(42, all flavours) Integer Division  how?

12112020, 10:48 AM
Post: #1




(42, all flavours) Integer Division  how?
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 2digit 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 halflength integers, or single digits in the case of the 2digit calculator) 42S equivalents (4e11+39) DIV 40 = 1e10 (/ IP = 1e10+1) 2e12 DIV 3 = 666666666666 (666666666667) 5e23 DIV (1e121) = 5e11 (5e11+1) (2e23+3e12) DIV 4e11 = 5e11+7 (5e11+8) (roundtoeven, 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 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

12112020, 03:12 PM
Post: #2




RE: (42, all flavours) Integer Division  how?
Code:
This gets only your first example right on the HP42S. So does INT÷ on the hp 33s. 

12112020, 04:47 PM
Post: #3




RE: (42, all flavours) Integer Division  how?
(12112020 10:48 AM)Werner Wrote: Anybody know how to implement integer division in a fast and reliable way? Have you tried BASE÷ ? 17bii  32s  32sii  41c  41cv  41cx  42s  48g  48g+  48gx  50g  30b 

12112020, 04:56 PM
Post: #4




RE: (42, all flavours) Integer Division  how?
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 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

12112020, 07:32 PM
Post: #5




RE: (42, all flavours) Integer Division  how?
(12112020 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+B1B1 @ B=BB1 40 Q1=Q*1000001 @ Q1=Q+Q1Q1 @ Q=QQ1 50 FND=FLOOR((AQ1*B1Q*B1B*Q1B*Q)/(B+B1))+Q+Q1 60 END DEF >RUN >FND(4E11+39,40) 10000000000 >FND(2E12,3) 666666666666 >FND(5E23,1E121) 500000000000 >FND(2E23+3E12,4E11) 500000000007 

12112020, 08:04 PM
Post: #6




RE: (42, all flavours) Integer Division  how?
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 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

12112020, 09:28 PM
Post: #7




RE: (42, all flavours) Integer Division  how?
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 {28Byte 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 

12112020, 09:46 PM
Post: #8




RE: (42, all flavours) Integer Division  how?
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 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

12122020, 10:36 AM
Post: #9




RE: (42, all flavours) Integer Division  how?
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 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

12122020, 09:31 PM
Post: #10




RE: (42, all flavours) Integer Division  how?
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 

12132020, 01:24 AM
Post: #11




RE: (42, all flavours) Integer Division  how?
There is the "cheating" way, by temporarily setting the rounding mode
I coded mathx.setround() for this purpose PHP Code: require 'mathx' 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 = (qQ)*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 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 

12132020, 08:12 AM
Post: #12




RE: (42, all flavours) Integer Division  how?
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 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

12132020, 12:09 PM
Post: #13




RE: (42, all flavours) Integer Division  how?
All that is left is to turn it in a routine..First try:
Code: { 38Byte Prgm } @ X Y Z T I can recover b too, if needed, but that's it. Werner 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

12132020, 01:18 PM
Post: #14




RE: (42, all flavours) Integer Division  how?
(12132020 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. 

12132020, 04:21 PM
(This post was last modified: 12132020 04:22 PM by Albert Chan.)
Post: #15




RE: (42, all flavours) Integer Division  how?
(12132020 01:18 PM)Albert Chan Wrote: We had assumed q never overflow, which results in r ≤ 1 Assuming q is correctly rounded (mode roundtonearest), r = 0 or 1 The code is simplified to correct, or not correct. PHP Code: function idiv3(a,b)  assumed b > 0 

12132020, 06:41 PM
Post: #16




RE: (42, all flavours) Integer Division  how?
(12132020 04:21 PM)Albert Chan Wrote: Assuming q is correctly rounded (mode roundtonearest), r = 0 or 1 This allowed optimization of my FMA (FusedMultiplyAdd) 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 It would be nice if Free42 exposed FMA(a,b,c) to the user ... 

12132020, 07:19 PM
Post: #17




RE: (42, all flavours) Integer Division  how?
(12132020 04:21 PM)Albert Chan Wrote: Assuming q is correctly rounded (mode roundtonearest), r = 0 or 1 Only for a and b positive? For a negative, it doesn’t work, eg (12 digits calc) a=4e116 b=4 DIV returns 1e112 Werner 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

12132020, 07:43 PM
Post: #18




RE: (42, all flavours) Integer Division  how?  
12132020, 08:51 PM
Post: #19




RE: (42, all flavours) Integer Division  how?
(12132020 07:19 PM)Werner Wrote:(12132020 04:21 PM)Albert Chan Wrote: Assuming q is correctly rounded (mode roundtonearest), r = 0 or 1Only for a and b positive? I defined IDIV matching MOD behavior: a = b * IDIV(a,b) + MOD(a,b) Free42, binary and decimal, uses floormod: (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**116, 4 >>> q, r = a//b, a%b >>> print q, r, q*b+r 100000000002 2 400000000006 

12132020, 08:59 PM
Post: #20




RE: (42, all flavours) Integer Division  how?
Ah yes, indeed!
Werner 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

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