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Shouldn´t you be adding that +1 to the x value before squaring?

Euclidean distance = sqrt( (x+1)^2 )

Manhattan distance = | x+1 |

my case is noise in another dimension ;)

however if noise is added to the main dimension,

it will be

sqrt((1 + x)^2 + 1)

vs

|1 + x | + 1

and if we put two curves together (shifted so that tey intersects on x=0)

http://robowiki.net/w/images/5/5a/C3BD3E15-EEB6-4F63-826F-7C1F5E54A78E.gif

euclidean looks terrible with large noise in one dimension, and manhattan looks robust.