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) 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"

#if FALCON_FPEMU // yyyFPEMU+1

/*
 * 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)

#if FALCON_ASM_CORTEXM4 // yyyASM_CORTEXM4+1

__attribute__((naked))
fpr
fpr_scaled(int64_t i __attribute__((unused)), int sc __attribute__((unused)))
{
	__asm__ (
	"push	{ r4, r5, r6, lr }\n\t"
	"\n\t"
	"@ Input i is in r0:r1, and sc in r2.\n\t"
	"@ Extract the sign bit, and compute the absolute value.\n\t"
	"@ -> sign bit in r3, with value 0 or -1\n\t"
	"asrs	r3, r1, #31\n\t"
	"eors	r0, r3\n\t"
	"eors	r1, r3\n\t"
	"subs	r0, r3\n\t"
	"sbcs	r1, r3\n\t"
	"\n\t"
	"@ Scale exponent to account for the encoding; if the source is\n\t"
	"@ zero or if the scaled exponent is negative, it is set to 32.\n\t"
	"addw	r2, r2, #1022\n\t"
	"orrs	r4, r0, r1\n\t"
	"bics	r4, r4, r2, asr #31\n\t"
	"rsbs	r5, r4, #0\n\t"
	"orrs	r4, r5\n\t"
	"ands	r2, r2, r4, asr #31\n\t"
	"adds	r2, #32\n\t"
	"\n\t"
	"@ Normalize value to a full 64-bit width, by shifting it left.\n\t"
	"@ The shift count is subtracted from the exponent (in r2).\n\t"
	"@ If the mantissa is 0, the exponent is set to 0.\n\t"
	"\n\t"
	"@ If top word is 0, replace with low word; otherwise, add 32 to\n\t"
	"@ the exponent.\n\t"
	"rsbs	r4, r1, #0\n\t"
	"orrs	r4, r1\n\t"
	"eors	r5, r0, r1\n\t"
	"bics	r5, r5, r4, asr #31\n\t"
	"eors	r1, r5\n\t"
	"ands	r0, r0, r4, asr #31\n\t"
	"lsrs	r4, r4, #31\n\t"
	"adds	r2, r2, r4, lsl #5\n\t"
	"\n\t"
	"@ Count leading zeros of r1 to finish the shift.\n\t"
	"clz	r4, r1\n\t"
	"subs	r2, r4\n\t"
	"rsbs	r5, r4, #32\n\t"
	"lsls	r1, r4\n\t"
	"lsrs	r5, r0, r5\n\t"
	"lsls	r0, r4\n\t"
	"orrs	r1, r5\n\t"
	"\n\t"
	"@ Clear the top bit; we know it's a 1 (unless the whole mantissa\n\t"
	"@ was zero, but then it's still OK to clear it)\n\t"
	"bfc	r1, #31, #1\n\t"
	"\n\t"
	"@ Now shift right the value by 11 bits; this puts the value in\n\t"
	"@ the 2^52..2^53-1 range. We also keep a copy of the pre-shift\n\t"
	"@ low bits in r5.\n\t"
	"movs	r5, r0\n\t"
	"lsrs	r0, #11\n\t"
	"orrs	r0, r0, r1, lsl #21\n\t"
	"lsrs	r1, #11\n\t"
	"\n\t"
	"@ Also plug the exponent at the right place. This must be done\n\t"
	"@ now so that, in case the rounding creates a carry, that carry\n\t"
	"@ adds to the exponent, which would be exactly what we want at\n\t"
	"@ that point.\n\t"
	"orrs	r1, r1, r2, lsl #20\n\t"
	"\n\t"
	"@ Rounding: we must add 1 to the mantissa in the following cases:\n\t"
	"@  - bits 11 to 9 of r5 are '011', '110' or '111'\n\t"
	"@  - bits 11 to 9 of r5 are '010' and one of the\n\t"
	"@    bits 0 to 8 is non-zero\n\t"
	"ubfx	r6, r5, #0, #9\n\t"
	"addw	r6, r6, #511\n\t"
	"orrs	r5, r6\n\t"
	"\n\t"
	"ubfx	r5, r5, #9, #3\n\t"
	"movs	r6, #0xC8\n\t"
	"lsrs	r6, r5\n\t"
	"ands	r6, #1\n\t"
	"adds	r0, r6\n\t"
	"adcs	r1, #0\n\t"
	"\n\t"
	"@ Put back the sign.\n\t"
	"orrs	r1, r1, r3, lsl #31\n\t"
	"\n\t"
	"pop	{ r4, r5, r6, pc}\n\t"
	);
}

#else // yyyASM_CORTEXM4+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);
}

#endif // yyyASM_CORTEXM4-

#if FALCON_ASM_CORTEXM4 // yyyASM_CORTEXM4+1

// yyyPQCLEAN+0
#if 0
/* Debug code -- To get a printout of registers from a specific point
   in ARM Cortex M4 assembly code, uncomment this code and add a
   "bl DEBUG" call where wished for. */

void
print_regs(uint32_t *rr, uint32_t flags)
{
	int i;
	extern int printf(const char *fmt, ...);

	printf("\nRegs:\n");
	for (i = 0; i < 7; i ++) {
		int j;

		j = i + 7;
		printf("  %2d = %08X    %2d = %08X\n", i, rr[i], j, rr[j]);
	}
	printf("  flags = %08X  ", flags);
	if ((flags >> 31) & 1) {
		printf("N");
	}
	if ((flags >> 30) & 1) {
		printf("Z");
	}
	if ((flags >> 29) & 1) {
		printf("C");
	}
	if ((flags >> 28) & 1) {
		printf("V");
	}
	if ((flags >> 27) & 1) {
		printf("Q");
	}
	printf("\n");
}

__attribute__((naked))
void
DEBUG(void)
{
	__asm__ (
	"push	{ r0, r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, lr }\n\t"
	"mov	r0, sp\n\t"
	"mrs	r1, apsr\n\t"
	"bl	print_regs\n\t"
	"pop	{ r0, r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, pc }\n\t"
	);
}
#endif
// yyyPQCLEAN-

