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

[Jacob Lifshay]

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, then
you just check if the output is within that distance of the transition
between two rounded outputs, if it is, then it could be rounded
incorrectly, otherwise it rounds correctly.

>
> or... oh hold on, are you saying that the exception would be raised (every
> time?) then the partially-accurate result would be dropped into Newton
> Raphson for (probably only one iteration) to get full accuracy?
>

Newton Raphson is basically useless for functions like sin or cos, since
you have to compute the inverse function (asin or acos) to a higher
accuracy which takes even more work than just computing the function to
enough accuracy in the first place.

Newton Raphson is useful for square-roots because computing the inverse
function (and its derivative) to high accuracy is easy (just multiplication
and addition).

Jacob