode == ROUNDING_TIES_AWAY && x_sign == y_sign)
+ || (rnd_mode == ROUNDING_UP && !y_sign)) {
+ // add 1 ulp and then check for overflow
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ x_exp = x_exp + EXP_P1;
+ if (x_exp == EXP_MAX_P1) { // overflow
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ x_exp = 0; // x_sign is preserved
+ // set overflow flag (the inexact flag was set too)
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ } else if ((rnd_mode == ROUNDING_DOWN && y_sign)
+ || (rnd_mode == ROUNDING_TO_ZERO
+ && x_sign != y_sign)) {
+ // subtract 1 ulp from C1
+ // Note: because delta >= P34 + 1 the result cannot be zero
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi = C1_hi - 1;
+ // if the coefficient is 10^33 - 1 then make it 10^34 - 1
+ // and decrease the exponent by 1 (because delta >= P34 + 1
+ // the exponent will not become less than e_min)
+ // 10^33 - 1 = 0x0000314dc6448d9338c15b09ffffffff
+ // 10^34 - 1 = 0x0001ed09bead87c0378d8e63ffffffff
+ if (C1_hi == 0x0000314dc6448d93ull
+ && C1_lo == 0x38c15b09ffffffffull) {
+ // make C1 = 10^34 - 1
+ C1_hi = 0x0001ed09bead87c0ull;
+ C1_lo = 0x378d8e63ffffffffull;
+ x_exp = x_exp - EXP_P1;
+ }
+ } else {
+ ; // the result is already correct
+ }
+ }
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // assemble the result
+ res.w[1] = x_sign | x_exp | C1_hi;
+ res.w[0] = C1_lo;
+ } else { // if C2 > halfulp128 ||
+ // (C2 == halfulp128 && q1 == P34 && ((C1 & 0x1) == 1)), i.e.
+ // 1/2 ulp(n1) < n2 < 1 ulp(n1) or n2 = 1/2 ulp(n1) and C1 odd
+ // res = x+1 ulp if n1*n2 > 0 and res = x-1 ulp if n1*n2 < 0
+ if (q1 < P34) { // then 1 ulp = 10^(e1+q1-P34) < 10^e1
+ // Note: if (q1 == P34) then 1 ulp = 10^(e1+q1-P34) = 10^e1
+ // because q1 < P34 we must first replace C1 by C1*10^(P34-q1),
+ // and must decrease the exponent by (P34-q1) (it will still be
+ // at least e_min)
+ scale = P34 - q1;
+ if (q1 <= 19) { // C1 fits in 64 bits
+ // 1 <= q1 <= 19 => 15 <= scale <= 33
+ if (scale <= 19) { // 10^scale fits in 64 bits
+ __mul_64x64_to_128MACH (C1, ten2k64[scale], C1_lo);
+ } else { // if 20 <= scale <= 33
+ // C1 * 10^scale = (C1 * 10^(scale-19)) * 10^19 where
+ // (C1 * 10^(scale-19)) fits in 64 bits
+ C1_lo = C1_lo * ten2k64[scale - 19];
+ __mul_64x64_to_128MACH (C1, ten2k64[19], C1_lo);
+ }
+ } else { //if 20 <= q1 <= 33=P34-1 then C1 fits only in 128 bits
+ // => 1 <= P34 - q1 <= 14 so 10^(P34-q1) fits in 64 bits
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ // C1 = ten2k64[P34 - q1] * C1
+ __mul_128x64_to_128 (C1, ten2k64[P34 - q1], C1);
+ }
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ x_exp = x_exp - ((UINT64) scale << 49);
+ }
+ if ((rnd_mode == ROUNDING_TO_NEAREST && x_sign != y_sign)
+ || (rnd_mode == ROUNDING_TIES_AWAY && x_sign != y_sign
+ && (C2_hi != halfulp128.w[1]
+ || C2_lo != halfulp128.w[0]))
+ || (rnd_mode == ROUNDING_DOWN && !x_sign && y_sign)
+ || (rnd_mode == ROUNDING_UP && x_sign && !y_sign)
+ || (rnd_mode == ROUNDING_TO_ZERO
+ && x_sign != y_sign)) {
+ // the result is x - 1
+ // for RN n1 * n2 < 0; underflow not possible
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi--;
+ // check if we crossed into the lower decade
+ if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
+ C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
+ C1_lo = 0x378d8e63ffffffffull;
+ x_exp = x_exp - EXP_P1; // no underflow, because n1 >> n2
+ }
+ } else
+ if ((rnd_mode == ROUNDING_TO_NEAREST
+ && x_sign == y_sign)
+ || (rnd_mode == ROUNDING_TIES_AWAY
+ && x_sign == y_sign)
+ || (rnd_mode == ROUNDING_DOWN && x_sign && y_sign)
+ || (rnd_mode == ROUNDING_UP && !x_sign
+ && !y_sign)) {
+ // the result is x + 1
+ // for RN x_sign = y_sign, i.e. n1*n2 > 0
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ x_exp = x_exp + EXP_P1;
+ if (x_exp == EXP_MAX_P1) { // overflow
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ x_exp = 0; // x_sign is preserved
+ // set the overflow flag
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ } else {
+ ; // the result is x
+ }
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // assemble the result
+ res.w[1] = x_sign | x_exp | C1_hi;
+ res.w[0] = C1_lo;
+ }
+ } // end q1 >= 20
+ // end case where C1 != 10^(q1-1)
+ } else { // C1 = 10^(q1-1) and x_sign != y_sign
+ // instead of C' = (C1 * 10^(e1-e2) + C2)rnd,P34
+ // calculate C' = C1 * 10^(e1-e2-x1) + (C2 * 10^(-x1))rnd,P34
+ // where x1 = q2 - 1, 0 <= x1 <= P34 - 1
+ // Because C1 = 10^(q1-1) and x_sign != y_sign, C' will have P34
+ // digits and n = C' * 10^(e2+x1)
+ // If the result has P34+1 digits, redo the steps above with x1+1
+ // If the result has P34-1 digits or less, redo the steps above with
+ // x1-1 but only if initially x1 >= 1
+ // NOTE: these two steps can be improved, e.g we could guess if
+ // P34+1 or P34-1 digits will be obtained by adding/subtracting
+ // just the top 64 bits of the two operands
+ // The result cannot be zero, and it cannot overflow
+ x1 = q2 - 1; // 0 <= x1 <= P34-1
+ // Calculate C1 * 10^(e1-e2-x1) where 1 <= e1-e2-x1 <= P34
+ // scale = (int)(e1 >> 49) - (int)(e2 >> 49) - x1; 0 <= scale <= P34-1
+ scale = P34 - q1 + 1; // scale=e1-e2-x1 = P34+1-q1; 1<=scale<=P34
+ // either C1 or 10^(e1-e2-x1) may not fit is 64 bits,
+ // but their product fits with certainty in 128 bits
+ if (scale >= 20) { //10^(e1-e2-x1) doesn't fit in 64 bits, but C1 does
+ __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
+ } else { // if (scale >= 1
+ // if 1 <= scale <= 19 then 10^(e1-e2-x1) fits in 64 bits
+ if (q1 <= 19) { // C1 fits in 64 bits
+ __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
+ } else { // q1 >= 20
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ __mul_128x64_to_128 (C1, ten2k64[scale], C1);
+ }
+ }
+ tmp64 = C1.w[0]; // C1.w[1], C1.w[0] contains C1 * 10^(e1-e2-x1)
+
+ // now round C2 to q2-x1 = 1 decimal digit
+ // C2' = C2 + 1/2 * 10^x1 = C2 + 5 * 10^(x1-1)
+ ind = x1 - 1; // -1 <= ind <= P34 - 2
+ if (ind >= 0) { // if (x1 >= 1)
+ C2.w[0] = C2_lo;
+ C2.w[1] = C2_hi;
+ if (ind <= 18) {
+ C2.w[0] = C2.w[0] + midpoint64[ind];
+ if (C2.w[0] < C2_lo)
+ C2.w[1]++;
+ } else { // 19 <= ind <= 32
+ C2.w[0] = C2.w[0] + midpoint128[ind - 19].w[0];
+ C2.w[1] = C2.w[1] + midpoint128[ind - 19].w[1];
+ if (C2.w[0] < C2_lo)
+ C2.w[1]++;
+ }
+ // the approximation of 10^(-x1) was rounded up to 118 bits
+ __mul_128x128_to_256 (R256, C2, ten2mk128[ind]); // R256 = C2*, f2*
+ // calculate C2* and f2*
+ // C2* is actually floor(C2*) in this case
+ // C2* and f2* need shifting and masking, as shown by
+ // shiftright128[] and maskhigh128[]
+ // the top Ex bits of 10^(-x1) are T* = ten2mk128trunc[ind], e.g.