__attribute__((naked))
fpr
fpr_add(fpr x __attribute__((unused)), fpr y __attribute__((unused)))
{
	__asm__ (
	"push	{ r4, r5, r6, r7, r8, r10, r11, lr }\n\t"
	"\n\t"
	"@ Make sure that the first operand (x) has the larger absolute\n\t"
	"@ value. This guarantees that the exponent of y is less than\n\t"
	"@ or equal to the exponent of x, and, if they are equal, then\n\t"
	"@ the mantissa of y will not be greater than the mantissa of x.\n\t"
	"@ However, if absolute values are equal and the sign of x is 1,\n\t"
	"@ then we want to also swap the values.\n\t"
	"ubfx	r4, r1, #0, #31  @ top word without sign bit\n\t"
	"ubfx	r5, r3, #0, #31  @ top word without sign bit\n\t"
	"subs	r7, r0, r2       @ difference in r7:r4\n\t"
	"sbcs	r4, r5\n\t"
	"orrs	r7, r4\n\t"
	"rsbs	r5, r7, #0\n\t"
	"orrs	r7, r5      @ bit 31 of r7 is 0 iff difference is zero\n\t"
	"bics	r6, r1, r7\n\t"
	"orrs	r6, r4      @ bit 31 of r6 is 1 iff the swap must be done\n\t"
	"\n\t"
	"@ Conditional swap\n\t"
	"eors	r4, r0, r2\n\t"
	"eors	r5, r1, r3\n\t"
	"ands	r4, r4, r6, asr #31\n\t"
	"ands	r5, r5, r6, asr #31\n\t"
	"eors	r0, r4\n\t"
	"eors	r1, r5\n\t"
	"eors	r2, r4\n\t"
	"eors	r3, r5\n\t"
	"\n\t"
	"@ Extract mantissa of x into r0:r1, exponent in r4, sign in r5\n\t"
	"ubfx	r4, r1, #20, #11   @ Exponent in r4 (without sign)\n\t"
	"addw	r5, r4, #2047 @ Get a carry to test r4 for zero\n\t"
	"lsrs	r5, #11       @ r5 is the mantissa implicit high bit\n\t"
	"bfc	r1, #20, #11  @ Clear exponent bits (not the sign)\n\t"
	"orrs	r1, r1, r5, lsl #20  @ Set mantissa high bit\n\t"
	"asrs	r5, r1, #31   @ Get sign bit (sign-extended)\n\t"
	"bfc	r1, #31, #1   @ Clear the sign bit\n\t"
	"\n\t"
	"@ Extract mantissa of y into r2:r3, exponent in r6, sign in r7\n\t"
	"ubfx	r6, r3, #20, #11   @ Exponent in r6 (without sign)\n\t"
	"addw	r7, r6, #2047 @ Get a carry to test r6 for zero\n\t"
	"lsrs	r7, #11       @ r7 is the mantissa implicit high bit\n\t"
	"bfc	r3, #20, #11  @ Clear exponent bits (not the sign)\n\t"
	"orrs	r3, r3, r7, lsl #20  @ Set mantissa high bit\n\t"
	"asrs	r7, r3, #31   @ Get sign bit (sign-extended)\n\t"
	"bfc	r3, #31, #1   @ Clear the sign bit\n\t"
	"\n\t"
	"@ Scale mantissas up by three bits.\n\t"
	"lsls	r1, #3\n\t"
	"orrs	r1, r1, r0, lsr #29\n\t"
	"lsls	r0, #3\n\t"
	"lsls	r3, #3\n\t"
	"orrs	r3, r3, r2, lsr #29\n\t"
	"lsls	r2, #3\n\t"
	"\n\t"
	"@ x: exponent=r4, sign=r5, mantissa=r0:r1 (scaled up 3 bits)\n\t"
	"@ y: exponent=r6, sign=r7, mantissa=r2:r3 (scaled up 3 bits)\n\t"
	"\n\t"
	"@ At that point, the exponent of x (in r4) is larger than that\n\t"
	"@ of y (in r6). The difference is the amount of shifting that\n\t"
	"@ should be done on y. If that amount is larger than 59 then\n\t"
	"@ we clamp y to 0. We won't need y's exponent beyond that point,\n\t"
	"@ so we store that shift count in r6.\n\t"
	"subs	r6, r4, r6\n\t"
	"subs	r8, r6, #60\n\t"
	"ands	r2, r2, r8, asr #31\n\t"
	"ands	r3, r3, r8, asr #31\n\t"
	"\n\t"
	"@ Shift right r2:r3 by r6 bits. The shift count is in the 0..59\n\t"
	"@ range. r11 will be non-zero if and only if some non-zero bits\n\t"
	"@ were dropped.\n\t"
	"subs	r8, r6, #32\n\t"
	"bics	r11, r2, r8, asr #31\n\t"
	"ands	r2, r2, r8, asr #31\n\t"
	"bics	r10, r3, r8, asr #31\n\t"
	"orrs	r2, r2, r10\n\t"
	"ands	r3, r3, r8, asr #31\n\t"
	"ands	r6, r6, #31\n\t"
	"rsbs	r8, r6, #32\n\t"
	"lsls	r10, r2, r8\n\t"
	"orrs	r11, r11, r10\n\t"
	"lsrs	r2, r2, r6\n\t"
	"lsls	r10, r3, r8\n\t"
	"orrs	r2, r2, r10\n\t"
	"lsrs	r3, r3, r6\n\t"
	"\n\t"
	"@ If r11 is non-zero then some non-zero bit was dropped and the\n\t"
	"@ low bit of r2 must be forced to 1 ('sticky bit').\n\t"
	"rsbs	r6, r11, #0\n\t"
	"orrs	r6, r6, r11\n\t"
	"orrs	r2, r2, r6, lsr #31\n\t"
	"\n\t"
	"@ x: exponent=r4, sign=r5, mantissa=r0:r1 (scaled up 3 bits)\n\t"
	"@ y: sign=r7, value=r2:r3 (scaled to same exponent as x)\n\t"
	"\n\t"
	"@ If x and y don't have the same sign, then we should negate r2:r3\n\t"
	"@ (i.e. subtract the mantissa instead of adding it). Signs of x\n\t"
	"@ and y are in r5 and r7, as full-width words. We won't need r7\n\t"
	"@ afterwards.\n\t"
	"eors	r7, r5    @ r7 = -1 if y must be negated, 0 otherwise\n\t"
	"eors	r2, r7\n\t"
	"eors	r3, r7\n\t"
	"subs	r2, r7\n\t"
	"sbcs	r3, r7\n\t"
	"\n\t"
	"@ r2:r3 has been shifted, we can add to r0:r1.\n\t"
	"adds	r0, r2\n\t"
	"adcs	r1, r3\n\t"
	"\n\t"
	"@ result: exponent=r4, sign=r5, mantissa=r0:r1 (scaled up 3 bits)\n\t"
	"\n\t"
	"@ Normalize the result with some left-shifting to full 64-bit\n\t"
	"@ width. Shift count goes to r2, and exponent (r4) is adjusted.\n\t"
	"clz	r2, r0\n\t"
	"clz	r3, r1\n\t"
	"sbfx	r6, r3, #5, #1\n\t"
	"ands	r2, r6\n\t"
	"adds	r2, r2, r3\n\t"
	"subs	r4, r4, r2\n\t"
	"\n\t"
	"@ Shift r0:r1 to the left by r2 bits.\n\t"
	"subs	r7, r2, #32\n\t"
	"lsls	r7, r0, r7\n\t"
	"lsls	r1, r1, r2\n\t"
	"rsbs	r6, r2, #32\n\t"
	"orrs	r1, r1, r7\n\t"
	"lsrs	r6, r0, r6\n\t"
	"orrs	r1, r1, r6\n\t"
	"lsls	r0, r0, r2\n\t"
	"\n\t"
	"@ The exponent of x was in r4. The left-shift operation has\n\t"
	"@ subtracted some value from it, 8 in case the result has the\n\t"
	"@ same exponent as x. However, the high bit of the mantissa will\n\t"
	"@ add 1 to the exponent, so we only add back 7 (the exponent is\n\t"
	"@ added in because rounding might have produced a carry, which\n\t"
	"@ should then spill into the exponent).\n\t"
	"adds	r4, #7\n\t"
	"\n\t"
	"@ If the mantissa new mantissa is non-zero, then its bit 63 is\n\t"
	"@ non-zero (thanks to the normalizing shift). Otherwise, that bit\n\t"
	"@ is zero, and we should then set the exponent to zero as well.\n\t"
	"ands	r4, r4, r1, asr #31\n\t"
	"\n\t"
	"@ Shrink back the value to a 52-bit mantissa. This requires\n\t"
	"@ right-shifting by 11 bits; we keep a copy of the pre-shift\n\t"
	"@ low word in r3.\n\t"
	"movs	r3, r0\n\t"
	"lsrs	r0, #11\n\t"
	"orrs	r0, r0, r1, lsl #21\n\t"
	"lsrs	r1, #11\n\t"
	"\n\t"
	"@ Apply rounding.\n\t"
	"ubfx	r6, r3, #0, #9\n\t"
	"addw	r6, r6, #511\n\t"
	"orrs	r3, r6\n\t"
	"ubfx	r3, r3, #9, #3\n\t"
	"movs	r6, #0xC8\n\t"
	"lsrs	r6, r3\n\t"
	"ands	r6, #1\n\t"
	"adds	r0, r6\n\t"
	"adcs	r1, #0\n\t"
	"\n\t"
	"@Plug in the exponent with an addition.\n\t"
	"adds	r1, r1, r4, lsl #20\n\t"
	"\n\t"
	"@ If the new exponent is negative or zero, then it underflowed\n\t"
	"@ and we must clear the whole mantissa and exponent.\n\t"
	"rsbs	r4, r4, #0\n\t"
	"ands	r0, r0, r4, asr #31\n\t"
	"ands	r1, r1, r4, asr #31\n\t"
	"\n\t"
	"@ Put back the sign. This is the sign of x: thanks to the\n\t"
	"@ conditional swap at the start, this is always correct.\n\t"
	"bfi	r1, r5, #31, #1\n\t"
	"\n\t"
	"pop	{ r4, r5, r6, r7, r8, r10, r11, pc }\n\t"
	);
}

