fpr.c

/*
 * 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) 2022 Hawk 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   Ludo Pulles <ludo.pulles@cwi.nl>
 *
 * ==========================(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 <thomas.pornin@nccgroup.com>
 */

#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 if 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..2^55-1, 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..2^55-1 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);
}



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,
	13826733675512664799U,  4605539368864982914U,
	 4595327571478659014U,  4607049811591515049U,
	13830421848446290857U,  4595327571478659014U,
	 4607114680469659603U,  4593485039402578702U,
	13816857076257354510U,  4607114680469659603U,
	 4603716733069447353U,  4605276012900672507U,
	13828648049755448315U,  4603716733069447353U,
	 4606012266443150634U,  4602550884377336506U,
	13825922921232112314U,  4606012266443150634U,
	 4598476289818621559U,  4606856142606846307U,
	13830228179461622115U,  4598476289818621559U,
	 4606727809065869586U,  4599322407794599425U,
	13822694444649375233U,  4606727809065869586U,
	 4601771097584682078U,  4606220668805321205U,
	13829592705660097013U,  4601771097584682078U,
	 4604995550503212910U,  4604058477489546729U,
	13827430514344322537U,  4604995550503212910U,
	 4589965306122607094U,  4607158013403433018U,
	13830530050258208826U,  4589965306122607094U,
	 4607158013403433018U,  4589965306122607094U,
	13813337342977382902U,  4607158013403433018U,
	 4604058477489546729U,  4604995550503212910U,
	13828367587357988718U,  4604058477489546729U,
	 4606220668805321205U,  4601771097584682078U,
	13825143134439457886U,  4606220668805321205U,
	 4599322407794599425U,  4606727809065869586U,
	13830099845920645394U,  4599322407794599425U,
	 4606856142606846307U,  4598476289818621559U,
	13821848326673397367U,  4606856142606846307U,
	 4602550884377336506U,  4606012266443150634U,
	13829384303297926442U,  4602550884377336506U,
	 4605276012900672507U,  4603716733069447353U,
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	13829082208465255112U,  4603110210506737381U,
	 4605618058006716661U,  4603248068256948438U,
	13826620105111724246U,  4605618058006716661U,
	 4595871363584150300U,  4607025146816593591U,
	13830397183671369399U,  4595871363584150300U,
	 4607130541380624519U,  4592387007752762956U,
	13815759044607538764U,  4607130541380624519U,
	 4603825001630339212U,  4605190175055178825U,
	13828562211909954633U,  4603825001630339212U,
	 4606079444829232727U,  4602309411551204896U,
	13825681448405980704U,  4606079444829232727U,
	 4598742041476147134U,  4606818271362779153U,
	13830190308217554961U,  4598742041476147134U,
	 4606770142132396069U,  4599059363095165615U,
	13822431399949941423U,  4606770142132396069U,
	 4602016966272225497U,  4606157602458368090U,
	13829529639313143898U,  4602016966272225497U,
	 4605084992581147553U,  4603953166845776383U,
	13827325203700552191U,  4605084992581147553U,
	 4591066993883984169U,  4607146792632922887U,
	13830518829487698695U,  4591066993883984169U,
	 4607167120476811757U,  4588556721781247689U,
	13811928758636023497U,  4607167120476811757U,
	 4604162403772767740U,  4604904503566677638U,
	13828276540421453446U,  4604162403772767740U,
	 4606281842017099424U,  4601523323048804569U,
	13824895359903580377U,  4606281842017099424U,
	 4599584122834874440U,  4606683463531482757U,
	13830055500386258565U,  4599584122834874440U,
	 4606891971185517504U,  4598209407597805010U,
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	 4602734543519989142U,  4605943243960030558U,
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	 4606627607157935956U,  4599896339047301634U,
	13823268375902077442U,  4606627607157935956U,
	 4601223560006786057U,  4606352730697093817U,
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	 4604793159020491611U,  4604285253548209224U,
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	 4585907115494236537U,  4607175255902437396U,
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	 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,
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	 4603651144395358093U,  4605326714874986465U,
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	 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,
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	 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,
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	 4604717681185626434U,  4604366005771528720U,
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	 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,
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	 4605426297151190466U,  4603518581031047189U,
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	 4594563856311064231U,  4607080832832247697U,
	13830452869687023505U,  4594563856311064231U,
	 4607088942243446236U,  4594345179472540681U,
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	 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
};