/* * Floating-point operations. * * This file implements the non-inline functions declared in * fpr.h, as well as the constants for FFT / iFFT. * * ==========================(LICENSE BEGIN)============================ * * Copyright (c) 2017-2019 Falcon Project * * Permission is hereby granted, free of charge, to any person obtaining * a copy of this software and associated documentation files (the * "Software"), to deal in the Software without restriction, including * without limitation the rights to use, copy, modify, merge, publish, * distribute, sublicense, and/or sell copies of the Software, and to * permit persons to whom the Software is furnished to do so, subject to * the following conditions: * * The above copyright notice and this permission notice shall be * included in all copies or substantial portions of the Software. * * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. * IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY * CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, * TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE * SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. * * ===========================(LICENSE END)============================= * * @author Thomas Pornin */ #include "inner.h" /* * Normalize a provided unsigned integer to the 2^63..2^64-1 range by * left-shifting it if necessary. The exponent e is adjusted accordingly * (i.e. if the value was left-shifted by n bits, then n is subtracted * from e). If source m is 0, then it remains 0, but e is altered. * Both m and e must be simple variables (no expressions allowed). */ #define FPR_NORM64(m, e) do { \ uint32_t nt; \ \ (e) -= 63; \ \ nt = (uint32_t)((m) >> 32); \ nt = (nt | -nt) >> 31; \ (m) ^= ((m) ^ ((m) << 32)) & ((uint64_t)nt - 1); \ (e) += (int)(nt << 5); \ \ nt = (uint32_t)((m) >> 48); \ nt = (nt | -nt) >> 31; \ (m) ^= ((m) ^ ((m) << 16)) & ((uint64_t)nt - 1); \ (e) += (int)(nt << 4); \ \ nt = (uint32_t)((m) >> 56); \ nt = (nt | -nt) >> 31; \ (m) ^= ((m) ^ ((m) << 8)) & ((uint64_t)nt - 1); \ (e) += (int)(nt << 3); \ \ nt = (uint32_t)((m) >> 60); \ nt = (nt | -nt) >> 31; \ (m) ^= ((m) ^ ((m) << 4)) & ((uint64_t)nt - 1); \ (e) += (int)(nt << 2); \ \ nt = (uint32_t)((m) >> 62); \ nt = (nt | -nt) >> 31; \ (m) ^= ((m) ^ ((m) << 2)) & ((uint64_t)nt - 1); \ (e) += (int)(nt << 1); \ \ nt = (uint32_t)((m) >> 63); \ (m) ^= ((m) ^ ((m) << 1)) & ((uint64_t)nt - 1); \ (e) += (int)(nt); \ } while (0) fpr fpr_scaled(int64_t i, int sc) { /* * To convert from int to float, we have to do the following: * 1. Get the absolute value of the input, and its sign * 2. Shift right or left the value as appropriate * 3. Pack the result * * We can assume that the source integer is not -2^63. */ int s, e; uint32_t t; uint64_t m; /* * Extract sign bit. * We have: -i = 1 + ~i */ s = (int)((uint64_t)i >> 63); i ^= -(int64_t)s; i += s; /* * For now we suppose that i != 0. * Otherwise, we set m to i and left-shift it as much as needed * to get a 1 in the top bit. We can do that in a logarithmic * number of conditional shifts. */ m = (uint64_t)i; e = 9 + sc; FPR_NORM64(m, e); /* * Now m is in the 2^63..2^64-1 range. We must divide it by 512; * if one of the dropped bits is a 1, this should go into the * "sticky bit". */ m |= ((uint32_t)m & 0x1FF) + 0x1FF; m >>= 9; /* * Corrective action: if i = 0 then all of the above was * incorrect, and we clamp e and m down to zero. */ t = (uint32_t)((uint64_t)(i | -i) >> 63); m &= -(uint64_t)t; e &= -(int)t; /* * Assemble back everything. The FPR() function will handle cases * where e is too low. */ return FPR(s, e, m); } fpr fpr_add(fpr x, fpr y) { uint64_t m, xu, yu, za; uint32_t cs; int ex, ey, sx, sy, cc; /* * Make sure that the first operand (x) has the larger absolute * value. This guarantees that the exponent of y is less than * or equal to the exponent of x, and, if they are equal, then * the mantissa of y will not be greater than the mantissa of x. * * After this swap, the result will have the sign x, except in * the following edge case: abs(x) = abs(y), and x and y have * opposite sign bits; in that case, the result shall be +0 * even if the sign bit of x is 1. To handle this case properly, * we do the swap is abs(x) = abs(y) AND the sign of x is 1. */ m = ((uint64_t)1 << 63) - 1; za = (x & m) - (y & m); cs = (uint32_t)(za >> 63) | ((1U - (uint32_t)(-za >> 63)) & (uint32_t)(x >> 63)); m = (x ^ y) & -(uint64_t)cs; x ^= m; y ^= m; /* * Extract sign bits, exponents and mantissas. The mantissas are * scaled up to 2^55..2^56-1, and the exponent is unbiased. If * an operand is zero, its mantissa is set to 0 at this step, and * its exponent will be -1078. */ ex = (int)(x >> 52); sx = ex >> 11; ex &= 0x7FF; m = (uint64_t)(uint32_t)((ex + 0x7FF) >> 11) << 52; xu = ((x & (((uint64_t)1 << 52) - 1)) | m) << 3; ex -= 1078; ey = (int)(y >> 52); sy = ey >> 11; ey &= 0x7FF; m = (uint64_t)(uint32_t)((ey + 0x7FF) >> 11) << 52; yu = ((y & (((uint64_t)1 << 52) - 1)) | m) << 3; ey -= 1078; /* * x has the larger exponent; hence, we only need to right-shift y. * If the shift count is larger than 59 bits then we clamp the * value to zero. */ cc = ex - ey; yu &= -(uint64_t)((uint32_t)(cc - 60) >> 31); cc &= 63; /* * The lowest bit of yu is "sticky". */ m = fpr_ulsh(1, cc) - 1; yu |= (yu & m) + m; yu = fpr_ursh(yu, cc); /* * If the operands have the same sign, then we add the mantissas; * otherwise, we subtract the mantissas. */ xu += yu - ((yu << 1) & -(uint64_t)(sx ^ sy)); /* * The result may be smaller, or slightly larger. We normalize * it to the 2^63..2^64-1 range (if xu is zero, then it stays * at zero). */ FPR_NORM64(xu, ex); /* * Scale down the value to 2^54..s^55-1, handling the last bit * as sticky. */ xu |= ((uint32_t)xu & 0x1FF) + 0x1FF; xu >>= 9; ex += 9; /* * In general, the result has the sign of x. However, if the * result is exactly zero, then the following situations may * be encountered: * x > 0, y = -x -> result should be +0 * x < 0, y = -x -> result should be +0 * x = +0, y = +0 -> result should be +0 * x = -0, y = +0 -> result should be +0 * x = +0, y = -0 -> result should be +0 * x = -0, y = -0 -> result should be -0 * * But at the conditional swap step at the start of the * function, we ensured that if abs(x) = abs(y) and the * sign of x was 1, then x and y were swapped. Thus, the * two following cases cannot actually happen: * x < 0, y = -x * x = -0, y = +0 * In all other cases, the sign bit of x is conserved, which * is what the FPR() function does. The FPR() function also * properly clamps values to zero when the exponent is too * low, but does not alter the sign in that case. */ return FPR(sx, ex, xu); } fpr fpr_mul(fpr x, fpr y) { uint64_t xu, yu, w, zu, zv; uint32_t x0, x1, y0, y1, z0, z1, z2; int ex, ey, d, e, s; /* * Extract absolute values as scaled unsigned integers. We * don't extract exponents yet. */ xu = (x & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); yu = (y & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); /* * We have two 53-bit integers to multiply; we need to split * each into a lower half and a upper half. Moreover, we * prefer to have lower halves to be of 25 bits each, for * reasons explained later on. */ x0 = (uint32_t)xu & 0x01FFFFFF; x1 = (uint32_t)(xu >> 25); y0 = (uint32_t)yu & 0x01FFFFFF; y1 = (uint32_t)(yu >> 25); w = (uint64_t)x0 * (uint64_t)y0; z0 = (uint32_t)w & 0x01FFFFFF; z1 = (uint32_t)(w >> 25); w = (uint64_t)x0 * (uint64_t)y1; z1 += (uint32_t)w & 0x01FFFFFF; z2 = (uint32_t)(w >> 25); w = (uint64_t)x1 * (uint64_t)y0; z1 += (uint32_t)w & 0x01FFFFFF; z2 += (uint32_t)(w >> 25); zu = (uint64_t)x1 * (uint64_t)y1; z2 += (z1 >> 25); z1 &= 0x01FFFFFF; zu += z2; /* * Since xu and yu are both in the 2^52..2^53-1 range, the * product is in the 2^104..2^106-1 range. We first reassemble * it and round it into the 2^54..2^56-1 range; the bottom bit * is made "sticky". Since the low limbs z0 and z1 are 25 bits * each, we just take the upper part (zu), and consider z0 and * z1 only for purposes of stickiness. * (This is the reason why we chose 25-bit limbs above.) */ zu |= ((z0 | z1) + 0x01FFFFFF) >> 25; /* * We normalize zu to the 2^54..s^55-1 range: it could be one * bit too large at this point. This is done with a conditional * right-shift that takes into account the sticky bit. */ zv = (zu >> 1) | (zu & 1); w = zu >> 55; zu ^= (zu ^ zv) & -w; /* * Get the aggregate scaling factor: * * - Each exponent is biased by 1023. * * - Integral mantissas are scaled by 2^52, hence an * extra 52 bias for each exponent. * * - However, we right-shifted z by 50 bits, and then * by 0 or 1 extra bit (depending on the value of w). * * In total, we must add the exponents, then subtract * 2 * (1023 + 52), then add 50 + w. */ ex = (int)((x >> 52) & 0x7FF); ey = (int)((y >> 52) & 0x7FF); e = ex + ey - 2100 + (int)w; /* * Sign bit is the XOR of the operand sign bits. */ s = (int)((x ^ y) >> 63); /* * Corrective actions for zeros: if either of the operands is * zero, then the computations above were wrong. Test for zero * is whether ex or ey is zero. We just have to set the mantissa * (zu) to zero, the FPR() function will normalize e. */ d = ((ex + 0x7FF) & (ey + 0x7FF)) >> 11; zu &= -(uint64_t)d; /* * FPR() packs the result and applies proper rounding. */ return FPR(s, e, zu); } fpr fpr_div(fpr x, fpr y) { uint64_t xu, yu, q, q2, w; int i, ex, ey, e, d, s; /* * Extract mantissas of x and y (unsigned). */ xu = (x & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); yu = (y & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); /* * Perform bit-by-bit division of xu by yu. We run it for 55 bits. */ q = 0; for (i = 0; i < 55; i ++) { /* * If yu is less than or equal xu, then subtract it and * push a 1 in the quotient; otherwise, leave xu unchanged * and push a 0. */ uint64_t b; b = ((xu - yu) >> 63) - 1; xu -= b & yu; q |= b & 1; xu <<= 1; q <<= 1; } /* * We got 55 bits in the quotient, followed by an extra zero. We * want that 56th bit to be "sticky": it should be a 1 if and * only if the remainder (xu) is non-zero. */ q |= (xu | -xu) >> 63; /* * Quotient is at most 2^56-1. Its top bit may be zero, but in * that case the next-to-top bit will be a one, since the * initial xu and yu were both in the 2^52..2^53-1 range. * We perform a conditional shift to normalize q to the * 2^54..2^55-1 range (with the bottom bit being sticky). */ q2 = (q >> 1) | (q & 1); w = q >> 55; q ^= (q ^ q2) & -w; /* * Extract