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/* Copyright (C) 2007, 2009 Free Software Foundation, Inc. This file is part of GCC. GCC is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. GCC is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. Under Section 7 of GPL version 3, you are granted additional permissions described in the GCC Runtime Library Exception, version 3.1, as published by the Free Software Foundation. You should have received a copy of the GNU General Public License and a copy of the GCC Runtime Library Exception along with this program; see the files COPYING3 and COPYING.RUNTIME respectively. If not, see <http://www.gnu.org/licenses/>. */ /***************************************************************************** * * BID128 fma x * y + z * ****************************************************************************/ #include "bid_internal.h" static void rounding_correction (unsigned int rnd_mode, unsigned int is_inexact_lt_midpoint, unsigned int is_inexact_gt_midpoint, unsigned int is_midpoint_lt_even, unsigned int is_midpoint_gt_even, int unbexp, UINT128 * ptrres, _IDEC_flags * ptrfpsf) { // unbiased true exponent unbexp may be larger than emax UINT128 res = *ptrres; // expected to have the correct sign and coefficient // (the exponent field is ignored, as unbexp is used instead) UINT64 sign, exp; UINT64 C_hi, C_lo; // general correction from RN to RA, RM, RP, RZ // Note: if the result is negative, then is_inexact_lt_midpoint, // is_inexact_gt_midpoint, is_midpoint_lt_even, and is_midpoint_gt_even // have to be considered as if determined for the absolute value of the // result (so they seem to be reversed) if (is_inexact_lt_midpoint || is_inexact_gt_midpoint || is_midpoint_lt_even || is_midpoint_gt_even) { *ptrfpsf |= INEXACT_EXCEPTION; } // apply correction to result calculated with unbounded exponent sign = res.w[1] & MASK_SIGN; exp = (UINT64) (unbexp + 6176) << 49; // valid only if expmin<=unbexp<=expmax C_hi = res.w[1] & MASK_COEFF; C_lo = res.w[0]; if ((!sign && ((rnd_mode == ROUNDING_UP && is_inexact_lt_midpoint) || ((rnd_mode == ROUNDING_TIES_AWAY || rnd_mode == ROUNDING_UP) && is_midpoint_gt_even))) || (sign && ((rnd_mode == ROUNDING_DOWN && is_inexact_lt_midpoint) || ((rnd_mode == ROUNDING_TIES_AWAY || rnd_mode == ROUNDING_DOWN) && is_midpoint_gt_even)))) { // C = C + 1 C_lo = C_lo + 1; if (C_lo == 0) C_hi = C_hi + 1; if (C_hi == 0x0001ed09bead87c0ull && C_lo == 0x378d8e6400000000ull) { // C = 10^34 => rounding overflow C_hi = 0x0000314dc6448d93ull; C_lo = 0x38c15b0a00000000ull; // 10^33 // exp = exp + EXP_P1; unbexp = unbexp + 1; exp = (UINT64) (unbexp + 6176) << 49; } } else if ((is_midpoint_lt_even || is_inexact_gt_midpoint) && ((sign && (rnd_mode == ROUNDING_UP || rnd_mode == ROUNDING_TO_ZERO)) || (!sign && (rnd_mode == ROUNDING_DOWN || rnd_mode == ROUNDING_TO_ZERO)))) { // C = C - 1 C_lo = C_lo - 1; if (C_lo == 0xffffffffffffffffull) C_hi--; // check if we crossed into the lower decade if (C_hi == 0x0000314dc6448d93ull && C_lo == 0x38c15b09ffffffffull) { // C = 10^33 - 1 if (exp > 0) { C_hi = 0x0001ed09bead87c0ull; // 10^34 - 1 C_lo = 0x378d8e63ffffffffull; // exp = exp - EXP_P1; unbexp = unbexp - 1; exp = (UINT64) (unbexp + 6176) << 49; } else { // if exp = 0 if (is_midpoint_lt_even || is_midpoint_lt_even || is_inexact_gt_midpoint || is_inexact_gt_midpoint) // tiny & inexact *ptrfpsf |= UNDERFLOW_EXCEPTION; } } } else { ; // the result is already correct } if (unbexp > expmax) { // 6111 *ptrfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); exp = 0; if (!sign) { // result is positive if (rnd_mode == ROUNDING_UP || rnd_mode == ROUNDING_TIES_AWAY) { // +inf C_hi = 0x7800000000000000ull; C_lo = 0x0000000000000000ull; } else { // res = +MAXFP = (10^34-1) * 10^emax C_hi = 0x5fffed09bead87c0ull; C_lo = 0x378d8e63ffffffffull; } } else { // result is negative if (rnd_mode == ROUNDING_DOWN || rnd_mode == ROUNDING_TIES_AWAY) { // -inf C_hi = 0xf800000000000000ull; C_lo = 0x0000000000000000ull; } else { // res = -MAXFP = -(10^34-1) * 10^emax C_hi = 0xdfffed09bead87c0ull; C_lo = 0x378d8e63ffffffffull; } } } // assemble the result res.w[1] = sign | exp | C_hi; res.w[0] = C_lo; *ptrres = res; } static void add256 (UINT256 x, UINT256 y, UINT256 * pz) { // *z = x + yl assume the sum fits in 256 bits UINT256 z; z.w[0] = x.w[0] + y.w[0]; if (z.w[0] < x.w[0]) { x.w[1]++; if (x.w[1] == 0x0000000000000000ull) { x.w[2]++; if (x.w[2] == 0x0000000000000000ull) { x.w[3]++; } } } z.w[1] = x.w[1] + y.w[1]; if (z.w[1] < x.w[1]) { x.w[2]++; if (x.w[2] == 0x0000000000000000ull) { x.w[3]++; } } z.w[2] = x.w[2] + y.w[2]; if (z.w[2] < x.w[2]) { x.w[3]++; } z.w[3] = x.w[3] + y.w[3]; // it was assumed that no carry is possible *pz = z; } static void sub256 (UINT256 x, UINT256 y, UINT256 * pz) { // *z = x - y; assume x >= y UINT256 z; z.w[0] = x.w[0] - y.w[0]; if (z.w[0] > x.w[0]) { x.w[1]--; if (x.w[1] == 0xffffffffffffffffull) { x.w[2]--; if (x.w[2] == 0xffffffffffffffffull) { x.w[3]--; } } } z.w[1] = x.w[1] - y.w[1]; if (z.w[1] > x.w[1]) { x.w[2]--; if (x.w[2] == 0xffffffffffffffffull) { x.w[3]--; } } z.w[2] = x.w[2] - y.w[2]; if (z.w[2] > x.w[2]) { x.w[3]--; } z.w[3] = x.w[3] - y.w[3]; // no borrow possible, because x >= y *pz = z; } static int nr_digits256 (UINT256 R256) { int ind; // determine the number of decimal digits in R256 if (R256.w[3] == 0x0 && R256.w[2] == 0x0 && R256.w[1] == 0x0) { // between 1 and 19 digits for (ind = 1; ind <= 19; ind++) { if (R256.w[0] < ten2k64[ind]) { break; } } // ind digits } else if (R256.w[3] == 0x0 && R256.w[2] == 0x0 && (R256.w[1] < ten2k128[0].w[1] || (R256.w[1] == ten2k128[0].w[1] && R256.w[0] < ten2k128[0].w[0]))) { // 20 digits ind = 20; } else if (R256.w[3] == 0x0 && R256.w[2] == 0x0) { // between 21 and 38 digits for (ind = 1; ind <= 18; ind++) { if (R256.w[1] < ten2k128[ind].w[1] || (R256.w[1] == ten2k128[ind].w[1] && R256.w[0] < ten2k128[ind].w[0])) { break; } } // ind + 20 digits ind = ind + 20; } else if (R256.w[3] == 0x0 && (R256.w[2] < ten2k256[0].w[2] || (R256.w[2] == ten2k256[0].w[2] && R256.w[1] < ten2k256[0].w[1]) || (R256.w[2] == ten2k256[0].w[2] && R256.w[1] == ten2k256[0].w[1] && R256.w[0] < ten2k256[0].w[0]))) { // 39 digits ind = 39; } else { // between 40 and 68 digits for (ind = 1; ind <= 29; ind++) { if (R256.w[3] < ten2k256[ind].w[3] || (R256.w[3] == ten2k256[ind].w[3] && R256.w[2] < ten2k256[ind].w[2]) || (R256.w[3] == ten2k256[ind].w[3] && R256.w[2] == ten2k256[ind].w[2] && R256.w[1] < ten2k256[ind].w[1]) || (R256.w[3] == ten2k256[ind].w[3] && R256.w[2] == ten2k256[ind].w[2] && R256.w[1] == ten2k256[ind].w[1] && R256.w[0] < ten2k256[ind].w[0])) { break; } } // ind + 39 digits ind = ind + 39; } return (ind); } // add/subtract C4 and C3 * 10^scale; this may follow a previous rounding, so // use the rounding information from ptr_is_* to avoid a double rounding error static void add_and_round (int q3, int q4, int e4, int delta, int p34, UINT64 z_sign, UINT64 p_sign, UINT128 C3, UINT256 C4, int rnd_mode, int *ptr_is_midpoint_lt_even, int *ptr_is_midpoint_gt_even, int *ptr_is_inexact_lt_midpoint, int *ptr_is_inexact_gt_midpoint, _IDEC_flags * ptrfpsf, UINT128 * ptrres) { int scale; int x0; int ind; UINT64 R64; UINT128 P128, R128; UINT192 P192, R192; UINT256 R256; int is_midpoint_lt_even = 0; int is_midpoint_gt_even = 0; int is_inexact_lt_midpoint = 0; int is_inexact_gt_midpoint = 0; int is_midpoint_lt_even0 = 0; int is_midpoint_gt_even0 = 0; int is_inexact_lt_midpoint0 = 0; int is_inexact_gt_midpoint0 = 0; int incr_exp = 0; int is_tiny = 0; int lt_half_ulp = 0; int eq_half_ulp = 0; // int gt_half_ulp = 0; UINT128 res = *ptrres; // scale C3 up by 10^(q4-delta-q3), 0 <= q4-delta-q3 <= 2*P34-2 = 66 scale = q4 - delta - q3; // 0 <= scale <= 66 (or 0 <= scale <= 68 if this // comes from Cases (2), (3), (4), (5), (6), with 0 <= |delta| <= 1 // calculate C3 * 10^scale in R256 (it has at most 67 decimal digits for // Cases (15),(16),(17) and at most 69 for Cases (2),(3),(4),(5),(6)) if (scale == 0) { R256.w[3] = 0x0ull; R256.w[2] = 0x0ull; R256.w[1] = C3.w[1]; R256.w[0] = C3.w[0]; } else if (scale <= 19) { // 10^scale fits in 64 bits P128.w[1] = 0; P128.w[0] = ten2k64[scale]; __mul_128x128_to_256 (R256, P128, C3); } else if (scale <= 38) { // 10^scale fits in 128 bits __mul_128x128_to_256 (R256, ten2k128[scale - 20], C3); } else if (scale <= 57) { // 39 <= scale <= 57 // 10^scale fits in 192 bits but C3 * 10^scale fits in 223 or 230 bits // (10^67 has 223 bits; 10^69 has 230 bits); // must split the computation: // 10^scale * C3 = 10*38 * 10^(scale-38) * C3 where 10^38 takes 127 // bits and so 10^(scale-38) * C3 fits in 128 bits with certainty // Note that 1 <= scale - 38 <= 19 => 10^(scale-38) fits in 64 bits __mul_64x128_to_128 (R128, ten2k64[scale - 38], C3); // now multiply R128 by 10^38 __mul_128x128_to_256 (R256, R128, ten2k128[18]); } else { // 58 <= scale <= 66 // 10^scale takes between 193 and 220 bits, // and C3 * 10^scale fits in 223 bits (10^67/10^69 has 223/230 bits) // must split the computation: // 10^scale * C3 = 10*38 * 10^(scale-38) * C3 where 10^38 takes 127 // bits and so 10^(scale-38) * C3 fits in 128 bits with certainty // Note that 20 <= scale - 38 <= 30 => 10^(scale-38) fits in 128 bits // Calculate first 10^(scale-38) * C3, which fits in 128 bits; because // 10^(scale-38) takes more than 64 bits, C3 will take less than 64 __mul_64x128_to_128 (R128, C3.w[0], ten2k128[scale - 58]); // now calculate 10*38 * 10^(scale-38) * C3 __mul_128x128_to_256 (R256, R128, ten2k128[18]); } // C3 * 10^scale is now in R256 // for Cases (15), (16), (17) C4 > C3 * 10^scale because C4 has at least // one extra digit; for Cases (2), (3), (4), (5), or (6) any order is // possible // add/subtract C4 and C3 * 10^scale; the exponent is e4 if (p_sign == z_sign) { // R256 = C4 + R256 // calculate R256 = C4 + C3 * 10^scale = C4 + R256 which is exact, // but may require rounding add256 (C4, R256, &R256); } else { // if (p_sign != z_sign) { // R256 = C4 - R256 // calculate R256 = C4 - C3 * 10^scale = C4 - R256 or // R256 = C3 * 10^scale - C4 = R256 - C4 which is exact, // but may require rounding // compare first R256 = C3 * 10^scale and C4 if (R256.w[3] > C4.w[3] || (R256.w[3] == C4.w[3] && R256.w[2] > C4.w[2]) || (R256.w[3] == C4.w[3] && R256.w[2] == C4.w[2] && R256.w[1] > C4.w[1]) || (R256.w[3] == C4.w[3] && R256.w[2] == C4.w[2] && R256.w[1] == C4.w[1] && R256.w[0] >= C4.w[0])) { // C3 * 10^scale >= C4 // calculate R256 = C3 * 10^scale - C4 = R256 - C4, which is exact, // but may require rounding sub256 (R256, C4, &R256); // flip p_sign too, because the result has the sign of z p_sign = z_sign; } else { // if C4 > C3 * 10^scale // calculate R256 = C4 - C3 * 10^scale = C4 - R256, which is exact, // but may require rounding sub256 (C4, R256, &R256); } // if the result is pure zero, the sign depends on the rounding mode // (x*y and z had opposite signs) if (R256.w[3] == 0x0ull && R256.w[2] == 0x0ull && R256.w[1] == 0x0ull && R256.w[0] == 0x0ull) { if (rnd_mode != ROUNDING_DOWN) p_sign = 0x0000000000000000ull; else p_sign = 0x8000000000000000ull; // the exponent is max (e4, expmin) if (e4 < -6176) e4 = expmin; // assemble result res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49); res.w[0] = 0x0; *ptrres = res; return; } } // determine the number of decimal digits in R256 ind = nr_digits256 (R256); // the exact result is (-1)^p_sign * R256 * 10^e4 where q (R256) = ind; // round to the destination precision, with unbounded exponent if (ind <= p34) { // result rounded to the destination precision with unbounded exponent // is exact if (ind + e4 < p34 + expmin) { is_tiny = 1; // applies to all rounding modes } res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | R256.w[1]; res.w[0] = R256.w[0]; // Note: res is correct only if expmin <= e4 <= expmax } else { // if (ind > p34) // if more than P digits, round to nearest to P digits // round R256 to p34 digits x0 = ind - p34; // 1 <= x0 <= 34 as 35 <= ind <= 68 if (ind <= 38) { P128.w[1] = R256.w[1]; P128.w[0] = R256.w[0]; round128_19_38 (ind, x0, P128, &R128, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); } else if (ind <= 57) { P192.w[2] = R256.w[2]; P192.w[1] = R256.w[1]; P192.w[0] = R256.w[0]; round192_39_57 (ind, x0, P192, &R192, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R128.w[1] = R192.w[1]; R128.w[0] = R192.w[0]; } else { // if (ind <= 68) round256_58_76 (ind, x0, R256, &R256, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R128.w[1] = R256.w[1]; R128.w[0] = R256.w[0]; } // the rounded result has p34 = 34 digits e4 = e4 + x0 + incr_exp; if (rnd_mode == ROUNDING_TO_NEAREST) { if (e4 < expmin) { is_tiny = 1; // for other rounding modes apply correction } } else { // for RM, RP, RZ, RA apply correction in order to determine tininess // but do not save the result; apply the correction to // (-1)^p_sign * significand * 10^0 P128.w[1] = p_sign | 0x3040000000000000ull | R128.w[1]; P128.w[0] = R128.w[0]; rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, 0, &P128, ptrfpsf); scale = ((P128.w[1] & MASK_EXP) >> 49) - 6176; // -1, 0, or +1 // the number of digits in the significand is p34 = 34 if (e4 + scale < expmin) { is_tiny = 1; } } ind = p34; // the number of decimal digits in the signifcand of res res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | R128.w[1]; // RN res.w[0] = R128.w[0]; // Note: res is correct only if expmin <= e4 <= expmax // set the inexact flag after rounding with bounded exponent, if any } // at this point we have the result rounded with unbounded exponent in // res and we know its tininess: // res = (-1)^p_sign * significand * 10^e4, // where q (significand) = ind <= p34 // Note: res is correct only if expmin <= e4 <= expmax // check for overflow if RN if (rnd_mode == ROUNDING_TO_NEAREST && (ind + e4) > (p34 + expmax)) { res.w[1] = p_sign | 0x7800000000000000ull; res.w[0] = 0x0000000000000000ull; *ptrres = res; *ptrfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); return; // BID_RETURN (res) } // else not overflow or not RN, so continue // if (e4 >= expmin) we have the result rounded with bounded exponent if (e4 < expmin) { x0 = expmin - e4; // x0 >= 1; the number of digits to chop off of res // where the result rounded [at most] once is // (-1)^p_sign * significand_res * 10^e4 // avoid double rounding error is_inexact_lt_midpoint0 = is_inexact_lt_midpoint; is_inexact_gt_midpoint0 = is_inexact_gt_midpoint; is_midpoint_lt_even0 = is_midpoint_lt_even; is_midpoint_gt_even0 = is_midpoint_gt_even; is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; if (x0 > ind) { // nothing is left of res when moving the decimal point left x0 digits is_inexact_lt_midpoint = 1; res.w[1] = p_sign | 0x0000000000000000ull; res.w[0] = 0x0000000000000000ull; e4 = expmin; } else if (x0 == ind) { // 1 <= x0 = ind <= p34 = 34 // this is <, =, or > 1/2 ulp // compare the ind-digit value in the significand of res with // 1/2 ulp = 5*10^(ind-1), i.e. determine whether it is // less than, equal to, or greater than 1/2 ulp (significand of res) R128.w[1] = res.w[1] & MASK_COEFF; R128.w[0] = res.w[0]; if (ind <= 19) { if (R128.w[0] < midpoint64[ind - 1]) { // < 1/2 ulp lt_half_ulp = 1; is_inexact_lt_midpoint = 1; } else if (R128.w[0] == midpoint64[ind - 1]) { // = 1/2 ulp eq_half_ulp = 1; is_midpoint_gt_even = 1; } else { // > 1/2 ulp // gt_half_ulp = 1; is_inexact_gt_midpoint = 1; } } else { // if (ind <= 38) { if (R128.w[1] < midpoint128[ind - 20].w[1] || (R128.w[1] == midpoint128[ind - 20].w[1] && R128.w[0] < midpoint128[ind - 20].w[0])) { // < 1/2 ulp lt_half_ulp = 1; is_inexact_lt_midpoint = 1; } else if (R128.w[1] == midpoint128[ind - 20].w[1] && R128.w[0] == midpoint128[ind - 20].w[0]) { // = 1/2 ulp eq_half_ulp = 1; is_midpoint_gt_even = 1; } else { // > 1/2 ulp // gt_half_ulp = 1; is_inexact_gt_midpoint = 1; } } if (lt_half_ulp || eq_half_ulp) { // res = +0.0 * 10^expmin res.w[1] = 0x0000000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // if (gt_half_ulp) // res = +1 * 10^expmin res.w[1] = 0x0000000000000000ull; res.w[0] = 0x0000000000000001ull; } res.w[1] = p_sign | res.w[1]; e4 = expmin; } else { // if (1 <= x0 <= ind - 1 <= 33) // round the ind-digit result to ind - x0 digits if (ind <= 18) { // 2 <= ind <= 18 round64_2_18 (ind, x0, res.w[0], &R64, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); res.w[1] = 0x0; res.w[0] = R64; } else if (ind <= 38) { P128.w[1] = res.w[1] & MASK_COEFF; P128.w[0] = res.w[0]; round128_19_38 (ind, x0, P128, &res, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); } e4 = e4 + x0; // expmin // we want the exponent to be expmin, so if incr_exp = 1 then // multiply the rounded result by 10 - it will still fit in 113 bits if (incr_exp) { // 64 x 128 -> 128 P128.w[1] = res.w[1] & MASK_COEFF; P128.w[0] = res.w[0]; __mul_64x128_to_128 (res, ten2k64[1], P128); } res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | (res.w[1] & MASK_COEFF); // avoid a double rounding error if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) && is_midpoint_lt_even) { // double rounding error upward // res = res - 1 res.w[0]--; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; // Note: a double rounding error upward is not possible; for this // the result after the first rounding would have to be 99...95 // (35 digits in all), possibly followed by a number of zeros; this // is not possible in Cases (2)-(6) or (15)-(17) which may get here is_midpoint_lt_even = 0; is_inexact_lt_midpoint = 1; } else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) && is_midpoint_gt_even) { // double rounding error downward // res = res + 1 res.w[0]++; if (res.w[0] == 0) res.w[1]++; is_midpoint_gt_even = 0; is_inexact_gt_midpoint = 1; } else if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // if this second rounding was exact the result may still be // inexact because of the first rounding if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) { is_inexact_gt_midpoint = 1; } if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) { is_inexact_lt_midpoint = 1; } } else if (is_midpoint_gt_even && (is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) { // pulled up to a midpoint is_inexact_lt_midpoint = 1; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else if (is_midpoint_lt_even && (is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) { // pulled down to a midpoint is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else { ; } } } // res contains the correct result // apply correction if not rounding to nearest if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e4, &res, ptrfpsf); } if (is_midpoint_lt_even || is_midpoint_gt_even || is_inexact_lt_midpoint || is_inexact_gt_midpoint) { // set the inexact flag *ptrfpsf |= INEXACT_EXCEPTION; if (is_tiny) *ptrfpsf |= UNDERFLOW_EXCEPTION; } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; *ptrres = res; return; } #if DECIMAL_CALL_BY_REFERENCE static void bid128_ext_fma (int *ptr_is_midpoint_lt_even, int *ptr_is_midpoint_gt_even, int *ptr_is_inexact_lt_midpoint, int *ptr_is_inexact_gt_midpoint, UINT128 * pres, UINT128 * px, UINT128 * py, UINT128 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT128 x = *px, y = *py, z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else static UINT128 bid128_ext_fma (int *ptr_is_midpoint_lt_even, int *ptr_is_midpoint_gt_even, int *ptr_is_inexact_lt_midpoint, int *ptr_is_inexact_gt_midpoint, UINT128 x, UINT128 y, UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} }; UINT64 x_sign, y_sign, z_sign, p_sign, tmp_sign; UINT64 x_exp = 0, y_exp = 0, z_exp = 0, p_exp; int true_p_exp; UINT128 C1, C2, C3; UINT256 C4; int q1 = 0, q2 = 0, q3 = 0, q4; int e1, e2, e3, e4; int scale, ind, delta, x0; int p34 = P34; // used to modify the limit on the number of digits BID_UI64DOUBLE tmp; int x_nr_bits, y_nr_bits, z_nr_bits; unsigned int save_fpsf; int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0; int is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0; int is_midpoint_lt_even0 = 0, is_midpoint_gt_even0 = 0; int is_inexact_lt_midpoint0 = 0, is_inexact_gt_midpoint0 = 0; int incr_exp = 0; int lsb; int lt_half_ulp = 0; int eq_half_ulp = 0; int gt_half_ulp = 0; int is_tiny = 0; UINT64 R64, tmp64; UINT128 P128, R128; UINT192 P192, R192; UINT256 R256; // the following are based on the table of special cases for fma; the NaN // behavior is similar to that of the IA-64 Architecture fma // identify cases where at least one operand is NaN BID_SWAP128 (x); BID_SWAP128 (y); BID_SWAP128 (z); if ((y.w[1] & MASK_NAN) == MASK_NAN) { // y is NAN // if x = {0, f, inf, NaN}, y = NaN, z = {0, f, inf, NaN} then res = Q (y) // 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]; // if z = SNaN or x = SNaN signal invalid exception if ((z.w[1] & MASK_SNAN) == MASK_SNAN || (x.w[1] & MASK_SNAN) == MASK_SNAN) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; } } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else if ((z.w[1] & MASK_NAN) == MASK_NAN) { // z is NAN // if x = {0, f, inf, NaN}, y = {0, f, inf}, z = NaN then res = Q (z) // check first for non-canonical NaN payload if (((z.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || (((z.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && (z.w[0] > 0x38c15b09ffffffffull))) { z.w[1] = z.w[1] & 0xffffc00000000000ull; z.w[0] = 0x0ull; } if ((z.w[1] & MASK_SNAN) == MASK_SNAN) { // z is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return quiet (z) res.w[1] = z.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] res.w[0] = z.w[0]; } else { // z is QNaN // return z res.w[1] = z.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] res.w[0] = z.w[0]; // if x = SNaN signal invalid exception if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // set invalid flag *pfpsf |= INVALID_EXCEPTION; } } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else if ((x.w[1] & MASK_NAN) == MASK_NAN) { // x is NAN // if x = NaN, y = {0, f, inf}, z = {0, f, inf} then res = Q (x) // check first for non-canonical NaN payload if (((x.w[1] & 0x00003fffffffffffull) > 0x0000314dc6448d93ull) || (((x.w[1] & 0x00003fffffffffffull) == 0x0000314dc6448d93ull) && (x.w[0] > 0x38c15b09ffffffffull))) { x.w[1] = x.w[1] & 0xffffc00000000000ull; x.w[0] = 0x0ull; } if ((x.w[1] & MASK_SNAN) == MASK_SNAN) { // x is SNAN // set invalid flag *pfpsf |= INVALID_EXCEPTION; // return quiet (x) res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out also G[6]-G[16] res.w[0] = x.w[0]; } else { // x is QNaN // return x res.w[1] = x.w[1] & 0xfc003fffffffffffull; // clear out G[6]-G[16] res.w[0] = x.w[0]; } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } // x, y, z are 0, f, or inf but not NaN => unpack the arguments and check // for non-canonical values x_sign = x.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative C1.w[1] = x.w[1] & MASK_COEFF; C1.w[0] = x.w[0]; if ((x.w[1] & MASK_ANY_INF) != MASK_INF) { // x != inf // if 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.w[1] = 0; // significand high C1.w[0] = 0; // significand low } else { // G0_G1 != 11 x_exp = x.w[1] & MASK_EXP; // biased and shifted left 49 bits if (C1.w[1] > 0x0001ed09bead87c0ull || (C1.w[1] == 0x0001ed09bead87c0ull && C1.w[0] > 0x378d8e63ffffffffull)) { // x is non-canonical if coefficient is larger than 10^34 -1 C1.w[1] = 0; C1.w[0] = 0; } else { // canonical ; } } } y_sign = y.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative C2.w[1] = y.w[1] & MASK_COEFF; C2.w[0] = y.w[0]; if ((y.w[1] & MASK_ANY_INF) != MASK_INF) { // y != inf // if 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.w[1] = 0; // significand high C2.w[0] = 0; // significand low } else { // G0_G1 != 11 y_exp = y.w[1] & MASK_EXP; // biased and shifted left 49 bits if (C2.w[1] > 0x0001ed09bead87c0ull || (C2.w[1] == 0x0001ed09bead87c0ull && C2.w[0] > 0x378d8e63ffffffffull)) { // y is non-canonical if coefficient is larger than 10^34 -1 C2.w[1] = 0; C2.w[0] = 0; } else { // canonical ; } } } z_sign = z.w[1] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative C3.w[1] = z.w[1] & MASK_COEFF; C3.w[0] = z.w[0]; if ((z.w[1] & MASK_ANY_INF) != MASK_INF) { // z != inf // if z is not infinity check for non-canonical values - treated as zero if ((z.w[1] & 0x6000000000000000ull) == 0x6000000000000000ull) { // G0_G1=11 // non-canonical z_exp = (z.w[1] << 2) & MASK_EXP; // biased and shifted left 49 bits C3.w[1] = 0; // significand high C3.w[0] = 0; // significand low } else { // G0_G1 != 11 z_exp = z.w[1] & MASK_EXP; // biased and shifted left 49 bits if (C3.w[1] > 0x0001ed09bead87c0ull || (C3.w[1] == 0x0001ed09bead87c0ull && C3.w[0] > 0x378d8e63ffffffffull)) { // z is non-canonical if coefficient is larger than 10^34 -1 C3.w[1] = 0; C3.w[0] = 0; } else { // canonical ; } } } p_sign = x_sign ^ y_sign; // sign of the product // identify cases where at least one operand is infinity if ((x.w[1] & MASK_ANY_INF) == MASK_INF) { // x = inf if ((y.w[1] & MASK_ANY_INF) == MASK_INF) { // y = inf if ((z.w[1] & MASK_ANY_INF) == MASK_INF) { // z = inf if (p_sign == z_sign) { res.w[1] = z_sign | MASK_INF; res.w[0] = 0x0; } else { // return QNaN Indefinite res.w[1] = 0x7c00000000000000ull; res.w[0] = 0x0000000000000000ull; // set invalid flag *pfpsf |= INVALID_EXCEPTION; } } else { // z = 0 or z = f res.w[1] = p_sign | MASK_INF; res.w[0] = 0x0; } } else if (C2.w[1] != 0 || C2.w[0] != 0) { // y = f if ((z.w[1] & MASK_ANY_INF) == MASK_INF) { // z = inf if (p_sign == z_sign) { res.w[1] = z_sign | MASK_INF; res.w[0] = 0x0; } else { // return QNaN Indefinite res.w[1] = 0x7c00000000000000ull; res.w[0] = 0x0000000000000000ull; // set invalid flag *pfpsf |= INVALID_EXCEPTION; } } else { // z = 0 or z = f res.w[1] = p_sign | MASK_INF; res.w[0] = 0x0; } } else { // y = 0 // return QNaN Indefinite res.w[1] = 0x7c00000000000000ull; res.w[0] = 0x0000000000000000ull; // set invalid flag *pfpsf |= INVALID_EXCEPTION; } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else if ((y.w[1] & MASK_ANY_INF) == MASK_INF) { // y = inf if ((z.w[1] & MASK_ANY_INF) == MASK_INF) { // z = inf // x = f, necessarily if ((p_sign != z_sign) || (C1.w[1] == 0x0ull && C1.w[0] == 0x0ull)) { // return QNaN Indefinite res.w[1] = 0x7c00000000000000ull; res.w[0] = 0x0000000000000000ull; // set invalid flag *pfpsf |= INVALID_EXCEPTION; } else { res.w[1] = z_sign | MASK_INF; res.w[0] = 0x0; } } else if (C1.w[1] == 0x0 && C1.w[0] == 0x0) { // x = 0 // z = 0, f, inf // return QNaN Indefinite res.w[1] = 0x7c00000000000000ull; res.w[0] = 0x0000000000000000ull; // set invalid flag *pfpsf |= INVALID_EXCEPTION; } else { // x = f and z = 0, f, necessarily res.w[1] = p_sign | MASK_INF; res.w[0] = 0x0; } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else if ((z.w[1] & MASK_ANY_INF) == MASK_INF) { // z = inf // x = 0, f and y = 0, f, necessarily res.w[1] = z_sign | MASK_INF; res.w[0] = 0x0; *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } true_p_exp = (x_exp >> 49) - 6176 + (y_exp >> 49) - 6176; if (true_p_exp < -6176) p_exp = 0; // cannot be less than EXP_MIN else p_exp = (UINT64) (true_p_exp + 6176) << 49; if (((C1.w[1] == 0x0 && C1.w[0] == 0x0) || (C2.w[1] == 0x0 && C2.w[0] == 0x0)) && C3.w[1] == 0x0 && C3.w[0] == 0x0) { // (x = 0 or y = 0) and z = 0 // the result is 0 if (p_exp < z_exp) res.w[1] = p_exp; // preferred exponent else res.w[1] = z_exp; // preferred exponent if (p_sign == z_sign) { res.w[1] |= z_sign; res.w[0] = 0x0; } else { // x * y and z have opposite signs if (rnd_mode == ROUNDING_DOWN) { // res = -0.0 res.w[1] |= MASK_SIGN; res.w[0] = 0x0; } else { // res = +0.0 // res.w[1] |= 0x0; res.w[0] = 0x0; } } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } // from this point on, we may need to know the number of decimal digits // in the significands of x, y, z when x, y, z != 0 if (C1.w[1] != 0 || C1.w[0] != 0) { // x = f (non-zero finite) // q1 = nr. of decimal digits in x // determine first the nr. of bits in x if (C1.w[1] == 0) { if (C1.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C1.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp.d = (double) (C1.w[0] >> 32); // exact conversion x_nr_bits = 33 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp.d = (double) (C1.w[0]); // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp.d = (double) C1.w[0]; // exact conversion x_nr_bits = 1 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C1.w[1] != 0 => nr. bits = 64 + nr_bits (C1.w[1]) tmp.d = (double) C1.w[1]; // exact conversion x_nr_bits = 65 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } q1 = nr_digits[x_nr_bits - 1].digits; if (q1 == 0) { q1 = nr_digits[x_nr_bits - 1].digits1; if (C1.w[1] > nr_digits[x_nr_bits - 1].threshold_hi || (C1.w[1] == nr_digits[x_nr_bits - 1].threshold_hi && C1.w[0] >= nr_digits[x_nr_bits - 1].threshold_lo)) q1++; } } if (C2.w[1] != 0 || C2.w[0] != 0) { // y = f (non-zero finite) if (C2.w[1] == 0) { if (C2.w[0] >= 0x0020000000000000ull) { // y >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C2.w[0] >= 0x0000000100000000ull) { // y >= 2^32 tmp.d = (double) (C2.w[0] >> 32); // exact conversion y_nr_bits = 32 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // y < 2^32 tmp.d = (double) C2.w[0]; // exact conversion y_nr_bits = ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if y < 2^53 tmp.d = (double) C2.w[0]; // exact conversion y_nr_bits = ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C2.w[1] != 0 => nr. bits = 64 + nr_bits (C2.w[1]) tmp.d = (double) C2.w[1]; // exact conversion y_nr_bits = 64 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } q2 = nr_digits[y_nr_bits].digits; if (q2 == 0) { q2 = nr_digits[y_nr_bits].digits1; if (C2.w[1] > nr_digits[y_nr_bits].threshold_hi || (C2.w[1] == nr_digits[y_nr_bits].threshold_hi && C2.w[0] >= nr_digits[y_nr_bits].threshold_lo)) q2++; } } if (C3.w[1] != 0 || C3.w[0] != 0) { // z = f (non-zero finite) if (C3.w[1] == 0) { if (C3.w[0] >= 0x0020000000000000ull) { // z >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C3.w[0] >= 0x0000000100000000ull) { // z >= 2^32 tmp.d = (double) (C3.w[0] >> 32); // exact conversion z_nr_bits = 32 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // z < 2^32 tmp.d = (double) C3.w[0]; // exact conversion z_nr_bits = ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if z < 2^53 tmp.d = (double) C3.w[0]; // exact conversion z_nr_bits = ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C3.w[1] != 0 => nr. bits = 64 + nr_bits (C3.w[1]) tmp.d = (double) C3.w[1]; // exact conversion z_nr_bits = 64 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } q3 = nr_digits[z_nr_bits].digits; if (q3 == 0) { q3 = nr_digits[z_nr_bits].digits1; if (C3.w[1] > nr_digits[z_nr_bits].threshold_hi || (C3.w[1] == nr_digits[z_nr_bits].threshold_hi && C3.w[0] >= nr_digits[z_nr_bits].threshold_lo)) q3++; } } if ((C1.w[1] == 0x0 && C1.w[0] == 0x0) || (C2.w[1] == 0x0 && C2.w[0] == 0x0)) { // x = 0 or y = 0 // z = f, necessarily; for 0 + z return z, with the preferred exponent // the result is z, but need to get the preferred exponent if (z_exp <= p_exp) { // the preferred exponent is z_exp