replacement - ReplaceRepeated multi-part pattern

This workaround does what I'd like to have happen:

{-(1/2) E^(-e t) 
(2 - 2 NormCDF[((-2 e + s^2) t - 2 Log[A])/(2 s Sqrt[t])])} //. {2 - 
    2 NormCDF[x___] -> 2 NormCDF[-x]}

I expected this to work, but it doesn't:

{-(1/2) E^(-e t) 
(2 - 2 NormCDF[((-2 e + s^2) t - 2 Log[A] + 2 Log[K])/(2 s Sqrt[t])])} //. {n_ - 
n_ NormCDF[x___] -> n NormCDF[-x]}

Appreciate any hints or tips.

Answer

As always with a replacement issue, look at the FullForm of your expressions. Here is your expression:

expr = (2-2 NormCDF[((-2 e+s^2) t-2 Log[A]+2 Log[K])/(2 s Sqrt[t])]);

expr //FullForm

Plus[2,Times[-2,NormCDF[Times[Rational[1,2],Power[s,-1],Power[t,Rational[-1,2]],Plus[Times[Plus[Times[-2,e],Power[s,2]],t],Times[-2,Log[A]],Times[2,Log[K]]]]]]]

And here is your rule:

rule = n_-n_ NormCDF[x___];
rule //FullForm

Plus[Pattern[n,Blank[]],Times[-1,NormCDF[Pattern[x,BlankNullSequence[]]],Pattern[n,Blank[]]]]

Notice how the product in your expression is of the form:

Times[-2, _NormCDF]

and your replacement rule uses:

Times[-1, _NormCDF, n_]

Since they don't match, no replacement occurs. An alternative that will work is:

expr /. n_ + m_ NormCDF[x_] /; n+m==0 :> n NormCDF[-x]

2 NormCDF[-(((-2 e + s^2) t - 2 Log[A] + 2 Log[K])/(2 s Sqrt[t]))]

Comments

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