Reason behind using Manhattan distance
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)
|1 + x | + 1
and if we put two curves together (shifted so that tey intersects on x=0)
euclidean looks terrible with large noise in one dimension, and manhattan looks robust.