Re: Floating point: sine, cosine etc.

From what I remember, any continuous function can be expressed as a power
series. This should, therefore, also be possible for LN and LOG as well as
the trigonometric and hyperbolic functions etc.

Dave

On Thu, 29 Sept 2022, 20:21 Michal Pleban, <lists_at_michau.name> wrote:

> ruud_at_baltissen.org wrote on 28.09.2022 21:21:
>
> > Hallo Michał,
>
> > The idea is to use tables instead of calculating the various factors. In
> > the final assembly version this will constants in FP format, not
> > variables. That reduces your formula to the first FOR loop.
> > Knowing, thanks to you, where to look for, I found Horner's method which
> > reduced the FOR loop even further. I even can reduce the number of steps
> > by skipping those steps where arFacSin[b] is zero (plus some other
> > needed changes).
>
> All very good ideas. One more: If the "s" variable from the previous
> step is the same as the one from the current step, stop the loop because
> this means that you have reached the maximum accuracy of your floating
> point numbers and any further steps will not improve the result any
> further. This is what CBM BASIC does, that's why calculating SIN(0)
> takes much less time than calculating SIN(PI/2).
>
> > Next step is tangens. I found the formula but here I ran into trouble:
> >
> https://en.wikipedia.org/wiki/Trigonometric_functions#Power_series_expansion
> > . The formula includes Bernoulli numbers and from here on I couldn't
> > follow it anymore.
>
> Hm, from what I can see it's basically the same formula that you have
> for SIN(X), except the table is different:
>
> arFacTan[0] := 0
> arFacTan[1] := 1
> arFacTan[2] := 0
> arFacTan[3] := 1/3
> arFacTan[4] := 0
> arFacTan[5] := 2/15
> arFacTan[6] := 0
> arFacTan[7] := 17/315
>
> What goes further, I would have to calculate since it's been like 20
> years since I have last done this stuff. But it shouldn't be hard.
>
> In fact, now that I look at it, all the other trigonometric functions
> can be also expressed by your mechanism just using a different table.
> That is kinda neat.
>
> > Next steps: EXP, LN, LOG, etc., etc. But know I know on who's door to
> > knock :)
>
> I'll be glad to help too!
>
> Regards,
> Michau.
>
>
>