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+--- Thread: (42, all flavours) Integer Division - how? (/thread-16039.html)
(42, all flavours) Integer Division - how? - Werner - 12-11-2020 10:48 AM
Anybody know how to implement integer division in a fast and reliable way?
And, no, / IP is not the answer..
Illustrating my point with a hypothetical 2-digit calculator, then:
79 DIV 40 = 1
200 DIV 3 = 66
890 DIV 99 = 8
2300 DIV 40 = 57
the goal is to be able to split Y=Q*X+R when possible, i.e. up to Cc00 = Qq*Xx + Rr with Cc and Rr < Xx, and Cc00 DIV Xx should give Qq (all single letters are half-length integers, or single digits in the case of the 2-digit calculator)
42S equivalents
(4e11+39) DIV 40 = 1e10 (/ IP = 1e10+1)
2e12 DIV 3 = 666666666666 (666666666667)
5e23 DIV (1e12-1) = 5e11 (5e11+1)
(2e23+3e12) DIV 4e11 = 5e11+7 (5e11+8) (round-to-even, and a particularly difficult one to get right)
Free42 equivalents
(4e33+39) DIV 40 = 1e32
2e34 DIV 3 = 666...666
I CAN do it, but it's not remotely pretty.
It would be a welcome addition to Free42 ;-)
Cheers, Werner
RE: (42, all flavours) Integer Division - how? - Gerson W. Barbosa - 12-11-2020 03:12 PM
Code:
00 { 17-Byte Prgm }
01▸LBL "INT÷"
02 RCL ST Y
03 X<>Y
04 MOD
05 STO- ST Y
06 X<> ST L
07 ÷
08 ENDThis gets only your first example right on the HP-42S. So does INT÷ on the hp 33s.
RE: (42, all flavours) Integer Division - how? - Allen - 12-11-2020 04:47 PM
(12-11-2020 10:48 AM)Werner Wrote: Anybody know how to implement integer division in a fast and reliable way?
And, no, / IP is not the answer..
Illustrating my point with a hypothetical 2-digit calculator, then:
79 DIV 40 = 1
200 DIV 3 = 66
890 DIV 99 = 8
2300 DIV 40 = 57
Have you tried BASE÷ ?
RE: (42, all flavours) Integer Division - how? - Werner - 12-11-2020 04:56 PM
Base/ doesn’t work for 7.2 DIV 5 for instance, but also not for the Free42 examples, as it is limited to 36 bits in the 42S and 64 in Free42.
Werner
RE: (42, all flavours) Integer Division - how? - Albert Chan - 12-11-2020 07:32 PM
(12-11-2020 10:48 AM)Werner Wrote: Anybody know how to implement integer division in a fast and reliable way?
I don't know if this is fast, but FMA should work.
10 DEF FND(A,B)
20 Q=FLOOR(A/B)
30 B1=B*1000001 @ B1=B+B1-B1 @ B=B-B1
40 Q1=Q*1000001 @ Q1=Q+Q1-Q1 @ Q=Q-Q1
50 FND=FLOOR((A-Q1*B1-Q*B1-B*Q1-B*Q)/(B+B1))+Q+Q1
60 END DEF
>RUN
>FND(4E11+39,40)
10000000000
>FND(2E12,3)
666666666666
>FND(5E23,1E12-1)
500000000000
>FND(2E23+3E12,4E11)
500000000007
RE: (42, all flavours) Integer Division - how? - Werner - 12-11-2020 08:04 PM
Yes that looks like Dekker’s double precision routines. That works, but in RPN it doesn’t look quite so elegant. I was hoping there would be a shorter, simpler way.
Werner
RE: (42, all flavours) Integer Division - how? - Gjermund Skailand - 12-11-2020 09:28 PM
Hi.
Here is one version fulfilling the test examples on the Free42.
It uses the DOT function to force calculation with all digits prior to rounding.
I am not sure how reliable it actually is.
It uses one extra stack level
00 {28-Byte Prgm }
01 LBL "IDIV2"
02 X<>Y
03 -0.5
04 RCLX ST Z
05 COMPLEX
06 X<>Y
07 1/X
08 ENTER
09 COMPLEX
10 DOT
11 IP
12 END
Best regards
Gjermund
RE: (42, all flavours) Integer Division - how? - Werner - 12-11-2020 09:46 PM
Never too old to learn! I didn’t know DOT worked on complex numbers, too. I will have to take a look at this, but one thing’s for sure: DOT does not use extended precision, as it does in the 42S.
Werner
RE: (42, all flavours) Integer Division - how? - Werner - 12-12-2020 10:36 AM
Too bad, Gjermund: on Free42,
4e33+6 DIV 4 should be 1e33+1, and your routine returns 1e33+2
But it does work on a 42S, where DOT uses 15 intermediate digits:
4e11+6 DIV 4 = 1e11 + 1
well, - for this particular example. But since it basically does Y/X - 1/2,
it fails for eg 7 DIV 3
Nevertheless, I learned something and it will be put to good use!