exponents to compute the scaling factor: * * - Each exponent is biased and we scaled them up by * 52 bits; but these biases will cancel out. * * - The division loop produced a 55-bit shifted result, * so we must scale it down by 55 bits. * * - If w = 1, we right-shifted the integer by 1 bit, * hence we must add 1 to the scaling. */ ex = (int)((x >> 52) & 0x7FF); ey = (int)((y >> 52) & 0x7FF); e = ex - ey - 55 + (int)w; /* * Sign is the XOR of the signs of the operands. */ s = (int)((x ^ y) >> 63); /* * Corrective actions for zeros: if x = 0, then the computation * is wrong, and we must clamp e and q to 0. We do not care * about the case y = 0 (as per assumptions in this module, * the caller does not perform divisions by zero). */ d = (ex + 0x7FF) >> 11; s &= d; e &= -d; q &= -(uint64_t)d; /* * FPR() packs the result and applies proper rounding. */ return FPR(s, e, q); } fpr fpr_sqrt(fpr x) { uint64_t xu, q, s, r; int ex, e; /* * Extract the mantissa and the exponent. We don't care about * the sign: by assumption, the operand is nonnegative. * We want the "true" exponent corresponding to a mantissa * in the 1..2 range. */ xu = (x & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); ex = (int)((x >> 52) & 0x7FF); e = ex - 1023; /* * If the exponent is odd, double the mantissa and decrement * the exponent. The exponent is then halved to account for * the square root. */ xu += xu & -(uint64_t)(e & 1); e >>= 1; /* * Double the mantissa. */ xu <<= 1; /* * We now have a mantissa in the 2^53..2^55-1 range. It * represents a value between 1 (inclusive) and 4 (exclusive) * in fixed point notation (with 53 fractional bits). We * compute the square root bit by bit. */ q = 0; s = 0; r = (uint64_t)1 << 53; for (int i = 0; i < 54; i ++) { uint64_t t, b; t = s + r; b = ((xu - t) >> 63) - 1; s += (r << 1) & b; xu -= t & b; q += r & b; xu <<= 1; r >>= 1; } /* * Now, q is a rounded-low 54-bit value, with a leading 1, * 52 fractional digits, and an additional guard bit. We add * an extra sticky bit to account for what remains of the operand. */ q <<= 1; q |= (xu | -xu) >> 63; /* * Result q is in the 2^54..2^55-1 range; we bias the exponent * by 54 bits (the value e at that point contains the "true" * exponent, but q is now considered an integer, i.e. scaled * up. */ e -= 54; /* * Corrective action for an operand of value zero. */ q &= -(uint64_t)((ex + 0x7FF) >> 11); /* * Apply rounding and back result. */ return FPR(0, e, q); } uint64_t fpr_expm_p63(fpr x, fpr ccs) { /* * Polynomial approximation of exp(-x) is taken from FACCT: * https://eprint.iacr.org/2018/1234 * Specifically, values are extracted from the implementation * referenced from the FACCT article, and available at: * https://github.com/raykzhao/gaussian * Here, the coefficients have been scaled up by 2^63 and * converted to integers. * * Tests over more than 24 billions of random inputs in the * 0..log(2) range have never shown a deviation larger than * 2^(-50) from the true mathematical value. */ static const uint64_t C[] = { 0x00000004741183A3u, 0x00000036548CFC06u, 0x0000024FDCBF140Au, 0x0000171D939DE045u, 0x0000D00CF58F6F84u, 0x000680681CF796E3u, 0x002D82D8305B0FEAu, 0x011111110E066FD0u, 0x0555555555070F00u, 0x155555555581FF00u, 0x400000000002B400u, 0x7FFFFFFFFFFF4800u, 0x8000000000000000u }; uint64_t z, y; unsigned u; uint32_t z0, z1, y0, y1; uint64_t a, b; y = C[0]; z = (uint64_t)fpr_trunc(fpr_mul(x, fpr_ptwo63)) << 1; for (u = 1; u < (sizeof