#Mathematica <s>92</s> 49



Calle deserves full credit for streamlining the code.

    f@l_:=2 N@Norm[Tr[l/.{"r"→1,"l"→-1,"u"→I,"d"→-I}]]

**Example**

    f[{"u", "u", "u", "u", "d", "r", "r", "r", "l"}]

>7.2111