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John Gustafson’s crusade to replace floating point with something better

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Re: John Gustafson’s crusade to replace floating point with something better

#21

Post by dnautics »

fhars wrote: post #17
savant_penguin wrote: post #16

"(even the same computation on the same system can produce different results for floats)" wait...what? Is this real?

Not really. I suspect they are referring to the original x87 FP functions with their odd 80 bit FP stack and operations, where the exact result depends on when intermediate results are written to RAM and when they are kept in the FP stack, which might depend on intransparent things like compiler optimisations.

it doesn't happen much in practice, but the official IEEE spec is silent about it, so you can toss it into whatever bin contains 'undefined' behaviour of C.

Re: John Gustafson’s crusade to replace floating point with something better

#22

Post by dnautics »

mrtnmcc wrote: post #7

How does the implementation of an FPU for this compare? I thought the existing IEEE 754 floating point standard was focused on reduced complexity of addition and multiplication hardware. This seems more complicated.

having implemented it, with unoptimized verilog (ok, i wrote a verilog generator to generate it), it requires about 30% fewer LUTs on a FPGA relative to berkeley hardfloat implementation.

Re: John Gustafson’s crusade to replace floating point with something better

#23

Post by dnautics »

jchw wrote: post #5

Obviously for any 32 bit value there’s only 2^32 possible values. So logically, a better number format is all about the distribution of values and reducing redundancy. It sounds like the basic idea here is that rather than a fixed amount of significant figures, you get more precision in the middle. Meanwhile, you can also get larger exponents. Is that right?

that's the general idea. There is an additional tradeoff - error analysis in non-extreme (aka subnormal) cases is easier for IEEE than posits.

The quire stuff has been thrown in there, too, because while we're boiling the ocean, we might as well encourage people to "do the right thing". think of the quire as a really large accumulator cache for things like matrix vector multiplication or tensor outer products, which I hear, AI uses a lot.

Re: John Gustafson’s crusade to replace floating point with something better

#24

Post by dragontamer »

savant_penguin wrote: post #16

"(even the same computation on the same system can produce different results for floats)" wait...what? Is this real?

Yes and no.

1. Yes -- On old x86 systems, the x87 registers were 80-bits and the bottom bits were undefined. You could theoretically have a 1-bit difference on some results depending on what the bottom 26 bits (that were undefined, because you had 64-bit floats most of the time, even though the machine did things 80-bits at a time).

2. No -- Modern x86 systems use SSE registers, which are 64-bits. There are a whole slew of configuration options, but if Windows / Linux does their job, you should have the same rounding-errors across your program (unless you manually change the rounding options, but that's your own fault if you go that route).

3. Kinda yes -- Floating point operations are NON-associative. (A+B)+C does NOT equal A+(B+C). The best example is Python:

>>> (1+1)+2.053 9007199254740994.0 >>> 1+(1+2.053) 9007199254740992.0

253 is the first value which starts to "round off" 1.0 in double-precision. So (1+2.053) == 2.053, because the value was rounded off.

Due to #3, even if you did everything correctly (ie: set the rounding flags so they were consistent), if your arithmetic happened in slightly different orders (very common in network-games, where different players may have their updates in slightly different orders), you end up with a 1-bit error between clients... which completely borks your simulation.

Because #3 is nonintuitive, many people think that you get different results when using floats. But its simply due to non-associativity.

Re: John Gustafson’s crusade to replace floating point with something better

#25

Post by dragontamer »

Iv wrote: post #18
fhars wrote: post #17

Earlier quoted context omitted.

Not really. I suspect they are referring to the original x87 FP functions with their odd 80 bit FP stack and operations, where the exact result depends on when intermediate results are written to RAM and when they are kept in the FP stack, which might depend on intransparent things like compiler optimisations.

They could refer to parallel systems, most likely GPUs, where it is common to have slightly different results even on the same operation, on the same system. This is due to the fact that several pipelines will do calculations in parallel and depending on which pipeline ends first (which can depend on many factors like current temperature) the sums can happen in a different order, leading to different rounding results…

> They could refer to parallel systems, most likely GPUs, where it is common to have slightly different results even on the same operation.

Parallel systems have great difficulty in defining an order of operations. Floating-point numbers are non-associative, so order matters for bitwise compatibility.

With that being said, you CAN define an order, and it CAN be done in parallel. The parallel-prefix sum is a fully defined order of operations for example, and if you sort all numbers (from smallest to largest), you'll get a more accurate result.

https://en.wikipedia.org/wiki/Prefix_sum#Algorithm_1:_Shorte...