C) / sizeof(C[0]); u ++) { /* * Compute product z * y over 128 bits, but keep only * the top 64 bits. * * TODO: On some architectures/compilers we could use * some intrinsics (__umulh() on MSVC) or other compiler * extensions (unsigned __int128 on GCC / Clang) for * improved speed; however, most 64-bit architectures * also have appropriate IEEE754 floating-point support, * which is better. */ uint64_t c; z0 = (uint32_t)z; z1 = (uint32_t)(z >> 32); y0 = (uint32_t)y; y1 = (uint32_t)(y >> 32); a = ((uint64_t)z0 * (uint64_t)y1) + (((uint64_t)z0 * (uint64_t)y0) >> 32); b = ((uint64_t)z1 * (uint64_t)y0); c = (a >> 32) + (b >> 32); c += (((uint64_t)(uint32_t)a + (uint64_t)(uint32_t)b) >> 32); c += (uint64_t)z1 * (uint64_t)y1; y = C[u] - c; } /* * The scaling factor must be applied at the end. Since y is now * in fixed-point notation, we have to convert the factor to the * same format, and do an extra integer multiplication. */ z = (uint64_t)fpr_trunc(fpr_mul(ccs, fpr_ptwo63)) << 1; z0 = (uint32_t)z; z1 = (uint32_t)(z >> 32); y0 = (uint32_t)y; y1 = (uint32_t)(y >> 32); a = ((uint64_t)z0 * (uint64_t)y1) + (((uint64_t)z0 * (uint64_t)y0) >> 32); b = ((uint64_t)z1 * (uint64_t)y0); y = (a >> 32) + (b >> 32); y += (((uint64_t)(uint32_t)a + (uint64_t)(uint32_t)b) >> 32); y += (uint64_t)z1 * (uint64_t)y1; return y; } const fpr fpr_gm_tab[] = { 0, 0, 9223372036854775808U, 4607182418800017408U, 4604544271217802189U, 4604544271217802189U, 13827916308072577997U, 4604544271217802189U, 4606496786581982534U, 4600565431771507043U, 13823937468626282851U, 4606496786581982534U, 4600565431771507043U, 4606496786581982534U, 13829868823436758342U, 4600565431771507043U, 4607009347991985328U, 4596196889902818827U, 13819568926757594635U, 4607009347991985328U, 4603179351334086856U, 4605664432017547683U, 13829036468872323491U, 4603179351334086856U, 4605664432017547683U, 4603179351334086856U, 13826551388188862664U, 4605664432017547683U, 4596196889902818827U, 4607009347991985328U, 13830381384846761136U, 4596196889902818827U, 4607139046673687846U, 4591727299969791020U, 13815099336824566828U, 4607139046673687846U, 4603889326261607894U, 4605137878724712257U, 13828509915579488065U, 4603889326261607894U, 4606118860100255153U, 4602163548591158843U, 13825535585445934651U, 4606118860100255153U, 4598900923775164166U, 4606794571824115162U, 13830166608678890970U, 4598900923775164166U, 4606794571824115162U, 4598900923775164166U, 13822272960629939974U, 4606794571824115162U, 4602163548591158843U, 4606118860100255153U, 13829490896955030961U, 4602163548591158843U, 4605137878724712257U, 4603889326261607894U, 13827261363116383702U, 4605137878724712257U, 4591727299969791020U, 4607139046673687846U, 13830511083528463654U, 4591727299969791020U, 4607171569234046334U, 4587232218149935124U, 13810604255004710932U, 4607171569234046334U, 4604224084862889120U, 4604849113969373103U, 13828221150824148911U, 4604224084862889120U, 4606317631232591731U, 4601373767755717824U, 13824745804610493632U, 4606317631232591731U, 4599740487990714333U, 4606655894547498725U, 13830027931402274533U, 4599740487990714333U, 4606912484326125783U, 4597922303871901467U, 13821294340726677275U, 4606912484326125783U, 4602805845399633902U, 4605900952042040894U, 13829272988896816702U, 4602805845399633902U, 4605409869824231233U, 4603540801876750389U, 13826912838731526197U, 4605409869824231233U, 4594454542771183930U, 4607084929468638487U, 13830456966323414295U, 4594454542771183930U, 4607084929468638487U, 4594454542771183930U, 