res.w[1] = z_sign | (z_exp & MASK_EXP) | C3.w[1]; res.w[0] = C3.w[0]; } else { // if (p_exp < z_exp) the preferred exponent is p_exp // return (C3 * 10^scale) * 10^(z_exp - scale) // where scale = min (p34-q3, (z_exp-p_exp) >> 49) scale = p34 - q3; ind = (z_exp - p_exp) >> 49; if (ind < scale) scale = ind; if (scale == 0) { res.w[1] = z.w[1]; // & MASK_COEFF, which is redundant res.w[0] = z.w[0]; } else if (q3 <= 19) { // z fits in 64 bits if (scale <= 19) { // 10^scale fits in 64 bits // 64 x 64 C3.w[0] * ten2k64[scale] __mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]); } else { // 10^scale fits in 128 bits // 64 x 128 C3.w[0] * ten2k128[scale - 20] __mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]); } } else { // z fits in 128 bits, but 10^scale must fit in 64 bits // 64 x 128 ten2k64[scale] * C3 __mul_128x64_to_128 (res, ten2k64[scale], C3); } // subtract scale from the exponent z_exp = z_exp - ((UINT64) scale << 49); res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else { ; // continue with x = f, y = f, z = 0 or x = f, y = f, z = f } e1 = (x_exp >> 49) - 6176; // unbiased exponent of x e2 = (y_exp >> 49) - 6176; // unbiased exponent of y e3 = (z_exp >> 49) - 6176; // unbiased exponent of z e4 = e1 + e2; // unbiased exponent of the exact x * y // calculate C1 * C2 and its number of decimal digits, q4 // the exact product has either q1 + q2 - 1 or q1 + q2 decimal digits // where 2 <= q1 + q2 <= 68 // calculate C4 = C1 * C2 and determine q C4.w[3] = C4.w[2] = C4.w[1] = C4.w[0] = 0; if (q1 + q2 <= 19) { // if 2 <= q1 + q2 <= 19, C4 = C1 * C2 fits in 64 bits C4.w[0] = C1.w[0] * C2.w[0]; // if C4 < 10^(q1+q2-1) then q4 = q1 + q2 - 1 else q4 = q1 + q2 if (C4.w[0] < ten2k64[q1 + q2 - 1]) q4 = q1 + q2 - 1; // q4 in [1, 18] else q4 = q1 + q2; // q4 in [2, 19] // length of C1 * C2 rounded up to a multiple of 64 bits is len = 64; } else if (q1 + q2 == 20) { // C4 = C1 * C2 fits in 64 or 128 bits // q1 <= 19 and q2 <= 19 so both C1 and C2 fit in 64 bits __mul_64x64_to_128MACH (C4, C1.w[0], C2.w[0]); // if C4 < 10^(q1+q2-1) = 10^19 then q4 = q1+q2-1 = 19 else q4 = q1+q2 = 20 if (C4.w[1] == 0 && C4.w[0] < ten2k64[19]) { // 19 = q1+q2-1 // length of C1 * C2 rounded up to a multiple of 64 bits is len = 64; q4 = 19; // 19 = q1 + q2 - 1 } else { // if (C4.w[1] == 0) // length of C1 * C2 rounded up to a multiple of 64 bits is len = 64; // else // length of C1 * C2 rounded up to a multiple of 64 bits is len = 128; q4 = 20; // 20 = q1 + q2 } } else if (q1 + q2 <= 38) { // 21 <= q1 + q2 <= 38 // C4 = C1 * C2 fits in 64 or 128 bits // (64 bits possibly, but only when q1 + q2 = 21 and C4 has 20 digits) // at least one of C1, C2 has at most 19 decimal digits & fits in 64 bits if (q1 <= 19) { __mul_128x64_to_128 (C4, C1.w[0], C2); } else { // q2 <= 19 __mul_128x64_to_128 (C4, C2.w[0], C1); } // if C4 < 10^(q1+q2-1) then q4 = q1 + q2 - 1 else q4 = q1 + q2 if (C4.w[1] < ten2k128[q1 + q2 - 21].w[1] || (C4.w[1] == ten2k128[q1 + q2 - 21].w[1] && C4.w[0] < ten2k128[q1 + q2 - 21].w[0])) { // if (C4.w[1] == 0) // q4 = 20, necessarily // length of C1 * C2 rounded up to a multiple of 64 bits is len = 64; // else // length of C1 * C2 rounded up to a multiple of 64 bits is len = 128; q4 = q1 + q2 - 1; // q4 in [20, 37] } else { // length of C1 * C2 rounded up to a multiple of 64 bits is len = 128; q4 = q1 + q2; // q4 in [21, 38] } } else if (q1 + q2 == 39) { // C4 = C1 * C2 fits in 128 or 192 bits // both C1 and C2 fit in 128 bits (actually in 113 bits) // may replace this by 128x128_to192 __mul_128x128_to_256 (C4, C1, C2); // C4.w[3] is 0 // if C4 < 10^(q1+q2-1) = 10^38 then q4 = q1+q2-1 = 38 else q4 = q1+q2 = 39 if (C4.w[2] == 0 && (C4.w[1] < ten2k128[18].w[1] || (C4.w[1] == ten2k128[18].w[1] && C4.w[0] < ten2k128[18].w[0]))) { // 18 = 38 - 20 = q1+q2-1 - 20 // length of C1 * C2 rounded up to a multiple of 64 bits is len = 128; q4 = 38; // 38 = q1 + q2 - 1 } else { // if (C4.w[2] == 0) // length of C1 * C2 rounded up to a multiple of 64 bits is len = 128; // else // length of C1 * C2 rounded up to a multiple of 64 bits is len = 192; q4 = 39; // 39 = q1 + q2 } } else if (q1 + q2 <= 57) { // 40 <= q1 + q2 <= 57 // C4 = C1 * C2 fits in 128 or 192 bits // (128 bits possibly, but only when q1 + q2 = 40 and C4 has 39 digits) // both C1 and C2 fit in 128 bits (actually in 113 bits); at most one // may fit in 64 bits if (C1.w[1] == 0) { // C1 fits in 64 bits // __mul_64x128_full (REShi64, RESlo128, A64, B128) __mul_64x128_full (C4.w[2], C4, C1.w[0], C2); } else if (C2.w[1] == 0) { // C2 fits in 64 bits // __mul_64x128_full (REShi64, RESlo128, A64, B128) __mul_64x128_full (C4.w[2], C4, C2.w[0], C1); } else { // both C1 and C2 require 128 bits // may use __mul_128x128_to_192 (C4.w[2], C4.w[0], C2.w[0], C1); __mul_128x128_to_256 (C4, C1, C2); // C4.w[3] = 0 } // if C4 < 10^(q1+q2-1) then q4 = q1 + q2 - 1 else q4 = q1 + q2 if (C4.w[2] < ten2k256[q1 + q2 - 40].w[2] || (C4.w[2] == ten2k256[q1 + q2 - 40].w[2] && (C4.w[1] < ten2k256[q1 + q2 - 40].w[1] || (C4.w[1] == ten2k256[q1 + q2 - 40].w[1] && C4.w[0] < ten2k256[q1 + q2 - 40].w[0])))) { // if (C4.w[2] == 0) // q4 = 39, necessarily // length of C1 * C2 rounded up to a multiple of 64 bits is len = 128; // else // length of C1 * C2 rounded up to a multiple of 64 bits is len = 192; q4 = q1 + q2 - 1; // q4 in [39, 56] } else { // length of C1 * C2 rounded up to a multiple of 64 bits is len = 192; q4 = q1 + q2; // q4 in [40, 57] } } else if (q1 + q2 == 58) { // C4 = C1 * C2 fits in 192 or 256 bits // both C1 and C2 fit in 128 bits (actually in 113 bits); at most one // may fit in 64 bits if (C1.w[1] == 0) { // C1 * C2 will fit in 192 bits __mul_64x128_full (C4.w[2], C4, C1.w[0], C2); // may use 64x128_to_192 } else if (C2.w[1] == 0) { // C1 * C2 will fit in 192 bits __mul_64x128_full (C4.w[2], C4, C2.w[0], C1); // may use 64x128_to_192 } else { // C1 * C2 will fit in 192 bits or in 256 bits __mul_128x128_to_256 (C4, C1, C2); } // if C4 < 10^(q1+q2-1) = 10^57 then q4 = q1+q2-1 = 57 else q4 = q1+q2 = 58 if (C4.w[3] == 0 && (C4.w[2] < ten2k256[18].w[2] || (C4.w[2] == ten2k256[18].w[2] && (C4.w[1] < ten2k256[18].w[1] || (C4.w[1] == ten2k256[18].w[1] && C4.w[0] < ten2k256[18].w[0]))))) { // 18 = 57 - 39 = q1+q2-1 - 39 // length of C1 * C2 rounded up to a multiple of 64 bits is len = 192; q4 = 57; // 57 = q1 + q2 - 1 } else { // if (C4.w[3] == 0) // length of C1 * C2 rounded up to a multiple of 64 bits is len = 192; // else // length of C1 * C2 rounded up to a multiple of 64 bits is len = 256; q4 = 58; // 58 = q1 + q2 } } else { // if 59 <= q1 + q2 <= 68 // C4 = C1 * C2 fits in 192 or 256 bits // (192 bits possibly, but only when q1 + q2 = 59 and C4 has 58 digits) // both C1 and C2 fit in 128 bits (actually in 113 bits); none fits in // 64 bits // may use __mul_128x128_to_192 (C4.w[2], C4.w[0], C2.w[0], C1); __mul_128x128_to_256 (C4, C1, C2); // C4.w[3] = 0 // if C4 < 10^(q1+q2-1) then q4 = q1 + q2 - 1 else q4 = q1 + q2 if (C4.w[3] < ten2k256[q1 + q2 - 40].w[3] || (C4.w[3] == ten2k256[q1 + q2 - 40].w[3] && (C4.w[2] < ten2k256[q1 + q2 - 40].w[2] || (C4.w[2] == ten2k256[q1 + q2 - 40].w[2] && (C4.w[1] < ten2k256[q1 + q2 - 40].w[1] || (C4.w[1] == ten2k256[q1 + q2 - 40].w[1] && C4.w[0] < ten2k256[q1 + q2 - 40].w[0])))))) { // if (C4.w[3] == 0) // q4 = 58, necessarily // length of C1 * C2 rounded up to a multiple of 64 bits is len = 192; // else // length of C1 * C2 rounded up to a multiple of 64 bits is len = 256; q4 = q1 + q2 - 1; // q4 in [58, 67] } else { // length of C1 * C2 rounded up to a multiple of 64 bits is len = 256; q4 = q1 + q2; // q4 in [59, 68] } } if (C3.w[1] == 0x0 && C3.w[0] == 0x0) { // x = f, y = f, z = 0 save_fpsf = *pfpsf; // sticky bits - caller value must be preserved *pfpsf = 0; if (q4 > p34) { // truncate C4 to p34 digits into res // x = q4-p34, 1 <= x <= 34 because 35 <= q4 <= 68 x0 = q4 - p34; if (q4 <= 38) { P128.w[1] = C4.w[1]; P128.w[0] = C4.w[0]; round128_19_38 (q4, x0, P128, &res, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); } else if (q4 <= 57) { // 35 <= q4 <= 57 P192.w[2] = C4.w[2]; P192.w[1] = C4.w[1]; P192.w[0] = C4.w[0]; round192_39_57 (q4, x0, P192, &R192, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); res.w[0] = R192.w[0]; res.w[1] = R192.w[1]; } else { // if (q4 <= 68) round256_58_76 (q4, x0, C4, &R256, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); res.w[0] = R256.w[0]; res.w[1] = R256.w[1]; } e4 = e4 + x0; if (incr_exp) { e4 = e4 + 1; } q4 = p34; // res is now the coefficient of the result rounded to the destination // precision, with unbounded exponent; the exponent is e4; q4=digits(res) } else { // if (q4 <= p34) // C4 * 10^e4 is the result rounded to the destination precision, with // unbounded exponent (which is exact) if ((q4 + e4 <= p34 + expmax) && (e4 > expmax)) { // e4 is too large, but can be brought within range by scaling up C4 scale = e4 - expmax; // 1 <= scale < P-q4 <= P-1 => 1 <= scale <= P-2 // res = (C4 * 10^scale) * 10^expmax if (q4 <= 19) { // C4 fits in 64 bits if (scale <= 19) { // 10^scale fits in 64 bits // 64 x 64 C4.w[0] * ten2k64[scale] __mul_64x64_to_128MACH (res, C4.w[0], ten2k64[scale]); } else { // 10^scale fits in 128 bits // 64 x 128 C4.w[0] * ten2k128[scale - 20] __mul_128x64_to_128 (res, C4.w[0], ten2k128[scale - 20]); } } else { // C4 fits in 128 bits, but 10^scale must fit in 64 bits // 64 x 128 ten2k64[scale] * CC43 __mul_128x64_to_128 (res, ten2k64[scale], C4); } e4 = e4 - scale; // expmax q4 = q4 + scale; } else { res.w[1] = C4.w[1]; res.w[0] = C4.w[0]; } // res is the coefficient of the result rounded to the destination // precision, with unbounded exponent (it has q4 digits); the exponent // is e4 (exact result) } // check for overflow if (q4 + e4 > p34 + expmax) { if (rnd_mode == ROUNDING_TO_NEAREST) { res.w[1] = p_sign | 0x7800000000000000ull; // +/-inf res.w[0] = 0x0000000000000000ull; *pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); } else { res.w[1] = p_sign | res.w[1]; rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e4, &res, pfpsf); } *pfpsf |= save_fpsf; *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } // check for underflow if (q4 + e4 < expmin + P34) { is_tiny = 1; // the result is tiny if (e4 < expmin) { // if e4 < expmin, we must truncate more of res x0 = expmin - e4; // x0 >= 1 is_inexact_lt_midpoint0 = is_inexact_lt_midpoint; is_inexact_gt_midpoint0 = is_inexact_gt_midpoint; is_midpoint_lt_even0 = is_midpoint_lt_even; is_midpoint_gt_even0 = is_midpoint_gt_even; is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; // the number of decimal digits in res is q4 if (x0 < q4) { // 1 <= x0 <= q4-1 => round res to q4 - x0 digits if (q4 <= 18) { // 2 <= q4 <= 18, 1 <= x0 <= 17 round64_2_18 (q4, x0, res.w[0], &R64, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); if (incr_exp) { // R64 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 1, 1 <= q4 - x0 <= 17 R64 = ten2k64[q4 - x0]; } // res.w[1] = 0; (from above) res.w[0] = R64; } else { // if (q4 <= 34) // 19 <= q4 <= 38 P128.w[1] = res.w[1]; P128.w[0] = res.w[0]; round128_19_38 (q4, x0, P128, &res, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); if (incr_exp) { // increase coefficient by a factor of 10; this will be <= 10^33 // R128 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 1, 1 <= q4 - x0 <= 37 if (q4 - x0 <= 19) { // 1 <= q4 - x0 <= 19 // res.w[1] = 0; res.w[0] = ten2k64[q4 - x0]; } else { // 20 <= q4 - x0 <= 37 res.w[0] = ten2k128[q4 - x0 - 20].w[0]; res.w[1] = ten2k128[q4 - x0 - 20].w[1]; } } } e4 = e4 + x0; // expmin } else if (x0 == q4) { // the second rounding is for 0.d(0)d(1)...d(q4-1) * 10^emin // determine relationship with 1/2 ulp if (q4 <= 19) { if (res.w[0] < midpoint64[q4 - 1]) { // < 1/2 ulp lt_half_ulp = 1; is_inexact_lt_midpoint = 1; } else if (res.w[0] == midpoint64[q4 - 1]) { // = 1/2 ulp eq_half_ulp = 1; is_midpoint_gt_even = 1; } else { // > 1/2 ulp // gt_half_ulp = 1; is_inexact_gt_midpoint = 1; } } else { // if (q4 <= 34) if (res.w[1] < midpoint128[q4 - 20].w[1] || (res.w[1] == midpoint128[q4 - 20].w[1] && res.w[0] < midpoint128[q4 - 20].w[0])) { // < 1/2 ulp lt_half_ulp = 1; is_inexact_lt_midpoint = 1; } else if (res.w[1] == midpoint128[q4 - 20].w[1] && res.w[0] == midpoint128[q4 - 20].w[0]) { // = 1/2 ulp eq_half_ulp = 1; is_midpoint_gt_even = 1; } else { // > 1/2 ulp // gt_half_ulp = 1; is_inexact_gt_midpoint = 1; } } if (lt_half_ulp || eq_half_ulp) { // res = +0.0 * 10^expmin res.w[1] = 0x0000000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // if (gt_half_ulp) // res = +1 * 10^expmin res.w[1] = 0x0000000000000000ull; res.w[0] = 0x0000000000000001ull; } e4 = expmin; } else { // if (x0 > q4) // the second rounding is for 0.0...d(0)d(1)...d(q4-1) * 10^emin res.w[1] = 0; res.w[0] = 0; e4 = expmin; is_inexact_lt_midpoint = 1; } // avoid a double rounding error if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) && is_midpoint_lt_even) { // double rounding error upward // res = res - 1 res.w[0]--; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; // Note: a double rounding error upward is not possible; for this // the result after the first rounding would have to be 99...95 // (35 digits in all), possibly followed by a number of zeros; this // not possible for f * f + 0 is_midpoint_lt_even = 0; is_inexact_lt_midpoint = 1; } else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) && is_midpoint_gt_even) { // double rounding error downward // res = res + 1 res.w[0]++; if (res.w[0] == 0) res.w[1]++; is_midpoint_gt_even = 0; is_inexact_gt_midpoint = 1; } else if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // if this second rounding was exact the result may still be // inexact because of the first rounding if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) { is_inexact_gt_midpoint = 1; } if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) { is_inexact_lt_midpoint = 1; } } else if (is_midpoint_gt_even && (is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) { // pulled up to a midpoint is_inexact_lt_midpoint = 1; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else if (is_midpoint_lt_even && (is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) { // pulled down to a midpoint is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else { ; } } else { // if e4 >= emin then q4 < P and the result is tiny and exact if (e3 < e4) { // if (e3 < e4) the preferred exponent is e3 // return (C4 * 10^scale) * 10^(e4 - scale) // where scale = min (p34-q4, (e4 - e3)) scale = p34 - q4; ind = e4 - e3; if (ind < scale) scale = ind; if (scale == 0) { ; // res and e4 are unchanged } else if (q4 <= 19) { // C4 fits in 64 bits if (scale <= 19) { // 10^scale fits in 64 bits // 64 x 64 res.w[0] * ten2k64[scale] __mul_64x64_to_128MACH (res, res.w[0], ten2k64[scale]); } else { // 10^scale fits in 128 bits // 64 x 128 res.w[0] * ten2k128[scale - 20] __mul_128x64_to_128 (res, res.w[0], ten2k128[scale - 20]); } } else { // res fits in 128 bits, but 10^scale must fit in 64 bits // 64 x 128 ten2k64[scale] * C3 __mul_128x64_to_128 (res, ten2k64[scale], res); } // subtract scale from the exponent e4 = e4 - scale; } } // check for inexact result if (is_inexact_lt_midpoint || is_inexact_gt_midpoint || is_midpoint_lt_even || is_midpoint_gt_even) { // set the inexact flag and the underflow flag *pfpsf |= INEXACT_EXCEPTION; *pfpsf |= UNDERFLOW_EXCEPTION; } res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | res.w[1]; if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e4, &res, pfpsf); } *pfpsf |= save_fpsf; *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } // no overflow, and no underflow for rounding to nearest res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | res.w[1]; if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e4, &res, pfpsf); // if e4 = expmin && significand < 10^33 => result is tiny (for RD, RZ) if (e4 == expmin) { if ((res.w[1] & MASK_COEFF) < 0x0000314dc6448d93ull || ((res.w[1] & MASK_COEFF) == 0x0000314dc6448d93ull && res.w[0] < 0x38c15b0a00000000ull)) { is_tiny = 1; } } } if (is_inexact_lt_midpoint || is_inexact_gt_midpoint || is_midpoint_lt_even || is_midpoint_gt_even) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; if (is_tiny) *pfpsf |= UNDERFLOW_EXCEPTION; } if ((*pfpsf & INEXACT_EXCEPTION) == 0) { // x * y is exact // need to ensure that the result has the preferred exponent p_exp = res.w[1] & MASK_EXP; if (z_exp < p_exp) { // the preferred exponent is z_exp // signficand of res in C3 C3.w[1] = res.w[1] & MASK_COEFF; C3.w[0] = res.w[0]; // the number of decimal digits of x * y is q4 <= 34 // Note: the coefficient fits in 128 bits // return (C3 * 10^scale) * 10^(p_exp - scale) // where scale = min (p34-q4, (p_exp-z_exp) >> 49) scale = p34 - q4; ind = (p_exp - z_exp) >> 49; if (ind < scale) scale = ind; // subtract scale from the exponent p_exp = p_exp - ((UINT64) scale << 49); if (scale == 0) { ; // leave res unchanged } else if (q4 <= 19) { // x * y fits in 64 bits if (scale <= 19) { // 10^scale fits in 64 bits // 64 x 64 C3.w[0] * ten2k64[scale] __mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]); } else { // 10^scale fits in 128 bits // 64 x 128 C3.w[0] * ten2k128[scale - 20] __mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]); } res.w[1] = p_sign | (p_exp & MASK_EXP) | res.w[1]; } else { // x * y fits in 128 bits, but 10^scale must fit in 64 bits // 64 x 128 ten2k64[scale] * C3 __mul_128x64_to_128 (res, ten2k64[scale], C3); res.w[1] = p_sign | (p_exp & MASK_EXP) | res.w[1]; } } // else leave the result as it is, because p_exp <= z_exp } *pfpsf |= save_fpsf; *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } // else we have f * f + f // continue with x = f, y = f, z = f delta = q3 + e3 - q4 - e4; delta_ge_zero: if (delta >= 0) { if (p34 <= delta - 1 || // Case (1') (p34 == delta && e3 + 6176 < p34 - q3)) { // Case (1''A) // check for overflow, which can occur only in Case (1') if ((q3 + e3) > (p34 + expmax) && p34 <= delta - 1) { // e3 > expmax implies p34 <= delta-1 and e3 > expmax is a necessary // condition for (q3 + e3) > (p34 + expmax) if (rnd_mode == ROUNDING_TO_NEAREST) { res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf res.w[0] = 0x0000000000000000ull; *pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); } else { if (p_sign == z_sign) { is_inexact_lt_midpoint = 1; } else { is_inexact_gt_midpoint = 1; } // q3 <= p34; if (q3 < p34) scale C3 up by 10^(p34-q3) scale = p34 - q3; if (scale == 0) { res.w[1] = z_sign | C3.w[1]; res.w[0] = C3.w[0]; } else { if (q3 <= 19) { // C3 fits in 64 bits if (scale <= 19) { // 10^scale fits in 64 bits // 64 x 64 C3.w[0] * ten2k64[scale] __mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]); } else { // 10^scale fits in 128 bits // 64 x 128 C3.w[0] * ten2k128[scale - 20] __mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]); } } else { // C3 fits in 128 bits, but 10^scale must fit in 64 bits // 64 x 128 ten2k64[scale] * C3 __mul_128x64_to_128 (res, ten2k64[scale], C3); } // the coefficient in res has q3 + scale = p34 digits } e3 = e3 - scale; res.w[1] = z_sign | res.w[1]; rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e3, &res, pfpsf); } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } // res = z if (q3 < p34) { // the preferred exponent is z_exp - (p34 - q3) // return (C3 * 10^scale) * 10^(z_exp - scale) // where scale = min (p34-q3, z_exp-EMIN) scale = p34 - q3; ind = e3 + 6176; if (ind < scale) scale = ind; if (scale == 0) { res.w[1] = C3.w[1]; res.w[0] = C3.w[0]; } else if (q3 <= 19) { // z fits in 64 bits if (scale <= 19) { // 10^scale fits in 64 bits // 64 x 64 C3.w[0] * ten2k64[scale] __mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]); } else { // 10^scale fits in 128 bits // 64 x 128 C3.w[0] * ten2k128[scale - 20] __mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]); } } else { // z fits in 128 bits, but 10^scale must fit in 64 bits // 64 x 128 ten2k64[scale] * C3 __mul_128x64_to_128 (res, ten2k64[scale], C3); } // the coefficient in res has q3 + scale digits // subtract scale from the exponent z_exp = z_exp - ((UINT64) scale << 49); e3 = e3 - scale; res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; if (scale + q3 < p34) *pfpsf |= UNDERFLOW_EXCEPTION; } else { scale = 0; res.w[1] = z_sign | ((UINT64) (e3 + 6176) << 49) | C3.w[1]; res.w[0] = C3.w[0]; } // use the following to avoid double rounding errors when operating on // mixed formats in rounding to nearest, and for correcting the result // if not rounding to nearest if ((p_sign != z_sign) && (delta == (q3 + scale + 1))) { // there is a gap of exactly one digit between the scaled C3 and C4 // C3 * 10^ scale = 10^(q3+scale-1) <=> C3 = 10^(q3-1) is special case if ((q3 <= 19 && C3.w[0] != ten2k64[q3 - 1]) || (q3 == 20 && (C3.w[1] != 0 || C3.w[0] != ten2k64[19])) || (q3 >= 21 && (C3.w[1] != ten2k128[q3 - 21].w[1] || C3.w[0] != ten2k128[q3 - 21].w[0]))) { // C3 * 10^ scale != 10^(q3-1) // if ((res.w[1] & MASK_COEFF) != 0x0000314dc6448d93ull || // res.w[0] != 0x38c15b0a00000000ull) { // C3 * 10^scale != 10^33 is_inexact_gt_midpoint = 1; // if (z_sign), set as if for abs. value } else { // if C3 * 10^scale = 10^(q3+scale-1) // ok from above e3 = (z_exp >> 49) - 6176; // the result is always inexact if (q4 == 1) { R64 = C4.w[0]; } else { // if q4 > 1 then truncate C4 from q4 digits to 1 digit; // x = q4-1, 1 <= x <= 67 and check if this operation is exact if (q4 <= 18) { // 2 <= q4 <= 18 round64_2_18 (q4, q4 - 1, C4.w[0], &R64, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); } else if (q4 <= 38) { P128.w[1] = C4.w[1]; P128.w[0] = C4.w[0]; round128_19_38 (q4, q4 - 1, P128, &R128, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R64 = R128.w[0]; // one decimal digit } else if (q4 <= 57) { P192.w[2] = C4.w[2]; P192.w[1] = C4.w[1]; P192.w[0] = C4.w[0]; round192_39_57 (q4, q4 - 1, P192, &R192, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R64 = R192.w[0]; // one decimal digit } else { // if (q4 <= 68) round256_58_76 (q4, q4 - 1, C4, &R256, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R64 = R256.w[0]; // one decimal digit } if (incr_exp) { R64 = 10; } } if (q4 == 1 && C4.w[0] == 5) { is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 1; is_midpoint_gt_even = 0; } else if ((e3 == expmin) || R64 < 5 || (R64 == 5 && is_inexact_gt_midpoint)) { // result does not change is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else { is_inexact_lt_midpoint = 1; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; // result decremented is 10^(q3+scale) - 1 if ((q3 + scale) <= 19) { res.w[1] = 0; res.w[0] = ten2k64[q3 + scale]; } else { // if ((q3 + scale + 1) <= 35) res.w[1] = ten2k128[q3 + scale - 20].w[1]; res.w[0] = ten2k128[q3 + scale - 20].w[0]; } res.w[0] = res.w[0] - 1; // borrow never occurs z_exp = z_exp - EXP_P1; e3 = e3 - 1; res.w[1] = z_sign | ((UINT64) (e3 + 6176) << 49) | res.w[1]; } if (e3 == expmin) { if (R64 < 5 || (R64 == 5 && !is_inexact_lt_midpoint)) { ; // result not tiny (in round-to-nearest mode) } else { *pfpsf |= UNDERFLOW_EXCEPTION; } } } // end 10^(q3+scale-1) // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } else { if (p_sign == z_sign) { // if (z_sign), set as if for absolute value is_inexact_lt_midpoint = 1; } else { // if (p_sign != z_sign) // if (z_sign), set as if for absolute value is_inexact_gt_midpoint = 1; } *pfpsf |= INEXACT_EXCEPTION; } // the result is always inexact => set the inexact flag // Determine tininess: // if (exp > expmin) // the result is not tiny // else // if exp = emin // if (q3 + scale < p34) // the result is tiny // else // if (q3 + scale = p34) // if (C3 * 10^scale > 10^33) // the result is not tiny // else // if C3 * 10^scale = 10^33 // if (xy * z > 0) // the result is not tiny // else // if (xy * z < 0) // if (z > 0) // if rnd_mode != RP // the result is tiny // else // if RP // the result is not tiny // else // if (z < 0) // if rnd_mode != RM // the result is tiny // else // if RM // the result is not tiny // endif // endif // endif // endif // endif // endif if ((e3 == expmin && (q3 + scale) < p34) || (e3 == expmin && (q3 + scale) == p34 && (res.w[1] & MASK_COEFF) == 0x0000314dc6448d93ull && // 10^33_high res.w[0] == 0x38c15b0a00000000ull && // 10^33_low z_sign != p_sign && ((!z_sign && rnd_mode != ROUNDING_UP) || (z_sign && rnd_mode != ROUNDING_DOWN)))) { *pfpsf |= UNDERFLOW_EXCEPTION; } if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e3, &res, pfpsf); } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else if (p34 == delta) { // Case (1''B) // because Case (1''A) was treated above, e3 + 6176 >= p34 - q3 // and C3 can be scaled up to p34 digits if needed // scale C3 to p34 digits if needed scale = p34 - q3; // 0 <= scale <= p34 - 1 if (scale == 0) { res.w[1] = C3.w[1]; res.w[0] = C3.w[0]; } else if (q3 <= 19) { // z fits in 64 bits if (scale <= 19) { // 10^scale fits in 64 bits // 64 x 64 C3.w[0] * ten2k64[scale] __mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]); } else { // 10^scale fits in 128 bits // 64 x 128 C3.w[0] * ten2k128[scale - 20] __mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]); } } else { // z fits in 128 bits, but 10^scale must fit in 64 bits // 64 x 128 ten2k64[scale] * C3 __mul_128x64_to_128 (res, ten2k64[scale], C3); } // subtract scale from the exponent z_exp = z_exp - ((UINT64) scale << 49); e3 = e3 - scale; // now z_sign, z_exp, and res correspond to a z scaled to p34 = 34 digits // determine whether x * y is less than, equal to, or greater than // 1/2 ulp (z) if (q4 <= 19) { if (C4.w[0] < midpoint64[q4 - 1]) { // < 1/2 ulp lt_half_ulp = 1; } else if (C4.w[0] == midpoint64[q4 - 1]) { // = 1/2 ulp eq_half_ulp = 1; } else { // > 1/2 ulp gt_half_ulp = 1; } } else if (q4 <= 38) { if (C4.w[2] == 0 && (C4.w[1] < midpoint128[q4 - 20].w[1] || (C4.w[1] == midpoint128[q4 - 20].w[1] && C4.w[0] < midpoint128[q4 - 20].w[0]))) { // < 1/2 ulp lt_half_ulp = 1; } else if (C4.w[2] == 0 && C4.w[1] == midpoint128[q4 - 20].w[1] && C4.w[0] == midpoint128[q4 - 20].w[0]) { // = 1/2 ulp eq_half_ulp = 1; } else { // > 1/2 ulp gt_half_ulp = 1; } } else if (q4 <= 58) { if (C4.w[3] == 0 && (C4.w[2] < midpoint192[q4 - 39].w[2] || (C4.w[2] == midpoint192[q4 - 39].w[2] && C4.w[1] < midpoint192[q4 - 39].w[1]) || (C4.w[2] == midpoint192[q4 - 39].w[2] && C4.w[1] == midpoint192[q4 - 39].w[1] && C4.w[0] < midpoint192[q4 - 39].w[0]))) { // < 1/2 ulp lt_half_ulp = 1; } else if (C4.w[3] == 0 && C4.w[2] == midpoint192[q4 - 39].w[2] && C4.w[1] == midpoint192[q4 - 39].w[1] && C4.w[0] == midpoint192[q4 - 39].w[0]) { // = 1/2 ulp eq_half_ulp = 1; } else { // > 1/2 ulp gt_half_ulp = 1; } } else { if (C4.w[3] < midpoint256[q4 - 59].w[3] || (C4.w[3] == midpoint256[q4 - 59].w[3] && C4.w[2] < midpoint256[q4 - 59].w[2]) || (C4.w[3] == midpoint256[q4 - 59].w[3] && C4.w[2] == midpoint256[q4 - 59].w[2] && C4.w[1] < midpoint256[q4 - 59].w[1]) || (C4.w[3] == midpoint256[q4 - 59].w[3] && C4.w[2] == midpoint256[q4 - 59].w[2] && C4.w[1] == midpoint256[q4 - 59].w[1] && C4.w[0] < midpoint256[q4 - 59].w[0])) { // < 1/2 ulp lt_half_ulp = 1; } else if (C4.w[3] == midpoint256[q4 - 59].w[3] && C4.w[2] == midpoint256[q4 - 59].w[2] && C4.w[1] == midpoint256[q4 - 59].w[1] && C4.w[0] == midpoint256[q4 - 59].w[0]) { // = 1/2 ulp eq_half_ulp = 1; } else { // > 1/2 ulp gt_half_ulp = 1; } } if (p_sign == z_sign) { if (lt_half_ulp) { res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; // use the following to avoid double rounding errors when operating on // mixed formats in rounding to nearest is_inexact_lt_midpoint = 1; // if (z_sign), as if for absolute value } else if ((eq_half_ulp && (res.w[0] & 0x01)) || gt_half_ulp) { // add 1 ulp to the significand res.w[0]++; if (res.w[0] == 0x0ull) res.w[1]++; // check for rounding overflow, when coeff == 10^34 if ((res.w[1] & MASK_COEFF) == 0x0001ed09bead87c0ull && res.w[0] == 0x378d8e6400000000ull) { // coefficient = 10^34 e3 = e3 + 1; // coeff = 10^33 z_exp = ((UINT64) (e3 + 6176) << 49) & MASK_EXP; res.w[1] = 0x0000314dc6448d93ull; res.w[0] = 0x38c15b0a00000000ull; } // end add 1 ulp res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; if (eq_half_ulp) { is_midpoint_lt_even = 1; // if (z_sign), as if for absolute value } else { is_inexact_gt_midpoint = 1; // if (z_sign), as if for absolute value } } else { // if (eq_half_ulp && !(res.w[0] & 0x01)) // leave unchanged res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; is_midpoint_gt_even = 1; // if (z_sign), as if for absolute value } // the result is always inexact, and never tiny // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; // check for overflow if (e3 > expmax && rnd_mode == ROUNDING_TO_NEAREST) { res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf res.w[0] = 0x0000000000000000ull; *pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e3, &res, pfpsf); z_exp = res.w[1] & MASK_EXP; } } else { // if (p_sign != z_sign) // consider two cases, because C3 * 10^scale = 10^33 is a special case if (res.w[1] != 0x0000314dc6448d93ull || res.w[0] != 0x38c15b0a00000000ull) { // C3 * 10^scale != 10^33 if (lt_half_ulp) { res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; // use the following to avoid double rounding errors when operating // on mixed formats in rounding to nearest is_inexact_gt_midpoint = 1; // if (z_sign), as if for absolute value } else if ((eq_half_ulp && (res.w[0] & 0x01)) || gt_half_ulp) { // subtract 1 ulp from the significand res.w[0]--; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; if (eq_half_ulp) { is_midpoint_gt_even = 1; // if (z_sign), as if for absolute value } else { is_inexact_lt_midpoint = 1; //if(z_sign), as if for absolute value } } else { // if (eq_half_ulp && !