divrem_1.asm
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上传日期:2022-08-06
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- dnl AMD K7 mpn_divrem_1, mpn_divrem_1c, mpn_preinv_divrem_1 -- mpn by limb
- dnl division.
- dnl Copyright 1999, 2000, 2001, 2002, 2004 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 K7: 17.0 cycles/limb integer part, 15.0 cycles/limb fraction part.
- C mp_limb_t mpn_divrem_1 (mp_ptr dst, mp_size_t xsize,
- C mp_srcptr src, mp_size_t size,
- C mp_limb_t divisor);
- C mp_limb_t mpn_divrem_1c (mp_ptr dst, mp_size_t xsize,
- C mp_srcptr src, mp_size_t size,
- C mp_limb_t divisor, mp_limb_t carry);
- C mp_limb_t mpn_preinv_divrem_1 (mp_ptr dst, mp_size_t xsize,
- C mp_srcptr src, mp_size_t size,
- C mp_limb_t divisor, mp_limb_t inverse,
- C unsigned shift);
- C
- C Algorithm:
- C
- C The method and nomenclature follow part 8 of "Division by Invariant
- C Integers using Multiplication" by Granlund and Montgomery, reference in
- C gmp.texi.
- C
- C The "and"s shown in the paper are done here with "cmov"s. "m" is written
- C for m', and "d" for d_norm, which won't cause any confusion since it's
- C only the normalized divisor that's of any use in the code. "b" is written
- C for 2^N, the size of a limb, N being 32 here.
- C
- C The step "sdword dr = n - 2^N*d + (2^N-1-q1) * d" is instead done as
- C "n-(q1+1)*d"; this rearrangement gives the same two-limb answer. If
- C q1==0xFFFFFFFF, then q1+1 would overflow. We branch to a special case
- C "q1_ff" if this occurs. Since the true quotient is either q1 or q1+1 then
- C if q1==0xFFFFFFFF that must be the right value.
- C
- C For the last and second last steps q1==0xFFFFFFFF is instead handled by an
- C sbbl to go back to 0xFFFFFFFF if an overflow occurs when adding 1. This
- C then goes through as normal, and finding no addback required. sbbl costs
- C an extra cycle over what the main loop code does, but it keeps code size
- C and complexity down.
- C
- C Notes:
- C
- C mpn_divrem_1 and mpn_preinv_divrem_1 avoid one division if the src high
- C limb is less than the divisor. mpn_divrem_1c doesn't check for a zero
- C carry, since in normal circumstances that will be a very rare event.
- C
- C The test for skipping a division is branch free (once size>=1 is tested).
- C The store to the destination high limb is 0 when a divide is skipped, or
- C if it's not skipped then a copy of the src high limb is used. The latter
- C is in case src==dst.
- C
- C There's a small bias towards expecting xsize==0, by having code for
- C xsize==0 in a straight line and xsize!=0 under forward jumps.
- C
- C Alternatives:
- C
- C If the divisor is normalized (high bit set) then a division step can
- C always be skipped, since the high destination limb is always 0 or 1 in
- C that case. It doesn't seem worth checking for this though, since it
- C probably occurs infrequently, in particular note that big_base for a
- C decimal mpn_get_str is not normalized in a 32-bit limb.
- dnl MUL_THRESHOLD is the value of xsize+size at which the multiply by
- dnl inverse method is used, rather than plain "divl"s. Minimum value 1.
- dnl
- dnl The inverse takes about 50 cycles to calculate, but after that the
- dnl multiply is 17 c/l versus division at 42 c/l.
- dnl
- dnl At 3 limbs the mul is a touch faster than div on the integer part, and
- dnl even more so on the fractional part.
