Create new column in data.frame based on if-else assessment of other columns

I am trying to add a new column (x_new)to my dataframe that is dependent on the value given in a 'definition' column. The definition column x_definition contains on of the following record types: - A constant number - A string describing the operation needed - NA

I want the resultant column, x_new, to look as follows: - If x_definition is NA, then x_new remains NA. - If x_definition is a string, then it requires a certain calculation. For example if it's 'equal_to_z' than the result should be z, or if its 'third_of_z', then x_new should be z/3. There are also more than just these definitions that indicate more complex functions of z are required. - If x_definition is any number, then x_new should just be that number.

I wrote the following code which works to handle those cases, but is a cumbersome group of nested ifelse statements. I am looking for a method that

data <- data %>% mutate(x_new = ifelse(
  is.na(x_definition), NA, ifelse(
    x_definition=='equal_to_z', z, ifelse(
      x_definition=='third_of_z', z/3, NA
      )
    )
  )
)

I also considered using switch but ran into the problem where I don't know how to say "if it's a number, leave it as a number"

a <- data %>% mutate(x_new = switch(x_definition,
  'equal_to_z' = z,
  'third_of_z' = z / 3,
  <number???> = x_definition
  )
)

What would be an appropriate process for addressing this?