#else // yyyASM_CORTEXM4+0

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);
}

#endif // yyyASM_CORTEXM4-

#if FALCON_ASM_CORTEXM4 // yyyASM_CORTEXM4+1

__attribute__((naked))
fpr
fpr_mul(fpr x __attribute__((unused)), fpr y __attribute__((unused)))
{
	__asm__ (
	"push	{ r4, r5, r6, r7, r8, r10, r11, lr }\n\t"
	"\n\t"
	"@ Extract mantissas: x.m = r4:r5, y.m = r6:r7\n\t"
	"@ r4 and r6 contain only 25 bits each.\n\t"
	"bics	r4, r0, #0xFE000000\n\t"
	"lsls	r5, r1, #7\n\t"
	"orrs	r5, r5, r0, lsr #25\n\t"
	"orrs	r5, r5, #0x08000000\n\t"
	"bics	r5, r5, #0xF0000000\n\t"
	"bics	r6, r2, #0xFE000000\n\t"
	"lsls	r7, r3, #7\n\t"
	"orrs	r7, r7, r2, lsr #25\n\t"
	"orrs	r7, r7, #0x08000000\n\t"
	"bics	r7, r7, #0xF0000000\n\t"
	"\n\t"
	"@ Perform product. Values are in the 2^52..2^53-1 range, so\n\t"
	"@ the product is at most 106-bit long. Of the low 50 bits,\n\t"
	"@ we only want to know if they are all zeros or not. Here,\n\t"
	"@ we get the top 56 bits in r10:r11, and r8 will be non-zero\n\t"
	"@ if and only if at least one of the low 50 bits is non-zero.\n\t"
	"umull	r8, r10, r4, r6      @ x0*y0\n\t"
	"lsls	r10, #7\n\t"
	"orrs	r10, r10, r8, lsr #25\n\t"
	"eors	r11, r11\n\t"
	"umlal	r10, r11, r4, r7     @ x0*y1\n\t"
	"umlal	r10, r11, r5, r6     @ x1*y0\n\t"
	"orrs	r8, r8, r10, lsl #7\n\t"
	"lsrs	r10, #25\n\t"
	"orrs	r10, r10, r11, lsl #7\n\t"
	"eors	r11, r11\n\t"
	"umlal	r10, r11, r5, r7     @ x1*y1\n\t"
	"\n\t"
	"@ Now r0, r2, r4, r5, r6 and r7 are free.\n\t"
	"@ If any of the low 50 bits was non-zero, then we force the\n\t"
	"@ low bit of r10 to 1.\n\t"
	"rsbs	r4, r8, #0\n\t"
	"orrs	r8, r8, r4\n\t"
	"orrs	r10, r10, r8, lsr #31\n\t"
	"\n\t"
	"@ r8 is free.\n\t"
	"@ r10:r11 contains the product in the 2^54..2^56-1 range. We\n\t"
	"@ normalize it to 2^54..2^55-1 (into r6:r7) with a conditional\n\t"
	"@ shift (low bit is sticky). r5 contains -1 if the shift was done,\n\t"
	"@ 0 otherwise.\n\t"
	"ands	r6, r10, #1\n\t"
	"lsrs	r5, r11, #23\n\t"
	"rsbs	r5, r5, #0\n\t"
	"orrs	r6, r6, r10, lsr #1\n\t"
	"orrs	r6, r6, r11, lsl #31\n\t"
	"lsrs	r7, r11, #1\n\t"
	"eors	r10, r10, r6\n\t"
	"eors	r11, r11, r7\n\t"
	"bics	r10, r10, r5\n\t"
	"bics	r11, r11, r5\n\t"
	"eors	r6, r6, r10\n\t"
	"eors	r7, r7, r11\n\t"
	"\n\t"
	"@ Compute aggregate exponent: ex + ey - 1023 + w\n\t"
	"@ (where w = 1 if the conditional shift was done, 0 otherwise)\n\t"
	"@ But we subtract 1 because the injection of the mantissa high\n\t"
	"@ bit will increment the exponent by 1.\n\t"
	"lsls	r0, r1, #1\n\t"
	"lsls	r2, r3, #1\n\t"
	"lsrs	r0, #21\n\t"
	"addw	r4, r0, #0x7FF   @ save ex + 2047 in r4\n\t"
	"lsrs	r2, #21\n\t"
	"addw	r8, r2, #0x7FF   @ save ey + 2047 in r8\n\t"
	"adds	r2, r0\n\t"
	"subw	r2, r2, #1024\n\t"
	"subs	r2, r5\n\t"
	"\n\t"
	"@ r5 is free.\n\t"
	"@ Also, if either of the source exponents is 0, or the result\n\t"
	"@ exponent is 0 or negative, then the result is zero and the\n\t"
	"@ mantissa and the exponent shall be clamped to zero. Since\n\t"
	"@ r2 contains the result exponent minus 1, we test on r2\n\t"
	"@ being strictly negative.\n\t"
	"ands	r4, r8    @ if bit 11 = 0 then one of the exponents was 0\n\t"
	"mvns	r5, r2\n\t"
	"ands	r5, r5, r4, lsl #20\n\t"
	"ands	r2, r2, r5, asr #31\n\t"
	"ands	r6, r6, r5, asr #31\n\t"
	"ands	r7, r7, r5, asr #31\n\t"
	"\n\t"
	"@ Sign is the XOR of the sign of the operands. This is true in\n\t"
	"@ all cases, including very small results (exponent underflow)\n\t"
	"@ and zeros.\n\t"
	"eors	r1, r3\n\t"
	"bfc	r1, #0, #31\n\t"
	"\n\t"
	"@ Plug in the exponent.\n\t"
	"bfi	r1, r2, #20, #11\n\t"
	"\n\t"
	"@ r2 and r3 are free.\n\t"
	"@ Shift back to the normal 53-bit mantissa, with rounding.\n\t"
	"@ Mantissa goes into r0:r1. For r1, we must use an addition\n\t"
	"@ because the rounding may have triggered a carry, that should\n\t"
	"@ be added to the exponent.\n\t"
	"movs	r4, r6\n\t"
	"lsrs	r0, r6, #2\n\t"
	"orrs	r0, r0, r7, lsl #30\n\t"
	"adds	r1, r1, r7, lsr #2\n\t"
	"ands	r4, #0x7\n\t"
	"movs	r3, #0xC8\n\t"
	"lsrs	r3, r4\n\t"
	"ands	r3, #1\n\t"
	"adds	r0, r3\n\t"
	"adcs	r1, #0\n\t"
	"\n\t"
	"pop	{ r4, r5, r6, r7, r8, r10, r11, pc }\n\t"
	);
}

#else // yyyASM_CORTEXM4+0

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);
}

#endif // yyyASM_CORTEXM4-

#if FALCON_ASM_CORTEXM4 // yyyASM_CORTEXM4+1

__attribute__((naked))
fpr
fpr_div(fpr x __attribute__((unused)), fpr y __attribute__((unused)))
{
	__asm__ (
	"push	{ r4, r5, r6, r7, r8, r10, r11, lr }\n\t"

	"@ Extract mantissas of x and y, in r0:r4 and r2:r5, respectively.\n\t"
	"@ We don't touch r1 and r3 as they contain the exponents and\n\t"
	"@ signs, which we'll need later on.\n\t"
	"ubfx	r4, r1, #0, #20\n\t"
	"ubfx	r5, r3, #0, #20\n\t"
	"orrs	r4, r4, #0x00100000\n\t"
	"orrs	r5, r5, #0x00100000\n\t"
	"\n\t"
	"@ Perform bit-by-bit division. We want a 56-bit result in r8:r10\n\t"
	"@ (low bit is 0). Bits come from the carry flag and are\n\t"
	"@ injected with rrx, i.e. in position 31; we thus get bits in\n\t"
	"@ the reverse order. Bits accumulate in r8; after the first 24\n\t"
	"@ bits, we move the quotient bits to r10.\n\t"
	"eors	r8, r8\n\t"
	"\n\t"

