Re: [Maxima-discuss] bfloat accuracy, %pi

[Richard F. <fa...@be...>]

On 6/4/2018 1:03 PM, Stavros Macrakis (Σταῦρος Μακράκης) wrote:
> I am not sure what the problem is that you want to solve.
>
> float(v) loses precision: v-rationalize(float(v)) => -1, so it's not 
> surprising that cos(v) is what it is:
>
>      fpprintprec:3$
>      fpprec:1000$
>      cos(bfloat(float(v))) => -0.84
>
> Similarly, for fpprec < 16, bfloat(v) loses digits. For example,
>
>      v:  18190586279576483$
>      fpprec:15$
>      v-rationalize(bfloat(v)) => 3
>
> So it's not surprising that cos(v) is not accurate.
>
> For fpprec > 15, cos(v) is consistently very accurate:
>
>      cv1000: block([fpprec:1000],bfloat(cos(v)))$
>  makelist(block([fpprec:3*i],[fpprec,(cv1000-bfloat(cos(v)))/10^-fpprec]),i,1,20) 
> =>
> [- 9.7b2,     3.12b5,   8.12b8, - 8.86b11, - 1.41b14,
>  - 2.63b-4, - 6.13b-4,  1.72b-4,  2.11b-4,  5.1b-4,
>    1.24b-4,   2.71b-4,  1.71b-4,  7.54b-5,  1.25b-4,
>    4.0b-4,  - 4.43b-5,  3.42b-4,  1.73b-4,  1.09b-4]
>
> So where exactly is the problem?
>
>          -s
>
>
<snip>
I get different results on
Maxima version: 5.40.0
Lisp: SBCL 1.3.12

running the same code as above, I get the error in the
cosine routine to seem more like 28 decimal digits after fpprec:30 or more.
I would expect the error to be more like 0.0b0;   try the same experiment
with, say,  v=123456789123456890...

(%i118)      cv1000: block([fpprec:1000],bfloat(cos(v)))$
makelist(block([fpprec:3*i],[fpprec,(cv1000-bfloat(cos(v)))/10^-fpprec]),i,1,20);

(%o118) 
[[3,4.65b2],[6,1.98b6],[9,6.46b8],[12,5.51b11],[15,2.84b14],[18,1.97b18],[21,-1.3b19],[24,6.82b23],[27,1.97b27],[30,1.4b29],[33,-3.93b28],[36,-8.57b28],[39,-5.4b28],[42,-2.38b28],[45,-3.94b28],[48,-1.26b29],[51,1.4b28],[54,-1.08b29],[57,-5.45b28],[60,-3.44b28]]