How to make this infinite decay envelope (y=1/(x+1)^c curve) more CPU efficient?

[mikejm]

Z1202
That's what I meant saying that b is the major source of precision loss.

Now I see what you mean. And I agree.

mikejm
What is the math to calculate k, a, b in this case? How do I graph it on desmos.com to confirm?

Roughly:
For `x[n] = a*x[n-1]` the impulse responce is `y = 1*a^n` (where n is number of iterations starting from 0)
For `x[n] = a*x[n-1] - b` it becomes `y = (1-b*n)*a^n`
and `n` -> `T in seconds` is obviously `n/sampleRate`.
E.g.:
https://www.desmos.com/calculator/cdjre0xvfq
https://www.desmos.com/calculator/2emxae2d6m

The rest depends on how exactly you're going to use it. Specifically if you need a separate "curve shape" parameter. In simple case we can for example choose values of the two curve points:
end point - to represent decay time,
some mid point - for its x or y value to specify the "curveness".
Then it becomes a usual "two-equations -> two-unknowns" solution.

Thanks max, but as noted by sonigen above those are c^x curves, and I want x^c, so recursion won't help me. Looks like the other optimizations already noted are as far as I can go.

Thanks all. This has been very instructive.

[mikejm]

Replace pow(x,c) with exp(log(x)*c).

Well I implemented all the changes reviewed in this thread and it caused a dramatic drop in CPU usage so hugely successful.

The only thing I don't understand is this one (even though I did implement it).

So:

x^c = e^(ln(x)*c)
ln(x^c) = ln(x)*c

I get that.

But why is it cheaper to do e^(ln(x)*c) than just plain x^c?

Is C++ or the computer in general better "optimized" for base e calculations such that two base e calculations are cheaper than one non base e calculation?

Thanks.

[2DaT]

But why is it cheaper to do e^(ln(x)*c) than just plain x^c?

Is C++ or the computer in general better "optimized" for base e calculations such that two base e calculations are cheaper than one non base e calculation?

Thanks.

In floating point it's less precise that way. In order to maintain precision, pow function has to do some calculations in extra precision.

[Z1202]

Thanks max, but as noted by sonigen above those are c^x curves, and I want x^c

Any particular reason for that? x^c curves for envelopes are rather uncommon I'd say (except for automation curves in some DAWs, no idea why they decided to do it that way)