#define DIVSTEP \
	"subs	r6, r0, r2\n\t" \
	"sbcs	r7, r4, r5\n\t" \
	"rrx	r8, r8\n\t" \
	"ands	r6, r2, r8, asr #31\n\t" \
	"ands	r7, r5, r8, asr #31\n\t" \
	"subs	r0, r6\n\t" \
	"sbcs	r4, r7\n\t" \
	"adds	r0, r0, r0\n\t" \
	"adcs	r4, r4, r4\n\t"

#define DIVSTEP4   DIVSTEP DIVSTEP DIVSTEP DIVSTEP
#define DIVSTEP8   DIVSTEP4 DIVSTEP4

	DIVSTEP8
	DIVSTEP8
	DIVSTEP8

	"\n\t"
	"@ We have the first 24 bits of the quotient, move them to r10.\n\t"
	"rbit	r10, r8\n\t"
	"\n\t"

	DIVSTEP8
	DIVSTEP8
	DIVSTEP8
	DIVSTEP4 DIVSTEP DIVSTEP DIVSTEP

#undef DIVSTEP
#undef DIVSTEP4
#undef DIVSTEP8

	"\n\t"
	"@ Lowest bit will be set if remainder is non-zero at this point\n\t"
	"@ (this is the 'sticky' bit).\n\t"
	"subs	r0, #1\n\t"
	"sbcs	r4, #0\n\t"
	"rrx	r8, r8\n\t"
	"\n\t"
	"@ We now have the next (low) 32 bits of the quotient.\n\t"
	"rbit	r8, r8\n\t"
	"\n\t"
	"@ Since both operands had their top bit set, we know that the\n\t"
	"@ result at this point is in 2^54..2^56-1. We scale it down\n\t"
	"@ to 2^54..2^55-1 with a conditional shift. We also write the\n\t"
	"@ result in r4:r5. If the shift is done, r6 will contain -1.\n\t"
	"ands	r4, r8, #1\n\t"
	"lsrs	r6, r10, #23\n\t"
	"rsbs	r6, r6, #0\n\t"
	"orrs	r4, r4, r8, lsr #1\n\t"
	"orrs	r4, r4, r10, lsl #31\n\t"
	"lsrs	r5, r10, #1\n\t"
	"eors	r8, r8, r4\n\t"
	"eors	r10, r10, r5\n\t"
	"bics	r8, r8, r6\n\t"
	"bics	r10, r10, r6\n\t"
	"eors	r4, r4, r8\n\t"
	"eors	r5, r5, r10\n\t"
	"\n\t"
	"@ Compute aggregate exponent: ex - ey + 1022 + w\n\t"
	"@ (where w = 1 if the conditional shift was done, 0 otherwise)\n\t"
	"@ But we subtract 1 because the injection of the mantissa high\n\t"
	"@ bit will increment the exponent by 1.\n\t"
	"lsls	r0, r1, #1\n\t"
	"lsls	r2, r3, #1\n\t"
	"lsrs	r0, r0, #21\n\t"
	"addw	r7, r0, #0x7FF  @ save ex + 2047 in r7\n\t"
	"subs	r0, r0, r2, lsr #21\n\t"
	"addw	r0, r0, #1021\n\t"
	"subs	r0, r6\n\t"
	"\n\t"
	"@ If the x operand was zero, then the computation was wrong and\n\t"
	"@ the result is zero. Also, if the result exponent is zero or\n\t"
	"@ negative, then the mantissa shall be clamped to zero. Since r0\n\t"
	"@ contains the result exponent minus 1, we test on r0 being\n\t"
	"@ strictly negative.\n\t"
	"mvns	r2, r0\n\t"
	"ands	r2, r2, r7, lsl #20\n\t"
	"ands	r0, r0, r2, asr #31\n\t"
	"ands	r4, r4, r2, asr #31\n\t"
	"ands	r5, r5, r2, asr #31\n\t"
	"\n\t"
	"@ Sign is the XOR of the sign of the operands. This is true in\n\t"
	"@ all cases, including very small results (exponent underflow)\n\t"
	"@ and zeros.\n\t"
	"eors	r1, r3\n\t"
	"bfc	r1, #0, #31\n\t"
	"\n\t"
	"@ Plug in the exponent.\n\t"
	"bfi	r1, r0, #20, #11\n\t"
	"\n\t"
	"@ Shift back to the normal 53-bit mantissa, with rounding.\n\t"
	"@ Mantissa goes into r0:r1. For r1, we must use an addition\n\t"
	"@ because the rounding may have triggered a carry, that should\n\t"
	"@ be added to the exponent.\n\t"
	"movs	r6, r4\n\t"
	"lsrs	r0, r4, #2\n\t"
	"orrs	r0, r0, r5, lsl #30\n\t"
	"adds	r1, r1, r5, lsr #2\n\t"
	"ands	r6, #0x7\n\t"
	"movs	r3, #0xC8\n\t"
	"lsrs	r3, r6\n\t"
	"ands	r3, #1\n\t"
	"adds	r0, r3\n\t"
	"adcs	r1, #0\n\t"
	"\n\t"
	"pop	{ r4, r5, r6, r7, r8, r10, r11, pc }\n\t"
	);
}

#else // yyyASM_CORTEXM4+0

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);
}

#endif // yyyASM_CORTEXM4-

#if FALCON_ASM_CORTEXM4 // yyyASM_CORTEXM4+1

__attribute__((naked))
fpr
fpr_sqrt(fpr x __attribute__((unused)))
{
	__asm__ (
	"push	{ r4, r5, r6, r7, r8, r10, r11, lr }\n\t"
	"\n\t"
	"@ Extract mantissa (r0:r1) and exponent (r2). We assume that the\n\t"
	"@ sign is positive. If the source is zero, then the mantissa is\n\t"
	"@ set to 0.\n\t"
	"lsrs	r2, r1, #20\n\t"
	"bfc	r1, #20, #12\n\t"
	"addw	r3, r2, #0x7FF\n\t"
	"subw	r2, r2, #1023\n\t"
	"lsrs	r3, r3, #11\n\t"
	"orrs	r1, r1, r3, lsl #20\n\t"
	"\n\t"
	"@ If the exponent is odd, then multiply mantissa by 2 and subtract\n\t"
	"@ 1 from the exponent.\n\t"
	"ands	r3, r2, #1\n\t"
	"subs	r2, r2, r3\n\t"
	"rsbs	r3, r3, #0\n\t"
	"ands	r4, r1, r3\n\t"
	"ands	r3, r0\n\t"
	"adds	r0, r3\n\t"
	"adcs	r1, r4\n\t"
	"\n\t"
	"@ Left-shift the mantissa by 9 bits to put it in the\n\t"
	"@ 2^61..2^63-1 range (unless it is exactly 0).\n\t"
	"lsls	r1, r1, #9\n\t"
	"orrs	r1, r1, r0, lsr #23\n\t"
	"lsls	r0, r0, #9\n\t"
	"\n\t"
	"@ Compute the square root bit-by-bit.\n\t"
	"@ There are 54 iterations; first 30 can work on top word only.\n\t"
	"@   q = r3 (bit-reversed)\n\t"
	"@   s = r5\n\t"
	"eors	r3, r3\n\t"
	"eors	r5, r5\n\t"

#define SQRT_STEP_HI(bit) \
	"orrs	r6, r5, #(1 << (" #bit "))\n\t" \
	"subs	r7, r1, r6\n\t" \
	"rrx	r3, r3\n\t" \
	"ands	r6, r6, r3, asr #31\n\t" \
	"subs	r1, r1, r6\n\t" \
	"lsrs	r6, r3, #31\n\t" \
	"orrs	r5, r5, r6, lsl #((" #bit ") + 1)\n\t" \
	"adds	r0, r0\n\t" \
	"adcs	r1, r1\n\t"

#define SQRT_STEP_HIx5(b)  \
		SQRT_STEP_HI((b)+4) \
		SQRT_STEP_HI((b)+3) \
		SQRT_STEP_HI((b)+2) \
		SQRT_STEP_HI((b)+1) \
		SQRT_STEP_HI(b)

	SQRT_STEP_HIx5(25)
	SQRT_STEP_HIx5(20)
	SQRT_STEP_HIx5(15)
	SQRT_STEP_HIx5(10)
	SQRT_STEP_HIx5(5)
	SQRT_STEP_HIx5(0)

