mode1o.asm
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- dnl Itanium-2 mpn_modexact_1c_odd -- mpn by 1 exact remainder.
- dnl Copyright 2003, 2004, 2005 Free Software Foundation, Inc.
- dnl
- dnl This file is part of the GNU MP Library.
- dnl
- dnl The GNU MP Library is free software; you can redistribute it and/or
- dnl modify it under the terms of the GNU Lesser General Public License as
- dnl published by the Free Software Foundation; either version 3 of the
- dnl License, or (at your option) any later version.
- dnl
- dnl The GNU MP Library is distributed in the hope that it will be useful,
- dnl but WITHOUT ANY WARRANTY; without even the implied warranty of
- dnl MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
- dnl Lesser General Public License for more details.
- dnl
- dnl You should have received a copy of the GNU Lesser General Public License
- dnl along with the GNU MP Library. If not, see http://www.gnu.org/licenses/.
- include(`../config.m4')
- C cycles/limb
- C Itanium: 15
- C Itanium 2: 8
- dnl Usage: ABI32(`code')
- dnl
- dnl Emit the given code only under HAVE_ABI_32.
- dnl
- define(ABI32,
- m4_assert_onearg()
- `ifdef(`HAVE_ABI_32',`$1')')
- C mp_limb_t mpn_modexact_1c_odd (mp_srcptr src, mp_size_t size,
- C mp_limb_t divisor, mp_limb_t carry);
- C
- C The modexact algorithm is usually conceived as a dependent chain
- C
- C l = src[i] - c
- C q = low(l * inverse)
- C c = high(q*divisor) + (src[i]<c)
- C
- C but we can work the src[i]-c into an xma by calculating si=src[i]*inverse
- C separately (off the dependent chain) and using
- C
- C q = low(c * inverse + si)
- C c = high(q*divisor + c)
- C
- C This means the dependent chain is simply xma.l followed by xma.hu, for a
- C total 8 cycles/limb on itanium-2.
- C
- C The reason xma.hu works for the new c is that the low of q*divisor is
- C src[i]-c (being the whole purpose of the q generated, and it can be
- C verified algebraically). If there was an underflow from src[i]-c, then
- C there will be an overflow from (src-c)+c, thereby adding 1 to the new c
- C the same as the borrow bit (src[i]<c) gives in the first style shown.
- C
- C Incidentally, fcmp is not an option for treating src[i]-c, since it
- C apparently traps to the kernel for unnormalized operands like those used
- C and generated by ldf8 and xma. On one GNU/Linux system it took about 1200
- C cycles.
- C
- C
- C First Limb:
- C
- C The first limb uses q = (src[0]-c) * inverse shown in the first style.
- C This lets us get the first q as soon as the inverse is ready, without
- C going through si=s*inverse. Basically at the start we have c and can use
- C it while waiting for the inverse, whereas for the second and subsequent
- C limbs it's the other way around, ie. we have the inverse and are waiting
- C for c.
- C
- C At .Lentry the first two instructions in the loop have been done already.
- C The load of f11=src[1] at the start (predicated on size>=2), and the
- C calculation of q by the initial different scheme.
- C
- C
- C Entry Sequence:
- C
- C In the entry sequence, the critical path is the calculation of the
- C inverse, so this is begun first and optimized. Apart from that, ar.lc is
- C established nice and early so the br.cloop's should predict perfectly.
- C And the load for the low limbs src[0] and src[1] can be initiated long
- C ahead of where they're needed.
- C
- C
- C Inverse Calculation:
- C
- C The initial 8-bit inverse is calculated using a table lookup. If it hits
- C L1 (which is likely if we're called several times) then it should take a
- C total 4 cycles, otherwise hopefully L2 for 9 cycles. This is considered
- C the best approach, on balance. It could be done bitwise, but that would
- C probably be about 14 cycles (2 per bit beyond the first couple). Or it
- C could be taken from 4 bits to 8 with xmpy doubling as used beyond 8 bits,
- C but that would be about 11 cycles.
- C
- C The table is not the same as binvert_limb_table, instead it's 256 bytes,
- C designed to be indexed by the low byte of the divisor. The divisor is
- C always odd, so the relevant data is every second byte in the table. The
- C padding lets us use zxt1 instead of extr.u, the latter would cost an extra
- C cycle because it must go down I0, and we're using the first I0 slot to get
- C ip. The extra 128 bytes of padding should be insignificant compared to
- C typical ia64 code bloat.
- C
- C Having the table in .text allows us to use IP-relative addressing,
- C avoiding a fetch from ltoff. .rodata is apparently not suitable for use
- C IP-relative, it gets a linker relocation overflow on GNU/Linux.
- C
- C
- C Load Scheduling:
- C
- C In the main loop, the data loads are scheduled for an L2 hit, which means
- C 6 cycles for the data ready to use. In fact we end up 7 cycles ahead. In
- C any case that scheduling is achieved simply by doing the load (and xmpy.l
- C for "si") in the immediately preceding iteration.
