PCM20210801_SICP_1.2.2_Tree_Recursion

### A Pluto.jl notebook ###
# v0.20.4

using Markdown
using InteractiveUtils

# ╔═╡ f2774a0c-968e-4e0c-a04a-47827ccafba2
begin 
	using Pluto
	using Plots	
	using LaTeXStrings
	using GraphRecipes
	#----------------------------------------------------------------
	println("pkgversion(Pluto)         = ", pkgversion(Pluto))
	println("pkgversion(Plots)         = ", pkgversion(Plots))
	println("pkgversion(LaTeXStrings)  = ", pkgversion(LaTeXStrings))
	println("pkgversion(GraphRecipes)  = ", pkgversion(GraphRecipes))
end # begin

# ╔═╡ 278e5870-f2d8-11eb-2a98-778e66257fe8
md"
==================================================================================
#### SICP: [1.2.2 Tree Recursion](https://sarabander.github.io/sicp/html/1_002e2.xhtml#g_t1_002e2_002e2)

##### file: PCM20210801\_SICP\_1.2.2\_Tree\_Recursion.jl

##### Julia/Pluto.jl-code (1.11.2/0.20.4) by PCM *** 2025/01/19 ***
==================================================================================
"

# ╔═╡ 63f4476e-4658-4850-ae51-e48877ceea7b
md"
##### 0. Introduction
*Another common pattern of computation is called tree recursion. As an example, consider computing the sequence of Fibonacci numbers, in which each number is the sum of the preceding two.*(SICP, 1996, 2016)
"

# ╔═╡ edfb9df5-cc53-47bb-8a3d-6aba2bf26519
md"
---
##### 1. Topics
- *tree* recursion, tree recursive process
- *Fibonacci* numbers
- *Generic* (*polymorphic*) method
- *Guard*
- *space* and *time complexity* 
- *big O*-notation: $O(g)$
- begin ... end
- *TCO* (= tail call optimization)
- *type* $Signed$
- function $subtypes(.)$
- *type* $Union$
- *State* variables
- *Parallel* Assignment
- *while*, *for* loop
- count of change
"

# ╔═╡ 748739c5-99b5-4fbd-8cad-acf71f2b9504
md"
---
##### 2. Libraries
"

# ╔═╡ 0ea20de0-b278-4341-a176-b42f57530174
md"
---
##### 3. SICP-Scheme-like *functional* Julia
###### 3.1 Formal Definition of $O(g)$
*Analysis of algorithms* is done by applying diverse measures of growth as a function of input length $n$. One of these measure is $O(g)$:

*Definition*: $f$ is $O(g)$ if there are constants $c$ and $n_0$ such that $|f(n)| \le cg(n)$ for all $n \ge n_0.$ (Sanella et al., 2021, p.191)
"

# ╔═╡ 249acfdd-6050-4451-862e-0437a6e18463
md"
---
###### 3.2 Definition: [*Fibonacci* Numbers](https://en.wikipedia.org/wiki/Fibonacci_sequence)

Each number is the *sum* of its *two* predessors starting with $0, 1$:

$0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...$

$fib(n) := \cases{ 
0 \;\;\;\;\; \text{ if } n = 0 \cr
1 \;\;\;\;\; \text{ if } n = 1 \cr
fib(n-1) + fib(n-2) \cr
}$

"

# ╔═╡ b8bc83b5-a720-4f33-a32a-8284129b5fd9
function fib(n) 
	n == 0 ? 0 :                   # n == 0 is a guard
    n == 1 ? 1 :                   # n == 1 is a guard
		+(fib(n-1), fib(n-2))
end # function fib

