332 lines
11 KiB
NASM
332 lines
11 KiB
NASM
dnl Itanium-2 mpn_modexact_1c_odd -- mpn by 1 exact remainder.
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dnl Copyright 2003, 2004, 2005 Free Software Foundation, Inc.
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dnl
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dnl This file is part of the GNU MP Library.
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dnl
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dnl The GNU MP Library is free software; you can redistribute it and/or
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dnl modify it under the terms of the GNU Lesser General Public License as
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dnl published by the Free Software Foundation; either version 2.1 of the
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dnl License, or (at your option) any later version.
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dnl
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dnl The GNU MP Library is distributed in the hope that it will be useful,
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dnl but WITHOUT ANY WARRANTY; without even the implied warranty of
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dnl MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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dnl Lesser General Public License for more details.
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dnl
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dnl You should have received a copy of the GNU Lesser General Public
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dnl License along with the GNU MP Library; see the file COPYING.LIB. If
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dnl not, write to the Free Software Foundation, Inc., 51 Franklin Street,
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dnl Fifth Floor, Boston, MA 02110-1301, USA.
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include(`../config.m4')
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C cycles/limb
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C Itanium: 14?
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C Itanium 2: 8.0
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dnl Usage: ABI32(`code')
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dnl
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dnl Emit the given code only under HAVE_ABI_32.
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dnl
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define(ABI32,
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m4_assert_onearg()
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`ifdef(`HAVE_ABI_32',`$1')')
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C mp_limb_t mpn_modexact_1c_odd (mp_srcptr src, mp_size_t size,
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C mp_limb_t divisor, mp_limb_t carry);
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C
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C The modexact algorithm is usually conceived as a dependent chain
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C
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C l = src[i] - c
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C q = low(l * inverse)
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C c = high(q*divisor) + (src[i]<c)
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C
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C but we can work the src[i]-c into an xma by calculating si=src[i]*inverse
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C separately (off the dependent chain) and using
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C
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C q = low(c * inverse + si)
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C c = high(q*divisor + c)
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C
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C This means the dependent chain is simply xma.l followed by xma.hu, for a
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C total 8 cycles/limb on itanium-2.
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C
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C The reason xma.hu works for the new c is that the low of q*divisor is
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C src[i]-c (being the whole purpose of the q generated, and it can be
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C verified algebraically). If there was an underflow from src[i]-c, then
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C there will be an overflow from (src-c)+c, thereby adding 1 to the new c
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C the same as the borrow bit (src[i]<c) gives in the first style shown.
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C
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C Incidentally, fcmp is not an option for treating src[i]-c, since it
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C apparently traps to the kernel for unnormalized operands like those used
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C and generated by ldf8 and xma. On one GNU/Linux system it took about 1200
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C cycles.
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C
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C
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C First Limb:
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C
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C The first limb uses q = (src[0]-c) * inverse shown in the first style.
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C This lets us get the first q as soon as the inverse is ready, without
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C going through si=s*inverse. Basically at the start we have c and can use
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C it while waiting for the inverse, whereas for the second and subsequent
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C limbs it's the other way around, ie. we have the inverse and are waiting
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C for c.
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C
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C At .Lentry the first two instructions in the loop have been done already.
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C The load of f11=src[1] at the start (predicated on size>=2), and the
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C calculation of q by the initial different scheme.
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C
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C
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C Entry Sequence:
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C
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C In the entry sequence, the critical path is the calculation of the
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C inverse, so this is begun first and optimized. Apart from that, ar.lc is
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C established nice and early so the br.cloop's should predict perfectly.
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C And the load for the low limbs src[0] and src[1] can be initiated long
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C ahead of where they're needed.
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C
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C
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C Inverse Calculation:
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C
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C The initial 8-bit inverse is calculated using a table lookup. If it hits
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C L1 (which is likely if we're called several times) then it should take a
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C total 4 cycles, otherwise hopefully L2 for 9 cycles. This is considered
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C the best approach, on balance. It could be done bitwise, but that would
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C probably be about 14 cycles (2 per bit beyond the first couple). Or it
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C could be taken from 4 bits to 8 with xmpy doubling as used beyond 8 bits,
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C but that would be about 11 cycles.
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C
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C The table is not the same as modlimb_invert_table, instead it's 256 bytes,
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C designed to be indexed by the low byte of the divisor. The divisor is
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C always odd, so the relevant data is every second byte in the table. The
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C padding lets us use zxt1 instead of extr.u, the latter would cost an extra
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C cycle because it must go down I0, and we're using the first I0 slot to get
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C ip. The extra 128 bytes of padding should be insignificant compared to
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C typical ia64 code bloat.
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C
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C Having the table in .text allows us to use IP-relative addressing,
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C avoiding a fetch from ltoff. .rodata is apparently not suitable for use
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C IP-relative, it gets a linker relocation overflow on GNU/Linux.