- C
- C The main loop requires size >= 2, and we handle size==1 by an initial
- C br.cloop to enter the loop only if size>1. Since ar.lc is established
- C early, this should predict perfectly.
- C
- C
- C Not done:
- C
- C Consideration was given to using a plain "(src[0]-c) % divisor" for
- C size==1, but cycle counting suggests about 50 for the sort of approach
- C taken by gcc __umodsi3, versus about 47 for the modexact. (Both assuming
- C L1 hits for their respective fetching.)
- C
- C Consideration was given to a test for high<divisor and replacing the last
- C loop iteration with instead c-=src[size-1] followed by c+=d if underflow.
- C Branching on high<divisor wouldn't be good since a mispredict would cost
- C more than the loop iteration saved, and the condition is of course data
- C dependent. So the theory would be to shorten the loop count if
- C high<divisor, and predicate extra operations at the end. That would mean
- C a gain of 6 when high<divisor, or a cost of 2 if not.
- C
- C Whether such a tradeoff is a win on average depends on assumptions about
- C how many bits in the high and the divisor. If both are uniformly
- C distributed then high<divisor about 50% of the time. But smallish
- C divisors (less chance of high<divisor) might be more likely from
- C applications (mpz_divisible_ui, mpz_gcd_ui, etc). Though biggish divisors
- C would be normal internally from say mpn/generic/perfsqr.c. On balance,
- C for the moment, it's felt the gain is not really enough to be worth the
- C trouble.
- C
- C
- C Enhancement:
- C
- C Process two source limbs per iteration using a two-limb inverse and a
- C sequence like
- C
- C ql = low (c * il + sil) quotient low limb
- C qlc = high(c * il + sil)
- C qh1 = low (c * ih + sih) quotient high, partial
- C
- C cl = high (ql * d + c) carry out of low
- C qh = low (qlc * 1 + qh1) quotient high limb
- C
- C new c = high (qh * d + cl) carry out of high
- C
- C This would be 13 cycles/iteration, giving 6.5 cycles/limb. The two limb
- C s*inverse as sih:sil = sh:sl * ih:il would be calculated off the dependent
- C chain with 4 multiplies. The bigger inverse would take extra time to
- C calculate, but a one limb iteration to handle an odd size could be done as
- C soon as 64-bits of inverse were ready.
- C
- C Perhaps this could even extend to a 3 limb inverse, which might promise 17
- C or 18 cycles for 3 limbs, giving 5.66 or 6.0 cycles/limb.
- C
- ASM_START()
- .explicit
- .text
- .align 32
- .Ltable:
- data1 0,0x01, 0,0xAB, 0,0xCD, 0,0xB7, 0,0x39, 0,0xA3, 0,0xC5, 0,0xEF
- data1 0,0xF1, 0,0x1B, 0,0x3D, 0,0xA7, 0,0x29, 0,0x13, 0,0x35, 0,0xDF
- data1 0,0xE1, 0,0x8B, 0,0xAD, 0,0x97, 0,0x19, 0,0x83, 0,0xA5, 0,0xCF
- data1 0,0xD1, 0,0xFB, 0,0x1D, 0,0x87, 0,0x09, 0,0xF3, 0,0x15, 0,0xBF
- data1 0,0xC1, 0,0x6B, 0,0x8D, 0,0x77, 0,0xF9, 0,0x63, 0,0x85, 0,0xAF
- data1 0,0xB1, 0,0xDB, 0,0xFD, 0,0x67, 0,0xE9, 0,0xD3, 0,0xF5, 0,0x9F
- data1 0,0xA1, 0,0x4B, 0,0x6D, 0,0x57, 0,0xD9, 0,0x43, 0,0x65, 0,0x8F
- data1 0,0x91, 0,0xBB, 0,0xDD, 0,0x47, 0,0xC9, 0,0xB3, 0,0xD5, 0,0x7F
- data1 0,0x81, 0,0x2B, 0,0x4D, 0,0x37, 0,0xB9, 0,0x23, 0,0x45, 0,0x6F
- data1 0,0x71, 0,0x9B, 0,0xBD, 0,0x27, 0,0xA9, 0,0x93, 0,0xB5, 0,0x5F