+ // if x1=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
+ // if (0 < f2* < 10^(-x1)) then
+ // if floor(C1+C2*) is even then C2* = floor(C2*) - logical right
+ // shift; C2* has p decimal digits, correct by Prop. 1)
+ // else if floor(C1+C2*) is odd C2* = floor(C2*)-1 (logical right
+ // shift; C2* has p decimal digits, correct by Pr. 1)
+ // else
+ // C2* = floor(C2*) (logical right shift; C has p decimal digits,
+ // correct by Property 1)
+ // n = C2* * 10^(e2+x1)
+
+ if (ind <= 2) {
+ highf2star.w[1] = 0x0;
+ highf2star.w[0] = 0x0; // low f2* ok
+ } else if (ind <= 21) {
+ highf2star.w[1] = 0x0;
+ highf2star.w[0] = R256.w[2] & maskhigh128[ind]; // low f2* ok
+ } else {
+ highf2star.w[1] = R256.w[3] & maskhigh128[ind];
+ highf2star.w[0] = R256.w[2]; // low f2* is ok
+ }
+ // shift right C2* by Ex-128 = shiftright128[ind]
+ if (ind >= 3) {
+ shift = shiftright128[ind];
+ if (shift < 64) { // 3 <= shift <= 63
+ R256.w[2] =
+ (R256.w[2] >> shift) | (R256.w[3] << (64 - shift));
+ R256.w[3] = (R256.w[3] >> shift);
+ } else { // 66 <= shift <= 102
+ R256.w[2] = (R256.w[3] >> (shift - 64));
+ R256.w[3] = 0x0ULL;
+ }
+ }
+ // redundant
+ is_inexact_lt_midpoint = 0;
+ is_inexact_gt_midpoint = 0;
+ is_midpoint_lt_even = 0;
+ is_midpoint_gt_even = 0;
+ // determine inexactness of the rounding of C2*
+ // (cannot be followed by a second rounding)
+ // if (0 < f2* - 1/2 < 10^(-x1)) then
+ // the result is exact
+ // else (if f2* - 1/2 > T* then)
+ // the result of is inexact
+ if (ind <= 2) {
+ if (R256.w[1] > 0x8000000000000000ull ||
+ (R256.w[1] == 0x8000000000000000ull
+ && R256.w[0] > 0x0ull)) {
+ // f2* > 1/2 and the result may be exact
+ tmp64A = R256.w[1] - 0x8000000000000000ull; // f* - 1/2
+ if ((tmp64A > ten2mk128trunc[ind].w[1]
+ || (tmp64A == ten2mk128trunc[ind].w[1]
+ && R256.w[0] >= ten2mk128trunc[ind].w[0]))) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // this rounding is applied to C2 only!
+ // x_sign != y_sign
+ is_inexact_gt_midpoint = 1;
+ } // else the result is exact
+ // rounding down, unless a midpoint in [ODD, EVEN]
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // this rounding is applied to C2 only!
+ // x_sign != y_sign
+ is_inexact_lt_midpoint = 1;
+ }
+ } else if (ind <= 21) { // if 3 <= ind <= 21
+ if (highf2star.w[1] > 0x0 || (highf2star.w[1] == 0x0
+ && highf2star.w[0] >
+ onehalf128[ind])
+ || (highf2star.w[1] == 0x0
+ && highf2star.w[0] == onehalf128[ind]
+ && (R256.w[1] || R256.w[0]))) {
+ // f2* > 1/2 and the result may be exact
+ // Calculate f2* - 1/2
+ tmp64A = highf2star.w[0] - onehalf128[ind];
+ tmp64B = highf2star.w[1];
+ if (tmp64A > highf2star.w[0])
+ tmp64B--;
+ if (tmp64B || tmp64A
+ || R256.w[1] > ten2mk128trunc[ind].w[1]
+ || (R256.w[1] == ten2mk128trunc[ind].w[1]
+ && R256.w[0] > ten2mk128trunc[ind].w[0])) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // this rounding is applied to C2 only!
+ // x_sign != y_sign
+ is_inexact_gt_midpoint = 1;
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // this rounding is applied to C2 only!
+ // x_sign != y_sign
+ is_inexact_lt_midpoint = 1;
+ }
+ } else { // if 22 <= ind <= 33
+ if (highf2star.w[1] > onehalf128[ind]
+ || (highf2star.w[1] == onehalf128[ind]
+ && (highf2star.w[0] || R256.w[1]
+ || R256.w[0]))) {
+ // f2* > 1/2 and the result may be exact
+ // Calculate f2* - 1/2
+ // tmp64A = highf2star.w[0];
+ tmp64B = highf2star.w[1] - onehalf128[ind];
+ if (tmp64B || highf2star.w[0]
+ || R256.w[1] > ten2mk128trunc[ind].w[1]
+ || (R256.w[1] == ten2mk128trunc[ind].w[1]
+ && R256.w[0] > ten2mk128trunc[ind].w[0])) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // this rounding is applied to C2 only!
+ // x_sign != y_sign
+ is_inexact_gt_midpoint = 1;
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // this rounding is applied to C2 only!
+ // x_sign != y_sign
+ is_inexact_lt_midpoint = 1;
+ }
+ }
+ // check for midpoints after determining inexactness
+ if ((R256.w[1] || R256.w[0]) && (highf2star.w[1] == 0)
+ && (highf2star.w[0] == 0)
+ && (R256.w[1] < ten2mk128trunc[ind].w[1]
+ || (R256.w[1] == ten2mk128trunc[ind].w[1]
+ && R256.w[0] <= ten2mk128trunc[ind].w[0]))) {
+ // the result is a midpoint
+ if ((tmp64 + R256.w[2]) & 0x01) { // MP in [EVEN, ODD]
+ // if floor(C2*) is odd C = floor(C2*) - 1; the result may be 0
+ R256.w[2]--;
+ if (R256.w[2] == 0xffffffffffffffffull)
+ R256.w[3]--;
+ // this rounding is applied to C2 only!
+ // x_sign != y_sign
+ is_midpoint_lt_even = 1;
+ is_inexact_lt_midpoint = 0;
+ is_inexact_gt_midpoint = 0;
+ } else {
+ // else MP in [ODD, EVEN]
+ // this rounding is applied to C2 only!
+ // x_sign != y_sign
+ is_midpoint_gt_even = 1;
+ is_inexact_lt_midpoint = 0;
+ is_inexact_gt_midpoint = 0;
+ }
+ }
+ } else { // if (ind == -1) only when x1 = 0
+ R256.w[2] = C2_lo;
+ R256.w[3] = C2_hi;
+ is_midpoint_lt_even = 0;
+ is_midpoint_gt_even = 0;
+ is_inexact_lt_midpoint = 0;
+ is_inexact_gt_midpoint = 0;
+ }
+ // and now subtract C1 * 10^(e1-e2-x1) - (C2 * 10^(-x1))rnd,P34
+ // because x_sign != y_sign this last operation is exact
+ C1.w[0] = C1.w[0] - R256.w[2];
+ C1.w[1] = C1.w[1] - R256.w[3];
+ if (C1.w[0] > tmp64)
+ C1.w[1]--; // borrow
+ if (C1.w[1] >= 0x8000000000000000ull) { // negative coefficient!