(res.w[0] & 0x01)) // leave unchanged res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; is_midpoint_lt_even = 1; // if (z_sign), as if for absolute value } // the result is always inexact, and never tiny // check for overflow for RN if (e3 > expmax) { if (rnd_mode == ROUNDING_TO_NEAREST) { res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf res.w[0] = 0x0000000000000000ull; *pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); } else { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e3, &res, pfpsf); } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e3, &res, pfpsf); } z_exp = res.w[1] & MASK_EXP; } else { // if C3 * 10^scale = 10^33 e3 = (z_exp >> 49) - 6176; if (e3 > expmin) { // the result is exact if exp > expmin and C4 = d*10^(q4-1), // where d = 1, 2, 3, ..., 9; it could be tiny too, but exact if (q4 == 1) { // if q4 = 1 the result is exact // result coefficient = 10^34 - C4 res.w[1] = 0x0001ed09bead87c0ull; res.w[0] = 0x378d8e6400000000ull - C4.w[0]; z_exp = z_exp - EXP_P1; e3 = e3 - 1; res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; } else { // if q4 > 1 then truncate C4 from q4 digits to 1 digit; // x = q4-1, 1 <= x <= 67 and check if this operation is exact if (q4 <= 18) { // 2 <= q4 <= 18 round64_2_18 (q4, q4 - 1, C4.w[0], &R64, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); } else if (q4 <= 38) { P128.w[1] = C4.w[1]; P128.w[0] = C4.w[0]; round128_19_38 (q4, q4 - 1, P128, &R128, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R64 = R128.w[0]; // one decimal digit } else if (q4 <= 57) { P192.w[2] = C4.w[2]; P192.w[1] = C4.w[1]; P192.w[0] = C4.w[0]; round192_39_57 (q4, q4 - 1, P192, &R192, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R64 = R192.w[0]; // one decimal digit } else { // if (q4 <= 68) round256_58_76 (q4, q4 - 1, C4, &R256, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R64 = R256.w[0]; // one decimal digit } if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // the result is exact: 10^34 - R64 // incr_exp = 0 with certainty z_exp = z_exp - EXP_P1; e3 = e3 - 1; res.w[1] = z_sign | (z_exp & MASK_EXP) | 0x0001ed09bead87c0ull; res.w[0] = 0x378d8e6400000000ull - R64; } else { // We want R64 to be the top digit of C4, but we actually // obtained (C4 * 10^(-q4+1))RN; a correction may be needed, // because the top digit is (C4 * 10^(-q4+1))RZ // however, if incr_exp = 1 then R64 = 10 with certainty if (incr_exp) { R64 = 10; } // the result is inexact as C4 has more than 1 significant digit // and C3 * 10^scale = 10^33 // example of case that is treated here: // 100...0 * 10^e3 - 0.41 * 10^e3 = // 0999...9.59 * 10^e3 -> rounds to 99...96*10^(e3-1) // note that (e3 > expmin} // in order to round, subtract R64 from 10^34 and then compare // C4 - R64 * 10^(q4-1) with 1/2 ulp // calculate 10^34 - R64 res.w[1] = 0x0001ed09bead87c0ull; res.w[0] = 0x378d8e6400000000ull - R64; z_exp = z_exp - EXP_P1; // will be OR-ed with sign & significand // calculate C4 - R64 * 10^(q4-1); this is a rare case and // R64 is small, 1 <= R64 <= 9 e3 = e3 - 1; if (is_inexact_lt_midpoint) { is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; } else if (is_inexact_gt_midpoint) { is_inexact_gt_midpoint = 0; is_inexact_lt_midpoint = 1; } else if (is_midpoint_lt_even) { is_midpoint_lt_even = 0; is_midpoint_gt_even = 1; } else if (is_midpoint_gt_even) { is_midpoint_gt_even = 0; is_midpoint_lt_even = 1; } else { ; } // the result is always inexact, and never tiny // check for overflow for RN if (e3 > expmax) { if (rnd_mode == ROUNDING_TO_NEAREST) { res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf res.w[0] = 0x0000000000000000ull; *pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); } else { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e3, &res, pfpsf); } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; res.w[1] = z_sign | ((UINT64) (e3 + 6176) << 49) | res.w[1]; if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e3, &res, pfpsf); } z_exp = res.w[1] & MASK_EXP; } // end result is inexact } // end q4 > 1 } else { // if (e3 = emin) // if e3 = expmin the result is also tiny (the condition for // tininess is C4 > 050...0 [q4 digits] which is met because // the msd of C4 is not zero) // the result is tiny and inexact in all rounding modes; // it is either 100...0 or 0999...9 (use lt_half_ulp, eq_half_ulp, // gt_half_ulp to calculate) // if (lt_half_ulp || eq_half_ulp) res = 10^33 stays unchanged // p_sign != z_sign so swap gt_half_ulp and lt_half_ulp if (gt_half_ulp) { // res = 10^33 - 1 res.w[1] = 0x0000314dc6448d93ull; res.w[0] = 0x38c15b09ffffffffull; } else { res.w[1] = 0x0000314dc6448d93ull; res.w[0] = 0x38c15b0a00000000ull; } res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; *pfpsf |= UNDERFLOW_EXCEPTION; // inexact is set later if (eq_half_ulp) { is_midpoint_lt_even = 1; // if (z_sign), as if for absolute value } else if (lt_half_ulp) { is_inexact_gt_midpoint = 1; //if(z_sign), as if for absolute value } else { // if (gt_half_ulp) is_inexact_lt_midpoint = 1; //if(z_sign), as if for absolute value } if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e3, &res, pfpsf); z_exp = res.w[1] & MASK_EXP; } } // end e3 = emin // set the inexact flag (if the result was not exact) if (is_inexact_lt_midpoint || is_inexact_gt_midpoint || is_midpoint_lt_even || is_midpoint_gt_even) *pfpsf |= INEXACT_EXCEPTION; } // end 10^33 } // end if (p_sign != z_sign) res.w[1] = z_sign | (z_exp & MASK_EXP) | res.w[1]; *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else if (((q3 <= delta && delta < p34 && p34 < delta + q4) || // Case (2) (q3 <= delta && delta + q4 <= p34) || // Case (3) (delta < q3 && p34 < delta + q4) || // Case (4) (delta < q3 && q3 <= delta + q4 && delta + q4 <= p34) || // Case (5) (delta + q4 < q3)) && // Case (6) !(delta <= 1 && p_sign != z_sign)) { // Case (2), (3), (4), (5) or (6) // the result has the sign of z if ((q3 <= delta && delta < p34 && p34 < delta + q4) || // Case (2) (delta < q3 && p34 < delta + q4)) { // Case (4) // round first the sum x * y + z with unbounded exponent // scale C3 up by scale = p34 - q3, 1 <= scale <= p34-1, // 1 <= scale <= 33 // calculate res = C3 * 10^scale scale = p34 - q3; x0 = delta + q4 - p34; } else if (delta + q4 < q3) { // Case (6) // make Case (6) look like Case (3) or Case (5) with scale = 0 // by scaling up C4 by 10^(q3 - delta - q4) scale = q3 - delta - q4; // 1 <= scale <= 33 if (q4 <= 19) { // 1 <= scale <= 19; C4 fits in 64 bits if (scale <= 19) { // 10^scale fits in 64 bits // 64 x 64 C4.w[0] * ten2k64[scale] __mul_64x64_to_128MACH (P128, C4.w[0], ten2k64[scale]); } else { // 10^scale fits in 128 bits // 64 x 128 C4.w[0] * ten2k128[scale - 20] __mul_128x64_to_128 (P128, C4.w[0], ten2k128[scale - 20]); } } else { // C4 fits in 128 bits, but 10^scale must fit in 64 bits // 64 x 128 ten2k64[scale] * C4 __mul_128x64_to_128 (P128, ten2k64[scale], C4); } C4.w[0] = P128.w[0]; C4.w[1] = P128.w[1]; // e4 does not need adjustment, as it is not used from this point on scale = 0; x0 = 0; // now Case (6) looks like Case (3) or Case (5) with scale = 0 } else { // if Case (3) or Case (5) // Note: Case (3) is similar to Case (2), but scale differs and the // result is exact, unless it is tiny (so x0 = 0 when calculating the // result with unbounded exponent) // calculate first the sum x * y + z with unbounded exponent (exact) // scale C3 up by scale = delta + q4 - q3, 1 <= scale <= p34-1, // 1 <= scale <= 33 // calculate res = C3 * 10^scale scale = delta + q4 - q3; x0 = 0; // Note: the comments which follow refer [mainly] to Case (2)] } case2_repeat: if (scale == 0) { // this could happen e.g. if we return to case2_repeat // or in Case (4) res.w[1] = C3.w[1]; res.w[0] = C3.w[0]; } else if (q3 <= 19) { // 1 <= scale <= 19; z fits in 64 bits if (scale <= 19) { // 10^scale fits in 64 bits // 64 x 64 C3.w[0] * ten2k64[scale] __mul_64x64_to_128MACH (res, C3.w[0], ten2k64[scale]); } else { // 10^scale fits in 128 bits // 64 x 128 C3.w[0] * ten2k128[scale - 20] __mul_128x64_to_128 (res, C3.w[0], ten2k128[scale - 20]); } } else { // z fits in 128 bits, but 10^scale must fit in 64 bits // 64 x 128 ten2k64[scale] * C3 __mul_128x64_to_128 (res, ten2k64[scale], C3); } // e3 is already calculated e3 = e3 - scale; // now res = C3 * 10^scale and e3 = e3 - scale // Note: C3 * 10^scale could be 10^34 if we returned to case2_repeat // because the result was too small // round C4 to nearest to q4 - x0 digits, where x0 = delta + q4 - p34, // 1 <= x0 <= min (q4 - 1, 2 * p34 - 1) <=> 1 <= x0 <= min (q4 - 1, 67) // Also: 1 <= q4 - x0 <= p34 -1 => 1 <= q4 - x0 <= 33 (so the result of // the rounding fits in 128 bits!) // x0 = delta + q4 - p34 (calculated before reaching case2_repeat) // because q3 + q4 - x0 <= P => x0 >= q3 + q4 - p34 if (x0 == 0) { // this could happen only if we return to case2_repeat, or // for Case (3) or Case (6) R128.w[1] = C4.w[1]; R128.w[0] = C4.w[0]; } else if (q4 <= 18) { // 2 <= q4 <= 18, max(1, q3+q4-p34) <= x0 <= q4 - 1, 1 <= x0 <= 17 round64_2_18 (q4, x0, C4.w[0], &R64, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); if (incr_exp) { // R64 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 1, 1 <= q4 - x0 <= 17 R64 = ten2k64[q4 - x0]; } R128.w[1] = 0; R128.w[0] = R64; } else if (q4 <= 38) { // 19 <= q4 <= 38, max(1, q3+q4-p34) <= x0 <= q4 - 1, 1 <= x0 <= 37 P128.w[1] = C4.w[1]; P128.w[0] = C4.w[0]; round128_19_38 (q4, x0, P128, &R128, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); if (incr_exp) { // R128 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 1, 1 <= q4 - x0 <= 37 if (q4 - x0 <= 19) { // 1 <= q4 - x0 <= 19 R128.w[0] = ten2k64[q4 - x0]; // R128.w[1] stays 0 } else { // 20 <= q4 - x0 <= 37 R128.w[0] = ten2k128[q4 - x0 - 20].w[0]; R128.w[1] = ten2k128[q4 - x0 - 20].w[1]; } } } else if (q4 <= 57) { // 38 <= q4 <= 57, max(1, q3+q4-p34) <= x0 <= q4 - 1, 5 <= x0 <= 56 P192.w[2] = C4.w[2]; P192.w[1] = C4.w[1]; P192.w[0] = C4.w[0]; round192_39_57 (q4, x0, P192, &R192, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); // R192.w[2] is always 0 if (incr_exp) { // R192 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 5, 1 <= q4 - x0 <= 52 if (q4 - x0 <= 19) { // 1 <= q4 - x0 <= 19 R192.w[0] = ten2k64[q4 - x0]; // R192.w[1] stays 0 // R192.w[2] stays 0 } else { // 20 <= q4 - x0 <= 33 R192.w[0] = ten2k128[q4 - x0 - 20].w[0]; R192.w[1] = ten2k128[q4 - x0 - 20].w[1]; // R192.w[2] stays 0 } } R128.w[1] = R192.w[1]; R128.w[0] = R192.w[0]; } else { // 58 <= q4 <= 68, max(1, q3+q4-p34) <= x0 <= q4 - 1, 25 <= x0 <= 67 round256_58_76 (q4, x0, C4, &R256, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); // R256.w[3] and R256.w[2] are always 0 if (incr_exp) { // R256 = 10^(q4-x0), 1 <= q4 - x0 <= q4 - 25, 1 <= q4 - x0 <= 43 if (q4 - x0 <= 19) { // 1 <= q4 - x0 <= 19 R256.w[0] = ten2k64[q4 - x0]; // R256.w[1] stays 0 // R256.w[2] stays 0 // R256.w[3] stays 0 } else { // 20 <= q4 - x0 <= 33 R256.w[0] = ten2k128[q4 - x0 - 20].w[0]; R256.w[1] = ten2k128[q4 - x0 - 20].w[1]; // R256.w[2] stays 0 // R256.w[3] stays 0 } } R128.w[1] = R256.w[1]; R128.w[0] = R256.w[0]; } // now add C3 * 10^scale in res and the signed top (q4-x0) digits of C4, // rounded to nearest, which were copied into R128 if (z_sign == p_sign) { lsb = res.w[0] & 0x01; // lsb of C3 * 10^scale // the sum can result in [up to] p34 or p34 + 1 digits res.w[0] = res.w[0] + R128.w[0]; res.w[1] = res.w[1] + R128.w[1]; if (res.w[0] < R128.w[0]) res.w[1]++; // carry // if res > 10^34 - 1 need to increase x0 and decrease scale by 1 if (res.w[1] > 0x0001ed09bead87c0ull || (res.w[1] == 0x0001ed09bead87c0ull && res.w[0] > 0x378d8e63ffffffffull)) { // avoid double rounding error is_inexact_lt_midpoint0 = is_inexact_lt_midpoint; is_inexact_gt_midpoint0 = is_inexact_gt_midpoint; is_midpoint_lt_even0 = is_midpoint_lt_even; is_midpoint_gt_even0 = is_midpoint_gt_even; is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; P128.w[1] = res.w[1]; P128.w[0] = res.w[0]; round128_19_38 (35, 1, P128, &res, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); // incr_exp is 0 with certainty in this case // avoid a double rounding error if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) && is_midpoint_lt_even) { // double rounding error upward // res = res - 1 res.w[0]--; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; // Note: a double rounding error upward is not possible; for this // the result after the first rounding would have to be 99...95 // (35 digits in all), possibly followed by a number of zeros; this // not possible in Cases (2)-(6) or (15)-(17) which may get here is_midpoint_lt_even = 0; is_inexact_lt_midpoint = 1; } else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) && is_midpoint_gt_even) { // double rounding error downward // res = res + 1 res.w[0]++; if (res.w[0] == 0) res.w[1]++; is_midpoint_gt_even = 0; is_inexact_gt_midpoint = 1; } else if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // if this second rounding was exact the result may still be // inexact because of the first rounding if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) { is_inexact_gt_midpoint = 1; } if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) { is_inexact_lt_midpoint = 1; } } else if (is_midpoint_gt_even && (is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) { // pulled up to a midpoint is_inexact_lt_midpoint = 1; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else if (is_midpoint_lt_even && (is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) { // pulled down to a midpoint is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else { ; } // adjust exponent e3 = e3 + 1; if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { if (is_midpoint_lt_even0 || is_midpoint_gt_even0 || is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0) { is_inexact_lt_midpoint = 1; } } } else { // this is the result rounded with unbounded exponent, unless a // correction is needed res.w[1] = res.w[1] & MASK_COEFF; if (lsb == 1) { if (is_midpoint_gt_even) { // res = res + 1 is_midpoint_gt_even = 0; is_midpoint_lt_even = 1; res.w[0]++; if (res.w[0] == 0x0) res.w[1]++; // check for rounding overflow if (res.w[1] == 0x0001ed09bead87c0ull && res.w[0] == 0x378d8e6400000000ull) { // res = 10^34 => rounding overflow res.w[1] = 0x0000314dc6448d93ull; res.w[0] = 0x38c15b0a00000000ull; // 10^33 e3++; } } else if (is_midpoint_lt_even) { // res = res - 1 is_midpoint_lt_even = 0; is_midpoint_gt_even = 1; res.w[0]--; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; // if the result is pure zero, the sign depends on the rounding // mode (x*y and z had opposite signs) if (res.w[1] == 0x0ull && res.w[0] == 0x0ull) { if (rnd_mode != ROUNDING_DOWN) z_sign = 0x0000000000000000ull; else z_sign = 0x8000000000000000ull; // the exponent is max (e3, expmin) res.w[1] = 0x0; res.w[0] = 0x0; *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } } else { ; } } } } else { // if (z_sign != p_sign) lsb = res.w[0] & 0x01; // lsb of C3 * 10^scale; R128 contains rounded C4 // used to swap rounding indicators if p_sign != z_sign // the sum can result in [up to] p34 or p34 - 1 digits tmp64 = res.w[0]; res.w[0] = res.w[0] - R128.w[0]; res.w[1] = res.w[1] - R128.w[1]; if (res.w[0] > tmp64) res.w[1]--; // borrow // if res < 10^33 and exp > expmin need to decrease x0 and // increase scale by 1 if (e3 > expmin && ((res.w[1] < 0x0000314dc6448d93ull || (res.w[1] == 0x0000314dc6448d93ull && res.w[0] < 0x38c15b0a00000000ull)) || (is_inexact_lt_midpoint && res.w[1] == 0x0000314dc6448d93ull && res.w[0] == 0x38c15b0a00000000ull)) && x0 >= 1) { x0 = x0 - 1; // first restore e3, otherwise it will be too small e3 = e3 + scale; scale = scale + 1; is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; incr_exp = 0; goto case2_repeat; } // else this is the result rounded with unbounded exponent; // because the result has opposite sign to that of C4 which was // rounded, need to change the rounding indicators if (is_inexact_lt_midpoint) { is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; } else if (is_inexact_gt_midpoint) { is_inexact_gt_midpoint = 0; is_inexact_lt_midpoint = 1; } else if (lsb == 0) { if (is_midpoint_lt_even) { is_midpoint_lt_even = 0; is_midpoint_gt_even = 1; } else if (is_midpoint_gt_even) { is_midpoint_gt_even = 0; is_midpoint_lt_even = 1; } else { ; } } else if (lsb == 1) { if (is_midpoint_lt_even) { // res = res + 1 res.w[0]++; if (res.w[0] == 0x0) res.w[1]++; // check for rounding overflow if (res.w[1] == 0x0001ed09bead87c0ull && res.w[0] == 0x378d8e6400000000ull) { // res = 10^34 => rounding overflow res.w[1] = 0x0000314dc6448d93ull; res.w[0] = 0x38c15b0a00000000ull; // 10^33 e3++; } } else if (is_midpoint_gt_even) { // res = res - 1 res.w[0]--; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; // if the result is pure zero, the sign depends on the rounding // mode (x*y and z had opposite signs) if (res.w[1] == 0x0ull && res.w[0] == 0x0ull) { if (rnd_mode != ROUNDING_DOWN) z_sign = 