Forgive me if this is the wrong thread area. My question is is it possible to generate very exact decimals of a sine wave? Short and long answers are both encouraged.

specifically I want to create sine waves of Equal Temperament to the closest precision possible for experiments in sensory/cognition

In standard pd, all numbers are single-precision floating point numbers. Floating point numbers have two parts: a mantissa (which defines the value) and an exponent (which defines the scale). For example, 5 is binary 1.01 * 2^2 -- mantissa is 5/4, and the exponent 2 is such that 2^exponent cancels out the "/4" leaving 5.

The mantissa determines the numeric precision. In single precision, you get 24 mantissa bits (23 are encoded, one is assumed), taking you up to 16777216. Usually the last digit is an approximation, where rounding error occurs, so let's call it 7 decimal digits.

That means 5000 Hz should be distinguishable from 5000.01 or even 5000.001 Hz, but 5000.0001 would be numerically the same as 5000 (by empirical testing, 5000.0005 is 1 bit off from 5000.0000 and that's as small as you can go). So there's a limit how precise you can go.

But the pitch ratio 16777216 / 16777215 is about 0.0001 cents, which no human will be able to hear (not even Jacob Collier) so it's kinda good enough.

hjh

internally pd uses a table for the [cos~] and [osc~] objects that is only 512 points with linear interpolation.
if you want to be more accurate you can make your own tables with [cos] object, or if you want to use the math.h functions directly at audio rate I think [expr~ sin($v1)] (or cos) does that

@seb-harmonik.ar 2048 points since Pd version 0.55 :
https://msp.ucsd.edu/Pd_documentation/resources/chapter5.htm#s5.1