- data1 0,0x61, 0,0x0B, 0,0x2D, 0,0x17, 0,0x99, 0,0x03, 0,0x25, 0,0x4F
- data1 0,0x51, 0,0x7B, 0,0x9D, 0,0x07, 0,0x89, 0,0x73, 0,0x95, 0,0x3F
- data1 0,0x41, 0,0xEB, 0,0x0D, 0,0xF7, 0,0x79, 0,0xE3, 0,0x05, 0,0x2F
- data1 0,0x31, 0,0x5B, 0,0x7D, 0,0xE7, 0,0x69, 0,0x53, 0,0x75, 0,0x1F
- data1 0,0x21, 0,0xCB, 0,0xED, 0,0xD7, 0,0x59, 0,0xC3, 0,0xE5, 0,0x0F
- data1 0,0x11, 0,0x3B, 0,0x5D, 0,0xC7, 0,0x49, 0,0x33, 0,0x55, 0,0xFF
- PROLOGUE(mpn_modexact_1c_odd)
- C r32 src
- C r33 size
- C r34 divisor
- C r35 carry
- .prologue
- .Lhere:
- { .mmi; add r33 = -1, r33 C M0 size-1
- mov r14 = 2 C M1 2
- mov r15 = ip C I0 .Lhere
- }{.mmi; setf.sig f6 = r34 C M2 divisor
- setf.sig f9 = r35 C M3 carry
- zxt1 r3 = r34 C I1 divisor low byte
- } ;;
- { .mmi; add r3 = .Ltable-.Lhere, r3 C M0 table offset ip and index
- sub r16 = 0, r34 C M1 -divisor
- .save ar.lc, r2
- mov r2 = ar.lc C I0
- }{.mmi; .body
- setf.sig f13 = r14 C M2 2 in significand
- mov r17 = -1 C M3 -1
- ABI32(` zxt4 r33 = r33') C I1 size extend
- } ;;
- { .mmi; add r3 = r3, r15 C M0 table entry address
- ABI32(` addp4 r32 = 0, r32') C M1 src extend
- mov ar.lc = r33 C I0 size-1 loop count
- }{.mmi; setf.sig f12 = r16 C M2 -divisor
- setf.sig f8 = r17 C M3 -1
- } ;;
- { .mmi; ld1 r3 = [r3] C M0 inverse, 8 bits
- ldf8 f10 = [r32], 8 C M1 src[0]
- cmp.ne p6,p0 = 0, r33 C I0 test size!=1
- } ;;
- C Wait for table load.
- C Hope for an L1 hit of 1 cycles to ALU, but could be more.
- setf.sig f7 = r3 C M2 inverse, 8 bits
- (p6) ldf8 f11 = [r32], 8 C M1 src[1], if size!=1
- ;;
- C 5 cycles
- C f6 divisor
- C f7 inverse, being calculated
- C f8 -1, will be -inverse
- C f9 carry
- C f10 src[0]
- C f11 src[1]
- C f12 -divisor
- C f13 2
- C f14 scratch
- xmpy.l f14 = f13, f7 C 2*i
- xmpy.l f7 = f7, f7 C i*i
- ;;
- xma.l f7 = f7, f12, f14 C i*i*-d + 2*i, inverse 16 bits
- ;;
- xmpy.l f14 = f13, f7 C 2*i
- xmpy.l f7 = f7, f7 C i*i
- ;;
- xma.l f7 = f7, f12, f14 C i*i*-d + 2*i, inverse 32 bits
- ;;
- xmpy.l f14 = f13, f7 C 2*i
- xmpy.l f7 = f7, f7 C i*i
- ;;
- xma.l f7 = f7, f12, f14 C i*i*-d + 2*i, inverse 64 bits
- xma.l f10 = f9, f8, f10 C sc = c * -1 + src[0]
- ;;
- ASSERT(p6, `
- xmpy.l f15 = f6, f7 ;; C divisor*inverse
- getf.sig r31 = f15 ;;
- cmp.eq p6,p0 = 1, r31 C should == 1
- ')
- xmpy.l f10 = f10, f7 C q = sc * inverse
- xmpy.l f8 = f7, f8 C -inverse = inverse * -1
- br.cloop.sptk.few.clr .Lentry C main loop, if size > 1
- ;;
- C size==1, finish up now
- xma.hu f9 = f10, f6, f9 C c = high(q * divisor + c)
- mov ar.lc = r2 C I0
- ;;
- getf.sig r8 = f9 C M2 return c
- br.ret.sptk.many b0
- .Ltop:
- C r2 saved ar.lc
- C f6 divisor
- C f7 inverse
- C f8 -inverse
- C f9 carry
- C f10 src[i] * inverse
- C f11 scratch src[i+1]
- add r16 = 160, r32
- ldf8 f11 = [r32], 8 C src[i+1]
- ;;
- C 2 cycles
- lfetch [r16]
- xma.l f10 = f9, f8, f10 C q = c * -inverse + si
- ;;
- C 3 cycles
- .Lentry:
- xma.hu f9 = f10, f6, f9 C c = high(q * divisor + c)
- xmpy.l f10 = f11, f7 C si = src[i] * inverse
- br.cloop.sptk.few.clr .Ltop
- ;;
- xma.l f10 = f9, f8, f10 C q = c * -inverse + si
- mov ar.lc = r2 C I0
- ;;
- xma.hu f9 = f10, f6, f9 C c = high(q * divisor + c)
- ;;
- getf.sig r8 = f9 C M2 return c
- br.ret.sptk.many b0
- EPILOGUE()