+ C1.w[0] = ~C1.w[0];
+ C1.w[0]++;
+ C1.w[1] = ~C1.w[1];
+ if (C1.w[0] == 0x0)
+ C1.w[1]++;
+ tmp_sign = y_sign; // the result will have the sign of y
+ } else {
+ tmp_sign = x_sign;
+ }
+ // the difference has exactly P34 digits
+ x_sign = tmp_sign;
+ if (x1 >= 1)
+ y_exp = y_exp + ((UINT64) x1 << 49);
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ // general correction from RN to RA, RM, RP, RZ; result uses y_exp
+ if (rnd_mode != ROUNDING_TO_NEAREST) {
+ if ((!x_sign
+ && ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint)
+ ||
+ ((rnd_mode == ROUNDING_TIES_AWAY
+ || rnd_mode == ROUNDING_UP)
+ && is_midpoint_gt_even))) || (x_sign
+ &&
+ ((rnd_mode ==
+ ROUNDING_DOWN
+ &&
+ is_inexact_lt_midpoint)
+ ||
+ ((rnd_mode ==
+ ROUNDING_TIES_AWAY
+ || rnd_mode ==
+ ROUNDING_DOWN)
+ &&
+ is_midpoint_gt_even))))
+ {
+ // C1 = C1 + 1
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ y_exp = y_exp + EXP_P1;
+ }
+ } else if ((is_midpoint_lt_even || is_inexact_gt_midpoint)
+ &&
+ ((x_sign
+ && (rnd_mode == ROUNDING_UP
+ || rnd_mode == ROUNDING_TO_ZERO))
+ || (!x_sign
+ && (rnd_mode == ROUNDING_DOWN
+ || rnd_mode == ROUNDING_TO_ZERO)))) {
+ // C1 = C1 - 1
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi--;
+ // check if we crossed into the lower decade
+ if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
+ C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
+ C1_lo = 0x378d8e63ffffffffull;
+ y_exp = y_exp - EXP_P1;
+ // no underflow, because delta + q2 >= P34 + 1
+ }
+ } else {
+ ; // exact, the result is already correct
+ }
+ }
+ // assemble the result
+ res.w[1] = x_sign | y_exp | C1_hi;
+ res.w[0] = C1_lo;
+ }
+ } // end delta = P34
+ } else { // if (|delta| <= P34 - 1)
+ if (delta >= 0) { // if (0 <= delta <= P34 - 1)
+ if (delta <= P34 - 1 - q2) {
+ // calculate C' directly; the result is exact
+ // in this case 1<=q1<=P34-1, 1<=q2<=P34-1 and 0 <= e1-e2 <= P34-2
+ // The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
+ // exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
+ // but their product fits with certainty in 128 bits (actually in 113)
+ scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
+
+ if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
+ __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ } else if (scale >= 1) {
+ // if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
+ if (q1 <= 19) { // C1 fits in 64 bits
+ __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
+ } else { // q1 >= 20
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ __mul_128x64_to_128 (C1, ten2k64[scale], C1);
+ }
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ } else { // if (scale == 0) C1 is unchanged
+ C1.w[0] = C1_lo; // C1.w[1] = C1_hi;
+ }
+ // now add C2
+ if (x_sign == y_sign) {
+ // the result cannot overflow
+ C1_lo = C1_lo + C2_lo;
+ C1_hi = C1_hi + C2_hi;
+ if (C1_lo < C1.w[0])
+ C1_hi++;
+ } else { // if x_sign != y_sign
+ C1_lo = C1_lo - C2_lo;
+ C1_hi = C1_hi - C2_hi;
+ if (C1_lo > C1.w[0])
+ C1_hi--;
+ // the result can be zero, but it cannot overflow
+ if (C1_lo == 0 && C1_hi == 0) {
+ // assemble the result
+ if (x_exp < y_exp)
+ res.w[1] = x_exp;
+ else
+ res.w[1] = y_exp;
+ res.w[0] = 0;
+ if (rnd_mode == ROUNDING_DOWN) {
+ res.w[1] |= 0x8000000000000000ull;
+ }
+ BID_SWAP128 (res);
+ BID_RETURN (res);
+ }
+ if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
+ C1_lo = ~C1_lo;
+ C1_lo++;
+ C1_hi = ~C1_hi;
+ if (C1_lo == 0x0)
+ C1_hi++;
+ x_sign = y_sign; // the result will have the sign of y
+ }
+ }
+ // assemble the result
+ res.w[1] = x_sign | y_exp | C1_hi;
+ res.w[0] = C1_lo;
+ } else if (delta == P34 - q2) {
+ // calculate C' directly; the result may be inexact if it requires
+ // P34+1 decimal digits; in this case the 'cutoff' point for addition
+ // is at the position of the lsb of C2, so 0 <= e1-e2 <= P34-1
+ // The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
+ // exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
+ // but their product fits with certainty in 128 bits (actually in 113)
+ scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
+ if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
+ __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
+ } else if (scale >= 1) {
+ // if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
+ if (q1 <= 19) { // C1 fits in 64 bits
+ __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
+ } else { // q1 >= 20
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ __mul_128x64_to_128 (C1, ten2k64[scale], C1);
+ }
+ } else { // if (scale == 0) C1 is unchanged
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo; // only the low part is necessary
+ }
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ // now add C2
+ if (x_sign == y_sign) {
+ // the result can overflow!
+ C1_lo = C1_lo + C2_lo;
+ C1_hi = C1_hi + C2_hi;
+ if (C1_lo < C1.w[0])
+ C1_hi++;
+ // test for overflow, possible only when C1 >= 10^34
+ if (C1_hi > 0x0001ed09bead87c0ull || (C1_hi == 0x0001ed09bead87c0ull && C1_lo >= 0x378d8e6400000000ull)) { // C1 >= 10^34
+ // in this case q = P34 + 1 and x = q - P34 = 1, so multiply
+ // C'' = C'+ 5 = C1 + 5 by k1 ~ 10^(-1) calculated for P34 + 1
+ // decimal digits
+ // Calculate C'' = C' + 1/2 * 10^x
+ if (C1_lo >= 0xfffffffffffffffbull) { // low half add has carry
+ C1_lo = C1_lo + 5;
+ C1_hi = C1_hi + 1;
+ } else {
+ C1_lo = C1_lo + 5;
+ }
+ // the approximation of 10^(-1) was rounded up to 118 bits
+ // 10^(-1) =~ 33333333333333333333333333333400 * 2^-129
+ // 10^(-1) =~ 19999999999999999999999999999a00 * 2^-128
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo; // C''
+ ten2m1.w[1] = 0x1999999999999999ull;
+ ten2m1.w[0] = 0x9999999999999a00ull;
+ __mul_128x128_to_256 (P256, C1, ten2m1); // P256 = C*, f*
+ // C* is actually floor(C*) in this case
+ // the top Ex = 128 bits of 10^(-1) are
+ // T* = 0x00199999999999999999999999999999
+ // if (0 < f* < 10^(-x)) then
+ // 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^(e2+x)
+ if ((P256.w[1] || P256.w[0])
+ && (P256.w[1] < 0x1999999999999999ull
+ || (P256.w[1] == 0x1999999999999999ull
+ && P256.w[0] <= 0x9999999999999999ull))) {
+ // the result is a midpoint
+ if (P256.w[2] & 0x01) {
+ is_midpoint_gt_even = 1;
+ // if floor(C*) is odd C = floor(C*) - 1; the result is not 0
+ P256.w[2]--;
+ if (P256.w[2] == 0xffffffffffffffffull)
+ P256.w[3]--;
+ } else {
+ is_midpoint_lt_even = 1;
+ }
+ }
+ // n = Cstar * 10^(e2+1)
+ y_exp = y_exp + EXP_P1;
+ // C* != 10^P because C* has P34 digits
+ // check for overflow
+ if (y_exp == EXP_MAX_P1
