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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/>. */ /***************************************************************************** * * BID64 encoding: * **************************************** * 63 62 53 52 0 * |---|------------------|--------------| * | S | Biased Exp (E) | Coeff (c) | * |---|------------------|--------------| * * bias = 398 * number = (-1)^s * 10^(E-398) * c * coefficient range - 0 to (2^53)-1 * COEFF_MAX = 2^53-1 = 9007199254740991 * ***************************************************************************** * * BID128 encoding: * 1-bit sign * 14-bit biased exponent in [0x21, 0x3020] = [33, 12320] * unbiased exponent in [-6176, 6111]; exponent bias = 6176 * 113-bit unsigned binary integer coefficient (49-bit high + 64-bit low) * Note: 10^34-1 ~ 2^112.945555... < 2^113 => coefficient fits in 113 bits * * Note: assume invalid encodings are not passed to this function * * Round a number C with q decimal digits, represented as a binary integer * to q - x digits. Six different routines are provided for different values * of q. The maximum value of q used in the library is q = 3 * P - 1 where * P = 16 or P = 34 (so q <= 111 decimal digits). * The partitioning is based on the following, where Kx is the scaled * integer representing the value of 10^(-x) rounded up to a number of bits * sufficient to ensure correct rounding: * * -------------------------------------------------------------------------- * q x max. value of a max number min. number * of bits in C of bits in Kx * -------------------------------------------------------------------------- * * GROUP 1: 64 bits * round64_2_18 () * * 2 [1,1] 10^1 - 1 < 2^3.33 4 4 * ... ... ... ... ... * 18 [1,17] 10^18 - 1 < 2^59.80 60 61 * * GROUP 2: 128 bits * round128_19_38 () * * 19 [1,18] 10^19 - 1 < 2^63.11 64 65 * 20 [1,19] 10^20 - 1 < 2^66.44 67 68 * ... ... ... ... ... * 38 [1,37] 10^38 - 1 < 2^126.24 127 128 * * GROUP 3: 192 bits * round192_39_57 () * * 39 [1,38] 10^39 - 1 < 2^129.56 130 131 * ... ... ... ... ... * 57 [1,56] 10^57 - 1 < 2^189.35 190 191 * * GROUP 4: 256 bits * round256_58_76 () * * 58 [1,57] 10^58 - 1 < 2^192.68 193 194 * ... ... ... ... ... * 76 [1,75] 10^76 - 1 < 2^252.47 253 254 * * GROUP 5: 320 bits * round320_77_96 () * * 77 [1,76] 10^77 - 1 < 2^255.79 256 257 * 78 [1,77] 10^78 - 1 < 2^259.12 260 261 * ... ... ... ... ... * 96 [1,95] 10^96 - 1 < 2^318.91 319 320 * * GROUP 6: 384 bits * round384_97_115 () * * 97 [1,96] 10^97 - 1 < 2^322.23 323 324 * ... ... ... ... ... * 115 [1,114] 10^115 - 1 < 2^382.03 383 384 * ****************************************************************************/ #include "bid_internal.h" void round64_2_18 (int q, int x, UINT64 C, UINT64 * ptr_Cstar, int *incr_exp, 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 P128; UINT128 fstar; UINT64 Cstar; UINT64 tmp64; int shift; int ind; // Note: // In round128_2_18() positive numbers with 2 <= q <= 18 will be // rounded to nearest only for 1 <= x <= 3: // x = 1 or x = 2 when q = 17 // x = 2 or x = 3 when q = 18 // However, for generality and possible uses outside the frame of IEEE 754R // this implementation works for 1 <= x <= q - 1 // assume *ptr_is_midpoint_lt_even, *ptr_is_midpoint_gt_even, // *ptr_is_inexact_lt_midpoint, and *ptr_is_inexact_gt_midpoint are // initialized to 0 by the caller // round a number C with q decimal digits, 2 <= q <= 18 // to q - x digits, 1 <= x <= 17 // C = C + 1/2 * 10^x where the result C fits in 64 bits // (because the largest value is 999999999999999999 + 50000000000000000 = // 0x0e92596fd628ffff, which fits in 60 bits) ind = x - 1; // 0 <= ind <= 16 C = C + midpoint64[ind]; // kx ~= 10^(-x), kx = Kx64[ind] * 2^(-Ex), 0 <= ind <= 16 // P128 = (C + 1/2 * 10^x) * kx * 2^Ex = (C + 1/2 * 10^x) * Kx // the approximation kx of 10^(-x) was rounded up to 64 bits __mul_64x64_to_128MACH (P128, C, Kx64[ind]); // calculate C* = floor (P128) and f* // Cstar = P128 >> Ex // fstar = low Ex bits of P128 shift = Ex64m64[ind]; // in [3, 56] Cstar = P128.w[1] >> shift; fstar.w[1] = P128.w[1] & mask64[ind]; fstar.w[0] = P128.w[0]; // the top Ex bits of 10^(-x) are T* = ten2mxtrunc64[ind], e.g. // if x=1, T*=ten2mxtrunc64[0]=0xcccccccccccccccc // if (0 < f* < 10^(-x)) then the result is a midpoint // if floor(C*) is even then C* = floor(C*) - logical right // shift; C* has q - x decimal digits, correct by Prop. 1) // else if floor(C*) is odd C* = floor(C*)-1 (logical right // shift; C* has q - x decimal digits, correct by Pr. 1) // else // C* = floor(C*) (logical right shift; C has q - x decimal digits, // correct by Property 1) // in the caling function n = C* * 10^(e+x) // determine inexactness of the rounding of C* // if (0 < f* - 1/2 < 10^(-x)) then // the result is exact // else // if (f* - 1/2 > T*) then // the result is inexact if (fstar.w[1] > half64[ind] || (fstar.w[1] == half64[ind] && fstar.w[0])) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[1] - half64[ind]; if (tmp64 || fstar.w[0] > ten2mxtrunc64[ind]) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } // check for midpoints (could do this before determining inexactness) if (fstar.w[1] == 0 && fstar.w[0] <= ten2mxtrunc64[ind]) { // the result is a midpoint if (Cstar & 0x01) { // Cstar is odd; MP in [EVEN, ODD] // if floor(C*) is odd C = floor(C*) - 1; the result may be 0 Cstar--; // Cstar is now even *ptr_is_midpoint_gt_even = 1; *ptr_is_inexact_lt_midpoint = 0; *ptr_is_inexact_gt_midpoint = 0; } else { // else MP in [ODD, EVEN] *ptr_is_midpoint_lt_even = 1; *ptr_is_inexact_lt_midpoint = 0; *ptr_is_inexact_gt_midpoint = 0; } } // check for rounding overflow, which occurs if Cstar = 10^(q-x) ind = q - x; // 1 <= ind <= q - 1 if (Cstar == ten2k64[ind]) { // if Cstar = 10^(q-x) Cstar = ten2k64[ind - 1]; // Cstar = 10^(q-x-1) *incr_exp = 1; } else { // 10^33 <= Cstar <= 10^34 - 1 *incr_exp = 0; } *ptr_Cstar = Cstar; } void round128_19_38 (int q, int x, UINT128 C, UINT128 * ptr_Cstar, int *incr_exp, int *ptr_is_midpoint_lt_even, int *ptr_is_midpoint_gt_even, int *ptr_is_inexact_lt_midpoint, int *ptr_is_inexact_gt_midpoint) { UINT256 P256; UINT256 fstar; UINT128 Cstar; UINT64 tmp64; int shift; int ind; // Note: // In round128_19_38() positive numbers with 19 <= q <= 38 will be // rounded to nearest only for 1 <= x <= 23: // x = 3 or x = 4 when q = 19 // x = 4 or x = 5 when q = 20 // ... // x = 18 or x = 19 when q = 34 // x = 1 or x = 2 or x = 19 or x = 20 when q = 35 // x = 2 or x = 3 or x = 20 or x = 21 when q = 36 // x = 3 or x = 4 or x = 21 or x = 22 when q = 37 // x = 4 or x = 5 or x = 22 or x = 23 when q = 38 // However, for generality and possible uses outside the frame of IEEE 754R // this implementation works for 1 <= x <= q - 1 // assume *ptr_is_midpoint_lt_even, *ptr_is_midpoint_gt_even, // *ptr_is_inexact_lt_midpoint, and *ptr_is_inexact_gt_midpoint are // initialized to 0 by the caller // round a number C with q decimal digits, 19 <= q <= 38 // to q - x digits, 1 <= x <= 37 // C = C + 1/2 * 10^x where the result C fits in 128 bits // (because the largest value is 99999999999999999999999999999999999999 + // 5000000000000000000000000000000000000 = // 0x4efe43b0c573e7e68a043d8fffffffff, which fits is 127 bits) ind = x - 1; // 0 <= ind <= 36 if (ind <= 18) { // if 0 <= ind <= 18 tmp64 = C.w[0]; C.w[0] = C.w[0] + midpoint64[ind]; if (C.w[0] < tmp64) C.w[1]++; } else { // if 19 <= ind <= 37 tmp64 = C.w[0]; C.w[0] = C.w[0] + midpoint128[ind - 19].w[0]; if (C.w[0] < tmp64) { C.w[1]++; } C.w[1] = C.w[1] + midpoint128[ind - 19].w[1]; } // kx ~= 10^(-x), kx = Kx128[ind] * 2^(-Ex), 0 <= ind <= 36 // P256 = (C + 1/2 * 10^x) * kx * 2^Ex = (C + 1/2 * 10^x) * Kx // the approximation kx of 10^(-x) was rounded up to 128 bits __mul_128x128_to_256 (P256, C, Kx128[ind]); // calculate C* = floor (P256) and f* // Cstar = P256 >> Ex // fstar = low Ex bits of P256 shift = Ex128m128[ind]; // in [2, 63] but have to consider two cases if (ind <= 