[libre-riscv-dev] [isa-dev] Re: FP transcendentals (trigonometry, root/exp/log) proposal

[Jacob Lifshay]

On Thu, Jan 16, 2020, 02:00 Luke Kenneth Casson Leighton <lkcl at lkcl.net>
wrote:

> On Thursday, January 16, 2020, Jacob Lifshay <programmerjake at gmail.com>
> wrote:
>
> > On Thu, Jan 16, 2020, 01:22 Luke Kenneth Casson Leighton <lkcl at lkcl.net>
> > wrote:
> >
> > > On Thursday, January 16, 2020, <whygee at f-cpu.org> wrote:
> > >
> > > > On 2020-01-16 04:04, lkcl . wrote:
> > >
> > >
> > >
> > > > the time (let alone rounding).  how do you *know* when the rounding
> (or
> > > the
> > > >> last bit of the mantissa) is wrong?
> > > >>
> > > >
> > > > That was my reaction :
> > > > if you *know* your LSB is wrong then you can correct it...
> > >
> > >
> > > ... or can you? this is what i'm fascinated to hear about, mitch.
> > >
> > > is there some mathematical way in which the accuracy can be computed
> yet
> > > the effort not known?
> > >
> >
> > Yes. you just compute the output to a few more bits where the output is
> > known to always be within a certain distance of the correct output,
>
>
> sorry to be sounding dumb: the question thus becomes, "how do you know when
> the output is always within a cer... etc etc " :)


when approximating a mathematical function, you get an approximate output
xa and you know by error analysis that the true mathematical output x is
within distance delta of xa: x is in [xa - delta, xa + delta]

So, to round, you just find the nearest spot xt to xa where the rounded
output would change (so, for round to nearest, xt is halfway between
rounded outputs), then:
If xt is in [xa - delta, xa + delta], then the rounded result could be
incorrect because you don't know if x is less, equal, or more than xt.
Otherwise, the rounded result is known to be correct because comparing xa
to xt always gives the same answer as comparing x to xt.

Jacob