Creating multiple new rows in a dataframe based on existing data

Suppose I have the following dataset:

data.frame(country = c("A", "A", "B", "B", "C", "C"),           type = c("x", "z", "x", "z", "x", "z"),           value = c(4, 2, 6, 10, 54, 10))

  country type value1       A    x     42       A    z     23       B    x     64       B    z    105       C    x    546       C    z    10

What I want to do is to create multiple new rows for each country (A, B, C) with a new type called "z-x" and a new value which is the difference between the values z and x. I'm looking for this output:

data.frame(country = c("A", "A", "A", "B", "B", "B", "C", "C", "C"),           type = c("x", "z", "z-x", "x", "z", "z-x", "x", "z", "z-x"),           value = c(4, 2, -2, 6, 10, 4, 54, 10, -44))

  country type value1       A    x     42       A    z     23       A  z-x    -24       B    x     65       B    z    106       B  z-x     47       C    x    548       C    z    109       C  z-x   -44