- deflit(MUL_THRESHOLD, 3)
- defframe(PARAM_PREINV_SHIFT, 28) dnl mpn_preinv_divrem_1
- defframe(PARAM_PREINV_INVERSE, 24) dnl mpn_preinv_divrem_1
- defframe(PARAM_CARRY, 24) dnl mpn_divrem_1c
- defframe(PARAM_DIVISOR,20)
- defframe(PARAM_SIZE, 16)
- defframe(PARAM_SRC, 12)
- defframe(PARAM_XSIZE, 8)
- defframe(PARAM_DST, 4)
- defframe(SAVE_EBX, -4)
- defframe(SAVE_ESI, -8)
- defframe(SAVE_EDI, -12)
- defframe(SAVE_EBP, -16)
- defframe(VAR_NORM, -20)
- defframe(VAR_INVERSE, -24)
- defframe(VAR_SRC, -28)
- defframe(VAR_DST, -32)
- defframe(VAR_DST_STOP,-36)
- deflit(STACK_SPACE, 36)
- TEXT
- ALIGN(32)
- PROLOGUE(mpn_preinv_divrem_1)
- deflit(`FRAME',0)
- movl PARAM_XSIZE, %ecx
- movl PARAM_DST, %edx
- subl $STACK_SPACE, %esp FRAME_subl_esp(STACK_SPACE)
- movl %esi, SAVE_ESI
- movl PARAM_SRC, %esi
- movl %ebx, SAVE_EBX
- movl PARAM_SIZE, %ebx
- leal 8(%edx,%ecx,4), %edx C &dst[xsize+2]
- movl %ebp, SAVE_EBP
- movl PARAM_DIVISOR, %ebp
- movl %edx, VAR_DST_STOP C &dst[xsize+2]
- movl %edi, SAVE_EDI
- xorl %edi, %edi C carry
- movl -4(%esi,%ebx,4), %eax C src high limb
- xor %ecx, %ecx
- C
- C
- cmpl %ebp, %eax C high cmp divisor
- cmovc( %eax, %edi) C high is carry if high<divisor
- cmovnc( %eax, %ecx) C 0 if skip div, src high if not
- C (the latter in case src==dst)
- movl %ecx, -12(%edx,%ebx,4) C dst high limb
- sbbl $0, %ebx C skip one division if high<divisor
- movl PARAM_PREINV_SHIFT, %ecx
- leal -8(%edx,%ebx,4), %edx C &dst[xsize+size]
- movl $32, %eax
- movl %edx, VAR_DST C &dst[xsize+size]
- shll %cl, %ebp C d normalized
- subl %ecx, %eax
- movl %ecx, VAR_NORM
- movd %eax, %mm7 C rshift
- movl PARAM_PREINV_INVERSE, %eax
- jmp L(start_preinv)
- EPILOGUE()
- ALIGN(16)
- PROLOGUE(mpn_divrem_1c)
- deflit(`FRAME',0)
- movl PARAM_CARRY, %edx
- movl PARAM_SIZE, %ecx
- subl $STACK_SPACE, %esp
- deflit(`FRAME',STACK_SPACE)
- movl %ebx, SAVE_EBX
- movl PARAM_XSIZE, %ebx
- movl %edi, SAVE_EDI
- movl PARAM_DST, %edi
- movl %ebp, SAVE_EBP
- movl PARAM_DIVISOR, %ebp
- movl %esi, SAVE_ESI
- movl PARAM_SRC, %esi
- leal -4(%edi,%ebx,4), %edi C &dst[xsize-1]
- jmp L(start_1c)
- EPILOGUE()
- C offset 0xa1, close enough to aligned
- PROLOGUE(mpn_divrem_1)
- deflit(`FRAME',0)
- movl PARAM_SIZE, %ecx
- movl $0, %edx C initial carry (if can't skip a div)
- subl $STACK_SPACE, %esp
- deflit(`FRAME',STACK_SPACE)
- movl %esi, SAVE_ESI
- movl PARAM_SRC, %esi
- movl %ebx, SAVE_EBX
- movl PARAM_XSIZE, %ebx
- movl %ebp, SAVE_EBP
- movl PARAM_DIVISOR, %ebp
- orl %ecx, %ecx C size
- movl %edi, SAVE_EDI
- movl PARAM_DST, %edi
- leal -4(%edi,%ebx,4), %edi C &dst[xsize-1]
- jz L(no_skip_div) C if size==0
- movl -4(%esi,%ecx,4), %eax C src high limb
- xorl %esi, %esi
- cmpl %ebp, %eax C high cmp divisor
- cmovc( %eax, %edx) C high is carry if high<divisor
- cmovnc( %eax, %esi) C 0 if skip div, src high if not
- movl %esi, (%edi,%ecx,4) C dst high limb
- sbbl $0, %ecx C size-1 if high<divisor
- movl PARAM_SRC, %esi C reload
- L(no_skip_div):
- L(start_1c):
- C eax
- C ebx xsize
- C ecx size
- C edx carry
- C esi src
- C edi &dst[xsize-1]
- C ebp divisor
- leal (%ebx,%ecx), %eax C size+xsize
- cmpl $MUL_THRESHOLD, %eax
- jae L(mul_by_inverse)
- C With MUL_THRESHOLD set to 3, the simple loops here only do 0 to 2 limbs.