18) { // if 0 <= ind <= 18 Cstar.w[0] = (P256.w[2] >> shift) | (P256.w[3] << (64 - shift)); Cstar.w[1] = (P256.w[3] >> shift); fstar.w[0] = P256.w[0]; fstar.w[1] = P256.w[1]; fstar.w[2] = P256.w[2] & mask128[ind]; fstar.w[3] = 0x0ULL; } else { // if 19 <= ind <= 37 Cstar.w[0] = P256.w[3] >> shift; Cstar.w[1] = 0x0ULL; fstar.w[0] = P256.w[0]; fstar.w[1] = P256.w[1]; fstar.w[2] = P256.w[2]; fstar.w[3] = P256.w[3] & mask128[ind]; } // the top Ex bits of 10^(-x) are T* = ten2mxtrunc64[ind], e.g. // if x=1, T*=ten2mxtrunc128[0]=0xcccccccccccccccccccccccccccccccc // if (0 < f* < 10^(-x)) then the result is a midpoint // if floor(C*) is even then C* = floor(C*) - logical right // shift; C* has q - x decimal digits, correct by Prop. 1) // else if floor(C*) is odd C* = floor(C*)-1 (logical right // shift; C* has q - x decimal digits, correct by Pr. 1) // else // C* = floor(C*) (logical right shift; C has q - x decimal digits, // correct by Property 1) // in the caling function n = C* * 10^(e+x) // determine inexactness of the rounding of C* // if (0 < f* - 1/2 < 10^(-x)) then // the result is exact // else // if (f* - 1/2 > T*) then // the result is inexact if (ind <= 18) { // if 0 <= ind <= 18 if (fstar.w[2] > half128[ind] || (fstar.w[2] == half128[ind] && (fstar.w[1] || fstar.w[0]))) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[2] - half128[ind]; if (tmp64 || fstar.w[1] > ten2mxtrunc128[ind].w[1] || (fstar.w[1] == ten2mxtrunc128[ind].w[1] && fstar.w[0] > ten2mxtrunc128[ind].w[0])) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } } else { // if 19 <= ind <= 37 if (fstar.w[3] > half128[ind] || (fstar.w[3] == half128[ind] && (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[3] - half128[ind]; if (tmp64 || fstar.w[2] || fstar.w[1] > ten2mxtrunc128[ind].w[1] || (fstar.w[1] == ten2mxtrunc128[ind].w[1] && fstar.w[0] > ten2mxtrunc128[ind].w[0])) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } } // check for midpoints (could do this before determining inexactness) if (fstar.w[3] == 0 && fstar.w[2] == 0 && (fstar.w[1] < ten2mxtrunc128[ind].w[1] || (fstar.w[1] == ten2mxtrunc128[ind].w[1] && fstar.w[0] <= ten2mxtrunc128[ind].w[0]))) { // the result is a midpoint if (Cstar.w[0] & 0x01) { // Cstar is odd; MP in [EVEN, ODD] // if floor(C*) is odd C = floor(C*) - 1; the result may be 0 Cstar.w[0]--; // Cstar is now even if (Cstar.w[0] == 0xffffffffffffffffULL) { Cstar.w[1]--; } *ptr_is_midpoint_gt_even = 1; *ptr_is_inexact_lt_midpoint = 0; *ptr_is_inexact_gt_midpoint = 0; } else { // else MP in [ODD, EVEN] *ptr_is_midpoint_lt_even = 1; *ptr_is_inexact_lt_midpoint = 0; *ptr_is_inexact_gt_midpoint = 0; } } // check for rounding overflow, which occurs if Cstar = 10^(q-x) ind = q - x; // 1 <= ind <= q - 1 if (ind <= 19) { if (Cstar.w[1] == 0x0ULL && Cstar.w[0] == ten2k64[ind]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k64[ind - 1]; // Cstar = 10^(q-x-1) *incr_exp = 1; } else { *incr_exp = 0; } } else if (ind == 20) { // if ind = 20 if (Cstar.w[1] == ten2k128[0].w[1] && Cstar.w[0] == ten2k128[0].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k64[19]; // Cstar = 10^(q-x-1) Cstar.w[1] = 0x0ULL; *incr_exp = 1; } else { *incr_exp = 0; } } else { // if 21 <= ind <= 37 if (Cstar.w[1] == ten2k128[ind - 20].w[1] && Cstar.w[0] == ten2k128[ind - 20].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k128[ind - 21].w[0]; // Cstar = 10^(q-x-1) Cstar.w[1] = ten2k128[ind - 21].w[1]; *incr_exp = 1; } else { *incr_exp = 0; } } ptr_Cstar->w[1] = Cstar.w[1]; ptr_Cstar->w[0] = Cstar.w[0]; } void round192_39_57 (int q, int x, UINT192 C, UINT192 * ptr_Cstar, int *incr_exp, int *ptr_is_midpoint_lt_even, int *ptr_is_midpoint_gt_even, int *ptr_is_inexact_lt_midpoint, int *ptr_is_inexact_gt_midpoint) { UINT384 P384; UINT384 fstar; UINT192 Cstar; UINT64 tmp64; int shift; int ind; // Note: // In round192_39_57() positive numbers with 39 <= q <= 57 will be // rounded to nearest only for 5 <= x <= 42: // x = 23 or x = 24 or x = 5 or x = 6 when