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2017-03-18 12:02

Repose

Registered: Oct 2010
Posts: 225

sets of add/sub

I want to do a-b+c-d+e-f. I wonder if you can save some cycles somewhere? I looked at the stats of the actual numbers, no hopes there, any pair can carry/borrow with about 50% chance.

First I'll look at adding 3x 16bit adds and generalizing

Standard version
-----------------
We will keep the running sum in a register
n1,n2,n3 - numbers to add
r, r+1,... - the sum

;assume n1 stored in r
2 clc
3 lda r
3 adc n2
2 tax;n2+n1 save low byte in X
2 lda #0;small optimization first time when you know r+1 is 0.
2 adc #0; basically convert carry to 1
2 tay;16c save high byte in Y
;-- repeat for n>=3 (n is number of numbers to add)
;clc not needed
2 txa
3 adc n3
3 sta r;r+n3=n1+n2+n3 or tax until final add
2 tya
2 adc #0
3 sta r+1;16+15=31 or tay until final add
rts

In general,
Time is 16+(n-2)*13+2=18+(n-2)*13

n time, cycles
2 18 =18
3 18+13 =31
4 18+13+13 =44

Fast Version 1
--------------

2 clc

3 lda n2
3 adc r;n2+n1
2 bcs s1;2+9=11 if taken
3 adc n3
3 sta r
2 lda #0
2 adc #0
3 sta r+1;2+8+13=23 if no carry in n1+n2
rts;or bcc s2 =26
;11c
2 s1 clc;there was a carry in first add, A=n1+n2
3 adc n3
3 sta r
2 lda #1;representing the carry from n1+n2>255
2 adc #0
3 sta r+1;11+15=26 if carry in n1+n2
s2 rts;23/26, average 24.5, saving 6.5 cycles from standard (31c)
realistically always 26 or saving 5 cycles

Conclusions
Adding 3 numbers, I went from 31 to 26 cycles.

The statistics of adding 3 numbers drawn from multiplies
--------------------------------------------------------

In a partial sums routine, only some bytes of the product are added

case carries % total
l l h 2908442244 67.7 4294967296
h h l 2044355014 47.6 4294967296

Conclusions
High bytes of products are always less likely to create a carry when added to other numbers. The differences are so little that it hardly makes any difference.

ps took a while to simulate 4 billion multiplies and gather stats. I'd hate to figure out stats for larger mults, would have to actually think through it instead. But I can tell you they change quite a bit, in some cases there's hardly two carries in a row for example.

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2017-04-13 21:01

Repose

Registered: Oct 2010
Posts: 225

Ok quick post yes I realized I need to keep the last carry which is why I did adc #0 before the subtract, I think we're on the page.

2017-04-14 15:54

ChristopherJam

Registered: Aug 2004
Posts: 1409

So, for the NxN multiply, as Repose pointed out it's only worthwhile skipping the CLCs for columns with a reasonable number of addends. Coincidentally, all of those columns also require the addition of a non-zero offset, to account for my excess $4000 representation of the g() table.

I end each of those columns with the following sequence:
sbc id+($ff-Kn),x
sta res+n
txa

where Kn is the additional constant for that column (must be at least as high as the maximum value X can reach, lest the indexing could cross the page boundary)

I've just run an exhaustive test in VICE, comparing a more traditionally computed reference value with the optimised code above, and it matches for all 16842752 valid sets of inputs. I've included the test code below, eliding the loop iterator and display routines for brevity.

; test parameters are
;  A  (00..ff)
;  C  (00..01)
;  X  (00..ff)
;  K  (X..ff)   ; routine requires that X+(0xff-K)<= 0xff,  ie X<=K
;
; result should be A+C+K+255*X

;-----------------------------
;  calculate reference value
;-----------------------------

    lda tC
    lsr
    lda tA
    adc tK
    sta ref+0
    lda tX
    adc#0
    sta ref+1  ; ref is now A+C+K+256*X

    sec
    lda ref+0
    sbc tX
    sta ref+0
    lda ref+1
    sbc#0
    sta ref+1   ; ref is now A+C+K+255*X
;-----------------------------
; prepare registers for test
;-----------------------------
    lda tK
    eor#$ff
    sta rn+1
    lda tC
    lsr
    lda tA
    ldx tX
;-----------------------------
; run test code
;-----------------------------
rn  sbc id+$3f,x
    sta res+0
    txa
    adc#0
    sta res+1

;-----------------------------
; compare result to reference value
;-----------------------------
    lda res+0
    cmp ref+0
    bne fail
    lda res+1
    cmp ref+1
    bne fail

2017-04-15 06:39

Repose

Registered: Oct 2010
Posts: 225

I got it to work now thanks :)
And good news, I found something even faster now! It always beats the clc version, even for 3 adds. I might be able to beat whatever you have :)

See if you can generate 16x16 your way, that will be my first working example for now.

2017-04-15 06:42

ChristopherJam

Registered: Aug 2004
Posts: 1409

And now proof that the adc:bcc:inx sequence retains sufficient information to calculate the sum of the column, regardless of when carries occur.

If we define T to be equal to the current value of A+C+255*X,
it's possible to prove that after each adc:bcc:inx sequence, T increases by the value added.