- C It'd be possible to write them out without the looping, but no speedup
- C would be expected.
- C
- C Using PARAM_DIVISOR instead of %ebp measures 1 cycle/loop faster on the
- C integer part, but curiously not on the fractional part, where %ebp is a
- C (fixed) couple of cycles faster.
- orl %ecx, %ecx
- jz L(divide_no_integer)
- L(divide_integer):
- C eax scratch (quotient)
- C ebx xsize
- C ecx counter
- C edx scratch (remainder)
- C esi src
- C edi &dst[xsize-1]
- C ebp divisor
- movl -4(%esi,%ecx,4), %eax
- divl PARAM_DIVISOR
- movl %eax, (%edi,%ecx,4)
- decl %ecx
- jnz L(divide_integer)
- L(divide_no_integer):
- movl PARAM_DST, %edi
- orl %ebx, %ebx
- jnz L(divide_fraction)
- L(divide_done):
- movl SAVE_ESI, %esi
- movl SAVE_EDI, %edi
- movl %edx, %eax
- movl SAVE_EBX, %ebx
- movl SAVE_EBP, %ebp
- addl $STACK_SPACE, %esp
- ret
- L(divide_fraction):
- C eax scratch (quotient)
- C ebx counter
- C ecx
- C edx scratch (remainder)
- C esi
- C edi dst
- C ebp divisor
- movl $0, %eax
- divl %ebp
- movl %eax, -4(%edi,%ebx,4)
- decl %ebx
- jnz L(divide_fraction)
- jmp L(divide_done)
- C -----------------------------------------------------------------------------
- L(mul_by_inverse):
- C eax
- C ebx xsize
- C ecx size
- C edx carry
- C esi src
- C edi &dst[xsize-1]
- C ebp divisor
- bsrl %ebp, %eax C 31-l
- leal 12(%edi), %ebx C &dst[xsize+2], loop dst stop
- leal 4(%edi,%ecx,4), %edi C &dst[xsize+size]
- movl %edi, VAR_DST
- movl %ebx, VAR_DST_STOP
- movl %ecx, %ebx C size
- movl $31, %ecx
- movl %edx, %edi C carry
- movl $-1, %edx
- C
- xorl %eax, %ecx C l
- incl %eax C 32-l
- shll %cl, %ebp C d normalized
- movl %ecx, VAR_NORM
- movd %eax, %mm7
- movl $-1, %eax
- subl %ebp, %edx C (b-d)-1 giving edx:eax = b*(b-d)-1
- divl %ebp C floor (b*(b-d)-1) / d
- L(start_preinv):
- C eax inverse
- C ebx size
- C ecx shift
- C edx
- C esi src
- C edi carry
- C ebp divisor
- C
- C mm7 rshift
- orl %ebx, %ebx C size
- movl %eax, VAR_INVERSE
- leal -12(%esi,%ebx,4), %eax C &src[size-3]
- jz L(start_zero)
- movl %eax, VAR_SRC
- cmpl $1, %ebx
- movl 8(%eax), %esi C src high limb
- jz L(start_one)
- L(start_two_or_more):
- movl 4(%eax), %edx C src second highest limb
- shldl( %cl, %esi, %edi) C n2 = carry,high << l
- shldl( %cl, %edx, %esi) C n10 = high,second << l
- cmpl $2, %ebx
- je L(integer_two_left)
- jmp L(integer_top)
- L(start_one):
- shldl( %cl, %esi, %edi) C n2 = carry,high << l
- shll %cl, %esi C n10 = high << l
- movl %eax, VAR_SRC
- jmp L(integer_one_left)
- L(start_zero):
- C Can be here with xsize==0 if mpn_preinv_divrem_1 had size==1 and
- C skipped a division.
