Years ago I found that 4096 was the right size for the cosine tables w/ linear interpolation in my externals, if I wanted to not hear any results of artifacts (usually). It was a while ago and I don't exactly remember what audio tests I did tho. anyways +1 for 4096.
-seb
Date: Thu, 6 Jun 2024 10:25:50 -0500
From: Miller Puckette mpuckette@cloud.ucsd.edu To: Matt Barber brbrofsvl@gmail.com, Alexandre Torres Porres porres@gmail.com Cc: Christof Ressi info@christofressi.com, Miller Puckette msp@ucsd.edu, pd-dev@lists.iem.at Subject: Re: [PD-dev] 64-bit (and/or ARM) Pd change default cosine table? Message-ID: 44fd2d44-3601-4c1a-8e88-44516df22797@cloud.ucsd.edu Content-Type: text/plain; charset=UTF-8; format=flowed
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:
- 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.
- 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 ?s 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 ?s 13:28, Alexandre Torres Porres porres@gmail.com escreveu:
Nice one Matt!
Em qua., 5 de jun. de 2024 ?s 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!!!
------------------------------
Message: 2 Date: Thu, 6 Jun 2024 12:51:45 -0400 From: Matt Barber brbrofsvl@gmail.com To: Miller Puckette mpuckette@cloud.ucsd.edu Cc: Alexandre Torres Porres porres@gmail.com, Christof Ressi info@christofressi.com, Miller Puckette msp@ucsd.edu, pd-dev@lists.iem.at Subject: Re: [PD-dev] 64-bit (and/or ARM) Pd change default cosine table? Message-ID: CAOrke7GRQ+j7_HV5mSz1hXFqh+BpHaAOZtppFaXM2gN7j8mcyA@mail.gmail.com Content-Type: text/plain; charset="utf-8"
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:
- 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.
- 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 ?s 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/37...
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/10...
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 ?s 13:28, Alexandre Torres Porres porres@gmail.com escreveu:
Nice one Matt!
Em qua., 5 de jun. de 2024 ?s 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!!!
-
Sebastian Shader