# ╔═╡ 739f010a-7623-411b-9376-9222b3183039
fib(0)                             # ==>    0 --> :)

# ╔═╡ 16dc2f1b-99f4-4a67-894a-523666117feb
fib(1)                             # ==>    1 --> :)

# ╔═╡ 90de1d05-0121-4e85-8674-00f18b4a0a9e
fib(4)                             # ==>    3 --> :)

# ╔═╡ 36b39dd5-076c-4d36-a8f7-ed6a47a5b870
fib(11)                            # ==>   89 --> :)

# ╔═╡ 1400c2fa-1c83-4aa9-baa6-4f319bd0bca2
fib(19)                            # ==> 4181 --> :)

# ╔═╡ 66bd9d13-4274-492e-8980-b9ea687e223a
let
	#------------------------------------------------------------------------------
	function plotUnaryTree!(markOfLeaf, markOfRoot, coordinateXOfRootMark, coordinateYOfRootMark; heightOfTree=2, fontSize=9)
		plot!([                                  #  plot of middle vertical arm '|'
			(coordinateXOfRootMark, coordinateYOfRootMark+heightOfTree/8), 
			(coordinateXOfRootMark, coordinateYOfRootMark+heightOfTree-heightOfTree/8)], 
			lw=1, linecolor=:black) 
		annotate!(                                               # mark of leaf 'x'
			(coordinateXOfRootMark, coordinateYOfRootMark+heightOfTree, text(markOfLeaf, fontSize, :blue))) 
		annotate!(                                               # mark of root 'x'
			(coordinateXOfRootMark, coordinateYOfRootMark, text(markOfRoot, fontSize, :blue))) 
	end # function plotUnaryTree!
	#------------------------------------------------------------------------------
	function plotBinaryTree!(markOfLeftLeaf, markOfRightLeaf, markOfRoot, coordinateXOfRootMark, coordinateYOfRootMark; widthOfTree=2.0, heightOfTree=2, fontSize=9) 
		annotate!(                                          # mark of left leaf 'x'
			(coordinateXOfRootMark-widthOfTree/2, 
			 coordinateYOfRootMark+heightOfTree, 
			 text(markOfLeftLeaf, fontSize, :blue))) 
		plot!([                                     # plot of left vertical arm '|'
			(coordinateXOfRootMark-widthOfTree/2,   
			coordinateYOfRootMark + heightOfTree-heightOfTree/8),  
			(coordinateXOfRootMark-widthOfTree/2, coordinateYOfRootMark+heightOfTree/2)], 
			lw=1, linecolor=:black) 
		annotate!(                                         # mark of right leaf 'x'
			(coordinateXOfRootMark+widthOfTree/2, 
			coordinateYOfRootMark+heightOfTree, 
			text(markOfRightLeaf, fontSize, :blue))) 
		plot!([                                    # plot of right vertical arm '|'
			(coordinateXOfRootMark+widthOfTree/2, 
			coordinateYOfRootMark+heightOfTree-heightOfTree/8), 
			(coordinateXOfRootMark+widthOfTree/2, coordinateYOfRootMark+heightOfTree/2)], 
			lw=1, linecolor=:black) 
		plot!([                                        # plot of horizontal bar '-'
			(coordinateXOfRootMark-widthOfTree/2, coordinateYOfRootMark+heightOfTree/2), 
			(coordinateXOfRootMark+widthOfTree/2, coordinateYOfRootMark+heightOfTree/2)], 
			lw=1, linecolor=:black) 
		plot!([                                   # plot of middle vertical arm '|'
			(coordinateXOfRootMark, coordinateYOfRootMark+heightOfTree/2), 
