julia> function fib1(n)
       if n==1
       return n
       else
       return n+fib1(n-1)
       end
       end
fib1 (generic function with 1 method)

julia> function fib2(n,s)
       if n == 1
       return s+1
       else
       return fib2(n-1,s+n)
       end
       end
fib2 (generic function with 1 method)

julia> fib1(10)
55

julia> fib2(10)
ERROR: MethodError: no method matching fib2(::Int64)
Closest candidates are:
  fib2(::Any, ::Any) at REPL[18]:2

julia> fib2(10,0)
55

julia> fib2(20)
ERROR: MethodError: no method matching fib2(::Int64)
Closest candidates are:
  fib2(::Any, ::Any) at REPL[18]:2

julia> fib2(20,0)
210

julia> fib1(20)
210

julia> time(fib1(100))
ERROR: MethodError: no method matching time(::Int64)
Closest candidates are:
  time() at libc.jl:206
  time(::Base.Libc.TmStruct) at libc.jl:205

julia> @time(fib1(100))
  0.000005 seconds (131 allocations: 7.734 KB)
5050

julia> @time(fib2(100,0))
  0.000007 seconds (5 allocations: 176 bytes)
5050

julia> @time(fib1(500))
  0.000005 seconds (5 allocations: 176 bytes)
125250

julia> @time(fib2(500,0))
  0.000010 seconds (5 allocations: 176 bytes)
125250

julia> @time(fib1(1000))
  0.000022 seconds (5 allocations: 176 bytes)
500500

julia> @time(fib2(1000,0))
  0.000009 seconds (5 allocations: 176 bytes)
500500

julia> @time(fib1(10000))
  0.000325 seconds (5 allocations: 176 bytes)
50005000

julia> @time(fib2(10000,0))
  0.000095 seconds (5 allocations: 176 bytes)
50005000

julia> code_native(fib1,(Int64,))
.text
Filename: REPL[17]
pushq %rbp
movq %rsp, %rbp
pushq %rsi
subq $40, %rsp
movq %rcx, %rsi
Source line: 2
cmpq $1, %rsi
jne L30
Source line: 3
movl $1, %eax
addq $40, %rsp
popq %rsi
popq %rbp
retq
Source line: 5
L30:
leaq -1(%rsi), %rcx
movabsq $fib1, %rax
callq *%rax
addq %rsi, %rax
addq $40, %rsp
popq %rsi
popq %rbp
retq
nopl (%rax,%rax)

julia> code_native(fib2,(Int64,Int64))
.text
Filename: REPL[18]
pushq %rbp
movq %rsp, %rbp
Source line: 2
subq $32, %rsp
cmpq $1, %rcx
jne L26
Source line: 3
incq %rdx
movq %rdx, %rax
addq $32, %rsp
popq %rbp
retq
Source line: 5
L26:
addq %rcx, %rdx
decq %rcx
movabsq $fib2, %rax
callq *%rax
addq $32, %rsp
popq %rbp
retq
nopw %cs:(%rax,%rax)