Another question: why is the cos table float* and not t_float *? With
Pd64 we basically throw away 29 bits of additional precision (23 bit vs.
52 bit). I assume this is done to reduce the table size for Pd64. Is 23
bit good enough for our purposes? I can imagine that the interpolation
error will be much larger than the difference between 23 bit and 52 bit
precision, but I didn't do the math.

Christof

On 06.06.2024 19:24, Miller Puckette wrote:
> Precisely that: cache pollution in general.  At some point the overall
> speed of the program will suffer, depending on CPU design, cache size,
> and probable other factors.
>
> If the input to a cos~ object (for example) is between 1 and 2 you'll
> get the same loss of accuracy but still there will be rounding
> behavior that will (probably) give unsymmetric behavior.
>
> Anyway, I don't remember hearing any reason why symmetry should be
> important in itself.
>
> cheers
>
> M
>
> On 6/6/24 6:51 PM, Matt Barber wrote:
>> Since cos~ wraps, one could theoretically take advantage of the equal
>> distribution of float values between 1.0 and 2.0.
>>
>> Profiling a larger table would be useful – I prefer accuracy over
>> performance in general, but I wonder where the performance hit would
>> come from, outside of unpredictable cache misses.
>>
>>
>>
>> On Thu, Jun 6, 2024, 11:25 AM Miller Puckette
>> <mpuckette@cloud.ucsd.edu> wrote:
>>
>>     Well, as far as I can tell making the table "symmetric" won't
>>     matter at
>>     all since, for instance, 0.1 and 0.9 won't give the same lookup
>>     values
>>     anyway because they can't themselves be represented exactly and
>>     will be
>>     truncated differently (0.1 will be more accurately represented than
>>     0.9).  On the other hand, values like 0.25 or -0.5 can be
>> represented
>>     exactly so it might be worthwhile to bash true 1s, -1,s, and 0s
>> where
>>     they belong in the table.
>>
>>     Hearing that Max defaults to a ridiculously big table makes me
>> wonder
>>     though... first, is 2048 really enough (and at what point is there a
>>     real performance penalty for bigger tables).  And: not for this
>>     release
>>     but later perhaps, should 64-bit Pd use a bigger table?
>>
>>     As I figure it, the 2048-point table differs from the true cosine,
>>     absolute worst case, by (2pi/2048)^2 / 8, or about 2(-19.7) -
>>     i.e., 19.7
>>     bit accuracy.  But the error is dominated by an amplitude change
>> (the
>>     best-matching cosine to the line-segment approximation has amplitude
>>     less than 1).  Accounting for that and taking RMS error instead of
>>     worst-case gives an error estimate 2.7 bits more optimistic: 22.4
>>     bits,
>>     which is close to the accuracy of a 32-bit number.
>>
>>     I don't have my RPI3 handy (I'm on the road) but I'm now
>> wondering if
>>     the default shouldn't be 4096, which would give us an additional 2
>>     bits
>>     of goodness.  Any opinions?
>>
>>     cheers
>>
>>     M
>>
>>     On 6/5/24 9:35 PM, Matt Barber wrote:
>>     > A couple of things:
>>     >
>>     > 1. I'm pretty sure any error in cos at pi and 2pi will be
>>     smaller in
>>     > double precision than float's epsilon, so I don't think that
>>     there's
>>     > any need to set -1.0 and 1.0 explicitly after all except to be
>>     extra
>>     > safe. However, at pi/2 and 3pi/2 the error is still larger than
>> the
>>     > minimum normal number, so it is worth setting the zero crossings
>>     to 0.0.
>>     >
>>     > 2. For garray_dofo() there isn't a great way of using explicit
>>     0.0 at
>>     > zero crossings without incurring an extra check, like don't add
>>     to the
>>     > sum if absolute value is less than e.g. 1.0e-10. For this,
>> probably
>>     > just using M_PI and incrementing integer phase like for the cosine
>>     > table is enough.
>>     >
>>     > MB
>>     >
>>     >
>>     > On Wed, Jun 5, 2024 at 2:20 PM Alexandre Torres Porres
>>     > <porres@gmail.com> wrote:
>>     >
>>     >     Em qua., 5 de jun. de 2024 às 14:31, Matt Barber
>>     >     <brbrofsvl@gmail.com> escreveu:
>>     >
>>     >         While we're at it, I think it would be worth tuning
>>     >         garray_dofo() to use the same so that sinesum and
>>     >         cosinesum have the same level of accuracy, guarantees of
>>     >         symmetry, etc.
>>     >
>>     >         MB
>>     >
>>     >
>>     >     Good catch! In fact, I think this is a great opportunity to
>> also
>>     >     fix this bug https://github.com/pure-data/pure-data/issues/371
>>     >
>>  <https://urldefense.com/v3/__https://github.com/pure-data/pure-data/issues/371__;!!Mih3wA!Gx7B-gwSgjsuIXmREh2__bBbYdt1d6pi29crpkLOOyltinVweZR3u6Q6vl9ItouugFy2oefgYhPlew$>
>>     >     which is totally related. I just reopened
>>     > https://github.com/pure-data/pure-data/issues/105
>>     >
>>  <https://urldefense.com/v3/__https://github.com/pure-data/pure-data/issues/105__;!!Mih3wA!Gx7B-gwSgjsuIXmREh2__bBbYdt1d6pi29crpkLOOyltinVweZR3u6Q6vl9ItouugFy2oedw4qUPfQ$>
>>     >     as well as I'm still considering the table could/should be
>> still
>>     >     "perfectly symmetric" considering 0 crossings and the
>> start/end
>>     >     points.
>>     >
>>     >
>>     >         On Wed, Jun 5, 2024 at 12:52 PM Alexandre Torres Porres
>>     >         <porres@gmail.com> wrote:
>>     >
>>     >             For the record and sake of comparison, Cyclone uses
>>     >             a 16384 points table, and linear interpolation,
>>     calculated
>>     >             with double precision. We did this because MAX
>> documents
>>     >             it uses such a table, and we made it (well, Matt did)
>>     >             simetric.
>>     >
>>     >             I see Pd is doing kind of the same, huh? linear
>>     >             interpolation on a table calculated with double
>>     precision.
>>     >
>>     >             I see SuperCollider mentions it uses 8192 points and
>>     >             linear interpolation on its oscillator.
>>     >
>>     >             I guess MAX is exaggerating its table size a bit :)
>>     but I
>>     >             wonder why Pd is still about to use a relatively
>> smaller
>>     >             table size. I'm curious to know how much an
>> increase in
>>     >             table size actually offers a better resolution and how
>>     >             much it ruins performance. For instance, I'm using the
>>     >             same as Cyclone in ELSE oscillators, could I just
>> reduce
>>     >             it at least to 8192 points or even less and down to
>> Pd's
>>     >             2048 size worry free?
>>     >
>>     >             Thanks
>>     >
>>     >
>>     >
>>     >             Em qua., 5 de jun. de 2024 às 13:28, Alexandre Torres
>>     >             Porres <porres@gmail.com> escreveu:
>>     >
>>     >                 Nice one Matt!
>>     >
>>     >                 Em qua., 5 de jun. de 2024 às 08:13, Christof
>> Ressi
>>     >                 <info@christofressi.com> escreveu:
>>     >
>>     >>                         @Miller: what do you think? IMO we should
>>     >>                         make the cos table as good as we can,
>> so we
>>     >>                         won't have any regrets :)
>>     >>
>>     >                 +1000!!!
>>     >
>>