			(coordinateXOfRootMark, coordinateYOfRootMark+heightOfTree/8)], 
			lw=1, linecolor=:black) 
		annotate!(                                               # mark of root 'x'
			(coordinateXOfRootMark, coordinateYOfRootMark, text(markOfRoot, fontSize, :blue))) 
	end # function plotBinaryTree!
	#------------------------------------------------------------------------------
	function plotTernaryTree!(markOfLeftLeaf, markOfMiddleLeaf, markOfRightLeaf, markOfRoot, coordinateXOfRootMark, coordinateYOfRootMark; widthOfTree=3.0,
	heightOfTree=2, fontSize=9)
		#---------------------------------------------------------
		plotBinaryTree!(markOfLeftLeaf, markOfRightLeaf, markOfRoot, coordinateXOfRootMark, coordinateYOfRootMark; widthOfTree=widthOfTree, heightOfTree=heightOfTree, fontSize=fontSize)
		#---------------------------------------------------------
		plotUnaryTree!(markOfMiddleLeaf, markOfRoot, coordinateXOfRootMark, coordinateYOfRootMark; heightOfTree=heightOfTree, fontSize=fontSize)
	end # function plotTernaryTree!
	#------------------------------------------------------------------------------
	function plotQuaternyTree!(markOfLeftMostLeaf, markOfLeftLeaf, markOfRightLeaf, markOfRightMostLeaf, markOfRoot, coordinateXOfRootMark, coordinateYOfRootMark; widthOfTree=4.0, heightOfTree=2, fontSize=9)
		#---------------------------------------------------------
		plotBinaryTree!(markOfLeftMostLeaf, markOfRightMostLeaf, markOfRoot, coordinateXOfRootMark, coordinateYOfRootMark; widthOfTree=widthOfTree, heightOfTree=heightOfTree, fontSize=fontSize)
		#---------------------------------------------------------
		plotBinaryTree!(markOfLeftLeaf, markOfRightLeaf, markOfRoot, coordinateXOfRootMark, coordinateYOfRootMark; widthOfTree=widthOfTree/3, heightOfTree=heightOfTree, fontSize=fontSize)
	end # function plotQuaternyTree!
	#------------------------------------------------------------------------------
	plot(size=(550, 400), xlim=(-3, 13), ylim=(0, 12), legend=:false, ticks=:none, xlabel=L"Computation\ time", ylabel=L"Space\ consumption", title=L"Tree-recursive process generated by $fib(5)$", titlefontsize=10)
	plotUnaryTree!("0", "fib(0)",  0.0, 9.0)
	plotUnaryTree!("1", "fib(1)", -2.0, 9.0)
	plotUnaryTree!("1", "fib(1)",  1.0, 7.0)
	plotUnaryTree!("1", "fib(1)",  3.0, 7.0)
	plotUnaryTree!("0", "fib(0)",  5.0, 7.0)
	plotUnaryTree!("1", "fib(1)",  7.0, 7.0)
	plotUnaryTree!("0", "fib(0)",  9.0, 7.0)
	plotBinaryTree!("fib(1)", "fib(0)", "fib(2)",-1.0, 7.0)
	plotUnaryTree!("1", "fib(1)", 12.0, 5.0)
	plotBinaryTree!("fib(1)", "fib(0)", "fib(2)",  8.0, 5.0)
	plotBinaryTree!("fib(1)", "fib(0)", "fib(2)",  4.0, 5.0)
	plotBinaryTree!("fib(2)", "fib(1)", "fib(3)",  0.0, 5.0)
	plotBinaryTree!("fib(3)", "fib(2)", "fib(4)",  2.0, 3.0; widthOfTree=4.0)
	plotBinaryTree!("fib(2)", "fib(1)", "fib(3)", 10.0, 3.0; widthOfTree=4.0)
	plotBinaryTree!("fib(4)", "fib(3)", "fib(5)=5",  6.0, 1.0; widthOfTree=8.0)
	#------------------------------------------------------------------------------
end # let

