Mill Computing, Inc. › Forums › The Mill › Architecture › Arbitrary precission integer arithmetic hardware help

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Hi, many programming languages (Python int, Erlang integer, Haskell Integer, Mathematica, Lisp, Smalltalk, and some PL/SQL implementations), does support naively arbitrary precision arithmetic on integers in the language, and these are base types. I.e. What you get from ‘5’, or ‘+’ is always arbitrary precission arithmetic integer. And in some cases it is essentially impossible to use fixed size integers in the language. Implementations of these languages will try to use native integers for small values, but in general every single operation will add additional branches, checks, use some registers, etc.

Of course in general case, big values will be handled by a library call, or for some ops inlined complex code. However, most of the time the implementation of the representation and inlined calls, handle the usual case of small arguments inline, and only switch to full machinery if the operands are big or marked with some flag. I was wondering if it would be beneficial to make a version of add/sub/mul/div/mod, that merges these concepts, and automatically do either real add/sub/mul/div/mod if some bit in the operand is not set, or call a library function if the bit is set, or the result would overflow and clobber the bit.

The fallback code, could be a code pointer in the opcode, or throw hardware exception, with exception vector handler under control of the user space software, so it can just run it transparently.

My guess, 64 bit value, would have top most bit indicating if it is big or small value. If both input operands, have this bit zero, then perform operand, and make sure there was no 64-bit overflow or overflow into 63 bit. If overflow happens or any of the bits is set, call library function with this two operands. Library function will then interpret the bitpatterns as pointers to heap allocated data for representing big integers, and is allowed to return both small native, and big (heap allocated) 64-bit data back into belt on return. The unpredictable memory allocation and memory deallocation doesn’t need to be dealt with, because all mentioned languages use automatic garbage collection mechanisms.

Or maybe, it can already be dealt with in single instruction in some way?

My wild guess:

myAdd(b0, b1): add(b0, b1), overflow(), test(b0, 63), test(b1, 63), or(), neq(), brtr(“AddFallback”, b0 b1);

?

Maybe some pick magic, or other trick to actually ignore add result (not put it on belt), if the branch is taken.

Similarly it would be nice, if one can use such integers in conversions to float (i.e. do normal conversion if it is marked as small, or call a function if it is not), and for indexing for loads (i.e. produce a NaR or hardware exception when it is not marked as small, because accessing so big offset is probably a bug anyway, in fact if the most significant bit is set, this should work almost transparently, because using so big index will almost always result in access to invalid location).

If I remember correctly from one of the talks, every math op had 4 versions:

1. saturating: 250 + 30 = 255 (8 bits);

1. excepting: 250 + 30 = 💣;

1. wrap-around: 250 + 30 = 24;

1. expanding: 250 + 30 = 280 (16×8 bits => 2×8×8 bits);I believe arbitrary-precision math can be easily implemented using the excepting versions (just handle the special case in an interrupt).

— nachokb

Given expanding arithmetic and splitting large words into 2 smaller words, arbitrary precision arithmetic can be implemented without too bad efficiency. That’s as much as you get on x86, and I believe x64 architectures. With the vector operations of higher end Mill processors, it gets better. Not sure what else you might want for such numeric representation.

I am not talking about implementing arbitrary arithmetic. This is easy. I am talking about optimization for small values. This can’t be done on Mill right now AFAIK (even when using expecting or widening operations).

Hi nachokb.

I am aware of expecting integer arithmetic in Mill. Unfortunately it is NOT enough to implement arbitrary precision arithmetic, especially with the properties I mentioned (efficient handling and encoding of small signed integers). Please prove me wrong, if you think otherwise.

Contrary to what everyone else is saying, this isn’t possible on the Mill as we know it.

However, it seems likely that it can be made to be possible. Unfortunately, it may be more trouble than it’s worth…maybe it’ll be in the GenASM but never actually implemented like Decimal. However, here’s my take:

- Have a new type of data that can be operated on aside from float, integer, and pointer: the int60. The int60 is like a pointer in that ops on it are only specified for pointer-size.
- To use int60, a SpecReg has to be set indicating one of the three-bit patterns in the reserved bits as the int60 pattern (default 000).
- SpecRegs also set the handlers for int60 overflow ops. There are ADD, SUB, MUL, DIV, MOD, DIVMOD, AND, OR, NOT, XOR, SHL, SHR, TOINT (with output width, overflow behavior, and output signedness arguments), FROMINT (with input signedness arguments), and DROP (explicitly-called destructor op).
- Basically, treat as 60-bit int when the flag bits are the three-bit “this is actually an integer” pattern and call routines as a pointer when otherwise.

Please accept my apologies, folks. Somehow a recent update of the Forum software removed me from the notify list, so I was unaware of the discussion. I’ll be working through the backlog promptly.

IvanEfficient representation of arbitrary precision math is a specific example of the more general problem of runtime typing, which is a problem we have long had on our backlog.

Of the various suggestions on this thread:

1) math ops do not trap, they fault: when they happen, you’re not coming back. Think C++ “throw”.

2) the “excepting” forms only fault on overflow, and while they could be used to catch a wide add for example, they then would not catch a subtract.

3) NaRs do not fault on speculable operations, they just pass through, and all the math ops are speculable.

4) We could indeed define a new “APinteger” domain. Currently the domains are signed and unsigned integer, signed and unsigned fixed point, logical, boolean, binary and decimal float, and some pseudos for various unique ops. The new domain would add the set of ops to the ISA, which would have entropy consequences in the encoding. However, implementing the behavior via traps is real problematic on a Mill because of the wide issue – a single instruction could have a ton of distinct traps. We currently do not have any result-replacing traps defined; all the traps are for things like page faults which are fundamentally asynchronous to the core and belt. Trap-based programming could be done, but it would thoroughly muck up the execution engine, and we haven’t and probably won’t put such a thing in.

5) AP integer is a special case; the Mill tends to do the general case of things. The general case is an efficient runtime-typing mechanism that is type-system agnostic.We have done some thinking about how to do runtime-typing in the Mill environment, but don’t have anything yet ready for publication, or even for patent filing. I wish I could be more informative; sorry.

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