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C
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C
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C Load Scheduling:
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C
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C In the main loop, the data loads are scheduled for an L2 hit, which means
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C 6 cycles for the data ready to use. In fact we end up 7 cycles ahead. In
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C any case that scheduling is achieved simply by doing the load (and xmpy.l
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C for "si") in the immediately preceding iteration.
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C
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C The main loop requires size >= 2, and we handle size==1 by an initial
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C br.cloop to enter the loop only if size>1. Since ar.lc is established
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C early, this should predict perfectly.
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C
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C
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C Not done:
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C
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C Consideration was given to using a plain "(src[0]-c) % divisor" for
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C size==1, but cycle counting suggests about 50 for the sort of approach
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C taken by gcc __umodsi3, versus about 47 for the modexact. (Both assuming
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C L1 hits for their respective fetching.)
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C
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C Consideration was given to a test for high<divisor and replacing the last
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C loop iteration with instead c-=src[size-1] followed by c+=d if underflow.
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C Branching on high<divisor wouldn't be good since a mispredict would cost
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C more than the loop iteration saved, and the condition is of course data
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C dependent. So the theory would be to shorten the loop count if
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C high<divisor, and predicate extra operations at the end. That would mean
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C a gain of 6 when high<divisor, or a cost of 2 if not.
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C
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C Whether such a tradeoff is a win on average depends on assumptions about
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C how many bits in the high and the divisor. If both are uniformly
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C distributed then high<divisor about 50% of the time. But smallish
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C divisors (less chance of high<divisor) might be more likely from
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C applications (mpz_divisible_ui, mpz_gcd_ui, etc). Though biggish divisors
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C would be normal internally from say mpn/generic/perfsqr.c. On balance,
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C for the moment, it's felt the gain is not really enough to be worth the
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C trouble.
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C
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C
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C Enhancement:
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C
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C Process two source limbs per iteration using a two-limb inverse and a
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C sequence like
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C
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C ql = low (c * il + sil) quotient low limb
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C qlc = high(c * il + sil)
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C qh1 = low (c * ih + sih) quotient high, partial
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C
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C cl = high (ql * d + c) carry out of low
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C qh = low (qlc * 1 + qh1) quotient high limb
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C
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C new c = high (qh * d + cl) carry out of high
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C
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C This would be 13 cycles/iteration, giving 6.5 cycles/limb. The two limb
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C s*inverse as sih:sil = sh:sl * ih:il would be calculated off the dependent
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C chain with 4 multiplies. The bigger inverse would take extra time to
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C calculate, but a one limb iteration to handle an odd size could be done as
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C soon as 64-bits of inverse were ready.
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C
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C Perhaps this could even extend to a 3 limb inverse, which might promise 17
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C or 18 cycles for 3 limbs, giving 5.66 or 6.0 cycles/limb.
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C
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ASM_START()
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.explicit
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.text
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.align 32
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.Ltable:
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data1 0,0x01, 0,0xAB, 0,0xCD, 0,0xB7, 0,0x39, 0,0xA3, 0,0xC5, 0,0xEF
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data1 0,0xF1, 0,0x1B, 0,0x3D, 0,0xA7, 0,0x29, 0,0x13, 0,0x35, 0,0xDF
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data1 0,0xE1, 0,0x8B, 0,0xAD, 0,0x97, 0,0x19, 0,0x83, 0,0xA5, 0,0xCF
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data1 0,0xD1, 0,0xFB, 0,0x1D, 0,0x87, 0,0x09, 0,0xF3, 0,0x15, 0,0xBF
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data1 0,0xC1, 0,0x6B, 0,0x8D, 0,0x77, 0,0xF9, 0,0x63, 0,0x85, 0,0xAF
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data1 0,0xB1, 0,0xDB, 0,0xFD, 0,0x67, 0,0xE9, 0,0xD3, 0,0xF5, 0,0x9F
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data1 0,0xA1, 0,0x4B, 0,0x6D, 0,0x57, 0,0xD9, 0,0x43, 0,0x65, 0,0x8F
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data1 0,0x91, 0,0xBB, 0,0xDD, 0,0x47, 0,0xC9, 0,0xB3, 0,0xD5, 0,0x7F
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data1 0,0x81, 0,0x2B, 0,0x4D, 0,0x37, 0,0xB9, 0,0x23, 0,0x45, 0,0x6F
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data1 0,0x71, 0,0x9B, 0,0xBD, 0,0x27, 0,0xA9, 0,0x93, 0,0xB5, 0,0x5F
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data1 0,0x61, 0,0x0B, 0,0x2D, 0,0x17, 0,0x99, 0,0x03, 0,0x25, 0,0x4F
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data1 0,0x51, 0,0x7B, 0,0x9D, 0,0x07, 0,0x89, 0,0x73, 0,0x95, 0,0x3F
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data1 0,0x41, 0,0xEB, 0,0x0D, 0,0xF7, 0,0x79, 0,0xE3, 0,0x05, 0,0x2F
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data1 0,0x31, 0,0x5B, 0,0x7D, 0,0xE7, 0,0x69, 0,0x53, 0,0x75, 0,0x1F
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data1 0,0x21, 0,0xCB, 0,0xED, 0,0xD7, 0,0x59, 0,0xC3, 0,0xE5, 0,0x0F
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data1 0,0x11, 0,0x3B, 0,0x5D, 0,0xC7, 0,0x49, 0,0x33, 0,0x55, 0,0xFF
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PROLOGUE(mpn_modexact_1c_odd)