0x0000000000000000ull; else z_sign = 0x8000000000000000ull; // the exponent is max (e3, expmin) res.w[1] = 0x0; res.w[0] = 0x0; *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } } else { ; } } else { ; } } // check for underflow if (e3 == expmin) { // and if significand < 10^33 => result is tiny if ((res.w[1] & MASK_COEFF) < 0x0000314dc6448d93ull || ((res.w[1] & MASK_COEFF) == 0x0000314dc6448d93ull && res.w[0] < 0x38c15b0a00000000ull)) { is_tiny = 1; } } else if (e3 < expmin) { // the result is tiny, so we must truncate more of res is_tiny = 1; x0 = expmin - e3; is_inexact_lt_midpoint0 = is_inexact_lt_midpoint; is_inexact_gt_midpoint0 = is_inexact_gt_midpoint; is_midpoint_lt_even0 = is_midpoint_lt_even; is_midpoint_gt_even0 = is_midpoint_gt_even; is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; // determine the number of decimal digits in res if (res.w[1] == 0x0) { // between 1 and 19 digits for (ind = 1; ind <= 19; ind++) { if (res.w[0] < ten2k64[ind]) { break; } } // ind digits } else if (res.w[1] < ten2k128[0].w[1] || (res.w[1] == ten2k128[0].w[1] && res.w[0] < ten2k128[0].w[0])) { // 20 digits ind = 20; } else { // between 21 and 38 digits for (ind = 1; ind <= 18; ind++) { if (res.w[1] < ten2k128[ind].w[1] || (res.w[1] == ten2k128[ind].w[1] && res.w[0] < ten2k128[ind].w[0])) { break; } } // ind + 20 digits ind = ind + 20; } // at this point ind >= x0; because delta >= 2 on this path, the case // ind = x0 can occur only in Case (2) or case (3), when C3 has one // digit (q3 = 1) equal to 1 (C3 = 1), e3 is expmin (e3 = expmin), // the signs of x * y and z are opposite, and through cancellation // the most significant decimal digit in res has the weight // 10^(emin-1); however, it is clear that in this case the most // significant digit is 9, so the result before rounding is // 0.9... * 10^emin // Otherwise, ind > x0 because there are non-zero decimal digits in the // result with weight of at least 10^emin, and correction for underflow // can be carried out using the round*_*_2_* () routines if (x0 == ind) { // the result before rounding is 0.9... * 10^emin res.w[1] = 0x0; res.w[0] = 0x1; is_inexact_gt_midpoint = 1; } else if (ind <= 18) { // check that 2 <= ind // 2 <= ind <= 18, 1 <= x0 <= 17 round64_2_18 (ind, x0, res.w[0], &R64, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); if (incr_exp) { // R64 = 10^(ind-x0), 1 <= ind - x0 <= ind - 1, 1 <= ind - x0 <= 17 R64 = ten2k64[ind - x0]; } res.w[1] = 0; res.w[0] = R64; } else if (ind <= 38) { // 19 <= ind <= 38 P128.w[1] = res.w[1]; P128.w[0] = res.w[0]; round128_19_38 (ind, x0, P128, &res, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); if (incr_exp) { // R128 = 10^(ind-x0), 1 <= ind - x0 <= ind - 1, 1 <= ind - x0 <= 37 if (ind - x0 <= 19) { // 1 <= ind - x0 <= 19 res.w[0] = ten2k64[ind - x0]; // res.w[1] stays 0 } else { // 20 <= ind - x0 <= 37 res.w[0] = ten2k128[ind - x0 - 20].w[0]; res.w[1] = ten2k128[ind - x0 - 20].w[1]; } } } // avoid a double rounding error if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) && is_midpoint_lt_even) { // double rounding error upward // res = res - 1 res.w[0]--; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; // Note: a double rounding error upward is not possible; for this // the result after the first rounding would have to be 99...95 // (35 digits in all), possibly followed by a number of zeros; this // not possible in Cases (2)-(6) which may get here is_midpoint_lt_even = 0; is_inexact_lt_midpoint = 1; } else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) && is_midpoint_gt_even) { // double rounding error downward // res = res + 1 res.w[0]++; if (res.w[0] == 0) res.w[1]++; is_midpoint_gt_even = 0; is_inexact_gt_midpoint = 1; } else if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // if this second rounding was exact the result may still be // inexact because of the first rounding if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) { is_inexact_gt_midpoint = 1; } if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) { is_inexact_lt_midpoint = 1; } } else if (is_midpoint_gt_even && (is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) { // pulled up to a midpoint is_inexact_lt_midpoint = 1; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else if (is_midpoint_lt_even && (is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) { // pulled down to a midpoint is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else { ; } // adjust exponent e3 = e3 + x0; if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { if (is_midpoint_lt_even0 || is_midpoint_gt_even0 || is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0) { is_inexact_lt_midpoint = 1; } } } else { ; // not underflow } // check for inexact result if (is_inexact_lt_midpoint || is_inexact_gt_midpoint || is_midpoint_lt_even || is_midpoint_gt_even) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; if (is_tiny) *pfpsf |= UNDERFLOW_EXCEPTION; } // now check for significand = 10^34 (may have resulted from going // back to case2_repeat) if (res.w[1] == 0x0001ed09bead87c0ull && res.w[0] == 0x378d8e6400000000ull) { // if res = 10^34 res.w[1] = 0x0000314dc6448d93ull; // res = 10^33 res.w[0] = 0x38c15b0a00000000ull; e3 = e3 + 1; } res.w[1] = z_sign | ((UINT64) (e3 + 6176) << 49) | res.w[1]; // check for overflow if (rnd_mode == ROUNDING_TO_NEAREST && e3 > expmax) { res.w[1] = z_sign | 0x7800000000000000ull; // +/-inf res.w[0] = 0x0000000000000000ull; *pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); } if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e3, &res, pfpsf); } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else { // we get here only if delta <= 1 in Cases (2), (3), (4), (5), or (6) and // the signs of x*y and z are opposite; in these cases massive // cancellation can occur, so it is better to scale either C3 or C4 and // to perform the subtraction before rounding; rounding is performed // next, depending on the number of decimal digits in the result and on // the exponent value // Note: overlow is not possible in this case // this is similar to Cases (15), (16), and (17) if (delta + q4 < q3) { // from Case (6) // Case (6) with 0<= delta <= 1 is similar to Cases (15), (16), and // (17) if we swap (C3, C4), (q3, q4), (e3, e4), (z_sign, p_sign) // and call add_and_round; delta stays positive // C4.w[3] = 0 and C4.w[2] = 0, so swap just the low part of C4 with C3 P128.w[1] = C3.w[1]; P128.w[0] = C3.w[0]; C3.w[1] = C4.w[1]; C3.w[0] = C4.w[0]; C4.w[1] = P128.w[1]; C4.w[0] = P128.w[0]; ind = q3; q3 = q4; q4 = ind; ind = e3; e3 = e4; e4 = ind; tmp_sign = z_sign; z_sign = p_sign; p_sign = tmp_sign; } else { // from Cases (2), (3), (4), (5) // In Cases (2), (3), (4), (5) with 0 <= delta <= 1 C3 has to be // scaled up by q4 + delta - q3; this is the same as in Cases (15), // (16), and (17) if we just change the sign of delta delta = -delta; } add_and_round (q3, q4, e4, delta, p34, z_sign, p_sign, C3, C4, rnd_mode, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, pfpsf, &res); *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } } else { // if delta < 0 delta = -delta; if (p34 < q4 && q4 <= delta) { // Case (7) // truncate C4 to p34 digits into res // x = q4-p34, 1 <= x <= 34 because 35 <= q4 <= 68 x0 = q4 - p34; if (q4 <= 38) { P128.w[1] = C4.w[1]; P128.w[0] = C4.w[0]; round128_19_38 (q4, x0, P128, &res, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); } else if (q4 <= 57) { // 35 <= q4 <= 57 P192.w[2] = C4.w[2]; P192.w[1] = C4.w[1]; P192.w[0] = C4.w[0]; round192_39_57 (q4, x0, P192, &R192, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); res.w[0] = R192.w[0]; res.w[1] = R192.w[1]; } else { // if (q4 <= 68) round256_58_76 (q4, x0, C4, &R256, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); res.w[0] = R256.w[0]; res.w[1] = R256.w[1]; } e4 = e4 + x0; if (incr_exp) { e4 = e4 + 1; } if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // if C4 rounded to p34 digits is exact then the result is inexact, // in a way that depends on the signs of x * y and z if (p_sign == z_sign) { is_inexact_lt_midpoint = 1; } else { // if (p_sign != z_sign) if (res.w[1] != 0x0000314dc6448d93ull || res.w[0] != 0x38c15b0a00000000ull) { // res != 10^33 is_inexact_gt_midpoint = 1; } else { // res = 10^33 and exact is a special case // if C3 < 1/2 ulp then res = 10^33 and is_inexact_gt_midpoint = 1 // if C3 = 1/2 ulp then res = 10^33 and is_midpoint_lt_even = 1 // if C3 > 1/2 ulp then res = 10^34-1 and is_inexact_lt_midpoint = 1 // Note: ulp is really ulp/10 (after borrow which propagates to msd) if (delta > p34 + 1) { // C3 < 1/2 // res = 10^33, unchanged is_inexact_gt_midpoint = 1; } else { // if (delta == p34 + 1) if (q3 <= 19) { if (C3.w[0] < midpoint64[q3 - 1]) { // C3 < 1/2 ulp // res = 10^33, unchanged is_inexact_gt_midpoint = 1; } else if (C3.w[0] == midpoint64[q3 - 1]) { // C3 = 1/2 ulp // res = 10^33, unchanged is_midpoint_lt_even = 1; } else { // if (C3.w[0] > midpoint64[q3-1]), C3 > 1/2 ulp res.w[1] = 0x0001ed09bead87c0ull; // 10^34 - 1 res.w[0] = 0x378d8e63ffffffffull; e4 = e4 - 1; is_inexact_lt_midpoint = 1; } } else { // if (20 <= q3 <=34) if (C3.w[1] < midpoint128[q3 - 20].w[1] || (C3.w[1] == midpoint128[q3 - 20].w[1] && C3.w[0] < midpoint128[q3 - 20].w[0])) { // C3 < 1/2 ulp // res = 10^33, unchanged is_inexact_gt_midpoint = 1; } else if (C3.w[1] == midpoint128[q3 - 20].w[1] && C3.w[0] == midpoint128[q3 - 20].w[0]) { // C3 = 1/2 ulp // res = 10^33, unchanged is_midpoint_lt_even = 1; } else { // if (C3 > midpoint128[q3-20]), C3 > 1/2 ulp res.w[1] = 0x0001ed09bead87c0ull; // 10^34 - 1 res.w[0] = 0x378d8e63ffffffffull; e4 = e4 - 1; is_inexact_lt_midpoint = 1; } } } } } } else if (is_midpoint_lt_even) { if (z_sign != p_sign) { // needs correction: res = res - 1 res.w[0] = res.w[0] - 1; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; // if it is (10^33-1)*10^e4 then the corect result is // (10^34-1)*10(e4-1) if (res.w[1] == 0x0000314dc6448d93ull && res.w[0] == 0x38c15b09ffffffffull) { res.w[1] = 0x0001ed09bead87c0ull; // 10^34 - 1 res.w[0] = 0x378d8e63ffffffffull; e4 = e4 - 1; } is_midpoint_lt_even = 0; is_inexact_lt_midpoint = 1; } else { // if (z_sign == p_sign) is_midpoint_lt_even = 0; is_inexact_gt_midpoint = 1; } } else if (is_midpoint_gt_even) { if (z_sign == p_sign) { // needs correction: res = res + 1 (cannot cross in the next binade) res.w[0] = res.w[0] + 1; if (res.w[0] == 0x0000000000000000ull) res.w[1]++; is_midpoint_gt_even = 0; is_inexact_gt_midpoint = 1; } else { // if (z_sign != p_sign) is_midpoint_gt_even = 0; is_inexact_lt_midpoint = 1; } } else { ; // the rounded result is already correct } // check for overflow if (rnd_mode == ROUNDING_TO_NEAREST && e4 > expmax) { res.w[1] = p_sign | 0x7800000000000000ull; res.w[0] = 0x0000000000000000ull; *pfpsf |= (OVERFLOW_EXCEPTION | INEXACT_EXCEPTION); } else { // no overflow or not RN p_exp = ((UINT64) (e4 + 6176) << 49); res.w[1] = p_sign | (p_exp & MASK_EXP) | res.w[1]; } if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e4, &res, pfpsf); } if (is_inexact_lt_midpoint || is_inexact_gt_midpoint || is_midpoint_lt_even || is_midpoint_gt_even) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else if ((q4 <= p34 && p34 <= delta) || // Case (8) (q4 <= delta && delta < p34 && p34 < delta + q3) || // Case (9) (q4 <= delta && delta + q3 <= p34) || // Case (10) (delta < q4 && q4 <= p34 && p34 < delta + q3) || // Case (13) (delta < q4 && q4 <= delta + q3 && delta + q3 <= p34) || // Case (14) (delta + q3 < q4 && q4 <= p34)) { // Case (18) // Case (8) is similar to Case (1), with C3 and C4 swapped // Case (9) is similar to Case (2), with C3 and C4 swapped // Case (10) is similar to Case (3), with C3 and C4 swapped // Case (13) is similar to Case (4), with C3 and C4 swapped // Case (14) is similar to Case (5), with C3 and C4 swapped // Case (18) is similar to Case (6), with C3 and C4 swapped // swap (C3, C4), (q3, q4), (e3, 34), (z_sign, p_sign), (z_exp, p_exp) // and go back to delta_ge_zero // C4.w[3] = 0 and C4.w[2] = 0, so swap just the low part of C4 with C3 P128.w[1] = C3.w[1]; P128.w[0] = C3.w[0]; C3.w[1] = C4.w[1]; C3.w[0] = C4.w[0]; C4.w[1] = P128.w[1]; C4.w[0] = P128.w[0]; ind = q3; q3 = q4; q4 = ind; ind = e3; e3 = e4; e4 = ind; tmp_sign = z_sign; z_sign = p_sign; p_sign = tmp_sign; tmp.ui64 = z_exp; z_exp = p_exp; p_exp = tmp.ui64; goto delta_ge_zero; } else if ((p34 <= delta && delta < q4 && q4 < delta + q3) || // Case (11) (delta < p34 && p34 < q4 && q4 < delta + q3)) { // Case (12) // round C3 to nearest to q3 - x0 digits, where x0 = e4 - e3, // 1 <= x0 <= q3 - 1 <= p34 - 1 x0 = e4 - e3; // or x0 = delta + q3 - q4 if (q3 <= 18) { // 2 <= q3 <= 18 round64_2_18 (q3, x0, C3.w[0], &R64, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); // C3.w[1] = 0; C3.w[0] = R64; } else if (q3 <= 38) { round128_19_38 (q3, x0, C3, &R128, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); C3.w[1] = R128.w[1]; C3.w[0] = R128.w[0]; } // the rounded result has q3 - x0 digits // we want the exponent to be e4, so if incr_exp = 1 then // multiply the rounded result by 10 - it will still fit in 113 bits if (incr_exp) { // 64 x 128 -> 128 P128.w[1] = C3.w[1]; P128.w[0] = C3.w[0]; __mul_64x128_to_128 (C3, ten2k64[1], P128); } e3 = e3 + x0; // this is e4 // now add/subtract the 256-bit C4 and the new (and shorter) 128-bit C3; // the result will have the sign of x * y; the exponent is e4 R256.w[3] = 0; R256.w[2] = 0; R256.w[1] = C3.w[1]; R256.w[0] = C3.w[0]; if (p_sign == z_sign) { // R256 = C4 + R256 add256 (C4, R256, &R256); } else { // if (p_sign != z_sign) { // R256 = C4 - R256 sub256 (C4, R256, &R256); // the result cannot be pure zero // because the result has opposite sign to that of R256 which was // rounded, need to change the rounding indicators lsb = C4.w[0] & 0x01; if (is_inexact_lt_midpoint) { is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; } else if (is_inexact_gt_midpoint) { is_inexact_gt_midpoint = 0; is_inexact_lt_midpoint = 1; } else if (lsb == 0) { if (is_midpoint_lt_even) { is_midpoint_lt_even = 0; is_midpoint_gt_even = 1; } else if (is_midpoint_gt_even) { is_midpoint_gt_even = 0; is_midpoint_lt_even = 1; } else { ; } } else if (lsb == 1) { if (is_midpoint_lt_even) { // res = res + 1 R256.w[0]++; if (R256.w[0] == 0x0) { R256.w[1]++; if (R256.w[1] == 0x0) { R256.w[2]++; if (R256.w[2] == 0x0) { R256.w[3]++; } } } // no check for rounding overflow - R256 was a difference } else if (is_midpoint_gt_even) { // res = res - 1 R256.w[0]--; if (R256.w[0] == 0xffffffffffffffffull) { R256.w[1]--; if (R256.w[1] == 0xffffffffffffffffull) { R256.w[2]--; if (R256.w[2] == 0xffffffffffffffffull) { R256.w[3]--; } } } } else { ; } } else { ; } } // determine the number of decimal digits in R256 ind = nr_digits256 (R256); // ind >= p34 // if R256 is sum, then ind > p34; if R256 is a difference, then // ind >= p34; this means that we can calculate the result rounded to // the destination precision, with unbounded exponent, starting from R256 // and using the indicators from the rounding of C3 to avoid a double // rounding error if (ind < p34) { ; } else if (ind == p34) { // the result rounded to the destination precision with // unbounded exponent // is (-1)^p_sign * R256 * 10^e4 res.w[1] = R256.w[1]; res.w[0] = R256.w[0]; } else { // if (ind > p34) // if more than P digits, round to nearest to P digits // round R256 to p34 digits x0 = ind - p34; // 1 <= x0 <= 34 as 35 <= ind <= 68 // save C3 rounding indicators to help avoid double rounding error is_inexact_lt_midpoint0 = is_inexact_lt_midpoint; is_inexact_gt_midpoint0 = is_inexact_gt_midpoint; is_midpoint_lt_even0 = is_midpoint_lt_even; is_midpoint_gt_even0 = is_midpoint_gt_even; // initialize rounding indicators is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; // round to p34 digits; the result fits in 113 bits if (ind <= 38) { P128.w[1] = R256.w[1]; P128.w[0] = R256.w[0]; round128_19_38 (ind, x0, P128, &R128, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); } else if (ind <= 57) { P192.w[2] = R256.w[2]; P192.w[1] = R256.w[1]; P192.w[0] = R256.w[0]; round192_39_57 (ind, x0, P192, &R192, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R128.w[1] = R192.w[1]; R128.w[0] = R192.w[0]; } else { // if (ind <= 68) round256_58_76 (ind, x0, R256, &R256, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); R128.w[1] = R256.w[1]; R128.w[0] = R256.w[0]; } // the rounded result has p34 = 34 digits e4 = e4 + x0 + incr_exp; res.w[1] = R128.w[1]; res.w[0] = R128.w[0]; // avoid a double rounding error if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) && is_midpoint_lt_even) { // double rounding error upward // res = res - 1 res.w[0]--; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; is_midpoint_lt_even = 0; is_inexact_lt_midpoint = 1; // Note: a double rounding error upward is not possible; for this // the result after the first rounding would have to be 99...95 // (35 digits in all), possibly followed by a number of zeros; this // not possible in Cases (2)-(6) or (15)-(17) which may get here // if this is 10^33 - 1 make it 10^34 - 1 and decrement exponent if (res.w[1] == 0x0000314dc6448d93ull && res.w[0] == 0x38c15b09ffffffffull) { // 10^33 - 1 res.w[1] = 0x0001ed09bead87c0ull; // 10^34 - 1 res.w[0] = 0x378d8e63ffffffffull; e4--; } } else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) && is_midpoint_gt_even) { // double rounding error downward // res = res + 1 res.w[0]++; if (res.w[0] == 0) res.w[1]++; is_midpoint_gt_even = 0; is_inexact_gt_midpoint = 1; } else if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // if this second rounding was exact the result may still be // inexact because of the first rounding if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) { is_inexact_gt_midpoint = 1; } if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) { is_inexact_lt_midpoint = 1; } } else if (is_midpoint_gt_even && (is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) { // pulled up to a midpoint is_inexact_lt_midpoint = 1; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else if (is_midpoint_lt_even && (is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) { // pulled down to a midpoint is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else { ; } } // determine tininess if (rnd_mode == ROUNDING_TO_NEAREST) { if (e4 < expmin) { is_tiny = 1; // for other rounding modes apply correction } } else { // for RM, RP, RZ, RA apply correction in order to determine tininess // but do not save the result; apply the correction to // (-1)^p_sign * res * 10^0 P128.w[1] = p_sign | 0x3040000000000000ull | res.w[1]; P128.w[0] = res.w[0]; rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, 0, &P128, pfpsf); scale = ((P128.w[1] & MASK_EXP) >> 49) - 6176; // -1, 0, or +1 // the number of digits in the significand is p34 = 34 if (e4 + scale < expmin) { is_tiny = 1; } } // the result rounded to the destination precision with unbounded exponent // is (-1)^p_sign * res * 10^e4 res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | res.w[1]; // RN // res.w[0] unchanged; // Note: res is correct only if expmin <= e4 <= expmax ind = p34; // the number of decimal digits in the signifcand of res // at this point we have the result rounded with unbounded exponent in // res and we know its tininess: // res = (-1)^p_sign * significand * 10^e4, // where q (significand) = ind = p34 // Note: res is correct only if expmin <= e4 <= expmax // check for overflow if RN if (rnd_mode == ROUNDING_TO_NEAREST && (ind + e4) > (p34 + expmax)) { res.w[1] = p_sign | 0x7800000000000000ull; res.w[0] = 0x0000000000000000ull; *pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } // else not overflow or not RN, so continue // from this point on this is similar to the last part of the computation // for Cases (15), (16), (17) // if (e4 >= expmin) we have the result rounded with bounded exponent if (e4 < expmin) { x0 = expmin - e4; // x0 >= 1; the number of digits to chop off of res // where the result rounded [at most] once is // (-1)^p_sign * significand_res * 10^e4 // avoid double rounding error is_inexact_lt_midpoint0 = is_inexact_lt_midpoint; is_inexact_gt_midpoint0 = is_inexact_gt_midpoint; is_midpoint_lt_even0 = is_midpoint_lt_even; is_midpoint_gt_even0 = is_midpoint_gt_even; is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; if (x0 > ind) { // nothing is left of res when moving the decimal point left x0 digits is_inexact_lt_midpoint = 1; res.w[1] = p_sign | 0x0000000000000000ull; res.w[0] = 0x0000000000000000ull; e4 = expmin; } else if (x0 == ind) { // 1 <= x0 = ind <= p34 = 34 // this is <, =, or > 1/2 ulp // compare the ind-digit value in the significand of res with // 1/2 ulp = 5*10^(ind-1), i.e. determine whether it is // less than, equal to, or greater than 1/2 ulp (significand of res) R128.w[1] = res.w[1] & MASK_COEFF; R128.w[0] = res.w[0]; if (ind <= 19) { if (R128.w[0] < midpoint64[ind - 1]) { // < 1/2 ulp lt_half_ulp = 1; is_inexact_lt_midpoint = 1; } else if (R128.w[0] == midpoint64[ind - 1]) { // = 1/2 ulp eq_half_ulp = 1; is_midpoint_gt_even = 1; } else { // > 1/2 ulp gt_half_ulp = 1; is_inexact_gt_midpoint = 1; } } else { // if (ind <= 38) if (R128.w[1] < midpoint128[ind - 20].w[1] || (R128.w[1] == midpoint128[ind - 20].w[1] && R128.w[0] < midpoint128[ind - 20].w[0])) { // < 1/2 ulp lt_half_ulp = 1; is_inexact_lt_midpoint = 1; } else if (R128.w[1] == midpoint128[ind - 20].w[1] && R128.w[0] == midpoint128[ind - 20].w[0]) { // = 1/2 ulp eq_half_ulp = 1; is_midpoint_gt_even = 1; } else { // > 1/2 ulp gt_half_ulp = 1; is_inexact_gt_midpoint = 1; } } if (lt_half_ulp || eq_half_ulp) { // res = +0.0 * 10^expmin res.w[1] = 0x0000000000000000ull; res.w[0] = 0x0000000000000000ull; } else { // if (gt_half_ulp) // res = +1 * 10^expmin res.w[1] = 0x0000000000000000ull; res.w[0] = 0x0000000000000001ull; } res.w[1] = p_sign | res.w[1]; e4 = expmin; } else { // if (1 <= x0 <= ind - 1 <= 33) // round the ind-digit result to ind - x0 digits if (ind <= 18) { // 2 <= ind <= 18 round64_2_18 (ind, x0, res.w[0], &R64, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); res.w[1] = 0x0; res.w[0] = R64; } else if (ind <= 38) { P128.w[1] = res.w[1] & MASK_COEFF; P128.w[0] = res.w[0]; round128_19_38 (ind, x0, P128, &res, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); } e4 = e4 + x0; // expmin // we want the exponent to be expmin, so if incr_exp = 1 then // multiply the rounded result by 10 - it will still fit in 113 bits if (incr_exp) { // 64 x 128 -> 128 P128.w[1] = res.w[1] & MASK_COEFF; P128.w[0] = res.w[0]; __mul_64x128_to_128 (res, ten2k64[1], P128); } res.w[1] = p_sign | ((UINT64) (e4 + 6176) << 49) | (res. w[1] & MASK_COEFF); // avoid a double rounding error if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) && is_midpoint_lt_even) { // double rounding error upward // res = res - 1 res.w[0]--; if (res.w[0] == 0xffffffffffffffffull) res.w[1]--; // Note: a double rounding error upward is not possible; for this // the result after the first rounding would have to be 99...95 // (35 digits in all), possibly followed by a number of zeros; this // not possible in this underflow case is_midpoint_lt_even = 0; is_inexact_lt_midpoint = 1; } else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) && is_midpoint_gt_even) { // double rounding error downward // res = res + 1 res.w[0]++; if (res.w[0] == 0) res.w[1]++; is_midpoint_gt_even = 0; is_inexact_gt_midpoint = 1; } else if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // if this second rounding was exact the result may still be // inexact because of the first rounding if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) { is_inexact_gt_midpoint = 1; } if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) { is_inexact_lt_midpoint = 1; } } else if (is_midpoint_gt_even && (is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) { // pulled up to a midpoint is_inexact_lt_midpoint = 1; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else if (is_midpoint_lt_even && (is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) { // pulled down to a midpoint is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else { ; } } } // res contains the correct result // apply correction if not rounding to nearest if (rnd_mode != ROUNDING_TO_NEAREST) { rounding_correction (rnd_mode, is_inexact_lt_midpoint, is_inexact_gt_midpoint, is_midpoint_lt_even, is_midpoint_gt_even, e4, &res, pfpsf); } if (is_midpoint_lt_even || is_midpoint_gt_even || is_inexact_lt_midpoint || is_inexact_gt_midpoint) { // set the inexact flag *pfpsf |= INEXACT_EXCEPTION; if (is_tiny) *pfpsf |= UNDERFLOW_EXCEPTION; } *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else if ((p34 <= delta && delta + q3 <= q4) || // Case (15) (delta < p34 && p34 < delta + q3 && delta + q3 <= q4) || //Case (16) (delta + q3 <= p34 && p34 < q4)) { // Case (17) // calculate first the result rounded to the destination precision, with // unbounded exponent add_and_round (q3, q4, e4, delta, p34, z_sign, p_sign, C3, C4, rnd_mode, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, pfpsf, &res); *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } else { ; } } // end if delta < 0 *ptr_is_midpoint_lt_even = is_midpoint_lt_even; *ptr_is_midpoint_gt_even = is_midpoint_gt_even; *ptr_is_inexact_lt_midpoint = is_inexact_lt_midpoint; *ptr_is_inexact_gt_midpoint = is_inexact_gt_midpoint; BID_SWAP128 (res); BID_RETURN (res) } #if DECIMAL_CALL_BY_REFERENCE void bid128_fma (UINT128 * pres, UINT128 * px, UINT128 * py, UINT128 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT128 x = *px, y = *py, z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT128 bid128_fma (UINT128 x, UINT128 y, UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif int is_midpoint_lt_even, is_midpoint_gt_even, is_inexact_lt_midpoint, is_inexact_gt_midpoint; UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} }; #if DECIMAL_CALL_BY_REFERENCE bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, &res, &x, &y, &z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #else res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, x, y, z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res); } #if DECIMAL_CALL_BY_REFERENCE void bid128ddd_fma (UINT128 * pres, UINT64 * px, UINT64 * py, UINT64 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px, y = *py, z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT128 bid128ddd_fma (UINT64 x, UINT64 y, UINT64 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0, is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0; UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} }; UINT128 x1, y1, z1; #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); bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, &res, &x1, &y1, &z1 _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); z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, x1, y1, z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res); } #if DECIMAL_CALL_BY_REFERENCE void bid128ddq_fma (UINT128 * pres, UINT64 * px, UINT64 * py, UINT128 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px, y = *py; UINT128 z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT128 bid128ddq_fma (UINT64 x, UINT64 y, UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0, is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0; 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_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, &res, &x1, &y1, &z _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_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, x1, y1, z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res); } #if DECIMAL_CALL_BY_REFERENCE void bid128dqd_fma (UINT128 * pres, UINT64 * px, UINT128 * py, UINT64 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px, z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT128 bid128dqd_fma (UINT64 x, UINT128 y, UINT64 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0, is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0; UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} }; UINT128 x1, z1; #if DECIMAL_CALL_BY_REFERENCE bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, &res, &x1, py, &z1 _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); z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, x1, y, z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res); } #if DECIMAL_CALL_BY_REFERENCE void bid128dqq_fma (UINT128 * pres, UINT64 * px, UINT128 * py, UINT128 * pz _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 bid128dqq_fma (UINT64 x, UINT128 y, UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0, is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0; 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_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, &res, &x1, py, pz _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_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, x1, y, z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res); } #if DECIMAL_CALL_BY_REFERENCE void bid128qdd_fma (UINT128 * pres, UINT128 * px, UINT64 * py, UINT64 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 y = *py, z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT128 bid128qdd_fma (UINT128 x, UINT64 y, UINT64 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0, is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0; UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} }; UINT128 y1, z1; #if DECIMAL_CALL_BY_REFERENCE bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, &res, px, &y1, &z1 _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); z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, x, y1, z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res); } #if DECIMAL_CALL_BY_REFERENCE void bid128qdq_fma (UINT128 * pres, UINT128 * px, UINT64 * py, UINT128 * pz _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 bid128qdq_fma (UINT128 x, UINT64 y, UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0, is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0; 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_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, &res, px, &y1, pz _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_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, x, y1, z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res); } #if DECIMAL_CALL_BY_REFERENCE void bid128qqd_fma (UINT128 * pres, UINT128 * px, UINT128 * py, UINT64 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT128 bid128qqd_fma (UINT128 x, UINT128 y, UINT64 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0, is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0; UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} }; UINT128 z1; #if DECIMAL_CALL_BY_REFERENCE bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, &res, px, py, &z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #else z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); res = bid128_ext_fma (&is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint, x, y, z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res); } // Note: bid128qqq_fma is represented by bid128_fma // Note: bid64ddd_fma is represented by bid64_fma #if DECIMAL_CALL_BY_REFERENCE void bid64ddq_fma (UINT64 * pres, UINT64 * px, UINT64 * py, UINT128 * pz _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 UINT64 bid64ddq_fma (UINT64 x, UINT64 y, UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res1 = 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); bid64qqq_fma (&res1, &x1, &y1, pz _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); res1 = bid64qqq_fma (x1, y1, z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res1); } #if DECIMAL_CALL_BY_REFERENCE void bid64dqd_fma (UINT64 * pres, UINT64 * px, UINT128 * py, UINT64 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x = *px, z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT64 bid64dqd_fma (UINT64 x, UINT128 y, UINT64 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res1 = 0xbaddbaddbaddbaddull; UINT128 x1, z1; #if DECIMAL_CALL_BY_REFERENCE bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid64qqq_fma (&res1, &x1, py, &z1 _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); z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); res1 = bid64qqq_fma (x1, y, z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res1); } #if DECIMAL_CALL_BY_REFERENCE void bid64dqq_fma (UINT64 * pres, UINT64 * px, UINT128 * py, UINT128 * pz _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 bid64dqq_fma (UINT64 x, UINT128 y, UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res1 = 0xbaddbaddbaddbaddull; UINT128 x1; #if DECIMAL_CALL_BY_REFERENCE bid64_to_bid128 (&x1, &x _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid64qqq_fma (&res1, &x1, py, pz _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); res1 = bid64qqq_fma (x1, y, z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res1); } #if DECIMAL_CALL_BY_REFERENCE void bid64qdd_fma (UINT64 * pres, UINT128 * px, UINT64 * py, UINT64 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 y = *py, z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT64 bid64qdd_fma (UINT128 x, UINT64 y, UINT64 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res1 = 0xbaddbaddbaddbaddull; UINT128 y1, z1; #if DECIMAL_CALL_BY_REFERENCE bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid64qqq_fma (&res1, px, &y1, &z1 _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); z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); res1 = bid64qqq_fma (x, y1, z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res1); } #if DECIMAL_CALL_BY_REFERENCE void bid64qdq_fma (UINT64 * pres, UINT128 * px, UINT64 * py, UINT128 * pz _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 bid64qdq_fma (UINT128 x, UINT64 y, UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res1 = 0xbaddbaddbaddbaddull; UINT128 y1; #if DECIMAL_CALL_BY_REFERENCE bid64_to_bid128 (&y1, &y _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid64qqq_fma (&res1, px, &y1, pz _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); res1 = bid64qqq_fma (x, y1, z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res1); } #if DECIMAL_CALL_BY_REFERENCE void bid64qqd_fma (UINT64 * pres, UINT128 * px, UINT128 * py, UINT64 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT64 bid64qqd_fma (UINT128 x, UINT128 y, UINT64 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT64 res1 = 0xbaddbaddbaddbaddull; UINT128 z1; #if DECIMAL_CALL_BY_REFERENCE bid64_to_bid128 (&z1, &z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); bid64qqq_fma (&res1, px, py, &z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #else z1 = bid64_to_bid128 (z _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); res1 = bid64qqq_fma (x, y, z1 _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif BID_RETURN (res1); } #if DECIMAL_CALL_BY_REFERENCE void bid64qqq_fma (UINT64 * pres, UINT128 * px, UINT128 * py, UINT128 * pz _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT128 x = *px, y = *py, z = *pz; #if !DECIMAL_GLOBAL_ROUNDING unsigned int rnd_mode = *prnd_mode; #endif #else UINT64 bid64qqq_fma (UINT128 x, UINT128 y, UINT128 z _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif int is_midpoint_lt_even0 = 0, is_midpoint_gt_even0 = 0, is_inexact_lt_midpoint0 = 0, is_inexact_gt_midpoint0 = 0; int is_midpoint_lt_even = 0, is_midpoint_gt_even = 0, is_inexact_lt_midpoint = 0, is_inexact_gt_midpoint = 0; int incr_exp; UINT128 res = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} }; UINT128 res128 = { {0xbaddbaddbaddbaddull, 0xbaddbaddbaddbaddull} }; UINT64 res1 = 0xbaddbaddbaddbaddull; unsigned int save_fpsf; // needed because of the call to bid128_ext_fma UINT64 sign; UINT64 exp; int unbexp; UINT128 C; BID_UI64DOUBLE tmp; int nr_bits; int q, x0; int scale; int lt_half_ulp = 0, eq_half_ulp = 0; // Note: for rounding modes other than RN or RA, the result can be obtained // by rounding first to BID128 and then to BID64 save_fpsf = *pfpsf; // sticky bits - caller value must be preserved *pfpsf = 0; #if DECIMAL_CALL_BY_REFERENCE bid128_ext_fma (&is_midpoint_lt_even0, &is_midpoint_gt_even0, &is_inexact_lt_midpoint0, &is_inexact_gt_midpoint0, &res, &x, &y, &z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #else res = bid128_ext_fma (&is_midpoint_lt_even0, &is_midpoint_gt_even0, &is_inexact_lt_midpoint0, &is_inexact_gt_midpoint0, x, y, z _RND_MODE_ARG _EXC_FLAGS_ARG _EXC_MASKS_ARG _EXC_INFO_ARG); #endif if ((rnd_mode == ROUNDING_DOWN) || (rnd_mode == ROUNDING_UP) || (rnd_mode == ROUNDING_TO_ZERO) || // no double rounding error is possible ((res.w[HIGH_128W] & MASK_NAN) == MASK_NAN) || //res=QNaN (cannot be SNaN) ((res.w[HIGH_128W] & MASK_ANY_INF) == MASK_INF)) { // result is infinity #if DECIMAL_CALL_BY_REFERENCE bid128_to_bid64 (&res1, &res _RND_MODE_ARG _EXC_FLAGS_ARG); #else res1 = bid128_to_bid64 (res _RND_MODE_ARG _EXC_FLAGS_ARG); #endif // determine the unbiased exponent of the result unbexp = ((res1 >> 53) & 0x3ff) - 398; // if subnormal, res1 must have exp = -398 // if tiny and inexact set underflow and inexact status flags if (!((res1 & MASK_NAN) == MASK_NAN) && // res1 not NaN (unbexp == -398) && ((res1 & MASK_BINARY_SIG1) < 1000000000000000ull) && (is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0 || is_midpoint_lt_even0 || is_midpoint_gt_even0)) { // set the inexact flag and the underflow flag *pfpsf |= (INEXACT_EXCEPTION | UNDERFLOW_EXCEPTION); } else if (is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0 || is_midpoint_lt_even0 || is_midpoint_gt_even0) { // set the inexact flag and the underflow flag *pfpsf |= INEXACT_EXCEPTION; } *pfpsf |= save_fpsf; BID_RETURN (res1); } // else continue, and use rounding to nearest to round to 16 digits // at this point the result is rounded to nearest (even or away) to 34 digits // (or less if exact), and it is zero or finite non-zero canonical [sub]normal sign = res.w[HIGH_128W] & MASK_SIGN; // 0 for positive, MASK_SIGN for negative exp = res.w[HIGH_128W] & MASK_EXP; // biased and shifted left 49 bits unbexp = (exp >> 49) - 6176; C.w[1] = res.w[HIGH_128W] & MASK_COEFF; C.w[0] = res.w[LOW_128W]; if ((C.w[1] == 0x0 && C.w[0] == 0x0) || // result is zero (unbexp <= (-398 - 35)) || (unbexp >= (369 + 16))) { // clear under/overflow #if DECIMAL_CALL_BY_REFERENCE bid128_to_bid64 (&res1, &res _RND_MODE_ARG _EXC_FLAGS_ARG); #else res1 = bid128_to_bid64 (res _RND_MODE_ARG _EXC_FLAGS_ARG); #endif *pfpsf |= save_fpsf; BID_RETURN (res1); } // else continue // -398 - 34 <= unbexp <= 369 + 15 if (rnd_mode == ROUNDING_TIES_AWAY) { // apply correction, if needed, to make the result rounded to nearest-even if (is_midpoint_gt_even) { // res = res - 1 res1--; // res1 is now even } // else the result is already correctly rounded to nearest-even } // at this point the result is finite, non-zero canonical normal or subnormal, // and in most cases overflow or underflow will not occur // determine the number of digits q in the result // q = nr. of decimal digits in x // determine first the nr. of bits in x if (C.w[1] == 0) { if (C.w[0] >= 0x0020000000000000ull) { // x >= 2^53 // split the 64-bit value in two 32-bit halves to avoid rounding errors if (C.w[0] >= 0x0000000100000000ull) { // x >= 2^32 tmp.d = (double) (C.w[0] >> 32); // exact conversion nr_bits = 33 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } else { // x < 2^32 tmp.d = (double) (C.w[0]); // exact conversion nr_bits = 1 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // if x < 2^53 tmp.d = (double) C.w[0]; // exact conversion nr_bits = 1 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } } else { // C.w[1] != 0 => nr. bits = 64 + nr_bits (C.w[1]) tmp.d = (double) C.w[1]; // exact conversion nr_bits = 65 + ((((unsigned int) (tmp.ui64 >> 52)) & 0x7ff) - 0x3ff); } q = nr_digits[nr_bits - 1].digits; if (q == 0) { q = nr_digits[nr_bits - 1].digits1; if (C.w[1] > nr_digits[nr_bits - 1].threshold_hi || (C.w[1] == nr_digits[nr_bits - 1].threshold_hi && C.w[0] >= nr_digits[nr_bits - 1].threshold_lo)) q++; } // if q > 16, round to nearest even to 16 digits (but for underflow it may // have to be truncated even more) if (q > 16) { x0 = q - 16; if (q <= 18) { round64_2_18 (q, x0, C.w[0], &res1, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); } else { // 19 <= q <= 34 round128_19_38 (q, x0, C, &res128, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); res1 = res128.w[0]; // the result fits in 64 bits } unbexp = unbexp + x0; if (incr_exp) unbexp++; q = 16; // need to set in case denormalization is necessary } else { // the result does not require a second rounding (and it must have // been exact in the first rounding, since q <= 16) res1 = C.w[0]; } // avoid a double rounding error if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) && is_midpoint_lt_even) { // double rounding error upward // res = res - 1 res1--; // res1 becomes odd is_midpoint_lt_even = 0; is_inexact_lt_midpoint = 1; if (res1 == 0x00038d7ea4c67fffull) { // 10^15 - 1 res1 = 0x002386f26fc0ffffull; // 10^16 - 1 unbexp--; } } else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) && is_midpoint_gt_even) { // double rounding error downward // res = res + 1 res1++; // res1 becomes odd (so it cannot be 10^16) is_midpoint_gt_even = 0; is_inexact_gt_midpoint = 1; } else if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // if this second rounding was exact the result may still be // inexact because of the first rounding if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) { is_inexact_gt_midpoint = 1; } if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) { is_inexact_lt_midpoint = 1; } } else if (is_midpoint_gt_even && (is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) { // pulled up to a midpoint is_inexact_lt_midpoint = 1; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else if (is_midpoint_lt_even && (is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) { // pulled down to a midpoint is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else { ; } // this is the result rounded correctly to nearest even, with unbounded exp. // check for overflow if (q + unbexp > P16 + expmax16) { res1 = sign | 0x7800000000000000ull; *pfpsf |= (INEXACT_EXCEPTION | OVERFLOW_EXCEPTION); *pfpsf |= save_fpsf; BID_RETURN (res1) } else if (unbexp > expmax16) { // q + unbexp <= P16 + expmax16 // not overflow; the result must be exact, and we can multiply res1 by // 10^(unbexp - expmax16) and the product will fit in 16 decimal digits scale = unbexp - expmax16; res1 = res1 * ten2k64[scale]; // res1 * 10^scale unbexp = expmax16; // unbexp - scale } else { ; // continue } // check for underflow if (q + unbexp < P16 + expmin16) { if (unbexp < expmin16) { // we must truncate more of res x0 = expmin16 - unbexp; // x0 >= 1 is_inexact_lt_midpoint0 = is_inexact_lt_midpoint; is_inexact_gt_midpoint0 = is_inexact_gt_midpoint; is_midpoint_lt_even0 = is_midpoint_lt_even; is_midpoint_gt_even0 = is_midpoint_gt_even; is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; // the number of decimal digits in res1 is q if (x0 < q) { // 1 <= x0 <= q-1 => round res to q - x0 digits // 2 <= q <= 16, 1 <= x0 <= 15 round64_2_18 (q, x0, res1, &res1, &incr_exp, &is_midpoint_lt_even, &is_midpoint_gt_even, &is_inexact_lt_midpoint, &is_inexact_gt_midpoint); if (incr_exp) { // res1 = 10^(q-x0), 1 <= q - x0 <= q - 1, 1 <= q - x0 <= 15 res1 = ten2k64[q - x0]; } unbexp = unbexp + x0; // expmin16 } else if (x0 == q) { // the second rounding is for 0.d(0)d(1)...d(q-1) * 10^emin // determine relationship with 1/2 ulp // q <= 16 if (res1 < midpoint64[q - 1]) { // < 1/2 ulp lt_half_ulp = 1; is_inexact_lt_midpoint = 1; } else if (res1 == midpoint64[q - 1]) { // = 1/2 ulp eq_half_ulp = 1; is_midpoint_gt_even = 1; } else { // > 1/2 ulp // gt_half_ulp = 1; is_inexact_gt_midpoint = 1; } if (lt_half_ulp || eq_half_ulp) { // res = +0.0 * 10^expmin16 res1 = 0x0000000000000000ull; } else { // if (gt_half_ulp) // res = +1 * 10^expmin16 res1 = 0x0000000000000001ull; } unbexp = expmin16; } else { // if (x0 > q) // the second rounding is for 0.0...d(0)d(1)...d(q-1) * 10^emin res1 = 0x0000000000000000ull; unbexp = expmin16; is_inexact_lt_midpoint = 1; } // avoid a double rounding error if ((is_inexact_gt_midpoint0 || is_midpoint_lt_even0) && is_midpoint_lt_even) { // double rounding error upward // res = res - 1 res1--; // res1 becomes odd is_midpoint_lt_even = 0; is_inexact_lt_midpoint = 1; } else if ((is_inexact_lt_midpoint0 || is_midpoint_gt_even0) && is_midpoint_gt_even) { // double rounding error downward // res = res + 1 res1++; // res1 becomes odd is_midpoint_gt_even = 0; is_inexact_gt_midpoint = 1; } else if (!is_midpoint_lt_even && !is_midpoint_gt_even && !is_inexact_lt_midpoint && !is_inexact_gt_midpoint) { // if this rounding was exact the result may still be // inexact because of the previous roundings if (is_inexact_gt_midpoint0 || is_midpoint_lt_even0) { is_inexact_gt_midpoint = 1; } if (is_inexact_lt_midpoint0 || is_midpoint_gt_even0) { is_inexact_lt_midpoint = 1; } } else if (is_midpoint_gt_even && (is_inexact_gt_midpoint0 || is_midpoint_lt_even0)) { // pulled up to a midpoint is_inexact_lt_midpoint = 1; is_inexact_gt_midpoint = 0; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else if (is_midpoint_lt_even && (is_inexact_lt_midpoint0 || is_midpoint_gt_even0)) { // pulled down to a midpoint is_inexact_lt_midpoint = 0; is_inexact_gt_midpoint = 1; is_midpoint_lt_even = 0; is_midpoint_gt_even = 0; } else { ; } } // else if unbexp >= emin then q < P (because q + unbexp < P16 + expmin16) // and the result is tiny and exact // check for inexact result if (is_inexact_lt_midpoint || is_inexact_gt_midpoint || is_midpoint_lt_even || is_midpoint_gt_even || is_inexact_lt_midpoint0 || is_inexact_gt_midpoint0 || is_midpoint_lt_even0 || is_midpoint_gt_even0) { // set the inexact flag and the underflow flag *pfpsf |= (INEXACT_EXCEPTION | UNDERFLOW_EXCEPTION); } } else if (is_inexact_lt_midpoint || is_inexact_gt_midpoint || is_midpoint_lt_even || is_midpoint_gt_even) { *pfpsf |= INEXACT_EXCEPTION; } // this is the result rounded correctly to nearest, with bounded exponent if (rnd_mode == ROUNDING_TIES_AWAY && is_midpoint_gt_even) { // correction // res = res + 1 res1++; // res1 is now odd } // else the result is already correct // assemble the result if (res1 < 0x0020000000000000ull) { // res < 2^53 res1 = sign | ((UINT64) (unbexp + 398) << 53) | res1; } else { // res1 >= 2^53 res1 = sign | MASK_STEERING_BITS | ((UINT64) (unbexp + 398) << 51) | (res1 & MASK_BINARY_SIG2); } *pfpsf |= save_fpsf; BID_RETURN (res1); }