- shll %cl, %edi C n2 = carry << l
- movl %edi, %eax C return value for zero_done
- cmpl $0, PARAM_XSIZE
- je L(zero_done)
- jmp L(fraction_some)
- C -----------------------------------------------------------------------------
- C
- C The multiply by inverse loop is 17 cycles, and relies on some out-of-order
- C execution. The instruction scheduling is important, with various
- C apparently equivalent forms running 1 to 5 cycles slower.
- C
- C A lower bound for the time would seem to be 16 cycles, based on the
- C following successive dependencies.
- C
- C cycles
- C n2+n1 1
- C mul 6
- C q1+1 1
- C mul 6
- C sub 1
- C addback 1
- C ---
- C 16
- C
- C This chain is what the loop has already, but 16 cycles isn't achieved.
- C K7 has enough decode, and probably enough execute (depending maybe on what
- C a mul actually consumes), but nothing running under 17 has been found.
- C
- C In theory n2+n1 could be done in the sub and addback stages (by
- C calculating both n2 and n2+n1 there), but lack of registers makes this an
- C unlikely proposition.
- C
- C The jz in the loop keeps the q1+1 stage to 1 cycle. Handling an overflow
- C from q1+1 with an "sbbl $0, %ebx" would add a cycle to the dependent
- C chain, and nothing better than 18 cycles has been found when using it.
- C The jump is taken only when q1 is 0xFFFFFFFF, and on random data this will
- C be an extremely rare event.
- C
- C Branch mispredictions will hit random occurrances of q1==0xFFFFFFFF, but
- C if some special data is coming out with this always, the q1_ff special
- C case actually runs at 15 c/l. 0x2FFF...FFFD divided by 3 is a good way to
- C induce the q1_ff case, for speed measurements or testing. Note that
- C 0xFFF...FFF divided by 1 or 2 doesn't induce it.
- C
- C The instruction groupings and empty comments show the cycles for a naive
- C in-order view of the code (conveniently ignoring the load latency on
- C VAR_INVERSE). This shows some of where the time is going, but is nonsense
- C to the extent that out-of-order execution rearranges it. In this case
- C there's 19 cycles shown, but it executes at 17.
- ALIGN(16)
- L(integer_top):
- C eax scratch
- C ebx scratch (nadj, q1)
- C ecx scratch (src, dst)
- C edx scratch
- C esi n10
- C edi n2
- C ebp divisor
- C
- C mm0 scratch (src qword)
- C mm7 rshift for normalization
- cmpl $0x80000000, %esi C n1 as 0=c, 1=nc
- movl %edi, %eax C n2
- movl VAR_SRC, %ecx
- leal (%ebp,%esi), %ebx
- cmovc( %esi, %ebx) C nadj = n10 + (-n1 & d), ignoring overflow
- sbbl $-1, %eax C n2+n1
- mull VAR_INVERSE C m*(n2+n1)
- movq (%ecx), %mm0 C next limb and the one below it
- subl $4, %ecx
- movl %ecx, VAR_SRC
- C
- addl %ebx, %eax C m*(n2+n1) + nadj, low giving carry flag
- leal 1(%edi), %ebx C n2+1
- movl %ebp, %eax C d
- C
- adcl %edx, %ebx C 1 + high(n2<<32 + m*(n2+n1) + nadj) = q1+1
- jz L(q1_ff)
- movl VAR_DST, %ecx
- mull %ebx C (q1+1)*d
- psrlq %mm7, %mm0
- leal -4(%ecx), %ecx
- C
- subl %eax, %esi
- movl VAR_DST_STOP, %eax
- C
- sbbl %edx, %edi C n - (q1+1)*d
- movl %esi, %edi C remainder -> n2
- leal (%ebp,%esi), %edx
- movd %mm0, %esi
- cmovc( %edx, %edi) C n - q1*d if underflow from using q1+1
- sbbl $0, %ebx C q
- cmpl %eax, %ecx
- movl %ebx, (%ecx)
- movl %ecx, VAR_DST
- jne L(integer_top)
- L(integer_loop_done):
- C -----------------------------------------------------------------------------
- C
- C Here, and in integer_one_left below, an sbbl $0 is used rather than a jz
- C q1_ff special case. This make the code a bit smaller and simpler, and
- C costs only 1 cycle (each).