+ && (rnd_mode == ROUNDING_TO_NEAREST
+ || rnd_mode == ROUNDING_TIES_AWAY)) {
+ // overflow for RN
+ res.w[1] = x_sign | 0x7800000000000000ull; // +/-inf
+ res.w[0] = 0x0ull;
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // set the overflow flag
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ BID_SWAP128 (res);
+ BID_RETURN (res);
+ }
+ // if (0 < f* - 1/2 < 10^(-x)) then
+ // the result of the addition is exact
+ // else
+ // the result of the addition is inexact
+ if (P256.w[1] > 0x8000000000000000ull || (P256.w[1] == 0x8000000000000000ull && P256.w[0] > 0x0ull)) { // the result may be exact
+ tmp64 = P256.w[1] - 0x8000000000000000ull; // f* - 1/2
+ if ((tmp64 > 0x1999999999999999ull
+ || (tmp64 == 0x1999999999999999ull
+ && P256.w[0] >= 0x9999999999999999ull))) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ is_inexact = 1;
+ } // else the result is exact
+ } else { // the result is inexact
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ is_inexact = 1;
+ }
+ C1_hi = P256.w[3];
+ C1_lo = P256.w[2];
+ if (!is_midpoint_gt_even && !is_midpoint_lt_even) {
+ is_inexact_lt_midpoint = is_inexact
+ && (P256.w[1] & 0x8000000000000000ull);
+ is_inexact_gt_midpoint = is_inexact
+ && !(P256.w[1] & 0x8000000000000000ull);
+ }
+ // general correction from RN to RA, RM, RP, RZ;
+ // result uses y_exp
+ if (rnd_mode != ROUNDING_TO_NEAREST) {
+ if ((!x_sign
+ &&
+ ((rnd_mode == ROUNDING_UP
+ && is_inexact_lt_midpoint)
+ ||
+ ((rnd_mode == ROUNDING_TIES_AWAY
+ || rnd_mode == ROUNDING_UP)
+ && is_midpoint_gt_even))) || (x_sign
+ &&
+ ((rnd_mode ==
+ ROUNDING_DOWN
+ &&
+ is_inexact_lt_midpoint)
+ ||
+ ((rnd_mode ==
+ ROUNDING_TIES_AWAY
+ || rnd_mode ==
+ ROUNDING_DOWN)
+ &&
+ is_midpoint_gt_even))))
+ {
+ // C1 = C1 + 1
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ y_exp = y_exp + EXP_P1;
+ }
+ } else
+ if ((is_midpoint_lt_even || is_inexact_gt_midpoint)
+ &&
+ ((x_sign
+ && (rnd_mode == ROUNDING_UP
+ || rnd_mode == ROUNDING_TO_ZERO))
+ || (!x_sign
+ && (rnd_mode == ROUNDING_DOWN
+ || rnd_mode == ROUNDING_TO_ZERO)))) {
+ // C1 = C1 - 1
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi--;
+ // check if we crossed into the lower decade
+ if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
+ C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
+ C1_lo = 0x378d8e63ffffffffull;
+ y_exp = y_exp - EXP_P1;
+ // no underflow, because delta + q2 >= P34 + 1
+ }
+ } else {
+ ; // exact, the result is already correct
+ }
+ // in all cases check for overflow (RN and RA solved already)
+ if (y_exp == EXP_MAX_P1) { // overflow
+ if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
+ (rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ } else { // RM and res > 0, RP and res < 0, or RZ
+ C1_hi = 0x5fffed09bead87c0ull;
+ C1_lo = 0x378d8e63ffffffffull;
+ }
+ y_exp = 0; // x_sign is preserved
+ // set the inexact flag (in case the exact addition was exact)
+ *pfpsf |= INEXACT_EXCEPTION;
+ // set the overflow flag
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ } // else if (C1 < 10^34) then C1 is the coeff.; the result is exact
+ } else { // if x_sign != y_sign the result is exact
+ C1_lo = C1_lo - C2_lo;
+ C1_hi = C1_hi - C2_hi;
+ if (C1_lo > C1.w[0])
+ C1_hi--;
+ // the result can be zero, but it cannot overflow
+ if (C1_lo == 0 && C1_hi == 0) {
+ // assemble the result
+ if (x_exp < y_exp)
+ res.w[1] = x_exp;
+ else
+ res.w[1] = y_exp;
+ res.w[0] = 0;
+ if (rnd_mode == ROUNDING_DOWN) {
+ res.w[1] |= 0x8000000000000000ull;
+ }
+ BID_SWAP128 (res);
+ BID_RETURN (res);
+ }
+ if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
+ C1_lo = ~C1_lo;
+ C1_lo++;
+ C1_hi = ~C1_hi;
+ if (C1_lo == 0x0)
+ C1_hi++;
+ x_sign = y_sign; // the result will have the sign of y
+ }
+ }
+ // assemble the result
+ res.w[1] = x_sign | y_exp | C1_hi;
+ res.w[0] = C1_lo;
+ } else { // if (delta >= P34 + 1 - q2)
+ // instead of C' = (C1 * 10^(e1-e2) + C2)rnd,P34
+ // calculate C' = C1 * 10^(e1-e2-x1) + (C2 * 10^(-x1))rnd,P34
+ // where x1 = q1 + e1 - e2 - P34, 1 <= x1 <= P34 - 1
+ // In most cases C' will have P34 digits, and n = C' * 10^(e2+x1)
+ // If the result has P34+1 digits, redo the steps above with x1+1
+ // If the result has P34-1 digits or less, redo the steps above with
+ // x1-1 but only if initially x1 >= 1
+ // NOTE: these two steps can be improved, e.g we could guess if
+ // P34+1 or P34-1 digits will be obtained by adding/subtracting just
+ // the top 64 bits of the two operands
+ // The result cannot be zero, but it can overflow
+ x1 = delta + q2 - P34; // 1 <= x1 <= P34-1
+ roundC2:
+ // Calculate C1 * 10^(e1-e2-x1) where 0 <= e1-e2-x1 <= P34 - 1
+ // scale = (int)(e1 >> 49) - (int)(e2 >> 49) - x1; 0 <= scale <= P34-1
+ scale = delta - q1 + q2 - x1; // scale = e1 - e2 - x1 = P34 - q1
+ // either C1 or 10^(e1-e2-x1) may not fit is 64 bits,
+ // but their product fits with certainty in 128 bits (actually in 113)
+ if (scale >= 20) { //10^(e1-e2-x1) doesn't fit in 64 bits, but C1 does
+ __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
+ } else if (scale >= 1) {
+ // if 1 <= scale <= 19 then 10^(e1-e2-x1) fits in 64 bits
+ if (q1 <= 19) { // C1 fits in 64 bits
+ __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
+ } else { // q1 >= 20
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ __mul_128x64_to_128 (C1, ten2k64[scale], C1);
+ }
+ } else { // if (scale == 0) C1 is unchanged
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ }
+ tmp64 = C1.w[0]; // C1.w[1], C1.w[0] contains C1 * 10^(e1-e2-x1)
+
+ // now round C2 to q2-x1 decimal digits, where 1<=x1<=q2-1<=P34-1
+ // (but if we got here a second time after x1 = x1 - 1, then
+ // x1 >= 0; note that for x1 = 0 C2 is unchanged)
+ // C2' = C2 + 1/2 * 10^x1 = C2 + 5 * 10^(x1-1)
+ ind = x1 - 1; // 0 <= ind <= q2-2<=P34-2=32; but note that if x1 = 0
+ // during a second pass, then ind = -1
+ if (ind >= 0) { // if (x1 >= 1)
+ C2.w[0] = C2_lo;
+ C2.w[1] = C2_hi;
+ if (ind <= 18) {
+ C2.w[0] = C2.w[0] + midpoint64[ind];
+ if (C2.w[0] < C2_lo)
+ C2.w[1]++;
+ } else { // 19 <= ind <= 32
+ C2.w[0] = C2.w[0] + midpoint128[ind - 19].w[0];
+ C2.w[1] = C2.w[1] + midpoint128[ind - 19].w[1];
+ if (C2.w[0] < C2_lo)
+ C2.w[1]++;
+ }
+ // the approximation of 10^(-x1) was rounded up to 118 bits
+ __mul_128x128_to_256 (R256, C2, ten2mk128[ind]); // R256 = C2*, f2*
+ // calculate C2* and f2*
+ // C2* is actually floor(C2*) in this case
+ // C2* and f2* need shifting and masking, as shown by
+ // shiftright128[] and maskhigh128[]
+ // the top Ex bits of 10^(-x1) are T* = ten2mk128trunc[ind], e.g.