# ╔═╡ ed4e99e9-69fc-4324-a2f9-28991de01953
md"
**Fig. 1.2.2.1**  *Tree-recursive* process with $O(2^n)$-*complexity* generated by computing $fib(5)$ (cf. Fig. 1.5 in SICP, 1996, 2016, p.49)

"

# ╔═╡ 14c0a735-b556-41fc-a5f9-e57ae206c046
md"
---
###### 3.3 *Space* and *Time* Complexity of Fibonacci Tree Recursive Process

From Fig. 1.2.2.1 we see that the *space complexity* is $O(n)$ and the *time complexity* is $O(2^n)$. *Space complexity* is a function of the *height* of the *process tree*, whereas *time complexity* a function of the *width* of the tree.

The computation process is highly inefficient. The right subtree $fib(3)$ appears identically as a subtree in the left subtree of $fib(4)$. Furthermore e.g. $fib(2)$ is computed three times.

"


# ╔═╡ 9d2d9e38-a34d-4df9-9b98-8bae8a251052
md"
---
###### 3.4 *Space* and *Time Complixity* of Closed Form Fibonacci Approximation

A *nonrecursive* approximation to $fib(n)$ is its nearest integer $f_{fib}(n)$:

$f : \mathbb N \rightarrow \mathbb N$
$f : n \mapsto f_{fib}(n)$
$f_{fib}(n) := \left\lceil\frac{\psi^n}{\sqrt{5}}\right\rfloor \text{ for } n \ge 0$

where:

$\psi := \frac{1+\sqrt{5}}{2} = 1.618033988749895 \text{ (golden ratio )}$

$\psi = \lim\limits_{x \to \infty}\frac{f_{fib}(n+1)}{f_{fib}(n)}$

more info can be found [here](https://en.wikipedia.org/wiki/Fibonacci_number). From the formula for $f_{fib}(n)$ we see that the *space complexity* is $O(1)$ and *time complexity* $O(\psi^n)$.

"

# ╔═╡ d18098af-8fbb-4579-baef-b8579a33c02b
ψ = (1 + √5) / 2.0                # golden ratio

# ╔═╡ 59f4bd16-f5fd-43a0-9130-733b318bb850
function fib2(n)                  # in infix notation
	sqrtOf5 = √5
	ϕ   = (1 + sqrtOf5) / 2.0     # golden ratio ≈ 1.6180 as a Float
	f_n = ϕ^n / sqrtOf5           # a Float number is generated
	convert(Int64, round(f_n))
end # function fib2

# ╔═╡ 45e8c4f6-c97a-4253-9913-5bdcb51ab6c9
fib2(0)

# ╔═╡ 7270c38a-d48b-46ae-a3cd-b713c45e57fb
fib2(1)

# ╔═╡ d5bff174-f4df-4478-9c3f-a4c387c1ae04
fib2(4)

# ╔═╡ 0ad1f4c3-a4aa-4c1b-ac1c-fe18b42f6cdb
fib2(4.0)

# ╔═╡ f3d63b59-55b7-48a6-94c4-8386c64d835a
fib2(11)                            # ==>   89 --> :)

# ╔═╡ 30cf0005-7427-498e-b8f0-81b30fc05978
fib2(19)                            # ==> 4181 --> :)

# ╔═╡ 966a7b04-9c19-4e0a-b1bb-f8ed1a5c53c3
md"
---
###### 3.5 Closed Form Fibonacci Approximation: [*Binet*'s Formula](https://en.wikipedia.org/wiki/Fibonacci_sequence)

Binet's formula is also a *nonrecursive* approximation to $fib(n)$ is its nearest integer $f_{fib}(n)$:

$f : \mathbb N \rightarrow \mathbb N$
$f : n \mapsto f_{fib}(n)$
$f_{fib}(n) := \left\lceil\frac{\psi^n - \phi^n}{\sqrt{5}}\right\rfloor \text{ for } n \ge 0$

where:

$\psi := \frac{1+\sqrt{5}}{2} =  1.618033988749895$
$\phi := \frac{1-\sqrt{5}}{2} = -0.6180339887498949$

more info can be found [here](https://en.wikipedia.org/wiki/Fibonacci_number). From the formula for $f_{fib}(n)$ we see that the *space complexity* is $O(1)$ and *time compexity* $O(n)$.

"

# ╔═╡ 7e5093aa-15f8-480b-a164-a7d73bd33486
ϕ = (1 - √5) / 2.0 

# ╔═╡ be5953f4-8ee2-4b8e-a6d4-f93eb8b9971e
function fib3new(n)                  # in infix notation
	sqrtOf5 = √5
	ψ   = (1 + sqrtOf5) / 2.0        # golden ratio ≈ 1.6180 as a Float
	ϕ   = (1 - sqrtOf5) / 2.0        # conjugate value of ψ ≈ -0.618034
	f_n = (ψ^n - ϕ^n) / sqrtOf5      # a Float number is generated
	convert(Int64, round(f_n))
end # function fib3new