13817826579625959738U, 4607084929468638487U, 4603540801876750389U, 4605409869824231233U, 13828781906679007041U, 4603540801876750389U, 4605900952042040894U, 4602805845399633902U, 13826177882254409710U, 4605900952042040894U, 4597922303871901467U, 4606912484326125783U, 13830284521180901591U, 4597922303871901467U, 4606655894547498725U, 4599740487990714333U, 13823112524845490141U, 4606655894547498725U, 4601373767755717824U, 4606317631232591731U, 13829689668087367539U, 4601373767755717824U, 4604849113969373103U, 4604224084862889120U, 13827596121717664928U, 4604849113969373103U, 4587232218149935124U, 4607171569234046334U, 13830543606088822142U, 4587232218149935124U, 4607179706000002317U, 4582730748936808062U, 13806102785791583870U, 4607179706000002317U, 4604386048625945823U, 4604698657331085206U, 13828070694185861014U, 4604386048625945823U, 4606409688975526202U, 4600971798440897930U, 13824343835295673738U, 4606409688975526202U, 4600154912527631775U, 4606578871587619388U, 13829950908442395196U, 4600154912527631775U, 4606963563043808649U, 4597061974398750563U, 13820434011253526371U, 4606963563043808649U, 4602994049708411683U, 4605784983948558848U, 13829157020803334656U, 4602994049708411683U, 4605539368864982914U, 4603361638657888991U, 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13823268375902077442U, 4606627607157935956U, 4601223560006786057U, 4606352730697093817U, 13829724767551869625U, 4601223560006786057U, 4604793159020491611U, 4604285253548209224U, 13827657290402985032U, 4604793159020491611U, 4585907115494236537U, 4607175255902437396U, 13830547292757213204U, 4585907115494236537U, 4607177290141793710U, 4585023436363055487U, 13808395473217831295U, 4607177290141793710U, 4604325745441780828U, 4604755543975806820U, 13828127580830582628U, 4604325745441780828U, 4606375745674388705U, 4601123065313358619U, 13824495102168134427U, 4606375745674388705U, 4599999947619525579U, 4606608350964852124U, 13829980387819627932U, 4599999947619525579U, 4606945027305114062U, 4597385183080791534U, 13820757219935567342U, 4606945027305114062U, 4602923807199184054U, 4605829012964735987U, 13829201049819511795U, 4602923807199184054U, 4605491322423429598U, 4603429196809300824U, 13826801233664076632U, 4605491322423429598U, 4595000592312171144U, 4607063608453868552U, 13830435645308644360U, 4595000592312171144U, 4607104153983298999U, 4593907249284540294U, 13817279286139316102U, 4607104153983298999U, 4603651144395358093U, 4605326714874986465U, 13828698751729762273U, 4603651144395358093U, 4605971073215153165U, 4602686793990243041U, 13826058830845018849U, 4605971073215153165U, 4598316292140394014U, 4606877885424248132U, 13830249922279023940U, 4598316292140394014U, 4606701442584137310U, 4599479600326345459U, 13822851637181121267U, 4606701442584137310U, 4601622657843474729U, 4606257600839867033U, 13829629637694642841U, 4601622657843474729U, 4604941113561600762U, 4604121000955189926U, 13827493037809965734U, 4604941113561600762U, 4589303678145802340U, 4607163731439411601U, 13830535768294187409U, 4589303678145802340U, 4607151534426937478U, 4590626485056654602U, 13813998521911430410U, 4607151534426937478U, 4603995455647851249U, 4605049409688478101U, 13828421446543253909U, 4603995455647851249U, 4606183055233559255U, 4601918851211878557U, 13825290888066654365U, 4606183055233559255U, 4599164736579548843U, 4606753451050079834U, 13830125487904855642U, 4599164736579548843U, 4606833664420673202U, 4598635880488956483U, 13822007917343732291U, 4606833664420673202U, 