+ // if x1=1, T*=ten2mk128trunc[0]=0x19999999999999999999999999999999
+ // if (0 < f2* < 10^(-x1)) then
+ // if floor(C1+C2*) is even then C2* = floor(C2*) - logical right
+ // shift; C2* has p decimal digits, correct by Prop. 1)
+ // else if floor(C1+C2*) is odd C2* = floor(C2*)-1 (logical right
+ // shift; C2* has p decimal digits, correct by Pr. 1)
+ // else
+ // C2* = floor(C2*) (logical right shift; C has p decimal digits,
+ // correct by Property 1)
+ // n = C2* * 10^(e2+x1)
+
+ if (ind <= 2) {
+ highf2star.w[1] = 0x0;
+ highf2star.w[0] = 0x0; // low f2* ok
+ } else if (ind <= 21) {
+ highf2star.w[1] = 0x0;
+ highf2star.w[0] = R256.w[2] & maskhigh128[ind]; // low f2* ok
+ } else {
+ highf2star.w[1] = R256.w[3] & maskhigh128[ind];
+ highf2star.w[0] = R256.w[2]; // low f2* is ok
+ }
+ // shift right C2* by Ex-128 = shiftright128[ind]
+ if (ind >= 3) {
+ shift = shiftright128[ind];
+ if (shift < 64) { // 3 <= shift <= 63
+ R256.w[2] =
+ (R256.w[2] >> shift) | (R256.w[3] << (64 - shift));
+ R256.w[3] = (R256.w[3] >> shift);
+ } else { // 66 <= shift <= 102
+ R256.w[2] = (R256.w[3] >> (shift - 64));
+ R256.w[3] = 0x0ULL;
+ }
+ }
+ if (second_pass) {
+ is_inexact_lt_midpoint = 0;
+ is_inexact_gt_midpoint = 0;
+ is_midpoint_lt_even = 0;
+ is_midpoint_gt_even = 0;
+ }
+ // determine inexactness of the rounding of C2* (this may be
+ // followed by a second rounding only if we get P34+1
+ // decimal digits)
+ // if (0 < f2* - 1/2 < 10^(-x1)) then
+ // the result is exact
+ // else (if f2* - 1/2 > T* then)
+ // the result of is inexact
+ if (ind <= 2) {
+ if (R256.w[1] > 0x8000000000000000ull ||
+ (R256.w[1] == 0x8000000000000000ull
+ && R256.w[0] > 0x0ull)) {
+ // f2* > 1/2 and the result may be exact
+ tmp64A = R256.w[1] - 0x8000000000000000ull; // f* - 1/2
+ if ((tmp64A > ten2mk128trunc[ind].w[1]
+ || (tmp64A == ten2mk128trunc[ind].w[1]
+ && R256.w[0] >= ten2mk128trunc[ind].w[0]))) {
+ // set the inexact flag
+ // *pfpsf |= INEXACT_EXCEPTION;
+ tmp_inexact = 1; // may be set again during a second pass
+ // this rounding is applied to C2 only!
+ if (x_sign == y_sign)
+ is_inexact_lt_midpoint = 1;
+ else // if (x_sign != y_sign)
+ is_inexact_gt_midpoint = 1;
+ } // else the result is exact
+ // rounding down, unless a midpoint in [ODD, EVEN]
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ // *pfpsf |= INEXACT_EXCEPTION;
+ tmp_inexact = 1; // just in case we will round a second time
+ // rounding up, unless a midpoint in [EVEN, ODD]
+ // this rounding is applied to C2 only!
+ if (x_sign == y_sign)
+ is_inexact_gt_midpoint = 1;
+ else // if (x_sign != y_sign)
+ is_inexact_lt_midpoint = 1;
+ }
+ } else if (ind <= 21) { // if 3 <= ind <= 21
+ if (highf2star.w[1] > 0x0 || (highf2star.w[1] == 0x0
+ && highf2star.w[0] >
+ onehalf128[ind])
+ || (highf2star.w[1] == 0x0
+ && highf2star.w[0] == onehalf128[ind]
+ && (R256.w[1] || R256.w[0]))) {
+ // f2* > 1/2 and the result may be exact
+ // Calculate f2* - 1/2
+ tmp64A = highf2star.w[0] - onehalf128[ind];
+ tmp64B = highf2star.w[1];
+ if (tmp64A > highf2star.w[0])
+ tmp64B--;
+ if (tmp64B || tmp64A
+ || R256.w[1] > ten2mk128trunc[ind].w[1]
+ || (R256.w[1] == ten2mk128trunc[ind].w[1]
+ && R256.w[0] > ten2mk128trunc[ind].w[0])) {
+ // set the inexact flag
+ // *pfpsf |= INEXACT_EXCEPTION;
+ tmp_inexact = 1; // may be set again during a second pass
+ // this rounding is applied to C2 only!
+ if (x_sign == y_sign)
+ is_inexact_lt_midpoint = 1;
+ else // if (x_sign != y_sign)
+ is_inexact_gt_midpoint = 1;
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ // *pfpsf |= INEXACT_EXCEPTION;
+ tmp_inexact = 1; // may be set again during a second pass
+ // rounding up, unless a midpoint in [EVEN, ODD]
+ // this rounding is applied to C2 only!
+ if (x_sign == y_sign)
+ is_inexact_gt_midpoint = 1;
+ else // if (x_sign != y_sign)
+ is_inexact_lt_midpoint = 1;
+ }
+ } else { // if 22 <= ind <= 33
+ if (highf2star.w[1] > onehalf128[ind]
+ || (highf2star.w[1] == onehalf128[ind]
+ && (highf2star.w[0] || R256.w[1]
+ || R256.w[0]))) {
+ // f2* > 1/2 and the result may be exact
+ // Calculate f2* - 1/2
+ // tmp64A = highf2star.w[0];
+ tmp64B = highf2star.w[1] - onehalf128[ind];
+ if (tmp64B || highf2star.w[0]
+ || R256.w[1] > ten2mk128trunc[ind].w[1]
+ || (R256.w[1] == ten2mk128trunc[ind].w[1]
+ && R256.w[0] > ten2mk128trunc[ind].w[0])) {
+ // set the inexact flag
+ // *pfpsf |= INEXACT_EXCEPTION;
+ tmp_inexact = 1; // may be set again during a second pass
+ // this rounding is applied to C2 only!
+ if (x_sign == y_sign)
+ is_inexact_lt_midpoint = 1;
+ else // if (x_sign != y_sign)
+ is_inexact_gt_midpoint = 1;
+ } // else the result is exact
+ } else { // the result is inexact; f2* <= 1/2
+ // set the inexact flag
+ // *pfpsf |= INEXACT_EXCEPTION;
+ tmp_inexact = 1; // may be set again during a second pass
+ // rounding up, unless a midpoint in [EVEN, ODD]
+ // this rounding is applied to C2 only!
+ if (x_sign == y_sign)
+ is_inexact_gt_midpoint = 1;
+ else // if (x_sign != y_sign)
+ is_inexact_lt_midpoint = 1;
+ }
+ }
+ // check for midpoints
+ if ((R256.w[1] || R256.w[0]) && (highf2star.w[1] == 0)
+ && (highf2star.w[0] == 0)
+ && (R256.w[1] < ten2mk128trunc[ind].w[1]
+ || (R256.w[1] == ten2mk128trunc[ind].w[1]
+ && R256.w[0] <= ten2mk128trunc[ind].w[0]))) {
+ // the result is a midpoint
+ if ((tmp64 + R256.w[2]) & 0x01) { // MP in [EVEN, ODD]
+ // if floor(C2*) is odd C = floor(C2*) - 1; the result may be 0
+ R256.w[2]--;
+ if (R256.w[2] == 0xffffffffffffffffull)
+ R256.w[3]--;
+ // this rounding is applied to C2 only!
+ if (x_sign == y_sign)
+ is_midpoint_gt_even = 1;
+ else // if (x_sign != y_sign)
+ is_midpoint_lt_even = 1;
+ is_inexact_lt_midpoint = 0;
+ is_inexact_gt_midpoint = 0;
+ } else {
+ // else MP in [ODD, EVEN]
+ // this rounding is applied to C2 only!
