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

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 END

This gets only your first example right on the HP-42S. So does INT÷ on the hp 33s.

(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÷ ?

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

(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

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

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

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

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

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

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

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

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

END

I can recover b too, if needed, but that's it.

Werner

(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.

(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

(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 ...

(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

(12-13-2020 06:41 PM)Albert Chan Wrote: [ -> ]It would be nice if Free42 exposed FMA(a,b,c) to the user ...

Noted...

(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 -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

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