# ╔═╡ ddf50581-6684-47f4-aafa-6fbb6b6965a1
fib3new(0)

# ╔═╡ cf93c252-5d12-4713-8fd1-d6211c064a59
fib3new(1)

# ╔═╡ 0c40e44a-d71c-472c-b436-ee33f2155d3f
fib3new(11)

# ╔═╡ aabe9aad-2097-4417-a6ac-a9c885a75923
fib3new(19)

# ╔═╡ 6e08bc0c-a556-4f7a-b656-7bcf3326a065
md"
---
###### 3.6 *Space* and *Time Complixity* of Tail Recursive Fibonacci Process
*Space* complexity of $fib3$ without TCO is $O(n)$ and *time* complexity is $O(n)$. With *TCO* (like in Scheme) both complexities are $O(1)$ and $O(n)$.
"

# ╔═╡ 584371d1-31bc-474b-b2bd-8bdfadecf398
#---------------------------------------------------------------------------
# definition with *two* functions in *one* cell blocked by 'begin ... end'
#---------------------------------------------------------------------------
begin
	#-----------------------------------------------------
	fib3(n) = fib_iter(1, 0, n)
	#-----------------------------------------------------
	fib_iter(a, b, count) = 
		count == 0 ? b : 
			fib_iter( +(a, b), a, -(count, 1))
	#-----------------------------------------------------
end # begin

# ╔═╡ b3bf7014-b05b-4bb4-8201-338763dccb77
fib3(0)

# ╔═╡ 61adc0e2-5c49-40bb-b64d-01072208c229
fib3(1)

# ╔═╡ ee6c42f5-39ae-4c1e-a07b-ef344329d33e
fib3(11)

# ╔═╡ ec4a7ae0-7a9b-476a-bcaf-24bda07b34c8
fib3(19)

# ╔═╡ 1fe5189b-ae8c-4d53-af87-9512738988b3
subtypes(Signed)

# ╔═╡ 757d1174-00d3-46c4-aa46-8a853360259a
function fib4(n::Signed)::Signed
  #----------------------------------------------------
  # local definition
  fib_iter(a, b, count) = 
  	count == 0 ? b : 
		fib_iter(+(a, b), a, -(count, 1))
  #----------------------------------------------------
  fib_iter(1, 0, n)
end # function fib4

# ╔═╡ 28d79756-1250-4980-a191-b4ffc587992f
fib4(0)

# ╔═╡ c2187fd6-0883-4e38-b6a9-deae30031b33
fib4(1)

# ╔═╡ da71e085-d3b1-403b-9ab3-05b376d88b6a
fib4(11)

# ╔═╡ 381d538b-43a5-443e-91b8-805c7789bf41
fib4(19)

# ╔═╡ 3374ddbc-5462-4d89-a36a-66366a2813db
fib4(19.0)                          # ==> blocked for Floats

# ╔═╡ 48e7629b-44f0-43a1-b94c-16df29f82ce6
function fib5(n::Signed)::Signed
  #----------------------------------------------------
  # local definition
  # localized scope of 'n'
  fib_iter(a, b, n) = 
  	n == 0 ? b : 
		fib_iter(+(a, b), a, -(n, 1))
  #----------------------------------------------------
  fib_iter(1, 0, n)
end # function fib5

# ╔═╡ 0b81b022-8ec5-4041-b3d5-4661cec2fdcb
fib5(0)

# ╔═╡ ed434b80-79f1-40e1-aec5-05f9055695ed
fib5(1)

# ╔═╡ 8d8a9763-b9d4-45b2-aba2-28faa57c78e8
fib5(11)

# ╔═╡ 89658567-4c8e-4f01-9a18-ffa2d8c4b037
fib5(19)

# ╔═╡ 3725f0ec-af0c-4c09-b124-0370d4bd8735
md"
---
##### 4. Example: Counting Change
*How many different ways can we make change of $ 1.00, given half-dollars, quarters, dimes, nickels, and pennies? More generally, can we write a procedure to compute the number of ways to change any given amount of money?* (SICP, 1996, 2016)
"