4602406247776385022U, 4606052795787882823U, 13829424832642658631U, 4602406247776385022U, 4605224709411790590U, 4603781852316960384U, 13827153889171736192U, 4605224709411790590U, 4592826452951465409U, 4607124449686274900U, 13830496486541050708U, 4592826452951465409U, 4607035262954517034U, 4595654028864046335U, 13819026065718822143U, 4607035262954517034U, 4603293641160266722U, 4605586791482848547U, 13828958828337624355U, 4603293641160266722U, 4605740310302420207U, 4603063884010218172U, 13826435920864993980U, 4605740310302420207U, 4596738097012783531U, 4606981354314050484U, 13830353391168826292U, 4596738097012783531U, 4606548680329491866U, 4600309328230211502U, 13823681365084987310U, 4606548680329491866U, 4600819913163773071U, 4606442934727379583U, 13829814971582155391U, 4600819913163773071U, 4604641218080103285U, 4604445825685214043U, 13827817862539989851U, 4604641218080103285U, 4579996072175835083U, 4607181359080094673U, 13830553395934870481U, 4579996072175835083U, 4607180341788068727U, 4581846703643734566U, 13805218740498510374U, 4607180341788068727U, 4604406033021674239U, 4604679572075463103U, 13828051608930238911U, 4604406033021674239U, 4606420848538580260U, 4600921238092511730U, 13824293274947287538U, 4606420848538580260U, 4600206446098256018U, 4606568886807728474U, 13829940923662504282U, 4600206446098256018U, 4606969576261663845U, 4596954088216812973U, 13820326125071588781U, 4606969576261663845U, 4603017373458244943U, 4605770164172969910U, 13829142201027745718U, 4603017373458244943U, 4605555245917486022U, 4603339021357904144U, 13826711058212679952U, 4605555245917486022U, 4595436449949385485U, 4607045045516813836U, 13830417082371589644U, 4595436449949385485U, 4607118021058468598U, 4593265590854265407U, 13816637627709041215U, 4607118021058468598U, 4603738491917026584U, 4605258978359093269U, 13828631015213869077U, 4603738491917026584U, 4606025850160239809U, 4602502755147763107U, 13825874792002538915U, 4606025850160239809U, 4598529532600161144U, 4606848731493011465U, 13830220768347787273U, 4598529532600161144U, 4606736437002195879U, 4599269903251194481U, 13822641940105970289U, 4606736437002195879U, 4601820425647934753U, 4606208206518262803U, 13829580243373038611U, 4601820425647934753U, 4605013567986435066U, 4604037525321326463U, 13827409562176102271U, 4605013567986435066U, 4590185751760970393U, 4607155938267770208U, 13830527975122546016U, 4590185751760970393U, 4607160003989618959U, 4589744810590291021U, 13813116847445066829U, 4607160003989618959U, 4604079374282302598U, 4604977468824438271U, 13828349505679214079U, 4604079374282302598U, 4606233055365547081U, 4601721693286060937U, 13825093730140836745U, 4606233055365547081U, 4599374859150636784U, 4606719100629313491U, 13830091137484089299U, 4599374859150636784U, 4606863472012527185U, 4598423001813699022U, 13821795038668474830U, 4606863472012527185U, 4602598930031891166U, 4605998608960791335U, 13829370645815567143U, 4602598930031891166U, 4605292980606880364U, 4603694922063032361U, 13827066958917808169U, 4605292980606880364U, 4593688012422887515U, 4607111255739239816U, 13830483292594015624U, 4593688012422887515U, 4607054494135176056U, 4595218635031890910U, 13818590671886666718U, 4607054494135176056U, 4603384207141321914U, 4605523422498301790U, 13828895459353077598U, 4603384207141321914U, 4605799732098147061U, 4602970680601913687U, 13826342717456689495U, 4605799732098147061U, 4597169786279785693U, 4606957467106717424U, 13830329503961493232U, 4597169786279785693U, 