q = 39 // x = 24 or x = 25 or x = 6 or x = 7 when q = 40 // ... // x = 41 or x = 42 or x = 23 or x = 24 when q = 57 // However, for generality and possible uses outside the frame of IEEE 754R // this implementation works for 1 <= x <= q - 1 // assume *ptr_is_midpoint_lt_even, *ptr_is_midpoint_gt_even, // *ptr_is_inexact_lt_midpoint, and *ptr_is_inexact_gt_midpoint are // initialized to 0 by the caller // round a number C with q decimal digits, 39 <= q <= 57 // to q - x digits, 1 <= x <= 56 // C = C + 1/2 * 10^x where the result C fits in 192 bits // (because the largest value is // 999999999999999999999999999999999999999999999999999999999 + // 50000000000000000000000000000000000000000000000000000000 = // 0x2ad282f212a1da846afdaf18c034ff09da7fffffffffffff, which fits in 190 bits) ind = x - 1; // 0 <= ind <= 55 if (ind <= 18) { // if 0 <= ind <= 18 tmp64 = C.w[0]; C.w[0] = C.w[0] + midpoint64[ind]; if (C.w[0] < tmp64) { C.w[1]++; if (C.w[1] == 0x0) { C.w[2]++; } } } else if (ind <= 37) { // if 19 <= ind <= 37 tmp64 = C.w[0]; C.w[0] = C.w[0] + midpoint128[ind - 19].w[0]; if (C.w[0] < tmp64) { C.w[1]++; if (C.w[1] == 0x0) { C.w[2]++; } } tmp64 = C.w[1]; C.w[1] = C.w[1] + midpoint128[ind - 19].w[1]; if (C.w[1] < tmp64) { C.w[2]++; } } else { // if 38 <= ind <= 57 (actually ind <= 55) tmp64 = C.w[0]; C.w[0] = C.w[0] + midpoint192[ind - 38].w[0]; if (C.w[0] < tmp64) { C.w[1]++; if (C.w[1] == 0x0ull) { C.w[2]++; } } tmp64 = C.w[1]; C.w[1] = C.w[1] + midpoint192[ind - 38].w[1]; if (C.w[1] < tmp64) { C.w[2]++; } C.w[2] = C.w[2] + midpoint192[ind - 38].w[2]; } // kx ~= 10^(-x), kx = Kx192[ind] * 2^(-Ex), 0 <= ind <= 55 // P384 = (C + 1/2 * 10^x) * kx * 2^Ex = (C + 1/2 * 10^x) * Kx // the approximation kx of 10^(-x) was rounded up to 192 bits __mul_192x192_to_384 (P384, C, Kx192[ind]); // calculate C* = floor (P384) and f* // Cstar = P384 >> Ex // fstar = low Ex bits of P384 shift = Ex192m192[ind]; // in [1, 63] but have to consider three cases if (ind <= 18) { // if 0 <= ind <= 18 Cstar.w[2] = (P384.w[5] >> shift); Cstar.w[1] = (P384.w[5] << (64 - shift)) | (P384.w[4] >> shift); Cstar.w[0] = (P384.w[4] << (64 - shift)) | (P384.w[3] >> shift); fstar.w[5] = 0x0ULL; fstar.w[4] = 0x0ULL; fstar.w[3] = P384.w[3] & mask192[ind]; fstar.w[2] = P384.w[2]; fstar.w[1] = P384.w[1]; fstar.w[0] = P384.w[0]; } else if (ind <= 37) { // if 19 <= ind <= 37 Cstar.w[2] = 0x0ULL; Cstar.w[1] = P384.w[5] >> shift; Cstar.w[0] = (P384.w[5] << (64 - shift)) | (P384.w[4] >> shift); fstar.w[5] = 0x0ULL; fstar.w[4] = P384.w[4] & mask192[ind]; fstar.w[3] = P384.w[3]; fstar.w[2] = P384.w[2]; fstar.w[1] = P384.w[1]; fstar.w[0] = P384.w[0]; } else { // if 38 <= ind <= 57 Cstar.w[2] = 0x0ULL; Cstar.w[1] = 0x0ULL; Cstar.w[0] = P384.w[5] >> shift; fstar.w[5] = P384.w[5] & mask192[ind]; fstar.w[4] = P384.w[4]; fstar.w[3] = P384.w[3]; fstar.w[2] = P384.w[2]; fstar.w[1] = P384.w[1]; fstar.w[0] = P384.w[0]; } // the top Ex bits of 10^(-x) are T* = ten2mxtrunc192[ind], e.g. if x=1, // T*=ten2mxtrunc192[0]=0xcccccccccccccccccccccccccccccccccccccccccccccccc // if (0 < f* < 10^(-x)) then the result is a midpoint // if floor(C*) is even then C* = floor(C*) - logical right // shift; C* has q - x decimal digits, correct by Prop. 1) // else if floor(C*) is odd C* = floor(C*)-1 (logical right // shift; C* has q - x decimal digits, correct by Pr. 1) // else // C* = floor(C*) (logical right shift; C has q - x decimal digits, // correct by Property 1) // in the caling function n = C* * 10^(e+x) // determine inexactness of the rounding of C* // if (0 < f* - 1/2 < 10^(-x)) then // the result is exact // else // if (f* - 1/2 > T*) then // the result is inexact if (ind <= 18) { // if 0 <= ind <= 18 if (fstar.w[3] > half192[ind] || (fstar.w[3] == half192[ind] && (fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[3] - half192[ind]; if (tmp64 || fstar.w[2] > ten2mxtrunc192[ind].w[2] || (fstar.w[2] == ten2mxtrunc192[ind].w[2] && fstar.w[1] > ten2mxtrunc192[ind].w[1]) || (fstar.w[2] == ten2mxtrunc192[ind].w[2] && fstar.w[1] == ten2mxtrunc192[ind].w[1] && fstar.w[0] > ten2mxtrunc192[ind].w[0])) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } } else if (ind <= 37) { // if 19 <= ind <= 37 if (fstar.w[4] > half192[ind] || (fstar.w[4] == half192[ind] && (fstar.w[3] || fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[4] - half192[ind]; if (tmp64 || fstar.w[3] || fstar.w[2] > ten2mxtrunc192[ind].w[2] || (fstar.w[2] == ten2mxtrunc192[ind].w[2] && fstar.w[1] > ten2mxtrunc192[ind].w[1]) || (fstar.w[2] == ten2mxtrunc192[ind].w[2] && fstar.w[1] == ten2mxtrunc192[ind].w[1] && fstar.w[0] > ten2mxtrunc192[ind].w[0])) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } } else { // if 38 <= ind <= 55 if (fstar.w[5] > half192[ind] || (fstar.w[5] == half192[ind] && (fstar.w[4] || fstar.w[3] || fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[5] - half192[ind]; if (tmp64 || fstar.w[4] || fstar.w[3] || fstar.w[2] > ten2mxtrunc192[ind].w[2] || (fstar.w[2] == ten2mxtrunc192[ind].w[2] && fstar.w[1] > ten2mxtrunc192[ind].w[1]) || (fstar.w[2] == ten2mxtrunc192[ind].w[2] && fstar.w[1] == ten2mxtrunc192[ind].w[1] && fstar.w[0] > ten2mxtrunc192[ind].w[0])) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } } // check for midpoints (could do this before determining inexactness) if (fstar.w[5] == 0 && fstar.w[4] == 0 && fstar.w[3] == 0 && (fstar.w[2] < ten2mxtrunc192[ind].w[2] || (fstar.w[2] == ten2mxtrunc192[ind].w[2] && fstar.w[1] < ten2mxtrunc192[ind].w[1]) || (fstar.w[2] == ten2mxtrunc192[ind].w[2] && fstar.w[1] == ten2mxtrunc192[ind].w[1] && fstar.w[0] <= ten2mxtrunc192[ind].w[0]))) { // the result is a midpoint if (Cstar.w[0] & 0x01) { // Cstar is odd; MP in [EVEN, ODD] // if floor(C*) is odd C = floor(C*) - 1; the result may be 0 Cstar.w[0]--; // Cstar is now even if (Cstar.w[0] == 0xffffffffffffffffULL) { Cstar.w[1]--; if (Cstar.w[1] == 0xffffffffffffffffULL) { Cstar.w[2]--; } } *ptr_is_midpoint_gt_even = 1; *ptr_is_inexact_lt_midpoint = 0; *ptr_is_inexact_gt_midpoint = 0; } else { // else MP in [ODD, EVEN] *ptr_is_midpoint_lt_even = 1; *ptr_is_inexact_lt_midpoint = 0; *ptr_is_inexact_gt_midpoint = 0; } } // check for rounding overflow, which occurs if Cstar = 10^(q-x) ind = q - x; // 1 <= ind <= q - 1 if (ind <= 19) { if (Cstar.w[2] == 0x0ULL && Cstar.w[1] == 0x0ULL && Cstar.w[0] == ten2k64[ind]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k64[ind - 1]; // Cstar = 10^(q-x-1) *incr_exp = 1; } else { *incr_exp = 0; } } else if (ind == 20) { // if ind = 20 if (Cstar.w[2] == 0x0ULL && Cstar.w[1] == ten2k128[0].w[1] && Cstar.w[0] == ten2k128[0].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k64[19]; // Cstar = 10^(q-x-1) Cstar.w[1] = 0x0ULL; *incr_exp = 1; } else { *incr_exp = 0; } } else if (ind <= 38) { // if 21 <= ind <= 38 if (Cstar.w[2] == 0x0ULL && Cstar.w[1] == ten2k128[ind - 20].w[1] && Cstar.w[0] == ten2k128[ind - 20].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k128[ind - 21].w[0]; // Cstar = 10^(q-x-1) Cstar.w[1] = ten2k128[ind - 21].w[1]; *incr_exp = 1; } else { *incr_exp = 0; } } else if (ind == 39) { if (Cstar.w[2] == ten2k256[0].w[2] && Cstar.w[1] == ten2k256[0].w[1] && Cstar.w[0] == ten2k256[0].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k128[18].w[0]; // Cstar = 10^(q-x-1) Cstar.w[1] = ten2k128[18].w[1]; Cstar.w[2] = 0x0ULL; *incr_exp = 1; } else { *incr_exp = 0; } } else { // if 40 <= ind <= 56 if (Cstar.w[2] == ten2k256[ind - 39].w[2] && Cstar.w[1] == ten2k256[ind - 39].w[1] && Cstar.w[0] == ten2k256[ind - 39].