aherve

my_fancy_func = minus.(square, plus_one) # defining a new function
my_fancy_func.(3) #=> 5 = 3*3 - ( 3 + 1)

aherve

derivative = -> f { # we take a function as argument
  -> x { # the function takes a real x as argument
    ( f.(x+1e-3) - f.(x-1e-3) ) / ( 2e-3 ) # apply the scheme
  }
}

# Let's try:
derivative_of_square = derivative.(square) => #<Proc:0x00000001b7c270@(irb):24 (lambda)> . Yay, a new function !

#Can we use it ?

aherve

second_derivative_of_square = derivative.( derivative.(square) ) # this should be a constant function that return 2
second_derivative_of_square.(2) #=> 1.999999999946489 
second_derivative_of_square.(3) #=> 2.000000000279556

aherve

n_times = -> n,f {
  n == 1 ? f : -> x { f.(n_times.(n-1,f).(x))}
}

aherve

nth_derivator = -> n {  n_times.(n,derivate) }

aherve

derivative_of_square = nth_derivator.(1).(square) #derivate one time
second_derivative_of_square = nth_derivator.(2).(square) #derivate two times
third_derivative_of_square = nth_derivator.(3).(square) #derivate three times
p derivative_of_square.(3) # => 5.999999999999339
p second_derivative_of_square.(3) # => 2.000000000279556
p third_derivative_of_square.(3) # => 0.0

aherve

minus = -> f,g { -> x { f.(x) - g.(x) } } # f - g 
div = -> f,g { -> x { f.(x) / g.(x) } } # f / g 
mult = -> f,g { -> x { f.(x) * g.(x) } } # f * g 
norm = -> f { -> x { f.(x).abs}} # absolute value 
const = -> const { -> x { const} } # That's right, the constant function !  

# You should recognize these: 
plus_eps = -> eps { -> f { -> x { f.(x+eps) } } } 
min_eps = -> eps { -> f { -> x { f.(x-eps) } } }  

aherve

lim = -> f,eps,tres {
  -> x {
    inf.(
      norm.(
        minus.(
          plus_eps.(eps/2.0).(f),
          plus_eps.(eps).(f)
        )
      ), const.(tres) ).(x) ? plus_eps.(eps).(f).(x) : lim.(f,eps/2.0,tres).(x)
    }

aherve

# derivative_scheme.(f).(x) shall be a function of epsilon
derivative_sheme = -> f {
  -> x {
    -> eps {
      div.( minus.(plus_eps.(eps).(f), min_eps.(eps).(f) ), mult.(const.(2), const.(eps))).(x)
    }
  }
} 

# And the derivative operator:

aherve

#!/usr/bin/env ruby
# Our fancy function. Could be exactly anything
square = -> x {x*x}

# some tools 
minus = -> f,g { -> x { f.(x) - g.(x) } } 
div = -> f,g { -> x { f.(x) / g.(x) } } 
mult = -> f,g { -> x { f.(x) * g.(x) } } 
norm = -> f { -> x { f.(x).abs}} 
const = -> const { -> x { const} }