+ if (x_sign == y_sign)
+ is_midpoint_lt_even = 1;
+ else // if (x_sign != y_sign)
+ is_midpoint_gt_even = 1;
+ is_inexact_lt_midpoint = 0;
+ is_inexact_gt_midpoint = 0;
+ }
+ }
+ // end if (ind >= 0)
+ } else { // if (ind == -1); only during a 2nd pass, and when x1 = 0
+ R256.w[2] = C2_lo;
+ R256.w[3] = C2_hi;
+ tmp_inexact = 0;
+ // to correct a possible setting to 1 from 1st pass
+ if (second_pass) {
+ is_midpoint_lt_even = 0;
+ is_midpoint_gt_even = 0;
+ is_inexact_lt_midpoint = 0;
+ is_inexact_gt_midpoint = 0;
+ }
+ }
+ // and now add/subtract C1 * 10^(e1-e2-x1) +/- (C2 * 10^(-x1))rnd,P34
+ if (x_sign == y_sign) { // addition; could overflow
+ // no second pass is possible this way (only for x_sign != y_sign)
+ C1.w[0] = C1.w[0] + R256.w[2];
+ C1.w[1] = C1.w[1] + R256.w[3];
+ if (C1.w[0] < tmp64)
+ C1.w[1]++; // carry
+ // if the sum has P34+1 digits, i.e. C1>=10^34 redo the calculation
+ // with x1=x1+1
+ if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] >= 0x378d8e6400000000ull)) { // C1 >= 10^34
+ // chop off one more digit from the sum, but make sure there is
+ // no double-rounding error (see table - double rounding logic)
+ // now round C1 from P34+1 to P34 decimal digits
+ // C1' = C1 + 1/2 * 10 = C1 + 5
+ if (C1.w[0] >= 0xfffffffffffffffbull) { // low half add has carry
+ C1.w[0] = C1.w[0] + 5;
+ C1.w[1] = C1.w[1] + 1;
+ } else {
+ C1.w[0] = C1.w[0] + 5;
+ }
+ // the approximation of 10^(-1) was rounded up to 118 bits
+ __mul_128x128_to_256 (Q256, C1, ten2mk128[0]); // Q256 = C1*, f1*
+ // C1* is actually floor(C1*) in this case
+ // the top 128 bits of 10^(-1) are
+ // T* = ten2mk128trunc[0]=0x19999999999999999999999999999999
+ // if (0 < f1* < 10^(-1)) then
+ // if floor(C1*) is even then C1* = floor(C1*) - logical right
+ // shift; C1* has p decimal digits, correct by Prop. 1)
+ // else if floor(C1*) is odd C1* = floor(C1*) - 1 (logical right
+ // shift; C1* has p decimal digits, correct by Pr. 1)
+ // else
+ // C1* = floor(C1*) (logical right shift; C has p decimal digits
+ // correct by Property 1)
+ // n = C1* * 10^(e2+x1+1)
+ if ((Q256.w[1] || Q256.w[0])
+ && (Q256.w[1] < ten2mk128trunc[0].w[1]
+ || (Q256.w[1] == ten2mk128trunc[0].w[1]
+ && Q256.w[0] <= ten2mk128trunc[0].w[0]))) {
+ // the result is a midpoint
+ if (is_inexact_lt_midpoint) { // for the 1st rounding
+ is_inexact_gt_midpoint = 1;
+ is_inexact_lt_midpoint = 0;
+ is_midpoint_gt_even = 0;
+ is_midpoint_lt_even = 0;
+ } else if (is_inexact_gt_midpoint) { // for the 1st rounding
+ Q256.w[2]--;
+ if (Q256.w[2] == 0xffffffffffffffffull)
+ Q256.w[3]--;
+ is_inexact_gt_midpoint = 0;
+ is_inexact_lt_midpoint = 1;
+ is_midpoint_gt_even = 0;
+ is_midpoint_lt_even = 0;
+ } else if (is_midpoint_gt_even) { // for the 1st rounding
+ // Note: cannot have is_midpoint_lt_even
+ is_inexact_gt_midpoint = 0;
+ is_inexact_lt_midpoint = 1;
+ is_midpoint_gt_even = 0;
+ is_midpoint_lt_even = 0;
+ } else { // the first rounding must have been exact
+ if (Q256.w[2] & 0x01) { // MP in [EVEN, ODD]
+ // the truncated result is correct
+ Q256.w[2]--;
+ if (Q256.w[2] == 0xffffffffffffffffull)
+ Q256.w[3]--;
+ is_inexact_gt_midpoint = 0;
+ is_inexact_lt_midpoint = 0;
+ is_midpoint_gt_even = 1;
+ is_midpoint_lt_even = 0;
+ } else { // MP in [ODD, EVEN]
+ is_inexact_gt_midpoint = 0;
+ is_inexact_lt_midpoint = 0;
+ is_midpoint_gt_even = 0;
+ is_midpoint_lt_even = 1;
+ }
+ }
+ tmp_inexact = 1; // in all cases
+ } else { // the result is not a midpoint
+ // determine inexactness of the rounding of C1 (the sum C1+C2*)
+ // if (0 < f1* - 1/2 < 10^(-1)) then
+ // the result is exact
+ // else (if f1* - 1/2 > T* then)
+ // the result of is inexact
+ // ind = 0
+ if (Q256.w[1] > 0x8000000000000000ull
+ || (Q256.w[1] == 0x8000000000000000ull
+ && Q256.w[0] > 0x0ull)) {
+ // f1* > 1/2 and the result may be exact
+ Q256.w[1] = Q256.w[1] - 0x8000000000000000ull; // f1* - 1/2
+ if ((Q256.w[1] > ten2mk128trunc[0].w[1]
+ || (Q256.w[1] == ten2mk128trunc[0].w[1]
+ && Q256.w[0] > ten2mk128trunc[0].w[0]))) {
+ is_inexact_gt_midpoint = 0;
+ is_inexact_lt_midpoint = 1;
+ is_midpoint_gt_even = 0;
+ is_midpoint_lt_even = 0;
+ // set the inexact flag
+ tmp_inexact = 1;
+ // *pfpsf |= INEXACT_EXCEPTION;
+ } else { // else the result is exact for the 2nd rounding
+ if (tmp_inexact) { // if the previous rounding was inexact
+ if (is_midpoint_lt_even) {
+ is_inexact_gt_midpoint = 1;
+ is_midpoint_lt_even = 0;
+ } else if (is_midpoint_gt_even) {
+ is_inexact_lt_midpoint = 1;
+ is_midpoint_gt_even = 0;
+ } else {
+ ; // no change
+ }
+ }
+ }
+ // rounding down, unless a midpoint in [ODD, EVEN]
+ } else { // the result is inexact; f1* <= 1/2
+ is_inexact_gt_midpoint = 1;
+ is_inexact_lt_midpoint = 0;
+ is_midpoint_gt_even = 0;
+ is_midpoint_lt_even = 0;
+ // set the inexact flag
+ tmp_inexact = 1;
+ // *pfpsf |= INEXACT_EXCEPTION;
+ }
+ } // end 'the result is not a midpoint'
+ // n = C1 * 10^(e2+x1)
+ C1.w[1] = Q256.w[3];
+ C1.w[0] = Q256.w[2];
+ y_exp = y_exp + ((UINT64) (x1 + 1) << 49);
+ } else { // C1 < 10^34
+ // C1.w[1] and C1.w[0] already set
+ // n = C1 * 10^(e2+x1)
+ y_exp = y_exp + ((UINT64) x1 << 49);
+ }
+ // check for overflow
+ if (y_exp == EXP_MAX_P1
+ && (rnd_mode == ROUNDING_TO_NEAREST
+ || rnd_mode == ROUNDING_TIES_AWAY)) {
+ res.w[1] = 0x7800000000000000ull | x_sign; // +/-inf
+ res.w[0] = 0x0ull;
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // set the overflow flag
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ BID_SWAP128 (res);
+ BID_RETURN (res);
+ } // else no overflow
+ } else { // if x_sign != y_sign the result of this subtract. is exact
+ C1.w[0] = C1.w[0] - R256.w[2];
+ C1.w[1] = C1.w[1] - R256.w[3];
+ if (C1.w[0] > tmp64)
+ C1.w[1]--; // borrow
+ if (C1.w[1] >= 0x8000000000000000ull) { // negative coefficient!
