The law that sets the price of ray tracing

Monte Carlo convergence

Monte Carlo error falls as 1/√N, and that single fact governs every render time you will ever wait for. It means the noise halves only when the sample count quadruples — so getting from visibly grainy to acceptable is cheap, and getting from acceptable to clean is not.

Error against sample count log–log
What the noise looks like —

A patch estimated with the current number of samples per pixel, using the selected strategy. The grain you can see is the error plotted on the left.

Controls the experiment
16
60
0.12
Samples per pixel
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RMS error, uniform
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Measured slope
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Samples for half the noise
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The cost of quality from the measured curve
QualitySamplesRelative costRender time

Assuming a frame takes one second at 16 samples. The last row is why studios rent render farms.

Why better sampling does not change the slope the important subtlety

Switch the three strategies on together. Importance sampling sits well below uniform on the chart — often by a factor of ten — but it runs parallel to it. It reduces the constant in front of the 1/√N, not the exponent. Every strategy in this family obeys the same law; they just start from a better place.

Stratified sampling does slightly better than parallel — spreading samples evenly rather than independently beats pure randomness for smooth integrands — but it degrades back towards 1/√N as the integrand gets spikier. Push the spikiness slider up and watch the three lines converge.

This is why denoising became standard. If you cannot beat 1/√N by sampling, the remaining option is to stop sampling early and repair the image afterwards — which is exactly what every production renderer and every RTX game now does.

Where 1/√N comes from. The Monte Carlo estimate is an average of N independent samples. The variance of an average of independent values is the variance of one value divided by N, and the error we perceive is the standard deviation — the square root of the variance. So the error goes as √(σ²/N), which is σ/√N. Nothing about rendering is special here; it is the same statistics that governs opinion polls, and it is equally immovable. The only lever it leaves you is σ, the variance of a single sample, which is precisely what importance sampling, stratification and next-event estimation all attack.