Unless someone can show me two waves that I can listen to and hear the difference.

I can gather by the amount of questions, and the series of tests we are putting the code through in order to demonstrate the difference...actually answers my question.

ORIGINAL: eikelbijter
...Much like with all of the 'cold medications' people love to take in this country, people are kidding themselves.

ORIGINAL: Ron Kuper [Cakewalk]

The program simulates what happens if you mix multiple audio streams using different gains, and scale the result, as shown below
( g1*[S1] + g2*[S2 ] – g1*[S1 ] ) / g2
You would expect the 1st and 3rd streams to cancel each other out, leaving the result of mix to be [S2 ].


. . .[/code]For nearly all values of gain1 and gain2 there is error.

More details to come when we post the slides from my talk. I will also post the actual source code used for my testing.

ORIGINAL: dcastle

Ah , finally a little algebra. So is the error in the division, the conversion, or somewhere else?

Right! The point has been made over and over, that floating-point is like the digits in your calculator. The more digits, the more accuracy. Accuracy is the opposite of error. For any given number of digits, there will be some "accuracy" and some "error". For twice as many digits, there will be a much greater "accuracy" and a much smaller "error". This is true for both multiplication and division. There is another kind of error, called round-off, that also affects addition and subtraction. All of these errors are reduced by increasing the number of bits of calculation. This is fundamental stuff!

The point I am making is that the "error" part of the 64-bit double-precision arithmetic is about -148db less than the "error" part of the 32-bit single-precision arithmetic . I conducted an experiment with 16 tracks of white noise and 16 tracks of sine waves mixed at random gains between +4.8 and -48 db. I have no idea what the actual error is for 32-bit single-precision arithmetic or for 64-bit double-precision arithmetic, but I have experimentally determined that there is -148db less error in the 64-bit double-precision arithmetic. I claim that this is unhearable!

I have yet to see any analysis or any experiment that proves differently, but I'm sure that there are panning differences or plugin differences that could easily account for all the wonderful stuff that is being reported. But on top of that, let's don't forget that Monster Cable is doing an incredible business selling stuff that is better than needed, because so many people need that. There is a psycho side of this that requires double-blind testing of controlled experiments, and I have not heard of anyone who has successfully implemented one yet.

Regards,
David

EDIT: Oops can't spell.

ORIGINAL: VariousArtist

Ron Kuper
That's 1 bit per multiply/summing stage. In other words, the more tracks and buses you have, the more these bit errors can accumulate and start to be come 2 bit errors, 4 bit errors, etc.

Isn't this the whole crux of the matter? The cumulative effects of truncation and or rounding cannot be underestimated, and sometimes a small, tiny, weeny error can manifest itself in ways that would surprise many.
...
Numerical analysis, of which all of this is a sub-topic, is not a trivial topic, but it's well worth investigating if you care to join me on the nerdy side of the fence...

Toto...I don't think we are in Kansas anymore.

The more accurate you are, even with the tiniest of details that seem ridiculous, the more chance that you will offset the downstream effect./

So I think you are saying you brought up the "butterfly effect" only to show that small things can be signifcant, but not because you were trying to imply that a LSB error on a bus will cause a hurricane --- because there is no such chaotic effect in digital audio.

I think a better example would be something like a cloud of butterflies causing an airplane to crash --- instead of one, by itself, just making a mess on the window.

Regards,
David

ORIGINAL: higain_guitar
You might be right, but I don't think one 24-bit LSB of error is generated for each 32-bit FLOP/MAC, such as a gain and summation of two tracks or buses. Here is what Ron Kuper said:

ORIGINAL: Ron Kuper [Cakewalk]
The verdict is that using 32-bit floats to mix 24-bit data introduces at most 6dB of error more than 25% of the time.

That is: at * maximum * a 1 bit error but only 25% of the time....

ORIGINAL: Ron Kuper [Cakewalk]
That's 1 bit per multiply/summing stage. In other words, the more tracks and buses you have, the more these bit errors can accumulate and start to be come 2 bit errors, 4 bit errors, etc.