This is just my personal notes on [[http://mitpress.mit.edu/sicp][Structure and interpretation of computer programs]].
I also studied with the [[https://archive.org/details/ucberkeley-webcast-PL3E89002AA9B9879E?sort=titleSorter][Brian Harvey's SICP lectures]] because I am a scrub. ;p
Before you can use this document, you need to do some prerequisite installation of [[https://racket-lang.org/][Racket]] and [[https://docs.racket-lang.org/sicp-manual/][SICP package]].
* Elements of programming
Programming often requires the following:
- Simple expressions with atomic value.
- A way to combine procedures into complex expressions.
In order to do programming, we must have a programming language.
A programming language often requires the following to have an effective way of expressing code:
- Expressions which varies from primitive expressions (e.g., ~42~, ~1.683~, ~53819492184~) to compound expressions (e.g., ~(+ 53 20)~, ~(- 464 254)~).
- An environment of objects which you can refer by name either with values (e.g., ~(define x 10)~, ~(define pi 3.14)~, ~(define e 2.71828)~) or procedures (e.g., ~(define (square x) (* x x))~, ~(define (my-formula height weight length) (* 23 (/ height weight) (+ 3500 length)))~).
- An evaluation model for expressions since certain procedures can have different output from the order of operations.
A programming language lets us abstract procedures as a black box.
Here's an example of implementing the square root given a number.
In order to do the square root extraction in our implementation, we define multiple procedures with each solving a part of the procedure: one procedure for indicating whether the guess is good enough, one for creating an improved guess for the next iteration, and one for actually doing the square root extraction.
In general cases, we don't implement things as one thing as it will result in a messy state of code.
Instead, we modularize those functions.
We classify these procedures as a *procedural abstraction*.
* Higher-order functions
Functions and data are often separated similarly to verbs and subjects.
We tend to think of them as different things relying on each other to do things: functions need data to manipulate while data are raw information to be arranged by a function.
However, the reality is that there is a blurry line to how distinct both of them are.
Functions can be treated similarly to data and vice versa.
The lesson of higher-order functions proves this.
It is one of the foundations of functional programming.
In order to learn about it, you need to know the key: *generalizing patterns*.
For example, say we have different functions for knowing the area of a shape.
What happens when Alyssa attempts to use this to compute square roots? Explain.
#+end_quote
The reason why ~if~ needs a special form is because of applicative-order evaluation.
Scheme (or rather Racket with the SICP package) interprets with applicative-order evaluation which it means it has to evaluate all of the arguments first before proceeding to evaluate the procedure.
As ~new-if~ is a procedure that we defined, it would cause an infinite loop of Racket trying to evaluate ~sqrt-iter~ inside of our ~new-if~ procedure.
** Exercise 1.7
#+begin_quote
The ~good-enough?~ test used in computing square roots will not be very effective for finding the square roots of very small numbers.
Also, in real computers, arithmetic operations are almost always performed with limited precision. This makes our test inadequate for very large numbers.
Explain these statements, with examples showing how the test fails for small and large numbers.
An alternative strategy for implementing ~good-enough?~ is to watch how ~guess~ changes from one iteration to the next and to stop when the change is a very small fraction of the guess.
Design a square-root procedure that uses this kind of end test.
Does this work better for small and large numbers?
#+end_quote
For Exercise 1.7, I'm afraid I cannot easily answer it since the results from the example implementation is already accurate due to the interpreter.
For this exercise, let's pretend the interpreter is not great.
For example, ~(sqrt 0.0001)~ results in ~.03230844833048122~ (should be ~0.01~).
[fn:: You can test how it really goes with the MIT Scheme interpreter.]
The reason varies from a combination of interpreter, hardware configurations, and implementation of arithmetics.
This is especially true with floating points arithmetics.
In implementing our improved square root implementation from the question, we start with editing the ~improve~ function.
I've modified the ~good-enough?~ function by making the tolerance as an argument.
Tested on the MIT Scheme v10.1.10, the results are more accurate closer to modern systems like Julia.
Bigger numbers are also calculated quicker than the previous implementation (for some reason that I don't know).
** Exercise 1.8
#+begin_quote
Newton's method for cube roots is based on the fact that if y is an approximation to the cube root of x, then a better approximation is given by the value
\begin{equation*}
\frac{x / y^2 + 2y}{3}
\end{equation*}
Use this formula to implement a cube-root procedure analogous to the square-root procedure.
(In section 1.3.4 we will see how to implement Newton's method in general as an abstraction of these square-root and cube-root procedures.)
Each of the following two procedures defines a method for adding two positive integers in terms of the procedures ~inc~, which increments its argument by 1, and ~dec~, which decrements its argument by 1.
Notwithstanding related to solving the entire problem, I'll include note on how I was able to create a procedure for the Pi value computation since it gave me the hardest time.
In order to start creating a procedure, I've simply observed the given formula with the induction that it can be separated into pairs like the following.
(I also simply didn't observe that each pair is also an iteration of a function.)
Show that ~sum~ and ~product~ (exercise 1.31) are both special cases of a still more general notion called ~accumulate~ that combines a collection of terms, using some general accumulation function:
~accumulate~ takes as arguments the same term and range specifications as sum and product, together with a combiner procedure (of two arguments) that specifies how the current term is to be combined with the accumulation of the preceding terms and a null-value that specifies what base value to use when the terms run out.
Write ~accumulate~ and show how ~sum~ and ~product~ can both be defined as simple calls to ~accumulate~.