+ C1.w[0] = ~C1.w[0];
+ C1.w[0]++;
+ C1.w[1] = ~C1.w[1];
+ if (C1.w[0] == 0x0)
+ C1.w[1]++;
+ tmp_sign = y_sign;
+ // the result will have the sign of y if last rnd
+ } else {
+ tmp_sign = x_sign;
+ }
+ // if the difference has P34-1 digits or less, i.e. C1 < 10^33 then
+ // redo the calculation with x1=x1-1;
+ // redo the calculation also if C1 = 10^33 and
+ // (is_inexact_gt_midpoint or is_midpoint_lt_even);
+ // (the last part should have really been
+ // (is_inexact_lt_midpoint or is_midpoint_gt_even) from
+ // the rounding of C2, but the position flags have been reversed)
+ // 10^33 = 0x0000314dc6448d93 0x38c15b0a00000000
+ if ((C1.w[1] < 0x0000314dc6448d93ull || (C1.w[1] == 0x0000314dc6448d93ull && C1.w[0] < 0x38c15b0a00000000ull)) || (C1.w[1] == 0x0000314dc6448d93ull && C1.w[0] == 0x38c15b0a00000000ull && (is_inexact_gt_midpoint || is_midpoint_lt_even))) { // C1=10^33
+ x1 = x1 - 1; // x1 >= 0
+ if (x1 >= 0) {
+ // clear position flags and tmp_inexact
+ is_midpoint_lt_even = 0;
+ is_midpoint_gt_even = 0;
+ is_inexact_lt_midpoint = 0;
+ is_inexact_gt_midpoint = 0;
+ tmp_inexact = 0;
+ second_pass = 1;
+ goto roundC2; // else result has less than P34 digits
+ }
+ }
+ // if the coefficient of the result is 10^34 it means that this
+ // must be the second pass, and we are done
+ if (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] == 0x378d8e6400000000ull) { // if C1 = 10^34
+ C1.w[1] = 0x0000314dc6448d93ull; // C1 = 10^33
+ C1.w[0] = 0x38c15b0a00000000ull;
+ y_exp = y_exp + ((UINT64) 1 << 49);
+ }
+ x_sign = tmp_sign;
+ if (x1 >= 1)
+ y_exp = y_exp + ((UINT64) x1 << 49);
+ // x1 = -1 is possible at the end of a second pass when the
+ // first pass started with x1 = 1
+ }
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ // general correction from RN to RA, RM, RP, RZ; result uses y_exp
+ if (rnd_mode != ROUNDING_TO_NEAREST) {
+ if ((!x_sign
+ && ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint)
+ ||
+ ((rnd_mode == ROUNDING_TIES_AWAY
+ || rnd_mode == ROUNDING_UP)
+ && is_midpoint_gt_even))) || (x_sign
+ &&
+ ((rnd_mode ==
+ ROUNDING_DOWN
+ &&
+ is_inexact_lt_midpoint)
+ ||
+ ((rnd_mode ==
+ ROUNDING_TIES_AWAY
+ || rnd_mode ==
+ ROUNDING_DOWN)
+ &&
+ is_midpoint_gt_even))))
+ {
+ // C1 = C1 + 1
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ y_exp = y_exp + EXP_P1;
+ }
+ } else if ((is_midpoint_lt_even || is_inexact_gt_midpoint)
+ &&
+ ((x_sign
+ && (rnd_mode == ROUNDING_UP
+ || rnd_mode == ROUNDING_TO_ZERO))
+ || (!x_sign
+ && (rnd_mode == ROUNDING_DOWN
+ || rnd_mode == ROUNDING_TO_ZERO)))) {
+ // C1 = C1 - 1
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi--;
+ // check if we crossed into the lower decade
+ if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
+ C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
+ C1_lo = 0x378d8e63ffffffffull;
+ y_exp = y_exp - EXP_P1;
+ // no underflow, because delta + q2 >= P34 + 1
+ }
+ } else {
+ ; // exact, the result is already correct
+ }
+ // in all cases check for overflow (RN and RA solved already)
+ if (y_exp == EXP_MAX_P1) { // overflow
+ if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
+ (rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ } else { // RM and res > 0, RP and res < 0, or RZ
+ C1_hi = 0x5fffed09bead87c0ull;
+ C1_lo = 0x378d8e63ffffffffull;
+ }
+ y_exp = 0; // x_sign is preserved
+ // set the inexact flag (in case the exact addition was exact)
+ *pfpsf |= INEXACT_EXCEPTION;
+ // set the overflow flag
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ // assemble the result
+ res.w[1] = x_sign | y_exp | C1_hi;
+ res.w[0] = C1_lo;
+ if (tmp_inexact)
+ *pfpsf |= INEXACT_EXCEPTION;
+ }
+ } else { // if (-P34 + 1 <= delta <= -1) <=> 1 <= -delta <= P34 - 1
+ // NOTE: the following, up to "} else { // if x_sign != y_sign
+ // the result is exact" is identical to "else if (delta == P34 - q2) {"
+ // from above; also, the code is not symmetric: a+b and b+a may take
+ // different paths (need to unify eventually!)
+ // calculate C' = C2 + C1 * 10^(e1-e2) directly; the result may be
+ // inexact if it requires P34 + 1 decimal digits; in either case the
+ // 'cutoff' point for addition is at the position of the lsb of C2
+ // The coefficient of the result is C1 * 10^(e1-e2) + C2 and the
+ // exponent is e2; either C1 or 10^(e1-e2) may not fit is 64 bits,
+ // but their product fits with certainty in 128 bits (actually in 113)
+ // Note that 0 <= e1 - e2 <= P34 - 2
+ // -P34 + 1 <= delta <= -1 <=> -P34 + 1 <= delta <= -1 <=>
+ // -P34 + 1 <= q1 + e1 - q2 - e2 <= -1 <=>
+ // q2 - q1 - P34 + 1 <= e1 - e2 <= q2 - q1 - 1 <=>
+ // 1 - P34 - P34 + 1 <= e1-e2 <= P34 - 1 - 1 => 0 <= e1-e2 <= P34 - 2
+ scale = delta - q1 + q2; // scale = (int)(e1 >> 49) - (int)(e2 >> 49)
+ if (scale >= 20) { // 10^(e1-e2) does not fit in 64 bits, but C1 does
+ __mul_128x64_to_128 (C1, C1_lo, ten2k128[scale - 20]);
+ } else if (scale >= 1) {
+ // if 1 <= scale <= 19 then 10^(e1-e2) fits in 64 bits
+ if (q1 <= 19) { // C1 fits in 64 bits
+ __mul_64x64_to_128MACH (C1, C1_lo, ten2k64[scale]);
+ } else { // q1 >= 20
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo;
+ __mul_128x64_to_128 (C1, ten2k64[scale], C1);
+ }
+ } else { // if (scale == 0) C1 is unchanged
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo; // only the low part is necessary
+ }
+ C1_hi = C1.w[1];
+ C1_lo = C1.w[0];
+ // now add C2
+ if (x_sign == y_sign) {
+ // the result can overflow!
+ C1_lo = C1_lo + C2_lo;
+ C1_hi = C1_hi + C2_hi;
+ if (C1_lo < C1.w[0])
+ C1_hi++;
+ // test for overflow, possible only when C1 >= 10^34
+ if (C1_hi > 0x0001ed09bead87c0ull || (C1_hi == 0x0001ed09bead87c0ull && C1_lo >= 0x378d8e6400000000ull)) { // C1 >= 10^34
+ // in this case q = P34 + 1 and x = q - P34 = 1, so multiply
+ // C'' = C'+ 5 = C1 + 5 by k1 ~ 10^(-1) calculated for P34 + 1
+ // decimal digits
+ // Calculate C'' = C' + 1/2 * 10^x
+ if (C1_lo >= 0xfffffffffffffffbull) { // low half add has carry
+ C1_lo = C1_lo + 5;
+ C1_hi = C1_hi + 1;
+ } else {
+ C1_lo = C1_lo + 5;
+ }
+ // the approximation of 10^(-1) was rounded up to 118 bits
+ // 10^(-1) =~ 33333333333333333333333333333400 * 2^-129
+ // 10^(-1) =~ 19999999999999999999999999999a00 * 2^-128
+ C1.w[1] = C1_hi;
+ C1.w[0] = C1_lo; // C''
+ ten2m1.w[1] = 0x1999999999999999ull;
+ ten2m1.w[0] = 0x9999999999999a00ull;
+ __mul_128x128_to_256 (P256, C1, ten2m1); // P256 = C*, f*
+ // C* is actually floor(C*) in this case
+ // the top Ex = 128 bits of 10^(-1) are
+ // T* = 0x00199999999999999999999999999999
+ // if (0 < f* < 10^(-x)) then
+ // 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^(e2+x)
+ if ((P256.w[1] || P256.w[0])
+ && (P256.w[1] < 0x1999999999999999ull
