Béarnaise

Friends are asking me the CDF formula of the sauce béarnaise.

Indeed, there are many recipes for this sauce.
For example, one recipes is :
1. Mix in a pan shallots, pepper, terragon, salt, vinegar
2. Boil until "dry"
3. Add egg yolks
4. Add butter
5. Heat while whipping
6. Add terragon and chervil.

Another recipe asks for :
1. white wine, vinegar, shallots, chervil, terragon, pepper
2. boil until "dry"
3. add water, egg yolks
4. cook
5. filter
6. add red pepper , terragon, chervil

In both cases, the whole formula would be ( O+ S1+ (W1/S2) + (W2/S3) + (W3/S4) + (W4/S5)) / W5, because the continuous water phase would disperse oil (melted butter), but also plant tissues, which are made of water dispersed into a solid.
But as all plant tissues are alike, this could be and it could be simplified into (O + S + (W/S) ) / W

If the product is carefully filtered so that no plant residues is present, then it can be further simplified into (O+S)/W.
Sometimes, also, a foam is made, which would introduce an additional phase G, and this leads to (G+O+S)/W.