#undef SQRT_STEP_HI
#undef SQRT_STEP_HIx5

	"@ Top 30 bits of the result must be reversed: they were\n\t"
	"@ accumulated with rrx (hence from the top bit).\n\t"
	"rbit	r3, r3\n\t"
	"\n\t"
	"@ For the next 24 iterations, we must use two-word operations.\n\t"
	"@   bits of q now accumulate in r4\n\t"
	"@   s is in r6:r5\n\t"
	"eors	r4, r4\n\t"
	"eors	r6, r6\n\t"
	"\n\t"
	"@ First iteration is special because the potential bit goes into\n\t"
	"@ r5, not r6.\n\t"
	"orrs	r7, r6, #(1 << 31)\n\t"
	"subs	r8, r0, r7\n\t"
	"sbcs	r10, r1, r5\n\t"
	"rrx	r4, r4\n\t"
	"ands	r7, r7, r4, asr #31\n\t"
	"ands	r8, r5, r4, asr #31\n\t"
	"subs	r0, r0, r7\n\t"
	"sbcs	r1, r1, r8\n\t"
	"lsrs	r7, r4, #31\n\t"
	"orrs	r5, r5, r4, lsr #31\n\t"
	"adds	r0, r0\n\t"
	"adcs	r1, r1\n\t"

#define SQRT_STEP_LO(bit) \
	"orrs	r7, r6, #(1 << (" #bit "))\n\t" \
	"subs	r8, r0, r7\n\t" \
	"sbcs	r10, r1, r5\n\t" \
	"rrx	r4, r4\n\t" \
	"ands	r7, r7, r4, asr #31\n\t" \
	"ands	r8, r5, r4, asr #31\n\t" \
	"subs	r0, r0, r7\n\t" \
	"sbcs	r1, r1, r8\n\t" \
	"lsrs	r7, r4, #31\n\t" \
	"orrs	r6, r6, r7, lsl #((" #bit ") + 1)\n\t" \
	"adds	r0, r0\n\t" \
	"adcs	r1, r1\n\t"

#define SQRT_STEP_LOx4(b) \
		SQRT_STEP_LO((b)+3) \
		SQRT_STEP_LO((b)+2) \
		SQRT_STEP_LO((b)+1) \
		SQRT_STEP_LO(b)

	SQRT_STEP_LO(30)
	SQRT_STEP_LO(29)
	SQRT_STEP_LO(28)
	SQRT_STEP_LOx4(24)
	SQRT_STEP_LOx4(20)
	SQRT_STEP_LOx4(16)
	SQRT_STEP_LOx4(12)
	SQRT_STEP_LOx4(8)

#undef SQRT_STEP_LO
#undef SQRT_STEP_LOx4

	"@ Put low 24 bits in the right order.\n\t"
	"rbit	r4, r4\n\t"
	"\n\t"
	"@ We have a 54-bit result; compute the 55-th bit as the 'sticky'\n\t"
	"@ bit: it is non-zero if and only if r0:r1 is non-zero. We put the\n\t"
	"@ three low bits (including the sticky bit) in r5.\n\t"
	"orrs	r0, r1\n\t"
	"rsbs	r1, r0, #0\n\t"
	"orrs	r0, r1\n\t"
	"lsls	r5, r4, #1\n\t"
	"orrs	r5, r5, r0, lsr #31\n\t"
	"ands	r5, #0x7\n\t"
	"\n\t"
	"@ Compute the rounding: r6 is set to 0 or 1, and will be added\n\t"
	"@ to the mantissa.\n\t"
	"movs	r6, #0xC8\n\t"
	"lsrs	r6, r5\n\t"
	"ands	r6, #1\n\t"
	"\n\t"
	"@ Put the mantissa (53 bits, in the 2^52..2^53-1 range) in r0:r1\n\t"
	"@ (rounding not applied yet).\n\t"
	"lsrs	r0, r4, #1\n\t"
	"orrs	r0, r0, r3, lsl #23\n\t"
	"lsrs	r1, r3, #9\n\t"
	"\n\t"
	"@ Compute new exponent. This is half the old one (then reencoded\n\t"
	"@ by adding 1023). Exception: if the mantissa is zero, then the\n\t"
	"@ encoded exponent is set to 0. At that point, if the mantissa\n\t"
	"@ is non-zero, then its high bit (bit 52, i.e. bit 20 of r1) is\n\t"
	"@ non-zero. Note that the exponent cannot go out of range.\n\t"
	"lsrs	r2, r2, #1\n\t"
	"addw	r2, r2, #1023\n\t"
	"lsrs	r5, r1, #20\n\t"
	"rsbs	r5, r5, #0\n\t"
	"ands	r2, r5\n\t"
	"\n\t"
	"@ Place exponent. This overwrites the high bit of the mantissa.\n\t"
	"bfi	r1, r2, #20, #11\n\t"
	"\n\t"
	"@ Apply rounding. This may create a carry that will spill into\n\t"
	"@ the exponent, which is exactly what should be done in that case\n\t"
	"@ (i.e. increment the exponent).\n\t"
	"adds	r0, r0, r6\n\t"
	"adcs	r1, r1, #0\n\t"
	"\n\t"
	"pop	{ r4, r5, r6, r7, r8, r10, r11, pc }\n\t"
	);
}

#else // yyyASM_CORTEXM4+0

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);
}

#endif // yyyASM_CORTEXM4-

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,
	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,
	13827088769924223161U,  4605276012900672507U,
	 4593485039402578702U,  4607114680469659603U,
	13830486717324435411U,  4593485039402578702U,
	 4607049811591515049U,  4595327571478659014U,
	13818699608333434822U,  4607049811591515049U,
	 4603361638657888991U,  4605539368864982914U,
	13828911405719758722U,  4603361638657888991U,
	 4605784983948558848U,  4602994049708411683U,
	13826366086563187491U,  4605784983948558848U,
	 4597061974398750563U,  4606963563043808649U,
	13830335599898584457U,  4597061974398750563U,
	 4606578871587619388U,  4600154912527631775U,
	13823526949382407583U,  4606578871587619388U,
	 4600971798440897930U,  4606409688975526202U,
	13829781725830302010U,  4600971798440897930U,
	 4604698657331085206U,  4604386048625945823U,
	13827758085480721631U,  4604698657331085206U,
	 4582730748936808062U,  4607179706000002317U,
	13830551742854778125U,  4582730748936808062U,
	 4607181740574479067U,  4578227681973159812U,
	13801599718827935620U,  4607181740574479067U,
	 4604465633578481725U,  4604621949701367983U,
	13827993986556143791U,  4604465633578481725U,
	 4606453861145241227U,  4600769149537129431U,
	13824141186391905239U,  4606453861145241227U,
	 4600360675823176935U,  4606538458821337243U,
	13829910495676113051U,  4600360675823176935U,
	 4606987119037722413U,  4596629994023683153U,
	13820002030878458961U,  4606987119037722413U,
	 4603087070374583113U,  4605725276488455441U,
	13829097313343231249U,  4603087070374583113U,
	 4605602459698789090U,  4603270878689749849U,
	13826642915544525657U,  4605602459698789090U,
	 4595762727260045105U,  4607030246558998647U,
	13830402283413774455U,  4595762727260045105U,
	 4607127537664763515U,  4592606767730311893U,
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	 4594126307716900071U,  4607096716058023245U,
	13830468752912799053U,  4594126307716900071U,
	 4607072388129742377U,  4594782329999411347U,
	13818154366854187155U,  4607072388129742377U,
	 4603473988668005304U,  4605458946901419122U,
	13828830983756194930U,  4603473988668005304U,
	 4605858005670328613U,  4602876755014813164U,
	13826248791869588972U,  4605858005670328613U,
	 4597600270510262682U,  4606932257325205256U,
	13830304294179981064U,  4597600270510262682U,
	 4606627607157935956U,  4599896339047301634U,
	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
};