- L(integer_two_left):
- C eax scratch
- C ebx scratch (nadj, q1)
- C ecx scratch (src, dst)
- C edx scratch
- C esi n10
- C edi n2
- C ebp divisor
- C
- C mm7 rshift
- cmpl $0x80000000, %esi C n1 as 0=c, 1=nc
- movl %edi, %eax C n2
- movl PARAM_SRC, %ecx
- leal (%ebp,%esi), %ebx
- cmovc( %esi, %ebx) C nadj = n10 + (-n1 & d), ignoring overflow
- sbbl $-1, %eax C n2+n1
- mull VAR_INVERSE C m*(n2+n1)
- movd (%ecx), %mm0 C src low limb
- movl VAR_DST_STOP, %ecx
- C
- addl %ebx, %eax C m*(n2+n1) + nadj, low giving carry flag
- leal 1(%edi), %ebx C n2+1
- movl %ebp, %eax C d
- adcl %edx, %ebx C 1 + high(n2<<32 + m*(n2+n1) + nadj) = q1+1
- sbbl $0, %ebx
- mull %ebx C (q1+1)*d
- psllq $32, %mm0
- psrlq %mm7, %mm0
- C
- subl %eax, %esi
- C
- sbbl %edx, %edi C n - (q1+1)*d
- movl %esi, %edi C remainder -> n2
- leal (%ebp,%esi), %edx
- movd %mm0, %esi
- cmovc( %edx, %edi) C n - q1*d if underflow from using q1+1
- sbbl $0, %ebx C q
- movl %ebx, -4(%ecx)
- C -----------------------------------------------------------------------------
- L(integer_one_left):
- C eax scratch
- C ebx scratch (nadj, q1)
- C ecx dst
- C edx scratch
- C esi n10
- C edi n2
- C ebp divisor
- C
- C mm7 rshift
- movl VAR_DST_STOP, %ecx
- cmpl $0x80000000, %esi C n1 as 0=c, 1=nc
- movl %edi, %eax C n2
- leal (%ebp,%esi), %ebx
- cmovc( %esi, %ebx) C nadj = n10 + (-n1 & d), ignoring overflow
- sbbl $-1, %eax C n2+n1
- mull VAR_INVERSE C m*(n2+n1)
- C
- C
- C
- addl %ebx, %eax C m*(n2+n1) + nadj, low giving carry flag
- leal 1(%edi), %ebx C n2+1
- movl %ebp, %eax C d
- C
- adcl %edx, %ebx C 1 + high(n2<<32 + m*(n2+n1) + nadj) = q1+1
- sbbl $0, %ebx C q1 if q1+1 overflowed
- mull %ebx
- C
- C
- C
- subl %eax, %esi
- C
- sbbl %edx, %edi C n - (q1+1)*d
- movl %esi, %edi C remainder -> n2
- leal (%ebp,%esi), %edx
- cmovc( %edx, %edi) C n - q1*d if underflow from using q1+1
- sbbl $0, %ebx C q
- movl %ebx, -8(%ecx)
- subl $8, %ecx
- L(integer_none):
- cmpl $0, PARAM_XSIZE
- jne L(fraction_some)
- movl %edi, %eax
- L(fraction_done):
- movl VAR_NORM, %ecx
- L(zero_done):
- movl SAVE_EBP, %ebp
- movl SAVE_EDI, %edi
- movl SAVE_ESI, %esi
- movl SAVE_EBX, %ebx
- addl $STACK_SPACE, %esp
- shrl %cl, %eax
- emms
- ret
- C -----------------------------------------------------------------------------
- C
- C Special case for q1=0xFFFFFFFF, giving q=0xFFFFFFFF meaning the low dword
- C of q*d is simply -d and the remainder n-q*d = n10+d
- L(q1_ff):
- C eax (divisor)
- C ebx (q1+1 == 0)
- C ecx
- C edx
- C esi n10
- C edi n2
- C ebp divisor
- movl VAR_DST, %ecx
- movl VAR_DST_STOP, %edx
- subl $4, %ecx
- psrlq %mm7, %mm0
- leal (%ebp,%esi), %edi C n-q*d remainder -> next n2
- movl %ecx, VAR_DST
- movd %mm0, %esi C next n10
- movl $-1, (%ecx)
- cmpl %ecx, %edx
- jne L(integer_top)
- jmp L(integer_loop_done)
- C -----------------------------------------------------------------------------
- C
- C Being the fractional part, the "source" limbs are all zero, meaning
- C n10=0, n1=0, and hence nadj=0, leading to many instructions eliminated.