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k256[ind - 40].w[0]; // Cstar = 10^(q-x-1) Cstar.w[1] = ten2k256[ind - 40].w[1]; Cstar.w[2] = ten2k256[ind - 40].w[2]; *incr_exp = 1; } else { *incr_exp = 0; } } ptr_Cstar->w[2] = Cstar.w[2]; ptr_Cstar->w[1] = Cstar.w[1]; ptr_Cstar->w[0] = Cstar.w[0]; } void round256_58_76 (int q, int x, UINT256 C, UINT256 * ptr_Cstar, int *incr_exp, int *ptr_is_midpoint_lt_even, int *ptr_is_midpoint_gt_even, int *ptr_is_inexact_lt_midpoint, int *ptr_is_inexact_gt_midpoint) { UINT512 P512; UINT512 fstar; UINT256 Cstar; UINT64 tmp64; int shift; int ind; // Note: // In round256_58_76() positive numbers with 58 <= q <= 76 will be // rounded to nearest only for 24 <= x <= 61: // x = 42 or x = 43 or x = 24 or x = 25 when q = 58 // x = 43 or x = 44 or x = 25 or x = 26 when q = 59 // ... // x = 60 or x = 61 or x = 42 or x = 43 when q = 76 // However, for generality and possible uses outside the frame of IEEE 754R // this implementation works for 1 <= x <= q - 1 // assume *ptr_is_midpoint_lt_even, *ptr_is_midpoint_gt_even, // *ptr_is_inexact_lt_midpoint, and *ptr_is_inexact_gt_midpoint are // initialized to 0 by the caller // round a number C with q decimal digits, 58 <= q <= 76 // to q - x digits, 1 <= x <= 75 // C = C + 1/2 * 10^x where the result C fits in 256 bits // (because the largest value is 9999999999999999999999999999999999999999 // 999999999999999999999999999999999999 + 500000000000000000000000000 // 000000000000000000000000000000000000000000000000 = // 0x1736ca15d27a56cae15cf0e7b403d1f2bd6ebb0a50dc83ffffffffffffffffff, // which fits in 253 bits) ind = x - 1; // 0 <= ind <= 74 if (ind <= 18) { // if 0 <= ind <= 18 tmp64 = C.w[0]; C.w[0] = C.w[0] + midpoint64[ind]; if (C.w[0] < tmp64) { C.w[1]++; if (C.w[1] == 0x0) { C.w[2]++; if (C.w[2] == 0x0) { C.w[3]++; } } } } else if (ind <= 37) { // if 19 <= ind <= 37 tmp64 = C.w[0]; C.w[0] = C.w[0] + midpoint128[ind - 19].w[0]; if (C.w[0] < tmp64) { C.w[1]++; if (C.w[1] == 0x0) { C.w[2]++; if (C.w[2] == 0x0) { C.w[3]++; } } } tmp64 = C.w[1]; C.w[1] = C.w[1] + midpoint128[ind - 19].w[1]; if (C.w[1] < tmp64) { C.w[2]++; if (C.w[2] == 0x0) { C.w[3]++; } } } else if (ind <= 57) { // if 38 <= ind <= 57 tmp64 = C.w[0]; C.w[0] = C.w[0] + midpoint192[ind - 38].w[0]; if (C.w[0] < tmp64) { C.w[1]++; if (C.w[1] == 0x0ull) { C.w[2]++; if (C.w[2] == 0x0) { C.w[3]++; } } } tmp64 = C.w[1]; C.w[1] = C.w[1] + midpoint192[ind - 38].w[1]; if (C.w[1] < tmp64) { C.w[2]++; if (C.w[2] == 0x0) { C.w[3]++; } } tmp64 = C.w[2]; C.w[2] = C.w[2] + midpoint192[ind - 38].w[2]; if (C.w[2] < tmp64) { C.w[3]++; } } else { // if 58 <= ind <= 76 (actually 58 <= ind <= 74) tmp64 = C.w[0]; C.w[0] = C.w[0] + midpoint256[ind - 58].w[0]; if (C.w[0] < tmp64) { C.w[1]++; if (C.w[1] == 0x0ull) { C.w[2]++; if (C.w[2] == 0x0) { C.w[3]++; } } } tmp64 = C.w[1]; C.w[1] = C.w[1] + midpoint256[ind - 58].w[1]; if (C.w[1] < tmp64) { C.w[2]++; if (C.w[2] == 0x0) { C.w[3]++; } } tmp64 = C.w[2]; C.w[2] = C.w[2] + midpoint256[ind - 58].w[2]; if (C.w[2] < tmp64) { C.w[3]++; } C.w[3] = C.w[3] + midpoint256[ind - 58].w[3]; } // kx ~= 10^(-x), kx = Kx256[ind] * 2^(-Ex), 0 <= ind <= 74 // P512 = (C + 1/2 * 10^x) * kx * 2^Ex = (C + 1/2 * 10^x) * Kx // the approximation kx of 10^(-x) was rounded up to 192 bits __mul_256x256_to_512 (P512, C, Kx256[ind]); // calculate C* = floor (P512) and f* // Cstar = P512 >> Ex // fstar = low Ex bits of P512 shift = Ex256m256[ind]; // in [0, 63] but have to consider four cases if (ind <= 18) { // if 0 <= ind <= 18 Cstar.w[3] = (P512.w[7] >> shift); Cstar.w[2] = (P512.w[7] << (64 - shift)) | (P512.w[6] >> shift); Cstar.w[1] = (P512.w[6] << (64 - shift)) | (P512.w[5] >> shift); Cstar.w[0] = (P512.w[5] << (64 - shift)) | (P512.w[4] >> shift); fstar.w[7] = 0x0ULL; fstar.w[6] = 0x0ULL; fstar.w[5] = 0x0ULL; fstar.w[4] = P512.w[4] & mask256[ind]; fstar.w[3] = P512.w[3]; fstar.w[2] = P512.w[2]; fstar.w[1] = P512.w[1]; fstar.w[0] = P512.w[0]; } else if (ind <= 37) { // if 19 <= ind <= 37 Cstar.w[3] = 0x0ULL; Cstar.w[2] = P512.w[7] >> shift; Cstar.w[1] = (P512.w[7] << (64 - shift)) | (P512.w[6] >> shift); Cstar.w[0] = (P512.w[6] << (64 - shift)) | (P512.w[5] >> shift); fstar.w[7] = 0x0ULL; fstar.w[6] = 0x0ULL; fstar.w[5] = P512.w[5] & mask256[ind]; fstar.w[4] = P512.w[4]; fstar.w[3] = P512.w[3]; fstar.w[2] = P512.w[2]; fstar.w[1] = P512.w[1]; fstar.w[0] = P512.w[0]; } else if (ind <= 56) { // if 38 <= ind <= 56 Cstar.w[3] = 