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C r32 src
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C r33 size
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C r34 divisor
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C r35 carry
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.prologue
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.Lhere:
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{ .mmi; add r33 = -1, r33 C M0 size-1
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mov r14 = 2 C M1 2
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mov r15 = ip C I0 .Lhere
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}{.mmi; setf.sig f6 = r34 C M2 divisor
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setf.sig f9 = r35 C M3 carry
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zxt1 r3 = r34 C I1 divisor low byte
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} ;;
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{ .mmi; add r3 = .Ltable-.Lhere, r3 C M0 table offset ip and index
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sub r16 = 0, r34 C M1 -divisor
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.save ar.lc, r2
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mov r2 = ar.lc C I0
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}{.mmi; .body
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setf.sig f13 = r14 C M2 2 in significand
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mov r17 = -1 C M3 -1
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ABI32(` zxt4 r33 = r33') C I1 size extend
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} ;;
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{ .mmi; add r3 = r3, r15 C M0 table entry address
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ABI32(` addp4 r32 = 0, r32') C M1 src extend
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mov ar.lc = r33 C I0 size-1 loop count
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}{.mmi; setf.sig f12 = r16 C M2 -divisor
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setf.sig f8 = r17 C M3 -1
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} ;;
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{ .mmi; ld1 r3 = [r3] C M0 inverse, 8 bits
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ldf8 f10 = [r32], 8 C M1 src[0]
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cmp.ne p6,p0 = 0, r33 C I0 test size!=1
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} ;;
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C Wait for table load.
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C Hope for an L1 hit of 1 cycles to ALU, but could be more.
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setf.sig f7 = r3 C M2 inverse, 8 bits
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(p6) ldf8 f11 = [r32], 8 C M1 src[1], if size!=1
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;;
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C 5 cycles
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C f6 divisor
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C f7 inverse, being calculated
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C f8 -1, will be -inverse
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C f9 carry
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C f10 src[0]
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C f11 src[1]
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C f12 -divisor
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C f13 2
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C f14 scratch
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xmpy.l f14 = f13, f7 C 2*i
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xmpy.l f7 = f7, f7 C i*i
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;;
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xma.l f7 = f7, f12, f14 C i*i*-d + 2*i, inverse 16 bits
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;;
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xmpy.l f14 = f13, f7 C 2*i
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xmpy.l f7 = f7, f7 C i*i
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;;
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xma.l f7 = f7, f12, f14 C i*i*-d + 2*i, inverse 32 bits
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;;
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xmpy.l f14 = f13, f7 C 2*i
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xmpy.l f7 = f7, f7 C i*i
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;;
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xma.l f7 = f7, f12, f14 C i*i*-d + 2*i, inverse 64 bits
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xma.l f10 = f9, f8, f10 C sc = c * -1 + src[0]
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;;
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ASSERT(p6, `
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xmpy.l f15 = f6, f7 ;; C divisor*inverse
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getf.sig r31 = f15 ;;
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cmp.eq p6,p0 = 1, r31 C should == 1
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')
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xmpy.l f10 = f10, f7 C q = sc * inverse
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xmpy.l f8 = f7, f8 C -inverse = inverse * -1
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br.cloop.sptk.few.clr .Lentry C main loop, if size > 1
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;;
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C size==1, finish up now
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xma.hu f9 = f10, f6, f9 C c = high(q * divisor + c)
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mov ar.lc = r2 C I0
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;;
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getf.sig r8 = f9 C M2 return c
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br.ret.sptk.many b0
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.Ltop:
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C r2 saved ar.lc
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C f6 divisor
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C f7 inverse
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C f8 -inverse
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C f9 carry
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C f10 src[i] * inverse
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C f11 scratch src[i+1]
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add r16 = 160, r32
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ldf8 f11 = [r32], 8 C src[i+1]
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;;
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C 2 cycles
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lfetch [r16]
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xma.l f10 = f9, f8, f10 C q = c * -inverse + si
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;;
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C 3 cycles
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.Lentry:
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xma.hu f9 = f10, f6, f9 C c = high(q * divisor + c)
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xmpy.l f10 = f11, f7 C si = src[i] * inverse
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br.cloop.sptk.few.clr .Ltop
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;;
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xma.l f10 = f9, f8, f10 C q = c * -inverse + si
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mov ar.lc = r2 C I0
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;;
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xma.hu f9 = f10, f6, f9 C c = high(q * divisor + c)
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;;
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getf.sig r8 = f9 C M2 return c
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br.ret.sptk.many b0
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EPILOGUE()
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