#elif FALCON_FPNATIVE // yyyFPEMU+0 yyyFPNATIVE+1

const fpr fpr_gm_tab[] = {
	{0}, {0}, /* unused */
	{-0.000000000000000000000000000}, { 1.000000000000000000000000000},
	{ 0.707106781186547524400844362}, { 0.707106781186547524400844362},
	{-0.707106781186547524400844362}, { 0.707106781186547524400844362},
	{ 0.923879532511286756128183189}, { 0.382683432365089771728459984},
	{-0.382683432365089771728459984}, { 0.923879532511286756128183189},
	{ 0.382683432365089771728459984}, { 0.923879532511286756128183189},
	{-0.923879532511286756128183189}, { 0.382683432365089771728459984},
	{ 0.980785280403230449126182236}, { 0.195090322016128267848284868},
	{-0.195090322016128267848284868}, { 0.980785280403230449126182236},
	{ 0.555570233019602224742830814}, { 0.831469612302545237078788378},
	{-0.831469612302545237078788378}, { 0.555570233019602224742830814},
	{ 0.831469612302545237078788378}, { 0.555570233019602224742830814},
	{-0.555570233019602224742830814}, { 0.831469612302545237078788378},
	{ 0.195090322016128267848284868}, { 0.980785280403230449126182236},
	{-0.980785280403230449126182236}, { 0.195090322016128267848284868},
	{ 0.995184726672196886244836953}, { 0.098017140329560601994195564},
	{-0.098017140329560601994195564}, { 0.995184726672196886244836953},
	{ 0.634393284163645498215171613}, { 0.773010453362736960810906610},
	{-0.773010453362736960810906610}, { 0.634393284163645498215171613},
	{ 0.881921264348355029712756864}, { 0.471396736825997648556387626},
	{-0.471396736825997648556387626}, { 0.881921264348355029712756864},
	{ 0.290284677254462367636192376}, { 0.956940335732208864935797887},
	{-0.956940335732208864935797887}, { 0.290284677254462367636192376},
	{ 0.956940335732208864935797887}, { 0.290284677254462367636192376},
	{-0.290284677254462367636192376}, { 0.956940335732208864935797887},
	{ 0.471396736825997648556387626}, { 0.881921264348355029712756864},
	{-0.881921264348355029712756864}, { 0.471396736825997648556387626},
	{ 0.773010453362736960810906610}, { 0.634393284163645498215171613},
	{-0.634393284163645498215171613}, { 0.773010453362736960810906610},
	{ 0.098017140329560601994195564}, { 0.995184726672196886244836953},
	{-0.995184726672196886244836953}, { 0.098017140329560601994195564},
	{ 0.998795456205172392714771605}, { 0.049067674327418014254954977},
	{-0.049067674327418014254954977}, { 0.998795456205172392714771605},
	{ 0.671558954847018400625376850}, { 0.740951125354959091175616897},
	{-0.740951125354959091175616897}, { 0.671558954847018400625376850},
	{ 0.903989293123443331586200297}, { 0.427555093430282094320966857},
	{-0.427555093430282094320966857}, { 0.903989293123443331586200297},
	{ 0.336889853392220050689253213}, { 0.941544065183020778412509403},
	{-0.941544065183020778412509403}, { 0.336889853392220050689253213},
	{ 0.970031253194543992603984207}, { 0.242980179903263889948274162},
	{-0.242980179903263889948274162}, { 0.970031253194543992603984207},
	{ 0.514102744193221726593693839}, { 0.857728610000272069902269984},
	{-0.857728610000272069902269984}, { 0.514102744193221726593693839},
	{ 0.803207531480644909806676513}, { 0.595699304492433343467036529},
	{-0.595699304492433343467036529}, { 0.803207531480644909806676513},
	{ 0.146730474455361751658850130}, { 0.989176509964780973451673738},
	{-0.989176509964780973451673738}, { 0.146730474455361751658850130},
	{ 0.989176509964780973451673738}, { 0.146730474455361751658850130},
	{-0.146730474455361751658850130}, { 0.989176509964780973451673738},
	{ 0.595699304492433343467036529}, { 0.803207531480644909806676513},
	{-0.803207531480644909806676513}, { 0.595699304492433343467036529},
	{ 0.857728610000272069902269984}, { 0.514102744193221726593693839},
	{-0.514102744193221726593693839}, { 0.857728610000272069902269984},
	{ 0.242980179903263889948274162}, { 0.970031253194543992603984207},
	{-0.970031253194543992603984207}, { 0.242980179903263889948274162},
	{ 0.941544065183020778412509403}, { 0.336889853392220050689253213},
	{-0.336889853392220050689253213}, { 0.941544065183020778412509403},
	{ 0.427555093430282094320966857}, { 0.903989293123443331586200297},
	{-0.903989293123443331586200297}, { 0.427555093430282094320966857},
	{ 0.740951125354959091175616897}, { 0.671558954847018400625376850},
	{-0.671558954847018400625376850}, { 0.740951125354959091175616897},
	{ 0.049067674327418014254954977}, { 0.998795456205172392714771605},
	{-0.998795456205172392714771605}, { 0.049067674327418014254954977},
	{ 0.999698818696204220115765650}, { 0.024541228522912288031734529},
	{-0.024541228522912288031734529}, { 0.999698818696204220115765650},
	{ 0.689540544737066924616730630}, { 0.724247082951466920941069243},
	{-0.724247082951466920941069243}, { 0.689540544737066924616730630},
	{ 0.914209755703530654635014829}, { 0.405241314004989870908481306},
	{-0.405241314004989870908481306}, { 0.914209755703530654635014829},
	{ 0.359895036534988148775104572}, { 0.932992798834738887711660256},
	{-0.932992798834738887711660256}, { 0.359895036534988148775104572},
	{ 0.975702130038528544460395766}, { 0.219101240156869797227737547},
	{-0.219101240156869797227737547}, { 0.975702130038528544460395766},
	{ 0.534997619887097210663076905}, { 0.844853565249707073259571205},
	{-0.844853565249707073259571205}, { 0.534997619887097210663076905},
	{ 0.817584813151583696504920884}, { 0.575808191417845300745972454},
	{-0.575808191417845300745972454}, { 0.817584813151583696504920884},
	{ 0.170961888760301226363642357}, { 0.985277642388941244774018433},
	{-0.985277642388941244774018433}, { 0.170961888760301226363642357},
	{ 0.992479534598709998156767252}, { 0.122410675199216198498704474},
	{-0.122410675199216198498704474}, { 0.992479534598709998156767252},
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	{-0.641481012808583151988739898}, { 0.767138911935820381181694573},
	{ 0.088853552582524596561586535}, { 0.996044700901251989887944810},
	{-0.996044700901251989887944810}, { 0.088853552582524596561586535},
	{ 0.998301544933892840738782163}, { 0.058258264500435759613979782},
	{-0.058258264500435759613979782}, { 0.998301544933892840738782163},
	{ 0.664710978203344868130324985}, { 0.747100605980180144323078847},
	{-0.747100605980180144323078847}, { 0.664710978203344868130324985},
	{ 0.900015892016160228714535267}, { 0.435857079922255491032544080},
	{-0.435857079922255491032544080}, { 0.900015892016160228714535267},
	{ 0.328209843579092526107916817}, { 0.944604837261480265659265493},
	{-0.944604837261480265659265493}, { 0.328209843579092526107916817},
	{ 0.967753837093475465243391912}, { 0.251897818154216950498106628},
	{-0.251897818154216950498106628}, { 0.967753837093475465243391912},
	{ 0.506186645345155291048942344}, { 0.862423956111040538690933878},
	{-0.862423956111040538690933878}, { 0.506186645345155291048942344},
	{ 0.797690840943391108362662755}, { 0.603066598540348201693430617},
	{-0.603066598540348201693430617}, { 0.797690840943391108362662755},
	{ 0.137620121586486044948441663}, { 0.990485084256457037998682243},
	{-0.990485084256457037998682243}, { 0.137620121586486044948441663},
	{ 0.987784141644572154230969032}, { 0.155828397654265235743101486},
	{-0.155828397654265235743101486}, { 0.987784141644572154230969032},
	{ 0.588281548222645304786439813}, { 0.808656181588174991946968128},