0x0ULL; Cstar.w[2] = 0x0ULL; Cstar.w[1] = P512.w[7] >> shift; Cstar.w[0] = (P512.w[7] << (64 - shift)) | (P512.w[6] >> shift); fstar.w[7] = 0x0ULL; fstar.w[6] = P512.w[6] & mask256[ind]; fstar.w[5] = P512.w[5]; fstar.w[4] = P512.w[4]; fstar.w[3] = P512.w[3]; fstar.w[2] = P512.w[2]; fstar.w[1] = P512.w[1]; fstar.w[0] = P512.w[0]; } else if (ind == 57) { Cstar.w[3] = 0x0ULL; Cstar.w[2] = 0x0ULL; Cstar.w[1] = 0x0ULL; Cstar.w[0] = P512.w[7]; fstar.w[7] = 0x0ULL; fstar.w[6] = P512.w[6]; fstar.w[5] = P512.w[5]; fstar.w[4] = P512.w[4]; fstar.w[3] = P512.w[3]; fstar.w[2] = P512.w[2]; fstar.w[1] = P512.w[1]; fstar.w[0] = P512.w[0]; } else { // if 58 <= ind <= 74 Cstar.w[3] = 0x0ULL; Cstar.w[2] = 0x0ULL; Cstar.w[1] = 0x0ULL; Cstar.w[0] = P512.w[7] >> shift; fstar.w[7] = P512.w[7] & mask256[ind]; fstar.w[6] = P512.w[6]; fstar.w[5] = P512.w[5]; fstar.w[4] = P512.w[4]; fstar.w[3] = P512.w[3]; fstar.w[2] = P512.w[2]; fstar.w[1] = P512.w[1]; fstar.w[0] = P512.w[0]; } // the top Ex bits of 10^(-x) are T* = ten2mxtrunc256[ind], e.g. if x=1, // T*=ten2mxtrunc256[0]= // 0xcccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc // if (0 < f* < 10^(-x)) then the result is a midpoint // if floor(C*) is even then C* = floor(C*) - logical right // shift; C* has q - x decimal digits, correct by Prop. 1) // else if floor(C*) is odd C* = floor(C*)-1 (logical right // shift; C* has q - x decimal digits, correct by Pr. 1) // else // C* = floor(C*) (logical right shift; C has q - x decimal digits, // correct by Property 1) // in the caling function n = C* * 10^(e+x) // determine inexactness of the rounding of C* // if (0 < f* - 1/2 < 10^(-x)) then // the result is exact // else // if (f* - 1/2 > T*) then // the result is inexact if (ind <= 18) { // if 0 <= ind <= 18 if (fstar.w[4] > half256[ind] || (fstar.w[4] == half256[ind] && (fstar.w[3] || fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[4] - half256[ind]; if (tmp64 || fstar.w[3] > ten2mxtrunc256[ind].w[2] || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] > ten2mxtrunc256[ind].w[2]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] > ten2mxtrunc256[ind].w[1]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] == ten2mxtrunc256[ind].w[1] && fstar.w[0] > ten2mxtrunc256[ind].w[0])) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } } else if (ind <= 37) { // if 19 <= ind <= 37 if (fstar.w[5] > half256[ind] || (fstar.w[5] == half256[ind] && (fstar.w[4] || fstar.w[3] || fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[5] - half256[ind]; if (tmp64 || fstar.w[4] || fstar.w[3] > ten2mxtrunc256[ind].w[3] || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] > ten2mxtrunc256[ind].w[2]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] > ten2mxtrunc256[ind].w[1]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] == ten2mxtrunc256[ind].w[1] && fstar.w[0] > ten2mxtrunc256[ind].w[0])) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } } else if (ind <= 57) { // if 38 <= ind <= 57 if (fstar.w[6] > half256[ind] || (fstar.w[6] == half256[ind] && (fstar.w[5] || fstar.w[4] || fstar.w[3] || fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[6] - half256[ind]; if (tmp64 || fstar.w[5] || fstar.w[4] || fstar.w[3] > ten2mxtrunc256[ind].w[3] || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] > ten2mxtrunc256[ind].w[2]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] > ten2mxtrunc256[ind].w[1]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] == ten2mxtrunc256[ind].w[1] && fstar.w[0] > ten2mxtrunc256[ind].w[0])) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } } else { // if 58 <= ind <= 74 if (fstar.w[7] > half256[ind] || (fstar.w[7] == half256[ind] && (fstar.w[6] || fstar.w[5] || fstar.w[4] || fstar.w[3] || fstar.w[2] || fstar.w[1] || fstar.w[0]))) { // f* > 1/2 and the result may be exact // Calculate f* - 1/2 tmp64 = fstar.w[7] - half256[ind]; if (tmp64 || fstar.w[6] || fstar.w[5] || fstar.w[4] || fstar.w[3] > ten2mxtrunc256[ind].w[3] || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] > ten2mxtrunc256[ind].w[2]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] > ten2mxtrunc256[ind].w[1]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] == ten2mxtrunc256[ind].w[1] && fstar.w[0] > ten2mxtrunc256[ind].w[0])) { // f* - 1/2 > 10^(-x) *ptr_is_inexact_lt_midpoint = 1; } // else the result is exact } else { // the result is inexact; f2* <= 1/2 *ptr_is_inexact_gt_midpoint = 1; } } // check for midpoints (could do this before determining inexactness) if (fstar.w[7] == 0 && fstar.w[6] == 0 && fstar.w[5] == 0 && fstar.w[4] == 0 && (fstar.w[3] < ten2mxtrunc256[ind].w[3] || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] < ten2mxtrunc256[ind].w[2]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] < ten2mxtrunc256[ind].w[1]) || (fstar.w[3] == ten2mxtrunc256[ind].w[3] && fstar.w[2] == ten2mxtrunc256[ind].w[2] && fstar.w[1] == ten2mxtrunc256[ind].w[1] && fstar.w[0] <= ten2mxtrunc256[ind].w[0]))) { // the result is a midpoint if (Cstar.w[0] & 0x01) { // Cstar is odd; MP in [EVEN, ODD] // if floor(C*) is odd C = floor(C*) - 1; the result may be 0 Cstar.w[0]--; // Cstar is now even if (Cstar.w[0] == 0xffffffffffffffffULL) { Cstar.w[1]--; if (Cstar.w[1] == 0xffffffffffffffffULL) { Cstar.w[2]--; if (Cstar.w[2] == 0xffffffffffffffffULL) { Cstar.w[3]--; } } } *ptr_is_midpoint_gt_even = 1; *ptr_is_inexact_lt_midpoint = 0; *ptr_is_inexact_gt_midpoint = 0; } else { // else MP in [ODD, EVEN] *ptr_is_midpoint_lt_even = 1; *ptr_is_inexact_lt_midpoint = 0; *ptr_is_inexact_gt_midpoint = 0; } } // check for rounding overflow, which occurs if Cstar = 10^(q-x) ind = q - x; // 1 <= ind <= q - 1 if (ind <= 19) { if (Cstar.w[3] == 0x0ULL && Cstar.w[2] == 0x0ULL && Cstar.w[1] == 0x0ULL && Cstar.w[0] == ten2k64[ind]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k64[ind - 1]; // Cstar = 10^(q-x-1) *incr_exp = 1; } else { *incr_exp = 0; } } else if (ind == 20) { // if ind = 20 if (Cstar.w[3] == 0x0ULL && Cstar.w[2] == 0x0ULL && Cstar.w[1] == ten2k128[0].w[1] && Cstar.w[0] == ten2k128[0].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k64[19]; // Cstar = 10^(q-x-1) Cstar.w[1] = 0x0ULL; *incr_exp = 1; } else { *incr_exp = 0; } } else if (ind <= 38) { // if 21 <= ind <= 38 if (Cstar.w[3] == 0x0ULL && Cstar.w[2] == 0x0ULL && Cstar.w[1] == ten2k128[ind - 20].w[1] && Cstar.w[0] == ten2k128[ind - 20].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k128[ind - 21].w[0]; // Cstar = 10^(q-x-1) Cstar.w[1] = ten2k128[ind - 21].w[1]; *incr_exp = 1; } else { *incr_exp = 0; } } else if (ind == 39) { if (Cstar.w[3] == 0x0ULL && Cstar.w[2] == ten2k256[0].w[2] && Cstar.w[1] == ten2k256[0].w[1] && Cstar.w[0] == ten2k256[0].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k128[18].w[0]; // Cstar = 10^(q-x-1) Cstar.w[1] = ten2k128[18].w[1]; Cstar.w[2] = 0x0ULL; *incr_exp = 1; } else { *incr_exp = 0; } } else if (ind <= 57) { // if 40 <= ind <= 57 if (Cstar.w[3] == 0x0ULL && Cstar.w[2] == ten2k256[ind - 39].w[2] && Cstar.w[1] == ten2k256[ind - 39].w[1] && Cstar.w[0] == ten2k256[ind - 39].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k256[ind - 40].w[0]; // Cstar = 10^(q-x-1) Cstar.w[1] = ten2k256[ind - 40].w[1]; Cstar.w[2] = ten2k256[ind - 40].w[2]; *incr_exp = 1; } else { *incr_exp = 0; } // else if (ind == 58) is not needed becauae we do not have ten2k192[] yet } else { // if 58 <= ind <= 77 (actually 58 <= ind <= 74) if (Cstar.w[3] == ten2k256[ind - 39].w[3] && Cstar.w[2] == ten2k256[ind - 39].w[2] && Cstar.w[1] == ten2k256[ind - 39].w[1] && Cstar.w[0] == ten2k256[ind - 39].w[0]) { // if Cstar = 10^(q-x) Cstar.w[0] = ten2k256[ind - 40].w[0]; // Cstar = 10^(q-x-1) Cstar.w[1] = ten2k256[ind - 40].w[1]; Cstar.w[2] = ten2k256[ind - 40].w[2]; Cstar.w[3] = ten2k256[ind - 40].w[3]; *incr_exp = 1; } else { *incr_exp = 0; } } ptr_Cstar->w[3] = Cstar.w[3]; ptr_Cstar->w[2] = Cstar.w[2]; ptr_Cstar->w[1] = Cstar.w[1]; ptr_Cstar->w[0] = Cstar.w[0]; }
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