+ || (P256.w[1] == 0x1999999999999999ull
+ && P256.w[0] <= 0x9999999999999999ull))) {
+ // the result is a midpoint
+ if (P256.w[2] & 0x01) {
+ is_midpoint_gt_even = 1;
+ // if floor(C*) is odd C = floor(C*) - 1; the result is not 0
+ P256.w[2]--;
+ if (P256.w[2] == 0xffffffffffffffffull)
+ P256.w[3]--;
+ } else {
+ is_midpoint_lt_even = 1;
+ }
+ }
+ // n = Cstar * 10^(e2+1)
+ y_exp = y_exp + EXP_P1;
+ // C* != 10^P34 because C* has P34 digits
+ // check for overflow
+ if (y_exp == EXP_MAX_P1
+ && (rnd_mode == ROUNDING_TO_NEAREST
+ || rnd_mode == ROUNDING_TIES_AWAY)) {
+ // overflow for RN
+ res.w[1] = x_sign | 0x7800000000000000ull; // +/-inf
+ res.w[0] = 0x0ull;
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ // set the overflow flag
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ BID_SWAP128 (res);
+ BID_RETURN (res);
+ }
+ // if (0 < f* - 1/2 < 10^(-x)) then
+ // the result of the addition is exact
+ // else
+ // the result of the addition is inexact
+ if (P256.w[1] > 0x8000000000000000ull || (P256.w[1] == 0x8000000000000000ull && P256.w[0] > 0x0ull)) { // the result may be exact
+ tmp64 = P256.w[1] - 0x8000000000000000ull; // f* - 1/2
+ if ((tmp64 > 0x1999999999999999ull
+ || (tmp64 == 0x1999999999999999ull
+ && P256.w[0] >= 0x9999999999999999ull))) {
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ is_inexact = 1;
+ } // else the result is exact
+ } else { // the result is inexact
+ // set the inexact flag
+ *pfpsf |= INEXACT_EXCEPTION;
+ is_inexact = 1;
+ }
+ C1_hi = P256.w[3];
+ C1_lo = P256.w[2];
+ if (!is_midpoint_gt_even && !is_midpoint_lt_even) {
+ is_inexact_lt_midpoint = is_inexact
+ && (P256.w[1] & 0x8000000000000000ull);
+ is_inexact_gt_midpoint = is_inexact
+ && !(P256.w[1] & 0x8000000000000000ull);
+ }
+ // general correction from RN to RA, RM, RP, RZ; result uses y_exp
+ if (rnd_mode != ROUNDING_TO_NEAREST) {
+ if ((!x_sign
+ && ((rnd_mode == ROUNDING_UP
+ && is_inexact_lt_midpoint)
+ || ((rnd_mode == ROUNDING_TIES_AWAY
+ || rnd_mode == ROUNDING_UP)
+ && is_midpoint_gt_even))) || (x_sign
+ &&
+ ((rnd_mode ==
+ ROUNDING_DOWN
+ &&
+ is_inexact_lt_midpoint)
+ ||
+ ((rnd_mode ==
+ ROUNDING_TIES_AWAY
+ || rnd_mode
+ ==
+ ROUNDING_DOWN)
+ &&
+ is_midpoint_gt_even))))
+ {
+ // C1 = C1 + 1
+ C1_lo = C1_lo + 1;
+ if (C1_lo == 0) { // rounding overflow in the low 64 bits
+ C1_hi = C1_hi + 1;
+ }
+ if (C1_hi == 0x0001ed09bead87c0ull
+ && C1_lo == 0x378d8e6400000000ull) {
+ // C1 = 10^34 => rounding overflow
+ C1_hi = 0x0000314dc6448d93ull;
+ C1_lo = 0x38c15b0a00000000ull; // 10^33
+ y_exp = y_exp + EXP_P1;
+ }
+ } else
+ if ((is_midpoint_lt_even || is_inexact_gt_midpoint) &&
+ ((x_sign && (rnd_mode == ROUNDING_UP ||
+ rnd_mode == ROUNDING_TO_ZERO)) ||
+ (!x_sign && (rnd_mode == ROUNDING_DOWN ||
+ rnd_mode == ROUNDING_TO_ZERO)))) {
+ // C1 = C1 - 1
+ C1_lo = C1_lo - 1;
+ if (C1_lo == 0xffffffffffffffffull)
+ C1_hi--;
+ // check if we crossed into the lower decade
+ if (C1_hi == 0x0000314dc6448d93ull && C1_lo == 0x38c15b09ffffffffull) { // 10^33 - 1
+ C1_hi = 0x0001ed09bead87c0ull; // 10^34 - 1
+ C1_lo = 0x378d8e63ffffffffull;
+ y_exp = y_exp - EXP_P1;
+ // no underflow, because delta + q2 >= P34 + 1
+ }
+ } else {
+ ; // exact, the result is already correct
+ }
+ // in all cases check for overflow (RN and RA solved already)
+ if (y_exp == EXP_MAX_P1) { // overflow
+ if ((rnd_mode == ROUNDING_DOWN && x_sign) || // RM and res < 0
+ (rnd_mode == ROUNDING_UP && !x_sign)) { // RP and res > 0
+ C1_hi = 0x7800000000000000ull; // +inf
+ C1_lo = 0x0ull;
+ } else { // RM and res > 0, RP and res < 0, or RZ
+ C1_hi = 0x5fffed09bead87c0ull;
+ C1_lo = 0x378d8e63ffffffffull;
+ }
+ y_exp = 0; // x_sign is preserved
+ // set the inexact flag (in case the exact addition was exact)
+ *pfpsf |= INEXACT_EXCEPTION;
+ // set the overflow flag
+ *pfpsf |= OVERFLOW_EXCEPTION;
+ }
+ }
+ } // else if (C1 < 10^34) then C1 is the coeff.; the result is exact
+ // assemble the result
+ res.w[1] = x_sign | y_exp | C1_hi;
+ res.w[0] = C1_lo;
+ } else { // if x_sign != y_sign the result is exact
+ C1_lo = C2_lo - C1_lo;
+ C1_hi = C2_hi - C1_hi;
+ if (C1_lo > C2_lo)
+ C1_hi--;
+ if (C1_hi >= 0x8000000000000000ull) { // negative coefficient!
+ C1_lo = ~C1_lo;
+ C1_lo++;
+ C1_hi = ~C1_hi;
+ if (C1_lo == 0x0)
+ C1_hi++;
+ x_sign = y_sign; // the result will have the sign of y
+ }
+ // the result can be zero, but it cannot overflow
+ if (C1_lo == 0 && C1_hi == 0) {
+ // assemble the result
+ if (x_exp < y_exp)
+ res.w[1] = x_exp;
+ else
+ res.w[1] = y_exp;
+ res.w[0] = 0;
+ if (rnd_mode == ROUNDING_DOWN) {
+ res.w[1] |= 0x8000000000000000ull;
+ }
+ BID_SWAP128 (res);
+ BID_RETURN (res);
+ }
+ // assemble the result
+ res.w[1] = y_sign | y_exp | C1_hi;
+ res.w[0] = C1_lo;
+ }
}
}
+ BID_SWAP128 (res);
BID_RETURN (res)
}
}
+
+
+// bid128_sub stands for bid128qq_sub
+
/*****************************************************************************
* BID128 sub
****************************************************************************/
#if DECIMAL_CALL_BY_REFERENCE
void
-__bid128_sub (UINT128 * pres, UINT128 * px,
- UINT128 *
- py _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
- _EXC_INFO_PARAM) {
+bid128_sub (UINT128 * pres, UINT128 * px, UINT128 * py
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
UINT128 x = *px, y = *py;
#if !DECIMAL_GLOBAL_ROUNDING
unsigned int rnd_mode = *prnd_mode;
#endif
#else
UINT128
-__bid128_sub (UINT128 x,
- UINT128 y _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
- _EXC_INFO_PARAM) {
+bid128_sub (UINT128 x, UINT128 y
+ _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM
+ _EXC_INFO_PARAM) {
#endif
UINT128 res;
UINT64 y_sign;
- if ((y.w[1] & MASK_NAN) != MASK_NAN) { // y is not NAN
+ if ((y.w[HIGH_128W] & MASK_NAN) != MASK_NAN) { // y is not NAN
// change its sign
- y_sign = y.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
+ y_sign = y.w[HIGH_128W] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative
if (y_sign)
- y.w[1] = y.w[1] & 0x7fffffffffffffffull;
+ y.w[HIGH_128W] = y.w[HIGH_128W] & 0x7fffffffffffffffull;
else
- y.w[1] = y.w[1] | 0x8000000000000000ull;
+ y.w[HIGH_128W] = y.w[HIGH_128W] | 0x8000000000000000ull;
}
#if DECIMAL_CALL_BY_REFERENCE
- __bid128_add (&res, &x,
- &y _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
- _EXC_INFO_ARG);
+ bid128_add (&res, &x, &y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
#else
- res =
- __bid128_add (x,
- y _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
- _EXC_INFO_ARG);
+ res = bid128_add (x, y
+ _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG
+ _EXC_INFO_ARG);
#endif
BID_RETURN (res);
}
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