The + operations always happen in the same order, even on a GPU. So you should get the same results every time with floating-point math. But it is non-intuitive for many people to work with non-associative floating-point math.

-----------

Note that writing a sequential algorithm with fully defined order is still grossly difficult! IMO, if anyone wants a bit-accurate simulation, use 64-bit integers instead.

If you absolutely must use floats, then define a proper order of all operations, and include sorting the numbers (!!) to force a fully defined order as much as possible. You may need to sort your pool of numbers many, many times per calculation. But that's what you have to do to get a defined ordering.

Re: John Gustafson’s crusade to replace floating point with something better

#26

Post by llukas »

shaklee3 wrote: post #20
Iv wrote: post #18

Earlier quoted context omitted.

They could refer to parallel systems, most likely GPUs, where it is common to have slightly different results even on the same operation, on the same system. This is due to the fact that several pipelines will do calculations in parallel and depending on which pipeline ends first (which can depend on many factors like current temperature) the sums can happen in a different order, leading to different rounding results…

I've never heard of that happening on a GPU. Can you give an example?

You could have this kind of behavior if you use atomic functions - but you need to explicitly opt-in actually using those and it is not limited to GPU.

Re: John Gustafson’s crusade to replace floating point with something better

#27

Post by deathanatos »

dragontamer wrote: post #24
savant_penguin wrote: post #16

"(even the same computation on the same system can produce different results for floats)" wait...what? Is this real?

Yes and no. 1. Yes -- On old x86 systems, the x87 registers were 80-bits and the bottom bits were undefined. You could theoretically have a 1-bit difference on some results depending on what the bottom 26 bits (that were undefined, because you had 64-bit floats most of the time, even though the machine did things 80-bits at a time). 2. No -- Modern x86 systems use SSE registers, which are 64-bits. There are a whole s…

In case you're like me, reading this comment and thinking "how the heck is 1 plus 1 plus 2-ish like 90 gazillion?", it's:

  >>> (1+1)+2.0**53
  9007199254740994.0
  >>> 1+(1+2.0**53)
  9007199254740992.0
The exponent operator (double star) got transformed into italics apparently.

Re: John Gustafson’s crusade to replace floating point with something better

#28

Post by martinpw »

dragontamer wrote: post #24
savant_penguin wrote: post #16

"(even the same computation on the same system can produce different results for floats)" wait...what? Is this real?

Yes and no. 1. Yes -- On old x86 systems, the x87 registers were 80-bits and the bottom bits were undefined. You could theoretically have a 1-bit difference on some results depending on what the bottom 26 bits (that were undefined, because you had 64-bit floats most of the time, even though the machine did things 80-bits at a time). 2. No -- Modern x86 systems use SSE registers, which are 64-bits. There are a whole s…

> On old x86 systems, the x87 registers were 80-bits and the bottom bits were undefined. You could theoretically have a 1-bit difference on some results

I spent a long time once debugging an issue arising from this. Code was basically:

  double x = y;
  [...]
  if (x>y) fail;
Nothing in between the assignment and the check changed the value of x or y, yet the check triggered the fail condition. Turned out to be that one of the value stayed in the 80 bit x87 register and one was written to RAM as a 64 bit value, then loaded back into an x87 register for the check, resulting in the inequality. Running in a debugger wrote both values out of the registers before the check, making the problem unreproducible.

Re: John Gustafson’s crusade to replace floating point with something better

#30

Post by theamk »

shaklee3 wrote: post #20
Iv wrote: post #18

Earlier quoted context omitted.

They could refer to parallel systems, most likely GPUs, where it is common to have slightly different results even on the same operation, on the same system. This is due to the fact that several pipelines will do calculations in parallel and depending on which pipeline ends first (which can depend on many factors like current temperature) the sums can happen in a different order, leading to different rounding results…

I've never heard of that happening on a GPU. Can you give an example?

We have this in our GPU computations, this is very annoying. The errors are usually too small to matter, but they make comparing binary results / reproducible builds impossible. Here is an example I just made:

    $ python3
    >>> s = [10**-x for x in range(16)]
    >>> s
    [1, 0.1, 0.01, 0.001, 0.0001, 1e-05, 1e-06, 1e-07, 1e-08, 1e-09, 1e-10, 1e-11, 1e-12, 1e-13, 1e-14, 1e-15]
    >>> sum(s)
    1.1111111111111112
    >>> sum(reversed(s))
    1.111111111111111
(it uses CPU for simplicity, but GPU has exactly the same math)
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