; before:
; T=A+C+255*X, so A=T-C-255*X

then we execute

    adc #V
    bcc *+3
    inx

Let C', X' and A' be the values of carry, accumulator and X after running the code sequence.

Now, either the add didn't set C,
in which case

  C'=0
  X'=X
  A'=A+V+C
    =(T-C-255*X)+V+C
    =T+V-255*X

  T'=A'+C'+255*X'
    =(T+V-255*X)+0+255*X
    =T+V

or, the add did set C, in which case

  C'=1
  X'=X+1          # as the increment happens
  A'=A+V+C-256    # as the add has overflowed into C
    =(T-C-255*X)+V+C-256
    =T+V-255*X-256

  T'=A'+C'+255*X'
    =(T+V-255*X-256)+1+255*(X+1)
    = T+V-255*X-255+255*X+255
    = T+V

Either way, T has increased by the correct amount

2017-04-15 06:48

Repose

Registered: Oct 2010
Posts: 225

Nice proof! Couldn't have said it better myself.

2017-04-15 06:53

Repose

Registered: Oct 2010
Posts: 225

So my adds are 14+9*(n-1) for column 0, 11+9*(n-1) for the middle columns, and something decent for the last column (overhead 14 or less). The point being it always beats 12+10*(n-1) for the clc version.

2017-04-15 09:28

Repose

Registered: Oct 2010
Posts: 225

I figured this out my way, and why the new fix is better and why the old version failed.

Follow along with the registers and code in two cases:

	 A C  X r1 r0
ldx #0	     00	
sec	   1 
lda #ff ff  
adc #2	02 1
bcc s1
inx	     01
adc #ff	02 1
bcc s2
inx	     02
stx r1	        02
sbc r1	00 1
sta r0	           00
txa	02
sbc #0  02 1
sta r1	        02

	 A C  X r1 r0
ldx #0	     00	
sec	   1 
lda #ff ff  
adc #1	01 1
bcc s1
inx	     01
adc #ff	01 1
bcc s2
inx	     02
stx r1	        02
sbc r1	ff 0
sta r0	           ff
txa	02
sbc #0	01 1
sta r1	        01

Essentially, if carry is 0 after the low bytes, decrement the high byte. It's that fix to the high byte which was missing in my first version. I didn't realize how to test for that particular case, but all I needed was hiding in the Carry after the low bytes. The workthrough above is assuming you are only adding one column of numbers, if you continued to another column, you'd end it differently:

s2 sbc id,x
sta r0
;column 2
txa
sbc #0;correction to high byte
ldx #0
adc n1h;add high bytes

2017-04-16 15:49

ChristopherJam

Registered: Aug 2004
Posts: 1409

Ok, I think I follow how that last one works now.

A slight optimisation first, going to replace

    stx r1
    sbc r1
with
    sbc id,x

It's two cycles faster, only needs as many bytes of table as you have INXs (so, under a dozen), and makes it easier to
present the proof in the rest of this comment. Don't need to touch r1, as it gets overwritten later anyway.

Now, again defining T to be A+C+255*X

    ldx #0
a)  sec
     ;  initialise T to 1

    lda n1
     ;        now T is A+C+255*X = n1+1+255*0 = n1+1

    adc n2
    bcc *+3
    inx
     ;        now T is 1+n1+n2, cf proof in comment about the adc:bcc:inx sequence earlier in this thread

    adc n3
    bcc *+3
    inx
     ;        now T is 1+n1+n2+n3

  ; so all that's required now is adding A+C+255*X-1 and storing them

b)  sbc id,x  ; now sbc nn is the same as adc $ff-nn, so here we are adding $ff and subtracting X
    sta r0

    txa
c)  sbc #0    ; again, equivalent to adc $ff-0, so here we are adding x, $ff, and any carry from the low byte
    sta r1    ; ie, adding 256*X and $ff00 to the total in r1:r0

So, between points (a) (b) and (c), you've added 0001, 00ff, and ff00 to your total, which should result in carry being set
at the end of the routine, but no other effect on the result. Lines (b) and (c) contribute 255*x, leaving the sum in r1:r0

Usefully, the set carry flag is just what's needed for the start of the next column. Overall, nice work!

2017-04-16 16:28

Fresh

Registered: Jan 2005
Posts: 101

Looks like you've already reached your target, just want to share a similar approach that would avoid messing with the carry.
This would clearly need to be executed on ZP to be really efficient but, in that case, the overhead would be (a bit) less than yours.

	lax eorval+1
	ldy #$00
val1:
	axs #$00
	
val2:
	axs #$00
	bcs val3
	iny
val3:
	axs #$00
	bcs val4
	iny
val4:
	axs #$00
	bcs storehi
	iny	
	
storehi:
	sty r1
	txa
eorval:
	eor #$ff
	sta r0

2017-04-17 02:20

ChristopherJam

Registered: Aug 2004
Posts: 1409

It's certainly a fast way of adding a bunch of constant values, but I fear any speed advantage would more than counterbalanced by having to fetch the addends in an earlier code segment and writing them into the routine.

There are no registers left for addressing (eg) table entries.

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