- C
- C The loop runs at 15 cycles. The dependent chain is the same as the
- C general case above, but without the n2+n1 stage (due to n1==0), so 15
- C would seem to be the lower bound.
- C
- C A not entirely obvious simplification is that q1+1 never overflows a limb,
- C and so there's no need for the sbbl $0 or jz q1_ff from the general case.
- C q1 is the high word of m*n2+b*n2 and the following shows q1<=b-2 always.
- C rnd() means rounding down to a multiple of d.
- C
- C m*n2 + b*n2 <= m*(d-1) + b*(d-1)
- C = m*d + b*d - m - b
- C = floor((b(b-d)-1)/d)*d + b*d - m - b
- C = rnd(b(b-d)-1) + b*d - m - b
- C = rnd(b(b-d)-1 + b*d) - m - b
- C = rnd(b*b-1) - m - b
- C <= (b-2)*b
- C
- C Unchanged from the general case is that the final quotient limb q can be
- C either q1 or q1+1, and the q1+1 case occurs often. This can be seen from
- C equation 8.4 of the paper which simplifies as follows when n1==0 and
- C n0==0.
- C
- C n-q1*d = (n2*k+q0*d)/b <= d + (d*d-2d)/b
- C
- C As before, the instruction groupings and empty comments show a naive
- C in-order view of the code, which is made a nonsense by out of order
- C execution. There's 17 cycles shown, but it executes at 15.
- C
- C Rotating the store q and remainder->n2 instructions up to the top of the
- C loop gets the run time down from 16 to 15.
- ALIGN(16)
- L(fraction_some):
- C eax
- C ebx
- C ecx
- C edx
- C esi
- C edi carry
- C ebp divisor
- movl PARAM_DST, %esi
- movl VAR_DST_STOP, %ecx C &dst[xsize+2]
- movl %edi, %eax
- subl $8, %ecx C &dst[xsize]
- jmp L(fraction_entry)
- ALIGN(16)
- L(fraction_top):
- C eax n2 carry, then scratch
- C ebx scratch (nadj, q1)
- C ecx dst, decrementing
- C edx scratch
- C esi dst stop point
- C edi (will be n2)
- C ebp divisor
- movl %ebx, (%ecx) C previous q
- movl %eax, %edi C remainder->n2
- L(fraction_entry):
- mull VAR_INVERSE C m*n2
- movl %ebp, %eax C d
- subl $4, %ecx C dst
- leal 1(%edi), %ebx
- C
- C
- C
- C
- addl %edx, %ebx C 1 + high(n2<<32 + m*n2) = q1+1
- mull %ebx C (q1+1)*d
- C
- C
- C
- negl %eax C low of n - (q1+1)*d
- C
- sbbl %edx, %edi C high of n - (q1+1)*d, caring only about carry
- leal (%ebp,%eax), %edx
- cmovc( %edx, %eax) C n - q1*d if underflow from using q1+1
- sbbl $0, %ebx C q
- cmpl %esi, %ecx
- jne L(fraction_top)
- movl %ebx, (%ecx)
- jmp L(fraction_done)
- EPILOGUE()