# ╔═╡ e57f1017-ce3c-4f87-bc53-32466fd6f42b
md"
###### 4.1 *Generic* (*Polymorphic*) Method of Function $count\_change$
"

# ╔═╡ b72f34e9-9106-4288-a5de-7488e83b6f16
first_denomination(kinds_of_coins) =
  kinds_of_coins == 1 ?  1 : # penny
  kinds_of_coins == 2 ?  5 : # nickel
  kinds_of_coins == 3 ? 10 : # dime
  kinds_of_coins == 4 ? 25 : # quarter
  kinds_of_coins == 5 ? 50 : # half_dollar
               "arg too big"
 

# ╔═╡ a60f0256-414b-4242-937f-b27b2d4bc18d
function cc(amount::Signed, kinds_of_coins::Signed)::Signed
	if amount == 0 
		1 
	elseif <(amount, 0) || ==(kinds_of_coins, 0) 
		0 
	else 
		cc(amount, -(kinds_of_coins, 1)) + 
		cc(-(amount, first_denomination(kinds_of_coins)), kinds_of_coins)
	end # if
end

# ╔═╡ 045cbbbc-161b-4460-8712-86b43d354418
count_change(amount) = cc(amount, 5)

# ╔═╡ 18f680ee-1bcf-4160-b0c4-543e40746d06
count_change(100)

# ╔═╡ 14028c89-9ca8-4d80-b84f-e3008b1e5dd3
md"
###### 4.2 An Alternative *Recursive* Solution
"

# ╔═╡ a76d6928-2136-4ac7-a1c4-dc50926005af
function myCountChange(amount, coin=50)        # parameter with default value
	#------------------------------------------------------------------------------
	next_smaller_coin(coin) =
	  coin == 50 ? 25 :
      coin == 25 ? 10 :
      coin == 10 ?  5 : 
      coin ==  5 ?  1 : 0
	#------------------------------------------------------------------------------
	amount == 0 ? 1 :
	coin   == 0 ? 0 :
  		<(amount, coin) ? myCountChange(amount, next_smaller_coin(coin)) :
  		≥(amount, coin) ? 
			+( 	myCountChange(-(amount, coin), coin),
	  			myCountChange(amount, next_smaller_coin(coin))) : "error"
	#------------------------------------------------------------------------------
end # function myCountChange

# ╔═╡ 5b73c7a1-2fd0-46cd-90d0-b7a3878b8656
myCountChange(4)

# ╔═╡ f6712d01-4c61-47e7-8061-2fac2584c7ef
myCountChange(5)

# ╔═╡ 5884909d-f49c-45e2-80e3-fb99521db0e4
myCountChange(10)

# ╔═╡ d0eb27b8-8903-4e42-ab91-8627e289e759
myCountChange(100)

# ╔═╡ 81aa817d-683e-4d13-bad6-2a9a5cb1959a
md"
---
##### 5. Idiomatic *Imperative* Julia 
... with *abstract* types 'Signed', 'AbstractFloat', *self-defined* abstract type 'FloatOrSigned', while', 'for', and *parallel* assignment
"

# ╔═╡ d55f4dd8-c5c5-4fab-8cbc-a56f6f6628ea
md"
###### 5.1 *Typed* Method with *State* Variables, Parallel Assignment, and $while$ Loop
"

# ╔═╡ 7efd9e26-7ede-49fe-8d06-0f30a3c76d56
#-------------------------------------------------------------------------
function fib6(n::Signed)::Signed
#-------------------------------------------------------------------------
  # fib_iter(a, b, count) = count == 0 ? b : fib_iter(a + b, a, count - 1)
#-------------------------------------------------------------------------
  # fib_iter(1, 0, n)
#-------------------------------------------------------------------------
	a, b, count  = 1, 0, n            # parallel assignment of state vars
	while !(count == 0)
		a, b, count = a + b, a, count - 1
	end
	# in the case 'count == 0'
	b
end # function fib6