4606588777269136769U, 4600103317933788342U, 13823475354788564150U, 4606588777269136769U, 4601022290077223616U, 4606398451906509788U, 13829770488761285596U, 4601022290077223616U, 4604717681185626434U, 4604366005771528720U, 13827738042626304528U, 4604717681185626434U, 4583614727651146525U, 4607178985458280057U, 13830551022313055865U, 4583614727651146525U, 4607172882816799076U, 4586790578280679046U, 13810162615135454854U, 4607172882816799076U, 4604244531615310815U, 4604830524903495634U, 13828202561758271442U, 4604244531615310815U, 4606329407841126011U, 4601323770373937522U, 13824695807228713330U, 4606329407841126011U, 4599792496117920694U, 4606646545123403481U, 13830018581978179289U, 4599792496117920694U, 4606919157647773535U, 4597815040470278984U, 13821187077325054792U, 4606919157647773535U, 4602829525820289164U, 4605886709123365959U, 13829258745978141767U, 4602829525820289164U, 4605426297151190466U, 4603518581031047189U, 13826890617885822997U, 4605426297151190466U, 4594563856311064231U, 4607080832832247697U, 13830452869687023505U, 4594563856311064231U, 4607088942243446236U, 4594345179472540681U, 13817717216327316489U, 4607088942243446236U, 4603562972219549215U, 4605393374401988274U, 13828765411256764082U, 4603562972219549215U, 4605915122243179241U, 4602782121393764535U, 13826154158248540343U, 4605915122243179241U, 4598029484874872834U, 4606905728766014348U, 13830277765620790156U, 4598029484874872834U, 4606665164148251002U, 4599688422741010356U, 13823060459595786164U, 4606665164148251002U, 4601423692641949331U, 4606305777984577632U, 13829677814839353440U, 4601423692641949331U, 4604867640218014515U, 4604203581176243359U, 13827575618031019167U, 4604867640218014515U, 4587673791460508439U, 4607170170974224083U, 13830542207828999891U, 4587673791460508439U, 4607141713064252300U, 4591507261658050721U, 13814879298512826529U, 4607141713064252300U, 4603910660507251362U, 4605120315324767624U, 13828492352179543432U, 4603910660507251362U, 4606131849150971908U, 4602114767134999006U, 13825486803989774814U, 4606131849150971908U, 4598953786765296928U, 4606786509620734768U, 13830158546475510576U, 4598953786765296928U, 4606802552898869248U, 4598848011564831930U, 13822220048419607738U, 4606802552898869248U, 4602212250118051877U, 4606105796280968177U, 13829477833135743985U, 4602212250118051877U, 4605155376589456981U, 4603867938232615808U, 13827239975087391616U, 4605155376589456981U, 4591947271803021404U, 4607136295912168606U, 13830508332766944414U, 4591947271803021404U, 4607014697483910382U, 4596088445927168004U, 13819460482781943812U, 4607014697483910382U, 4603202304363743346U, 4605649044311923410U, 13829021081166699218U, 4603202304363743346U, 4605679749231851918U, 4603156351203636159U, 13826528388058411967U, 4605679749231851918U, 4596305267720071930U, 4607003915349878877U, 13830375952204654685U, 4596305267720071930U, 4606507322377452870U, 4600514338912178239U, 13823886375766954047U, 4606507322377452870U, 4600616459743653188U, 4606486172460753999U, 13829858209315529807U, 4600616459743653188U, 4604563781218984604U, 4604524701268679793U, 13827896738123455601U, 4604563781218984604U, 4569220649180767418U, 4607182376410422530U, 13830554413265198338U, 4569220649180767418U }; const fpr fpr_p2_tab[] = { 4611686018427387904U, 4607182418800017408U, 4602678819172646912U, 4598175219545276416U, 4593671619917905920U, 4589168020290535424U, 4584664420663164928U, 4580160821035794432U, 4575657221408423936U, 4571153621781053440U, 4566650022153682944U };