	{-0.808656181588174991946968128}, { 0.588281548222645304786439813},
	{ 0.852960604930363657746588082}, { 0.521975292937154342694258318},
	{-0.521975292937154342694258318}, { 0.852960604930363657746588082},
	{ 0.234041958583543423191242045}, { 0.972226497078936305708321144},
	{-0.972226497078936305708321144}, { 0.234041958583543423191242045},
	{ 0.938403534063108112192420774}, { 0.345541324963989065539191723},
	{-0.345541324963989065539191723}, { 0.938403534063108112192420774},
	{ 0.419216888363223956433010020}, { 0.907886116487666212038681480},
	{-0.907886116487666212038681480}, { 0.419216888363223956433010020},
	{ 0.734738878095963464563223604}, { 0.678350043129861486873655042},
	{-0.678350043129861486873655042}, { 0.734738878095963464563223604},
	{ 0.039872927587739811128578738}, { 0.999204758618363895492950001},
	{-0.999204758618363895492950001}, { 0.039872927587739811128578738},
	{ 0.999430604555461772019008327}, { 0.033741171851377584833716112},
	{-0.033741171851377584833716112}, { 0.999430604555461772019008327},
	{ 0.682845546385248068164596123}, { 0.730562769227827561177758850},
	{-0.730562769227827561177758850}, { 0.682845546385248068164596123},
	{ 0.910441292258067196934095369}, { 0.413638312238434547471944324},
	{-0.413638312238434547471944324}, { 0.910441292258067196934095369},
	{ 0.351292756085567125601307623}, { 0.936265667170278246576310996},
	{-0.936265667170278246576310996}, { 0.351292756085567125601307623},
	{ 0.973644249650811925318383912}, { 0.228072083170885739254457379},
	{-0.228072083170885739254457379}, { 0.973644249650811925318383912},
	{ 0.527199134781901348464274575}, { 0.849741768000852489471268395},
	{-0.849741768000852489471268395}, { 0.527199134781901348464274575},
	{ 0.812250586585203913049744181}, { 0.583308652937698294392830961},
	{-0.583308652937698294392830961}, { 0.812250586585203913049744181},
	{ 0.161886393780111837641387995}, { 0.986809401814185476970235952},
	{-0.986809401814185476970235952}, { 0.161886393780111837641387995},
	{ 0.991310859846115418957349799}, { 0.131540028702883111103387493},
	{-0.131540028702883111103387493}, { 0.991310859846115418957349799},
	{ 0.607949784967773667243642671}, { 0.793975477554337164895083757},
	{-0.793975477554337164895083757}, { 0.607949784967773667243642671},
	{ 0.865513624090569082825488358}, { 0.500885382611240786241285004},
	{-0.500885382611240786241285004}, { 0.865513624090569082825488358},
	{ 0.257831102162159005614471295}, { 0.966190003445412555433832961},
	{-0.966190003445412555433832961}, { 0.257831102162159005614471295},
	{ 0.946600913083283570044599823}, { 0.322407678801069848384807478},
	{-0.322407678801069848384807478}, { 0.946600913083283570044599823},
	{ 0.441371268731716692879988968}, { 0.897324580705418281231391836},
	{-0.897324580705418281231391836}, { 0.441371268731716692879988968},
	{ 0.751165131909686411205819422}, { 0.660114342067420478559490747},
	{-0.660114342067420478559490747}, { 0.751165131909686411205819422},
	{ 0.064382630929857460819324537}, { 0.997925286198596012623025462},
	{-0.997925286198596012623025462}, { 0.064382630929857460819324537},
	{ 0.996571145790554847093566910}, { 0.082740264549375693111987083},
	{-0.082740264549375693111987083}, { 0.996571145790554847093566910},
	{ 0.646176012983316364832802220}, { 0.763188417263381271704838297},
	{-0.763188417263381271704838297}, { 0.646176012983316364832802220},
	{ 0.889048355854664562540777729}, { 0.457813303598877221904961155},
	{-0.457813303598877221904961155}, { 0.889048355854664562540777729},
	{ 0.304929229735402406490728633}, { 0.952375012719765858529893608},
	{-0.952375012719765858529893608}, { 0.304929229735402406490728633},
	{ 0.961280485811320641748659653}, { 0.275571819310958163076425168},
	{-0.275571819310958163076425168}, { 0.961280485811320641748659653},
	{ 0.484869248000791101822951699}, { 0.874586652278176112634431897},
	{-0.874586652278176112634431897}, { 0.484869248000791101822951699},
	{ 0.782650596166575738458949301}, { 0.622461279374149972519166721},
	{-0.622461279374149972519166721}, { 0.782650596166575738458949301},
	{ 0.113270952177564349018228733}, { 0.993564135520595333782021697},
	{-0.993564135520595333782021697}, { 0.113270952177564349018228733},
	{ 0.983662419211730274396237776}, { 0.180022901405699522679906590},
	{-0.180022901405699522679906590}, { 0.983662419211730274396237776},
	{ 0.568258952670131549790548489}, { 0.822849781375826332046780034},
	{-0.822849781375826332046780034}, { 0.568258952670131549790548489},
	{ 0.839893794195999504583383987}, { 0.542750784864515906586768661},
	{-0.542750784864515906586768661}, { 0.839893794195999504583383987},
	{ 0.210111836880469621717489972}, { 0.977677357824509979943404762},
	{-0.977677357824509979943404762}, { 0.210111836880469621717489972},
	{ 0.929640895843181265457918066}, { 0.368466829953372331712746222},
	{-0.368466829953372331712746222}, { 0.929640895843181265457918066},
	{ 0.396809987416710328595290911}, { 0.917900775621390457642276297},
	{-0.917900775621390457642276297}, { 0.396809987416710328595290911},
	{ 0.717870045055731736211325329}, { 0.696177131491462944788582591},
	{-0.696177131491462944788582591}, { 0.717870045055731736211325329},
	{ 0.015339206284988101044151868}, { 0.999882347454212525633049627},
	{-0.999882347454212525633049627}, { 0.015339206284988101044151868},
	{ 0.999769405351215321657617036}, { 0.021474080275469507418374898},
	{-0.021474080275469507418374898}, { 0.999769405351215321657617036},
	{ 0.691759258364157774906734132}, { 0.722128193929215321243607198},
	{-0.722128193929215321243607198}, { 0.691759258364157774906734132},
	{ 0.915448716088267819566431292}, { 0.402434650859418441082533934},
	{-0.402434650859418441082533934}, { 0.915448716088267819566431292},
	{ 0.362755724367397216204854462}, { 0.931884265581668106718557199},
	{-0.931884265581668106718557199}, { 0.362755724367397216204854462},
	{ 0.976369731330021149312732194}, { 0.216106797076219509948385131},
	{-0.216106797076219509948385131}, { 0.976369731330021149312732194},
	{ 0.537587076295645482502214932}, { 0.843208239641845437161743865},
	{-0.843208239641845437161743865}, { 0.537587076295645482502214932},
	{ 0.819347520076796960824689637}, { 0.573297166698042212820171239},
	{-0.573297166698042212820171239}, { 0.819347520076796960824689637},
	{ 0.173983873387463827950700807}, { 0.984748501801904218556553176},
	{-0.984748501801904218556553176}, { 0.173983873387463827950700807},
	{ 0.992850414459865090793563344}, { 0.119365214810991364593637790},
	{-0.119365214810991364593637790}, { 0.992850414459865090793563344},
	{ 0.617647307937803932403979402}, { 0.786455213599085757522319464},
	{-0.786455213599085757522319464}, { 0.617647307937803932403979402},
	{ 0.871595086655951034842481435}, { 0.490226483288291154229598449},
	{-0.490226483288291154229598449}, { 0.871595086655951034842481435},
	{ 0.269668325572915106525464462}, { 0.962953266873683886347921481},
	{-0.962953266873683886347921481}, { 0.269668325572915106525464462},
	{ 0.950486073949481721759926101}, { 0.310767152749611495835997250},
	{-0.310767152749611495835997250}, { 0.950486073949481721759926101},
	{ 0.452349587233770874133026703}, { 0.891840709392342727796478697},
	{-0.891840709392342727796478697}, { 0.452349587233770874133026703},