# ╔═╡ 5cb9879c-1b3a-401b-8df6-ddb969b47e7e
fib6(0)

# ╔═╡ 31c99f8f-ed87-4ce8-a185-6a867e730390
fib6(1)

# ╔═╡ 3f0ed8a8-c55a-4b25-9be3-0c937a82f2c7
fib6(11)

# ╔═╡ ec9121dd-b356-4abf-bb66-cb5a3d2135a8
fib6(19)

# ╔═╡ d86a5807-c5a9-4013-b9b2-34ff55cdbdb3
md"
---
###### 5.2 *Typed* Method with *State* Variables, Parallel Assignment, and $for$ Loop
"

# ╔═╡ cca61987-5ff2-43d5-a13d-850786905127
#---------------------------------------------------------------------------
# idiomatic Julia: 'for' and parallel assignment
#---------------------------------------------------------------------------
function fib7(n::Signed)::Signed
	a, b = 1, 0
	for count = n : -1 : 1
		a, b = a + b, a
	end
	b
end

# ╔═╡ 38913d58-3f78-4039-bb93-ab64b4e5e1c4
fib7(0)

# ╔═╡ aca4a8a1-bc10-4abb-af38-941f3de29769
fib7(1)

# ╔═╡ c0786dd7-c833-4218-9fc1-3c05ee85df0e
fib7(11)

# ╔═╡ 39f932a8-6d42-4c25-9f30-e19e0b262a6c
fib7(19)

# ╔═╡ 5c44c7be-fd22-4859-8a03-434eb6564942
md"
---
###### 5.3 *Union-Typed* Method
"

# ╔═╡ a863f97c-18b0-4e89-b2d8-e3e83eab9c0b
FloatOrSigned = Union{AbstractFloat, Signed}

# ╔═╡ e333e3b5-66cf-4d59-b031-5e0676d4fbad
function fib8(n::FloatOrSigned)::AbstractFloat
	sqrtOf5 = √5                 # a Float number is generated
	ϕ = (1.0 + sqrtOf5) / 2.0    # a Float number is generated
	round(ϕ^n/sqrtOf5)
end

# ╔═╡ 7fe95fc6-8704-432c-b101-55dcb0bd8777
fib8(0)

# ╔═╡ c1aa2447-bc4d-482a-887a-4ece1a2c72e9
fib8(1)

# ╔═╡ 7b75e150-69a0-4754-b47c-55f20396c2a1
fib8(11)

# ╔═╡ 3d3f7a93-dfa9-4424-8032-e4eeb792fece
fib8(11.0)

# ╔═╡ fc9eebe4-9c4a-478b-94bb-5e0c01017122
fib8(19)

# ╔═╡ ae8463f2-09d6-442d-bbc7-cd4faf310fd9
fib8(19.0)

# ╔═╡ 01193090-2008-4ebf-a72a-9bb6a13b4c00
md"
---
##### References
- **Abelson, H., Sussman, G.J. & Sussman, J.**, [*Structure and Interpretation of Computer Programs*](https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/6515/sicp.zip/full-text/book/book.html), Cambridge, Mass.: MIT Press, (2/e), 1996; last visit 2025/01/08
- **Abelson, H., Sussman, G.J. & Sussman, J.**, [*Structure and Interpretation of Computer Programs*](https://web.mit.edu/6.001/6.037/sicp.pdf); Cambridge, Mass.: MIT Press, (2/e), 2016; last visit 2025/01/08
- **Wikipedia**, [*Fibonacci Sequence*](https://en.wikipedia.org/wiki/Fibonacci_sequence), last visit, 2025/01/08
"

# ╔═╡ 641439d6-9442-4e2a-9696-3153d1fd27f7
md"
---
##### end of ch. 1.2.2
===================================================================================

This is a **draft** under the [Attribution-NonCommercial-ShareAlike 4.0 International **(CC BY-NC-SA 4.0)**](https://creativecommons.org/licenses/by-nc-sa/4.0/) license. Comments, suggestions for improvement and bug reports are welcome: **claus.moebus(@)uol.de**

===================================================================================
"

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deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8a22cf860a7d27e4f3498a0fe0811a7957badb38"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.3+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "b7bfd3ab9d2c58c3829684142f5804e4c6499abc"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.45+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "490376214c4721cdaca654041f635213c6165cb3"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+2"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "63406453ed9b33a0df95d570816d5366c92b7809"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+2"
"""

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