[D] Why does the diffuser library change the original sampling algorithm?

I have been recently analyzing the inner workings of the hugging face diffusers library, and I have realised what seems to be a change in the sampling method inside the step() function:

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https://preview.redd.it/9zz4yzm37rjc1.png?width=2041&format=png&auto=webp&s=de825eb3d57bbffa103998368f60401dc9f6f304

If I have understood this correctly, first, they create a prediction of x₀ with the following fórmula:

https://preview.redd.it/2cy3dp257rjc1.png?width=1483&format=png&auto=webp&s=3126d7b6ddb26743f4c0545275d0dc3a12f154a0

Next, they create xₜ₋₁ with these equations:

https://preview.redd.it/72hbpzn57rjc1.png?width=1936&format=png&auto=webp&s=3f839aa6ad96d3c81812256e33573859a30b8f9b

But I think that this is a huge change compared with line 4 of the original equation:

https://preview.redd.it/upexate77rjc1.png?width=1060&format=png&auto=webp&s=4c6d17715f74fa4b72ecf63b4a82598634a36372

Why do …

algorithm change diffusers face function hugging face inside library machinelearning prediction sampling