From: terje.mathisen@tmsw.no
Michael S wrote:
> On Sun, 2 Nov 2025 16:09:10 +0100
> Terje Mathisen wrote:
>
>> Michael S wrote:
>>> On Sun, 2 Nov 2025 11:36:36 +0100
>>> Terje Mathisen wrote:
>>>
>>>> Thomas Koenig wrote:
>>>>> MitchAlsup schrieb:
>>>>>>
>>>>>> Terje Mathisen posted:
>>>>>
>>>>>>> Actually, for the five required basic operations, you can always
>>>>>>> do the op in the next higher precision, then round again down to
>>>>>>> the target, and get exactly the same result.
>>>>>>
>>>>>> https://guillaume.melquiond.fr/doc/05-imacs17_1-article.pdf
>>>>>
>>>>> The PowerISA version 3.0 introduced rounding to odd for its
>>>>> 128-bit floating point arithmetic, for that very reason (I
>>>>> assume).
>>>>
>>>> Rounding to odd is basically the same as rounding to sticky, i.e if
>>>> there are any trailing 1 bits in the exact result, then put that in
>>>> the ulp position.
>>>>
>>>> We have known since before the 1978 ieee754 standard that
>>>> guard+sticky (plus sign and ulp) is enough to get the rounding
>>>> correct in all modes.
>>>>
>>>> The single exception is when rounding up from the maximum magnitude
>>>> value to inf should be suppressed, there you do in fact need to
>>>> check all the bits.
>>>>
>>>> Terje
>>>>
>>>
>>> People use names like guard and sticky bits and sometimes also
>>> rounding bit (e.g. in Wikipedia article) without explanation, as if
>>> everybody had agreed about what they mean. But I don't think that
>>> everybody really agree.
>>
>> Within the 754 working group the definition is totally clear:
>>
>
> I could believe that there is consensus about these names between
> current members of 754 working group. But nothing of that sort is
> mentioned in the text of the Standard. Which among other things means
> that you can not rely on being understood even by new members of 754
> working group.
>
>> Guard is the first bit after the normal mantissa.
>>
>> Sticky is the bit following the guard bit, it is generated by OR'ing
>> together all subsequent bits in the exact/infinitely precise result.
>>
>> I.e if an exact result is exactly halfway between two representable
>> numbers, the Guard bit will be set and Sticky unset.
>>
>> Ulp (Unit in Last Place)) is the final mantissa bit
>>
>> Sign is of course the sign in the Sign-Magnitude format used for all
>> fp numbers.
>>
>> This means that those four bits in combination suffices to separate
>> between rounding directions:
>>
>> Default rounding is nearest or even: (In this case Sign does not
>> matter.)
>>
>> Ulp | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
>> Guard | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
>> Sticky | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
>>
>> Round | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
>>
>> Terje
>>
>
> I mostly use ULP/Guard/Sticky in the same meaning. Except when I use
> them, esp. Guard, differently.
> Given the choice, [in the context of binary floating point] I'd rather
> not use the term 'guard' at all. Names like 'rounding bit' or
> 'half-ULP' are far more self-describing.
Guard also works for decimal FP, where you need a single Sticky bit if
the Guard digit is equal to 5. If you work with the binary
representation for decimal, then you just need two extra bits, just like
BFP.
Correct rounding also work when Guard temporarily contains more than one
bit, possibly due to normalization, but you would normally squash this
down (Guard, Sticky) by OR'ing any secondary guard bits into Sticky.
Terje
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