Werner
RE: (42, all flavours) Integer Division - how? - Gjermund Skailand - 12-12-2020 09:31 PM
Yeah, it turned out to be a bad idea. I also found out that the HP50g and the 42S behaves differently for the DOT and CROSS when using complex numbers for 2D. HP50g will not allow it.
best regards
Gjermund
RE: (42, all flavours) Integer Division - how? - Albert Chan - 12-13-2020 01:24 AM
There is the "cheating" way, by temporarily setting the rounding mode
I coded mathx.setround() for this purpose
PHP Code:
require 'mathx'
function idiv1(a,b)
mathx.setround(-1) -- round downwards
a = mathx.rint(a/b)
mathx.setround(0) -- restore round-to-nearest
return a
end
Another way is to correct the quotient of a and b (assumed both integers)
Here, we assumed q, Q may have errors of ±1
a = q*b + r = q*c - q + r , where c = b+1
a = Q*c + R
0 = (q-Q)*c - q + r - R → -q + r - R ≡ 0 (mod c)
Since r and R can be calculated with MOD, we can correct for q
Assuming |q| < 2^53, this is the code:
PHP Code:
function idiv2(a,b) -- assumed b > 0
local q, r, c = floor(a/b), a%b, b+1
r = (r+1 - q%c - a%c) % c - 1
return q + r
end
lua> a = 0x1p72
lua> for b=1e6+1, 1e6+9 do
: q = idiv1(a,b) -- reference
: print(b, q - floor(a/b), q - idiv2(a,b))
: end
1000001 -1 0
1000002 -1 0
1000003 -1 0
1000004 0 0
1000005 0 0
1000006 0 0
1000007 0 0
1000008 0 0
1000009 -1 0
RE: (42, all flavours) Integer Division - how? - Werner - 12-13-2020 08:12 AM
Yes. YES!
Thanks a million, Albert, this is what I was looking for!
(the first part, cheating, doesn't apply to 41,42,Free42 of course)
Werner
RE: (42, all flavours) Integer Division - how? - Werner - 12-13-2020 12:09 PM
All that is left is to turn it in a routine..First try:
Code:
{ 38-Byte Prgm } @ X Y Z T
LBL "DIV" @ b a
RCL ST Y
RCL ST Y
/
IP @ q b a a
Rv
MOD @ b r a q q
X<>Y
1
STO+ ST Z
RCL+ ST L @ c a r+1 q
MOD
STO- ST Y @ c R r+1-R q q
X<> ST T
LASTX
MOD @ c q%c r+1-R q q
STO- ST Y
X<> ST L @ c r' q q
MOD
+
DSE ST X
ENDI can recover b too, if needed, but that's it.
Werner
RE: (42, all flavours) Integer Division - how? - Albert Chan - 12-13-2020 01:18 PM
(12-13-2020 01:24 AM)Albert Chan Wrote: return q + r
We had assumed q never overflow, which results in |r| ≤ 1
However, if q already overflowed, calculated r is basically garbage.
We should return q + sign(r), to limit the damage.
In other words, if q overflow calculator precision, don't correct the quotient.
RE: (42, all flavours) Integer Division - how? - Albert Chan - 12-13-2020 04:21 PM
(12-13-2020 01:18 PM)Albert Chan Wrote: We had assumed q never overflow, which results in |r| ≤ 1
Assuming q is correctly rounded (mode round-to-nearest), r = 0 or -1
The code is simplified to correct, or not correct.
PHP Code:
function idiv3(a,b) -- assumed b > 0
local q, c = floor(a/b), b+1
a, b = a%b - a%c, q%c -- a-b = [2-2c, c-2]
if a==b or a==b-c then return q end
return q-1
end
RE: (42, all flavours) Integer Division - how? - Albert Chan - 12-13-2020 06:41 PM
(12-13-2020 04:21 PM)Albert Chan Wrote: Assuming q is correctly rounded (mode round-to-nearest), r = 0 or -1
This allowed optimization of my FMA (Fused-Multiply-Add) code.
Without using MOD, this is the fastest of all.
Bonus: it removed the integer arguments requirement.
PHP Code:
function idiv4(a,b) -- asuumed b > 0
local q = floor(a/b)
local b1 = b * (0x1p27+1)
local q1 = q * (0x1p27+1)
b1 = b + b1 - b1 -- hi bits
q1 = q + q1 - q1
local b0 = b - b1 -- lo bits
local q0 = q - q1
if a - q1*b1 - q1*b0 - q0*b1 >= q0*b0 then return q end
return q - 1
end
It would be nice if Free42 exposed FMA(a,b,c) to the user ...
RE: (42, all flavours) Integer Division - how? - Werner - 12-13-2020 07:19 PM
(12-13-2020 04:21 PM)Albert Chan Wrote: Assuming q is correctly rounded (mode round-to-nearest), r = 0 or -1
Only for a and b positive?
For a negative, it doesn’t work, eg
(12 digits calc)
a=-4e11-6
b=4
DIV returns -1e11-2
Werner
RE: (42, all flavours) Integer Division - how? - Thomas Okken - 12-13-2020 07:43 PM
(12-13-2020 06:41 PM)Albert Chan Wrote: It would be nice if Free42 exposed FMA(a,b,c) to the user ...
Noted...
RE: (42, all flavours) Integer Division - how? - Albert Chan - 12-13-2020 08:51 PM
(12-13-2020 07:19 PM)Werner Wrote:(12-13-2020 04:21 PM)Albert Chan Wrote: Assuming q is correctly rounded (mode round-to-nearest), r = 0 or -1Only for a and b positive?
For a negative, it doesn’t work, eg
(12 digits calc)
a=-4e11-6
b=4
DIV returns -1e11-2
I defined IDIV matching MOD behavior: a = b * IDIV(a,b) + MOD(a,b)
Free42, binary and decimal, uses floor-mod: (a MOD b) has sign of b
To match it, IDIV(a,b) = floor(a/b)
So, above DIV is correct: floor((-4e11 - 6)/4) = floor(-1e11 - 1.5) = -1e11 - 2
Python also define it this way, see Why Python Integer Division Floors
>>> a, b = -4*10**11-6, 4
>>> q, r = a//b, a%b
>>> print q, r, q*b+r
-100000000002 2 -400000000006
RE: (42, all flavours) Integer Division - how? - Werner - 12-13-2020 08:59 PM
Ah yes, indeed!
Werner