	{ 0.759209188978388033485525443}, { 0.650846684996380915068975573},
	{-0.650846684996380915068975573}, { 0.759209188978388033485525443},
	{ 0.076623861392031492278332463}, { 0.997060070339482978987989949},
	{-0.997060070339482978987989949}, { 0.076623861392031492278332463},
	{ 0.997511456140303459699448390}, { 0.070504573389613863027351471},
	{-0.070504573389613863027351471}, { 0.997511456140303459699448390},
	{ 0.655492852999615385312679701}, { 0.755201376896536527598710756},
	{-0.755201376896536527598710756}, { 0.655492852999615385312679701},
	{ 0.894599485631382678433072126}, { 0.446868840162374195353044389},
	{-0.446868840162374195353044389}, { 0.894599485631382678433072126},
	{ 0.316593375556165867243047035}, { 0.948561349915730288158494826},
	{-0.948561349915730288158494826}, { 0.316593375556165867243047035},
	{ 0.964589793289812723836432159}, { 0.263754678974831383611349322},
	{-0.263754678974831383611349322}, { 0.964589793289812723836432159},
	{ 0.495565261825772531150266670}, { 0.868570705971340895340449876},
	{-0.868570705971340895340449876}, { 0.495565261825772531150266670},
	{ 0.790230221437310055030217152}, { 0.612810082429409703935211936},
	{-0.612810082429409703935211936}, { 0.790230221437310055030217152},
	{ 0.125454983411546238542336453}, { 0.992099313142191757112085445},
	{-0.992099313142191757112085445}, { 0.125454983411546238542336453},
	{ 0.985797509167567424700995000}, { 0.167938294974731178054745536},
	{-0.167938294974731178054745536}, { 0.985797509167567424700995000},
	{ 0.578313796411655563342245019}, { 0.815814410806733789010772660},
	{-0.815814410806733789010772660}, { 0.578313796411655563342245019},
	{ 0.846490938774052078300544488}, { 0.532403127877197971442805218},
	{-0.532403127877197971442805218}, { 0.846490938774052078300544488},
	{ 0.222093620973203534094094721}, { 0.975025345066994146844913468},
	{-0.975025345066994146844913468}, { 0.222093620973203534094094721},
	{ 0.934092550404258914729877883}, { 0.357030961233430032614954036},
	{-0.357030961233430032614954036}, { 0.934092550404258914729877883},
	{ 0.408044162864978680820747499}, { 0.912962190428398164628018233},
	{-0.912962190428398164628018233}, { 0.408044162864978680820747499},
	{ 0.726359155084345976817494315}, { 0.687315340891759108199186948},
	{-0.687315340891759108199186948}, { 0.726359155084345976817494315},
	{ 0.027608145778965741612354872}, { 0.999618822495178597116830637},
	{-0.999618822495178597116830637}, { 0.027608145778965741612354872},
	{ 0.998941293186856850633930266}, { 0.046003182130914628814301788},
	{-0.046003182130914628814301788}, { 0.998941293186856850633930266},
	{ 0.673829000378756060917568372}, { 0.738887324460615147933116508},
	{-0.738887324460615147933116508}, { 0.673829000378756060917568372},
	{ 0.905296759318118774354048329}, { 0.424779681209108833357226189},
	{-0.424779681209108833357226189}, { 0.905296759318118774354048329},
	{ 0.339776884406826857828825803}, { 0.940506070593268323787291309},
	{-0.940506070593268323787291309}, { 0.339776884406826857828825803},
	{ 0.970772140728950302138169611}, { 0.240003022448741486568922365},
	{-0.240003022448741486568922365}, { 0.970772140728950302138169611},
	{ 0.516731799017649881508753876}, { 0.856147328375194481019630732},
	{-0.856147328375194481019630732}, { 0.516731799017649881508753876},
	{ 0.805031331142963597922659282}, { 0.593232295039799808047809426},
	{-0.593232295039799808047809426}, { 0.805031331142963597922659282},
	{ 0.149764534677321517229695737}, { 0.988721691960323767604516485},
	{-0.988721691960323767604516485}, { 0.149764534677321517229695737},
	{ 0.989622017463200834623694454}, { 0.143695033150294454819773349},
	{-0.143695033150294454819773349}, { 0.989622017463200834623694454},
	{ 0.598160706996342311724958652}, { 0.801376171723140219430247777},
	{-0.801376171723140219430247777}, { 0.598160706996342311724958652},
	{ 0.859301818357008404783582139}, { 0.511468850437970399504391001},
	{-0.511468850437970399504391001}, { 0.859301818357008404783582139},
	{ 0.245955050335794611599924709}, { 0.969281235356548486048290738},
	{-0.969281235356548486048290738}, { 0.245955050335794611599924709},
	{ 0.942573197601446879280758735}, { 0.333999651442009404650865481},
	{-0.333999651442009404650865481}, { 0.942573197601446879280758735},
	{ 0.430326481340082633908199031}, { 0.902673318237258806751502391},
	{-0.902673318237258806751502391}, { 0.430326481340082633908199031},
	{ 0.743007952135121693517362293}, { 0.669282588346636065720696366},
	{-0.669282588346636065720696366}, { 0.743007952135121693517362293},
	{ 0.052131704680283321236358216}, { 0.998640218180265222418199049},
	{-0.998640218180265222418199049}, { 0.052131704680283321236358216},
	{ 0.995480755491926941769171600}, { 0.094963495329638998938034312},
	{-0.094963495329638998938034312}, { 0.995480755491926941769171600},
	{ 0.636761861236284230413943435}, { 0.771060524261813773200605759},
	{-0.771060524261813773200605759}, { 0.636761861236284230413943435},
	{ 0.883363338665731594736308015}, { 0.468688822035827933697617870},
	{-0.468688822035827933697617870}, { 0.883363338665731594736308015},
	{ 0.293219162694258650606608599}, { 0.956045251349996443270479823},
	{-0.956045251349996443270479823}, { 0.293219162694258650606608599},
	{ 0.957826413027532890321037029}, { 0.287347459544729526477331841},
	{-0.287347459544729526477331841}, { 0.957826413027532890321037029},
	{ 0.474100214650550014398580015}, { 0.880470889052160770806542929},
	{-0.880470889052160770806542929}, { 0.474100214650550014398580015},
	{ 0.774953106594873878359129282}, { 0.632018735939809021909403706},
	{-0.632018735939809021909403706}, { 0.774953106594873878359129282},
	{ 0.101069862754827824987887585}, { 0.994879330794805620591166107},
	{-0.994879330794805620591166107}, { 0.101069862754827824987887585},
	{ 0.981379193313754574318224190}, { 0.192080397049892441679288205},
	{-0.192080397049892441679288205}, { 0.981379193313754574318224190},
	{ 0.558118531220556115693702964}, { 0.829761233794523042469023765},
	{-0.829761233794523042469023765}, { 0.558118531220556115693702964},
	{ 0.833170164701913186439915922}, { 0.553016705580027531764226988},
	{-0.553016705580027531764226988}, { 0.833170164701913186439915922},
	{ 0.198098410717953586179324918}, { 0.980182135968117392690210009},
	{-0.980182135968117392690210009}, { 0.198098410717953586179324918},
	{ 0.925049240782677590302371869}, { 0.379847208924051170576281147},
	{-0.379847208924051170576281147}, { 0.925049240782677590302371869},
	{ 0.385516053843918864075607949}, { 0.922701128333878570437264227},
	{-0.922701128333878570437264227}, { 0.385516053843918864075607949},
	{ 0.709272826438865651316533772}, { 0.704934080375904908852523758},
	{-0.704934080375904908852523758}, { 0.709272826438865651316533772},
	{ 0.003067956762965976270145365}, { 0.999995293809576171511580126},
	{-0.999995293809576171511580126}, { 0.003067956762965976270145365}
};

const fpr fpr_p2_tab[] = {
	{ 2.00000000000 },
	{ 1.00000000000 },
	{ 0.50000000000 },
	{ 0.25000000000 },
	{ 0.12500000000 },
	{ 0.06250000000 },
	{ 0.03125000000 },
	{ 0.01562500000 },
	{ 0.00781250000 },
	{ 0.00390625000 },
	{ 0.00195312500 }
};

#else // yyyFPNATIVE+0 yyyFPEMU+0

#error No FP implementation selected

#endif // yyyFPNATIVE- yyyFPEMU-